Split (1)

train · ~4.74M rows (showing the first 1.13M)

text stringlengths 256 16.4k
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than be $v(t) = 3t^2-12t+9$ how could I find the intervals Fix $c\in\{0,1,\dots\}$, let $K\geq c$ be an integer, and define $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$.I believe I have numerically discovered that$$\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n \quad \text{ as } K\to\infty$$but cannot ... So, the whole discussion is about some polynomial $p(A)$, for $A$ an $n\times n$ matrix with entries in $\mathbf{C}$, and eigenvalues $\lambda_1,\ldots, \lambda_k$. Anyways, part (a) is talking about proving that $p(\lambda_1),\ldots, p(\lambda_k)$ are eigenvalues of $p(A)$. That's basically routine computation. No problem there. The next bit is to compute the dimension of the eigenspaces $E(p(A), p(\lambda_i))$. Seems like this bit follows from the same argument. An eigenvector for $A$ is an eigenvector for $p(A)$, so the rest seems to follow. Finally, the last part is to find the characteristic polynomial of $p(A)$. I guess this means in terms of the characteristic polynomial of $A$. Well, we do know what the eigenvalues are... The so-called Spectral Mapping Theorem tells us that the eigenvalues of $p(A)$ are exactly the $p(\lambda_i)$. Usually, by the time you start talking about complex numbers you consider the real numbers as a subset of them, since a and b are real in a + bi. But you could define it that way and call it a "standard form" like ax + by = c for linear equations :-) @Riker "a + bi where a and b are integers" Complex numbers a + bi where a and b are integers are called Gaussian integers. I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd... Does anyone know if $T: V \to R^n$ is an inner product space isomorphism if $T(v) = (v)_S$, where $S$ is a basis for $V$? My book isn't saying so explicitly, but there was a theorem saying that an inner product isomorphism exists, and another theorem kind of suggesting that it should work. @TobiasKildetoft Sorry, I meant that they should be equal (accidently sent this before writing my answer. Writing it now) Isn't there this theorem saying that if $v,w \in V$ ($V$ being an inner product space), then $\|\|v\|\| = \|\|(v)_S\|\|$? (where the left norm is defined as the norm in $V$ and the right norm is the euclidean norm) I thought that this would somehow result from isomorphism @AlessandroCodenotti Actually, such a $f$ in fact needs to be surjective. Take any $y \in Y$; the maximal ideal of $k[Y]$ corresponding to that is $(Y_1 - y_1, \cdots, Y_n - y_n)$. The ideal corresponding to the subvariety $f^{-1}(y) \subset X$ in $k[X]$ is then nothing but $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$. If this is empty, weak Nullstellensatz kicks in to say that there are $g_1, \cdots, g_n \in k[X]$ such that $\sum_i (f^* Y_i - y_i)g_i = 1$. Well, better to say that $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$ is the trivial ideal I guess. Hmm, I'm stuck again O(n) acts transitively on S^(n-1) with stabilizer at a point O(n-1) For any transitive G action on a set X with stabilizer H, G/H $\cong$ X set theoretically. In this case, as the action is a smooth action by a Lie group, you can prove this set-theoretic bijection gives a diffeomorphism
I just noticed Joseph's comment in his question about Markov chains. My observations about the correctness of trying to use Markov chains to describe the rolling of a die, fair or unfair: If by state in the Markov chains, you mean just the "face" it is currently on or the "face" which is lower-most in attitude at a particular point in time, then it is inappropriate to use Markov chains because the likelihood of transitioning from die face $F_i$ to die face $F_j$ is not purely dependent upon the current state. If $F_j$ and $F_k$ are two "faces" adjacent to face $F_i$, then the likelihoods of transitioning $F_i \to F_j$ vs. $F_i \to F_k$ is not just dependent on the "current state" being $F_i$, but also dependent upon the velocity, position, and orientation of the die. The "faces" are necessary but not sufficient to encode state in such a way for Markov chains to be applicable: that the Bayesian requirement that "current state" at time $t$ is all that is needed to be known in order to be able to predict the likelihood of the state at time $t+1$ (if you talk about discrete time) or time $t+\varepsilon$ if you talk about continous time. If by "state", you try to get around this factor that only current state be considered and not the history of how you came to currently be in that state, then you could try to add the vectors of position, velocity, and orientation as extra "states", which is valid in numerical simulation, because ultimately all reals are still encoded into limited precision "floating point" representations of reals. However, the transition table would be huge if you allowed even for 16-bit floating point representation. I do not think that history-less "Markov chains" can be applied in this situation. older answer components below To answer Benoît Kloeckner's comment that <<[then] the solid angle under which each face is seen from the center of gravity alone would determine the probability of the dice landing on that face. But to determine all polyhedra for which this solid angle is constant is already a nice problem.>> I don't believe that having similar solid angles is sufficient to determine the equal probabilities of the die landing on faces with similar solid angles. Here is a construction for 2-d die (which can easily be converted into prismatic die, disregard if the die lands on "top" or "bottom" face, and look at the relative probabilities of landing on the prismatic faces) Using polar coordinates $(r,\theta)$ , let's define a fair hexagonal die's profile as the closed path determined by the six vertices at $(1,\frac{\pi}{3}), (1,\frac{2\pi}{3}), (1,\pi), (1,\frac{4\pi}{3}), (1,\frac{5\pi}{3}), (1,{2\pi})$ Now let us define an unfair hexagonal die's profile as the path defined by the polar coordinates $(1,\frac{\pi}{3}), (1,\frac{2\pi}{3}), (100,\pi), (1,\frac{4\pi}{3}), (1,\frac{5\pi}{3}), (100,{2\pi})$ Now this die's center of mass (center of gravity) remains at $(0,0)$ since the material the die is composed of has uniformly homogeneous density. This unfair die also has each prismatic face subtending equal solid angles (and equal angles of $\pi/3$ for each edge in the $2$-dimensional case), however the this unfair die is highly biased towards landing on two faces to the detriment of the other four faces probabilities. Thus Benoît Kloeckner's conjecture that "the solid angle under which each face is seen from the center of gravity alone would determine the probability of the dice landing on that face" is incorrect. In fact, using this polar coordinate approach, it can be seen that using any three radii greater than $0$ in length yields a rotationally symmetric die profile with equiangular faces ( edges which subtend equal angles in $2$-d, prismatic faces which subtend equal steradians of solid angle in $3$-d) and with center of mass still at $(0,0)$: $(r_1,\pi/3), (r_2,2\pi/3), (r_3, \pi), (r_1,4\pi/3), (r_2,5\pi/3), (r_3,2\pi)$ but very few of these would be fair. Particularly the non-convex profiles, which also are equiangular, but make it possible to land on pairs of vertices/edges without landing on a specific face
$ L^{p, q} $ estimates on the transport density Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France In this paper, we show a new regularity result on the transport density $ \sigma $ in the classical Monge-Kantorovich optimal mass transport problem between two measures, $ \mu $ and $ \nu $, having some summable densities, $ f^+ $ and $ f^- $. More precisely, we prove that the transport density $ \sigma $ belongs to $ L^{p,q}(\Omega) $ as soon as $ f^+,\,f^- \in L^{p,q}(\Omega) $. Mathematics Subject Classification:35B65, 46N10, 49N60. Citation:Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134 References: [1] L. Ambrosio, Lecture notes on optimal transport problems, in [2] [3] [4] [5] [6] [7] [8] [9] [10] F. Santambrogio, Absolute continuity and summability of transport densities: simpler proofs and new estimates, [11] F. Santambrogio, [12] show all references References: [1] L. Ambrosio, Lecture notes on optimal transport problems, in [2] [3] [4] [5] [6] [7] [8] [9] [10] F. Santambrogio, Absolute continuity and summability of transport densities: simpler proofs and new estimates, [11] F. Santambrogio, [12] [1] Giuseppe Buttazzo, Eugene Stepanov. Transport density in Monge-Kantorovich problems with Dirichlet conditions. [2] [3] Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. [4] [5] [6] [7] [8] [9] Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. [10] Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. [11] Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. [12] [13] Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. [14] [15] Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. [16] [17] [18] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. [19] Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. [20] Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. 2018 Impact Factor: 0.925 Tools Metrics Other articles by authors [Back to Top]
Ivan Raikov About me I am a researcher at Stanford University. My CV is available here. You can reach me at ivan (dot) g (dot) raikov (at) gmail (dot) com. Software I have a Github page. Most of my recent software is written in Scheme for the Chicken Scheme compiler, or in Standard ML for the MLton compiler. A list of Scheme libraries that I have developed and/or maintain is available on my Chicken user page. Previous work in which I am no longer involved includes MRCI and RTXI. Papers and publications Photos My Google Plus photos Literature Master and Margarita The Master and Margarita by Mikhail Bulgakov original in Russian The Master and Margarita by Mikhail Bulgakov 1967 translation by Michael Glenny, based on the Soviet-censored Russian text. The Master and Margarita by Mikhail Bulgakov 1997 translation by Richard Pevear and Larissa Volokhonsky. Based on the uncensored original. Bulgakov's Master and Margarita An annotation to Bulgakov's novel, created by Kevin Moss at Middlebury College. Atanas Dalchev Collected works of Bulgarian poet Atanas Dalchev Yordan Yovkov The Harvester, a novel by Bulgarian author Yordan Yovkov. Notes Definitions We define an arithmetic expression as a combination of variables, numerical values and operations over these values. For example, the arithmetic expression 2 + 3 uses two numeric values 2 and 3 and an operation +. The syntax of an expression specifies the allowed combinations of expressions. Each expression has a meaning (or value), which is defined by the evaluation of that expression. Evaluation is a process where expressions composed of various components get simplified until eventually we get a value. For example, evaluating 2 + 3resultsin5. Syntax The conceptual structure of an expression is called the abstract syntax. The particular details and rules for writing expressions as strings of characters is called the concrete syntax, e.g. MathML. The abstract syntax for arithmetic expressions is very simple, while the concrete syntax can include additional punctuation and other lexical features. Abstract Syntax The abstract syntax of expressions can be represented by the following EBNF grammar: Expr ::= Number Real \| BinOp B Expr Expr \| UnOp U Expr B ::= + \| - \| * \| / \| ^ U ::= neg \| abs \| atan \| asin \| acos \| sin \| cos \| exp \| ln \| sqrt \| tan \| cosh \| sinh \| tanh \| gamma \| lgamma \| log10 \| log2 where Number, BinOp, UnOp indicate the different kinds of expressions: numeric value, binary operation and unary operation, respectively, and the set of binary and unary operations are the the usual arithmetic operators on real numbers. Variables and Substitution Arithmetic expressions contain variables in addition to constants and arithmetic operations. Thus we will need to extend the abstract syntax of expressions to include variables. We can represent variables as an additional kind of expression. The following data definition modifies Expr to include a Variable case: Expr ::= Number Real \| BinOp B Expr Expr \| UnOp U Expr \| Variable Symbol (B and U as before) Further, in order to assign a meaning to a variable in a particular context, we will define the association of a variable x with a value v as a binding, which can be written as x \mapsto v. Bindings can be represented as pair. For example, the binding of x \mapsto 5 can be represented as (symbol\;"x",\;5). Substitution The substitution operation replaces a variable with a value in an expression. Here are some examples of substitution: substitute (x \mapsto 5) in (x + 2) \rightarrow{} (5 + 2) substitute (x \mapsto 5) in (2) \rightarrow{} (2) substitute (x \mapsto 5) in (x * x + x) \rightarrow{} (5 * 5 + 5) substitute (x \mapsto 5) in (x + y) \rightarrow{} (5 + y) If the variable names don't match, they are not substituted. Given the syntax defined for expressions, the process of substitution can be defined by cases: simpleSubstitute (var, newVar) exp = subst exp where subst (Number r) = Number r subst (BinOp B a b) = BinOp B (subst a) (subst b) subst (UnOp U a b) = UnOp U (subst a) (subst b) subst (Variable name) = if var == name then Variable newVar else Variable name Environments There can be multiple variables in a single expression. For example, evaluating (2 * x + y) where x = 3 and y = -2 results in 4. A collection of bindings is called an environment. Since a binding is represented as a tuple, an environment can be represented as a list of tuples. The environment from the example above would be env = [ ("x", 3), ("y", -1) ] An important operation on environments is variable lookup. Variable lookup is an operation that given a variable name and an environment looks up that variable in the environment. For example: lookup (x) in (env)\rightarrow3 lookup (y) in (env)\rightarrow-1 The substitution function can then be redefined to use environments rather than single bindings: substitute env exp = subst exp where subst (Number r) = Number r subst (BinOp B a b) = BinOp B (subst a) (subst b) subst (UnOp U a) = UnOp U (subst a) subst (Variable name) = case lookup name env of Some r => Number r \| None => Variable name Local Variables It is also useful to allow variables to be defined within an expression. We will define local variables with a let expression: let x = 1 in 2x + 3 Local variable declarations are themselves expressions, cand can be used inside other expressions: 2 (let x = 3 in x + 5) Local variable declarations can be represented in the abstract syntax by adding another clause: Expr ::= ... \| Let Symbol Expr Expr When substituting a variable into an expression, it must correctly take into account the scope of the variable. In particular, when substituting for x in an expression, if the expression is of the form let x = e in body then x should be substituted within e but not in body. simpleSubstitute (var, val) exp = subst exp subst (Let x exp body) = Let x (subst exp) body1 where body1 = if x == var then body else subst body In the Let case for subst, the variable is always substituted into the bound expression e. But the substitution is only performed on the body b if the variable var is not the same as x. Evaluation TODO Summary Here is the complete definition, substitution and evaluation using an expression language with local variables. Expr ::= Number Real \| BinOp B Expr Expr \| UnOp U Expr \| Variable Symbol \| Let Symbol Expr Expr simpleSubstitute (var, val) exp = subst exp where subst (Number r) = Number r subst (BinOp B a b) = BinOp B (subst a) (subst b) subst (UnOp U a b) = UnOp U (subst a) (subst b) subst (Variable name) = if var == name then Number val else Variable name subst (Let x exp body) = Let x (subst exp) body1 where body1 = if x == var then body else subst body evaluate (Number r) = r evaluate (BinOp + a b) = (evaluate a) + (evaluate b) ... evaluate (UnOp sqrt a) = sqrt (evaluate a) ... evaluate (Let x exp body) = evaluate (simpleSubstitute (x, evaluate exp) body) evaluate (Variable x) = Error The simulation of a neuronal network model with N neurons can be represented as a sequence of transformations. We assume the constant matrix W of size N by N contains the synaptic weights of each connection in the network. We further assume that the global delay of synaptic events is represented by a delay matrix D of dimensions N by T, where T is the global delay as number of time steps. The first transformation computes the current state of all neurons given an initial state vector A^{t-1} of size N and initial input vector I^{t-1}, also of size N, A_{i}^{t} = f_{i} (A_{i}^{t-1}, I_{i}^{t-1}) where f is the function that computes neuronal and synaptic dynamics. The next transformation computes the set S of neurons that have spiked during the current time step, S = \forall{i}(spike(A_{i}^{t}) == true) where spike is a function that determines whether a spike has ocurred given the current state of a neuron. The next transformation computes weighted sums of the inputs, J_{j} = \sum_{i \in S} W_{i,j} spikecount(A_{j}^{t}) where J is a of the same dimensionality as I, and spikecount is a function that returns the number of spikes that have occurred given the current state of a neuron. The final transformation computes the new delay matrix and input vector, I_{i}^{t} = D_{i,1}^{t-1} D^{t} = nshift (cat (D, J, 2), 1) where function cat concatenates J to the right side of D, and function nshift shifts the columns of the resulting matrix to the left by one. The algorithm outlined above assumes a single type of synaptic connection. This restriction can be lifted by extending W to be of dimensions N by N by K, where K is the number of synaptic connection types, and extending I to be of dimensions N by K. The assumption that W is constant can be lifted by introducing a function that computesthe new value of W for each time step. Sparse Matrix Technique for solving chemical kinetic equations The solution of any chemical reaction mechanism requires the useof the stoichiometric coefficients to calculate the right-hand sidesof the kinetic equations. If the number of species participating inany chemical reaction is always much less than the total number ofchemical species involved, then the matrices of stoichiometriccoefficients will be sparse. A sparse matrix is one in which there aremany more zero elements than nonzero ones. In such a case, one canexploit the prevalence of zero matrix elements with a sparse matrixtechnique in which only nonzero matrix elements are stored andprocessed. This technique involves three one-dimensional arrays: twointeger index arrays, IA, JA,and a real one, A. Nonzero elements of the originalmatrix M are stored row-by-row in A. To identify the nonzero elements in a row, we need to know thecolumn of each entry. The arrayJA is used to store the column indices whichcorrespond to the nonzero entries of M, i.e., if A(k)=M(i,j), then JA(k)=j. We also need to know the number of nonzero elements in eachrow. The index positions in JA and A where each row of M begins are stored in the IA array. If M(i,j) is the leftmost entry ofthe i-th row and A(k)=M(i,j), then IA(k)=k. Moreover, IA(N+1) is defined as the index in JA and A of the first location following thelast (N-th), element in the row. Thus, the number of entries in thei-th row is given by the difference IA(i+1)-IA(i), and the nonzero elements of the i-th row are stored in a sequence A(IA(i)), A(IA(i)+1), \ldots, A(IA(i+1)-1) while the corresponding column indices are stored in JA(IA(i)), JA(IA(i)+1), \ldots, JA(IA(i+1)-1) An algorithm for computing the equations of a compartmental neuron model for a general tree structure: For each cylinder i with radius and length a_i and L_i in micrometers, let the surface area, A_i = 2 \pi a_i L_iand the axial resistance, Q_i = L_i / (\pi a_i^2) Given specific membrane capacitance and resistance c_i and r_{m_i}, respectively, the compartmental capacitance is C_i = c_i A_i 10^{-8} and the compartmental resistance is R_i = (r_{m_i} / A_i) 10^8 Given longitudinal (intracellular) resistivity r_l, the coupling resistance between compartments i and j is R_{ij} = 0.5 r_l (Q_i + Q_j) 10^4 The membrane potential equation for compartment i is then C_i \frac{dV_i}{dt} = -\frac{V_i}{R_i} + \sum_{j,i} \frac{(V_j - V_i)}{R_{ij}} + I_i A_i {10^{-2}} The factors of 10^-8, 10^8 and 10^4 are conversion factors from micrometers to centimeters. Older articles The main idea of Brep is to combine experimental data on cell morphology and synthetic cell geometry. Cell geometry is described as sets of 3D points and can be one of the following: Neurolucida data cylinders described by start point, end point, diameter Synthetic geometry points along a line, polynomial curve, or angular displacement Cell placement can be defined by rectangular grids, regular tilings (e.g. bricks or hexagons), random point distributions. Connectivity currently is simply projections based on Euclidean distances between cell objects. Layers In Brep terminology, a layer describes the placement of cell objects in 3D space. The placement is determined by a set of points, which can be generated by one of the following methods: sampling from typical distributions such as uniform or exponential point coordinates loaded from a file Regular grid defined by spacing in x, y and z direction Regular 2D tiling, such as hexagons or bricks with a given side length. In this case, position x and y coordinates are determined by the center point of each generated tile, and the z coordinate is set to zero. A layer is defined by a point set described by one of the methods above, and one or more cell objects that are to be placed at positions determined by the point set. If there are less cell objects than there are points in the point set, then the cell objects are copied until an object is placed at each point. Cell objects A cell object consists of a bounding box and a set of connection points. The connection points can be loaded from a Neurolucida file, or can be sampled from the trajectory of a polynomial curve. Currently the types of polynomial curves supported are lines and angular displacements of the form L * cos (theta) * cos (phi) where theta and phi are chosen randomly from a given range of angles. The set of points that belong to a cell object can be further divided into compartments, e.g. dendrites and axon, which can be later used when specifying connectivity. In addition, random perturbations can be applied to the points of a cell or one of its compartments. Parallel fibers can then be described as lines with small random perturbations, and artificial Golgi cell morphologies can be desribed with angular displacements of their dendritic and axon points. Projections Once layers are defined with the different cell types, connectivity is specified in Brep as projections between layers. The single type of projection currently supported by BRep is based on Euclidean distances between the points of the different cell object. Given a maximum distance, BRep can determine all cell points in two layers that are close enough to be considered connected.
