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# Abstracts

### Monday, September 11th, 2017

9:30 am10:30 am

We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of size roughly n^d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program.

11:00 am11:30 am

We consider Multiway Cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set  of special vertices called terminals. Multiway Cut admits a well-known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best known solutions to the problem are comprised of a mix of several  different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically and its approximation factor was verified using a computer. We propose a new approach to simplex partitioning and the Multiway Cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms, and in addition yields an approximation guarantee for the Multiway Cut problem that (roughly) matches the current best computer verified approximation factor.

11:30 am12:00 pm

We will survey some ways that linear programming relaxations can capture optimal strategies in (discrete) stochastic optimization
problems. These problems are often NP-hard or worse, and we will discuss techniques used to develop approximation algorithms for
them. Examples include  stochastic knapsack and some budgeted multi-armed bandit problems.

2:00 pm3:00 pm

Combinatorial discrepancy has recently found applications in approximation algorithms, and, in particular, in designing new LP rounding algorithms. It can be seen as a common generalization to randomized and iterated rounding, as well as to total unimodularity. In this talk we will give an introduction to combinatorial discrepancy, focusing on the discrepancy of matrices, and its connections to LP rounding. We will describe several beautiful discrepancy minimization algorithms based on geometric random walks. Finally, we will also touch on approximation algorithms for discrepancy itself.

3:30 pm4:00 pm

We show that the problem of approximating the covering radius of a lattice, i.e. approximating the largest distance of any point in space from the lattice, up to a constant factor is in coNP assuming the slicing conjecture for Voronoi cells of so-called stable lattices. This improves on the approximation factor of O(sqrt{log n} x stable voronoi slicing) (or O(log^{3/2} n) unconditionally) achieved by Regev & Stephens-Davidowitz (STOC 17), based on their resolution of the reverse Minkowski conjecture.

Our characterization relies upon the canonical filtration of a lattice, as does that of Regev & Stephens-Davidowitz, i.e. canonical way to decompose a lattice along dense lattice subspaces, combined with a convex relaxation of the covering radius due to Dadush & Regev (FOCS 2016). From the algorithmic perspective, we show how to compute an O(log n) approximate version of the canonical filtration in 2^{n+o(n)} time, using discrete Gaussian sampling and a new form of basis reduction that we call filtration reduction.

As a companion contribution, we show that the slicing constant of any Voronoi cell is bounded by O(log^{3/2} n), which complements the O(log n) achieved by Regev & Stephens-Davidowitz for Voronoi cell of stable lattices.

4:00 pm4:30 pm

We show that any integer linear program (ILP) that is bimodular, which includes ILPs defined by a constraint matrix whose subdeterminants are all within {-2,-1,0,1,2}, can be solved efficiently; even in strongly polynomial time. This is a natural extension of the well-known fact that ILPs with totally unimodular (TU) constraint matrices are strongly polynomial-time solvable.

To derive our result we combine several techniques. In particular, starting from an optimal vertex solution v to the natural linear relaxation, we show how v can be rounded in several steps through combinatorial techniques to an optimal integral solution. More precisely, we first reduce the rounding problem to a particular parity-constrained ILP over a TU constraint matrix. We then leverage Seymour's decomposition of TU matrices to break this parity-constrained ILP into simpler base problems, which can be rephrased as combinatorial optimization problems and solved with combinatorial techniques, including parity-constrained submodular minimization. Finally, we show how efficient procedures for the base problems can be propagated up through Seymour's TU decomposition to round v to an optimal integral solution.

### Tuesday, September 12th, 2017

9:30 am10:30 am
Speaker:

Recent years have shown remarkable progress in the design and analysis of approximation algorithms for the graphical metric TSP and the general metric case of the s-t-path TSP. We will survey a number of these results, focusing in particular on the s-t-path TSP, where these results have seen the improvements to Christifides' algorithm yield a performance guarantee essentially matching the integrality gap of the standard LP relaxation.

11:00 am12:00 pm

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation.

Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem.

This is joint work with Ola Svensson and Jakub Tarnawski.

2:00 pm2:30 pm

We develop polynomial-size LP-relaxations for orienteering and the regret-bounded vehicle routing problem (RVRP) and devise suitable LP-rounding algorithms that lead to various new insights and approximation results for these problems. In orienteering, the goal is to find a maximum-reward r-rooted path, possibly ending at a specied node, of length at most some given budget B. In RVRP, the goal is to find the minimum number of r-rooted paths of regret at most a given bound R that cover all nodes, where the regret of an r-v path is its length - (distance of v from r).

For orienteering without a specified end-node, we introduce a natural bidirected LP-relaxation and obtain a simple 3-approximation algorithm via LP-rounding. This is the first LP-based guarantee for this problem. We also show that point-to-point orienteering (where the end-node is also specified) can be reduced to a regret-version of rooted orienteering at the expense of a factor-2 loss in approximation. For RVRP, we propose two compact LPs that lead to signicant improvements, in both approximation ratio and running time, over the previous O(1)-factor approximation algorithm. One of these LPs is a rather unconventional formulation that leverages various structural properties of an RVRP-solution.

This is joint work with Zachary Friggstad.

2:30 pm3:00 pm

The high-level goal of survivable network design is to design cheap networks that satisfy giving connectivity requirements even in the preserve of node or edge faults. While many approximation algorithms are known for a variety of such problems in undirected graphs, not much is known for directed ones. In this work we present the first non-trivial approximation algorithm for the 2-edge connectivity generalization of Directed Steiner Tree (DST). Analogously to DST, our approach gives a polylogarithmic approximation in quasi-polynomial time, or an $n^{\epsilon}$ approximation in polynomial time. Our approach is based on a complex LP-rounding algorithm that combines the notion of divergent trees and Zelikosky’s height reduction via a probabilistic mapping.

Joint Work with Bundit Laekhanukit

3:30 pm4:00 pm

In this talk, I will cover some recent progresses on approximating non-preemptive scheduling problems with the total weighted completion time objective, under different machine settings. For many such problems, we give approximation algorithms improving upon the previous state-of-art results that stood for 15 to 20 years. Surprisingly, the improvements are achieved via some natural time-indexed linear programming relaxations, that were not explicitly studied in the literature.

4:00 pm4:30 pm

The poise of a spanning tree in an undirected graph is the sum of its diameter and its maximum degree. Finding Steiner trees of minimum poise have applications for the design of fast multicast schemes among the terminals. We present an O(log k) approximation algorithm for minimum poise Steiner tree on k terminals with respect to its natural LP relaxation. This in turn finds use in the first poly-logarithmic approximation algorithms for multicommodity multicast problems in planar graphs. (Details in https://arxiv.org/abs/1612.01492)

Joint work with Jennifer Iglesias (Carnegie Mellon University), Rajmohan Rajaraman (Northeastern) and Ravi Sundaram (Northeastern)

### Wednesday, September 13th, 2017

9:30 am10:30 am

The study of combinatorial optimization problems with a submodular objective has attracted much attention in recent years. Submodular functions are commonly used as utility functions in economics and algorithmic game theory, and they play a major role in combinatorial optimization.
In this talk, I will survey several approaches for maximizing submodular functions. In particular, I will show several recent results that are based on solving multilinear relaxations. These relaxations play the role of linear programming relaxations in linear optimization problems.

11:00 am11:30 am

We study the Unsplittable Flow problem (UFP) on trees with a submodular objective function. The input to this problem is a tree with edge capacities and a collection of tasks, each characterized by a source node, a sink node, and a demand. A subset of the tasks is feasible if the tasks can simultaneously send their demands from the source to the sink without violating the edge capacities. The goal is to select a feasible subset of the tasks that maximizes a submodular objective function.

Our main result is an O(k log n)-approximation algorithm for Submodular UFP on trees where k denotes the pathwidth of the given tree. Since every tree has pathwidth O(log n), we obtain an O(log^2 n) approximation for arbitrary trees. This is the first non-trivial approximation guarantee for the problem and it matches the best approximation known for UFP on trees with a linear objective function.

Our main technical contribution is a new geometric relaxation for UFP on trees that builds on the recent work of [Bonsma et al., FOCS 2011; Anagnostopoulos et al., SODA 2014] for UFP on paths with a linear objective. Our relaxation is very structured and we can combine it with the contention resolution framework of [Chekuri et al., STOC 2011]. Our approach is robust and extends to several related problems, such as UFP with bag constraints and the Storage Allocation Problem.

Joint work with Parinya Chalermsook, Alina Ene, and Andreas Wiese.

11:30 am12:00 pm

We consider the problem of maximizing a monotone submodular function subject to a single knapsack constraint. We describe an algorithm that achieves a nearly-optimal 1 - 1/e - eps approximation using (1/eps)^O(1/eps^4) n log^2 n queries to the evaluation oracle for the function. Our algorithm solves the multilinear relaxation for the problem but it avoids a quadratic dependency on the number of value queries by ensuring that the solution has only O(1/eps^4) entries that are strictly fractional (not equal to 0 or 1).

