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Spacetime & Substance
International Physical Journal

CONTENTS of No. 2(22)-2004

W.B.Belayev D.Yu.Tsipenyuk. GRAVY-ELECTROMAGNETISM IN FIVE DIMENSIONS AND MOVING BODIES IN GALAXY AREA (49)

Angelo Loinger. ON THE ORIGIN OF THE NOTION OF GW ET CETERA (53)

N.K. Lohani and Lalan Prasad. HEATING OF SOLAR CORONAL LOOPS BY PHASE-MIXING (57)

Dumitru Pricopi. A COLLAPSING POLYTPOPE (61)

Walter Cabrera-Febola. ON NATURAL STRUCTURES: COMPLEMENTARY ISSUES (65)

V.R.Terrovere. DIONIC THEORY OF MESON (72)

I.M.Galitsky. ABOUT THE EXPERIMENTAL CONFIRMATIONS OF SOME PROGNOSES WHICH FOLLOW FROM THE SOLUTION OF THE TIME PROBLEM (75)

V.R.Terrovere. STRUCTURE OF PHOTON & LEPTON (78)

Rasulkhozha S. Sharafiddinov. MASS OF THE NEUTRINO AND ITS CHARGE (83)

V.F.Barybin. THE CONTINUUM IS STRUCTURED (85)

V.V.Balyberdindag, A.V.Nechayevddag. MAGNETIC-GRAVITATIONAL EXPERIMENTS (87)

V.V.Balyberdin, V.T.Glyanko, N.A.Zhuck, N.D.Kolpakov, A.V.Nechaev, S.I.Chernyshov. PENETRATING RADIATION OF CAVITATION CAVITIES (91)

Jorge Guala-Valverde; M.F.Najar. BRAKING FORCE AND ENERGETICS IN HOMOPOLAR GENERATORS (95)

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ABSTRACTS

W.B.Belayev D.Yu.Tsipenyuk. GRAVY-ELECTROMAGNETISM IN FIVE DIMENSIONS AND MOVING BODIES IN GALAXY AREA

In Klein geometric model of space the mass is manifestation of the quantized charges oscillations in additional compactified dimension. We analyze model in which common in four-dimensional space-time for mass and electric charge of the particle trajectory is disintegrated in five dimensions on movement of the mass along null geodesic line and trajectory of the charge corresponding to the time-like interval in 5D volume. We find relation between five-velocity vector of electric charge and mass. This scheme is regarded to have concern with many worlds theory. Considered approach is applied to the model of rotating space having four-dimensional spherical symmetry. One proposed appearance additional force in included 4D space-time, which may be explanation of the Pioneer-effect. We analyze also possible part of this force in conservation of the substance in Galaxy area.

Angelo Loinger. ON THE ORIGIN OF THE NOTION OF GW emphET CETERA

The notion of gravitational wave (GW) came forth originally as a by-product of the \emph{\textbf{linear}} approximation of general relativity (GR). Now, it can be proved that this approximation is quite \emph{\textbf{inadequate}} to a proper study of the hypothetic GW's. The significant role of the approximations beyond the linear stage is emphasized.

N.K. Lohani and Lalan Prasad. HEATING OF SOLAR CORONAL LOOPS BY PHASE-MIXING

This paper present an analytical method for the heating of solar coronal loops by phase-mixing. We also discuss herewith the non-linear mode of phase mixing by Alfv\'en waves. Under typical coronal heating conditions by ohmic dissipation due to phase-mixing can provide magnetic energy on a time scale comparable with the coronal radiative time. For large Lundquist number, it is possible that phase-mixing can attain a hot coronal loop. We introduce two model of loops; i.e., flat symmetric loop model and cylindrical symmetric loop model. The magnetic field assumed to be static and associated with only in inhomogeneities plasma density. The solution under initial boundary condition and the ohmic dissipation have been discussed.\\

Dumitru Pricopi. A COLLAPSING POLYTPOPE

The general results for the gravitational collapse of a polytrope are obtained for whatever polytropic configuration with index $n\in \left( 0.5\right).$ The time-dependence of the solution is analyzed and the collapsing time is determined. This is important for the determination of the time-scale for collapse of polytropes, which will be compared with the stellar free-fall time.

Walter Cabrera-Febola. ON NATURAL STRUCTURES: COMPLEMENTARY ISSUES

In this paper the propositions developed in a previous paper [30] are complemented by the identification of some of the kinds of natural structures which develop from their dependence on NE-connection and from other inherent characteristics pertaining to them. Also it is stated that, although natural structure theory has some resemblance to other theoretical developments, it is clearly differentiated from them. This is complemented by a demonstration of the possibilities for further research and applications that the theory offers, perhaps to improve all this a generalization of the definition of a NE is given and it is noted that NEs have no scale.

V.R.Terrovere. DIONIC THEORY OF MESON

Dynamical properties of Meson gave way to the experimental Exploration. But a meson's Structure isn't decoded for the present, although a theory predicts that it consists of two Quarks. It is remarkable that Meson's Quarks need not endow with three Colours. Author identified a Quark with only Colour with the Dion. It is a particle that has an electric \& magnetic Charge. The offered Theory permits to calculate a mass of the fundamental Quark and to calculate Masses of the well-known Quarks on the Basis of experimental Data for Masses of Mesons. It is showed that a formula for the Hadron's Mass is equivalent to a Condition of the Quantization of Dirac's Monopole.

I.M.Galitsky. ABOUT THE EXPERIMENTAL CONFIRMATIONS OF SOME PROGNOSES WHICH FOLLOW FROM THE SOLUTION OF THE TIME PROBLEM

The experimental confirmations of prognoses which were made by the author in 1992: about new, beyond gravitational interaction (dark energy'') by intensity $\sim 10^{ - 52}$ of unities of the strong; about new, a neutrino interaction by intensity $\sim 10^{ - 26}$ of unities of the strong; about presence at a neutrino the rest mass $\sim 10^{ - 32}$ g; about a substantiation of charging asymmetry of the Universe by a corollary of its masses-oscillations in coordinates substance''---antimatter'' are given.

V.R.Terrovere. STRUCTURE OF PHOTON & LEPTON

The present author develops a new theory of particles on the basis of fractals and clusters. The theory contains two radical suggestions. 1) Any particle is divisible without end. 2) Any charge is a power of a source. According to Author a Photon consists of Neutrino and Antineutrino. This idea has allowed to calculate a Neutrino Mass spectrum. In Author's opinion the published experimental data 2.2 eV (V.M. Lobashev a.o.) are understated since the Experimenters leave out of account $\alpha$---effect. According to Author electron consists of the Preons. The Mass of Preon is equal to 7.3 meV. Preons radiate Photons. A wave-length this Radiation is equal to 0.662 mm. A "Relic" Radiation has the same wave-length. Thus both euristic ideas have an Experimental corroboration.

Rasulkhozha S. Sharafiddinov. MASS OF THE NEUTRINO AND ITS CHARGE

Both electron and its neutrino possess not only the anomalous magnetic moment but each of existing types of the electric charges and their dipole moments. Any of them can interact with field of emission leading to the elastic scattering on a spinless nucleus. We list some implications implied from the analysis of these phenomena in the case of one - photon exchange. The processes cross sections give the possibility to establish the individual as well as the united equation between the Dirac and the Pauli form factors of light leptons. They define the electronic neutrino electric charge as a consequence of the availability of a kind of mass.

V.F.Barybin. THE CONTINUUM IS STRUCTURED

Multiplicities of masses, time (frequencies), the dimensional parameters were considered in the previous paper Multiplicities of fundamental physical constants. Cosmological model''. The deduction is obtained that a continuum -- space -- time -- field -- substance are structured. Correlation of a series of multiplicities has enabled to interpret, the obtained results, as cosmological model. It has been put forward a series of theses. One of theses treated a compendency of parameters of space, time and a multiplicity of masses, as structurization of space -- time -- substances (empty space -- fields -- substances). It is a corollary of existence of a uniform, quantized continuum at all levels which has blanket laws of the organization, and as a whole, with the mechanism of regulation of processes forms a blanket Universum.

V.V.Balyberdindag, A.V.Nechayevddag. MAGNETIC-GRAVITATIONAL EXPERIMENTS

The preliminary qualitative experimental results of the influence investigation upon the weight of the laboratory container of thermal magnetic chemical process, made in the container, are presented in the paper.

V.V.Balyberdin, V.T.Glyanko, N.A.Zhuck, N.D.Kolpakov, A.V.Nechaev, S.I.Chernyshov. PENETRATING RADIATION OF CAVITATION CAVITIES

The schemes and descriptions of experimental devices as well as the experiment results of radiation recording transiting through metal screens and having not even precisely obscure nature are given in the article.

Jorge Guala-Valverde, M.F.Najar. BRAKING FORCE AND ENERGETICS IN HOMOPOLAR GENERATORS

We analyze, for the first time, energy conservation as applied to homopolar electromagnetic devices.

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# Electronic – Using single resistor to receive 5v signal on the 3.3v rated pin

input-impedancemicrocontrollerpinsvoltage divider

I have been investigating ways to safely receive 5v signal on the input pin that can only support 3.3v input signal. I came across an article that states that it's possible to only use a single 10k Ohm resistor to achieve that. I can understand how this can prevent too much current from flowing through the schottky diode, but I am still skeptical this resistor would produce the necessary voltage drop for the signal that goes further into the MCU since the resistance of the resistor that is placed after diodes is much higher than 10k.

Will the 10k resistor work for this purpose and if so, what voltage drop would it produce?

Thanks a lot.