Of course not. But after reading a bit, some points make me believe it should be: Let $S$ be a nice$^{\}$ surface defined over $Spec\ \mathbb{Z}$. The Brauer group $Br(S\otimes \bar{\mathbb{Q}})$ is an abelian divisible group, It is also a $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ module, For good primes there are reductions $Br(S)\rightarrow Br(S\otimes \mathbb{F}_q)$, These $Br(S\otimes \mathbb{F}_q)$ are finite, There is a formal Brauer group $\hat{Br}(S)$ of dimension 1, The coefficients of $\hat{Br}$, in suitable natural coordinates, relate to $\|Br(S\otimes \mathbb{F}_q)\|$. There are some examples where the associated L-function comes from a modular form (of weight 3). I'm not sure if this is conjectured (let alone known) in general. Since the Brauer group observes many characteristics of an abelian variety (all properties) of dimension 1 (properties 5 and 7 [weight isn't two, but it's the right space]), my vague question is: how far is it from actually being a variety? There are some easy examples of $S$ with $\|Br(S\otimes \mathbb{F}_q)\|$ varying between $1$ and $4(q-4)$, as $q$ varies over the primes. This is a clear point of departure from elliptic curves and varieties in general. Maybe there's a family of natural galois-module homomorphisms into certain abelian varieties defined over $\mathbb{Q}$, commuting with the reduction maps and restriction (or some other appropriate term) to formal groups? What's going on with these Brauer groups? $^\$ say a K3 surface. Something that (1) is true for (so not a rational surface) and (4) is proven for.
To interpolate between data points $ \fvec{x}_i $ which are not somehow ordered in a grid structure the scattered data interpolation (Shepard-Interpolation) algorithm can be used\begin{equation} f(\fvec{x}) = \sum_{i=1}^n \frac{w_i(\fvec{x}) f(\fvec{x}_i)}{\sum_{i=1}^n w_i(\fvec{x})}. \end{equation} The algorithm creates a smooth interpolation between the points based on their distances. At each interpolating position $ \fvec{x} $ all data points are taken into account but weighted accordingly to their distances (nominator, higher weight for nearer distances) with the weighting function\begin{equation} w_i(\fvec{x}) = \frac{1}{\left\\| \fvec{x} - \fvec{x}_i \right\\|^{\alpha}}. \end{equation} In the end, the total result is normalized by all weights (denominator). The weighting is controlled by the $\alpha$ parameter. If this parameter is increased up to infinity, the result is a Voronoi decomposition. List of attached files: ← Back to the overview page
To prove $I=f^{-1}(I’)$ is an ideal of $R$, we need to check the following two conditions: For any $a, b\in I$, we have $a-b\in I$. For any $a\in I$ and $r\in R$, we have $ra\in I$. Let us first prove condition 1. Let $a, b\in I$. Then it follows from the definition of $I$ that $f(a), f(b)\in I’$.Since $I’$ is an ideal (and hence an additive abelian group) we have $f(a)-f(b)\in I’$.Since $f$ is a ring homomorphism, it yields that\[f(a-b)=f(a)-f(b)\in I’.\]Thus we have $a-b \in I$, and condition 1 is met.This implies that $I$ is an additive abelian group of $R$. Next, we check condition 2. Let $a\in I$ and $r\in R$. Since $a\in I$, we have $f(a)\in I’$.Since $I’$ is an ideal of $R’$ and $f(r)\in R’$, we have $f(r)f(a)\in I’$.Since $f$ is a ring homomorphism, it follows that\begin{align}f(ra)=f(r)f(a)\in I’,\end{align}and hence $ra\in I$. So condition 2 is also met and we conclude that $I$ is an ideal of $R$. Comment. Instead of condition 1, we could have used Condition 1′: For any $a, b\in I$, we have $a+b\in I$. The reason is that condition 2 guarantee the existence of the additive inverses, and hence condition 1 and 2 are equivalent to condition 1′ and 2. The Preimage of Prime ideals are Prime IdealsLet $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$.Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.Proof.The preimage of an ideal by a ring homomorphism is an ideal.(See the post "The inverse image of an ideal by […] Generators of the Augmentation Ideal in a Group RingLet $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […] Characteristic of an Integral Domain is 0 or a Prime NumberLet $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.Definition of the characteristic of a ring.The characteristic of a commutative ring $R$ with $1$ is defined as […]
Hi, Can someone provide me some self reading material for Condensed matter theory? I've done QFT previously for which I could happily read Peskin supplemented with David Tong. Can you please suggest some references along those lines? Thanks @skullpatrol The second one was in my MSc and covered considerably less than my first and (I felt) didn't do it in any particularly great way, so distinctly average. The third was pretty decent - I liked the way he did things and was essentially a more mathematically detailed version of the first :) 2. A weird particle or state that is made of a superposition of a torus region with clockwise momentum and anticlockwise momentum, resulting in one that has no momentum along the major circumference of the torus but still nonzero momentum in directions that are not pointing along the torus Same thought as you, however I think the major challenge of such simulator is the computational cost. GR calculations with its highly nonlinear nature, might be more costy than a computation of a protein. However I can see some ways approaching it. Recall how Slereah was building some kind of spaceitme database, that could be the first step. Next, one might be looking for machine learning techniques to help on the simulation by using the classifications of spacetimes as machines are known to perform very well on sign problems as a recent paper has shown Since GR equations are ultimately a system of 10 nonlinear PDEs, it might be possible the solution strategy has some relation with the class of spacetime that is under consideration, thus that might help heavily reduce the parameters need to consider to simulate them I just mean this: The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system. @ooolb Even if that is really possible (I always can talk about things in a non joking perspective), the issue is that 1) Unlike other people, I cannot incubate my dreams for a certain topic due to Mechanism 1 (consicous desires have reduced probability of appearing in dreams), and 2) For 6 years, my dream still yet to show any sign of revisiting the exact same idea, and there are no known instance of either sequel dreams nor recurrence dreams @0celo7 I felt this aspect can be helped by machine learning. You can train a neural network with some PDEs of a known class with some known constraints, and let it figure out the best solution for some new PDE after say training it on 1000 different PDEs Actually that makes me wonder, are the space of all coordinate choices more than all possible moves of Go? enumaris: From what I understood from the dream, the warp drive showed here may be some variation of the alcuberrie metric with a global topology that has 4 holes in it whereas the original alcuberrie drive, if I recall, don't have holes orbit stabilizer: h bar is my home chat, because this is the first SE chat I joined. Maths chat is the 2nd one I joined, followed by periodic table, biosphere, factory floor and many others Btw, since gravity is nonlinear, do we expect if we have a region where spacetime is frame dragged in the clockwise direction being superimposed on a spacetime that is frame dragged in the anticlockwise direction will result in a spacetime with no frame drag? (one possible physical scenario that I can envision such can occur may be when two massive rotating objects with opposite angular velocity are on the course of merging) Well. I'm a begginer in the study of General Relativity ok? My knowledge about the subject is based on books like Schutz, Hartle,Carroll and introductory papers. About quantum mechanics I have a poor knowledge yet. So, what I meant about "Gravitational Double slit experiment" is: There's and gravitational analogue of the Double slit experiment, for gravitational waves? @JackClerk the double slits experiment is just interference of two coherent sources, where we get the two sources from a single light beam using the two slits. But gravitational waves interact so weakly with matter that it's hard to see how we could screen a gravitational wave to get two coherent GW sources. But if we could figure out a way to do it then yes GWs would interfere just like light wave. Thank you @Secret and @JohnRennie . But for conclude the discussion, I want to put a "silly picture" here: Imagine a huge double slit plate in space close to a strong source of gravitational waves. Then like water waves, and light, we will see the pattern? So, if the source (like a Black Hole binary) are sufficent away, then in the regions of destructive interference, space-time would have a flat geometry and then with we put a spherical object in this region the metric will become schwarzschild-like. if Pardon, I just spend some naive-phylosophy time here with these discussions The situation was even more dire for Calculus and I managed! This is a neat strategy I have found-revision becomes more bearable when I have The h Bar open on the side. In all honesty, I actually prefer exam season! At all other times-as I have observed in this semester, at least-there is nothing exciting to do. This system of tortuous panic, followed by a reward is obviously very satisfying. My opinion is that I need you kaumudi to decrease the probabilty of h bar having software system infrastructure conversations, which confuse me like hell and is why I take refugee in the maths chat a few weeks ago (Not that I have questions to ask or anything; like I said, it is a little relieving to be with friends while I am panicked. I think it is possible to gauge how much of a social recluse I am from this, because I spend some of my free time hanging out with you lot, even though I am literally inside a hostel teeming with hundreds of my peers) that's true. though back in high school ,regardless of code, our teacher taught us to always indent your code to allow easy reading and troubleshooting. We are also taught the 4 spacebar indentation convention @JohnRennie I wish I can just tab because I am also lazy, but sometimes tab insert 4 spaces while other times it inserts 5-6 spaces, thus screwing up a block of if then conditions in my code, which is why I had no choice I currently automate almost everything from job submission to data extraction, and later on, with the help of the machine learning group in my uni, we might be able to automate a GUI library search thingy I can do all tasks related to my work without leaving the text editor (of course, such text editor is emacs). The only inconvenience is that some websites don't render in a optimal way (but most of the work-related ones do) Hi to all. Does anyone know where I could write matlab code online(for free)? Apparently another one of my institutions great inspirations is to have a matlab-oriented computational physics course without having matlab on the universities pcs. Thanks. @Kaumudi.H Hacky way: 1st thing is that $\psi\left(x, y, z, t\right) = \psi\left(x, y, t\right)$, so no propagation in $z$-direction. Now, in '$1$ unit' of time, it travels $\frac{\sqrt{3}}{2}$ units in the $y$-direction and $\frac{1}{2}$ units in the $x$-direction. Use this to form a triangle and you'll get the answer with simple trig :) @Kaumudi.H Ah, it was okayish. It was mostly memory based. Each small question was of 10-15 marks. No idea what they expect me to write for questions like "Describe acoustic and optic phonons" for 15 marks!! I only wrote two small paragraphs...meh. I don't like this subject much :P (physical electronics). Hope to do better in the upcoming tests so that there isn't a huge effect on the gpa. @Blue Ok, thanks. I found a way by connecting to the servers of the university( the program isn't installed on the pcs on the computer room, but if I connect to the server of the university- which means running remotely another environment, i found an older version of matlab). But thanks again. @user685252 No; I am saying that it has no bearing on how good you actually are at the subject - it has no bearing on how good you are at applying knowledge; it doesn't test problem solving skills; it doesn't take into account that, if I'm sitting in the office having forgotten the difference between different types of matrix decomposition or something, I can just search the internet (or a textbook), so it doesn't say how good someone is at research in that subject; it doesn't test how good you are at deriving anything - someone can write down a definition without any understanding, while someone who can derive it, but has forgotten it probably won't have time in an exam situation. In short, testing memory is not the same as testing understanding If you really want to test someone's understanding, give them a few problems in that area that they've never seen before and give them a reasonable amount of time to do it, with access to textbooks etc.