This is joint work with Huy Nguyen (Northeastern University).

2:00 pm3:00 pm

In Online Optimization, the data in an optimization problem is revealed over time, and at each step a decision variable needs to be set without knowing the future data. We consider an online optimization setup that includes problems such as online resource allocation with a fixed inventory and the `Adwords' problem popular in online advertising. We consider two broad primal-dual algorithms, with a focus on the competitive ratio, i.e., the ratio of the objective achieved by the algorithm to that of the optimal offline sequence of decisions. We give a bound on this ratio and show how certain smoothing of the objective function can improve the bound; and for separable functions, how to seek the optimal smoothing by solving a convex design problem. This allows us to design effective smoothing customized for a given cost function and problem structure.

3:30 pm4:00 pm

Given a finite metric space, the usual k-center problem asks where to place the centers of k unit radius "inflatable" balls such that they cover all the points of the space with as small dilation as possible. This problem has been well understood for more than 3 decades. In this talk we will look at the same question when the balls have different radii. I will try to convince you that this is a natural generalization to study; in particular it generalizes the problem with outliers. Technically, one interesting aspect will be connections to the so-called "firefighter" problem.

Joint work with Prachi Goyal and Ravishankar Krishnaswamy

4:00 pm4:30 pm

A classical problem in scheduling is to optimize load balancing objectives in an online assignment of jobs to machines. If the machines have non-uniform speed, then the problem is called scheduling on related machines. In this talk, I will describe a new algorithm for this problem that achieves a constant-competitive assignment for any l-p norm objective. Previous to this work, a constant competitive ratio was known only for the makespan objective (l-infinity norm), via the well-known "slowest-fit" algorithm. Our new algorithm uses gradient descent to solve a convex relaxation of the problem, followed by a rounding technique based on a new framework called machine smoothing. In contrast, most previous relax-and-round algorithms for online problems have employed multiplicative weights to solve the convex relaxation, which incurs a super-constant loss in the competitive ratio. Our techniques also give new results for more general vector scheduling problems. This is joint work with Sungjin Im, Nat Kell, and Maryam Shadloo.

### Thursday, September 14th, 2017

9:30 am10:30 am

For a given NP-hard combinatorial optimization problem, one may identify interesting subclasses of instances with approximation ratios that are better than those that are known for worst case instances. Such a study can serve two purposes. One is to circumvent hardness of approximation results (in cases where they exist). The other is to guide us towards better worst case approximation algorithms (in cases where hardness of approximation results do not exist). The talk will present examples for these two themes.

11:00 am11:30 am

We study the notion of stability and perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A combinatorial optimization problem is α-stable or α-perturbation-resilient if the optimal solution does not change when we perturb all parameters of the problem by a factor of at most α. We present improved algorithms for stable instances of various clustering and optimization problems, including k-means, k-median, Max Cut, and Minimum Multiway Cut. We also show several hardness results.

The talk is based on joint papers with Haris Angelidakis, Yury Makarychev, and Aravindan Vijayaraghavan.

11:30 am12:00 pm

This talk consists of two different but related results. First, we show how to approximate unique games in minor-closed graph families using linear programming and standard low diameter graph decomposition. We will illustrate the proof through the min-uncut problem. Motivated by this result, we consider graph decompositions using effective resistance as the distance measure, and show that every graph (not just minor-free graphs) has a low effective-resistance diameter graph decomposition.

2:00 pm3:00 pm

In a classical problem in scheduling, one has n unit size jobs with a precedence order and the goal is to find a schedule of those jobs on m identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is NP-hard for m=3.

We prove that for any fixed epsilon > 0 and m, an LP-hierarchy lift of the time-index LP with a slightly super poly-logarithmic number of rounds provides a (1 + epsilon)-approximation. For example Sherali-Adams suffices as hierarchy. This implies an algorithm that yields a (1+epsilon)-approximation in almost quasi-polynomial time. The previously best approximation algorithms guarantee only a 2-approximation for large m.

This is joint work with Elaine Levey.

3:30 pm4:00 pm

We present LP relaxations for uniform and non-uniform reordering buffer management, and their application to the design and analysis of competitive online algorithms and polynomial time approximation algorithms for these problems. The talk is based on several joint papers with Noa Avigdor-Elgrabli, Sungjin Im, and Benjamin Moseley.

4:00 pm4:30 pm

Given a graph and a set of paths we want to cut (usually implicitly defined), cut problems ask to find the smallest number of edges of vertices that intersect all given paths. We survey recent developments on approximation algorithms and hardness of approximation (assuming the Unique Games Conjecture) for various cut problems, including Directed Multicut, Length-Bounded Cut, and their non-fixed terminal counterparts. We will highlight the efforts to find the optimal LP relaxation for each problem, and introduce a framework that converts their integrality gaps to hardness results.

Based on joint work with Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Chao Xu, and Vivek Madan.

### Friday, September 15th, 2017

9:30 am10:30 am

Dependent rounding, originating from the "pipage rounding" work of Ageev and Sviridenko, is now a fundamental algorithmic approach in combinatorial optimization. We survey this area, with an emphasis on the types of correlation inequalities that are achievable/desirable, and applications thereof. Topics covered include connections with matroids, matroid intersection, and scheduling, as well as how to handle situations where some amount of positive correlation is inevitable: the last part is recent joint work with David Harris, Thomas Pensyl, and Khoa Trinh, and arises in facility-location problems.

This talk will be a remote presentation.

11:00 am11:30 am

The problem of subdeterminant maximization subject to combinatorial constraints arises in various algorithmic and machine learning settings and, recently, convex-programming based estimation algorithms for it have been proposed. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms -- that also output a set -- remained open.

We present novel nonconvex formulations for the constrained subdeterminant maximization problem and prove a new anti-concentration inequality for dependent random variables that together allow us to obtain approximation algorithms for this problem in a variety of settings.

Based on a joint work with Javad Ebrahimi and Damian Straszak available https://arxiv.org/abs/1707.02757

11:30 am12:00 pm
Speaker:

Despite superficial similarities with the determinant, the permanent is #P-hard to compute, even for {0,1}-matrices. A long line of work has focused on designing approximation algorithms for the permanent of matrices with nonnegative entries, culminating with an FPRAS due to Jerrum, Sinclair, and Vigoda.

The permanent has been known to also be nonnegative for another class of matrices, namely positive semidefinite (PSD) matrices. In this talk I will show how to get a c^n approximation algorithm for the permanent of PSD matrices, where c is a universal constant.

Our algorithm is based on a semidefinite program with a convex nonlinear objective. We show that the input matrix can be sandwiched between a diagonal matrix and a rank-1 matrix whose permanents differ by at most a factor of c^n. To obtain the lower bound matrix we use a randomized rounding technique based on projections onto complex gaussian vectors.

If time permits, I will talk about the implications of our result for the recently refuted permanent-on-top conjecture as well as some open problems.

Joint work with Leonid Gurvits, Shayan Oveis Gharan, Amin Saberi.

12:00 pm12:30 pm

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min s − t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled + or − and the goal is to produce a clustering that agrees with the labels as much as possible: + edges within clusters and − edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max s − t Cut: find an s − t cut minimizing the largest number of cut edges incident on any node. We present the following results:
(1) an O(sqrt{n})-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems),
(2) a remarkably simple 7-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 48), and
(3) a 1/(2+ε)-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the 1/(4+ε)-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.


text:
Percent error calculator is used to find out the error in a calculation in terms of percentage. It is an online calculator for the easy calculation. The equations used for this calculation is given below.

Percent Error = $\frac{True\ value-observed\ value}{True\ value}$$\times$ 100

## Percent Error Steps

Following steps are used for the calculation of percent error.

Step1: Find out the true value and observed value from the question.

Step2: Substitute these values in the above equation and find out the percent error.

### Percentage Difference Calculator

 How to Calculate Relative Error How do you Calculate Percent of Increase Percentage Error Decimal to Percent Fraction to Percent Cumulative Percent Frequency
 Calculate Percent off Adding Percent Calculator Calculate Percent Change Convert Decimal to Percent Calculator Convert Fraction to Percent Calculator Percent Decrease Calculator

format: plain


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MathSciNet bibliographic data MR907153 32M15 (31C10 46E15 46J15) Chen, Wu Young Hardy and Lipschitz spaces of holomorphic functions and pluriharmonic functions over bounded symmetric domains in ${\bf C}\sp N$${\bf C}\sp N$. Chinese J. Math. 15 (1987), no. 1, 31–41. Links to the journal or article are not yet available

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


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# Radio Frequency Probe Post #2: 2019 Update

(Written by Jonathan Sheppard)

### Intro

This summer the team took the previous year team’s RF probe design and both updated and upgraded it, in an attempt to optimize its functions. As with last year’s design, we used the text Experimental Pulse NMR: A Nuts and Bolts Approach by Eiichi Fukushima and Stephen B. W. Roeder as a guide. The main sections used will be noted throughout this post.