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# Weak convergence in the Hilbert space

Is it true that, if $$\{u_n\}$$ is a sequence in a Hilbert space $$H$$ that converges weakly to its limit $$u \in H$$ and the sequence satisfies $$\limsup_{n \rightarrow \infty} \|u_n \| \leq \|u\|$$ then $$\lim_{n \rightarrow \infty} \|u_n \| = \|u\|.$$

My attempt:

I wanted to show $$\|u\| \le \liminf_{n \to \infty} \|u_n\|,$$ if this is true, then combined with $$\liminf \|u\| \le \limsup \|u\|$$ the desired result follows.

If the forward implication is true, then to my understanding this would in fact be an "if and only if" relation because the converse implication is trivial.

## 1 Answer

It is in fact true that $$\|u\| \le \liminf_{n \to \infty} \|u_n\|$$, and so the claim at the top of the question does indeed follow.

Hint to get you started on the proof: $$\|u\|^2 = \langle u,u \rangle = \lim_{n \to \infty} \langle u_n, u \rangle$$

Incidentally, when $$u_n \to u$$ weakly and $$\limsup_{n \to \infty} \|u_n\| \le \|u\|$$, then you can prove even more: in fact $$u_n \to u$$ in norm.

• I did a slightly different approach that is sort of based on what you said: If $e_k$ is an orthonormal basis of $H$, then $$\|u\|^2=\sum_k |\langle u,e_k \rangle|^2=\lim_{n \to \infty} \sum_k |\langle u_n,e_k\rangle|^2 \le \liminf_{n \to \infty} \|u_n\|^2=(\liminf_{n \to \infty}\|u_n\|)^2,$$ where I used Bessel's inequality on $u_n$ and took infimum over $n$ on the right-hand side, then send $n \to \infty$ on both sides. Is this what you intended me to think about? – Cookie Sep 23 '19 at 3:26
• @Cookie: I had in mind something a little simpler. By Cauchy-Schwarz, we have $\langle u_n, u \rangle \le \|u_n\| \|u\|$. Now take the liminf of both sides. – Nate Eldredge Sep 23 '19 at 12:10
• Then I would eventually have to divide both sides by $\|u\|$, which only makes sense if $u$ is nonzero. But I think this is fine because there is a separate trivial argument for the case $u=0$. – Cookie Sep 23 '19 at 17:49
• Also, to prove $u_n \to u$ in norm, I think that means to prove $\|u_n-u\| \to 0$ as $n \to \infty$. So I did $\|u_n-u\|^2=\langle u_n-u,u_n-u \rangle = \|u_n\|^2-2\operatorname{Re}\langle u_n,u\rangle +\|u\|^2 \to \|u\|^2-2\langle u,u\rangle+\|u\|^2=0$ as $n \to \infty$. – Cookie Sep 23 '19 at 17:51
• @Cookie: Yes, exactly. Indeed, all you really need to make that argument work is the hypothesis on the limsup, so your lemma about the liminf is not really needed for this (but it's interesting in its own right). – Nate Eldredge Sep 23 '19 at 18:25


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2 JeatsKg) 4. which will be the set of things that eat everything { which is what we want. In this example, $func is an instance of Closure and echo$func($x) is equivalent to echo$func->__invoke($x). 3. lambda-abstractie. Instead, an anonymous function, that does not return a value, can be used: The function statement in the first (outer) pair of parentheses declares an anonymous function, which is then executed when used with the last pair of parentheses. The following example squares every element in an array with an anonymous function. Interpretation Translation  lambda abstraction. [37]. This will cause total to be stored as a reference, but value will be stored as a copy. The following example binds the variable "threshold" in an anonymous function that compares the input to the threshold. An anonymous function is already a value. lambda abstraction A term in lambda-calculus denoting a function. Anonymous functions are important in programming the latter. Clang supports anonymous functions, called blocks, which have the form: The type of the blocks above is return_type (^)(parameters). The divisor function also forms a closure by binding the variable d. A higher-order function is a function that takes a function as an argument. The lower case letter lambda (λάμδα), the eleventh letter of the modern Greek alphabet. Lambdas do not support currying, unless paired with an apply to form a command prefix. Command names can be bound to command prefixes by means of the interp alias command. If arguments are to be passed, a "value:...value:" message with a corresponding number of value arguments must be used. The returned function is a function that adds X to its argument: Python supports simple anonymous functions through the lambda form. (where, as usual, \and ->can be replaced by λand →). PHP 5.3 mimics anonymous functions but it does not support true anonymous functions because PHP functions are still not first-class objects. 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# lambda abstraction definition

Abstraction definition, an abstract or general idea or term. // returns one output can also be implicitly declared with the Func<> type. However, because most lambda functions are small and local in scope, they are likely candidates for inlining, and thus need no added storage for references. This. Anonymous functions have been a feature of programming languages since Lisp in 1958, and a growing number of modern programming languages support anonymous functions. Lambda Abstraction ¶ Pattern matching ... Internally this is translated into a function definition of the following form:. Python supports anonymous functions through the lambda syntax, Ruby's anonymous functions, inherited from, Smalltalk's anonymous functions are called. The first provides no syntactic hint of any resident function but the second MUST evaluate the first parenthetical as a function to be legal JavaScript. PHP 5.3 added a new class called Closure and magic method __invoke() that makes a class instance invocable.[39]. Since version 2.0, D allocates closures on the heap unless the compiler can prove it is unnecessary; the scope keyword can be used for forcing stack allocation. Snippet from CS3110, Cornell, taught by Michael R. Clarkson, Robert L. Constable, Nate Foster, Michael D. George, Dan Grossman, Daniel P. Huttenlocher, Dexter Kozen, Greg Morrisett, Andrew C. Myers, Radu Rugina, and Ramin Zabih. are anonymous functions. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus. [42] They are defined using the following syntax: With type inference, however, the compiler is able to infer the type of each parameter and the return type, so the above form can be written as: With closures with a single expression (i.e. Lambda expressions can be used inside lambda forms and with the special operator "function". This is the part which caches the compiled form of the anonymous function, but it can only be invoked by being passed to the apply command. 0. "function" can be abbreviated as #'. CFML supports recursive anonymous functions: CFML anonymous functions implement closure. A lambda term is either a variable, an application of a function term to a variable term or a lambda abstraction. Lambda-abstraction to move from the flat perspective, whereby for instance determiners were conceived of as relations between sets and predication as the expression of set membership, to one whereby these types of meaning can be represented functionally, a way … $^\dagger$ Edit: As Derek Elkins points out, the rules that would make the category of types strictly Cartesian (closed) are flagged in the HoTT book as optional, so the above statement is actually inaccurate. Using the aforementioned blocks extension and Grand Central Dispatch (libdispatch), the code could look simpler: The code with blocks should be compiled with -fblocks and linked with -lBlocksRuntime. However, statically typed languages can support anonymous functions. Define Lambda abstraction. f(x) = x + 1. PascalABC.NET supports anonymous functions using lambda syntax, Elixir uses the closure fn for anonymous functions. b. then λxβ EXP. They are implemented using the Fn, FnMut, and FnOnce traits:[43], With these traits, the compiler will capture variables in the least restrictive manner possible. Keywords modular genetic programming lambda abstraction modules higher-order functions financial trading rules buy-and-hold S&P 500 index automatically defined functions PolyGP system stock market technical analysis constrained syntactic structure strongly typed genetic programming financial time series lambda abstraction GP The lambda will have the same access as the member that created it, in terms of protected/private members. this can only be captured if the closest enclosing function is a non-static member function. CFML supports any statements within the function's definition, not simply expressions. In this example, the built-in power function can be said to have been curried: In R the anonymous functions are defined using the syntax function(argument-list)expression . Introduced by A. In Maxima anonymous functions are defined using the syntax lambda(argument-list,expression). References [1] A. FOL and Gödel's Incompleteness Theorem 1. To invoke the function, you must apply it. A term in lambda-calculus denoting a function. Compiling Lambda Expressions: Scala vs. Java 8, Lambda functions in various programming languages, https://en.wikipedia.org/w/index.php?title=Anonymous_function&oldid=1000604671, Articles needing additional references from February 2018, All articles needing additional references, Articles needing cleanup from December 2018, Copied and pasted articles and sections from December 2019, All copied and pasted articles and sections, Creative Commons Attribution-ShareAlike License, Expression functions are a part of Ada2012. C# names them lambda expressions, following the original version of anonymous functions, the lambda calculus.[32]. Lambda-abstraction rule, first version. a body with one line), the curly braces may be omitted: Closures with no input parameter are written like so: Closures may be passed as input parameters of functions that expect a function pointer: However, one may need complex rules to describe how values in the body of the closure are captured. ARS is a generalized form of the basic operations of the Lambda Calculus in that respect, that the definition of abstraction retains the full meaning of the word (give something a name') and that all three operations can be applied anywhere in a program without restrictions.. Lambda expressions can be stored in variables if the variable's type is an interface which has only one method. lambda abstraction to give us as the semantic value of the VP the set of things which are the semantic value of x(for all u2 U, x2 fy:2 JeatsKg) 4. which will be the set of things that eat everything { which is what we want. In this example, $func is an instance of Closure and echo$func($x) is equivalent to echo$func->__invoke($x). 3. lambda-abstractie. Instead, an anonymous function, that does not return a value, can be used: The function statement in the first (outer) pair of parentheses declares an anonymous function, which is then executed when used with the last pair of parentheses. The following example squares every element in an array with an anonymous function. Interpretation Translation  lambda abstraction. [37]. This will cause total to be stored as a reference, but value will be stored as a copy. The following example binds the variable "threshold" in an anonymous function that compares the input to the threshold. An anonymous function is already a value. lambda abstraction A term in lambda-calculus denoting a function. Anonymous functions are important in programming the latter. Clang supports anonymous functions, called blocks, which have the form: The type of the blocks above is return_type (^)(parameters). The divisor function also forms a closure by binding the variable d. A higher-order function is a function that takes a function as an argument. The lower case letter lambda (λάμδα), the eleventh letter of the modern Greek alphabet. Lambdas do not support currying, unless paired with an apply to form a command prefix. Command names can be bound to command prefixes by means of the interp alias command. If arguments are to be passed, a "value:...value:" message with a corresponding number of value arguments must be used. The returned function is a function that adds X to its argument: Python supports simple anonymous functions through the lambda form. (where, as usual, \and ->can be replaced by λand →). PHP 5.3 mimics anonymous functions but it does not support true anonymous functions because PHP functions are still not first-class objects. Anonymous functions originate in the work of Alonzo Church in his invention of the lambda calculus, in which all functions are anonymous, in 1936, before electronic computers. For anonymous functions, called lambda abstraction ( read: function expression ), JavaScript defines a, comma... X ) in w ] ] ” no type annotations on the parameters of other functions: anonymous! Matching lambda... Internally this is translated into a Proc in some circumstances behave similarly to closures, them! Stop and a lambda term of the λ-operator was given as follows: functional abstraction: if VAR! Supports simple anonymous functions is a reference to a method, a direct reference to the first is... Body must be escaped associated with if, while, etc. > can be used to simulate any machine. Is undefined VB.NET function - it must return a value at multiple places looping. Lambda forms and with the ` lambda '' construct, which is a variable term or a block... Those elements that match the given condition a shorthand for anonymous functions through lambda... Name for an anonymous function the Wolfram language is the construct from which the calculus takes name. Abstraction in programming is e.g of abstraction object state ( by capturing this ) for in! Functions and statement expressions a closure ), a dialect of object Pascal, supports anonymous.. Perhaps not common with currying, unless paired with an apply to form a variety in the list then the! Λ is used to customize the behavior of a fold need not be compound statements not be compound statements of. Eleventh letter of the functional interface haskell uses a concise syntax for anonymous functions ( general. Using lambda syntax, Elixir uses the closure the inner block, which have the,! As with Python, in allowing access to variables in the proofs work. \Lambda$ abstraction and explicit definition that does the comparing 5.3 mimics anonymous functions. [ 32 ] capture! The contained expression to make recursive anonymous functions. [ 32 ] possible lambda terms the function... Underscore character to introduce unnamed parameters we will use names partial anonymous function accepts an argument and it. Even those associated with if, while, etc. third curry the divide function with lot! Body must be in single quotes, or something similar a library has been made to these. Are inherited from, Smalltalk 's anonymous functions are a few anonymous functions, either as dfns, in of. Same access as the theoretical foundation of functional programming ' # 0 ' refers to the apply )... Non-Local functions ) two nested lambda expressions and anonymous functions. [ 35 ] '' in object... X.T ) } where x is inferred by the λ x, Dictionary of Computing Legal. ( new in Tcl 8.5 ) syntax, Elixir uses the closure for... Is inferred by the lambda abstraction definition x i don ’ t think we have gotten to the bottom of.! In the constructs above, they become ( ( ) method to be stored as a.... The.NET Framework v3.5, has full support of anonymous functions are still not first-class objects element a. # 0 ' refers to the threshold a VB.NET function - it return. Value returned by the λ x change its value to create a lambda on an existing method before its... An Identity function ) do not have to have names ; they can be! Can not be compound statements as follows: functional abstraction: if x VAR ocaml functions do not anonymous. Called lambda, and returns its name ( as a reference to lambda calculus is error!

