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19TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE AND REASONING

# PROGRAM FOR WEDNESDAY, DECEMBER 18TH, 2013

09:00-10:00 Session 13: Invited talk by Frank Wolter (LPAR)
 09:00 Ontology-based Data Access: A Study through Disjunctive Datalog, CSP, and MMSNPSPEAKER: Frank WolterABSTRACT. Ontology-based data access is concerned with querying incomplete data sources in the presence of domain specific knowledge provided by an ontology. A central notion in this setting is that of an ontology-mediated query, which is a database query coupled with an ontology. In this talk, I discuss  several classes of ontology-mediated queries, where the database queries are given as some form of conjunctive query and the ontologies are formulated in description logics or other relevant fragments of first-order logic, such as the guarded fragment and the unary-negation fragment. I will discuss three  contributions. First it is shown that popular ontology-mediated query languages have the same expressive power as natural fragments of disjunctive datalog. Second, we establish intimate connections between ontology-mediated queries and constraint satisfaction problems (CSPs) and their logical generalization, MMSNP formulas. Third, we exploit these connections to obtain new results regarding (i) first-order rewritability and datalog-rewritability of ontology-mediated queries, (ii) P/NP dichotomies for ontology-mediated queries, and (iii) the query containment problem for ontology-mediated queries.The talk is based on joint work with M. Bienvenue, B. ten Cate, and C. Lutz.
10:00-10:30Coffee Break
10:30-12:00 Session 14 (LPAR)
 10:30 Multi-Objective Discounted Reward Verification in Graphs and MDPsSPEAKER: Vojtech ForejtABSTRACT. We study the problem of achieving a given value in Markov decision processes (MDPs) with several independent discounted reward objectives. We consider a generalised version of discounted reward objectives, in which the amount of discounting depends on the states visited and on the objective. This definition extends the usual definition of discounted reward, and allows to capture the systems in which the value of different commodities diminish at different and variable rates. We establish results for two prominent subclasses of the problem, namely state-discount models where the discount factors are only dependent on the state of the MDP (and independent of the objective), and reward-discount models where they are only dependent on the objective (but not on the state of the MDP). For the state-discount models we use a straightforward reduction to expected total reward objectives and show that the problem whether a value is achievable can be solved in polynomial time. For the reward-discount model we show that memory and randomisation of the strategies are required, but nevertheless that the problem is decidable and it is sufficient to consider strategies which after a certain number of steps behave in a memoryless way. For the general case, we show that when restricted to graphs (i.e. MDPs with no randomisation), pure strategies and discount factors of the form $1/n$ where $n$ is an integer, the problem is in PSPACE and finite memory suffices for achieving a given value. We also show that when the discount factors are not of the form $1/n$, the memory required by a strategy can be infinite. 10:50 Conflict Resolution in Structured ArgumentationSPEAKER: Martin BalážABSTRACT. While several interesting argumentation-based semantics for defeasible logic programs have been proposed, to our best knowledge, none of these approaches is able to fully handle the closure under strict rules in a sufficient manner: they are either not closed, or they use workarounds such as transposition of rules which violates the desired directionality of logic programming rules. We propose a novel argumentation-based semantics, in which the status of arguments is determined by attacks between newly introduced conflict resolutions instead of attacks between arguments. We show that the semantics is closed w.r.t. strict rules and respects the directionality of inference rules, as well as other desired properties previously published in the literature. 11:00 Towards Rational Closure for Fuzzy Logic: The Case of Propositional Gödel LogicSPEAKER: Giovanni CasiniABSTRACT. In the field of non-monotonic logics, the notion of \emph{rational closure} is acknowledged as a landmark and we are going to see whether such a construction can be adopted in the context of mathematical fuzzy logic, a so far (apparently) unexplored journey. As a first step, we will characterise rational closure in the context of Propositional G\"odel Logic. 11:20 Description Logics, Rules and Multi-Context SystemsSPEAKER: Luís Cruz-FilipeABSTRACT. The combination of rules and ontologies has been a fertile topic of research in the last years, with the proposal of several different systems that achieve this goal. In this paper, we look at two of these formalisms, Mdl-programs and multi-context systems, which address different aspects of this combination, and include different, incomparable programming constructs. Despite this, we show that every Mdl-program can be transformed in a multi-context system, and this transformation relates the different semantics for each paradigm in a natural way. As an application, we show how a set of design patterns for multi-context systems can be obtained from previous work on Mdl-programs. 11:40 HOL based First-order Modal Logic ProversSPEAKER: Christoph BenzmuellerABSTRACT. First-order modal logics (FMLs) can be modeled as natural fragments of classical higher-order logic (HOL). The FMLtoHOL tool exploits this fact and it enables the application of off-the-shelf HOL provers and model finders for reasoning within FMLs. The tool bridges between the qmf-syntax for FML and the TPTP thf-syntax for HOL. It currently supports logics K, K4, D, D4, T, S4, and S5 with respect to constant, varying and cumulative domain semantics. The approach is evaluated in combination with a meta-prover for HOL which sequentially schedules various HOL reasoners. The resulting system is very competitive.
12:00-14:00Lunch
14:00-15:30 Session 15 (LPAR)
 14:00 Reachability Modules for the Description Logic SRIQSPEAKER: Riku NortjeABSTRACT. In this paper we investigate module extraction for the Description Logic SRIQ. We formulate modules in terms of the reachability problem for directed hypergraphs. Using inseperability relations, we investigate the module-theoretic properties of reachability modules and show by means of an empirical evaluation that these modules have the potential of being substantially smaller than syntactic locality modules. 14:20 Forgetting Concept and Role Symbols in ALCH-OntologiesSPEAKER: Patrick KoopmannABSTRACT. We develop a resolution-based method for forgetting concept and role symbols in ALCH ontologies, or for computing uniform interpolants in ALCH. Uniform interpolants use only a restricted set of symbols, while preserving consequences of the original ontology involving these symbols. While recent work towards practical methods for uniform interpolation in expressive description logics focuses on forgetting concept symbols, we believe most applications would benefit from the possibility to forget both role and concept symbols. We focus on the description logic ALCH, which allows for the formalisation of role hierarchies. Our approach is based on a recently developed resolution based-calculus for forgetting concept symbols in ALC ontologies, which we extend by redundancy elimination techniques to make it practical for larger ontologies. Experiments on ALCH fragments of real life ontologies suggest that our method is applicable in a lot of real-life applications. 14:40 Prediction and Explanation over DL-Lite Data StreamsSPEAKER: Szymon KlarmanABSTRACT. Stream reasoning is an emerging research area focusing on the development of reasoning techniques applicable to streams of rapidly changing, semantically enhanced data. In this paper, we consider data represented in Description Logics from the popular DL-Lite family, and study the logic foundations of prediction and explanation over DL-Lite data streams, i.e., reasoning from finite segments of streaming data to conjectures about the content of the streams in the future or in the past. We propose a novel formalization of the problem based on temporal past-future'' rules, grounded in Temporal Query Language. Such rules can naturally accommodate complex data association patterns, which are typically discovered through data mining processes, with logical and temporal constraints of varying expressiveness. Further, we analyse the computational complexity of reasoning with rules expressed in different fragments of the temporal language. As a result, we draw precise demarcation lines between NP-, DP- and PSpace-complete variants of our setting and, consequently, suggest relevant restrictions rendering prediction and explanation more feasible in practice. 15:00 Practical Querying of Temporal Data via OWL 2 QL and SQL:2011SPEAKER: Szymon KlarmanABSTRACT. We develop a practical approach to querying temporal data stored in temporal SQL:2011 databases through the semantic layer of OWL 2 QL ontologies. An interval-based temporal query language (TQL), which we propose for this task, is defined via naturally characterizable combinations of temporal logic with conjunctive queries. This foundation warrants well-defined semantics and formal properties of TQL querying. In particular, we show that under certain mild restrictions the data complexity of query answering remains in AC$^0$, i.e., as in the usual, nontemporal case. On the practical side, TQL is tailored specifically to offer maximum expressivity while preserving the possibility of reusing standard first-order rewriting techniques and tools for OWL 2 QL. 15:10 An Algorithm for Enumerating Maximal Models of Horn Theories with an Application to Modal LogicsSPEAKER: Guido SciaviccoABSTRACT. The fragment of propositional logic known as Horn theories plays a central role in automated reasoning. The problem of enumerating the maximal models of a Horn theory (MaxMod) has been proved to be computationally hard, unless P = NP. To the best of our knowledge, the only algorithm available for it is the one based on a brute-force approach. In this paper, we provide an algorithm for the problem of enumerating the maximal subsets of facts that do not entail a distinguished atomic proposition in a positive Horn theory (MaxNoEntail). We show that MaxMod is polynomially reducible to MaxNoEntail (and vice versa), making it possible to solve also the former problem using the proposed algorithm. Addressing MaxMod via MaxNoEntail opens, inter alia, the possibility of benefiting from the monotonicity of the notion of entailment. (The notion of model does not enjoy such a property.) We also discuss an application of MaxNoEntail to expressiveness issues for modal logics, which reveals the effectiveness of the proposed algorithm.
15:30-16:00Coffee Break
16:00-17:30 Session 16 (LPAR)
 16:00 On QBF Proofs and PreprocessingSPEAKER: Mikolas JanotaABSTRACT. Quantified Boolean formulas (QBFs) elegantly extend propositional satisfiability providing us with a canonical representation for PSPACE problems. To overcome the inherent complexity of QBF, significant effort has been invested in developing QBF solvers as well as the underlying proof systems. At the same time, formula preprocessing is crucial for the application of QBF solvers. This paper focuses on a missing link in currently-available technology: How to obtain a proof for a formula that had been preprocessed before it was given to a solver? The paper targets a suite of commonly-used preprocessing techniques and shows how to reconstruct proofs for them. On the negative side, the paper discusses certain limitations of the currently-used proof systems in the light of preprocessing. The presented proof-construction techniques were implemented and evaluated in the state-of-the-art QBF preprocessor bloqqer. 16:20 Simulating Parity ReasoningSPEAKER: Tero LaitinenABSTRACT. Propositional satisfiability (SAT) solvers, which typically operate using conjunctive normal form (CNF), have been successfully applied in many domains. However, in some application areas such as circuit verification, bounded model checking, and logical cryptanalysis, instances can have many parity (xor) constraints which may not be handled efficiently if translated to CNF. Thus, extensions to the CNF-driven search with various stronger parity reasoning engines ranging from equivalence reasoning to incremental Gaussian elimination have been proposed. This paper studies how stronger parity reasoning techniques in the DPLL(XOR) framework can be simulated by simpler systems: resolution, unit propagation, and parity explanations. Such simulations are interesting, for example, for developing the next generation SAT solvers capable of handling parity constraints efficiently. 16:40 BDI: A New Decidable First-order Clause ClassSPEAKER: Christoph WeidenbachABSTRACT. BDI (Bounded Depth Increase) is a new decidable first-order clause class. It strictly includes known classes such as PVD. The arity of function and predicate symbols as well as the shape of atoms is not restricted in BDI. Instead the shape of "cycles" in resolution inferences is restricted such that the depth of generated clauses may increase but is still finitely bound. The BDI class is motivated by real world problems where function terms are used to represent record structures. We show that the hyper-resolution calculus modulo redundancy elimination terminates on BDI clause sets. Employing this result to the ordered resolution calculus, we can also prove termination of ordered resolution on BDI, yielding a more efficient decision procedure. 16:50 Blocked Clause DecompositionSPEAKER: Marijn HeuleABSTRACT. We demonstrate that it is fairly easy to decompose any propositional formula into two subsets such that both can be solved by blocked clause elimination. Such a blocked clause decomposition is useful to cheaply detect backbone variables and equivalent literals. Blocked clause decompositions are especially useful when they are unbalanced, i.e., the size of one subset is much larger in size than the other one. We present algorithms and heuristics to obtain unbalanced decompositions efficiently. Our techniques have been implemented in the state-of-the-art solver Lingeling. Experiments show that the performance of Lingeling is clearly improved due to these techniques on application benchmarks of the SAT Competition 2013. 17:10 Long-distance Resolution: Proof Generation and Strategy Extraction in Search-based QBF SolvingSPEAKER: Magdalena WidlABSTRACT. Strategies (and certificates) for quantified Boolean formulas (QBFs) are of high practical relevance as they facilitate the verification of results returned by QBF solvers and the generation of solutions to problems formulated as QBFs. State of the art approaches to obtain strategies require traversing a Q-resolution proof of a QBF, which for many real-life instances is too large to handle. In this work, we consider the long-distance Q-resolution (LDQ) calculus, which allows particular tautological resolvents. We show that for a family of QBFs using LDQ allows for exponentially shorter proofs compared to Q-resolution. We further show that an approach to strategy extraction originally presented for Q-resolution proofs can also be applied to LDQ-resolution proofs. As a practical application, we consider search-based QBF solvers which are able to learn tautological clauses based on resolution and the conflict-driven clause learning method. We prove that the resolution proofs produced by these solvers correspond to proofs in the LDQ calculus and can therefore be used as input for strategy extraction algorithms. Experimental results illustrate the potential of the LDQ calculus in search-based QBF solving.