As noted in the previous post, below are some general guidelines on the inductor coil specifications (from the section “V.C.4. Probes” (p. 373-385)): * A good coil for RF probe has a large inductance, which is given by the equation: $$L=\frac{n^2a^2}{23a\ +\ 25b},$$ where $n$ is the number of loops, and $a$ and $b$ are the radius of the loop and the length of the coil measured in centimeters. * The spacing between the turns of the coil should approximately equal to the diameter of the wire. * AWG14 wire is about the smallest that can be recommended for a free-standing coil.

To begin the process, we had to choose a desired resonance frequency for our coil. We brought it down from last summer’s 21 MHz to 18.75 MHz. Then, we had to note the range of capacitances the capacitors in our lab have: 10-1000 pF. Using these values, we used the formula for angular resonance: $$\omega_0=\frac{1}{\sqrt{LC}},$$ solving for $L$, to find the theoretical maximum inductance our could have: 7.2 µH.

We wanted to fit the coil inside the bore of our magnet, which has an inner radius of 1.7 cm, and also we wanted the length of the coil, $b$, to be at least 2.5 cm. Variable $a$ is the inner radius, so we subtracted the wire width (thickness) from our total 1.7 cm diameter. In theory, this would have been 1.37 cm. In practice, this became 1.1 cm. Thus, 1.1 is also the diameter of the tube we will be using to wrap our wire around. So, our $a$, which is the inner radius, is 1.1 / 2, giving $a$ = 0.55 cm.

We then chose our $b$ to simply be the length of our bore, 5 cm. Following the advice of A Nuts and Bolts Approach, we experimented with using both AWG14 and AWG12 copper wire, as our inductor coil would be free-standing. We settled on AWG14 (with nominal wire diameter 1.628 mm) due to the lowered requirements of our $b$ to achieve the highest possible inductance, $L$.

Using a python script, we created, we discovered the rest of our values needed to create an inductor coil. These values included $L$ (estimated inductance), $n$ (number of turns), $l$ (length of copper wire needed), $R$ (resistance of our length of copper wire), $Q$ (quality factor of our inductor coil), $C_t$ (tuning capacitance), and $C_m$ (matching capacitance). We’ll continue this post by walking through the equations used to get these values.

To begin, we used the following function: $$n=\frac{\sqrt{(23a\ +\ 25b)\ \ast\ L}}{a},$$ which you’ll notices is the equation we used to calculate the maximum inductance, solved for number of turns, $n$.

   1. def findL_n(a, b, d):
2.     wireWidth = 1000
3.     L = 0
4.     while wireWidth > d:                  #width in cm
5.         L += .01
6.         n = sqrt(((23 * a + 25 * b) * L)) / a
7.         wireWidth = b / (2 * n)
8.     return L, n                             #L in μH

We started with a ridiculously high $wireWidth$ and an inductance, $L$, of zero. Then, we increment $L$ up in each loop, updating the $wireWidth$ each time, until our wireWidth is equal to $d$ (1.628 mm, for our AWG14 copper wire). When the wireWidth is equal (or greater than) our actual wire width, the loop breaks, returning the $L$ given by the parameters we’ve chosen. In this case, we are given our number of turns as well. Using this function, we got back $L$ = 0.52 µH and n = 15.5.

The next step was to find out how much copper wire we’d need to create the inductor coil. To do that, we needed to use our inner radius $a$ and the number of turns $n$.

   1. def findCoilLength(a, n):
2.     return 2 * pi * (a / 100) * n

This function makes use of multiply the circumference of each loop (converting a to meters so that our output will be given in meters), by the number of loops. The output here was $l$ = 0.53 m.

Here we needed to make a quick check on our calculation. The length of copper wire used must be lower than an eighth of a wavelength at resonance.

   1. def isCoilLengthOkay(a, f_0, n):
2.     eighthWavelength = c / (8 * f_0)
3.     l = findCoilLength(a, n)
4.     return l < eighthWavelength

To do this, we used the formula for wavelength, multiplied by 18 to get, you guessed it, an eighth of a wavelength. Then we simply called the function we defined before, and compared the two, and out l passed the check.

Next, we needed to find the resistance of our length of wire in order to calculate a quality factor, $Q$.

   1. def findCopperResistance(l, d):
2.     return 4 * l * 0.0171 / (pi * d**2)   #d in mm
3.
4. def findQualityFactor(f_0, L, R):       #L input μH, converted to H
5.     w_0 = 2 * pi * f_0
6.     return w_0 * L * 10**(-6) / R

Using the formula for copper resistance: $$R=\frac{4l\rho}{\pi d^2},$$ where $l$ is the length of wire used in meters (0.53 m in our case), $d$ is the nominal diameter of wire in millimeters (1.628 mm in our case), and $ρ$ is the resistivity (0.0171 Ω*mm2/m for copper). The function estimated our coil to have an $R$ = 0.0044 Ω. Then, plugging our $R$ into the second function alongside our angular resonance frequency, ω0, and inductance, $L$, using the equation $$Q=\frac{\omega_0L}{R},$$ we get a $Q$ = 14,028. In practice, this is much lower, as will be shown later.

Finally, we use all the information collected so far to calculate how much capacitance our tuning and matching capacitors ($C_t$ and $C_m$ respectively) should have.

   1. def findC_t(f_0, L):                    #L input μH, but converted to H
2.             w_0 = 2 * pi * f_0
3.             return 1 / (L * 10**(-6) * w_0**2)
4.
5. def findC_m(f_0, L, R):                 #L input μH, but converted to H
6.             Q = findQualityFactor(f_0, L, R)
7.             C_t = findC_t(f_0, L)
8.             w_0 = 2 * pi * f_0
9.             return sqrt((Q * w_0 * L * 10**(-6) * C_t**2) / 50) - C_t

Our tuning capacitor simply required us to solve our angular resonance equation for $C$, this time, instead of $L$. Our matching capacitor required a bit more work, but it was simple enough. We used an equation given by A Nuts and Bolts Approach, solved for $C_m$: $$R\ =\ \frac{Q\omega_0L{C_t}^2}{{(C_m\ +\ C_t)}^2}.$$ This function gave us a $C_t$ of 138 pF and a $C_m$ of 180 nF.

### Summary of Coil Design Process

Thus, we had all the specifications we needed to create an inductor coil to suit our needs. Below is a quick summary of all of the values above, so they’re all in one place and also to refresh your mind of them. Keep in mind these are theoretical values received from our python script. Not all of them stayed the same, as theory and practice are separate beasts.

$f_0$ = 18.75 MHz $L$ = 0.52 μH $Q$ = 14,028
$\textrm{Max }L$ = 7.2 μH $b$ = 5 cm $n$ = 15.5
$R$ = 0.0044 Ω $C_t$ = 138 pF $d$ = 1.628 mm
$a$ = 0.55 cm $l$ = .53 m $C_m$ = 180 nF

### RF Probe Circuit Design, Building, & Testing

In accordance with A Nuts and Bolts Approach, we used the parallel tank circuit suggested on page 381.

This design is a bit different than last summer’s, due to the fact that we more closely followed the route suggested in A Nuts and Bolts Approach. Another difference, and one that is a bit more fundamental to the design, was to future-proof it. This came in the form of making the circuit Plug-and-Play. Before, everything, including the inductor coil and BNC cable, were hardwired into the circuit. This required the BNC cable to be spliced and soldered into the circuit. The new design made use of a BNC connector (no more destroying BNC cables) and banana clips for the inductor coil. This meant we could use varying lengths of BNC cable, depending on the situation, and that we could also try out different coils if necessary. In the second version of the design, we even added a clip for the static matching capacitors, so those could be clipped in and out at will with ease.

Another design change came in the form of the coil design. The previous year’s design created an inductor coil inside of a test tube, which in turn would fit into the bore of the magnetic field. This updated design puts the inductor coil on the outside, with no tube to fit in other than the bore itself. This would have increased the room we had for samples, if not for the fact that we upgraded to AWG14 wire from AWG18; hence, any gains were offset with the thicker wire.

And finally, the last design change came by turn the coil perpendicular to the rest of the circuit. This was due to the fact that a final form of our machine would ideally have a flow system in place to let a live Zebrafish be imaged. To do this, the flow system would have to run through the bore (and hence through our inductor coil) on both sides. So, twisting the inductor coil perpendicular allowed us to craft a tube (or simply two holes in the second design variation) through which the flow system would snake through, without hitting any circuitry and impeding our RF Probe.

Pictured to below is the first complete design using these new thoughts. It was completely 3D-printed with the exception of the white cap we salvaged and bolted our own variable capacitor to. We found a good tuning capacitance at 72 pF, and continued to test different matching capacitances.

The graph below is one of the resonance curves for matching capacitance of 100 nF, with the accompanying quality factor.

When the testing began, the resonance frequency was quite a bit lower than we wanted. By changing the matching capacitance, we managed to get it around 18.75 MHz, which was our desired resonance frequency. In this design, the matching capacitors were hard-wired into the circuit, so each test required soldering to modify the capacitors. In the picture above, you’ll notice the snap connector. After building an aluminum version (more on this in the next), we learned that this was a much simpler method of changing our static matching capacitors.