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# Solve real-world problems involving division of unit fractions by non-zero whole numbers and

### Cluster

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.


text:
# Question #71a0a

##### 1 Answer
Mar 2, 2017

Equation of the line is $x = 3$

#### Explanation:

Look at the graph -

Equation of the line is $x = 3$


text:
# forecasting for data excluding holiday dummies

I used tbats to fit a model to a set of 3 years of historic data for daily number of shipments moved by a trucking company. my data included double seasonality so I used tbats. However, tbats did not consider holidays and the point forecast for holidays of the test data were very off from the actual data. This makes total sense as I did not include holidays in my forecast. Is it reasonable that I use this forecasting model but for holidays, I basically use the average value of that holiday day over the historic data as the point forecast? I tried arima with fourier series to include seasonality and regressors to include holiday but the fitted model was much worse than the tbats one. It worked better for holidays but was not accurate enough for other days.

• Did you try using dummies for the holidays? Use the xreg parameter to arima() - this also works for tbats(), as it is passed through to auto.arima(), which models the errors. – S. Kolassa - Reinstate Monica Sep 25 '14 at 16:27
• do you mean that I can use xreg for tbats as well? could you please give me a reference which shows the related command for that?Thank you – user12 Sep 25 '14 at 16:29
• Here you go. – S. Kolassa - Reinstate Monica Sep 25 '14 at 16:30


text:
# Organizing point cloud from HDL-32E

Hi there,

in order to accelerate the data processing on my HLD-32E I would like to organized the data. To this point, I could, under Octave, compute manually an point cloud. But I do actually have no idea how to implement it in C++ and the suggested functions, such as KDTree, don't seems to be fast enough for this task...

Is there somewhere a topic giving information about how to work with those Point clouds in ROS?

edit retag close merge delete

Sort by » oldest newest most voted

Are you using the ROS velodyne stack to provide the data?

If so, the velodyne_pointcloud package handles converting the raw data packets received from the device into ROS sensor_msgs/PointCloud2 messages. The HDL-32E spews out about 700,000 points per second in a highly permuted order, the HDL-64E provides about 1,000,000 points per second in a very different and less regular order. So, velodyne_pointcloud keeps things simpler by always producing "unordered" point clouds as the data arrive from the device.

To prevent that unordered representation from losing useful information, velodyne_pointcloud publishes its data using a custom PCL point definition: velodyne_pointcloud::PointXYZIR, defined in the velodyne_pointcloud/point_types.h header. A PointXYZIR differs from the standard PCL PointXYZI (X, Y, Z, and intensity) in adding an additional field containing the "laser ring number" from which the point was measured.

Each ring number corresponds to a device-specific laser number, but the numbers are reordered so that ring 0 corresponds to the innermost laser (at the lowest angle) and ring 31 corresponds to the outermost laser (at the highest angle) of a 32E (that would be ring 63 on a 64E). That encoding ensures that adjacent lasers are labelled with adjacent ring numbers, which is not at all true of the hardware laser numbers. The actual mapping is defined in the velodyne_driver/ring_sequence.h header, provided by the velodyne_driver package. The permutation for a 32E is the same as the first 32 lasers of the 64E, so the same conversions work for both.

So, it should theoretically be possible to produce an ordered point cloud from the unordered PointXYZIR data by using the ring numbers as rows in a 2D representation. I have not heard of anyone actually doing that, and I suspect that the results might become somewhat confused when the device is in motion.

more

Sorry for the long delay, but it could finaly organize the pointcloud and now I can segment and classify the point cloud in less than 0.2s ;)

( 2013-07-03 07:54:50 -0500 )edit

@Dexter111 Do you mind describing how you were able to do that? Did you use PCL at all?

( 2013-07-18 10:22:19 -0500 )edit

Hi. I need the pointcloud from Velodyne VLP16. You mentioned that "velodyne_pointcloud publishes its data using a custom PCL point definition: velodyne_pointcloud::PointXYZIR". Do you mean that is a topic? After I do roslaunch "velodyne_pointclound VLP16_points.launch", nothing like that are found

( 2016-10-23 05:56:08 -0500 )edit

For Velodyne VLP16, nothing like velodyne_pointcloud::PointXYZIR are published. Please tell me why

( 2016-10-24 06:01:28 -0500 )edit

The topic name is /velodyne_points

( 2016-10-24 10:16:41 -0500 )edit
1

I don't know if this is still useful for you, but I wrote some code with PCL to organize a VLP16 point cloud and now I have a 16xN point cloud. I am still doing some tests, but you can find my code here: https://github.com/claydergc/vlp16_po...

( 2017-05-26 16:34:20 -0500 )edit

@claydergc, did you ever get that code fully functional? I've tried it out but haven't been able to get it working so far.

( 2019-03-07 14:16:42 -0500 )edit

TL;DR: Use the function I wrote here: organize_velodyne_cloud

I had to solve this problem recently in order to use the cloud for mesh creation with the Point Cloud Library (PCL). At the time I couldn't find anyone else who had solved this problem, so I had to write the code myself following the suggestions here from @joq. I've added to core functions to the repo linked above, as they're a bit too long to put here. Hopefully this will save anyone after me ~1-2 weeks effort.

How it works is this: every Velodyne LiDAR unit fires its lasers in a pre-defined sequence that repeats. E.g. 0, 16, 1, 17, 2, 18... But not every laser beam returns to the unit, so some parts of the cloud need to be filled in with points whose coordinates are NaN to preserve the organized nature of the cloud. My function creates a 2D organized cloud where there is one row for each "ring" of velodyne data. It then fills in each row, tracking how many points are in each row, i.e. what "column" of data is currently being filled in. When it moves from one column of data to the next, it fills in any gaps with NaN points.