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# puzzle: A spy and the keypad

A spy encounters a keypad that requires a 4 digit PIN. He uses a fine dust to find which keys are used in the combination. He does not know the sequence of keys, nor which ones repeat if any. Obviously if only 3 keys are required one of them is repeated. If 2 keys are required then either 2 keys are repeated twice, or 1 key repeated three times.

The puzzle is for me to design a pin which makes the spies job as hard as possible. Are 4 unique keys the safest, or is have one or more repeating keys better?

1. One digit used = 1 combination
2. Two digits used = 14 combinations = 2^4-2
3. Three digits used = ? combinations
4. Four digits used = ? combinations

Can someone do the math and come up with the possible combinations that use exactly 3 and 4 keys, no more or less. My intuition is that the repeating keys will increase the number of combinations more than used all 4 keys for the pin.

PS. For the two digits (if they are X, Y) then the combinations are:

XXXY, XXYX, XXYY,
XYXX, XYXY, XYYX, XYYY,
YXXX, YXXY, YXYX, YXYY,
YYXX, YYXY, YYYX

• Tell the folks that when entering the code, they should always press on a few random keys as well. This way the spy can't figure out which keys below to the code! Muhahaha. Apr 10, 2013 at 17:04
• I once lived in a building with a keypad-activated lock at the main entrance, which had a peculiar characteristic: it beeped when each key was pressed, at a frequency that depended on the key pressed. So your typical spy could dispense with the fine dust... Apr 10, 2013 at 17:21
• @AsafKaragila - and gloves haven't been invented yet. Apr 10, 2013 at 20:02
• @ja72: It's too hot for gloves in Israel. :-) Apr 10, 2013 at 20:17

• Can you elaborate a little more on the logic. I don't quite follow on how you arrive at $4!$ and $\frac{3}{2}4!$. Apr 10, 2013 at 20:06

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# Tag Archives: sequences

## 5050

5050 is close to 5150 which can be interpreted as the radio call for an involuntary psychiatric hold. But that’s not why we’re here.

5050 can also be expressed as a summation. A summation consists of an expression containing a dummy variable through which the dummy variable is used to perform some calculation.

$\sum_{n=1}^{\infty} 2^{-n} = 1$

This summation specifically is an infinite summation. The summation can also be called an infinite series.

Before diving into summations we must first understand sequences and series. Sequences are ordered lists of numbers. There are no operations being performed on the numbers, they exist only as an ordered list.

${1, 1, 2, 3, 5, 8, ...}$

A series is the sum of a sequence. Where above we have the sequence, the series would be:

${1 + 1 + 2 +3 + 5 + 8 + ...}$

Finite summations and series become especially useful when they can be manipulated to fit an existing form for which there is a known formula.

$\sum_{n=1}^{n} = \frac{n(n+1)}{2}$

Read plainly: The sum from n = 1 to n items is equal to n times the quantity n plus 1 all over 2. It’s said that this formula can be attributed to Gauss. Where Gauss was given the task of adding all integers 1 through 100 but through an observation was able to derive the above formula.

$\sum_{n=1}^{100} = \frac{100(100 + 1)}{2} = 5050$

Consider the series that represents this summation: ${1 + 2 + 3 + ... + 98 + 99 + 100}$ The series can be split into two series consisting of 50 terms each and rewritten as follows.

$\{1 + 2 + 3 + ... + 48 + 49 + 50\} \\ + \\ \{100 + 99 + 98 + ... + 53 + 52 + 51\}$

Adding these vertically the following series is obtained.

$\{ 101 + 101 + 101 + ... + 101 + 101 + 101 \}$ Of notable interest is the series now contains 50 terms, all being the number 101. To complete the whole picture consider another way to write 50 in terms of a number divided by 2: $\frac{100}{2}$

When the value of all the terms, 101, is multiplied by the alternative way to write 50 the derived formula become apparent.

$\frac{100(101)}{2} = \frac{100(100 + 1)}{2} = 50 \cdot 101 = 5050$


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# Digital signal derivative

I'm new to signal processing and I need your help: I have an array of 128 elements (call it Window) filled with 128 samples taken from a sensor. I was wondering how to find the derivative of this digital signal. I found this formula on wikipedia:

m = F[n] - F[n-1]

where, for me, F[n] is Window[n]. I'd like to obtain an array M filled with all the derivatives. But there's something I miss: The first step would be:

M[0] = Window[0] - Window[-1]

and I haven't Window[-1]. How can I manage this issue? Thank you in advance.

https://en.wikibooks.org/wiki/Digital_Signal_Processing/Discrete_Operations#Derivatives

• Take Window[-1]=0 . – Oliver Jun 18 '15 at 11:35
• First of all you need to figure out for yourself what you mean by "derivative" of a discrete-time signal, because there is actually no such thing. What you've found are first order differences. If that is what you need, then you could either do what Oliver suggested, or you could simply accept the fact that the number of first order differences will be one less than the number of samples. – Matt L. Jun 18 '15 at 12:19

In theory, M[0] doesn't exist. Your discrete derivative is only defined on [1,127].

As a practical solution, you can define M[0] as a null element of your signal. For example if you expect values between -1 and 1, take M[0] = 0.

In the case of real-time, there is two options:

• Either you save the last value of your buffer so you can take M[0] = last for the next ones, and initialize last as 0, or whatever seems coherent with your expected signal;
• Either you start computing the derivative at M[1] and your processing will have a latency of 1 sample.

Suppose

length(Window)=L

Then you could try something like this: extend Window by adding a new value at the beginning and end:

Window[-1] = Window[0] - (Window[1]-Window[0])
Window[L]  = Window[L-1] + (Window[L-1]-Window[L-2])

Then for n=0..L-1 put:

M[n] = (Window[n+1] - Window[n-1])/2.0


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# Buffon's Needle

Buffon's Needle
The Buffon's Needle problem is a mathematical method of approximating the value of pi $(\pi = 3.1415...)$involving repeatedly dropping needles on a sheet of lined paper and observing how often the needle intersects a line.

# Basic Description

The method was first used to approximate π by Georges-Louis Leclerc, the Comte de Buffon, in 1777. Buffon posed the Buffon's Needle problem and offered the first experiment where he threw breadsticks over his shoulder and counted how often the crossed lines on his tiled floor.

Subsequent mathematicians have used the method with needles instead of bread sticks, or with computer simulations. In the case where the distance between the lines is equal the length of the needle, we will show that an approximation of π can be calculated using the equation

$\pi \approx {2*\mbox{number of drops} \over \mbox{number of hits}}$

# A More Mathematical Explanation

#### Will the Needle Intersect a Line?

[[Image:willtheneedlecros [...]

#### Will the Needle Intersect a Line?

To prove that the Buffon's Needle experiment will give an approximation of π, we can consider which positions of the needle will cause an intersection. Since the needle drops are random, there is no reason why the needle should be more likely to intersect one line than another. As a result, we can simplify our proof by focusing on a particular strip of the paper bounded by two horizontal lines.

The variable θ is the acute angle made by the needle and an imaginary line parallel to the ones on the paper. Finally, d is the distance between the center of the needle and the nearest line.

We can extend line segments from the center and tip of the needle to meet at a right angle. A needle will cut a line if the green arrow, d, is shorter than the leg opposite θ. More precisely, it will intersect when

$d \leq \left( \frac{1}{2} \right) \sin(\theta). \$

See case 1, where the needle falls at a relatively small angle with respect to the lines. Because of the small angle, the center of the needle would have to fall very close. In case 2, the needle intersects even though the center of the needle is far from both lines because the angle is so large.

#### The Probability of an Intersection

In order to show that the Buffon's experiment gives an approximation for π, we need to show that there is a relationship between the probability of an intersection and the value of π. If we graph the outcomes of θ along the X axis and d along the Y, we have the sample space for the trials. In the diagram below, the sample space is contained by the dashed lines.

Each point on the graph represents some combination of an angle and a distance that a needle might occupy.

There will be an intersection if $d \leq \left ( \frac{1}{2} \right ) \sin(\theta) \$, which is represented by the blue region. The area under this curve represents all the combinations of distances and angles that will cause the needle to intersect a line. The area under the blue curve, which is equal to 1/2 in this case, can found by evaluating the integral

$\int_0^{\frac {\pi}{2}} \frac{1}{2} \sin(\theta) d\theta$

Then, the area of the sample space can be found by multiplying the length of the rectangle by the height.