As a first design test, things went surprisingly well. However, we were seeing some odd background noise. To combat this, we decided to try again, replacing our plastic, 3D-printed enclosure for a shielding aluminum one, pictured below. The circuit was essentially the same, and we used more variable capacitors for both the tuning and matching, in addition to the static capacitors in parallel with the matching. The only change came in the form of the matching capacitor. We added a small spring connector (in keeping with our Plug-and-Play mentality), so that if we wanted to further experiment with the matching capacitance, we wouldn’t have to solder over and over again, improving the quality control of subsequent tests. An important thing to note is that, due to the banana clips, we were able to use the exact same inductor coil as our last model was tested with.

We also noticed that our matching capacitance was much too high. The TeachSpin system had a static matching capacitor of approximately 900 pF, with variable capacitors equaling 330 pF. Knowing that our system was similar to the TeachSpin (and that it was the TeachSpin we were trying to emulate) we lowered our static matching capacitor from nanofarads to picofarads, and began to see better peak-peak voltages. This went against our python program, and against our working theory. On possible explanation is that the resistance in reality is larger than the calculated resistance of the copper wire. This is exemplified below in the resonance curves, the first has a large matching capacitance and the second has a lower one. The third curve was taken with an Analog Discovery 2 System Analyzer using the SpinCore RF processor to generate a fake signal (rather than the TeachSpin synthesizer and an oscilloscope). This fake signal consisted of a 5 Vpp pulse that was 25 μs long and repeated every 50 μs. The last curve was generated using the same setup, except we used the TeachSpin magnet and RF tank circuit instead of our own.

NOTE: The final two curves are not resonance curves, they are the amplitude of the fake signal at the tuned frequency is being detected. (To get a resonance curve, one would need to change the frequency of the fake signal and plot the amplitude of the response curve relative to the change in frequency.)

We tried using the TeachSpin receiver to measure the response to a fake signal generated by the SpinCore RF processor, but kept on getting very noisy signal with a large baseline and barely visible pulse. We hypothesize that this bad data was due to the receiver on the TeachSpin `subtracting out’ the RF frequency before output. The SpinCore RF processor is not synced with the TeachSpin so this subtraction process would not be very successful. However, the fact that our tank circuit’s sensitivity is comparable to the TeachSpin’s is notable.

Before we began to use the Analog Discovery system, we were still seeing the same noise that we noticed with our 3D-printed version. In another attempt to lessen this noise, we crafted some rudimentary shielding to encompass our system. Importantly, we also had to ground the actually body of our enclosure, and, when connected to the shielding system, this allowed our shield to become grounded as well. Without grounding, our shield wouldn’t give us much noise reduction. As in all things with this project, our goal was to match or surpass the TeachSpin system. Surprisingly, our shielding (shown below surpasses the TeachSpin system ever so slightly in its ability to reduce noise.

The big, end goal of this summer was to get some sort of signal from our system. Unfortunately, we were unable to achieve this goal. However, the sheer amount of progress we were able to make in other areas more than makes up for this. We have several designs for new circuit enclosures. Said designs are consistent and reproducible, with implementations for ease of access, such as adding banana clips to the inductor coil and using a snap connector for the static matching capacitors. We made real progress with figuring out how to use both the RF Processor and Analog Discovery System Analyzer. Used in conjunction with each other, we are well on our way to acquiring signal not dissimilar from the TeachSpin System.

One of the things we’re looking forward to working on next summer is obtaining a directional coupler to further our utilization of our RF Processor and Analog Discovery Analyzer. We will also be building a transmit/receive circuit so that we can use our magnet, RF probe, and the RF processor to be completely independent of the TeachSpin system.


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# Numerical method for wave equation

1. Nov 20, 2014

### semc

Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic solution for both the linear and nonlinear part. So how do I solve numerically any arbitrary PDE? Specifically, my PDE is

$\partial_zE(r_⊥,z,\tau)-∇^2_⊥\int^{\tau}_{-\infty}d\tau 'E(r_⊥,z,\tau ')=\int^{\tau}_{-\infty}d\tau '\omega^2(r_⊥,z,\tau ')-\frac{\partial_{\tau}n(r_⊥,z,\tau)}{E(r_⊥,z,\tau)}$

and I want to solve for the evolution of E. Any help would be greatly appreciated. Thanks!

2. Nov 21, 2014

### UVCatastrophe

From my experience with PDE, you only need boundary conditions when solving for specific solutions. I am looking for a way to solve the PDE numerically. Would it be possible to give me the general method? For example you can $\bf{explain}$ the split step Fourier method without knowing the boundary conditions?


text:
# Kollektion: Geometrics

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Experiments demonstrate the possibility of engineering stable, nanometer-scale magnetic structures in thin magnetic films by firing laser pulses at them.

Magnetic spins in a material can swirl around a single point in a stable particlelike configuration. These magnetic vortices, called “skyrmions,” may find use in future forms of computer memory. New experiments have generated these magnetic structures using laser pulses directed at a thin magnetic film. By varying the strength of the pulse, the researchers created individual skyrmions and even “molecules” made from multiple skyrmions, as detailed in Physical Review Letters.

A skyrmion can be thought of as a twist in a field, which can’t easily be unwound. The idea originated fifty years ago as a way to explain baryons, but it was superseded by the quark model. However, researchers have identified skyrmion states in other systems, including Bose-Einstein condensates and superconductors, where the field equations exhibit the characteristic skyrmion topology. Over the last few years, physicists have also demonstrated several types of magnetic skyrmions by exposing magnetic thin films to a strong external magnetic field.

The skyrmions produced by Marco Finazzi of the Polytechnic Institute of Milan, Italy, and his collaborators are unique in not requiring an applied magnetic field, neither for generation nor for stabilization. The team fired laser pulses at thin films of the ferrimagnetic alloy terbium-iron-cobalt (TbFeCo). They then mapped out the resulting magnetic spin configuration with optical microscopy. At relatively low laser energy, they observed $100$-nanometer-wide cylindrically shaped skyrmions, in which the magnetization of the interior was opposite that of the exterior. Higher energy produced donut-shaped structures, which could be described as a skyrmion overlapping with an antiskyrmion (with opposite vortex circulation). The team called this moleculelike combination skyrmonium. These structures, as well as other more complex ones, did not noticeably change over one year of monitoring. The authors attribute this stability to the film’s own internal magnetic field. – Michael Schirber

### Announcements

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Atomic and Molecular Physics

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Researchers have verified experimentally that the reflection and refraction of spin waves at an interface follow a Snell’s-like law. Read More »

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By bathing a superconducting qubit in squeezed light, researchers have been able to confirm a decades-old prediction for the resulting phase-dependent spectrum of resonance fluorescence. Read More »

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### Focus: Tiny Digital Bits in Ferroelectric Material

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# State-Feedback Controller

Discrete-time state-feedback controller with integral action

• Library:
• Simscape / Electrical / Control / General Control

## Description

The State-Feedback Controller block implements a discrete-time state-feedback controller with integral action. Use this block to control linear systems with single or multiple inputs and single or multiple outputs. The integral action serves to eliminate steady-state error in the controlled outputs. You can define the controller using a precomputed optimal gain or use the state-space model of your system to generate this gain using pole placement.

### Equations

The integral of the tracking error, xi, is an additional state that ensures zero steady-state error for the closed-loop system. The extended state vector is

`${x}_{e}=\left[\begin{array}{c}x\\ {x}_{i}\end{array}\right],$`

Where:

• x is the state vector.

• xi is the integral of the tracking error.

• xe is the extended state vector.

Therefore, the control action is

`$u=K{x}_{e},$`

Where:

• K is the feedback matrix, that is, the pole placement.

• u is the controller output.

### Assumptions

System state measurement and estimation occur outside the controller.

## Ports

### Input

expand all

Plant system reference signal.

Data Types: `single` | `double`

Measured or estimated system state vector.

Data Types: `single` | `double`

External reset signal (rising edge) for the integrator.

Data Types: `Boolean`

Plant system output signal.

Data Types: `single` | `double`

### Output

expand all

Control system output signal.

Data Types: `single` | `double`

## Parameters

expand all

Select the strategy for parameterizing controller gain:

• `State-feedback gain` — Specify the controller gain directly

• `Desired eigenvalues` — Specify the plant model and desired eigenvalues from which to generate the controller gain

Select the strategy for parameterizing the state-space matrices and desired poles for the controller. The block implementation is discrete regardless of this parameterization.

#### Dependencies

To enable this parameter, set State-feedback design to ```Desired eigenvalues```.

Controller feedback matrix. To determine the controller matrix, if you have a license for Control System Toolbox™, use the `lqr` or `lqi` function.

#### Dependencies

To enable this parameter, set State-feedback design to ```State-feedback gain```.

State matrix of the discrete-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Discrete-time`.

Input matrix of the discrete-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Discrete-time`.

Output matrix of the discrete-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Discrete-time`.