Basic tutorial¶ This basic tutorial gives a brief overview of some of functionality of the particles package. Details are deferred to more advanced tutorials. First steps: defining a state-space model¶ We start by importing some standard libraries, plus some modules from the package. [12]: %matplotlib inlineimport warnings; warnings.simplefilter('ignore') # hide warnings# standard librariesfrom matplotlib import pyplot as pltimport numpy as npimport seaborn as sb# modules from particlesimport particles # core modulefrom particles import distributions as dists # where probability distributions are definedfrom particles import state_space_models as ssm # where state-space models are defined Let’s define our first state-space model class. We consider a basic stochastic volalitility model, that is: Note that this model depends on fixed parameter $\theta=(\mu, \rho, \sigma)$. In case you are not familiar with the notations above: a state-space model is a model for a joint process $(X_t, Y_t)$, where $(X_t)$ is an unobserved Markov process (the state of the system), and $Y_t$ is some noisy measurement of $X_t$ (hence it is observed). For instance, in stochastic volatility, $Y_t$ is typically the log-return of some asset, and $X_t$ its (unobserved) volatility. The code below is hopefully transparent. [13]: class StochVol(ssm.StateSpaceModel): def PX0(self): # Distribution of X_0 return dists.Normal(loc=self.mu, scale=self.sigma / np.sqrt(1. - self.rho*2)) def PX(self, t, xp): # Distribution of X_t given X_{t-1}=xp (p=past) return dists.Normal(loc=self.mu + self.rho (xp - self.mu), scale=self.sigma) def PY(self, t, xp, x): # Distribution of Y_t given X_t=x (and possibly X_{t-1}=xp) return dists.Normal(loc=0., scale=np.exp(x))my_model = StochVol(mu=-1., rho=.9, sigma=.1) # actual modeltrue_states, data = my_model.simulate(100) # we simulate from the model 100 data pointsplt.style.use('ggplot')plt.figure()plt.plot(data) [13]: [<matplotlib.lines.Line2D at 0x7f55124bf9b0>] Methods PX0, PX and PY return objects that represent probability distributions (defined in module distributions. Parameters $\mu$, $\rho$ and $\sigma$ are defined as attributes of a class instance: i.e. self.mu and so on. ( self is the generic name for an instance of a class in Python.) If your are not very familiar with OOP (object oriented programming) and related concepts (classes, instances, etc.), here is a simple way to understand them in our context: The class StochVolrepresents the parametric class of stochastic volatility models. The object my_model(a class instanceof StochVol) defines a particular model, where parameters $\mu$, $\rho$ and $\sigma$ are fixed to certain values. In particular, we can inspect the attributes of my_model that store the parameter values. [14]: my_model.mu, my_model.rho, my_model.sigma [14]: (-1.0, 0.9, 0.1) Class StochVol is a sub-class of StateSpaceModel. (You can see that from the first line of its definition.) For instance, it inherits a method called simulate that generates states and datapoints from the considered model. Particle filtering¶ There are several particle algorithms that one may associate to a given state-space model. Here we consider the simplest option: the bootstrap filter. (See next tutorial for how to implement a guided or auxiliary filter.) The code below runs such a boostrap filter for $N=100$ particles, using stratified resampling. [15]: fk_model = ssm.Bootstrap(ssm=my_model, data=data) # we use the Bootstrap filterpf = particles.SMC(fk=fk_model, N=100, resampling='stratified', moments=True, store_history=True) # the algorithmpf.run() # actual computation# plotplt.figure()plt.plot([yt**2 for yt in data], label='data-squared')plt.plot([m['mean'] for m in pf.summaries.moments], label='filtered volatility')plt.legend() [15]: <matplotlib.legend.Legend at 0x7f55124ac7b8> Recall that a particle filter is a Monte Carlo algorithm: each execution returns a random, slightly different result. Thus, it is useful to run a particle filter multiple times to assess how stable are the results. Say, for instance, that we would like to compare the variability of the log-likelihood estimate provided by a particle filter, when either a standard Monte Carlo algorithm is used, or its QMC variant, called SQMC. The following command runs 30 times each of these two algorithms. [16]: results = particles.multiSMC(fk=fk_model, N=100, nruns=30, qmc={'SMC':False, 'SQMC':True})plt.figure()sb.boxplot(x=[r['output'].logLt for r in results], y=[r['qmc'] for r in results]) [16]: <matplotlib.axes._subplots.AxesSubplot at 0x7f55126c50f0> As expected, the variance of SQMC estimates is quite lower. Command multiSMC makes it possible to run several particle filters, with varying options. Parallel execution is also possible, as explained in next tutorial. Smoothing¶ So far, we have only considered filtering; let’s try smoothing, that is, approximating the distribution of the whole trajectory $X_{0:T}$, given data $Y_{0:T}=y_{0:T}$, for some fixed time horizon $T=100$. In particular, we are going to sample smoothing trajectories from the output of the first particle filter we ran a few steps above. [17]: smooth_trajectories = pf.hist.backward_sampling(10)plt.figure()plt.plot(smooth_trajectories); Here, we used the standard version of the FFBS (forward filtering, backward sampling) algorithm, which generates smoothing trajectories from the particle history (which we generated when running pf above). Other smoothing algorithms are available, see the next tutorial. (Bayesian) Parameter estimation¶ Finally, we consider how to estimate the parameter $\theta=(\alpha, \rho, \sigma)$ from a given data-set. First, we set up a prior as follows. [18]: prior_dict = {'mu':dists.Normal(), 'sigma': dists.Gamma(a=1., b=1.), 'rho':dists.Beta(9., 1.)}my_prior = dists.StructDist(prior_dict) Again, the code above should be fairly readable: the prior for $\mu$ is $N(0,1)$, the one for $\sigma$ is a Gamma(1,1) and so on. (As before, probability distributions are represented by objects defined in the distributions module.) This may not be a sensible prior distribution; for instance it restricts $\rho$ to $[0,1]$. However, this will suffice for our purposes. A popular way to simulate from the posterior distribution of the parameters of a state-space model is PMMH, a particular instance of the PMMC framework. Basically, this is a MCMC algorithm that runs at each iteration a particle filter so as to evaluate the likelihood. [19]: %%capturefrom particles import mcmc # where the MCMC algorithms (PMMH, Particle Gibbs, etc) livepmmh = mcmc.PMMH(ssm_cls=StochVol, prior=my_prior, data=data, Nx=50, niter = 1000)pmmh.run() # Warning: takes a few seconds Again, the code above is hopefully readable: PMMH is run for 1000 iterations ( niter), the particle filter run at each iteration have $N_x=50$ particles, and so on. One point worth mentioning is that we pass as an argument the class StochVol. Again, remember that this class indeed represents the considered parametric class (as opposed to my_model, which was a stochastic volatility model, for certain fixed parameter values). OK, we have waited long enough, let’s plot the results. [20]: # plot the marginalsburnin = 100 # discard the 100 first iterationsfor i, param in enumerate(prior_dict.keys()): plt.subplot(2, 2, i+1) sb.distplot(pmmh.chain.theta[param][burnin:], 40) plt.title(param) These results might not be very reliable, given that we used a fairly small number of MCMC iterations. If you are familiar with MCMC, you may wonder how the Metropolis proposal was set: by default, PMMH uses an adaptive Gaussian random walk proposal, such that the covariance matrix of the random step is iteratively adapted to the running simulation. The library also implements SMC$^2$, and Particle Gibbs, but that will be covered in the next tutorial. The end¶ That’s all, folks! This very basic tutorial is over. If you crave for more, head to the other tutorials: Advanced tutorial for state-space models: covers the same topics as above (filtering, smoothing, parameter estimation for state-space models) but with more details. Tutorial on Bayesian estimation: covers PMCMC and SMC^2 algorithms that may be used to estimate the parameters of a state-space model. Tutorial on SMC samplers: a tutorial on how to run a SMC sampler (such as IBIS or tempering SMC) for a given static target distribution. How to define manually Feynman-Kac models: an advanced tutorial on how to define your own Feynman-Kac models.
If a matrix is diagonalizable, then the algebraic multiplicity of an eigenvalue is the same as the geometric multiplicity of the eigenvalue. Recall that the geometric multiplicity of an eigenvalue is the dimension of the eigenspace of the eigenvalue. Solution. (a) Find the size of the matrix $A$ If $A$ is an $n\times n$ matrix, then its characteristic polynomial is of degree $n$. Since the degree of $f_A$ is $9$, the size of $A$ is $9 \times 9$. (b) Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$. The algebraic multiplicity of the eigenvalue $\lambda =4$ is $3$. (This is the number of times that the factor $\lambda -4$ appears in $f_A$.)Since $A$ is diagonalizable, the algebraic multiplicity is the same as the geometric multiplicity for each eigenvalue.The geometric multiplicity is the dimension of the eigenspace by definition. Thus the dimension of $E_4$ is $3$. (c) Find the dimension of the kernel(nullspace) of $A$. Note that $\ker(A)=\{x\in \R^9 \mid Ax=0\}=\{x\in \R^9 \mid (A-0I_9)x=0\}=E_0$. Thus the kernel of $A$ is the same as the eigenspace $E_0$ corresponding to the eigenvalue $\lambda=0$.Since $A$ is diagonalizable, the geometric multiplicity of $\lambda=0$, which is the dimension of the eigenspace $E_0=\ker(A)$, is the same as the algebraic multiplicity of $\lambda=0$.Thus the dimension of $\ker(A)$ is $2$. How to Diagonalize a Matrix. Step by Step Explanation.In this post, we explain how to diagonalize a matrix if it is diagonalizable.As an example, we solve the following problem.Diagonalize the matrix\[A=\begin{bmatrix}4 & -3 & -3 \\3 &-2 &-3 \\-1 & 1 & 2\end{bmatrix}\]by finding a nonsingular […] Maximize the Dimension of the Null Space of $A-aI$Let\[ A=\begin{bmatrix}5 & 2 & -1 \\2 &2 &2 \\-1 & 2 & 5\end{bmatrix}.\]Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.Your score of this problem is equal to that […]
It looks so. We start with inductive proof of Steinitz Theorem. Let $A$ be a finite set of rays starting from the origin in $\mathbb{R}^d$. Assume that positive span of these rays is the whole $\mathbb{R}^d$. Then there exists a subset of at most $2d$ rays from $A$ with the same property. Proof. Induction in $d$. Base $d=1$ is clear. Induction step. Choose minimal $k$ such that some $k$ vectors $a_1,\dots,a_k$ which generate rays from $A$ are linearly dependent with positive coefficients: $\sum t_i a_i=0$, $t_i>0$. Then $0$ lies in interior of a $(k-1)$-dimensional simplex with vertices $a_1,\dots,a_k$. Let $X={\rm span}\,(a_1,\dots,a_k)$, $\dim X=k-1$. Factor everything modulo $X$, we get a space of dimension $d-k+1$, 0 lies in an interior of a convex hull of the image of $A\setminus {\mathbb R}_+\cdot \{a_1,\dots,a_k\}$, and it suffices to use $2(d-k+1)$ rays by induction proposition, add to them rays generated by $a_1,\dots,a_k$ to get totally at most $2(d-k+1)+k=2d+(2-k)\leqslant 2d$ rays. If on the first step it was $k>2$, we get improved bound $2d-1$ on the number of used rays. If $k=2$, then we find a pair of opposite rays. If we proceed by induction proving your statement, we may think that two rays have generators $\pm e_d$ and generators of others are partitioned onto pairs $(x_k,\alpha_d)$, $(-x_k,\beta_d)$ for some $x_1,\dots,x_{2d-2}\in \mathbb{R}^{d-1}$. If $\alpha_d\ne \beta_d$, then considering above 4 rays, which lie in a 2-plane, it is easy to see that the whole 2-plane is generated by some three of them, so safely remove one ray.
I’m extremely agitated today. I dunno why. Maybe because there was some convulsion in the peaceful tidings of the house I live in, or the fact that I’m kinda hungry at the moment. Anyways, I don’t have time for chitchat. Let’s get to the studying. The following is taken from Foundations of Machine Learning by Rostamyar, et al. Support Vector Machines are the most theoretically well motivated and practically most effective classification algorithms in modern machine learning. Consider an input space $\mathcal{X}$ that is a subset of $\mathbb{R}^N$ with $N \geq 1$, and the output or target space $\mathcal{Y}=\{-1, +1\}$, and let $f : \mathcal{X} \rightarrow \mathcal{Y} $ be the target function. Given a hypothesis set $\mathcal{H}$ of functions mapping $\mathcal{X}$ to $\mathcal{Y}$, the binary classification task is formulated as follows: The learner receives a training sample $S$ of size $m$ drawn independently and identically from $\mathcal{X}$ to some unknown distribution $\mathcal{D}$, $S = ((x_1, y_1), \ldots, (x_m, y_m)) \in (\mathcal{X}\times\mathcal{Y})^m$, with $y_i = f(x_i) $ for all $i \in [m]$. The problem consists of determining a hypothesis $ h \in \mathcal{H}$, a binary classifier, with small generalization error : The probability that hypothesis set is not the target function is our error rate. \[ R_{\mathcal{D}} = \underset{x\sim\mathcal{D}}{\mathbb{P}} [h(x) \neq f(x)]. \] Different hypothesis sets $\mathcal{H}$ can be selected for this task. Hypothesis sets with smaller complexity provide better learning guarantees, everything else being equal. A natural hypothesis set with relatively small complexity is that of a linear classifier, or hyperplanes, which can be defined as follows: \[ \mathcal{H}= \{x \rightarrow sign(w.x+b) : w \in \mathbb{R}^N, b \in r\} \] The learning problem is then referred to as a linear classification problem. The general equation of a hyperplane in $\mathbb{R}^N$is $w.x+b=0$ where $w\in\mathbb{R}^N$ is a non-zero vector normal to the hyperplane $b\in\mathbb{R}$ a scalar. A hypothesisol. of the form $x\rightarrow sign(w.x+b)$ thus labels positively all points falling on one side of the hyperplane $w.x+b=0$ and negatively all others. From now until we say so, we’ll assume that the training sample $S$ can be linearly separated, that is, we assume the existence of a hyperplane that perfectly separates the training samples into two populations of positively and negatively labeled points, as illustrated by the left panel of figure below. This is equivalent to the existence of $ (\boldsymbol{w}, b) \in (\mathbb{R}^N – \boldsymbol{\{0\}}) \times \mathbb{R}$such that: \[ \forall i \in [m], \quad y_i(\boldsymbol{w}.x_i + b) \geq 0 \] But, as you can see above, there are then infinitely many such separating hyperplane. Which hyperplane should a learning algorithm select? The definition of SVM solution is based on the notion of geometric margin. Let’s define what we just came up with: The geometric margin $\rho_h(x)$ pf a linear classifier $h:\rightarrow \boldsymbol{w.x} + b $ at a point $x$ is its Euclidean distance to the hyperplane $\boldsymbol{w.x}+b=0$: \[ \rho_h(x) = \frac{w.x+b}{\|\|w\|\|_2} \] The geometric margin of $\rho_h$ of a linear classifier h for a sample $S = (x_1, …, x_m) $ is the minimum geometric margin over the points in the sample, $\rho_h = min_{i\in[m]} \rho_h(x_i)$, that is the distance of hyperplane defining h to the closest sample points. So what is the solution? It is that, the separating hyperplane with the maximum geometric margin is thus known as maximum-margin hyperplane. The right panel of the figure above illustrates the maximum-margin hyperplane returned by SVM algorithm is the separable case. We will present later in this chapter a theory that provides a strong justification for the solution. We can observe already, however, that the SVM solution can also be viewed as the safest choice in the following sense: a test point is classified correctly by separating hyperplanes with geometric margin $\rho$ even when it falls within a distance $\rho$ of the training samples sharing the same label: for the SVM solution, $\rho$ is the maximum geometric margin and thus the safest value. We now derive the equations nd optimization problem that define the SVM solution. By definition of the geometric margin, the maximum margin of $\rho$ of a separating hyperplane is given by: \[ \rho = \underset{w,b : y_i(w.x_i+b) \geq 0}{max}\underset{i\in[m]}{min}\frac{\|w.x_i=b}{\|\|w\|\|} = \underset{w,b}{max}{min}\frac{y_(w.x_i+b)}{\|\|w\|\|} \] The second quality follows from the fact that, since sample is linearly separable, for the maximizing pair $(w, b), y_i(w.x_i+b)$ must be non-negative for al $i\in[m]$. Now, observe that the last expression is invariant to multiplication of $(w, b)$ by a positive scalar. Thus, we can restrict ourselves to pairs $(\boldsymbol{w},b)$ scaled such that $min_{i\in[m]}(\boldsymbol{w}.x_i+b) = 1$: \[ \rho = \underset{min_{i\in[m]}y_i(w.x_i+b)=1}{max}\frac{1}{\|\|w\|\|} = \underset{\forall i \in[m],y_i(w.x_i+b) \geq }{max}\frac{1}{\|\|w\|\|} \] Figure below illustrates the solution $(w, b)$ of the maximization we just formalized. In addition to the maximum-margin hyperplane, it also shows the marginal hyperplanes, which are the hyperplanes parallel to the separating hyperplane and passing through the closest points on the negative or positive sides. Since maximizing $1/\|\|w\|\|$ is equivalent to minimizing $\frac{1}{2}\|\|w\|\|^2$, in view of the equation above, the pair $(\boldsymbol{w}, b)$ returned by the SVM in the separable case is the solution of the following convex optimization problem: \[ \underset{w, b}{min}\frac{1}{2}\|\|w\|\|^2 \]\[ \text{subject to}: y_i(\boldsymbol{w}.x_i+b) \geq 1, \forall i \in[m] \] Since the objective function is quadratic and the constraints are affine (meaning they are greater or equal to) the optimization problem above is in fact a specific instance of quadratic programming (QP), a family of problems extensively studied in optimization. A variety of commercial and open-source solvers are available for solving convex QP problems. Additionally, motivated by the empirical success of SVMs along with its rich theoretical underpinnings, specialized methods have been developed to more efficiently solve this particular convex QP problem, notably the block coordinate descent algorithms with blocks of just two coordinates. So what are support vectors? See the formula above, we note that constraints tare affine We introduce Lagrange variables $\alpha_i \geq 0, i\in[m]$, associated to the m constrains and denoted by $\boldsymbol{\alpha}$ the vector $(\alpha_1, \ldots, \alpha_m)^T$. The Lagrangian can then be defiend for all $\boldsymbol{w}\in\mathbb{R}^N,b\in\mathbb{R}$, and $\boldsymbol{\alpha}\in\mathbb{R}_+^m$ by: \[ \mathcal{L}(\boldsymbol{w},b,\boldsymbol{\alpha} = \frac{1}{2}\|\|w\|\|^2 – \sum_{i = 1}^{m}\alpha_i[y_i(w.x_i+b) -1] \] Support vectors fully define the maximum-margin hyperplane or SVM solution, which justifies the name of the algorithm. By definition, vectors not lying on the marginal hyperplanes do not affect the definiton of these hyperplanse – in their absence, the solution the solution to the SVM problem is unique, the support vectors are not. In dimensiosn $N, N+1$ points are sufficient to define a hyperplane. Thus when more then $N+1$ points lie on them marginal hyperplane, different choices are possible for $N+1$ support vectors. But the points in the space are not always separable. In most practical settings, the training data is not linearly separable, which implies that for any hyperplane $\boldsymbol{w.x}+b=0$, there exists $x_i \in S$ such that: \[ y_i[\boldsymbol{w.x_i}+b] \ngeq 1 \] Thus, the constrains imposed on the linearly separable case cannot be hold simultaneously. However, a relaxed version of these constraints can indeed hold, that is, for each $i\in[m]$, there exists $\xi_i \geq 0$ such that: \[ y_i[\boldsymbol{w.x_i}+b] \ngeq 1-\xi_i \] The variables $\xi_i$ are known as slack variables and are commonly used in optimization to define relaxed versions of constraints. Here, a slack variable $\xi_i$ measures the distance by which vector $x_i$ violates the desires inequality, $y_i(\boldsymol{w.x_i} + b) \geq 1 . This figure illustrates this situation: For hyperplane $y_i(w.x_i+b) = 1 $, a vector $x_i$ with $x_i > 0$ can be viewed as an outlier. Each $x_i$ must be positioned on the correct side the appropriate marginal hyperplane. Here’s the formula we use to optimize the non-separable cases: \[ \underset{w, b, \xi}{min} \frac{1}{2}\|\|w\|\|^2 + C\sum_{i=1}^{m}\xi_i^p \]\[ \text{subject to} \quad y_i(w.x_i+b) \geq 1-\xi_i \wedge \xi_i \geq 0, i\in[m] \] Okay! Alright! I think I understand it now. That’s enough classification for today. I’m going to study something FUN next. Altough I’m a bit drowsy… No matter! I have some energy drinks at home. Plus I have some methamphetamine which I have acquired to boost my eenrgy… Nah, kidding. I’m a cocaine man!