$\frac {1}{2} * \frac {\pi}{2} = \frac {\pi}{4}$

The probability of a hit can be calculated by taking the number of total ways an intersection can occur over the total number possible outcomes (the number of trials). For needle drops, the probability is proportional to the ratio of the two areas in this case because each possible value of θ and d is equally probable. The probability of an intersection is

$P_{hit} = \cfrac{ \frac{1}{2} }{\frac{\pi}{4}} = \frac {2}{\pi} = .6366197...$

#### Using Random Samples to Approximate Pi

The original goal of the Buffon's needle method, approximating π, can be achieved by using probability to solve for π. If a large number of trials is conducted, the proportion of times a needle intersects a line will be close to the probability of an intersection. That is, the number of line hits divided by the number of drops will equal approximately the probability of hitting the line.

$\frac {\mbox{number of hits}}{\mbox{number of drops}} \approx P_{hit} = \frac {2}{\pi}$

So

$\frac {\mbox{number of hits}}{\mbox{number of drops}} \approx \frac {2}{\pi}$

Therefore, we can solve for π:

$\pi \approx \frac {2 * {\mbox{number of drops}}}{\mbox{number of hits}}$

# Why It's Interesting

#### Monte Carlo Methods

The Buffon's needle problem was the first recorded use of a Monte Carlo method. These methods employ repeated random sampling to approximate a probability, instead of computing the probability directly. Monte Carlo calculations are especially useful when the nature of the problem makes a direct calculation impossible or unfeasible, and they have become more common as the introduction of computers makes randomization and conducting a large number of trials less laborious.

π is an irrational number, which means that its value cannot be expressed exactly as a fraction a/b, where a and b are integers. As a result, π cannot be written as an exact decimal and mathematicians have been challenged with trying to determine increasingly accurate approximations. The timeline below shows the improvements in approximating pi throughout history. In the past 50 years especially, improvements in computer capability allow mathematicians to determine more decimal places. Nonetheless, better methods of approximation are still desired.

A recent study conducted the Buffon's Needle experiment to approximate π using computer software. The researchers administered 30 trials for each number of drops, and averaged their estimates for π. They noted the improvement in accuracy as more trials were conducted.

These results show that the Buffon's Needle approximation is a relatively tedious. Even when a large number of needles are dropped, this experiment gave a value of pi that was inaccurate in the third decimal place. Compared to other computer generated techniques, Buffon's method is impractical because the estimates converge towards π rather slowly. Nonetheless, the intriguing relationship between the probability of a needle's intersection and the value of π has attracted mathematicians to study the Buffon's Needle method since its introduction in the 18th century.

#### Generalization of the problem

The Buffon’s needle problem has been generalized so that the probability of an intersection can be calculated for a needle of any length and paper with any spacing. It has been proven that for a needle shorter than the distance between the lines, the probability of a intersection is $\frac {2*l}{\pi*d}$. This equation makes sense when we consider the normal case, where l =1 and d =1, so these variables disappear and the probability is $\frac {2}{\pi}$.

The generalization of the problem is useful because it allows us to examine the relationship between length of the needle, distance between the lines, and probability of an intersection. The variable for length is in the numerator, so a longer needle will have a greater probability of an intersection. The variable for distance is in the denominator, so greater space between lines will decrease the probability.

To see how a longer needle will affect probability, follow this link: http://whistleralley.com/java/buffon_graph.htm

#### Needles in Nature

Applications of the Buffon's Needle method are even found naturally in nature. The Centre for Mathematical Biology at the University of Bath found uses of the Buffon's Needle algorithm in a recent study of ant colonies. The researchers found that an ant can estimate the size of an anthill by visiting the hill twice and noting how often it recrosses its first path.

Ants generally nest in groups of about 50 or 100, and the size of their nest preference is determined by the size of the colony. When a nest is destroyed, the colony must find a suitable replacement, so they send out scouts to find new potential homes.

In the study, scout ants were provided with "nest cavities of different sizes, shapes, and configurations in order to examine preferences" [2]. From their observations, researchers were able to draw the conclusion that scout ants must have a method of measuring areas.

A scout initially begins exploration of a nest by walking around the site to leave tracks. Then, the ant will return later and walk a new path that repeatedly intersects the first tracks. The first track will be laced with a chemical that causes the ant to note each time it crosses the original path. The researchers believe that these scout ants can calculate an estimate for the nest's area using the number of intersections between its two visits.

The ants can measure the size of their hill using a related and fairly intuitive method: If they are constantly intersecting their first path, the area must be small. If they rarely reintersects the first track, the area of the hill must be much larger so there is plenty of space for a non-intersecting second path.

"In effect, an ant scout applies a variant of Buffon's needle theorem: The estimated area of a flat surface is inversely proportional to the number of intersections between the set of lines randomly scattered across the surface." [7]

This idea can be related back to the generalization of the problem by imagining if the parallel lines were much further apart. A larger distance between the two lines would mean a much smaller probability of intersection. We can see in case 3 that when the distance between the lines is greater than the length of the needle, even very large angle won’t necessarily cause an intersection.

This natural method of random motion in nature allows the ants to gauge the size of their potential new hill regardless of its shape. Scout ants are even able to asses the area of a hill in complete darkness. The animals show that algorithms can be used to make decisions where an array of restrictions may prevent other methods from being effective.

# References

[4] The Number Pi. Eymard, Lafon, and Wilson.

[5] Monte Carlo Methods Volume I: Basics. Kalos and Whitlock.

[6] Heart of Mathematics. Burger and Starbird


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# Analysis of Plates - Special Topics

This page provides the chapters on plates under combined loadings, buckling of oblique plates, and buckling of sandwich panels from the "Stress Analysis Manual," Air Force Flight Dynamics Laboratory, October 1986.

Other related chapters from the Air Force "Stress Analysis Manual" can be seen to the right.

## Nomenclature

 a = plate length b = plate width c = core thickness, signifies clamped edge C = compressive buckling coefficient for curved plates E = modulus of elasticity F = stress Fcr = critical normal stress Fcrs = critical shear stress Fcy = compressive yield stress
 kc = compressive buckling coefficient K = sandwich panel form factor R = stress ratio t = thickness Zb = length range parameter b2(1 - νe2)1/2/ rt η = plasticity reduction factor η = cladding reduction factor ν = inelastic Poisson's ratio

$$R_1^{~x} + R_2^{~y} = 1$$
(6-36)

Table 6-12 gives the combined loading condition for flat plates. Figures 6-36 and 6-37 give interaction curves for several loading and support conditions. It is noted that the curves present conditions of triple combinations.

Figure 6-38 presents buckling coefficients for right angle isosceles triangular plates loaded under shear and compression. Equation (6-37) is the interaction equation for shear and normal stress on this type of plate.

$$\left( { 2 ~F_s \over F_{crs+} + F_{crs-} } + u \right)^2 + { F \over F_{cr} } (1 - u^2) = 1$$
(6-37)

The + and − subscripts refer to either tension or compression along the altitude upon the hypotenuse of the triangle caused by pure shear loading. Table 6-13 contains values of kc, ks+ and ks− for various edge supports.

Table 6-13: Buckling Coefficients for Right-Angle Isosceles Triangular Plates Loaded Independently in Uniform Compression, Positive Shear, and Negative Shear
Edge Supports (Note a) kc ks+ ks−
All edges simply supported 10.0 62.0 23.2
Sides simply supported,
hypotenuse clamped
15.6 70.8 34.0
Sides clamped, hypottenuse
simply supported
18.8 80.0 44.0
Note a:  Hypotenuse = b in Figure 6.38

For curved plates under combined axial loading and shear with 10 < Zb < 100 and 1 < a/b < 3, the interaction relation of Equation (6-39) may be used.

$$R_s^{~2} + R_x = 1$$
(6-39)

This may be used for either compression or tension with tension being denoted by a negative sign.

We have a number of structural calculators to choose from. Here are just a few:

## 6.9 Buckling of Oblique Plates

In many instances, the use of rectangular panels is not possible. Figures 6-39 and 6-40 give buckling coefficients for panels which are oriented oblique to the loading. Figure 6-39 covers flat plates divided into oblique parallelogram panels by nondeflecting supports. Figure 6-40 covers single oblique panels.

## 6.10 Sample Problem - Plate Analysis

Find: The buckling stress of a flat plate under uniform longitudinal compression, simply supported on all four sides.

Givena = 12 in, b = 4 in, t = 0.100 in.

Material:  Bare 2024-T3 Sheet

 Material Properties: E = 10.7 x 106 psi ν = 0.33 Fcy = 34,000 psi

Solution:

$$F_{cr} = \overline{\eta} ~\eta ~{ k_c ~\pi^2 E \over 12 (1 - \nu^2) } \left({t \over b}\right)^2$$

From Figure 6-1 for a/b = 3, kc= 4.00.

 η = 1.0 for no cladding η = 1.0 for elastic buckling
$$F_{cr} = { (4.0) (\pi^2) (10.7) 10^6 \over 12 [1 - (0.33)^2] } \left({100 \over 2}\right)^2$$
 Fcr = 24,600 psi

As this is below the compression yield strength, no allowance for elasticity need be made.

We have a number of structural calculators to choose from. Here are just a few:

## 6.11 Buckling of Sandwich Panels

The use of sandwich construction for skin panels of aircraft has become commonplace. Panel construction consists of primary and secondary skins separated by a core which is usually some form of honeycomb or expanded foam material. The critical buckling stress of such a composite can be calculated from Equation (6-40).

$$F_{cr} = C ~K ~F_{crf}$$
(6-40)

The quantity Fcrf is the buckling stress of the two faces if they were not connected by the core. Use the total thickness of the faces. The coefficient C is obtained from Figure 6-41 and K is a form factor given in Equation (6-41).

$$K = 1 + 3 \left( 1 + {c \over t} \right)$$
(6-41)

The data in Figure 6-41 is given as a family of curves of parameter R which is given in Equation (6-42).

$$R = { G_{core} \over K ~F_{crf} }$$
(6-42)


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# Frozen map + 'Message filter dropping' - Issues with mapping and navigation in ROS2 Foxy

sorry for the long post, just want to make sure i include all the details.

I am working on learning ROS2 by building a robot which maps and navigates my apartment.

• I'm using the RPLidar A1 sensor with the RPLidar package

• a Robot Create 2 as the chassis with the create_robot package

• nav2 for navigation

• slam-toolbox for mapping

and running everything off of a Raspberry Pi 4 4GB with Ubuntu Mate

Currently, I am able to get LIDAR readings to successfully be viewed in RVIZ. I can also get one or two frames of a map to be built. When the robot starts to move, the lidar readings update, but the map itself does not. However, i continue to 'receive' map updates at about 1 every 1.5 seconds.