Feedthrough matrix of the discrete-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Discrete-time`.

State matrix of the continuous-time state-space model. The A matrix must be square, with the number of rows and columns equal to the order of the system.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Continuous-time`.

Input matrix of the continuous-time state-space model. The B matrix must have the number of rows equal to the order of the system, and the number of columns equal to the number of system inputs.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Continuous-time`.

Output matrix of the continuous-time state-space model. The C matrix must have the number of rows equal the number of outputs of the system, and the number of columns equal to the order of the system.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Continuous-time`.

Feedthrough matrix of the continuous-time state-space model. The D matrix must have the number of rows equal to the number of system outputs, and the number of columns equal to the number of system inputs.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Continuous-time`.

Value used to discretize the state space matrices and also approximate the discrete-time eigenvalues.

#### Dependencies

To enable this parameter, set State-feedback parameterization to `Continuous-time` and Sample time (-1 for inherited) to `-1`.

Specify the location of the eigenvalues to lie within the unit circle. The controller gain is then calculated based on these eigenvalues. The size of the vector must be equal to the system order plus the number of outputs.

Upper limit for the control output signal.

Lower limit for the control output signal.

Time interval between samples. If the block is inside a triggered subsystem, inherit the sample time by setting this parameter to `-1`. If this block is in a continuous variable-step model, specify the sample time explicitly. For more information, see What Is Sample Time? and Specify Sample Time.


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A metric sphere not a quasisphere but for which every weak tangent is Euclidean

# A metric sphere not a quasisphere but for which every weak tangent is Euclidean

Angela WU Angela Wu
Department of Mathematics
University of California
Los Angeles
CA90095
USA
June 7, 2018
###### Abstract.

We show that for all , there exists a doubling linearly locally contractible metric space that is topologically a -sphere such that every weak tangent is isometric to but is not quasisymmetrically equivalent to the standard -sphere. The same example shows that -Ahlfors regularity in Theorem 1.1 of [BK02] on quasisymmetric uniformization of metric -spheres is optimal.

The author was partially supported by NSF grants DMS-1506099 and DMS-1266164

## 1. Introduction

Let and be two metric spaces. A homeomorphism is a quasisymmetry if there exists a homeomorphism such that for all , with , we have

 dY(f(x),f(y))dY(f(x),f(z))≤η(dX(x,y)dX(x,z)).

In this case we say that and are quasisymmetrically equivalent.

In [Kin17] Kinneberg characterized metric circles that are quasisymmetric to the standard circle in terms of weak tangents (defined in Section 4 below):

###### Theorem 1.1 (Kinneberg, [Kin17]).

A doubling metric circle is quasisymmetrically equivalent to the standard circle if and only if every weak tangent of quasisymmetrically equivalent to the real line based at .

Here a metric space is said to be doubling if for every , every ball of radius can be covered by balls of radius , where is a positive integer that does not depend on the choice of the ball . In this paper we prove that Kinneberg’s result cannot be extended to higher dimensions:

###### Theorem 1.2.

For every , there exists a doubling, linearly locally contractible metric space that is topologically an -sphere such that every weak tangent is isometric to but is not quasisymmetrically equivalent to the standard -sphere.

When , one can compare our result with the following Theorem:

###### Theorem 1.3 (Bonk and Kleiner, [Bk02]).

Let be an 2-Ahlfors regular metric space homeomorphic to . Then is quasisymmetric to if and only if is linearly locally contractible.

Our Theorem 1.2 shows that the conclusion of Theorem 1.3 is false if we replace -Ahlfors regularity with -Ahlfors regularity for .

Our study is also related to the following theorem, proven in [Lin17]:

###### Theorem 1.4.

Let be a doubling metric space homeomorphic to . The following are equivalent:

1. is quasisymmetrically equivalent to the standard -sphere.

2. For every , there exists an open neighborhood of in such that is quasisymmetrically equivalent to .

An alternative proof of Theorem 1.4 can be given using ideas in [GW18]. Roughly speaking, Theorem 1.4 says local geometry properties promote to global property. Since weak tangents are local, one could ask the following question:

###### Question 1.5.

Suppose is doubling and linearly locally connected. Are the following two statements equivalent?

1. is quasisymmetrically equivalent to the standard -sphere

2. Every weak tangent of is quasisymmetrically equivalent to .

When is a doubling and linearly locally connected metric sphere, statement (i) implies statement (ii). However, our construction shows that statement (ii) does not imply statement (i).

###### Acknowledgement.

The author would like to thank Mario Bonk and John Garnett for their support and many conversations. The author thanks Jeff Lindquist for helpful discussions and Wenbo Li for his comments and corrections.

## 2. The Modification of the Metric on R

We first construct a metric on so that every weak tangent of is isometric to , where is the Euclidean metric on , and so that there are segments of so that . Here, and in the remaining of the paper, denotes the length of the curve with respect to the metric . Once the metric is constructed, we show that for all , the weak tangents of the product space are isometric to the Euclidean space , but because , the product space cannot be quasisymmetric to the Euclidean space .

To construct , we glue together segments that look like snowflake curves of Hausdorff dimension i.e. intervals equipped with the metric . Such snowflake curves will have infinite length, so we modify the metric on the segment to get a metric so that looks like the Euclidean metric when two points are close together, but it looks like when the two points are far apart.

Note that our construction of only modifies the geometry of a bounded subset of , and therefore we can embed our -dimensional construction into a topological -sphere.

To begin the construction let and and define

 φα,c(x)={L(α,c)x,x∈[0,c](x−c(1−α)1−c(1−α))α,x∈[c,1],

where

 L(α,c)=1c(cα1−c(1−α))α.

The function is the only function that has the following properties:

1. ,

2. ,

3. is linear on ,

4. There exists , with , such that when ,

5. is continous and differentiable at .

In addition to the above properties that uniquely define , we observe that has the following extra properties:

1. is a homeomorphism,

2. is concave on .

###### Lemma 2.1.

For any , , and , we have

 φα,c(ab)φα,c(a)φα,c(b).
###### Proof.

The function is decreasing in to , therefore if , then

 φα,c1≥φα,c2.

Let

 φ∗(x)=φα,c(ax)φα,c(a).

The function meets condition (1)-(5) listed above for some , therefore . This implies that for all . Taking , we have

 φα,c(ab)≤φα,c(a)φα,c(b).

###### Lemma 2.2.

For any fixed , for any , and whenever , we have

 0≤φα,c(t)+φα,c(x−t)−φα,c(x)≤(2−2α)φα,c(x).
###### Proof.

When , for all , we have . When is fixed, is maximized when

 φ′α,c(t)−φ′α,c(x−t)=0,

which is possible only if i.e. . Then we have

 φα,c(t)+φα,c(x−t)−φα,c(x) ≤2φα,c(x/2)−φα,c(x) ≤2φα,c(x/2)−2αφα,c(x/2) =(2−2α)φα,c(x/2).

When , we have

 φα,c(t)+φα,c(x−t)≤L(α,c)x=xcφα,c(c).

By the concavity of , we have

 φα,c(x)x≥φα,c(2c)2c.

Therefore

 φα,c(t)+φα,c(x−t)−φα,c(x) ≤xcφα,c(c)−xφα,c(2c)2c=x2c(2φα,c(c)−φα,c(2c)).

But

 2φα,c(c)−φα,c(2c)≤(2−2α)φα,c(c)=(2−2α)cL(α,c).

Therefore

 φα,c(t)+φα,c(x−t)−φα,c(x)≤x2c(2−2α)cL(α,c)=(2−2α)φα,c(x/2).

In any case, we have

 0≤φα,c(t)+φα,c(x−t)−φα,c(x)≤(2−2α)φα,c(x/2)≤(2−2α)φα,c(x).

## 3. The One Dimensional Construction

For any , we have

 (1) limc→0+L(α,c)=+∞.

Let be an increasing sequence in such that . By (1), we can choose such that , and

 limn→∞L(αn,cn)=+∞.

Let . Choose a sequence so that for all , , such that is decreasing and .

For all , let . Let , and equip with the metric , where is the usual Euclidean metric on . Note that

1. The distance between the two endpoints of is .

2. is rectifiable and the length of is .

We construct a metric on so that if ,

1. when restricted to ,

2. when ,

3. when and ,

4. when and , and

5. when , .

## 4. The Weak Tangents of (R,δ)

A pointed metric space is a a triplet , where is a metric space and is a point in . Let . A map between two pointed metric spaces and is a -rough isometry if

1. ,

2. , and

3. for all ,

 |dY(ψ(x1),ψ(x2))−dX(x1,x2)|≤ε.

Note that a -rough isometry may not be continuous. The pointed Gromov-Hausdorff distance between 2 pointed metric spaces, denoted , is defined as the infimum of for which we can find a -rough isometry .

A pointed metric space is called a weak tangent of another metric space if there exist points and positive integers such that converges to in pointed Gromov-Hausdorff sense, i.e. for all , and for all there exists such that for all ,

 dGH(¯¯¯¯BλndX(xn,R+ε),¯¯¯¯BdT(p,R))≤ε.