Search Now showing items 1-10 of 26 Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider (American Physical Society, 2016-02) The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has been studied using the ALICE detector at the LHC. The spectra exhibit a significant hardening with ... Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV (Elsevier, 2016-02) Two-particle angular correlations between trigger particles in the forward pseudorapidity range ($2.5 < \|\eta\| < 4.0$) and associated particles in the central range ($\|\eta\| < 1.0$) are measured with the ALICE detector in ... Measurement of D-meson production versus multiplicity in p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV (Springer, 2016-08) The measurement of prompt D-meson production as a function of multiplicity in p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with the ALICE detector at the LHC is reported. D$^0$, D$^+$ and D$^{*+}$ mesons are reconstructed ... Measurement of electrons from heavy-flavour hadron decays in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV (Elsevier, 2016-03) The production of electrons from heavy-flavour hadron decays was measured as a function of transverse momentum ($p_{\rm T}$) in minimum-bias p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC for $0.5 ... Direct photon production in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV (Elsevier, 2016-03) Direct photon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV was studied in the transverse momentum range $0.9 < p_{\rm T} < 14$ GeV/$c$. Photons were detected via conversions in the ALICE ... Multi-strange baryon production in p-Pb collisions at $\sqrt{s_\mathbf{NN}}=5.02$ TeV (Elsevier, 2016-07) The multi-strange baryon yields in Pb--Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, $\Xi$ and $\Omega$ production rates have been measured with the ALICE experiment as a ... $^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV (Elsevier, 2016-03) The production of the hypertriton nuclei $^{3}_{\Lambda}\mathrm H$ and $^{3}_{\bar{\Lambda}} \overline{\mathrm H}$ has been measured for the first time in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV with the ALICE ... Multiplicity dependence of charged pion, kaon, and (anti)proton production at large transverse momentum in p-Pb collisions at $\sqrt{s_{\rm NN}}$= 5.02 TeV (Elsevier, 2016-09) The production of charged pions, kaons and (anti)protons has been measured at mid-rapidity ($-0.5 < y < 0$) in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE detector at the LHC. Exploiting particle ... Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV (Elsevier, 2016-12) We present measurements of two-particle correlations with neutral pion trigger particles of transverse momenta $8 < p_{\mathrm{T}}^{\rm trig} < 16 \mathrm{GeV}/c$ and associated charged particles of $0.5 < p_{\mathrm{T}}^{\rm ... Centrality dependence of charged jet production in p-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 5.02 TeV (Springer, 2016-05) Measurements of charged jet production as a function of centrality are presented for p-Pb collisions recorded at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector. Centrality classes are determined via the energy ...
Let $a, b$ be arbitrary elements of the group $G$. We want to show that $ab=ba$. By the given relation $(ab)^3=a^3b^3$, we have\begin{align}ababab=a^3b^3.\end{align}Multiplying by $a^{-1}$ on the left and $b^{-1}$ on the right, we obtain\[baba=a^2b^2,\]or equivalently we have\[(ba)^2=a^2b^2 \tag{}\]for any $a, b\in G$. Now we consider $aba^{-1}b^{-1}$ (such an expression is called the commutator of $a, b$).We have\begin{align}(aba^{-1}b^{-1})^2&=(a^{-1}b^{-1})^2(ab)^2 && \text{by ()}\\&=b^{-2}a^{-2}b^2a^2 && \text{by ()}\\&=b^{-2}(ba^{-1})^2a^2 && \text{by ()}\\&=b^{-2}ba^{-1}ba^{-1}a^2\\&=b^{-1}a^{-1}ba.\end{align}Hence we have obtained\[(aba^{-1}b^{-1})^2=b^{-1}a^{-1}ba \tag{}\]for any $a, b\in G$.Taking the square of (), we obtain\begin{align}(aba^{-1}b^{-1})^4&=(b^{-1}a^{-1}ba)^2\\&=aba^{-1}b^{-1}. && \text{by ()}\end{align}It follows that we have\[(aba^{-1}b^{-1})^3=e,\]where $e$ is the identity element of $G$.Since the group $G$ does not have an element of order $3$, this yields that\[aba^{-1}b^{-1}=e.\](Otherwise, the order of the element $aba^{-1}b^{-1}$ would be $3$.) This is equivalent to\[ab=ba.\]Thus, we have obtained $ab=ba$ for any elements $a, b$ in $G$.Therefore, the group $G$ is abelian. Related Question. I came up with this problem when I solved the previous problem: Problem. Prove that if a group $G$ satisfies $(ab)^2=a^2b^2$ for $a, b \in G$, then $G$ is an abelian group. Prove a Group is Abelian if $(ab)^2=a^2b^2$Let $G$ be a group. Suppose that\[(ab)^2=a^2b^2\]for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.Proof.To prove that $G$ is an abelian group, we need\[ab=ba\]for any elements $a, b$ in $G$.By the given […] Non-Abelian Simple Group is Equal to its Commutator SubgroupLet $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.Definitions/Hint.We first recall relevant definitions.A group is called simple if its normal subgroups are either the trivial subgroup or the group […] Commutator Subgroup and Abelian Quotient GroupLet $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.Let $N$ be a subgroup of $G$.Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.Definitions.Recall that for any $a, b \in G$, the […] Quotient Group of Abelian Group is AbelianLet $G$ be an abelian group and let $N$ be a normal subgroup of $G$.Then prove that the quotient group $G/N$ is also an abelian group.Proof.Each element of $G/N$ is a coset $aN$ for some $a\in G$.Let $aN, bN$ be arbitrary elements of $G/N$, where $a, b\in […] Normal Subgroups Intersecting Trivially Commute in a GroupLet $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.Hint.Consider the commutator of $a$ and $b$, that […]
Nontrivial Action of a Simple Group on a Finite Set Problem 112 Let $G$ be a simple group and let $X$ be a finite set.Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.Then show that $G$ is a finite group and the order of $G$ divides $\|X\|!$. Since $G$ acts on $X$, it induces a permutation representation\[\rho: G \to S_{X}.\] Let $N=\ker(\rho)$ be the kernel of $\rho$.Since a kernel is normal in $G$ and $G$ is simple, we have either $N=\{e\}$ or $N=G$. If $N=G$, then for any $g\in G$ we have $\rho(g)$ is a trivial action, that is, $g\cdot x=x$ for any $X$.This contradicts the assumption that $G$ acts nontrivially on $X$.Hence we have $N=\{e\}$, and it follows that the homomorphism $\rho$ is injective. Thus we have\[G \cong \mathrm{im} (\rho) < S_{X}.\]Since $S_{X}$ is a finite group and $G$ is isomorphic to its subgroup, the group $G$ is finite.By Lagrange’s theorem, the order $\|G\|=\|\mathrm{im}(\rho)\|$ of $G$ divides the order $\|S_{X}\|=\|X\|!$ of $S_{X}$. Subgroup of Finite Index Contains a Normal Subgroup of Finite IndexLet $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.Proof.The group $G$ acts on the set of left cosets $G/H$ by left multiplication.Hence […] Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.Proof.Let $G$ be a group of order $24$.Note that $24=2^3\cdot 3$.Let $P$ be a Sylow $2$-subgroup of $G$. Then $\|P\|=8$.Consider the action of the group $G$ on […] A Group Homomorphism is Injective if and only if MonicLet $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […] Group Homomorphism, Preimage, and Product of GroupsLet $G, G'$ be groups and let $f:G \to G'$ be a group homomorphism.Put $N=\ker(f)$. Then show that we have\[f^{-1}(f(H))=HN.\]Proof.$(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$.It follows that there exists $h\in H$ […] Any Finite Group Has a Composition SeriesLet $G$ be a finite group. Then show that $G$ has a composition series.Proof.We prove the statement by induction on the order $\|G\|=n$ of the finite group.When $n=1$, this is trivial.Suppose that any finite group of order less than $n$ has a composition […]
Article Непрерывные потоки Морса–Смейла с тремя состояниями равновесия We consider the class of continuous Morse-Smale flows defined on a topological closed manifold $M^n$ of dimension n which is not less than three, and such that the stable and unstable manifolds of saddle equilibrium states do not have intersection. We establish a relationship between the existence of such flows and topology of closed trajectories and topology of ambient manifold. Namely, it is proved that if $f^t$ (that is a continuous Morse-Smale flow from considered class) has mu sink and source equilibrium states and $\nu$ saddles of codimension one, and the fundamental group $\pi_{1}(M^n$) does not contain a subgroup isomorphic to the free product $g = 1/ 2 ( \nu−\mu + 2)$ copies of the group of integers Z , then the flow $f^t$ has at least one periodic trajectory. We introduce the denition of consistent equivalence of Meyer ξ -functions for Morse- Smale ows on surfaces (that are Lyapunov funñtions) and state that consistent equivalence of ξ -functions is necessary and sucient condition for such ows. This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study. A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station. Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself. Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.
Isomorphism Criterion of Semidirect Product of Groups Problem 113 Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism. The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation \[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\] where $a_i \in A, b_i \in B$ for $i=1, 2$. Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$. \[\require{AMScd} \begin{CD} B @>{\phi}>> \Aut(A)\\ @A{g^{-1}}AA @VV{\sigma_f}V \\ B’ @>{\phi’}>> \Aut(A’) \end{CD}\] Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $ \alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$. Then show that \[A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.\] Proof. Define $\Psi: A \rtimes_{\phi} B \to A’ \rtimes_{\phi’} B’$ by \[(a,b) \mapsto (f(a), g(b))\] for $(a,b)\in A \rtimes_{\phi} B$. We show that this is a group isomorphism. Since $f, g$ are group isomorphisms, it suffices to show that $\Psi$ is a group homomorphism. Let $(a_1,b_1), (a_2, b_2) \in A \rtimes_{\phi} B$. We compute the product in $A\rtimes_{\phi} B$ is \[(a_1,b_1)\cdot(a_2, b_2)=(a_1\phi(b_1)(a_2), b_1 b_2).\] Thus we have \begin{align} \Psi\left( (a_1,b_1)\cdot(a_2, b_2) \right) &= \Psi\left( (a_1\phi(b_1)(a_2), b_1 b_2) \right)\\ &=\bigg(f \big(a_1\phi(b_1)(a_2) \big), g(b_1 b_2) \bigg). \tag{} \end{align} On the other hand, we have \begin{align} \Psi \left( (a_1, b_1) \right) \cdot \Psi \left( (a_2, b_2) \right) &= (f(a_1), g(b_1)) \cdot (f(a_2), g(b_2)).\\ &=\bigg(f(a_1) \phi’\big(g(b_1))(f(a_2) \big), g(b_1)g(b_2) \bigg) \tag{*} \end{align} Here we used group operation in $A’\rtimes_{\phi’} B’$ in the second equality. Now by the definition of $\phi’$ we have \[\phi'(g(b_1))=f\circ \phi(g^{-1}(g(b_1))) \circ f^{-1}=f\circ \phi(b_1) \circ f^{-1}.\] Thus we have \[\phi'(g(b_1))(f(a_2))=f\circ \phi(b_1) a_2.\] Hence we have \begin{align} () &= \bigg(f(a_1)f \big(\phi(b_1)a_2 \big), g(b_1)g(b_2) \bigg)\\ &=\bigg(f \big(a_1\phi(b_1)(a_2) \big), g(b_1 b_2)\bigg), \end{align} where the last equality follows since $f, g$ are group homomorphisms. Comparing this with (*), we see that \[\Psi\left( (a_1,b_1)\cdot(a_2, b_2) \right)=\Psi \left( (a_1, b_1) \right) \cdot \Psi \left( (a_2, b_2) \right) \] and thus $\Psi$ is a group homomorphism, hence it is a group isomorphism. Corollary In particular, taking $A=A’$ and $B=B’$, we have the following corollary. Let $A \rtimes_{\phi} B$ be the semidirect product of groups $A$ and $B$ with respect to a homomorphism $\phi: B \to \Aut(A)$. If $\phi’ :B \to \Aut(A)$ is defined by the following diagram, then we have \[A \rtimes_{\phi} \cong A \rtimes_{\phi’} B.\] \[\require{AMScd} \begin{CD} B @>{\phi}>> \Aut(A)\\ @A{g^{-1}}AA @VV{\sigma_f}V \\ B @>{\phi’}>> \Aut(A) \end{CD}\] Application Determine all isomorphism classes of semidirect product groups $(C_2 \times C_2) \rtimes C_3$, where $C_i$ denotes a cyclic group of order $i$. Proof. We first determine all homomorphism $\phi: C_3 \to \Aut(C_2 \times C_2)$. Note that $\Aut(C_2 \times C_2)\cong S_3$. ($C_2 \times C_2$ has three degree $2$ elements and an automorphism permutes these elements.) Let $g$ be a generator of $C_3$. Then since $g$ is of order $3$, the image $\phi(g)$ is one of $1$, $(123)$, $(132)$. Thus there are three homomorphism from $C_3 \to \Aut(C_2 \times C_2)$ defined by \[\phi_0(g)=1, \phi_1(g)=(123), \phi_2(g)=(132).\] Since $\phi_0$ is a trivial homomorphism, the semidirect product is actually a direct product. Thus \[(C_2\times C_2)\rtimes_{\phi_0} C_3 =(C_2\times C_2)\times C_3,\] which is an abelian group. For $\phi_1$ and $\phi_2$, we obtain nonabelian groups \[(C_2\times C_2)\rtimes_{\phi_1} C_3 \text{ and } (C_2\times C_2)\rtimes_{\phi_2} C_3 .\] We claim that these two groups are isomorphic. Note that we have \[\phi_1(x)=\tau \phi_2 (x) \tau^{-1}\] for all $x \in C_3$, where $\tau=(23)\in S_3$. Thus the claim follows from Corollary. (In the notations in Corollary, $A=C_2 \times C_2$, $B=C_3$, $f\in \Aut(A)\cong S_3$ is $(23)$ and $g:B\to B$ is the identity.) Therefore we have two isomorphism classes for $(C_2 \times C_2) \rtimes C_3$, one is abelian, the other is nonabelian. (In fact, the nonabelian group is isomorphic to $A_4$.) Add to solve later
Event detail BLISS Seminar: Phase Transitions in Generalized Linear Models Seminar \| April 30 \| 3-4 p.m. \| 540 Cory Hall Leo Miolane, INRIA We consider generalized linear models (GLMs) where an unknown $n$-dimensional signal vector is observed through the application of a random matrix and a non-linear (possibly probabilistic) componentwise output function. We study the models in the high-dimensional limit, where the observation consists of $m$ points, and $m/n \to \alpha > 0$ as $n \to \infty$. This situation is ubiquitous in applications ranging from supervised machine learning to signal processing. We will analyze the model-case when the observation matrix has i.i.d.\ elements and the components of the ground-truth signal are taken independently from some known distribution. We will compute the limit of the mutual information between the signal and the observations in the large system limit. This quantity is particularly interesting because it is related to the free energy (i.e. the logarithm of the partition function) of the posterior distribution of the signal given the observations. Therefore, the study of the asymptotic mutual information allows to deduce the limit of important quantities such as the minimum mean squared error for the estimation of the signal. We will observe some phase transition phenomena. Depending on the noise level, the distribution of the signal and the non-linear function of the GLM we may encounter various scenarios where it may be impossible / hard (only with exponential-time algorithms) / easy (with polynomial-time algorithms) to recover the signal. This is joint work with Jean Barbier, Florent Krzakala, Nicolas Macris and Lenka Zdeborova.