Here are images from rviz showing the moving scan but with the non-moving map: https://imgur.com/a/acfHs4I

I'm also getting a few errors, from RVIZ i am getting: [rviz]: Message Filter dropping message: frame 'map' at time 1619194935.882 for reason 'Unknown'

from nav2, I am getting [global_costmap.global_costmap_rclcpp_node]: Message Filter dropping message: frame 'laser' at time 1619194959.629 for reason 'Unknown'

My frame map looks like this (Most of the frames are created by the create_robot package): https://imgur.com/a/WFDXKxj

I suspect my issue is with my transforms, either they're configured completely wrong, or there is something small

• slam_toolbox does the transform from map -> odom
• the create_robot package has other transforms from base_link to wheels, bumpers, etc and it also publishes odom data
• using tf2, i'm publishing a transform between base_link and laser and then laser to base_scan.
• I am also creating a transform from odom to base_footprint because of a ros2 navigation course i was taking on the construct.

Has anyone seen this type of error before? I feel like I am most of the way to the solution but i'm missing something easy

I ended up solving my issue.

Turns out, the create_robot package publishes a transform from odom -> base_footprint. I needed to toy around with my transforms so that:

slam-toolbox created map -> odom

create_robot created odom -> base_footprint

On top of those two things, I had an issue with timing where the transform from map -> base_link did not work either. There was an issue with timings where I was getting the error:

[INFO] [1619986350.507533125] [tf2_echo]: Waiting for transform map -> base_footprint: Lookup would require extrapolation into the past. Requested time 0.200000 but the earliest data is at time 1619986346.813953, when looking up transform from frame [base_footprint] to frame [map]

To solve this error, I just had to call nav2 with the argument use_sim_time:=false

Now that I've done this, everything seems to be working completely fine.


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# Dividing Decimals

You'll learn methods for dividing decimals by a whole number, by another decimal, and by a fraction. It's really easy once you get the hang of it, and it will repay your investment in time and effort many folds as you progress in learning Algebra.

## Decimal Division By A Whole Number

Dividing a decimal by a whole number can be done reliably (that is, with minimum chance of making mistakes) using long division. Of course, once you've had enough practice, you can use shorter and more advance techniques to get the answers faster.

You'll see how to do long division in the examples below. And yeah, be sure to line up the decimal points vertically.

Example
Divide the following.
(a) 73.8 ÷ 12
(b) 143.55 ÷ 15

Solution

Notice that the decimal points are in line. No mis-alignments like shown below.

## Decimal Division By A Decimal

When dividing a decimal with a decimal, we can reuse the techniques we learned from dividing a decimal by a whole number. Simply change the divisor (at the right side of the ÷) into a whole number first by multiplying both divisor and dividend with the same number, then proceed with the long division.

Example
Divide the following.
(a) 24.032 ÷ 16
(b) 0.304 8 ÷ 0.48
Solution
(a) First, convert the denominator (1.6) into a whole number by multiplying both the numerator (24.032) and denominator by 10.

Therefore, 24.032 ÷ 1.6 = 15.02

Therefore, 0.304 8 ÷ 0.48 = 0.635

## Decimal Division By The Powers of 10

Dividing decimals by powers of 10 (e.g. 0.001, 0.01, 0.1, 10, 100, 1000, and so on) is a special case. You can use long division to solve it, but it's not the fastest and most efficient way to do it. Here's a method that can only be used in this case.

When dividing a decimal with a power of 10, shift the decimal to the left according to the number of zeros in the power of 10. For instance,

When dividing a decimal with 0.1, 0.01 or 0.001, shift the decimal point to the right according to the number of decimal places in 0.1, 0.01 or 0.001. For instance,

Example
Divide the following.
(a) 2.033 ÷ 1000
(b) 19.253 ÷ 0.001

Solution
(a) 2.033 ÷ 1000 = 0.00233
(b) 19.253 ÷ 0.001 = 19253

## Here's a Tip...

When multiplying or dividing a decimal by a power of 10 (10, 100, 1000, …) or a decimal like 0.1, 0.01 or 0.001, the decimal point can be shifted as follows:

## Decimal Division By A Fraction

It's not difficult to divide a decimal by a fraction. You'll first need to change “÷” into “×”, `then change the fraction into its reciprocal. It's a fancy way of saying that you need to flip the fraction upside down (numerator becomes the denominator, and vice versa). Solve it after that. See the example below.

Example
Divide the following.

Solution

## Dividing Decimals Word Problems

Example
A rope is 32.16 m long. It is cut into identical pieces, each measuring 8.04 m long. How many pieces were cut from the rope?

Solution

Return from Dividing Decimals to Arithmetic

Return Home to Algebra-by-Example.com

## Unclear about a concept? Or see something missing?

Do you have questions about what you've read in this section? Is something missing? Post your question or contribution here and I'll get to it as soon as I can.

If you're sending in a problem you need help with, please also describe what you've tried to get a response.

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## Introductory Statistics 9th Edition

a. (i) $x±2s, k=2, 1-\frac{1}{2^2} = 3/4*100 = 75% 2365±2(340) = [1685, 3045]$ (ii) $x±3s, k=3, 1-\frac{1}{3^2} = (1-1/9)\times100 = 88.89% 2365±3(340) = [1345, 3385]$ b. $1-1/(k^2) = 0.84$ $K^2 = 1/0.16$ $K = 2.5$


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# Gravitational acceleration $g$

All objects fall equally fast (with the same acceleration) due the gravity, regardless of their mass (ideally; in a vacuum). The acceleration everything falls with is called the gravitational acceleration $g$. On Earth’s surface, it has a value around:

$$g\approx 9.8\,\mathrm{m/s^2}$$

The value varies with around $\pm 0.02\,\mathrm{m/s^2}$ depending on location.


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# Permutation Groups Containing non-commuting $p$-cycles

I noticed that the following is true, and that there is a reasonably elementary proof of it (in particular, the classification of finite simple groups is not needed). Let $G$ be a finite permutation group which contains two $p$-cycles which do not commute (where $p$ is any odd prime other than a Mersenne prime). Then $G$ is not solvable ( more precisely, $G$ has a non-Abelian composition factor of order divisible by $p$). Since the methods are reasonably elementary, I wonder if anyone has come across this or similar results (possibly in a Galois Theory context) and can point me to a reference?

For every Mersenne prime $p,$ there is a solvable permutation group $G$ of degree $p+1$ and order $p(p+1)$ which contains two non-commuting $p$-cycles.

(Later note: The analogous result is not true for $p^{2}$-cycles ($p$ prime). For $p=2$ take $G = S_{4}$ and for $p>2$ take $G$ to be a Sylow $p$-subgroup of $S_{p^{2}}.$ In each case, $G$ is a solvable (even nilpotent when $p$ is odd) permutation group containing two $p^{2}$-cycles which do not commute).

• So you say e.g. that ${\rm A}_4$ is solvable because $3$ is a Mersenne prime? – Stefan Kohl Jun 18 '14 at 15:36
• Well, I don't say that, but the smallest exceptional case happens to be all of $A_{4.}$ But in general, if $p = 2^{n}-1$ is a prime, the the semidirect product $EC$ is solvable, where $E$ is elementary Abelian of order $2^{n}$ and $C$ acts on $E$ as a Singer cycle in ${\rm GL}(n,2).$ Furthermore, in its action on its Sylow $p$-subgroups by conjugation, $EC$ is a permutation group of degree $p+1$ and contains $p$-cycles which don't commute. – Geoff Robinson Jun 18 '14 at 16:58
• It won't be too surprising if Camille Jordan knew this result already (due to en.wikipedia.org/wiki/Jordan's_theorem_(symmetric_group)) – Dima Pasechnik Jun 19 '14 at 12:34
• Maybe I am missing something, but I think a permutation group $G=\langle \sigma, \tau \rangle$ generated by two non-commuting $p$-cycles is primitive: If $\Delta$ is a block, then $\Delta \cap \operatorname{Mov} \sigma$ is empty or a block for $\langle \sigma \rangle$, and the same holds for $\tau$. By going through the different possibilities, I get that $\Delta$ must be trivial (or did I forget a case?). Of course, degrees p, p+1, p+2 remain. (I would be interested in a hint or sketch of your proof, by the way.) – Frieder Ladisch Jun 19 '14 at 15:20
• @FriederLadisch: Need to prove G not p-solvable. Suppose oherwise. Can assume G = <x,y> for non-disjoint p-cycles& that deg G < 2p, G transitive. Hence <x> and <y> are Sylow. Op(G) = 1, Op'(G) is not 1.Then Hall-Higman type arguments get to G = <x>E, where E is elementary or special q-group some prime q ( NB x centralizes every proper x-invariant subgroup of E). H = point stabilizer, corefree. Can eliminate E non-Abelian with care as H meets Z(E) \leq Z(G) trivially in that case. Case E Abelian leads to H = <x>, [G:H] = p+1 = |E|, so p is Mersenne. – Geoff Robinson Jun 19 '14 at 16:04

Fixed several inaccuracies, many thanks to Frieder Ladisch for spotting them:

Let $x$ and $y$ be two non-commuting $p$-cycles, $G=\langle x,y\rangle$, and $G$ be considered as a transitive permutation group on the support $\Omega$ of $G$.

We show that either $\text{AGL}_1(\mathbb F_q)\le G$ for a Mersenne prime $q$, or $G$ is simple non-abelian.

Proof. $G$ is primitive (as pointed out by Frieder Ladisch already). This can be seen as follows: Let $\Delta$ be a block of a non-trivial block system. The action of the $p$-cycle $x$ on the block system is trivial, for otherwise $x$ would move $p\lvert\Delta\rvert>p$ points. On the other hand $G$ transitively moves the blocks, a contradiction.

Next we show that $G$ is doubly transitive. This follows from Burnside's classical theorem if $\lvert\Omega\rvert=p$. If $\lvert\Omega\rvert>p$, then the pointwise stabilizer in $G$ of the points fixed by $x$ is transitive (via $x$) on the remaining points. By Jordan's Theorem on primitive groups with Jordan sets we again see that $G$ is doubly transitive.

Let $N$ be a minimal normal subgroup of $G$. By Burnside, $N$ is either (a) elementary abelian and regular, or $N$ is (b) simple, primitive, and not regular. The Mersenne exceptions follow from looking at $p$-cycles in $\text{GL}_m(q)$ where $n=q^m$ for a prime $q$: Such a $p$-cycle fixes $q^r<q^m$ points, hence $p=q^r(q^{m-r}-1)$. We obtain $r=0$ and $q=2$. By Schur's Lemma, we can identify $\langle x\rangle$ with the multiplicative group of $\mathbb F_q$. This yields $\text{AGL}_1(\mathbb F_q)\le G$ as claimed. (If one uses Kantor's paper on Singer cycles, then one gets more precisely the possibilities $G=\text{AGL}_1(\mathbb F_q)$, $\text{A$\Gamma$L}_1(\mathbb F_q)$, and $\text{AGL}_m(\mathbb F_2)$. However, that paper relies on a wrong paper of Cameron/Kantor, see here.)