In particular, we get if for all , and for all there exists such that for all ,

 dGH(¯¯¯¯BλndX(xn,R),¯¯¯¯BdT(p,R))≤ε.

Our notion of pointed Gromov-Hausdorff convergence is adopted from [HKST15, Definition 11.3.1]. Also see [BBI01] for detailed discussion on Gromov-Hausdorff distance and Gromov-Hausdorff convergence.

Let M be a set of separable, uniformly doubling, and uniformly linearly locally contractible pointed metric spaces. The pointed Gromov Hausdorff convergence on the set induces a topology on that is metrizable[Don11]. In particular, if a sequence of pointed metric space converges to a Gromov-Hausdorff limit, then the limit is unique.

The goal of this section is to prove the following proposition that describe all the weak tangents of .

###### Proposition 4.1.

For all , and for all positive integers , converges in pointed Gromov-Hausdorff sense to .

Note that Proposition 4.1 guarentees the existence of weak tangents of . To prove the above proposition we will use the following three lemmas:

###### Lemma 4.2.

Suppose are three points so that . Suppose . Then

 0≤δ(x,y)+δ(y,z)−δ(z,x)≤(2−2αN)min{sN,δ(x,y)}
###### Proof.

If , then . Otherwise, let

 a =sup{an≤y:n∈N}∨sup{bn≤y:n∈N}∨x b =inf{an≥y:n∈N}∧inf{bn≥y:n∈N}∧z

We have . By definition of , we have

 δ(x,y)+δ(y,z)−δ(x,z) =(δ(x,a)+δ(a,y))+(δ(y,b)+δ(b,z)) −(δ(x,a)+δ(a,b)+δ(b,z)) =δ(a,y)+δ(y,b)−δ(a,b).

Bu our choice of and , either for some , or . In the latter case, we have

 δ(x,y)+δ(y,z)−δ(z,x)=δ(a,y)+δ(y,b)−δ(a,b)=0.

In the former case, Lemma 2.2 gives

 0 ≤δ(x,y)+δ(y,z)−δ(z,x)=δ(a,y)+δ(y,b)−δ(a,b) ≤(2−2αn)δ(a,b)=(2−2αN)min{sN,δ(x,y)}.

Since , we have our desired conclusion. ∎

###### Lemma 4.3.

Let and be arbitrary. Suppose . Let and . The map

 ψ:([a,b],p,δ)→([−r,r],0,d) ψ(x)={−δ(p,x),x≤pδ(p,x),x>p.

is a -rough isometry, where .

###### Proof.

Since is surjective and fixes , it remains to check that for all , we have

 (d(ψ(x),ψ(y))−δ(x,y))≤ε.

Suppose . If , then

 |d(ψ(x),ψ(y))−δ(x,y)| =||δ(p,x)−δ(p,y)|−δ(x,y)| =|δ(p,y)−δ(p,x)−δ(x,y)| ≤(2−2αN)min{sN,δ(p,x)} =(2−2αN)min{sN,2r}.

The second to last inequality is a consequence of Lemma 4.2. If , then

 |d(ψ(x),ψ(y))−δ(x,y)| =||δ(p,x)−δ(p,y)|−δ(x,y)| =|δ(p,y)+δ(p,x)−δ(x,y)| ≤(2−2αN)min{sN,δ(x,y)} =(2−2αN)min{sN,2r}.

If , then following a similar arguement as when , we get

 |d(ψ(x),ψ(y))−δ(x,y)|≤(2−2αN)min{sN,2r}

This verifies that is a -rough isometry. ∎

###### Lemma 4.4.

For , we have

 supp∈RdGH((¯¯¯¯Bδ(p,r),p,δ),(¯¯¯¯Bd(0,r),0,d))=o(r)

as .

###### Proof.

Let and be arbitrary. Let . If , then on . We have

 dGH((¯¯¯¯Bδ(p,r),p,δ),(¯¯¯¯Bd(0,r),0,d))=0.

If , we consider 2 cases:

Case 1: . As , , therefore . In this case Lemma 4.3 implies

 dGH((¯¯¯¯Bδ(p,r),p,δ),(¯¯¯¯Bd(0,r),0,d))≤2(2−2αN)r=o(r).

Case 2: . In this case, is a length metric on , and is an isometry between two length spaces. We have

 dGH((¯¯¯¯Bδ(p,r),p,δ),(¯¯¯¯Bd(0,r),0,d))=0.

###### Proof of Proposition 4.1.

Let be a sequence in and be a sequence of positive numbers that diverges to . By Lemma 4.4,

 dGH((¯¯¯¯Bλnδ(an,R),an,λnδ),(¯¯¯¯Bλnd(0,R),0,λnd)) =λndGH((¯¯¯¯Bδ(an,λ−1nR),an,δ),(¯¯¯¯Bd(0,λ−1nR),0,d))=λno(λ−1nR).

But by symmetry of . As , . We have

 (¯¯¯¯Bλnδ(an,R),an,δ)→¯¯¯¯Bd(0,R),0,d).

This is true for all . We conclude that . ∎

## 5. Linear Local Contractibility and Assouad Dimension of (R,δ)

In this section we establish two properties of the space . These properties often appear in the study of quasisymmetry classes of metric spheres. Both properties are discussed in detail in [Hei01].

###### Definition 5.1.

Let be a constant. A metric space is -linearly locally contractible if every small ball is contractible inside a ball whose radius is times larger. A metric space is linearly locally contractible if it is -linearly locally contractible for some .

###### Definition 5.2.

Let . A metric space is -doubling if for all , every open ball of can be covered by balls of radius . A metric space is doubling if it is -doubling for some .

Both doubling and linear local contractibility are preserved under quasisymmetry. The Euclidean spaces like or are doubling and linearly locally contractible. The doubling property also ensures the existence of weak tangents.

###### Proposition 5.3.

The space is -linearly locally contractible.

###### Proof.

Any open ball in is an open interval . Denote the center of (in .) Note that the map is increasing on , and decreasing on . Therefore the map is a homotopy of to in . This proves that is -linearly locally contractible. ∎

###### Proposition 5.4.

The space is doubling.

If a metric space is doubling, then there exists and such that for all and , any set of diameter in can be covered by at most subsets of diameter at most . The function is called the covering function of . The Assouad dimension of is defined to be the infimum of all so that a covering function of the form of exists. Conversely, any metric space of finite Assouad dimension is doubling. Proposition 5.4 will follow from the stronger proposition below.

###### Lemma 5.5.

For each , the function is a covering function of .

###### Proof.

Every subinterval of of -diameter has -diameter . Thus our goal is to show that for every , and every , every subset of of -diameter can be covered by no more than -many subintervals of of -diameter at most . The number of subintervals we need can be bounded from above by

 snφ−1n(s−1nr)snφ−1n(s−1nεr)+1≤supy∈(0,1]φ−1n(y)φ−1n(εy)+1=supy∈[φn(cn),1]φ−1n(y)φ−1n(εy)+1.

We claim that the last supremum is attained when . This is equivalent to

 (2) φn(φ−1n(εy))φn(φ−1n(y))≤φn(φ−1n(εy)φ−1n(y))

Inequality (2) is true by Lemma 2.1.

Suppose . When , we have , and

 εα−1n=x0−c(1−αn)1−c(1−αn)≤x0.

When , we have and . Since , we have

 1x0=φ(c)cε=ε−α−1n(εφ(c))α−1n−1φ(c)α−1nc≤ε−α−1n.

In any case, we can take the covering function of to be

 ε−α−1n+1≤2ε−α−1n.

###### Proposition 5.6.

The Assouad dimension of is .

###### Proof.

Let be arbitrary. There exists such that when , . Let . Then the function , where , is a covering function of for all . Thus is a covering function of .

Since has Assouad dimension , the proposition follows. ∎

## 6. Higher Dimension Construction

Let . We will denote by the Euclidean metric on . Let

 Xd=(R×Rd−1,√δ2+d2Euclid)

be the product of and . Write . Here are some facts about .

###### Proposition 6.1.
1. Every weak tangent of is isometric to .

2. is doubling and linearly locally contractible.

###### Proof.
1. Every weak tangent of is of the form , where is a weak tangent of .

2. Recall that is doubling (Proposition 5.4). is product of doubling metric space, hence doubling. By Proposition 5.3, is -linearly localy contractible for some . Let be any point in , and be arbitrary. The ball in can first be contracted to within a , which can then be contracted to the point .

Every finite segment in is rectifiable. Let be a the measure on given by length. For , let be the product measure on , where is the -dimensional Lebesgue measure on .

In the remaining of this section we show that is not quasisymmetrically equivalent to . To do that we consider a geometric quantity that is roughly preserved under quasisymmetry called modulus. Given a family of curves in a measured metric space , we say that a function is admissible if for all ,

 ∫γρ(x)ds≥1.

Let . We define the -modulus of as

Let be two disjoint nondegerate continua in . Let to be the collection of all rectifiable curves joining and . We write .