What are the consequences of having complete problems in $NP\cap coNP$? This is a (wide) open problem; as in, we know almost nothing. Specifically, because of the trickiness in proving $NP \cap coNP$-complete problems, we need very different proof techniques than currently exist. As such, a discussion of consequences should reasonably include a tangent on "What would it mean to have such powerful, new proof techniques?" For a relatively recent discussion of the topic, there is David Johnson's 26th NP-Completeness column in the ACM Transactions on Algorithms from 2007 (PDF). Allow me to paraphrase some of what David says regarding the question of proving $NP \cap coNP$-complete problems' existence and add my thoughts: Currently, we only have "weak," natural candidates for membership in $NP \cap coNP - P$ in the sense that the strongest evidence for their membership is that we haven't managed to find a polynomial time algorithm for them yet. He lists a couple candidates: SMALL FACTOR, SIMPLE STOCHASTIC GAME, and MEAN PAYOFF GAME. Some of the extra "weirdness" of these problems comes from the best heuristic run times for solving them, e.g. SMALL FACTOR, aka INTEGER FACTOR $\le k$, has a randomized time complexity of $poly(n) 2^{\sqrt{k log(k)}}$. (If complete problems exist in $NP \cap coNP - P$, then is such sub-exponential (neither purely exponential, nor quite polynomial) runtime endemic of the class?) So specifically, we would want to prove something like: problem A is only in $P$ iff $NP \cap coNP = P$, i.e. a completeness result like Cook's theorem for 3SAT and $NP$. For $NP$, such proofs universally involve polynomial-time reductions (and fix your favorite, additional restrictions, e.g. Cook-reductions, Karp-reductions). As a result, under polynomial-time reduction techniques, it must be the case that there exists a polynomial-time recognizable representation of the class. For $NP$, we can use non-deterministic Turing machines that halt within a polynomial, $p(\|x\|)$, number of steps. As David points out, we have similar representations for other classes (where the status is more clear) such as $PSPACE$ and #$P$. The difficulty, however, with providing a similar representation for $NP \cap coNP$ is that the "natural" analog allows us to embed the Halting Problem within the representation and is therefore undecidable. That is, consider the following attempt to represent $NP \cap coNP$ with two non-deterministic Turing machines that, purportedly, recognize complementary languages: Question: Does a Turing Machine $M^$ halt on input $x \in {0,1}^n$? Construct two linear-time Turing machines $M_1$ and $M_2$ as follows. On input $y$, $M_1$ reads the input and always accepts. $M_2$ always rejects unless $\|y\| \ge \|x\|$ and $M^$ accepts $x$ in steps $\le \|y\|$. Therefore, $M_1$ and $M_2$ accept complementary languages iff $M^*$ does not halt on input $x$. Therefore, by contradiction, deciding if two polynomial-time Turing machines accept complementary languages is undecidable. So, the "natural" representation of $NP \cap coNP$ problems is not polynomial-time recognizable. The question remains: How do you represent $NP \cap coNP$ problems such that they are polynomial-time recognizable? There has been no significant work done on this issue, but its successful resolution is necessary to prove completeness in $NP \cap coNP$. Hence, I claim that the existence of a proof technique that can resolve the completeness of $NP \cap coNP$ will be the bigger story here -- not the "automatic" results of $NP \cap coNP$-complete problems (e.g. complexity classes, perhaps, collapsing) that we are already aware of (or rather, will be aware of, hypothetically in the future).
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than be $v(t) = 3t^2-12t+9$ how could I find the intervals Fix $c\in\{0,1,\dots\}$, let $K\geq c$ be an integer, and define $z_K=K^{-\alpha}$ for some $\alpha\in(0,2)$.I believe I have numerically discovered that$$\sum_{n=0}^{K-c}\binom{K}{n}\binom{K}{n+c}z_K^{n+c/2} \sim \sum_{n=0}^K \binom{K}{n}^2 z_K^n \quad \text{ as } K\to\infty$$but cannot ... So, the whole discussion is about some polynomial $p(A)$, for $A$ an $n\times n$ matrix with entries in $\mathbf{C}$, and eigenvalues $\lambda_1,\ldots, \lambda_k$. Anyways, part (a) is talking about proving that $p(\lambda_1),\ldots, p(\lambda_k)$ are eigenvalues of $p(A)$. That's basically routine computation. No problem there. The next bit is to compute the dimension of the eigenspaces $E(p(A), p(\lambda_i))$. Seems like this bit follows from the same argument. An eigenvector for $A$ is an eigenvector for $p(A)$, so the rest seems to follow. Finally, the last part is to find the characteristic polynomial of $p(A)$. I guess this means in terms of the characteristic polynomial of $A$. Well, we do know what the eigenvalues are... The so-called Spectral Mapping Theorem tells us that the eigenvalues of $p(A)$ are exactly the $p(\lambda_i)$. Usually, by the time you start talking about complex numbers you consider the real numbers as a subset of them, since a and b are real in a + bi. But you could define it that way and call it a "standard form" like ax + by = c for linear equations :-) @Riker "a + bi where a and b are integers" Complex numbers a + bi where a and b are integers are called Gaussian integers. I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd... Does anyone know if $T: V \to R^n$ is an inner product space isomorphism if $T(v) = (v)_S$, where $S$ is a basis for $V$? My book isn't saying so explicitly, but there was a theorem saying that an inner product isomorphism exists, and another theorem kind of suggesting that it should work. @TobiasKildetoft Sorry, I meant that they should be equal (accidently sent this before writing my answer. Writing it now) Isn't there this theorem saying that if $v,w \in V$ ($V$ being an inner product space), then $\|\|v\|\| = \|\|(v)_S\|\|$? (where the left norm is defined as the norm in $V$ and the right norm is the euclidean norm) I thought that this would somehow result from isomorphism @AlessandroCodenotti Actually, such a $f$ in fact needs to be surjective. Take any $y \in Y$; the maximal ideal of $k[Y]$ corresponding to that is $(Y_1 - y_1, \cdots, Y_n - y_n)$. The ideal corresponding to the subvariety $f^{-1}(y) \subset X$ in $k[X]$ is then nothing but $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$. If this is empty, weak Nullstellensatz kicks in to say that there are $g_1, \cdots, g_n \in k[X]$ such that $\sum_i (f^* Y_i - y_i)g_i = 1$. Well, better to say that $(f^* Y_1 - y_1, \cdots, f^* Y_n - y_n)$ is the trivial ideal I guess. Hmm, I'm stuck again O(n) acts transitively on S^(n-1) with stabilizer at a point O(n-1) For any transitive G action on a set X with stabilizer H, G/H $\cong$ X set theoretically. In this case, as the action is a smooth action by a Lie group, you can prove this set-theoretic bijection gives a diffeomorphism
In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. Ellipse The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. $ b = a \sqrt{1-e^2}\,\! $ $ \ell=a(1-e^2)\,\! $. $ a\ell=b^2\,\! $. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping $ \ell\,\! $ fixed. Thus $ a\,\! $ and $ b\,\! $ tend to infinity, $ a\,\! $ faster than $ b\,\! $. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, $ r(1-e\cos\theta)=l\,\! $ The mean value of $ r={\ell\over{1+e}}\,\! $ and $ r={\ell\over{1-e}}\,\! $, is $ a={\ell\over{1-e^2}}\,\! $. Hyperbola The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is a in the x-direction the equation is: $ \frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1 $ In terms of the semi-latus rectum and the eccentricity we have $ a={\ell\over e^2-1 } $ Astronomy Orbital period $ T = 2\pi\sqrt{a^3/\mu} $ where: $ a\, $ is the length of the orbit's semi-major axis $ \mu $ is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), $ T^2=a^3\, $ $ T^2= \frac{4\pi^2}{G(M+m)}a^3\, $ where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value. Average distance It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken. averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis. averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis $ b = a \sqrt{1-e^2}\,\! $. averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman): $ a (1 + \frac{e^2}{2})\,\! $. The time-average of the inverse of the radius, $ r^{-1}\,\! $, is $ a^{-1}\,\! $. Energy; calculation of semi-major axis from state vectors and $ \epsilon = { v^2 \over {2} } - {\mu \over \left \| \mathbf{r} \right \|} $ (specific orbital energy) and $ \mu = GM \, $ (standard gravitational parameter), where: $ v\, $ is orbital velocity from velocity vector of an orbiting object, $ \mathbf{r }\, $ is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun), $ G \, $ is the gravitational constant, $ M \, $ the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same. Example The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [1]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.
Difference between revisions of "Printing with FLTK" m (Minor format changes) (One intermediate revision by one other user not shown) Line 1: Line 1: Printing with graphics_toolkit FLTK has some known limitations: Printing with graphics_toolkit FLTK has some known limitations: − * Tex/Latex symbols won't show up in the generated print even if they are visible in the plot window. See bugs #42988, [http://savannah.gnu.org/bugs/?42320 + * Tex/Latex symbols won't show up in the generated print even if they are visible in the plot window. See bugs #42988, [http://savannah.gnu.org/bugs/?42320 #42320], #42340 which are mostly duplicate entries for the same problem. * Can't print multiline text objects: [http://savannah.gnu.org/bugs/?31468 bug#31468] * Can't print multiline text objects: [http://savannah.gnu.org/bugs/?31468 bug#31468] Latest revision as of 23:30, 3 June 2017 Printing with graphics_toolkit FLTK has some known limitations: Tex/Latex symbols won't show up in the generated print even if they are visible in the plot window. See bugs #42988, #42320, #42340 which are mostly duplicate entries for the same problem. Should be solved in the meanwhile, see bug report Can't print multiline text objects: bug#31468 However there are some ways to overcome these: Use print [ps\|eps\|pdf] latex [standalone] for symbols and formulas[edit] See "help print" for a description of 'pslatex', 'epslatex' 'pdflatex', 'pslatexstandalone', 'epslatexstandalone', 'pdflatexstandalone' Code: print with fltk and epslatexstandalone" close allgraphics_toolkit fltksombrero ();title ("The sombrero function:")fcn = "$z = \\frac{\\sin\\left(\\sqrt{x^2 + y^2}\\right)}{\\sqrt{x^2 + y^2}}$";text (0.5, -10, 1.8, fcn, "fontsize", 20);print -depslatexstandalone sombrero## process generated files with pdflatexsystem ("latex sombrero.tex");## dvi to pssystem ("dvips sombrero.dvi");## convert to png for wiki pagesystem ("gs -dNOPAUSE -dBATCH -dSAFER -sDEVICE=png16m -dTextAlphaBits=4 -dGraphicsAlphaBits=4 -r100x100 -dEPSCrop -sOutputFile=sombrero.png sombrero.ps") Use psfrag[edit] TODO: Fill me!
Answer The longest three wavelengths in the Lyman series are $121.5~nm$, $102.6~nm$, and $97.23~nm$ Work Step by Step The longest three wavelengths in the Lyman series would be the three transitions with the least amount of energy, that is, the transitions from level 2 to the ground state, from level 3 to the ground state, and from level 4 to the ground state. We can find the wavelength of the photon emitted by a transition from level 2 to the ground state: $\frac{1}{\lambda} = R~(\frac{1}{n_f^2}-\frac{1}{n_i^2})$ $\frac{1}{\lambda} = R~(\frac{1}{1^2}-\frac{1}{2^2})$ $\frac{1}{\lambda} = R~(\frac{3}{4})$ $\lambda = \frac{4}{3~R}$ $\lambda = \frac{4}{(3)~(1.097\times 10^7~m^{-1})}$ $\lambda = 1.215\times 10^{-7}~m$ $\lambda = 121.5~nm$ We can find the wavelength of the photon emitted by a transition from level 3 to the ground state: $\frac{1}{\lambda} = R~(\frac{1}{n_f^2}-\frac{1}{n_i^2})$ $\frac{1}{\lambda} = R~(\frac{1}{1^2}-\frac{1}{3^2})$ $\frac{1}{\lambda} = R~(\frac{8}{9})$ $\lambda = \frac{9}{8~R}$ $\lambda = \frac{9}{(8)~(1.097\times 10^7~m^{-1})}$ $\lambda = 1.026\times 10^{-7}~m$ $\lambda = 102.6~nm$ We can find the wavelength of the photon emitted by a transition from level 4 to the ground state: $\frac{1}{\lambda} = R~(\frac{1}{n_f^2}-\frac{1}{n_i^2})$ $\frac{1}{\lambda} = R~(\frac{1}{1^2}-\frac{1}{4^2})$ $\frac{1}{\lambda} = R~(\frac{15}{16})$ $\lambda = \frac{16}{15~R}$ $\lambda = \frac{16}{(15)~(1.097\times 10^7~m^{-1})}$ $\lambda = 0.9723\times 10^{-7}~m$ $\lambda = 97.23~nm$ The longest three wavelengths in the Lyman series are $121.5~nm$, $102.6~nm$, and $97.23~nm$.
The Inhomogeneous Short Integer Solution (ISIS) problem is as follows: given an integer $q$, a matrix $A \in \mathbb Z^{n\times m}_q$, a vector $b\in \mathbb Z^n_q$, and a real $\beta$, find an integer vector $e\in \mathbb Z^m$ such that $Ae = b \mod q $ and $\|\|e\|\|_2 \leq \beta$. Now assume $e_0$ is a solution to the ISIS instance $(q,A,b,\beta)$, so $A e_0 = b \mod q $ and $\|\|e_0\|\|_2 \leq \beta$. A clue to this solution is provided to me, so I receive $M e_0$, for a matrix $M \in \mathbb Z^{m\times m}_q$ of my choice. How could I choose $M$ in order to be able to retrieve $e_0$ from the clue $Me_0$, or more generally, to compute some short solution $e$ for $Ae = b$? Solving ISIS on $(q,M,Me_0,\beta)$ doesn't help me, since that only finds a $e_1$ so that $Me_1=Me_0$, but not necessarily $Ae_1=b$ (and assuming I am able to solve that instance of ISIS...). A possible solution, If I were able to solve ISIS, would be to solve $\begin{bmatrix}A\\ M\end{bmatrix}e = \begin{bmatrix}b\\ Me_0\end{bmatrix}$ , but if I could solve that... I could just solve $Ae=b$, right? A trivial solution could be to choose $M = I$, so I receive $I e_0 = e_0$, which is a valid solution, since $Ae_0 = b$. However, in that case the clue would be "trivial", and I am interested in knowing other possible choices of $M$. Permutation matrices are also considered trivial (the identity is a particular case of permutation matrix). My question is: What choices of $M$, different to permutation matrices, would enable me to calculate a solution to $Ae=b$, assuming I receive a clue $Me_0$?