Be careful how you use the organized cloud after this, as some PCL functions will crash if you input a cloud with NaNs, or worse, silently destroy the organized nature of the cloud without warning you.

more

@M@t: would this be something to contribute to some more visible package? I'm thinking laser_filters or something similar, but then for lidars. Perhaps even the velodyne package itself?

( 2019-03-28 11:00:29 -0500 )edit

That might certainly be a good idea, the velodyne package would be the best fit as the function depends on the specific operation of Velodyne LiDARs. How time consuming is that process likely to be? Unfortunately I have precious little spare time right now.

( 2019-03-28 15:27:34 -0500 )edit

re: time: no idea. I'd suggest just opening an issue on the velodyne tracker and seeing whether there is any interest.

( 2019-03-28 15:29:09 -0500 )edit

An organized point cloud is just a point cloud where some information about the sensor that captured the "depth image" is given. E.g. for a kinect with a resolution of 640 x 480 pixels, the properties of an organized point cloud would be set to

width: 640
height 480
isOrganized = true.

Internally, the point cloud stores all 3D points in a vector that contains

point(0,0), point(1,0), ..., point(638, 0), point(639,0),
point(0,1), point(1,1), ..., point(638, 1), point(639,1),
...
point(0,479), point(1,479), ..., point(638, 479), point(639,479)

The important bit is that all measured points have to be stored in the point cloud. As soon as some (invalid, ...) points are filtered, the cloud is no longer organized.

So if you have a laser that e.g. provides 1.000 points per line and scans 500 lines per "picture" (?), you'd just push_back the data points into the point cloud in the order which they were acquired (starting with the first one of the first line, ending with the last on eof the last line). Then set width to 1.000, height to 500, and voila!

Disclaimer: I haven't worked with lasers much, so apologies for the poor terminology! The above of course only makes sense when there's no motion in the scene during the time all lines are captured...

more


text:
# Why is the band gap of graphene “opened” in this VASP calculation?

I used VASP to do the band structure calculation on Graphene. Originally I would expect that the Dirac cone can be observed in the K point. If we watched qualitatively, we can see this behavior. But when I zoom in the region where the conduction band touches with the valences band, a bandgap can be observed as shown in the figure. That's weird because I've already adjusted my K point meshing to be an odd number and increase the number of meshing, it doesn't help. The upper band was still lifted.

• Can you please clarify what you mean by "adjusting" you k-point grid? Does this mean that in the self-consistent part of the calculation the K symmetry point is explicitly calculated? – ProfM Jun 27 '20 at 8:08
• In the self-consistent calculation, I use Monkhorst-Pack to do the meshing with around 279 meshing grid – JensenPang Jun 27 '20 at 9:10
• @JensenPang, thanks for the clarification, but I need a little more: if you look at the coordinates of the K point, is that point in the list of 279 points that you have in your mesh? – ProfM Jun 27 '20 at 10:45
• @JensenPang have you implemented smearing? Since graphene doesn't have a gap, you need to implement smearing to deal with some BZ integrations that can cause trouble. You should look into that or post your vasp input files here - I remember calculating band structure of graphene long back with vasp, so I should be able to help – Xivi76 Jun 27 '20 at 19:17
• Re Xivi76: I implemented the gaussian smearing when I do my calculation, I think it is probably caused by other reason. Anyway thanks your help – JensenPang Jun 28 '20 at 1:48

The reason why graphene has Dirac points at K and $$-$$K is because of a combination of time reversal and inversion symmetries. Therefore, if you impose these symmetries in your VASP calculation, there is no reason why there should be a gap at K. Even if you do not impose these symmetries explicitly, you should be able to get the degeneracy to very high accuracy (better than what you are showing) if you do a well converged calculation in terms of the number of $$\mathbf{k}$$-points that you are using to sample the Brillouin zone.

So what could be going wrong? From the partial information that you provide, my best guess is that: (i) you don't include the symmetries in the calculation, (ii) the $$\mathbf{k}$$-point grid you are using for the self-consistent part of the calculation is not fine enough, and in particular it does not explicitly include the K point, and (iii) when you then do the non-self-consistent calculation to obtain the band structure along the high-symmetry lines, then you have very few points along the lines and the code simply naively interpolates between them to generate the plot.

So what would I suggest you do? The first thing is to impose the relevant symmetries. The second is to make sure that the $$\mathbf{k}$$-point you are using in the self-consistent calculations is large enough, and in particular that it explicitly includes the K point, and (iii) that in the non-self-consistent calculation you include a large number of $$\mathbf{k}$$-points along the high symmetry points near K.

All of this is based on my interpretation of the partial information you provide.

The discussion above is somewhat different if you included spin-orbit coupling. In that case, a gap develops at the K point in graphene, an observation that led to the field of topological insulators. However, I don't think that this explains your results, because spin-orbit coupling in graphene is very small, on the $$\mu$$eV scale, orders of magnitude smaller than what you are seeing. For this reason spin-orbit coupling can be neglected when doing calculations for graphene, so I imagine that this is what you are doing.

• Agree with the discussion about K point mesh, but I have a hunch that OP has not implemented smearing. Absence of smearing can easily cause this issue in my opinion. – Xivi76 Jun 27 '20 at 19:15
• As ProfM and others explained very well, it could also be that you can try Gamma centered k-mesh to check the results. Slight buckling can also cause, check your POSCAR also to see if everything is alright. – Shahid Sattar Jun 27 '20 at 20:16
• Re ProfM and Shahid Sattar : Thank you guys helps. Now Im trying to follow the guideline provided by ProM and check whether there is some mistake in my calculation. I will post my mistake once I found the problem. I am very appreciate you guys help! – JensenPang Jun 28 '20 at 1:51
• Ive solved the problem completely! Actually I made 2 mistake. 1. As ProfM said, The K point didn't lying on the K-meshing grid point. 2. When I do the calculate in the band structure , I made a low level mistake which the parameter of KPOINTS file are not good enough. I didn't take enough decimal place for the location of KPOINTS. Therefore the sampling location are deviated from the exact location. – JensenPang Jun 28 '20 at 11:20


text:
# associativity of multiplication

It’s important to know the following interpretation of the associative law

 $\displaystyle a\cdot(b\cdot c)=(a\cdot b)\cdot c$ (1)

of multiplication in arithmetics and elementary algebra:

A product ($b\cdot c$) is multiplied by a number ($a$) so that only one ($b$) of the product is multiplied by that number.

This rule is sometimes violated even in high school e.g. like

 $10\cdot 2.5\cdot 0.3=25\cdot 3=75,$

which is wrong.  Or when solving an equation like

 $x\cdot\frac{2x-1}{3}=1$

one would like to multiply both sides by 3 for removing the denominator, getting perhaps

 $3x(2x-1)=3;$

then the both of left side have incorrectly been multiplied by 3.

The reason of such mistakes is very likely that one confuses the associative law with the distributive law;  cf. (1) with this latter

 $\displaystyle a\cdot(b+c)=a\cdot b+a\cdot c,$ (2)

which two different operations, multiplication and addition; both addends must be multiplied separately.

Title associativity of multiplication AssociativityOfMultiplication 2013-03-22 15:09:22 2013-03-22 15:09:22 pahio (2872) pahio (2872) 5 pahio (2872) Application msc 12D99 msc 00A35


text:
## Custom WordPress theme not displaying posts from category

I am creating a custom WordPress theme and as far as I can remember WordPress is supposed to do the heavy lifting for filtering of post types by their category, author, etc. on it’s own. This is just an altered query from my index.php specifically for category types.

Another question… do I even need an archive.php? I thought WordPress was supposed to do all the heavy lifting as far as filtering by categories, tags, authors, etc. My index.php works great, but when I tried to go to a specific category it was showing all of my posts and not just the selected category so I tried implementing an archive.php

EDIT 1: I should have mentioned I’m not using the main loops because I’m using custom post types

archive.php

<?php /**  * @author    Mark Abel  * @package   client-name/archive  * @version   1.0  */ if ( ! defined( 'ABSPATH' ) ) exit; // Exit if accessed directly get_header(); ?>  <?php get_template_part( 'partials/partial', 'sidenav' );?>

     <?php       $category = get_category( get_query_var( 'cat' ) );$  cat_id = $category->cat_ID;$  loop = new WP_Query( array( 'post_type' => 'project', 'cat' => $cat_id ) ); if ($  loop->have_posts() ) :         while ( $loop->have_posts() ) :$  loop->the_post(); ?>           <?php $project_for = get_post_meta($  post->ID, "project_for", true); ?>
             <img src=<?php the_post_thumbnail_url(); ?> alt="an image" />

                 <?php echo get_the_title() ?>

                 <?php echo $project_for ?>

<a href="<?php echo get_post_permalink(); ?>">Click to view
 Play video

 <?php endwhile; endif; wp_reset_postdata(); ?>

   <?php get_footer(); ?>   partial imported by index.php <?php /** * @author Mark Abel * @package client-name/partials/partial-masonry * @version 1.0 */ if ( ! defined( 'ABSPATH' ) ) exit; // Exit if accessed directly ?>

 <?php$  loop = new WP_Query( array( 'post_type' => 'project' ) );       if ( $loop->have_posts() ) : while ($  loop->have_posts() ) : $loop->the_post(); ?> <?php$  project_for = get_post_meta($post->ID, "project_for", true); ?>
 <img src=<?php the_post_thumbnail_url(); ?> alt="an image" />

 <?php echo get_the_title() ?>

 <?php echo$  project_for ?>

<a href="<?php echo get_post_permalink(); ?>">Click to view
               Play video

         <?php endwhile;       endif;       wp_reset_postdata();     ?>

## Is Python’s Grammar in a known Category between CFG and CSG?

I have a high level understanding of formal languages and grammars, and I’m familiar with the four major types of grammars in Chomsky hierarchy. I was interested in knowing the classification of Python’s grammar. A quick search yielded some quick, but incomplete answers.

http://trevorjim.com/python-is-not-context-free/

Is Python a context-free language?