Moduli behave nicely under quasisymmetry, as illustrated by the following theorem.

###### Theorem 6.2 (Tyson, [Tys98]).

Let be locally compact, connected, -Ahlfors regular metric spaces, where , and be a quasisymmetric homeomorphism. Then there exists such that for all curve family , we have

 1CmodQ(Γ)≤modQ(f(Γ))≤CmodQ(Γ).

See the next section for the definition of -Ahlfors regularity.

We can now prove that is not quasisymmetrically equivalent to . The idea is that if the two spaces are quasisymmetrically equivalent, then for any curve family , and should have comparable moduli. We know that has the property that any disjoint nondegerate continua in satisfy

 (3) modd(E,F)≤ϕ(d(E,F)diam(E)∧diam(F)).

for some non-increasing function . However, Proposition 6.3 shows that some sequence disjoint nondegerate continua in do not behaviour as in (3). The only problem is that is not -Ahlfors regular, so we cannot apply Theorem 6.2 directly. In Proposition 6.4, however, we will show that the inequality

 modd(En,Fn)≤Cmodd(f(En),f(Fn))

holds for some independent of .

From now on, we will denote .

###### Proposition 6.3.

There exists such that and .

###### Proof.

Take . Then , and , therefore

 Δ(En,Fn)=d(En,Fn)diamEn∧diamFn=1.

For each , and for and , let be the path

 t↦(x,t,v2,…,vd−1),t∈[0,sn].

Let

 Γn={γx,v2,…,vd−1:x∈In,(v2,…,vd−1


text:
dimod.generators.random_2in4sat¶

random_2in4sat(variables: , num_clauses: int, *, plant_solution: bool = False, seed: = None) [source]

Generate a random 2-in-4 satisfiability problem as a binary quadratic model.

2-in-4 satisfiability 1 is an NP-complete problem class that consists in satisfying a number of conjunctive clauses of four literals (variables, or their negations). For valid solutions, two of the literals in each clause should +1 and the other two should be -1.

Each clause contributes -2 to the energy when the clause is satisfied, and at least 0 when unsatisfied. The energy $$H(s)$$ for a spin assignment $$s$$ is thus lower bounded by $$E_{SAT}=-2$$num_clauses, this lower bound matches the ground state energy in satisfiable instances. The number of violated clauses is at most $$(H(s) - E_{SAT})/2$$.

Parameters
• num_variables – The number of variables in the problem.

• num_clauses – The number of clauses. Each clause contains three literals.

• plant_solution – Create literals uniformly subject to the constraint that the all 1 (and all -1) are ground states satisfying all clauses.

• seed – Passed to numpy.random.default_rng(), which is used to generate the clauses and the variable negations.

Returns

A binary quadratic model with spin variables.

Note

The clauses are randomly sampled from the space of 4-variable clauses with replacement which can result in collisions. However, collisions are allowed in standard problem definitions, are absent with high probability in interesting cases, and are almost always harmless when they do occur.

Planting of a not all 1 ground state can be achieved with a spin-reversal transform without loss of generality. Planting can significantly modify the hardness of optimization problems. However, for large problems planting of a solution can be achieved (in the SAT phase) without modification of the hardness qualities of the instance class. 2

1

Adam Douglass, Andrew D. King & Jack Raymond, “Constructing SAT Filters with a Quantum Annealer”, https://link.springer.com/chapter/10.1007/978-3-319-24318-4_9

2

Lenka Zdeborová and Florent Krzakala, “Quiet Planting in the Locked Constraint Satisfaction Problems”, https://epubs.siam.org/doi/10.1137/090750755


text:
# Zero to the Power of Zero/As a Limit

## Example of Zero to the Power of Zero

Consider the real function:

$y = x^x$

This function is well defined for $x > 0$.

It is not obvious whether or not the right hand limit:

$\ds \lim_{x \mathop \to 0^+} y$

exists.

If it does, it would be nice if:

$\ds \lim_{x \mathop \to 0^+} x^x = 0^0$

Indeed, by Limit of x to the x, we have:

$\ds \lim_{x \mathop \to 0^+} x^x = 1$

We see that defining $0^0 = 1$ allows $x^x$ to be right-continuous at $x = 0$.


text:
# Diamond and Graphite

Chemistry Level pending

Carbon is capable of forming many allotropes due to its valency. The above is a simplified phase diagram of carbon. Which of following explanations is correct?

a) At $$25$$°C and $$1$$ atm, graphite is the most stable phase of carbon.

b) If we put more and more pressure on graphite while maintaining the temperature at $$25$$°C, then graphite can become a diamond.

c) If we increase the temperature while maintaining the pressure below T in the diagram, then graphite sublimates into vapor.

×


text:
### Home > INT2 > Chapter 8 > Lesson 8.2.2 > Problem8-47

8-47.

Always a romantic, Marris decides to bake his girlfriend a cookie in the shape of a regular dodecagon ($12$‑gon) for their $12$-day anniversary.

1. If the edge of the dodecagon is $6$ cm, what is the area of the top of the cookie?

2. His girlfriend decides to divide the cookie into $12$ congruent pieces.
After nine of the pieces have been eaten, what area of the cookie is left?

Try splitting the cookie into triangles.
What information do you already know about these triangles' angles and sides?
Can you use this to find the area of a single triangle?

Multiply the area of a single triangle by the number of triangles remaining after some have been eaten.


text:
Home > CCA2 > Chapter Ch4 > Lesson 4.2.3 > Problem4-92

4-92.

Solve the system of equations below algebraically and explain what the solution tells you about the graphs of the two equations.

$3x+2=y$
$−9x+3y=11$

Use substitution to solve the system of equations.

There is no solution. If the lines never intersect, what does it say about their slopes?


text:
# Electric Dipole

Electric Dipole moment is the measure of the polarity of the system. It is the separation of positive charge and a negative charge by a distance. The SI unit of electric dipole moment is Coulomb-meter. The simplest example is a pair electric charge of two opposite sign and equal magnitude separated by distance.

## What is Electric Dipole?

Suppose there are two charges of equal magnitude ‘q’, separated by a distance ‘d’. Let the first charge be negative and the second charge be positive. This combination can be called as an electric dipole. Therefore, we can say that an electric dipole is created by the combination of equal and opposite charges by a separation of a certain distance.

Now, the electric dipole moment for this pair of equal and opposite charges is equal to the magnitude of the charges multiplied by the distance between them.

### Magnitude of Electric Dipole Moment

The magnitude of electric dipole moment can be written as:

 $\overrightarrow{p}$ = $q \overrightarrow {d}$

## Direction of Electric Dipole Moment

The electric dipole moment is a vector quantity; it has a defined direction which is from the negative charge to the positive charge. Though, it is important to remember that this convention of direction is only followed in Physics. In Chemistry, the convention is taken to be opposite i.e. from positive to negative. The line along the direction of an electric dipole is called the axis of the dipole.

You may also want to check out these topics given below!

### Electric potential due to a Dipole (V)

Suppose there are two charges –q, placed at A, and +q placed a B, separated by a distance d, forming a dipole. Suppose the midpoint of AB is O.

The Electric potential due to a dipole at any point P, such that OP = r will be:

 V = $\frac{1}{4\pi \epsilon } \frac{p cos \theta }{r^{2}}$

Case 1: If θ = 90°

 Electric potential = V = 0

Case 2: If θ = 0°

 Electric potential = V = $\frac{1}{4\pi \epsilon } \frac{p }{r^{2}}$

### Electric Dipole Questions

Q1) Define electric dipole.

An electric dipole is the measure of a system’s overall polarity. It is defined as the measure of positive and negative electrical charges within a system.

Q2) Which of the following is true of an electric dipole.

1. A pair of equal and unlike charges separated by a small distance
2. A pair of unequal and like charges located at the origin
3. A pair of equal and like charges located at the origin
4. A pair of unequal and unlike charges separated by a small distance

Answer: a. A pair of equal and unlike charges separated by a small distance

An electric dipole is a pair of equal and unlike charges separated by a small distance.

Q3) Find the potential due to the dipole when the angle subtended by the two charges at the point P is perpendicular.

Answer: The potential due to the dipole when the angle subtended by the two charges at the point P is perpendicular is zero.

The potential due to a dipole is given by the formula V = m cos θ/(4πεr2). When the angle is perpendicular θ = 90, therefore cos 90 = 0 as result of which the potential is zero.

Q4) Dipole moments are used to calculate the _____.

1. susceptibility
2. electric field intensity
3. strength of the dipole in the field
4. polarisation patterns

Dipole moments are used to calculate the polarisation pattern.

Q5) Calculate the distance between two charges of 6 C with a dipole moment of 8 units.

The dipole moment of a pair of charges is given by the formula

M = Q x d

Rearranging the formula, we get

d = M/Q

Substituting the values in the above equation, we get

d = 8/6 = 1.33 units

Q6) Calculate the dipole moment of equal charges of 7C separated by a distance of 4 cm.