This interactive simulation estimates the value of the fundamental constant, pi (π), by drawing lots of random points to estimate the relative areas of a square and an inscribed circle. Pi, (π), is used in a number of math equations related to circles, including calculating the area, circumference, etc. and is widely used in geometry, trigonometry and physics. This app estimates the value of pi by comparing the area of a square and an inscribed circle. The areas are calculated by randomly placing dots into the square and then counting how many of them are also inside the circle. If you do this enough times, you will get a rough ratio of the relative areas of the two shapes. These points are plotted on the graph (red if the fall inside the circle and blue if the fall outside). Also shown on the graph is the value of our estimate of pi as the simulation progresses, from a few points to many thousands, to millions of points. We can see that when we have only a few points, the value may not be very accurate but as the number of points increases the value of our estimate gets closer to the true value. Running the simulation will add and plot 1 million points. After the first 100 points are added, the rate at which points are added increases. You’ll notice this as the speed at which dots fills up the square increases and because the plot is shown with a logarithmic x-axis. Here is the math: Length of side of square: $2 \times r$ radius of circle: $r$ Area of square: $A_{square} = 4r^2$ Area of circle: $A_{circle} = \pi r^2$ The ratio of areas is $A_{circle}/A_{square} = \pi r^2 / 4r^2 = \pi / 4$ Solving for pi: $\pi = 4 \times A_{circle}/A_{square} \approx 4 \times N_{dots_{circle}}/N_{dots_{square}}$ So pi is estimated as 4 times the ratio of dots in the circle vs square Tools: This was programmed in javascript, canvas and plotted using the open source plotly javascript plotting library. Related Posts No Comments »
A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals.That is, whenever we have ideals $I_n$ of $R$ satisfying\[I_1\supset I_2 \supset \cdots \supset I_n \supset \cdots,\]there is an integer $N$ such that\[I_N=I_{N+1}=I_{N+2}=\cdots.\] Proof. Let $x\in R$ be a nonzero element. To prove $R$ is a field, we show that the inverse of $x$ exists in $R$.Consider the ideal $(x)=xR$ generated by the element $x$. Then we have a descending chain of ideals of $R$:\[(x) \supset (x^2) \supset \cdots \supset (x^i) \supset (x^{i+1})\supset \cdots.\] In fact, if $r\in (x^{i+1})$, then we write it as $r=x^{i+1}s$ for some $s\in R$.Then we have\[r=x^i\cdot xs\in (x^i)\]since $(x^i)$ is an ideal and $xs\in R$.Hence $(x^{i+1})\subset (x^i)$ for any positive integer $i$. Since $R$ is an Artinian ring by assumption, the descending chain of ideals terminates.That is, there is an integer $N$ such that we have\[(x^N)=(x^{N+1})=\cdots.\] It follows from the equality $(x^N)=(x^{N+1})$ that there is $y\in R$ such that\[x^N=x^{N+1}y.\]It yields that\[x^N(1-xy)=0.\] Since $R$ is an integral domain, we have either $x^N=0$ or $1-xy=0$.Since $x$ is a nonzero element and $R$ is an integral domain, we know that $x^N\neq 0$. Thus, we must have $1-xy=0$, or equivalently $xy=1$.This means that $y$ is the inverse of $x$, and hence $R$ is a field. Every Maximal Ideal of a Commutative Ring is a Prime IdealLet $R$ be a commutative ring with unity.Then show that every maximal ideal of $R$ is a prime ideal.We give two proofs.Proof 1.The first proof uses the following facts.Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […] Finite Integral Domain is a FieldShow that any finite integral domain $R$ is a field.Definition.A commutative ring $R$ with $1\neq 0$ is called an integral domain if it has no zero divisors.That is, if $ab=0$ for $a, b \in R$, then either $a=0$ or $b=0$.Proof.We give two proofs.Proof […]
As I understand from the survey "Progress on Polynomial Identity Testing - II" a polynomial-time algorithm for solving PIT for $\Sigma \Pi \Sigma \Pi (2, r)$ is unknown. However, there exists paper of Ankit Gupta "Algebraic Geometric Techniques for Depth-4 PIT & Sylvester-Gallai Conjectures for Varieties" that gives a polynomial-time algorithm for this case modulo the following conjecture from this paper. Conjecture Let $F_1, F_2, \ldots, F_k$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $r$ such that $ \cap F_i = \varnothing$ and for every $Q_1, \ldots , Q_{k-1}$ from distinct sets there are polynomial $P_1, \ldots, P_c$ in the remaining set such that $V(Q_1, \ldots, Q_{k-1}) \subseteq \cup_i V(P_i)$. Then $\text{trdeg}_{\mathbb{C}}(\cup_i F_i) = \lambda(k,r,c)$ for some function $\lambda$. This conjecture impies the existense of polynomial-time for PIT for $\Sigma \Pi \Sigma \Pi (k, r)$. As I understand for $k=2$ this conjecture holds. Note first that $V(Q_1)\subseteq \cup_i V(P_i)$ implies $V(Q_1)\subseteq V(P_j)$ for some $j$. Indeed, by Nullstellensatz $(P_1 \cdot P_2 \ldots \cdot P_c)^m = Q_1 \cdot H$ for some $m$ and $H$. Since $Q_1$ is irreducible this means that some $P_j$ is divided by $Q_1$. So, now we have two sets of irreducible polynomials $F_1$ and $F_2$ such that $F_1 \cap F_2 = \varnothing$ and for every $P$ from one of this set there exists $Q$ from another set such that $V(P) \subseteq V(Q)$. Of course this means that both sets are trivial. So, there exists a polynomial-time algortihm for solving PIT for $\Sigma \Pi \Sigma \Pi(2,r)$-circuits. Am I wrong?
Search Now showing items 1-2 of 2 Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE (Elsevier, 2017-11) Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions, as well as give insight into the initial state of such collisions and ... Investigations of anisotropic collectivity using multi-particle correlations in pp, p-Pb and Pb-Pb collisions (Elsevier, 2017-11) Two- and multi-particle azimuthal correlations have proven to be an excellent tool to probe the properties of the strongly interacting matter created in heavy-ion collisions. Recently, the results obtained for multi-particle ...
The L-band (1-2GHz) and S-band (2-4GHz) are traditionally used either as direct input or as an intermediate frequency (IF) when downconverting from a higher RF frequency (like in a Ku-band radar, for example). Rather than designing a heterodyne receiver to sample these bands at a lower frequency, system designers are eager to simplify their receiver signal chain using a direct RF sampling analog-to-digital converter (ADC), resulting in reduced size, weight and power. Clock phase-noise performance is a key parameter to overall receiver performance – both for a traditional heterodyne receiver as well as for a direct RF sampling implementation. During a blocking condition with an in-band interferer, the clock phase-noise performance determines the minimum detectable signal power. In radar systems, the receiver close-in phase-noise performance directly impacts the accuracy of the speed and direction of the target. Since the data converter sampling clock is such a critical element in a direct RF receiver (and I’m often asked about it), in this article I’ll examine the common assumption that the clock-performance requirement for an RF sampling ADC is much more stringent compared to a heterodyne approach. Heterodyne receiver A traditional downconversion (heterodyne) receiver employs a mixer together with a local oscillator (LO) to frequency shift the RF input signal to a lower IF, which the data converter digitizes (see Figure 1). Figure 1. Heterodyne receiver approach. The mixing operation is a convolution of the LO and the RF input in the frequency domain (multiplication in the time domain). As a result of mixing, the phase noise from the LO gets added to the input signal and increases the overall noise floor. This is typically referred to as reciprocal mixing. During the ADC sampling process, the clock phase noise, which includes ADC aperture jitter and external clock phase noise, gets added to the IF input signal as well. However, the clock phase-noise amplitude scales based on the relationship between the input and the clock frequency $$\text{20Log}\left( \frac{\text{IF}}{\text{CLK}} \right)$$. Additionally, you need to consider the ADC thermal noise needs as well. Equation 1 calculates the resulting phase-noise amplitude of the sampled IF signal: $$\text{P}{{\text{N}}_{\text{IF}}}\text{=20Log}\left( \sqrt{\text{PN}_{\text{LO}}^{\text{2}}\text{+PN}_{\text{ADC }\!\!\_\!\!\text{ PN}}^{\text{2}}\text{+PN}_{\text{ADC }\!\!\_\!\!\text{ THERMAL}}^{\text{2}}} \right) $$ Equation 1 where $$[\text{P}{{\text{N}}_{\text{ADC }\!\!\_\!\!\text{ PN}}}\text{=P}{{\text{N}}_{\text{CLK}}}\text{ }\!\!\times\!\!\text{ 20LOG }\!\!~\!\!\text{ }\left( \frac{\text{IF}}{\text{CLK}} \right)$$ and PN CLK is the combination of external clock phase noise and internal ADC aperture jitter (phase noise). Both components experience the amplitude scaling of 20LOG[IF/CLK]. RF Sampling Receiver In a modern receiver architecture, the RF sampling ADC digitizes the input signal directly at RF without the use of a downconversion stage, as illustrated in Figure 2. Removing a downconversion stage can significantly simplify the receiver signal chain and save power, cost, and printed circuit board (PCB) area. Figure 2. Direct RF sampling approach. As shown earlier, the ADC aperture jitter and clock phase-noise amplitudes first scale by $$\text{P}{{\text{N}}_{\text{CLK}}}$$ before adding to the input signal together with the ADC thermal noise (Equation 2): $$\text{P}{{\text{N}}_{\text{RF}}}\text{=20Log}\left( \sqrt{\text{PN}_{\text{ADC }\!\!\_\!\!\text{ PN}}^{\text{2}}\text{+PN}_{\text{ADC }\!\!\_\!\!\text{ Thermal}}^{\text{2}}} \right)$$ Equation 2 where CLK again the combination of external clock phase noise and internal ADC aperture jitter (phase noise). Assuming equivalent thermal noise (in decibels relative to full scale/hertz) for both IF and RF sampling data converters, the receiver noise degradation ultimately comes down to LO and clock phase noise. In order to better compare the clock phase-noise requirements for the two receiver architectures, you can normalize the phase noise of the frequency sources (LO, ADC CLK) by deriving them from the same source, as shown in Figure 3. For example, an LO at half the frequency of the source would have 6dB lower phase noise than the source itself. In order to keep things simple, I picked the same RF frequency for the frequency source. Figure 3. Deriving frequency sources from the same "RF" source. Normalizing the Clock Phase Noise Requirement for the Downconversion Receiver Equation 3 calculates the frequency of the local oscillator (LO) as: $$\text{LO=RF-IF=RF }\!\!\times\!\!\text{ (1-a)}$$ Equation 3 where $$\text{IF=a }\!\!\times\!\!\text{ RF}$$. Assuming low-side injection of the mixer, Equation 4 calculates the phase noise of the LO from a common clock source at RF as: $$\text{P}{{\text{N}}_{\text{LO}}}\text{=P}{{\text{N}}_{\text{RF}}}\text{ }\!\!\times\!\!\text{ 20Log}\left( \frac{\text{LO}}{\text{RF}} \right)\text{=P}{{\text{N}}_{\text{RF}}}\text{ }\!\!\times\!\!\text{ 20Log}\left( \text{1-a} \right)$$ Equation 4 Equation 5 calculates the phase noise of the sampled ADC output as a function of the clock phase noise as: $$\text{P}{{\text{N}}_{\text{ADC}}}\text{=P}{{\text{N}}_{\text{CLK}}}\text{ }\!\!\times\!\!\text{ 20Log}\left( \frac{\text{IF}}{\text{CLK}} \right)$$ Equation 5 You can derive the phase noise of the external ADC sampling clock from the common RF clock source with Equations 6 and 7 (not including aperture jitter for simplicity): $$\text{P}{{\text{N}}_{\text{CLK}}}\text{=P}{{\text{N}}_{\text{RF}}}\times \text{20Log}\left( \frac{\text{CLK}}{\text{RF}} \right)$$ Equation 6 $$\text{P}{{\text{N}}_{\text{ADC}}}\text{=P}{{\text{N}}_{\text{RF}}}\times 2\text{0Log}\left( \frac{\text{IF}}{\text{CLK}} \right)\times \text{20Log}\left( \frac{\text{CLK}}{\text{RF}} \right)\text{=P}{{\text{N}}_{\text{RF}}}\times \text{20Log}\left( \frac{\text{IF}}{\text{RF}} \right)\text{=P}{{\text{N}}_{\text{RF}}}\text{ }\!\!~\!\!\text{ }\times \text{20Log}\left( \text{a} \right)$$ Equation 7 Combine the LO and ADC phase-noise contributions using Equation 8: $$\begin{align}& \text{P}{{\text{N}}_{\text{total}}}\text{=20Log}\left( \sqrt{\text{PN}_{\text{LO}}^{\text{2}}\text{+PN}_{\text{ADC}}^{\text{2}}} \right) \\ & \text{=20Log}\left( \sqrt{{{\left( \text{P}{{\text{N}}_{\text{RF}}}\times \left( \text{1-a}\right) \right)}^{\text{2}}}\text{+}{{\left( \text{P}{{\text{N}}_{\text{RF}}}\times \text{a}) \right)}^{\text{2}}}} \right)\\ & \text{=P}{{\text{N}}_{\text{RF}}}\text{ }\!\!~\!\!\text{ }\times \text{10log}\left( {{\left( \text{1-a} \right)}^{\text{2}}}\text{+}{{\text{a}}^{\text{2}}} \right) \end{align}$$ Equation 8 Where $$\text{a=}\frac{\text{IF}}{\text{RF}}$$. This shows that the amplitude of the phase noise, which gets added to the input signal, is dependent on the IF frequency. Assuming that the IF is ~10% of the RF frequency (for example, if IF = 180MHz, RF = 1.8GHz), the phase-noise contribution of the LO normalized to the RF frequency changes by 20LOG(0.9) = -0.91dB. This means that the LO phase noise improves by 0.91dB. The clock phase-noise contribution of the ADC changes by 20LOG(a = 10%) = -20dB. As a result, the added phase noise of the sampled signal is about 0.86dB lower than the phase noise of the RF clock source. The ADC aperture jitter, assuming a similar level as the external clock phase noise, would have only a minor impact on the overall result. Normalizing the Clock Phase Noise Requirement for the RF ADC Equation 9 calculates the phase noise of the sampled ADC output as a function of the clock phase noise: $$\text{P}{{\text{N}}_{\text{ADC}}}\text{=P}{{\text{N}}_{\text{CLK}}}\text{ }\!\text{ }\!\times\!\text{ }\!\text{ 20log}\left( \frac{\text{RF}}{\text{CLK}} \right)$$ Equation 9 You can derive the phase noise of the ADC clock from the common RF clock source using Equations 10 and 11: $$\text{P}{{\text{N}}_{\text{CLK}}}\text{=P}{{\text{N}}_{\text{RF}}}\text{ }\!\text{ }\!\times\!\text{ }\!\text{ 20log}\left( \frac{\text{CLK}}{\text{RF}} \right)$$ Equation 10 $$P{{N}_{ADC}}=P{{N}_{RF}}~\times ~20log\left( \frac{RF}{CLK} \right)\times ~20log\left( \frac{CLK}{RF} \right)=P{{N}_{RF}}$$ Equation 11 When normalized to the RF frequency, there is no change in amplitude at all. The additive phase noise of the RF ADC is the same as the phase noise of the RF clock source. Unlike the heterodyne receiver, the aperture jitter of an RF ADC can cause significant phase-noise degradation. The heterodyne receiver shows a small improvement, which is dependent on the IF frequency. The larger the IF frequency (in percentage of the RF frequency), the larger the difference in the normalized phase-noise requirements for the two cases. Thus, the two architectures in fact show very minor external clock phase-noise requirements. A typical use case with IF = 0.1 x RF shows only about a 0.9dB normalized phase-noise relaxation compared to the direct RF sampling approach. So the clock-noise requirements for both architectures are pretty comparable. The major difference is the internal ADC aperture jitter, which becomes more dominant as the RF input frequency increases. Industry Articles are a form of content that allows industry partners to share useful news, messages, and technology with All About Circuits readers in a way editorial content is not well suited to. All Industry Articles are subject to strict editorial guidelines with the intention of offering readers useful news, technical expertise, or stories. The viewpoints and opinions expressed in Industry Articles are those of the partner and not necessarily those of All About Circuits or its writers.