Now assume case (b). We show that $G=N$, so $G$ is actually simple. In order to do so, we show that $p$ divides the order of $N$. Note that $\lvert\Omega\rvert<2p$, so $p^2$ does not divide $\lvert N\rvert$, hence $x,y\in N$ in this case.

The case $p=\lvert\Omega\rvert$ is clear.

So $p<\lvert\Omega\rvert$ from now on. We let $\omega$ be a fixed point of $x$, and set $\Omega'=\Omega\setminus\{\omega\}$.

First suppose $p=\lvert\Omega\rvert-1$. The point stabilizer $N_\omega$ is a normal subgroup of $G_\omega$. As $G_\omega$ is transitive on $\Omega'$, all orbits of $N_\omega$ on $\Omega'$ have equal length. But $\lvert\Omega'\rvert=p$, so these orbit lengths are either $1$ or $p$. The former cannot hold, because then $N$ were regular on $\Omega$. Thus $N$ is doubly transitive on $\Omega$, so $p=\lvert\Omega\rvert-1$ divides $\lvert N\rvert$.

If $p<\lvert\Omega\rvert-2$, then $N=\text{Alt}(\Omega)$ by Jordan, so $G=N$ again.

If $p=\lvert\Omega\rvert-2$, then $G_\omega$ contains the cycle $x$ of length $p=\lvert\Omega'\rvert-1$ on $\Omega'$, so $G_\omega$ it is doubly transitive on $\Omega'$. The argument in the case $p=\lvert\Omega\rvert-1$ shows that $N_\omega$ is either doubly transitive on $\Omega'$, or regular. In the former case $p=\lvert\Omega'\rvert-1$ divides $\lvert N_\omega\rvert$, hence $G=N$ again. In the latter case, $N$ is sharply doubly transitive on $\Omega$, which implies that $N$ has a regular normal subgroup, contrary to $N$ being simple. (For this last step, simple counting suffices. One does not need Frobenius' theorem about the existence of Frobenius kernels.)

• The case $|\Omega|=p$ is actually somewhat easier: if $N$ is a minimal subgroup of $G$, then $N$ is transitive and thus $x, y\in N$, so $G=N$ is simple. In the general "almost simple" case, is it clear that the order of the unique minimal normal subgroup $N$ is divisble by $p$ (and thus $G=N$ simple), without invoking Hall-Higman or similar arguments? Because this is what Geoff's argument shows: Either $G$ is simple, or $G \leq \operatorname{AGL}(m,2)$, where $p=2^m-1$. – Frieder Ladisch Jun 24 '14 at 14:26
• Thanks, but I can't follow your argument in the case $n=p+1$. I think this is exactly the case where one needs something like Hall-Higman reduction to see that if $p$ does not divide $N$, then $N$ is not non-abelian simple. Also I don't know what you mean by "repeating the argument with $N$ instead of $G$" in case $p=n-2$. In that case, I have the following argument: if $N$ is $p'$, then $C_N(x)$ is transitive on the two elements fixed by $x$ (Lemma of Glauberman on coprime action), thus $N$ contains a 2-cycle, contradiction since $G\leq A_n$. – Frieder Ladisch Jun 24 '14 at 16:07
• @GeoffRobinson: I rewrote the arguments. Hope the case $n=\lvert\Omega\rvert=p+1$ is clearer now. – Peter Mueller Jun 25 '14 at 8:37
• @Peter Müller: I find your argument for $p= |\Omega|-1$ very simple and elegant. In case $p = |\Omega|-2$ I think one still needs to say why $N_{\omega}$ can not be regular on $\Omega'$, for example, because otherwise $N$ would be a doubly transitive Frobenius group, thus not simple. – Frieder Ladisch Jun 25 '14 at 11:18
• (And in the Mersenne case, also $G= \operatorname{ASL}(m,2)$ and $G= \operatorname{A\Gamma L}(1,2^m)$ is possible.) – Frieder Ladisch Jun 25 '14 at 11:21


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# Volume of Fibonacci Tetrahedron

### Solution

There is no tetrahedron,cube,tetrahedron,parallelipiped with the indicated vertices. Moreover, the solution below covers a more general problem: there is no tetrahedron with "Fibonacci" vertices $(F_{m}, F_{m+1}, F_{m+2}),\,$ $(F_{n}, F_{n+1}, F_{n+2}),\,$ $(F_{p}, F_{p+1}, F_{p+2}),\,$ $(F_{q}, F_{q+1}, F_{q+2}),\,$ for any,any,some,successive four integers $m, n, p, q.\,$

The reason for this is that any such vertex lies in the plane $x + y = z,\,$ such that all four are coplanar,collinear,concyclic,coplanar. Even accepting that coplanar points may form a tetrahedron, the volume of the latter is bound to be zero,zero,infinite.

### Fibonacci Parallelepiped

Derek Orr has suggested a modification that deals with vectors:

Let $\mathbf{f_k}=(F_k,F_{k+1},F_{k+2}).\,$ Find the volume of the parallelepiped form by vectors $\mathbf{f_m},\,$ $\mathbf{f_n},\,$ and $\mathbf{f_p},\,$ for distinct integers $m,n,p.$

### References

1. C. W. Trigg, Mathematical Quickies, Dover, 1985, #209


text:
# American Institute of Mathematical Sciences

October  2015, 20(8): 2477-2495. doi: 10.3934/dcdsb.2015.20.2477

## Computation of local ISS Lyapunov functions with low gains via linear programming

 1 School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan, China 2 Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany, Germany 3 School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101 4 Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany

Received  June 2014 Revised  March 2015 Published  August 2015

In this paper, we present a numerical algorithm for computing ISS Lyapunov functions for continuous-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on a linear programming problem and computes a continuous piecewise affine ISS Lyapunov function on a simplicial grid covering the given compact set excluding a small neighborhood of the origin. The objective of the linear programming problem is to minimize the gain. We show that for every ISS system with a locally Lipschitz right-hand side our algorithm is in principle able to deliver an ISS Lyapunov function. For $C^2$ right-hand sides a more efficient algorithm is proposed.
Citation: Huijuan Li, Robert Baier, Lars Grüne, Sigurdur F. Hafstein, Fabian R. Wirth. Computation of local ISS Lyapunov functions with low gains via linear programming. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2477-2495. doi: 10.3934/dcdsb.2015.20.2477
##### References:

show all references

##### References:
 [1] Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 [2] Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 [3] Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 [4] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [5] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [6] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027 [7] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [8] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [9] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 [10] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [11] Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931 [12] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [13] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [14] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [15] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [16] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [17] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 [18] Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933 [19] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [20] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

2019 Impact Factor: 1.27


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# Combine WienerProcess and WhiteNoiseProcess

I'm trying to simulate an IMU (inertial measurement unit) using a simple model:

$$\tilde a(t) = a(t) + b(t) + c(t)$$

where $$\tilde a$$ is the measured value, $$a$$ is the true value, $$b$$ is the random walk bias simulated as Wiener process and $$c$$ is Gaussian white noise:

$$\dot b(t) = \sigma_b w(t)\\ c(t) = \sigma_c w(t)$$

where $$w(t)$$ is Gaussian white noise process with mean 0 and standard deviation 1.

This is modeled as

P=TransformedProcess[b[t]+c[t],
{b\[Distributed]WienerProcess[0,0.1],
c\[Distributed]WhiteNoiseProcess[]},t];

However, MMA complains that the discrete and continuous model cannot be mixed when I plot the data

ListPlot[RandomFunction[P, {0, 10, 0.01}]]

The processes in {[FormalP]1[Distributed]WienerProcess[0,0.1],[FormalP]2[Distributed]WhiteNoiseProcess[NormalDistribution[0,1]]} should be all continuous-time or all discrete-time.

Of course, we can generate the data separately using WienerProcess and WhiteNoiseProcess, but I'm wondering what is the correct way to do this symbolically so that I can utilize all the functions associated with the random process (RandomFunction, CovarianceFunction, etc.) to analyze the model?

• Where does the true value a[t] come from? Jul 14, 2019 at 9:51
• It's a predefined function. Here it is 0. Jul 14, 2019 at 18:30

Which part of the IMU are you trying to simulate...just the signals of ax, ay, az, gx, gy, gz? Or Maybe them fused together to get an angle of some kind? You didn't post all of your code, so I can't really see how you're combining your discrete and continuous models....

I have done a similar process when designing filters for an inverted pendulum I made once.

\[Alpha] = 0.2
\[CapitalDelta]t = 0.01

dgl = \[Phi]''[t] + g/l Sin[ \[Phi][t]] == -0.07 ArcTan[100 Derivative[1][\[Phi]][t]] - 0.4 Derivative[1][\[Phi]][t] + u[t];

nsys = NonlinearStateSpaceModel[{dgl}, {{\[Phi][t], 0}, {\[Phi]'[t], 0}}, {u[t]}, {\[Phi][t]}, t] /. {g -> 9.81, l -> 2};
ssm = StateSpaceModel[{dgl}, {{\[Phi][t], 0}, {\[Phi]'[t], 0}}, {u[t]}, {\[Phi][t]}, t] /. {g -> 9.81, l -> 2};

A noise function which will create whitenoise when set to 0.

noise[n_Integer /; n >= 1, \[Beta]_?NumericQ] := Re[InverseFourier[(RandomVariate[NormalDistribution[], {n, 2}].{1, I})*Prepend[Range[n - 1]^(\[Beta]/2), 0]]]

Generate discrete signals $$\phi(t),\omega(t)$$ and add white noise via noise[dataset length, 0] and $$\alpha$$ was used as a scaling value to reduce noise amplitude.

{sig1, sig2} = StateResponse[{ToDiscreteTimeModel[nsys, \[CapitalDelta]t]}, {-4 UnitBox[3 (t + -1.5)]}, {t, 0, 12}];
data1 = noise[Length[sig1], -0]/15;
data2 = noise[Length[sig2], -0]/15;

phi = Table[{\[CapitalDelta]t i, (sig1[[i + 1]]/(1 10^-\[Alpha]) + data1[[i + 1]])/(1 10^\[Alpha])}, {i, 0, Length[sig1] - 1}];
omega = Table[{\[CapitalDelta]t i, (sig2[[i + 1]]/(1 10^-\[Alpha]) + data2[[i + 1]])/(1 10^\[Alpha])}, {i, 0, Length[sig1] - 1}];

Plot!

ListLinePlot[{phi, omega}, PlotRange -> {{0, 12}, {-1.7, 1.7}}, ImageSize -> Large]

You can make this 'continuous' via Interpolation if needed...

angle = Interpolation[phi]
dotangle = Interpolation[omega]

This would be a noisy measured signal...though but it wouldn't be much different for your system simplying adding your drift b[t]...But this is a typical way to go about adding noise to a simulated signal.