The dipole moment of charge can be calculated as follows:

M = Q x d

Substituting the values, we get

M = 7 x 0.04 = 0.28 C-m

The dipole moment of the given pair of charge is 0.28 C-m.

Q7) Dipole under the influence of an electric field will undergo _____.

1. Polarisation
2. Electromagnetism
3. Magnetisation
4. Magnetism

Dipole under the influence of an electric field will undergo polarisation.

Q8) The electric potential due to a dipole is given by the formula _____.

$V=\frac{1}{4\pi \varepsilon }\frac{p\cos \Theta}{r^2}$

The electric potential due to a dipole is given by the formula $V=\frac{1}{4\pi \varepsilon }\frac{p\cos \Theta}{r^2}$.

Q9: What is the net force acting on a dipole placed in a uniform electric field?

Ans: The forces on the two charges constituting dipole are equal and opposite. Hence the net force is zero.

Q10: What is the SI unit of dipole moment?

Ans: The SI unit of dipole moment is Coulomb.meter

Q11: Define the dipole axis of a dipole.

Ans: The line passing through the two opposite charges of a dipole is called dipole axis

Q12: Define the equatorial axis of a dipole?

Ans: The line passing through the centre of the dipole axis and perpendicular to the dipole axis is called the equatorial axis.

Q13: When is the torque on a dipole is maximum?

Ans: When the axis of the dipole is perpendicular to the field.

Q14: When is the torque on a dipole is minimum?

Ans: When the axis of the dipole is parallel to the field.

Q15: When an electric dipole is in stable equilibrium in an electric field?

Ans: When dipole moment is parallel to the electric field.

Q16: What is the electric dipole moment?

Ans:  It is the measure of the strength of electric dipole. Its magnitude is equal to the product of the magnitude of any one charge and the distance between them.

Q17: Give an example for the electric dipole.

Ans: A pair electric charge of two opposite sign and equal magnitude separated by a distance.

Q18: Give the expression for the magnitude of an electric dipole.

Ans: p = 2qa

Stay tuned with BYJU’S to learn more Physics concepts with the help of interactive video lessons.

#### Practise This Question

A coil of 100 turns of dimensions 25cm×4cm carries a current of 2A. It makes an angle of 37 with a Magnetic field of 10T. Find the torque acting on it. Also find the work done to rotate it such that the angle increases to 53.


text:
# Knowledge Bank

## University Libraries and the Office of the Chief Information Officer

The Knowledge Bank is scheduled for regular maintenance on Sunday, April 20th, 8:00 am to 12:00 pm EDT. During this time users will not be able to register, login, or submit content.

# VIBRATION-ROTATION INTERACTION IN THE FUNDAMENTAL BAND OF HF

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/8365

Files Size Format View
1970-B-12.jpg 129.8Kb JPEG image

 Title: VIBRATION-ROTATION INTERACTION IN THE FUNDAMENTAL BAND OF HF Creators: Lovell, R. J. Issue Date: 1970 Abstract: Measurements of line intensities in the fundamental band of HF have been extended from $m = -10$ to $m = +12$. The rotational correction factors $F^{1}_{u}$(m) have been calculated and the experimental curve of $F^{1}_{0}$(m) versus m has been compared with that predicted by Herman and Wallace. The experimental curve drops below predicted values in both wings of the band, though not so abruptly in the P-branch as in the R-branch. An attempt has been made by the method of Herman and Rubin to fit the experimental curve taking both a linear and a quadratic expression for the dipole moment expansion. It is possible with the linear dipole moment expression to fit the experimental curve through the center of the band, but not in the wings. No better fit is obtained with a quadratic expression for the dipole moment. Another approach seems desirable that will explicitly incorporate the effect of the large vibration-rotation term present in the experimentally determined general potential function for the molecule. Waggoner and Shaffer have applied the contact transformation method to the effect of vibration-rotation interaction on the line intensities of the linear symmetric molecule. An adaptation of this approach is being applied to HF to take explicit account of the large distortions in this light molecule. URI: http://hdl.handle.net/1811/8365 Other Identifiers: 1970-B-12


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# Complicated “functional integral”

I came across the following "functional" at work:

$$\Pi [b]=\iint_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda$$

it's part of an optimization problem that tries to find $b$, subject to some constraints on $b$.

I'm not familiar with that type of integral, where the solution function is actually in one of the bounds of the integral. Is there a specific name for that type of integral? Would the calculus of variations address that type of optimization problem? Or is there a field of functional analysis (calculus?) that would address it?

Thanks!

-
 Are you integrating with respect to $v$ or $\lambda$? Surely it can't be with respect to both? – Bruno Dec 1 '11 at 3:51 Yeah, I'm afraid it's with respect to both... But you are right, I need to fix the formula :-) Thanks! - the second integral is from 0 to +infinity. – Frank Dec 1 '11 at 4:04


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# Try to solve it 1

$\large \lim_{x\to\frac13} \dfrac{\sqrt3 - \tan(\pi x)}{3x-1} = \ ?$

Note by A K
2 years, 9 months ago

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We could just use L'Hopital's rule, but I'm assuming that you are looking for an approach that does not use this rule.

Let $$u = x - \dfrac{1}{3}.$$ Then $$u \rightarrow 0$$ as $$x \rightarrow \dfrac{1}{3}$$ and $$x = u + \dfrac{1}{3},$$ in which case

$$\sqrt{3} - \tan(\pi x) = \sqrt{3} - \tan\left(\pi u + \dfrac{\pi}{3}\right) = \sqrt{3} - \dfrac{\tan(\pi u) + \tan(\frac{\pi}{3})}{1 - \tan(\pi u)\tan(\frac{\pi}{3})} =$$

$$\sqrt{3} - \dfrac{\tan(\pi u) + \sqrt{3}}{1 - \sqrt{3}\tan(\pi u)} = \dfrac{\sqrt{3} - 3\tan(\pi u) - \tan(\pi u) - \sqrt{3}}{1 - \sqrt{3}\tan(\pi u)} = \dfrac{-4\tan(\pi u)}{1 - \sqrt{3}\tan(\pi u)}.$$

Thus the original limit can be written as

$$\lim_{u \rightarrow 0} \dfrac{-4\tan(\pi u)}{3u*(1 - \sqrt{3}\tan(\pi u))} =$$

$$\lim_{u \rightarrow 0} \dfrac{-4\pi \tan(\pi u)}{3*(\pi u)} * \lim_{u \rightarrow 0} \dfrac{1}{1 - \sqrt{3}\tan(\pi u)} = -\dfrac{4\pi}{3} * \lim_{w \rightarrow 0} \dfrac{\tan(w)}{w} * 1 = \boxed{-\dfrac{4 \pi}{3}},$$

where $$w = \pi u \rightarrow 0$$ as $$u \rightarrow 0.$$ Note also that $$(1 - \sqrt{3}\tan(\pi u)) \rightarrow 1$$ as $$u \rightarrow 0.$$

- 2 years, 9 months ago

Nice! Approaches without using L'hopital's rule are always really good.

- 2 years, 9 months ago

Thanks! L'Hopital's is really useful if you want a quick result, but I like the challenge of trying to find an alternative approach. :)

- 2 years, 9 months ago

Another way is to put $$\pi x = y$$ and by first principles we can say the limit equals negative of the derivative of $$\tan y$$ at $$y = \dfrac{ \pi}{3}$$. This is similar to L'Hospital but not completely.

- 2 years, 9 months ago

-4π/3

- 1 year, 12 months ago

$$\frac{-4\pi}{3}$$

- 2 years, 9 months ago


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# fastavro /

Filename Size Date modified Message
docs
fastavro
tests
128 B
1.7 KB
68 B
2.0 KB
45 B
484 B
563 B
2.0 KB
221 B
329 B
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## fastavro

The current Python avro package is packed with features but dog slow.

On a test case of about 10K records, it takes about 14sec to iterate over all of them. In comparison the JAVA avro SDK does it in about 1.9sec.

fastavro is less feature complete than avro, however it's much faster. It iterates over the same 10K records in 2.9sec, and if you use it with PyPy it'll do it in 1.5sec (to be fair, the JAVA benchmark is doing some extra JSON encoding/decoding).

If the optional C extension (generated by Cython) is available, then fastavro will be even faster. For the same 10K records it'll run in about 1.7sec.

## Usage

import fastavro as avro

with open('weather.avro', 'rb') as fo:

process_record(record)

You can also use the fastavro script from the command line to dump avro files. Each record will be dumped to standard output in one line of JSON.

fastavro weather.avro

You can also dump the avro schema:

fastavro --schema weather.avro

## Limitations

• Support only iteration
• No writing for you!

## Hacking

As recommended by Cython, the C files output is distributed. This has the advantage that the end user does not need to have Cython installed. However it means that every time you change fastavro/pyfastavro.py you need to run make.

For make to succeed you need both python and python3 installed, cython on both of them. For ./test-install.sh you'll need virtualenv.

## Builds

We're currently using travis.ci

## Changes

See the ChangeLog