The immediate takeaway from most resources it that Python’s grammar is not a context free grammar (CFG). But that doesn’t answer the question of what it is. Looking deeper I found that its a complete context sensitive grammar (CSG) either. But still no classification.

At this point the conclusion was that there must exist classes of grammars between CFGs and CSGs, but I had never heard anything about these.

What I do understand is Python’s lexer (which transforms character sequences into tokens) does something that a CFG cannot do: it tracks the level of indentation and yields special INDENT DEDENT tokens. After this transformation resulting tokens are context-free and can be parsed into an abstract syntax tree. Thus the grammar on tokens is a CFG, but the grammar on characters uses slightly power power than a CFG can provide alone. I want to know if there is a classification for this type of grammar. What is between a CFG and a CSG?

After a bit more searching I stumbled on this table at the bottom of a Wikpedia article: https://en.wikipedia.org/wiki/Linear_bounded_automaton#External_links

Here is an image of that table:

Cool, I found that there are known grammars between CFG and CSGs. But I’m not an expert on formal languages, so I don’t know how I would go about determining which if any of these categories Python’s grammar belonged to.

Is it a “Positive range concatenation”, “Indexed”, “Thread automation’s Grammar”, “Linear context-free rewriting system”, “Tree-adjoining”? Is it none of these; if so does what it is have a classification?

Note: The full grammar specification of Python 3 can be found here: https://docs.python.org/3/reference/grammar.html

## What size category is a reduced Tarrasque?

The rule for a creature’s size is as follows on Monster Manual page 6:

$$\begin{array}{lll} \textbf{Size} & \textbf{Space} & \textbf{Examples} \ \hline \text{Tiny} & \text{2½ by 2½ ft.} & \text{Imp, sprite} \ \text{Small} & \text{5 by 5 ft.} & \text{Giant rat, goblin} \ \text{Medium} & \text{5 by 5 ft.} & \text{Orc, werewolf} \ \text{Large} & \text{10 by 10 ft.} & \text{Hippogriff, ogre} \ \text{Huge} & \text{15 by 15 ft.} & \text{Fire giant, treant} \ \text{Gargantuan} & \text{20 by 20 ft. or larger} & \text{Kraken, purple worm} \ \end{array}$$

The Monster Manual page 268 states:

A scaly biped, the tarrasque is fifty feet tall and seventy feet long, weighing hundreds of tons.

And the Reduce use of the Enlarge/Reduce spell states:

The target’s size is halved in all dimensions, and its weight is reduced to one-eighth of normal. This reduction decreases its size by one category – from Medium to Small, for example.

Emphasis mine.

In the case of our Tarrasque, these statements in Reduce are contradictory. Its new size is now 25′×35′, which means it’s still classified as Gargantuan.

Which rule takes priority?

We can break the rules even more here. If we assume the Tarrasque is now huge rather than gargantuan, PHB 192 states:

Squeezing into a Smaller Space: A creature can squeeze through a space that is large enough for a creature one size smaller than it. Thus, a Large creature can squeeze through a passage that’s only 5 feet wide. While squeezing through a space, a creature must spend 1 extra foot for every foot it moves there, and it has disadvantage on attack rolls and Dexterity saving throws. Attack rolls against the creature have advantage while it’s in the smaller space.

This means our 25′×35′ beastie can now get through a 10 foot door.

## Inserting Ultimate member username as WooCommerce category into Ultimate member page

The website in question are using the Ultimate User plugin.

I am interested in installing WooCommerce and create a WooCommerce category for each Ultimate Member (Subscriber). (If there also is an easy way to auto-generate the category when users register, just yell it out!).

Is there a way to insert (based on the WooCommerce shortcodes):

[products limit="12" columns="4" orderby="popularity" category="CURRENT USER PROFILE"]

And maybe a button for “see all products” (in the same category).

So that if a user is selling something, their products will displayed on the profile.

I´m not sure what the best approach for this. I was thinking about stripping the current URL /user/exampleuser/ – so that exampleuser is extracted and inserted into the shortcode.

Thank you for any input!

## Should I add category page number to breadcrumbs on product page

I am currently working on a website which has paginated content. It is a list of products. There are many categories, but for simplicity, let’s assume that I have just one category with many pages.

I have implemented pagination in such a way that search engines can index it. Here is my implementation:

1. prev/next meta tags.
2. canonical URL pointing to itself (i.e. first page points to itself, and second page points to itself as well).
3. there are no blocks on indexing of ?page= pages.
4. Breadcrumbs for the first page are Main > Category, but for all other pages they are Main > Category > Page X.
5. Breadcrumbs are also added to structured data, and I also have an ItemList structured data item representing the list of products which is displayed.

It all works as expected. However, currently, all my products have the following breadcrumbs: Main > Category > Product X regardless of whether my product appears on the first page or not. I was wondering whether there is any SEO benefit in changing these to Main > Category > Page X > Product for the products which do not appear on the first page of my category?

I think that this would be a welcome change from a UI point of view, but I am worried that it might negatively affect my SEO. Also, I am worried that it might make my structured data breadcrumbs somewhat worse. Google does show them in the search results, and I am not sure that Page X would be a great detail to put there for a product path 🙂

## Category archive rewrite rule to include category/subcategory?

I’d like to have my category archive (category.php) accessible using via a url structure like:

mysite.com/mycategory/mysubcategory

For example, I’ve got “health” as a category and “smoking” as its subcategory. The following:

• mysite.com/health
• mysite.com/smoking
• (and even) mysite.com/category/health/smoking

all work fine.

But mysite.com/health/smoking gets a 404.

Is there something under Settings->Permalinks that will fix this? Do I need to put a rewrite rule of some kind in functions.php?

Thanks.

## How can I display a specific category on Front-Page

Please how can I display a specific category on Front-Page page, I want to display music category on my WordPress front page at least 3post on Front-Page, I don’t know how I can do this

## SharePoint Online : Search webpart category missing

Search webpart category is missing in SharePoint Online WebPart page (E3 Plan)

I am trying to create Search Result, Search Box and Refiners webparts in Webpart page.

I am added into Designer security group of the site as well.

Help me to get it displayed there.

## Show all posts of sub category in a page like: foo.com/category/subcategory/ using UNCODE Theme

I’ll be upfront and say that this is a difficult question to phrase for achieving the “correct” answer as it seems it can be done a few ways.

I am using the WP Theme Uncode from undsgn.com that uses the WP Bakery Page Builder / Visual Composer. Here is a brief how-to video that shows the module as part of the theme to query each archive type. https://www.youtube.com/watch?v=M8_H2neBRRA


text:
# Benjamin Doerr

## D-Index & Metrics

Discipline name D-index Citations Publications World Ranking National Ranking
Computer Science D-index 41 Citations 6,657 226 World Ranking 4312 National Ranking 198
Mathematics D-index 41 Citations 5,900 214 World Ranking 952 National Ranking 57

## What is he best known for?

### The fields of study he is best known for:

• Algorithm
• Combinatorics
• Algebra

His scientific interests lie mostly in Evolutionary algorithm, Algorithm, Combinatorics, Mathematical optimization and Function. The concepts of his Evolutionary algorithm study are interwoven with issues in Discrete mathematics, Heuristics, Mutation rate, Evolutionary computation and Upper and lower bounds. His study in the field of Time complexity also crosses realms of Test functions for optimization.

His Binary logarithm study in the realm of Combinatorics interacts with subjects such as Value. His study focuses on the intersection of Mathematical optimization and fields such as Shortest path problem with connections in the field of Human-based evolutionary computation and Interactive evolutionary computation. His Function research includes themes of Linear function, Spanning tree and Constant.

### His most cited work include:

• Theory of Randomized Search Heuristics: Foundations and Recent Developments (256 citations)
• Why rumors spread so quickly in social networks (248 citations)
• Multiplicative Drift Analysis (194 citations)

## What are the main themes of his work throughout his whole career to date?

The scientist’s investigation covers issues in Combinatorics, Evolutionary algorithm, Discrete mathematics, Algorithm and Upper and lower bounds. He interconnects Rounding and Constant in the investigation of issues within Combinatorics. His Evolutionary algorithm research incorporates themes from Function, Genetic algorithm, Evolutionary computation and Heuristics.

His Function research integrates issues from Black box, Simple, Mutation rate, Applied mathematics and Benchmark. His Discrete mathematics research is multidisciplinary, relying on both Linear programming, Expected value and Random walk. His work on Asymptotically optimal algorithm and Randomized algorithm as part of general Algorithm study is frequently linked to Test functions for optimization, therefore connecting diverse disciplines of science.

### He most often published in these fields:

• Combinatorics (46.47%)
• Evolutionary algorithm (33.58%)
• Discrete mathematics (25.06%)

## What were the highlights of his more recent work (between 2018-2021)?

• Evolutionary algorithm (33.58%)
• Genetic algorithm (11.19%)
• Function (16.79%)

### In recent papers he was focusing on the following fields of study:

His primary areas of study are Evolutionary algorithm, Genetic algorithm, Function, Upper and lower bounds and Combinatorics. His Evolutionary algorithm study necessitates a more in-depth grasp of Mathematical optimization. His studies in Genetic algorithm integrate themes in fields like Algorithm, Distribution, Mutation and Exponential function.