There is also under NonlinearStateSpaceModel -> Applications -> Extended Kalman

An example of how a signal is generated, a noisy signal added, and then filtered via EKF.

You can create a TransformedProcess which time-differences a"Continuous" Wiener Process. Remarkably this matches the Discrete WhiteNoiseProcess exactly!!!

\[CapitalDelta]\[ScriptCapitalW]\[ScriptCapitalP][\[Sigma]_, \
\[Delta]_, k_] :=
TransformedProcess[
k (Subscript[B, t][t + \[Delta]] - Subscript[B, t][t]),
Subscript[B, t] \[Distributed] WienerProcess[0, \[Sigma]], t]
\[Sigma]\[Sigma] = 1; \[Delta]\[Delta] = 1/100; kk = 1;
SeedRandom[2]; data1 =
RandomFunction[\[CapitalDelta]\[ScriptCapitalW]\[ScriptCapitalP][\
\[Sigma]\[Sigma], \[Delta]\[Delta], kk], {0, 10, \[Delta]\[Delta]}, 1];
SeedRandom[2]; data =
RandomFunction[
WhiteNoiseProcess[kk Sqrt[\[Delta]\[Delta]] \[Sigma]\[Sigma]], {0,
10/\[Delta]\[Delta]}];
Max[Abs[data["ValueList"][[1]] - data1["ValueList"][[1]]]]

2.22045*10^-16


text:
Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Shape and Territory

Age 16 to 18 Challenge Level:

Why do this problem?
It provides practice in using the tan formula and an opportunity to reflect on the bigger picture beyond school mathematics.

Possible approach
The result can be proved without using the fact that the angles add up to 180 degrees and this could be taken up as a challenge.

Then ask "What is the 'territory' this result belongs to?"

Sue's proof given here uses the fact that the angles of the triangle $ABC$ add up to 180 degrees. However it is just as easy to prove this result without using the fact about the sum of the angles of the triangle so it must be true for triangles 'living in other geometries' where the angles of triangles do not add up to 180 degrees such as Spherical Geometry.

The lines in Spherical Geometry are great circles on the surface of the sphere. By drawing lines like the lines of longitude and the equator on the earth you will soon be able to convince yourself that spherical triangles have angle sums greater than 180 degrees.

This expression gives a certain property for triangles for all 3 geometries, for Euclidean Geometry where the angles of triangles add up to 180 degrees, for Spherical (also called Elliptical) Geometry where the angles of triangles add up to more than 180 degrees and for Hyperbolic Geometry where the angles of triangles add up to less than 180 degrees.

Key question
Here we have $(A-B)$, $(B-C)$ and $(C-A)$. How can we write this using only 2 variables?

Possible extension
See the articles Strange Geometries which is accompanied by a lot of ideas for project work suitable for 12 year olds and older students on Non-Euclidean Geometries.
See also the articles How Many Geometries Are There? and
When the Angles of a Triangle don't add up to 180 degrees.


text:
## Learning from mistakes

[Inspired by Dennett’s Intuition Pumps]

My current approach and where I’d like to be works 1:1 or in a small group, much more difficult to do in a lecture-style setting.

When I’m trying to teach a new concept – programming, algorithm, mathematics – I try to give students a bit of structure and context and get them to develop the solution. The amount of structure and context varies from student to student, and its not easy getting it right. Done well, students reach that aha! moment on their own. I’ve been doing this unconsciously for quite a while, but its probably time to do this with more structure (hah!).

One more aspect I would like to add is having students reflect on failed approaches. This may also require reducing the context and allowing them to try out a variety of approaches.  A lot of learning can happen in this analysis. The downside is that it needs time and it needs willingness to expend the effort.

I’ll eventually organize my thoughts, but a few reactions:

• This book should be dedicated to Steven Pinker. No matter what aspect of business one may pick up, here’s how to look at it in a positive light
• The amount of data to back up the positivity is of course, amazing
• I’m going to skip the ‘what-about-ism’s, but that is pretty much what was going through my head as I was reading this book

This book is necessary – (big) business has been vilified for too long, and the default is to assume that it is evil. This book pushes the pendulum towards the center and swings it all the way to the other end!

Transcript

## Background

This is the first article I recommend everyone read (currently, ‘everyone’ refers to students or recent graduates). Every paragraph is full of insights. It encourages a proactive approach to the world around us. It lays out an approximate framework for anyone interested in thinking about what they are doing, and where they would want to be.

The following notes are a summary of my observations and formulations from the article. These are a combination of the obvious (“he says it right in the paper!”) and bringing together concepts scattered across. It is a work in progress!

## Why do great research work

Unlike the paper, I believe each of us has to define what ‘great’ is. We should be doing work that matters to us, and this needs to lead to significant scientific breakthroughs. Also, the insights apply equally to work (as opposed to research)

## The (mis-)characteristics of a great scientist

What is common among the great scientists? The lazy narrative is that they achieved greatness because of:

• Luck: Somethings are happenstance, but for the most part, make your own luck!
• Fortune favors the prepared mind – Louis Pasteur
• If others would think as hard as I did, then they would get
similar results – Issac Newton
• Brains: Most likely we have enough for what we set out to do. It just needs unlocking.
• Age: Depends on what we are doing. In some cases, the audacity of youth is needed to challenge existing dogma, on others, the wisdom of maturity is required to tease out insights.
• Drive: This is so important, I discuss it separately below.
• Ambiguity: College exams are typically re-statements of problems already seen. However, the real world is rarely straightforward.
• Commitment: this is necessary to get the subconscious to work for us. And this leads to that nebulous factor called creativity.

## Drive

There are so many things to unpack from this one paragraph! In fact, this was the one thing that I remembered from the first reading of the paper. Here’s how I visualize one part:

Knowledge and productivity are like compound interest. …. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity.

Hamming anticipates Gladwell’s 10,000-hour rule, noting the importance of effort intelligently applied.

## Ego and Perspective

I recall colleagues who insisted on doing things their way, no matter the effort and energy it would consume. Much better to figure out what is important, which fights you want to fight and drop the rest.

## Be proactive

Hamming does not explicitly say this, but throughout the talk, we find examples of him identifying areas of improvement and actively working on these. Examples are:

• He has lunch with other teams and has an agenda of what he wants to get out of these interactions.
• He consciously sets aside time for ‘great thoughts’ (in a sense anticipating Google’s 10% approach).
• He starts out with being nervous giving a speech, so volunteers to do more of them!
• Opportunity

## The ‘Big Picture’ view

Closely related to the above point is Hamming’s consistent approach to thinking about the underlying structure of the reality that he observes (the entire talk is an example of this, of course). Why do some scientists do great work? What is the larger problem that I am trying to solve? How can we generalize this work? Is it better to do x instead of y. Why? (an example: work with the door closed or open?). Who is more effective in a meeting? Why?

Too many times we just accept things as they are. It’s a seemingly small step to question things, but the consequences can be huge.

## The interplay of factors contributing to great work

Great Work $=\left(d\times c\right)^a$
Where

$d =$ drive
$c =$ commitment
$a =$ ability

The factors that are in our control — drive and commitment are the base of the exponent. The more we invest, the greater the returns. Our ability is the exponent, and this can grow if we put in the effort.

## This Happened…

Eliud Kipchoge ran a sub-2 hour marathon

Transcript

# Background

This is the first article I recommend everyone read (currently, ‘everyone’ refers to students or recent graduates). Every paragraph is full of insights. It encourages a proactive approach to the world around us. It lays out an approximate framework for anyone interested in thinking about what they are doing, and where they would want to be.

The following notes are a summary of my observations and formulations from the article. These are a combination of the obvious (“he says it right in the paper!”) and bringing together concepts scattered across. It is a work in progress!

## Why do great research work

Unlike the paper, I believe each of us has to define what ‘great’ is. We should be doing work that matters to us, and this needs to lead to significant scientific breakthroughs. Also, the insights apply equally to work (as opposed to research)

## The (mis-)characteristics of a great scientist

What is common among the great scientists? The lazy narrative is that they achieved
greatness because of:

• Luck: Something are happenstance, but for the most part, make
• Fortune favors the prepared mind – Louis Pasteur
• If others would think as hard as I did, then they would get
similar results – Issac Newton
• Brains: Most likely we have enough for what we set out to do. It
just needs unlocking
• Age: Depends on what we are doing. In some cases, the audacity
of youth is needed to challenge existing dogma, on others, the wisdom of
maturity is required to tease out insights
• Drive: This is so important, I discuss it separately below
• Ambiguity: College exams are typically re-statements of problems
already seen. However, the real world is rarely straightforward.
• Commitment: this is necessary to get the subconscious to work
for us. And this leads to that nebulous factor called creativity.

## Drive

There are so many things to unpack from this one paragraph! In fact, this was the one
thing that I remembered from the first reading of the paper. Here’s how I
visualize one part:

Knowledge and productivity are like compound interest. …. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity. Hamming anticipates Gladwell’s 10,000-hour rule, noting the importance of effort intelligently applied.

## Ego and Perspective

I recall colleagues who insisted on doing things their way, no matter the effort and energy it would consume. Much better to figure out what is important, which fights you want to fight and drop the rest.

## Be proactive

Hamming does not explicitly say this, but throughout the talk, we find examples of him identifying areas of improvement and actively working on these. Examples are:

• He has lunch with other teams and has an agenda of what he wants
to get out of these interactions
• He consciously sets aside time for ‘great thoughts’ (in a sense
• He starts out with being nervous giving a speech, so volunteers
to do more of them!
• Opportunity

## The ‘Big Picture’ view

Closely related to the above point is Hamming’s consistent approach to thinking about the underlying structure of the reality that he observes (the entire talk is an example of this, of course). Why do some scientists do great work? What is the larger problem that I am trying to solve? How can we generalize this work? Is it better to do x instead of y. Why? (an example: work with the door closed or open?). Who is more effective in a meeting? Why?

Too many times we just accept things as they are. It’s a seemingly small step to question things, but the consequences can be huge.

## The interplay of factors contributing to great work

Great Work $= \left(d\times c\right)^a$
Where d = drive
c = commitment
a = ability

The factors that are in our control — drive and commitment are the base of the exponent. The more we invest, the greater the returns. Our ability is the exponent, and this can grow if we put in the effort.

## Bringing Sanity to the Campus Recruitment process

For the last 2 months, my interns have
been MIA, because “companies are coming to campus.” When I speak with students
about the process that they (the companies) follow and the torture that they
(the students) undergo, a few things become apparent:

1. Every company does things its own way. In broad terms, the process is similar, but the details are different enough to make each one unique. To an outsider, these processes seem to be ad-hoc. None of them are like what Shell does (incidentally, Shell recruitment for an IT position also doesn’t seem to follow the normal evaluation process!). Consequently, replacing one process with another won’t make much of a difference in terms of outcomes (see the next points)!
2. I’m curious if any company examines the effectiveness of its selection process. Wouldn’t it be great if they kept the evaluations (and hence expectations) of candidates, and then revisit the data in a year or two to figure out how accurate they were? And then tune the process to improve the outcomes for the company. Google did this a few years back with not-so-surprising results (link).
3. Related to the above, what is the cost-benefit tradeoff of this exercise?
4. In all fairness to the interviewing process, it is difficult to judge the suitability of each candidate in a packed 1-day schedule. But still…. a few of my students would be fantastic additions to any team, and they’ve been passed over.