Summary Let $T : \Omega \rightarrow \mathbb{C}^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset \mathbb{C}$. We call a point $\lambda \in \Omega$ an eigenvalue if the matrix $T(\lambda)$ is singular. Nonlinear eigenvalue problems arise in many applications, often from applying transform methods to analyze differential and difference equations. We describe new localization results for nonlinear eigenvalues that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, the Bauer-Fike theorem, and others. Papers @techreport{2016-transient-tr, author = {Hood, Amanda and Bindel, David}, title = {Pseudospectral bounds on transient growth for higher order and constant delay differential equations}, month = nov, year = {2016}, arxiv = {1611.05130}, link = {http://arxiv.org/pdf/1611.05130}, status = {unrefereed}, submit = {Submitted to SIAM Journal on Matrix Analysis and Applictions.} } SIAM Review, vol. 57, no. 4, pp. 585–607, Dec. 2015. SIGEST feature article. @article{2015-sirev, author = {Bindel, David and Hood, Amanda}, title = {Localization Theorems for Nonlinear Eigenvalues}, journal = {SIAM Review}, publisher = {SIAM}, volume = {57}, number = {4}, pages = {585--607}, month = dec, year = {2015}, notable = {SIGEST feature article.}, doi = {10.1137/15M1026511} } Abstract: Let $T : \Omega \rightarrow {\Bbb C}^{n\times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation. SIAM Journal on Matrix Analysis, vol. 34, no. 4, pp. 1728–1749, 2013. 2015 SIAG/LA award (best journal paper in applied LA in three years). @article{2013-simax, author = {Bindel, David and Hood, Amanda}, title = {Localization Theorems for Nonlinear Eigenvalues}, journal = {SIAM Journal on Matrix Analysis}, volume = {34}, number = {4}, pages = {1728--1749}, year = {2013}, doi = {10.1137/130913651}, arxiv = {http://arxiv.org/abs/1303.4668}, notable = {2015 SIAG/LA award (best journal paper in applied LA in three years).} } Abstract: Let $T : \Omega \rightarrow {\Bbb C}^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation. Talks SIAM Computational Science and Engineering 2019, Spokane nep eigenbounds • minisymposium external invited Foundations of Computational Math 2017 nep eigenbounds • minisymposium external invited University of Arizona Math Colloquium nep eigenbounds • colloquium external invited Cornell Applied Math Colloquium nep eigenbounds • colloquium local SIAM LA 2015 (Prize Lecture) nep eigenbounds • meeting external invited plenary NEP14 Workshop eigenbounds nep • meeting external invited Workshop on Dissipative Spectral Theory, Cardiff eigenbounds matscat nep pml resonance • meeting external invited Weyl at 100 Workshop (Fields Institute) eigenbounds matscat nep pml resonance • meeting external invited ICIAM eigenbounds matscat nep resonance • minisymposium external invited Cornell SCAN Seminar eigenbounds matscat nep resonance • seminar local Workshop in honor of Pete Stewart at UT Austin eigenbounds matscat nep resonance • meeting external invited Simon Fraser University NA Seminar eigenbounds matscat nep resonance pml • seminar external invited NYCAM eigenbounds matscat nep resonance • meeting external Cornell SCAN Seminar eigenbounds matscat nep pml resonance • seminar local SIAM Annual Meeting eigenbounds matscat nep pml resonance • minisymposium external invited MSRI Workshop on Resonances eigenbounds matscat nep resonance • meeting external invited Berkeley Applied Math Seminar eigenbounds matscat nep pml resonance • seminar external invited Householder Symposium eigenbounds nep • minisymposium external invited NYU NA Seminar eigenbounds • seminar local Stanford NA Seminar eigenbounds • seminar external invited UC Berkeley LAPACK Seminar cis eigenbounds • seminar local
Observe the prime factorization $72=2^3\cdot 3^2$.Let $G$ be a group of order $72$. Let $n_3$ be the number of Sylow $3$-subgroups in $G$.By Sylow’s theorem, we know that $n_3$ satisfies\begin{align}&n_3\equiv 1 \pmod{3} \text{ and }\\&n_3 \text{ divides } 8.\end{align}The first condition gives $n_3$ could be $1, 4, 7, \dots$.Only $n_3=1, 4$ satisfy the second condition. Now if $n_3=1$, then there is a unique Sylow $3$-subgroup and it is a normal subgroup of order $9$.Hence, in this case, the group $G$ is not simple. It remains to consider the case when $n_3=4$.So there are four Sylow $3$-subgroups of $G$.Note that these subgroups are not normal by Sylow’s theorem. The group $G$ acts on the set of these four Sylow $3$-subgroups by conjugation.Hence it affords a permutation representation homomorphism\[f:G\to S_4,\]where $S_4$ is the symmetric group of degree $4$. By the first isomorphism theorem, we have\begin{align}G/\ker f < S_4.\end{align}Thus, the order of $G/\ker f$ divides the order of $S_4$.Since $\|S_4\|=4!=2^3\cdot 3$, the order $\|\ker f\|$ must be divisible by $3$ (otherwise $\|G/\ker f$\|$ does not divide $\|S_4\|$), hence $\ker f$ is not the trivial group. We claim that $\ker f \neq G$.If $\ker f=G$, then it means that the action given by the conjugation by any element $g\in G$ is trivial. That is, $gPg^{-1}=P$ for any $g\in G$ and for any Sylow $3$-subgroup $P$.Since those Sylow $3$-subgroups are not normal, this is a contradiction.Thus, $\ker f \neq G$. Since a kernel of a homomorphism is a normal subgroup, this yields that $\ker f$ is a nontrivial proper normal subgroup of $G$, hence $G$ is not a simple group. Group of Order $pq$ Has a Normal Sylow Subgroup and SolvableLet $p, q$ be prime numbers such that $p>q$.If a group $G$ has order $pq$, then show the followings.(a) The group $G$ has a normal Sylow $p$-subgroup.(b) The group $G$ is solvable.Definition/HintFor (a), apply Sylow's theorem. To review Sylow's theorem, […] Non-Abelian Group of Order $pq$ and its Sylow SubgroupsLet $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.Hint.Use Sylow's theorem. To review Sylow's theorem, check […] Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.Hint.Use Sylow's theorem.(See Sylow’s Theorem (Summary) for a review of Sylow's theorem.)Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […] Sylow Subgroups of a Group of Order 33 is Normal SubgroupsProve that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.Hint.We use Sylow's theorem. Review the basic terminologies and Sylow's theorem.Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […] Are Groups of Order 100, 200 Simple?Determine whether a group $G$ of the following order is simple or not.(a) $\|G\|=100$.(b) $\|G\|=200$.Hint.Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$.Check out the post Sylow’s Theorem (summary) for a review of Sylow's […] A Group of Order $20$ is SolvableProve that a group of order $20$ is solvable.Hint.Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem.See the post summary of Sylow’s Theorem to review Sylow's theorem.Proof.Let $G$ be a group of order $20$. The […]
The Set of Vectors Perpendicular to a Given Vector is a Subspace Problem 659 Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define\[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\]Prove that $W$ is a vector subspace of $\R^3$. Because, again, $\mathbf{b} \mathbf{v} = \mathbf{0}$, we have\[\mathbf{b} ( c \mathbf{v} ) = c \mathbf{b} \mathbf{v} = c \mathbf{0} = \mathbf{0}.\]Thus $c \mathbf{v} \in W$. These three criteria show that $W$ is a vector subspace of $\R^3$. Comment. We can generalize the problem with an arbitrary $1\times 3$ row vector $\mathbf{b}$. The proof is almost identical.(Look at the proof. We didn’t use components of the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$.) Note that vectors $\mathbf{u}, \mathbf{v}\in \R^3$ is said to be perpendicular if\[\mathbf{u}\cdot \mathbf{v}=\mathbf{u}^{\trans}\mathbf{v}=0.\] Thus, the result of the problem says that for a fixed vector $\mathbf{u}\in \R^3$, the set of vectors $\mathbf{v}$ that are perpendicular to $\mathbf{u}$ is a subspace in $\R^3$.(Note that we appy the problem to $\mathbf{b}=\mathbf{u}^{\trans}$.) Subset of Vectors Perpendicular to Two Vectors is a SubspaceLet $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]Prove that the subset $W$ is a subspace of […] The Centralizer of a Matrix is a SubspaceLet $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define\[W = \{ A \in V \mid AM = MA \}.\]The set $W$ here is called the centralizer of $M$ in $V$.Prove that $W$ is a subspace of $V$.Proof.First we check that the zero […] Prove that the Center of Matrices is a SubspaceLet $V$ be the vector space of $n \times n$ matrices with real coefficients, and define\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]The set $W$ is called the center of $V$.Prove that $W$ is a subspace […] Subspaces of Symmetric, Skew-Symmetric MatricesLet $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.(a) The set $S$ consisting of all $n\times n$ symmetric matrices.(b) The set $T$ consisting of […] Determine the Values of $a$ so that $W_a$ is a SubspaceFor what real values of $a$ is the set\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?Solution.The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]
Geometry and Topology Seminar Contents 1 Fall 2016 2 Spring 2017 3 Fall Abstracts 4 Spring Abstracts 5 Archive of past Geometry seminars Fall 2016 Spring 2017 date speaker title host(s) Jan 20 Carmen Rovi (University of Indiana Bloomington) "The mod 8 signature of a fiber bundle" Maxim Jan 27 Feb 3 Rafael Montezuma (University of Chicago) "TBA" Lu Wang Feb 10 Feb 17 Yair Hartman (Northwestern University) "TBA" Dymarz Feb 24 Lucas Ambrozio (University of Chicago) "TBA" Lu Wang March 3 Mark Powell (Université du Québec à Montréal) "TBA" Kjuchukova March 10 Autumn Kent (Wisconsin) Analytic functions from hyperbolic manifolds local March 17 March 24 Spring Break March 31 Xiangwen Zhang (University of California-Irvine) "TBA" Lu Wang April 7 April 14 April 21 Joseph Maher (CUNY) "TBA" Dymarz April 28 Bena Tshishiku (Harvard) "TBA" Dymarz Fall Abstracts Ronan Conlon New examples of gradient expanding K\"ahler-Ricci solitons A complete K\"ahler metric $g$ on a K\"ahler manifold $M$ is a \emph{gradient expanding K\"ahler-Ricci soliton} if there exists a smooth real-valued function $f:M\to\mathbb{R}$ with $\nabla^{g}f$ holomorphic such that $\operatorname{Ric}(g)-\operatorname{Hess}(f)+g=0$. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universit\'e Paris-Sud). Jiyuan Han Deformation theory of scalar-flat ALE Kahler surfaces We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kahler surfaces. This is used to prove that for any scalar-flat Kahler ALE surfaces, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of \C^2/\Gamma, where \Gamma is a finite subgroup of U(2) without complex reflections. This is a joint work with Jeff Viaclovsky. Sean Howe Representation stability and hypersurface sections We give stability results for the cohomology of natural local systems on spaces of smooth hypersurface sections as the degree goes to \infty. These results give new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. The stabilization occurs in point-counting and in the Grothendieck ring of Hodge structures, and we give explicit formulas for the limits using a probabilistic interpretation. These results have natural geometric analogs -- for example, we show that the "average" smooth hypersurface in \mathbb{P}^n is \mathbb{P}^{n-1}! Nan Li Quantitative estimates on the singular sets of Alexandrov spaces The definition of quantitative singular sets was initiated by Cheeger and Naber. They proved some volume estimates on such singular sets in non-collapsed manifolds with lower Ricci curvature bounds and their limit spaces. On the quantitative singular sets in Alexandrov spaces, we obtain stronger estimates in a collapsing fashion. We also show that the (k,\epsilon)-singular sets are k-rectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber. Yu Li In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem. A classification of diffeomorphism types is also given for all AE 3-manifolds with nonnegative scalar curvature. Peyman Morteza We develop a procedure to construct Einstein metrics by gluing the Calabi metric to an Einstein orbifold. We show that our gluing problem is obstructed and we calculate the obstruction explicitly. When our obstruction does not vanish, we obtain a non-existence result in the case that the base orbifold is compact. When our obstruction vanishes and the base orbifold is non-degenerate and asymptotically hyperbolic we prove an existence result. This is a joint work with Jeff Viaclovsky. Caglar Uyanik Geometry and dynamics of free group automorphisms A common theme in geometric group theory is to obtain structural results about infinite groups by analyzing their action on metric spaces. In this talk, I will focus on two geometrically significant groups; mapping class groups and outer automorphism groups of free groups.We will describe a particular instance of how the dynamics and geometry of their actions on various spaces provide deeper information about the groups. Bing Wang The extension problem of the mean curvature flow We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3. A key ingredient of the proof is to show a two-sided pseudo-locality property of the mean curvature flow, whenever the mean curvature is bounded. This is a joint work with Haozhao Li. Ben Weinkove Gauduchon metrics with prescribed volume form Every compact complex manifold admits a Gauduchon metric in each conformal class of Hermitian metrics. In 1984 Gauduchon conjectured that one can prescribe the volume form of such a metric. I will discuss the proof of this conjecture, which amounts to solving a nonlinear Monge-Ampere type equation. This is a joint work with Gabor Szekelyhidi and Valentino Tosatti. Jonathan Zhu Entropy and self-shrinkers of the mean curvature flow The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set. Yu Zeng Short time existence of the Calabi flow with rough initial data Calabi flow was introduced by Calabi back in 1950’s as a geometric flow approach to the existence of extremal metrics. Analytically it is a fourth order nonlinear parabolic equation on the Kaehler potentials which deforms the Kaehler potential along its scalar curvature. In this talk, we will show that the Calabi flow admits short time solution for any continuous initial Kaehler metric. This is a joint work with Weiyong He. Spring Abstracts Lucas Ambrozio "TBA" Rafael Montezuma "TBA" Carmen Rovi The mod 8 signature of a fiber bundle In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4k-dimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the Brown-Kervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect non-trivial signatures modulo 8 of surface bundles. Bena Tshishiku "TBA" Autumn Kent Analytic functions from hyperbolic manifolds At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps." These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known about their behavior. The ideas involved form a mix of geometry, algebra, and analysis. Xiangwen Zhang "TBA" Archive of past Geometry seminars 2015-2016: Geometry_and_Topology_Seminar_2015-2016 2014-2015: Geometry_and_Topology_Seminar_2014-2015 2013-2014: Geometry_and_Topology_Seminar_2013-2014 2012-2013: Geometry_and_Topology_Seminar_2012-2013 2011-2012: Geometry_and_Topology_Seminar_2011-2012 2010: Fall-2010-Geometry-Topology
I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is a much simpler setting where showing hardness should be easier. Please let me know if I should amend the original question instead. Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. We will restrict $G$ to be a cycle where every vertex has degree $2$ (and $m = n$). The challenge is to assign weights $w_e$ to each edge to maximize the objective function $E = \sum_{e = \{i,j\}} w_e \Pr[X_i + X_j \geq w_e]$ where $\Pr[X_i + X_j \geq w_e]$ denotes the probability that that the sum of values taken by vertex $i$ and $j$ is greater than $w_e$. The additional constraint is that the weights $w_e$ need to be sub-additive, i.e., for any two edges $e'$ and $e''$ that "cover" edge $e$ meaning $e'$ and $e''$ include the vertices that make $e$, it holds that $w_e \leq w_{e'} + w_{e''}$. Observe that the deterministic version where $p_i = 1$ is trivial. Any suggestions on possible directions for hardness or PTIME algorithm would be very helpful!