His Function study incorporates themes from Evolutionary computation, Simple and Polynomial. Benjamin Doerr combines subjects such as Symmetry, Logarithm and Applied mathematics with his study of Upper and lower bounds. His study in the fields of Binary logarithm under the domain of Combinatorics overlaps with other disciplines such as Lambda.

### Between 2018 and 2021, his most popular works were:

• Analyzing randomized search heuristics via stochastic domination (46 citations)
• A tight runtime analysis for the cGA on jump functions: EDAs can cross fitness valleys at no extra cost (33 citations)
• Theory of evolutionary computation : recent developments in discrete optimization (27 citations)

## In his most recent research, the most cited papers focused on:

• Algorithm
• Algebra
• Combinatorics

His primary areas of investigation include Evolutionary algorithm, Function, Genetic algorithm, Mathematical optimization and Benchmark. His work carried out in the field of Evolutionary algorithm brings together such families of science as Discrete mathematics, Multiplicative function, Probabilistic logic, Mutation rate and Constant. His Genetic algorithm research incorporates elements of EDAS, Estimation of distribution algorithm, Algorithm, Applied mathematics and Upper and lower bounds.

His work deals with themes such as Black box and Unary operation, which intersect with Algorithm. Benjamin Doerr conducts interdisciplinary study in the fields of Test functions for optimization and Combinatorics through his research. His studies deal with areas such as Matching and Order as well as Combinatorics.

This overview was generated by a machine learning system which analysed the scientist’s body of work. If you have any feedback, you can contact us here.

## Best Publications

Why rumors spread so quickly in social networks

Benjamin Doerr;Mahmoud Fouz;Tobias Friedrich.
Communications of The ACM (2012)

389 Citations

Theory of Randomized Search Heuristics: Foundations and Recent Developments

Anne Auger;Benjamin Doerr.
(2011)

340 Citations

Multiplicative Drift Analysis

Benjamin Doerr;Daniel Johannsen;Carola Winzen.
Algorithmica (2012)

194 Citations

Social Networks Spread Rumors in Sublogarithmic Time

Benjamin Doerr;Mahmoud Fouz;Tobias Friedrich.
Electronic Notes in Discrete Mathematics (2011)

163 Citations

Crossover can provably be useful in evolutionary computation

Benjamin Doerr;Edda Happ;Christian Klein.
Theoretical Computer Science (2012)

134 Citations

From black-box complexity to designing new genetic algorithms

Benjamin Doerr;Carola Doerr;Franziska Ebel.
Theoretical Computer Science (2015)

129 Citations

Süntje Böttcher;Benjamin Doerr;Frank Neumann.
parallel problem solving from nature (2010)

127 Citations

Optimal Static and Self-Adjusting Parameter Choices for the $$(1+(\lambda ,\lambda ))$$ Genetic Algorithm

Benjamin Doerr;Carola Doerr.
Algorithmica (2018)

116 Citations

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics.

Benjamin Doerr.
arXiv: Data Structures and Algorithms (2018)

109 Citations

Stabilizing consensus with the power of two choices

Benjamin Doerr;Leslie Ann Goldberg;Lorenz Minder;Thomas Sauerwald.
acm symposium on parallel algorithms and architectures (2011)

99 Citations

## Editorial Boards

Journal of Complexity
(Impact Factor: 1.333)

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]> No Title

1.2. THE GAUSS MAP AND THE WEINGARTEN MAP.

13

1.2 The Gauss map and the Weingarten map.

In order to state the general formula, we make the following definitions: Let $\mathit{Y}$ be an (immersed) oriented hypersurface. At each $\mathit{x}\mathrm{\in }\mathit{Y}$ there is a unique (positive) unit normal vector, and hence a well defined Gauss map

$\mathit{l}\mathit{\nu }\mathrm{:}\mathit{Y}\mathrm{\to }{\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$

assigning to each point $\mathit{x}\mathrm{\in }\mathit{Y}$ its unit normal vector, $\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ . Here ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ denotes the unit sphere, the set of all unit vectors in ${\mathrm{R}}^{\mathit{n}}\mathrm{.}$

The normal vector, $\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is orthogonal to the tangent space to $\mathit{Y}$ at $\mathit{x}$. We will denote this tangent space by $\mathit{T}{\mathit{Y}}_{\mathit{x}}$. For our present purposes, we can regard $\mathit{T}{\mathit{Y}}_{\mathit{x}}$ as a subspace of ${\mathrm{R}}^{\mathit{n}}$: If $\mathit{t}\mathrm{↦}\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is a differentiable curve lying on the hypersurface $\mathit{Y}$, (this means that $\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathit{Y}$ for all t) and if $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{x}$, then $\mathit{\gamma }\mathrm{\prime }\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ belongs to the tangent space $\mathit{T}{\mathit{Y}}_{\mathit{x}}$. Conversely, given any vector $\mathit{v}\mathrm{\in }\mathit{T}{\mathit{Y}}_{\mathit{x}}$, we can always find a differentiable curve $\mathit{\gamma }$ with $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{x}\mathrm{,}$ $\mathit{\gamma }\mathrm{\prime }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{v}$. So a good way to think of a tangent vector to $\mathit{Y}$ at $\mathit{x}$ is as an ""infinitesimal curve“ on $\mathit{Y}$ passing through $\mathit{x}\mathrm{.}$

Examples:

1. Suppose that $\mathit{Y}$ is a portion of an $\mathrm{\left(}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\right)}$ dimensional linear or affine sub- space space sitting in ${\mathrm{R}}^{\mathit{n}}$. For example suppose that $\mathit{Y}\mathrm{=}{\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ consisting of those points in ${\mathrm{R}}^{\mathit{n}}$ whose last coordinate vanishes. Then the tangent space to $\mathit{Y}$ at every point isjust this same subspace, and hence the normal vector is a constant. The Gauss map is thus a constant, mapping all of $\mathit{Y}$ onto a single point in ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{.}$

2. Suppose that $\mathit{Y}$ is the sphere of radius $\mathit{R}$ (say centered at the origin). The Gauss map carries every point of $\mathit{Y}$ into the corresponding (parallel) point of ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$. In other words, it is multiplication by $\mathrm{1}\mathrm{/}\mathit{R}$:

$\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{R}}\mathit{y}\mathrm{.}$

3. Suppose that $\mathit{Y}$ is a right circular cylinder in ${\mathrm{R}}^{\mathrm{3}}$ whose base is the circle of radius $\mathit{r}$ in the ${\mathit{x}}^{\mathrm{1}}\mathrm{,}$ ${\mathit{x}}^{\mathrm{2}}$ plane. Then the Gauss map sends $\mathit{Y}$ onto the equator of the unit sphere, ${\mathit{S}}^{\mathrm{2}}$, sending a point $\mathit{x}$ into $\mathrm{\left(}\mathrm{1}\mathrm{/}\mathit{r}\mathrm{\right)}\mathit{\pi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ where $\mathit{\pi }$ : ${\mathrm{R}}^{\mathrm{3}}\mathrm{\to }{\mathrm{R}}^{\mathrm{2}}$ is projection onto the ${\mathit{x}}^{\mathrm{1}}\mathrm{,}$ ${\mathit{x}}^{\mathrm{2}}$ plane.

Another good way to think of the tangent space is in terms of a local

parameterization which means that we are given a map $\mathit{X}\mathrm{:}\mathit{M}\mathrm{↦}{\mathrm{R}}^{\mathit{n}}$ where $\mathit{M}$ is some open subset of ${\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ and such that $\mathit{X}\mathrm{\left(}\mathit{M}\mathrm{\right)}$ is some neighborhood of $\mathit{x}$ in $\mathit{Y}$. Let ${\mathit{y}}^{\mathrm{1}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{y}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ be the standard coordinates on ${\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$. Part of the requirement that goes into the definition of parameterization is that the map $\mathit{X}$ be regular, in the sense that its Jacobian matrix

$\mathit{d}\mathit{X}\mathrm{:}\mathrm{=}\mathrm{\left(}\frac{\mathrm{\partial }\mathit{X}}{\mathrm{\partial }{\mathit{y}}^{\mathrm{1}}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{}\frac{\mathrm{\partial }\mathit{X}}{\mathrm{\partial }{\mathit{y}}^{\mathit{n}\mathrm{-}\mathrm{1}}}\mathrm{\right)}$


text:
# Math Help - Help please!!

How do I start this problem? do i use the distance formula in finding the equation... please help me solve this.

do you know what the distance from the point $(x_p,y_p)$ to the point $(a,b)$ is?

do you know what the distance from the point $(x_p,y_p)$ to the line $y=mx+c$ is?


text:
# The angle between the hour hand and the minute hand of a clock when time is 3 : 25 will be

30 views

closed
The angle between the hour hand and the minute hand of a clock when time is 3 : 25 will be
1. 47 $\frac{1}{2}$°
2. 42 $\frac{1}{2}$°
3. 60°
4. 45°

by (53.7k points)
selected

Correct Answer - Option 1 : 47 $\frac{1}{2}$°

Given Time: 3 : 25

Formula:

|(60H – 11M ) /  2|

Angle at 3 : 25

|(60 × 3 – 11 × 25) / 2|

|180 – 275 / 2| = 95 / 2 = 47 $\frac{1}{2}$°

TRICK:

Hour hand → 1 hour = 30°

3 hour → 90°

1 minute = 1/2°

25 minute = 12.5°

Minute hand → 1 minute = 6°

25 minute = 150°

Now, Angle made by the hour hand – Angle made by the minute hand

Required Angle = |90° + 12.5° – 150°| = |102.5° – 150°| = 47 $\frac{1}{2}$°

Hence, 47 $\frac{1}{2}$° is correct.


text:
# Complex Numbers – Properties, Graph, and Examples

You may have encountered numbers that are said to be imaginary. You may have been asked to disregard the square root of a negative number in the past, but we’ll focus on these types of complex numbers in this article.  Along with the rest of the real numbers, these are part of a larger group of complex numbers.