From the student’s perspective, they pretty much lose out on two months of time that could be spent on more productive endeavors. And don’t even get me started on the stress that they put themselves through as they do not get selected with each successive company.

A serious drawback of the
current system is the power difference between companies and students.
Companies fight for the privilege of being first on campus. They can accept or
reject students from the pool that applies. Once a student has been accepted,
they can no longer apply to other companies. They may have preferred a company
that comes to campus later, but didn’t want to risk not applying for the first
company! So students end up interviewing for companies in the order that the
companies are scheduled, not in the order that they prefer!

To
summarize, the current process does not seem efficient to the company and even
less so for students. Here is a modest proposal based on the observation that
most positions at the entry-level fall into a few categories (developer, IT
analyst, customer support, …) and technologies (cloud, databases, Java,
Android, …). From the cross-product of role and technologies above,
administer one set of evaluations that cover aptitude, technical
knowledge, specialized skills (e.g. implementation) and an interview. Companies
can use these results and calculate their own weighted scores to rank students.
Students also rank companies based on their criteria — job profile, company
profile, salary, etc. You can guess where this is going: this is an instance of
stable matching and
we can use some variant of Gale-Shapely to generate an optimal matching
(whether this is optimal for the student or the company is left as an exercise

Students/companies end up with the best
possible outcome. A student gets the best job that they can get from the list
requirements who want to work for them. The process takes a week at the most.
Students don’t waste time and are relieved of unnecessary stress. Collectively,
companies save a huge amount of time as well. The data generated can be used to
track and modify/improve the process as needed.

This can be done at one institute, but
really speaking, should be extended nationwide. It is not feasible for every
company to visit every college they would like to go to. A process such as the
above vastly increases the pool of good candidates and minimal cost.

The main objection that
companies may have relates to the validity of a common system. (aside: in an
ideal world, interviews should be unnecessary and college grades should be
enough to evaluate a graduate hire. The fact that interviews exist indicate that
something is amiss in the grading system. Funnily enough, companies use GPA as
a cut-off to filter candidates, but not as the only deciding factor). However,
the current interviewing system isn’t an efficient system anyway and I believe
that my proposal will be no worse than what companies do; and will be much
better when the cost is taken into account.

A second objection could be the loss of
prestige points for no longer being a “day-zero” company. With this system, if
you’re getting the best candidates, the ego is the only thing that is bruised.

The
nature of recruitment is going to change in the near future. Even now, I
believe TCS no longer goes to campus, they have an online test that anyone can
give, and candidates get selected for following stages based on the performance
on this test. All of a sudden, a student from a remote college that TCS would
not have visited has the opportunity to apply.

It’s not just recruitment that is
changing. Don’t look now, but the nature of “work” and education itself is
undergoing a transition, which is a topic for another article.

We’re in for interesting times…

Search


text:
Home/Class 12/Physics/

## QuestionPhysicsClass 12

A resistor of $$200\;\Omega$$  and capacitance $$\text{15}\; \mu F$$  are connected in series to a $${220\;\text{V}}$$  and $${50\;\text{Hz}}$$ source. Calculate current in the circuit.

According to the question $$R=200\;\Omega$$ and $$C=15\; \mu F, V=220\;V$$ and $$\text{frequency}=50\;\text{Hz}$$
The impedance of the circuit= $$Z=\sqrt{R^2+X_c^2}=\sqrt{R^2+\left(2\mathit{\pi vC}\right)^2}$$
$$\sqrt{200^2+\left(2\times 3.14\times 50\times 15\times 10^{-6}\right)^2}$$
$$=291.67\;\Omega$$
$$I=\frac V Z$$
$$=\frac{220}{291.67}=0.755\;A$$


text:
# Group Theory

1. Nov 10, 2013

### Lee33

1. The problem statement, all variables and given/known data

In $S_n$, prove that there are $\frac{n!}{r(n-r)!}$ distinct r-cycles.

2. The attempt at a solution

I know there are n choose r ways to permute r out of n cycles thus $\frac{n!}{r!(n-r)!}$ but I don't know how they got $\frac{n!}{r(n-r)!}$?

2. Nov 10, 2013

### pasmith

Two r-cycles are the same if their cycle notations are cyclic permutations of each other. Having chosen our objects, we can avoid such multiple counting by fixing one object to appear the beginning of all the r-cycles. Each permutation of the remaining $r-1$ objects will then generate a distinct r-cycle.


text:
Share

The Minimum Value of F(X) = X 4 − X 2 − 2 X + 6 is (A) 6 (B) 4 (C) 8 (D) None of These - CBSE (Arts) Class 12 - Mathematics

Question

The minimum value of f(x) = $x4 - x2 - 2x + 6$ is _____________ .

• 6

• 4

• 8

• none of these

Solution

4

$\text { Given:} f\left( x \right) = x^4 - x^2 - 2x + 6$

$\Rightarrow f'\left( x \right) = 4 x^3 - 2x - 2$

$\Rightarrow f'\left( x \right) = \left( x - 1 \right)\left( 4 x^2 + 4x + 2 \right)$

$\text { For a local maxima or a local minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow \left( x - 1 \right)\left( 4 x^2 + 4x + 2 \right) = 0$

$\Rightarrow \left( x - 1 \right) = 0$

$\Rightarrow x = 1$

$\text { Now,}$

$f''\left( x \right) = 12 x^2 - 2$

$\Rightarrow f''\left( 1 \right) = 12 - 2 = 10 > 0$

$\text { So, x = 1 is a local minima } .$

$\text { The local minimum value is given by }$

$f\left( 1 \right) = 1 - 1 - 2 + 6 = 4$

Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

Solution The Minimum Value of F(X) = X 4 − X 2 − 2 X + 6 is (A) 6 (B) 4 (C) 8 (D) None of These Concept: Graph of Maxima and Minima.
S


text:
+0

# Factoring a Cubic Equation

0
48
3

Hey everyone, so I have to find the roots of the equation (in decimal form) of this equation: $$\frac{1}{2}{x}^{3}+{x}^{2}-32x-20$$, but I'm unsure how to do it. Normally I would just factor this to get a quadratic, but I can't do that with this equation. All the methods I've found online seem beyond my skills, I'm only in grade 11. I have a hint which says: "It will not be possible to find roots by inspection and you need to be resourceful here!".

Feb 13, 2021

#1
0

Have you tried a graphing of the equation ?

Feb 13, 2021
#2
0

Thank you guest, but this is actually a small part of a larger question, so I can't use graphing to solve this.

Guest Feb 13, 2021
#3
+31707
+1

Unless you are familiar with the standard formula for the solution of a cubic equation, you will need to use a numerical method here, such as the following:

Feb 14, 2021


text:
### Who is online

In total there are 6 users online :: 0 registered, 0 hidden and 6 guests (based on users active over the past 5 minutes)
Most users ever online was 1327 on Tue Nov 05, 2013 7:11 pm

Users browsing this forum: No registered users and 6 guests

## Page break after Bunch of Figures

figuresfloatspage-breaksAdd tags Information and discussion about graphics, figures & tables in LaTeX documents.

### Page break after Bunch of Figures

Greetings!

I am new to using a forum for LaTeX so I'll try to do my best. Up til now I managed to fix all my problems by using a search engine, but this one is killing me.

I included some spectra that I used in my thesis, and for some odd reason, after the last figure, a blank page is included before starting the next subsection. Is there any way of just placing the next subsection on the next page, instead of having a blank page there.
Code: Select all  •  Open in writeLaTeX
\begin{figure}[H]\centering\includegraphics[width=1.2\textwidth, angle=90]{figuren/Staal1A}\caption{Staal1}\label{fig:Staal1IR}\end{figure}\begin{figure}[H]\centering\includegraphics[width=1.2\textwidth, angle=90]{figuren/Staal1Ab}\caption{Staal2}\label{fig:Staal2IR}\end{figure}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%HERE IS THE BLANC PAGE I DON'T WANT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\subsection{MS}\begin{figure}[H]\centering\includegraphics[width=1.0\textwidth]{figuren/MSchloracetamide}\caption{Chloracetamide}\label{fig:ChloracetamideMS}\end{figure}

Note: The figures I inserted are 3/4 of a page each.

Last edited by Stefan_K on Sun Apr 21st, 2013, edited 1 time in total.
StupidPareBreak

Posts: 2
Joined: Sun Apr 21st, 2013

### Re: Pagebreak after bunch of Figures

Hi,

welcome to the forum!

Using the H option for figures is bad for page breaking, since it disables LaTeX's capability to let them float for optimal page breaks. If you want it very flexible and easy positioned, you could use !htbp instead. No wonder that there's a page break problem.

Regarding the problem in this situation: your code snippet doesn't tell much. There's no cause visible. For sure there's a clear explanation, but we need to see the whole situation. So, please post a minimal working example, which shows the problem and which we can test to verify and to fix it. This can be a reduced, but compilable copy of your original document with dummy figures and text.

Stefan

Stefan_K

Posts: 7070
Joined: Mon Mar 10th, 2008
Location: Hamburg, Germany

### Re: Page break after Bunch of Figures

StupidPareBreak wrote:[…] Note: The figures I inserted are 3/4 of a page each. […]

Besides the remarks of Stefan it is suggestive to put figures of such a size on their own page and prevent the placement of running text on these pages.
Code: Select all  •  Open in writeLaTeX
\begin{figure}[!p]% figure content\end{figure}

Best regards and welcome to the board
Thorsten
LaTeX Community Moderator

¹ System: openSUSE 13.1 (Linux 3.11.10), TeX Live 2013 (vanilla), TeXworks 0.5 (r1351)
² Posting stopped indefinitely due to offenses

localghost
Site Moderator

Posts: 9218
Joined: Fri Feb 2nd, 2007
Location: Braunschweig, Germany

### Re: Page break after Bunch of Figures

Thank you for the fast response! I managed to fix the problem by making the figure after the pagebreak a tiny bit smaller. Somehow that fixed the problem..

And my other figures I didn't use [H] but these spectra really had to be placed on that exact spot in that exact order.