Complex numbers are numbers that may contain real and imaginary parts.

• The parts that make up complex numbers.
• Properties and operation rules for complex numbers.
• What the form $a + bi$ represents in a complex plane.

Since complex numbers encompass an extensive range of numbers and are crucial elements in higher math, we must familiarize ourselves with all of the concepts mentioned. Why don’t we begin with the most fundamental of them all – the definition of complex numbers?

## What is a complex number?

A complex number is a number that belongs to either the imaginary or real number groups. In short, they encompass all numbers belonging to the two mentioned groups. The general form of a complex number is $z = a + bi$, where $i$ represents $\sqrt{-1}$.

Let’s inspect the form that we’re given to understand the two crucial parts: real and imaginary.

\begin{aligned} \text{Complex Number } &= {\color{red} \boldsymbol{a}} + {\color{blue} \boldsymbol{bi}}\\ {\color{red} \boldsymbol{a}} &: \text{real number part}\\ {\color{blue} \boldsymbol{bi}} &: \text{imaginary number part}\end{aligned}

This chart can help you visualize the group of numbers that belong to the entire complex number system.

• Real numbers such as $4$, $\sqrt{5}$, $\dfrac{4}{2}$, and $\pi$ all belong to the complex number group as well.
• Imaginary numbers such as $\sqrt{-2}$, $-3 + 2i$, and $5 + 2\sqrt{-6}$ also belong under the complex number system group.

As long as the number can be expressed in the form $a + bi$, it’s considered part of the complex number group.

• For example, $4$ can be expressed as $4 + 0i$, showing that $4$ can be considered a complex number.
• Similarly, $\dfrac{2}{3}$ can be rewritten as $\dfrac{2}{3} + 0i$.
• Meanwhile, $4i$ can be written as $0 + 4i$, further confirming that imaginary numbers like $4i$ are part of the complex number group.

Notice how $i$ is often used to represent complex numbers? Let’s go ahead and review what we know about $i$ – an essential component of imaginary numbers.

Understanding the Imaginary Unit, $\boldsymbol{i}$

The number $i$ represents $\sqrt{-1}$ and consequently, $i^2 = -1$. We normally use $i$ to represent the square root of a negative number.

This also means that we can rewrite $\sqrt{-36}$ in terms of $i$.

• We can factor out $-1$ inside the square root and evaluate $\sqrt{36}$.
• Rewrite $\sqrt{-1}$ as $i$.

\begin{aligned}\sqrt{-36}&= \sqrt{36 \cdot -1}\\&=\sqrt{36} \cdot \sqrt{-1}\\&=6 \cdot i\\&= 6i\end{aligned}

We have another important property to keep in mind when dealing with complex numbers: complex numbers equality.

Equality Property

The equality property will still apply for complex numbers: $a + bi = m + ni$ only when $a = m$ and $b = n$. What does this mean? Two complex numbers are equal if and only if the real parts are equal and the imaginary part is equal.

Now that we have thoroughly discussed the definition of complex numbers let’s go ahead and dive into how the rules of operations may differ or may be alike with the rules we have for real numbers.

## How to solve complex numbers?

It’s helpful to think of complex numbers, $z = a + bi$, as binomials when we need to perform arithmetic operations. We consider the real number and imaginary parts as two separate variables. Here are some articles we’ve prepared for a much deeper discussion of the following topics.

### Complex number rules

We can start by reviewing how to add and subtract complex numbers – again, and it helps to treat the two parts of the complex numbers as two different variables.

When given two complex numbers, we can find their sum or difference by combining the real numbers and the imaginary numbers then simplifying the result.

• $(a + bi) + (m + ni) = (a + m) + (b + n)i$
• This means that to add two complex numbers, we add the two real parts and do the same for the imaginary parts.
• $(a + bi) – (m + ni) = (a – m) + (b – n)i$
• Similarly, if we want to subtract two complex numbers, we can subtract the real parts and combine them with the difference between the two imaginary parts.

Let’s go ahead and try finding the sum and difference of the two complex numbers, $(-3 + 5i)$ and $(4 – 2i)$.

• Remove the parenthesis and double-check the signs, especially when finding the difference between the two complex numbers.
• Group the real and imaginary number parts and combine each pair of terms.
 Sum Difference \begin{aligned}(-3 + 5i) + (4 – 2i)&= -3 + 5i+ 4 – 2i\\&= -3 + 4 + 5i -2i\\&= (-3 + 4) + (5 – 2)i\\&= 1 + 3i\end{aligned} \begin{aligned}(-3 + 5i) – (4 – 2i)&= -3 + 5i- 4 + 2i\\&= -3 – 4 + 5i + 2i\\&= (-3 -4) + (5 + 2)i\\&= -7 + 7i\end{aligned}

Multiplying Complex Numbers

If one of the complex numbers we want to multiply contains a single term, we can distribute this into the remaining complex number.

Let’s say we want to multiply $6i$ and $(2 – i)$. We distribute $6i$ into each of the terms inside the parenthesis. Keep in mind that $i^2 = -1$, so make sure to replace $i^2$ with $-1$ whenever possible.

\begin{aligned}6i(2 – i)&= (6i)(2) – (6i)(i)\\&=12i – 6i^2\\&= 12i -6(-1)\\&=12i + 6\end{aligned}

We use the FOIL method when we want to distribute two binomials, so we also apply the same process if we want to multiply two complex numbers of the form $(a+ bi)$ and $(m + ni)$.

Let’s review the FOIL method to multiply the two complex numbers, $(2 – 3i)$ and $(2 + 5i)$. We multiply the first terms, the terms outside and inside the parentheses, and the last term of each binomial.

\begin{aligned}(2 – 3i)(2 + 5i)\\\text{First} = 2 \cdot 2 \phantom{xx}\\\text{Outside} = 2 \cdot 5i \phantom{x}\\\text{Inside} = -3i \cdot 2 \\\text{Last} = -3i \cdot 5i \end{aligned}

Combine these four terms to find the product of two complex numbers.

\begin{aligned}(2 – 3i)(2 + 5i) &= 4 + 10i- 6i – 15i^2\\&= 4 + 10i – 6i – 15(-1)\\&=4 + 10i – 6i + 15\\&= (4 + 15) + (10 – 6)i\\&= 19+ 4i \end{aligned}

In general, the product of $(a+ bi)$ and $(m + ni)$ is equal to $(am –bn)(an + bm)i$.

Conjugates of Complex Numbers

We’ve learned about conjugates in the past, so if you need a quick refresher, make sure to check out this article we wrote about conjugates in math.

This means that the conjugate of $a + bi$ will be equal to $a – bi$. Here’s another interesting part – why don’t we multiply the two conjugates?

\begin{aligned}(a – bi)(a + bi)&= (a \cdot a – b \cdot b) + (a \cdot b – b \cdot a)i\\&=(a^2 + b^2) + 0i \\&= a^2 + b^2\end{aligned}

This means that the product of a complex number and its conjugate is equal to the sum of the squares of their imaginary and real parts.

Dividing Complex Numbers

When we divide two complex numbers, we might end up with a denominator having an imaginary part. When this happens, we’ll apply a similar process when we want to rationalize a radical expression.

Let’s say we want to divide $(a + bi)$ by $(m +ni)$, we’ll end up with a quotient of $\dfrac{a + bi}{m + ni}$. If we want to eliminate the imaginary number found in the denominator, we can:

• Multiply both the numerator and denominator by $m – ni$.
• Simplify the numerator using the FOIL method.
• Simplify the numerator given that the product of a complex number and its conjugate is equal to the sum of the square of their real and imaginary parts.

\begin{aligned}\dfrac{a + bi}{m + ni}&= \dfrac{a + bi}{m + ni} \cdot \dfrac{\color{blue} m – ni}{\color{blue} m – ni}\\&= \dfrac{(a + bi){\color{blue}( m – ni)}}{(m + ni){\color{blue}( m – ni)}}\\&= \dfrac{(am -bn)+(mb -an)i}{(m +ni)(m -ni)}, \color{green}\text{FOIL method}\\&= \dfrac{(am -bn)+(mb -an)i}{m^2 + n^2}, \color{green}(a-bi)(a +bi) = a^2 + b^2\\&= \left( \dfrac{am – bn}{m^2 + n^2}\right )+ \left( \dfrac{mb – an}{m^2 + n^2}\right )i \end{aligned}

This shows the general form of two complex numbers’ quotient, where $m + ni \neq 0$. We can apply similar steps when finding the quotient of other pairs of complex numbers.

We may need to use three or more of these rules for some problems, so make sure to review all these rules and check out the links we’ve provided earlier in this section.

### Solving equations that involve complex numbers

Complex numbers can appear in equations. When this happens, it will be helpful to make use of the equality property whenever possible.

Keep in mind that we can only equate the real numbers part and the imaginary number parts based on the equality property.

Let’s try to solve for $x$ and $y$ in the equation, $5x – (2y + 3)i = 18 – 12i$.

Through the equality property, the real and imaginary parts must be equal to each other for the equation to be true.

 Equation Real Part Imaginary Part $5x – (2y + 3)i$ $5x$ $-(2y + 3)$ $18 – 12i$ $18$ $-2$

Let’s equate the real parts to solve for $x$:

\begin{aligned} 5x &= 18\\ x&= \dfrac{18}{5}\end{aligned}

We do the same for the imaginary parts so we can solve for $y$:

\begin{aligned} -(2y + 3)&= -2\\ 2y + 3 &= 2\\2y &=-1\\y&= -\dfrac{1}{2}\end{aligned}

Hence, we have $x = \dfrac{18}{5}$ and $y = -\dfrac{1}{2}$.

## How to graph complex numbers?

Now that we’ve learned about complex numbers and how we can manipulate their expressions let’s learn how to represent complex numbers on a graph.

When we talk about graphing complex numbers, it’s equally important for us to review what we know about complex planes.

Graphing Complex Numbers in Complex Planes

The complex plane is a coordinate system where the horizontal coordinate represents the real numbers. The vertical coordinates represent the imaginary parts.

This is also why the horizontal and vertical axes are called the real axis and imaginary axis.

Comparing this with our regular coordinate system, we can write the coordinate of $a + bi$ as $(x, y)$. This means that the way we plot the complex number will be similar.

• We’ll plot the point by moving $a$ units to the right if it’s positive or $a$ units to the left when it’s negative.
• Then we also move $b$ upward when it’s positive or downwards when it’s negative.

Here are some complex numbers graphed in one complex planes as examples:

Let’s check each of the complex numbers graphed:

• $4 + 4i$: Graph the point $(4, 4)$ or a point that is $4$ units to the right and upward.
• $-2 + i$: Similarly, we can graph $(-2, 1)$ or plotting a point that’s $2$ to the left from the origin and one unit upward.
• $3$: Recall that $3 = 3 + 0i$, so that’s $(3, 0)$ or $3$ units along the real axis.
• $4i$: We can rewrite $4i$ as $0 + 4i$, so that’s $(0, 4)$. The point is expected to lie $4$ along the positive imaginary axis.

Understanding Absolute Values of $\boldsymbol{a + bi}$

The absolute value of a complex number represents the distance of the complex number from $0$. We use Pythagorean theorem ($c^2 = a^2 + b^2$, remember?) to derive the formula for the $a + bi$’s absolute value.

The graph above shows the absolute value of the complex number, $|z|$, by constructing a right triangle on the complex plane.

Using the Pythagorean theorem, we have $|z| = \sqrt{a^2 | b^2}$. This applies to all complex numbers as well.

### Summary of complex numbers definition and properties

We’ll learn about complex numbers and more properties involving them in the next articles. For now, we should wrap this up and summarize what we’ve learned so far about complex numbers.

• The general form of complex numbers is $a + bi$, where $a$ and $b$ represent the real and imaginary parts, respectively.
• The number $i$ represents $\sqrt{-1}$, so $i^2 = -1$.
• To solve equations that involve complex numbers, always go back to the equality property.
• To graph $a + bi$ on a complex plane, we move $a$ units horizontally and $b$ units vertically.

Here’s a summary of the operations that can be done on two complex numbers, $a + bi$ and $m + ni$.

 Addition $(a + bi) + (m + ni) = (a + m) + (b + n)i$ Subtraction $(a + bi) – (m + ni) = (a – m) + (b – n)i$ Multiplication $(a + bi) \cdot (m + ni) = (am – bn) + (bm + an)i$ Division $\dfrac{a + bi}{m + ni} = \left( \dfrac{am – bn}{m^2 + n^2}\right )+ \left( \dfrac{mb – an}{m^2 + n^2}\right )i$ Product of Two Conjugates $(a + bi)(a – bi) = a^2 – b^2$ Absolute Value $|a + bi| = \sqrt{a^2 + b^2}$

### Example 1

Simplify $(2 – 4i)(3 + i) – (2 – 3i)^2$.

Solution

Before we can simplify the expression, we’ll have to expand each group of binomials first:

• Applying the FOIL method on the first pair of binomials.
• Apply the perfect square trinomial property, (a – b)^2 = a^2 -2ab + b^2, to expand the last binomial.

\begin{aligned} (2 – 4i)(3 + i) – (2 – 3i)^2 &= [(2\cdot 3 + 4\cdot1) -(-4 \cdot 3 + 2 \cdot 1)i] – (2 – 3i)^2\\&=(10 +12 i) – (2 – 3i)^2\\&= (10 + 12i) -[2^2 – 2 \cdot 3i + (3i)^2]\\&=(10 + 12i)- (4 -6i – 9) \end{aligned}

Combine the real and imaginary parts to simplify the expression further.

\begin{aligned} (10 + 12i)- (4 -6i – 9) &= 10 + 12i – 4 + 6i + 9\\&= (10 – 4 + 6) + (12 + 6)i\\&=12 + 18i \end{aligned}

Hence, we have $(2 – 4i)(3 + i) – (2 – 3ui)^2 = 12 + 18i$.

### Example 2

Simplify $\dfrac{(2 – i)^3}{(1 + i)^2}$.

Solution

What we can do is expand the numerator and denominator of the rational expression. Let’s begin by expanding $(2 – i)^3$ as shown below. Take note of the blue texts if you need a refresher on the property used in each step.

\begin{aligned} (2-i)^3 &= (2- i)(2 -i)^2\\&= (2- i)[2^2 – 2(2)(i) +(i)^2], \phantom{xxx}\color{green} (a -b)^2 = a^2 -2ab + b^2\\&= (2 – i)[4 – 4i – 1], \phantom{xxxxxxxxxxx}\color{green} i^2 = -1\\&= (2 – i)(3 – 4i)\\&=(2\cdot3 – 4\cdot1) + [-1(3)+2(-4)]i, \color{green} \text{FOIL method}\\&= 2 – 11i\end{aligned}

We can expand the denominator using the perfect square trinomial property as well.

\begin{aligned} (1 + i)^2&= (1)^2 +2(1)(i) + (i)^2, \phantom{xx}\color{green} (a + b)^2 = a^2 + 2ab + b^2 \\&= 1 +- 2i – 1, \phantom{xxxxxxxxxxxxxx}\color{green} i^2 = -1\\&= 2i\end{aligned}

Let’s replace the denominator and numerator with these simplified expressions and see what we have.

\begin{aligned} \dfrac{(2 – i)^3}{(1 + i)^2} &= \dfrac{2 – 11i}{2i}\end{aligned}

To eliminate the imaginary part found in the denominator, we can multiply both denominator and numerator by $i$.

\begin{aligned} \dfrac{2 – 11i}{2i} \cdot \dfrac{\color{blue}i}{\color{blue}i}&= \dfrac{2{\color{blue}i} -(11i)({\color{blue}i})}{(2i)({\color{blue}i})}\\&=\dfrac{2i – 11i^2}{2i^2}\\&=\dfrac{2i + 11}{-2}\\&= \dfrac{2i}{-2}- \dfrac{11}{2}\\ &= -i- \dfrac{11}{2}\\&= -\dfrac{11}{2} -i\end{aligned}

Hence, $\dfrac{(2 – i)^3}{(1 + i)^2} = -\dfrac{11}{2} -i$.

#### Example 3

Let’s observe the power of $i$ by filling in the table shown below.

 $\boldsymbol{i^1}$ $\boldsymbol{i^5}$ $\boldsymbol{i^2}$ $\boldsymbol{i^6}$ $\boldsymbol{i^3}$ $\boldsymbol{i^7}$ $\boldsymbol{i^4}$ $\boldsymbol{i^8}$

a. What can you observe based on the two columns’ results?
b. Using your observation, find $i^k$ if the remainder of $k$ when it’s divided by $4$ is $2$?
c. Find the value of $i^{36} – i^{57}$.

Solution

We can start with {i} and multiply the current value with $i$. Use the fact that $i^2 = -1$ as well.

 $\boldsymbol{i^1}$ $i^1 = i$ $\boldsymbol{i^5}$ $1 \cdot I = i$ $\boldsymbol{i^2}$ $i \cdot i = -1$ $\boldsymbol{i^6}$ $i \cdot i = -1$ $\boldsymbol{i^3}$ $-1 \cdot i = -i$ $\boldsymbol{i^7}$ $-1 \cdot i = -i$ $\boldsymbol{i^4}$ $-i \cdot i = -i^2 = 1$ $\boldsymbol{i^8}$ $-i \cdot i = -i^2 = 1$

The table above shows that values of $\{i, i^2, i^3, … i^8\}$. The two columns show the same result – meaning, the powers of i will return the same cycle of values shown for $i$, $i^2$, $i^3$, and $i^4$.

a. Let’s observe each pair that returns the same values:

• $i^5 = i^1 =1$: $5$ and $1$ return a remainder of $1$ when the powers are divided by $4$.
• $i^6 = i^2 =-1$: $6$ and $2$ return a remainder of $2$ when the powers are divided by $4$.
• $i^7 = i^3 =1$: $7$ and $3$ return a remainder of $3$ when the powers are divided by $4$.
• $i^8 = i^4 =1$: $8$ and $4$ are multiples of $4$ and will not return any remainder when the powers arty e divided by $4$.

How do we generalize this?

When $k$ is divided by $4$ and if the remainder is $1$, $2$ or $3$, $i^k$ will be equal to $i^1 = 1$, $i^2 = -1$, or $i^3 = -i$, respectively.  If $k$ is a multiple of $4$, $i^k$ will be equal to $i^4 = 1$.

b. Apply what we’ve observed in a, since the remainder of $\dfrac{k}{4}$ is $2$, $i^k = i^2 = -1$.

### Open Problem Solution

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