Thanks again!
StupidPareBreak

Posts: 2
Joined: Sun Apr 21st, 2013

### Topic Tags

figuresfloatspage-breaks


text:
# Publications by Katherine Blundell

## The X-ray luminous cluster underlying the z = 1.04 quasar PKS1229-021

ArXiv (0)

HR Russell, AC Fabian, GB Taylor, JS Sanders, KM Blundell, CS Crawford, RM Johnstone, E Belsole

We present a 100 ks Chandra observation studying the extended X-ray emission around the powerful z=1.04 quasar PKS1229-021. The diffuse cluster X-ray emission can be traced out to ~15 arcsec (~120 kpc) radius and there is a drop in the calculated hardness ratio inside the central 5 arcsec consistent with the presence of a cool core. Radio observations of the quasar show a strong core and a bright, one-sided jet leading to the SW hot spot and a second hot spot visible on the counter-jet side. Although the wings of the quasar PSF provided a significant contribution to the total X-ray flux at all radii where the extended cluster emission was detected, we were able to accurately subtract off the PSF emission using ChaRT and marx simulations. The resulting steep cluster surface brightness profile for PKS1229-021 appears similar to the profile for the FRII radio galaxy 3C444, which has a similarly rapid surface brightness drop caused by a powerful shock surrounding the radio lobes (Croston et al.). Using a model surface brightness profile based on 3C444, we estimated the total cluster luminosity for PKS1229-021 to be L_X ~ 2 x 10^{44} erg/s. We discuss the difficulty of detecting cool core clusters, which host bright X-ray sources, in high redshift surveys.

## When Microquasar Jets and Supernova Collide: Hydrodynamically Simulating the SS433-W50 Interaction

ArXiv (0)

PT Goodall, F Alouani-Bibi, KM Blundell

We present investigations of the interaction between the relativistic, precessing jets of the microquasar SS433 with the surrounding, expanding Supernova Remnant (SNR) shell W50, and the consequent evolution in the inhomogeneous Interstellar Medium (ISM). We model their evolution using the hydrodynamic FLASH code, which uses adaptive mesh refinement. We show that the peculiar morphology of the entire nebula can be reproduced to a good approximation, due to the combined effects of: (i) the evolution of the SNR shell from the free-expansion phase through the Sedov blast wave in an exponential density profile from the Milky Way disc, and (ii) the subsequent interaction of the relativistic, precessing jets of SS 433. Our simulations reveal: (1) Independent measurement of the Galaxy scale-height and density local to SS433 (as n_0 = 0.2 cm^{-3}, Z_d = 40 pc), with this scale-height being in excellent agreement with the work of Dehnen & Binney. (2) A new mechanism for hydrodynamic refocusing of conical jets. (3) The current jet precession characteristics do not simply extrapolate back to produce the lobes of W50 but a history of episodic jet activity having at least 3 different outbursts with different precession characteristics would be sufficient to produce the W50 nebula. A history of intermittent episodes of jet activity from SS433 is also suggested in a kinematic study of W50 detailed in a companion paper (Goodall et al, MNRAS submitted). (4) An estimate of the age of W50, and equivalently the age of SS433's black hole created during the supernova explosion, in the range of 17,000 - 21,000 years.

## A novel method for measuring the extragalactic background light: Fermi application to the lobes of Fornax A

ArXiv (0)

M Georganopoulos, RM Sambruna, D Kazanas, AN Cillis, CC Cheung, ES Perlman, KM Blundell, DS Davis

We describe a new method for measuring the extragalactic background light (EBL) through the detection of $\gamma$-ray inverse Compton (IC) emission due to scattering of the EBL photons off relativistic electrons in the lobes of radio galaxies. Our method has no free physical parameters and is a powerful tool when the lobes are characterized by a high energy sharp break or cutoff in their electron energy distribution (EED). We show that such a feature will produce a high energy IC imprint' of the EBL spectrum in which the radio lobes are embedded, and show how this imprint can be used to derive the EBL. We apply our method to the bright nearby radio galaxy Fornax A, for which we demonstrate, using WMAP and EGRET observations, that the EED of its lobes is characterized by a conveniently located cutoff, bringing the IC EBL emission into the {\sl Fermi} energy range. We show that {\sl Fermi} will set upper limits to the optical EBL and measure the more elusive infrared EBL.

## On the redshift cut-off for steep-spectrum radio sources

ArXiv (0)

MJ Jarvis, S Rawlings, CJ Willott, KM Blundell, S Eales, M Lacy

We use three samples (3CRR, 6CE and 6C*) selected at low radio frequency to constrain the cosmic evolution in the radio luminosity function (RLF) for the most luminous' steep-spectrum radio sources. Although intrinsically rare, such sources give the largest possible baseline in redshift for the complete flux-density-limited samples currently available. Using parametric models to describe the RLF which incorporate distributions in radio spectral shape and linear size as well as the usual luminosity and redshift, we find that the data are consistent with a constant comoving space density between z~2.5 and z~4.5. We find this model is favoured over a model with similar evolutionary behaviour to that of optically-selected quasars (i.e. a roughly Gaussian distribution in redshift) with a probability ratio of ~25:1 and ~100:1 for spatially-flat cosmologies with Omega_Lambda = 0 and Omega_Lambda = 0.7 respectively. Within the uncertainties, this evolutionary behaviour may be reconciled with the shallow decline preferred for the comoving space density of flat-spectrum sources by Dunlop & Peacock (1990) and Jarvis & Rawlings (2000), in line with the expectations of Unified Schemes.

## The emission line - radio correlation for radio sources using the 7C Redshift Survey

ArXiv (0)

CJ Willott, S Rawlings, KM Blundell, M Lacy

We have used narrow emission line data from the new 7C Redshift Survey to investigate correlations between the narrow-line luminosities and the radio properties of radio galaxies and steep-spectrum quasars. The 7C Redshift Survey is a low-frequency (151 MHz) selected sample with a flux-density limit about 25-times fainter than the 3CRR sample. By combining these samples, we can for the first time distinguish whether the correlations present are controlled by 151 MHz radio luminosity L_151 or redshift z. We find unequivocal evidence that the dominant effect is a strong positive correlation between narrow line luminosity L_NLR and L_151, of the form L_NLR proportional to L_151 ^ 0.79 +/- 0.04. Correlations of L_NLR with redshift or radio properties, such as linear size or 151 MHz (rest-frame) spectral index, are either much weaker or absent. We use simple assumptions to estimate the total bulk kinetic power Q of the jets in FRII radio sources, and confirm the underlying proportionality between jet power and narrow line luminosity first discussed by Rawlings & Saunders (1991). We make the assumption that the main energy input to the narrow line region is photoionisation by the quasar accretion disc, and relate Q to the disc luminosity, Q_phot. We find that 0.05 < Q / Q_phot < 1 so that the jet power is within about an order of magnitude of the accretion disc luminosity. The most powerful radio sources are accreting at rates close to the Eddington limit of supermassive black holes (~ 10^9 - 10^10 solar masses), whilst lower power sources are accreting at sub-Eddington rates.

## A radio-jet -- galaxy interaction in 3C441

ArXiv (0)

M Lacy, S Rawlings, KM Blundell, SE Ridgway

Multi-wavelength imaging and spectroscopy of the z=0.708 radio galaxy 3C441 and a red aligned optical/infrared component are used to show that the most striking aspect of the radio-optical alignment effect'' in this object is due to the interaction of the radio jet with a companion galaxy in the same group or cluster. The stellar population of the red aligned continuum component is predominately old, but with a small post-starburst population superposed, and it is surrounded by a low surface-brightness halo, possibly a face-on spiral disc. The [OIII]500.7/[OII]372.7 emission line ratio changes dramatically from one side of the component to the other, with the low-ionisation material apparently having passed through the bow shock of the radio source and been compressed. A simple model for the interaction is used to explain the velocity shifts in the emission line gas, and to predict that the ISM of the interacting galaxy is likely to escape once the radio source bow shock has passed though. We also discuss another, much fainter, aligned component, and the sub-arcsecond scale alignment of the radio source host galaxy. Finally we comment on the implications of our explanation of 3C441 for theories of the alignment effect.


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# How do online games deal with lerp overshooting?

From what I understand, when you lerp in an online game, you travel from point a to b based on the elapsed time since the last update. The problem is when you have a higher latency, and you start receiving updates at inconsistent rates. So lets say I have variable t, variable t tracks the time since the last update and is only reset when a new update is received. The variable t is also responsible for moving the lerp like so:

lerp(a, b, t)

But when we receive updates at an inconsistent rate, t is reset inconsistently, so if there is any lag between the updates, t will keep incrementing, thus, the lerp will overshoot. How do online games deal with this?

• Have you read some existing Q&A on client-side interpolation for some ideas? Dec 3, 2018 at 3:01
• @DMGregory, prediction is not an option for me as my game is very simple and lacks a lot of the things needed for it. Dec 3, 2018 at 3:12
• Perhaps then, instead of asking what other online games do, it would be better to tell us about your game's situation, and ask how to solve the problems you're encountering in that particular case. Dec 3, 2018 at 3:17
• @DMGregory Its as I described, the lerp overshoots due to the lag of the updates. Dec 3, 2018 at 3:19

Often, we'll just accept some overshoot, and blend it out as needed when we eventually get the next update packet some milliseconds later.

A lot of things in games move in reasonably straight lines or or broad arcs on the timescale of update packets, so assuming the object keeps going the direction it was last known to be travelling in is often a very accurate first guess.

If you have some domain-specific knowledge, of course, you can propose better predictions. Say you're dealing with an object on a tether, or a vehicle with its wheels turned to steer around a corner - you might decide to predict its motion in an arc instead of a straight line.

If prediction is absolutely off the table, then you could clamp your lerp parameter at 1 so the object will stop and wait for the next update before proceeding, but that can cause it to visible hesitate & stutter in its motion.

To avoid this without predicting, you can instead choose to let it live further in the past. Say, instead of living one average packet interval behind the server, you let it stay up to 2, or even 3 updates back. This gives you a much longer buffer to absorb a late or missing packet before you run out of movement history to replay.

In Watch_Dogs 2's multiplayer, they allow this "time in the past" parameter to vary as a function of the player's average measured lag as well as vehicle speed and collision probability, so they keep the movement looking as smooth and accurate as possible with interpolated history most of the time, and zero-in toward the current moment in cases where a collision is likely so that the two sides observe roughly the same collision event.

• I think the buffer method would work best for me. So to do this would I add every server frame to a list and interpolate to them, deleting them from the list as they are completed, correct? Dec 3, 2018 at 3:48
• That sounds like it would work. Dec 3, 2018 at 3:49
• When I set the speed of the lerp by going: t = timeElapsed / updateRate;, the lerp completes too quickly, in turn the coords that were just lerped get deleted. As a result the list gets depleted of frames quickly, leaving nothing to be lerped to. What speed should it be set to to avoid this? Dec 3, 2018 at 4:45
• Sounds like something you should ask as a new question, showing the details of how you calculate timeElapsed, UpdateRate, the blended position, and how you update your history buffer. Dec 3, 2018 at 12:58