A variational approach to resonance for asymmetric oscillators 1. Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve $x'' + \alpha x^+ - \beta x^- + g(x) =p(t),$ where $x^+ =$ max{$x,0$} is the positive part of $x$, $x^- $ =max{$-x,0$} its negative part and $\alpha,\beta$ are positive parameters. We assume that $g :\mathbb R \to \mathbb R$ is continuous and bounded on $\mathbb R$, $p:\mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of $2\pi$-periodic solutions when $(\alpha,\beta)$ belongs to one of the curves of the Fučík spectrum corresponding to $2\pi$-periodic boundary conditions. Keywords:periodic solutions, Ahmad-Lazer-Paul type conditions., nonlinear resonance, Asymmetric oscillators. Mathematics Subject Classification:Primary: 34B15, 34C25; Secondary: 70K3. Citation:D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure & Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163 [1] [2] Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. [3] [4] Carlos Garca-Azpeitia, Jorge Ize. Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators. [5] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. [6] Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. [7] José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. [8] Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. [9] [10] [11] Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. [12] Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. [13] Jian Wu, Jiansheng Geng. Almost periodic solutions for a class of semilinear quantum harmonic oscillators. [14] Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. [15] Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. [16] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. [17] [18] Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. [19] Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. [20] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. 2018 Impact Factor: 0.925 Tools Metrics Other articles by authors [Back to Top]
Tagged: matrix Problem 38 Let $A$ be an $m \times n$ real matrix. Then the of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$. kernel The kernel is also called the of $A$. null space Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible. ( Stanford University Linear Algebra Exam) Problem 35 Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix. ( UCB-University of California, Berkeley, Exam) Problem 34 (a) Let \[A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}\] be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$. (Such a matrix is called (right) (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix).) stochastic matrix Then prove that the matrix $A$ has an eigenvalue $1$. (b) Find all the eigenvalues of the matrix \[B=\begin{bmatrix} 0.3 & 0.7\\ 0.6& 0.4 \end{bmatrix}.\] Add to solve later (c) For each eigenvalue of $B$, find the corresponding eigenvectors. Problem 33 Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space. ( MIT-Massachusetts Institute of Technology Exam) Problem 26 In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility. That is, we will prove that: (a)Show that if $A$ is invertible, then $A$ is nonsingular. (b)Let $A, B, C$ be $n\times n$ matrices such that $AB=C$. Prove that if either $A$ or $B$ is singular, then so is $C$. (c)Show that if $A$ is nonsingular, then $A$ is invertible. Add to solve later Problem 25 An $n \times n$ matrix $A$ is called if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$. nonsingular Otherwise $A$ is called . singular (a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular. (b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent. (c) Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$. Add to solve later Restriction Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible. Problem 19 Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$. (b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$. Add to solve later (c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$. Problem 12 Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.) Add to solve later (d) All the eigenvalues of $AA^{\trans}$ is non-negative. Problem 9 Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$. Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues. Read solution
Tips and Tricks for Modeling Induction Furnaces Today, we are pleased to introduce a new guest author, Vincent Bruyere of SIMTEC, who shares insight into the modeling of induction furnaces. Induction heating has become an important process in many applications, from cooking meals to manufacturing. It is valued for its precision and efficiency along with being a non-contact form of heating. In this guest post, I will describe how to build an induction furnace model in COMSOL Multiphysics and demonstrate how it can enhance your design. The Principles Behind Induction Heating The physical principles that govern the process of induction heating are quite simple: An alternating current flows in a solenoid (coil), which generates a transient magnetic field. Following Maxwell’s equations, this magnetic field induces electric currents (eddy currents) in nearby conductor materials. If the application is a furnace and due to the Joule effect, heat is generated and the melting point of the charge (metal) can be reached. By adjusting the current parameters, the molten metal can be maintained as a liquid or its solidification can be precisely controlled. Induction heating. (In the public domain, via Wikimedia Commons). Modeling with COMSOL Multiphysics When building the model, we begin by describing the geometry and the associated materials. As is often the case for such industrial applications, an axisymmetric assumption can be considered. The chosen geometry (shown in the figure below) is composed of the classical components of an induction furnace: the crucible that contains the charge (metal), a thermal screen that controls the heat radiation, and a water-cooled coil in which the electrical power is applied. The model’s geometry. By using the Induction Heating multiphysics interface, two physic interfaces — Magnetic Field and Heat Transfer in Solids — are automatically added to the component. The multiphysics couplings add the electromagnetic power dissipation as a heat source, while the electromagnetic material properties can depend on the temperature. A strong coupling is then ensured by applying the preselected study step, which can be the Frequency-Stationary or the Frequency-Transient study. In these cases, Ampere’s law is solved for each time step for a given frequency, and then the thermal problem is solved for a transient or stationary state. The Electromagnetic Problem Through considering the axisymmetric assumption, only the component of the magnetic vector potential that is perpendicular to the geometry plane (A\Phi) is non-zero. In order to apply boundary conditions, we can assume that a state of magnetic insulation is apparent quite a “far” distance from the furnace. It is important to ensure that this state of insulation is far enough away to guarantee that it does not affect the solution. An efficient technique is to use the Infinite Elements domain, available in the Definition item of the component. This method allows you to limit the size of the problem by applying a coordinate scaling to a layer of virtual domains surrounding the physical region of interest. The Infinite Elements domain. To apply the electromagnetic source, different methods are available. The selected method depends on the type of geometry and how well the electric properties are known. In our case, the geometry of the coil is truly represented (by four turns), and a Single-Turn Coil condition is thus added to these copper surfaces. Concerning our knowledge of the coil excitation, we consider a case where the coil power is known. To apply this quantity to the entire coil, the Coil Group mode has to be activated to ensure that the voltage used to compute the global coil power is the sum of the voltages of all the turns. By using this kind of excitation, the problem becomes nonlinear, and COMSOL Multiphysics automatically adds the related equations to compute the correct power (see here). The Thermal Problem The heat equation is solved for only the solid parts by neglecting the effect of the surrounding air. Indeed, heat is essentially transferred by radiation in this problem. Therefore, the Surface-to-Surface Radiation boundary condition is added to the Heat Transfer in Solids physics interface by selecting the external boundaries of each component. Surface-to-surface radiation boundaries. A circulation of water is classically used in industrial furnaces for cooling the coil. The design of the coil enables this channel flow by using a circular hollow section (see the geometry of the coil in the next figure). A convective volume loss term is then added to each turn by considering the mass flow rate \dot{m} and the heat capacity of water C_p, the inlet temperature of water T_{in}, and the internal radius of the coil r_{int}: Numerical Aspects For each computation, an important parameter has to be quantified: the skin depth, as most of the electric current will flow through this skin depth. This parameter is dependent on the vacuum permeability, \mu_0, the relative permeability of the material, \mu_R, the electrical conductivity, \sigma, and the frequency f by the following formula: The higher the frequency, the thinner the skin depth. Consequently, by modulating the current frequency, the location of the heat source can be precisely controlled. Numerically, it means that for each conductor material, the mesh has to be sufficiently fine to ensure precision. Convention requires that at least four elements cover this area. This can be easily achieved by using the Boundary Layers mesh type, as illustrated here: The Boundary Layers mesh type for the external coil boundaries. The figure also shows the inner tube where cooling water flows. The model can now be solved by specifying the frequency in the study step. In our case, a frequency of 1000 Hz is used and a stationary solution is obtained in less than a minute with a laptop. Electromagnetic and Thermal Results The magnitude (norm) of the resulting current density is plotted in the following figure, together with the magnetic field fluxlines. We can see that the maximum current density is located in the coil domains. The distribution of the current density is not uniform throughout the coil section and the current tends to flow in the inner part of the turns. In the charge (metal), the magnetic field fluxlines are highly deformed and an eddy current flowing in the opposite direction is induced. Global and local plots of the current density norm. As it flows into the resistive charge, this current dissipates energy in the material. The resulting temperature in each part of the furnace is highlighted below. We can observe that, even if the current is very intense in the coil, the temperature is close to ambient thanks to the water cooling system. On the contrary, the temperature in the charge is high and close to the melting point of the material due to eddy currents and the Joule effect. The other parts of the furnace are heated by radiation. A model illustrating the temperature distribution in the furnace. The furnace geometry can now be customized according to different design constraints. The coil characteristics (frequency, power, type of geometry, number of turns, etc.) and the geometry of all the parts can now be optimized to reduce the energy consumption and ensure a controlled melting of the material. How Does the Electromagnetic Field Affect the Melted Pool Behavior? To take things a step further and understand how the melted metal behaves in the furnace, hydrodynamics equations can easily be added to the model. By considering a homogeneous temperature of the melted pool, surface tension and buoyancy effects can be neglected, with only the Lorentz forces remaining. An additional source term is then added to the fluid momentum equation, with j representing the current density and B representing the magnetic flux density: Both vectors are complex entities and were previously obtained for a given frequency. The time-averaged Lorentz force contribution, given in COMSOL Multiphysics by the “mf.FLtzavr” and “mf.FLtzavz” parameters, must be used. Through neglecting the effect of the fluid on the magnetic field, the hydraulic problem can then be solved in an uncoupled way. The image below shows the melted metal behavior at a stationary state. Two typical recirculation zones are generated in the fluid. Stirring can be controlled by adapting the frequency or the power of the current. This can have both positive and negative effects. On the one hand, it is a way to improve the homogeneity of the bath. On the other hand, stirring may lead to a rapid erosion of the refractory walls. As for the heating phase, and depending on the design constraints, parametric studies can now be computed to improve the process. Velocity vectors in the melted pool. About the Guest Author Vincent Bruyere received his PhD in mechanical engineering from the National Institute of Applied Science (Lyon), with a research topic related to lubricated contacts. Following a post-doctoral position at the Atomic Energy and Alternative Energies Commission (CEA), Bruyere now works at SIMTEC as a modeling engineer. He develops numerical models applied predominantly to fluid dynamics as well as to thermal and electromagnetic applications. Comments (4) CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science TAGS CATEGORIES Chemical COMSOL Now Electrical Fluid General Interfacing Mechanical Today in Science
Answer The total mass of matter and antimatter required is $1.35\times 10^4~kg$ Work Step by Step We can find the kinetic energy of the starship: $K = (\gamma-1)mc^2$ $K = (\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1)mc^2$ $K = (\frac{1}{\sqrt{1-\frac{(0.3500~c)^2}{c^2}}}-1)(2.0\times 10^5~kg)(3.0\times 10^8~m/s)^2$ $K = (1.0675-1)(2.0\times 10^5~kg)(3.0\times 10^8~m/s)^2$ $K = 1.215\times 10^{21}~J$ We can find the total mass $M$ of matter and antimatter required to produce this amount of energy: $E = Mc^2$ $M = \frac{E}{c^2}$ $M = \frac{1.215\times 10^{21}~J}{(3.0\times 10^8~m/s)^2}$ $M = 1.35\times 10^4~kg$ The total mass of matter and antimatter required is $1.35\times 10^4~kg$.
In the last post I have presented some historical context about programming and mathematical methodology. If you read it, then you should have an idea when and why programmers started to investigate on mathematical methodology. However, I haven’t mentioned any aspects of mathematical methodology that can help us to improve our programming or mathematical skills. In this post, I’ll talk about mathematical proofs. And what’s the relevance of this topic to programmers? Well, computer programs are mathematical formulae, with a precise formal meaning and embodying constructive theorems about the systems they implement (as well-written in “Mathematics and Programming – A Revolution in the Art of Effective Reasoning”, by Roland Backhouse). The difference between theorems embodied by computer programs and the ones usually studied in mathematics is that they are applied by an unforgiving machine, with the effect that the smallest error can cause a huge damage. This means that computer programmers must create trustworth designs, i.e., the constructive theorems embodied by their programs must be programmed correctly. Mathematical Proofs Mathematicians job is to do mathematics, i.e., to design and present theorems, arguments, algorithms and in some cases whole theories. However, the traditional mathematical curriculum is more concerned with teaching mathematical facts — existing theories and concepts — than with the doing of mathematics. And even when design and presentation get some attention, they are treated separately: design of solutions is viewed as a psychological issue, while presentation is viewed as a matter of personal style (words from this Dijkstra’s note on Mathematical Methodology). A proof of a theorem should demonstrate, using certain facts (also known as axioms) or previously proved theorems, why it is true. Additionally, a good proof should explain clearly how the facts are combined and it should express the design considerations so that readers can understand it better, explain it to others and prove other theorems in a similar fashion. Now, look at the following conventional proof of $A{\cup}(B{\cap}C) = (A{\cup}B){\cap}(A{\cup}C)$, which is similar to the majority of proofs we can find in math textbooks (the proof is actually from a math textbook, but it was extracted from [GS95]). We first show that $A{\cup}(B{\cap}C){\subseteq}(A{\cup}B){\cap}(A{\cup}C)$. If $x{\in}(A{\cup}(B{\cap}C))$ , then either $x{\in}A$ or $x{\in}(B{\cap}C)$. If $x{\in}A$, then certainly $x{\in}(A{\cup}B)$ and $x{\in}(A{\cup}C)$, so $x{\in}((A{\cup}B){\cap}(A{\cup}C))$. On the other hand, if $x{\in}(B{\cap}C)$, then $x{\in}B$ and $x{\in}C$, so $x{\in}(A{\cup}B)$ and $x{\in}(A{\cup}C)$, so $x{\in}((A{\cup}B){\cap}(A{\cup}C))$. Hence, $A{\cup}(B{\cap}C) {\subseteq} (A{\cup}B){\cap}(A{\cup}C)$. Conversely, if $y{\in}((A{\cup}B){\cap}(A{\cup}C))$, then $y{\in}(A{\cup}B)$ and $y{\in}(A{\cup}C)$. We consider two cases: $y{\in}A$ and $y{\notin}A$. If $y{\in}A$, then $y{\in}(A{\cup}(B{\cap}C))$, and this part is done. If $y{\notin}A$, then, since $y{\in}(A{\cup}B)$ we must have $y{\in}B$. Similarly, since $y{\in}(A{\cup}C)$ and $y{\notin}A$, we have $y{\in}C$. Thus, $y{\in}(B{\cap}C)$, and this implies $y{\in}(A{\cup}(B{\cap}C))$. Hence $(A{\cup}B){\cap}(A{\cup}C) {\subseteq} A{\cup}(B{\cap}C)$. The theorem follows. Proofs are usually like this one: written using natural language, which, by nature, admits ambiguity. These proofs, also called informal proofs, place a large burden on the reader since it is difficult to see precisely how facts interact. In order to achieve the crispness we aspire to, we must deviate from informal proofs. That is why most of our mathematical reasoning is carried out in what is known as calculational method. This method reduces proof obligations to targets to be reached by formula manipulation. Although there are some criticisms and objections to the method, the experience shows that arguments become more clear and systematic, since we are (usually) restricting ourselves to a very modest repertoire and to simple syntactic manipulations. Besides, with the development of the method, some heuristics have been created, allowing the writer to prefer certain decisions (almost according with the principle “There is really only one thing one can do.”). Proof Format Note that calculational arguments usually start with a boolean expression that, by value-preserving transformations, is massaged according to our needs (we can evaluate it to true or false, or we can transform it into an equivalent boolean expression). So, if we want to evaluate a boolean expression A to true and it takes 3 steps, then, for general expressions B and C, we would have: the first step would establish A is B ; the second step would establish B is C; the third step would establish C is true . This means that we would have the intermediate expressions B and C repeated twice. In general, B and C can be long expressions, so we need a proof format that allows us to omit intermediate expressions. We use Wim Feijen’s proof format (described in detail in EWD999: Our Proof Format, Predicate Calculus and Program Semantics and in Chapter 3 of Program Construction: Calculating Implementations from Specifications), which for this small example would render: \[ \beginproof \pexp{A} \hint{=}{hint why $A{\equiv}B$} \pexp{B} \hint{=}{hint why $B{\equiv}C$} \pexp{C} \hint{=}{hint why $C{\equiv}true$} \pexp{true ~~.} \endproof \] The advantages of this format are more than just brevity. It allows us to conclude immediately that A is true without reading the intermediate expressions and we also can use any transitive relation between the steps. Also, the use of a systematic proof format allows us to compare two different proofs of the same theorem more effectively. Now, look how the proof presented above in an informal style can be rewritten in a calculational style: Below, we prove $v{\in}(A{\cup}(B{\cap}C)) {\equiv} v{\in}((A{\cup}B){\cap}(A{\cup}C))$. By Extensionality (the definition of equality of sets), we then conclude $A{\cup}(B{\cap}C) {=} (A{\cup}B){\cap}(A{\cup}C)$. \[ \beginproof \pexp{v{\in}(A{\cup}(B{\cap}C))} \hint{=}{Definition of $\cup$} \pexp{v{\in}A {~\vee~} v{\in}(B{\cap}C)} \hint{=}{Definition of $\cap$} \pexp{v{\in}A {~\vee~} (v{\in}B \wedge v{\in}C)} \hint{=}{Distributivity of $\vee$ over $\wedge$} \pexp{(v{\in}A {\,\vee\,} v{\in}B) \wedge (v{\in}A {\,\vee\,} v{\in}C)} \hint{=}{Definition of $\cup$, twice} \pexp{(v{\in}(A {\cup} B)) \wedge (v{\in} (A {\cup} C))} \hint{=}{Definition of $\cap$} \pexp{v{\in}((A{\cup}B){\cap}(A{\cup}C)) ~~.} \endproof \] In this second version, the proof format makes all the ingredients much more evident; every step is well identified and we can see it uses a common strategy: it eliminates the union and disjunction operators using their definition, performs some manipulation and reintroduces the operators. Also, this second argument eliminates the “if” and “only if” argument. Final notes To conclude I’d just like to quote a paragraph from Roland’s Backhouse article I’ve mentioned above: “The vast complexity of computer software makes its unreliability understandable, but not excusable. The implementation of reliable computer systems, carrying a guarantee of fitness-for-purpose, imposes major intellectual challenges that can only be met by a science of computing whose hallmark is the avoidance of error.”
SMC samplers¶ SMC samplers are SMC algorithms that sample from a sequence of target distributions. In this tutorial, these target distributions will be Bayesian posterior distributions of static models. SMC samplers are covered in Chapter 17 of the book. Defining a static model¶ A static model is a Python object that represents a Bayesian model with static parameter $\theta$. One may define a static model by subclassing base class StaticModel, and defining method logpyt, which evaluates the log-likelihood of datapoint $Y_t$ (given $\theta$ and past datapoints $Y_{0:t-1}$). Here is a simple example: [1]: %matplotlib inlineimport warnings; warnings.simplefilter('ignore') # hide warningsfrom matplotlib import pyplot as pltimport seaborn as sbfrom scipy import statsimport particlesfrom particles import smc_samplers as sspfrom particles import distributions as distsclass ToyModel(ssp.StaticModel): def logpyt(self, theta, t): # density of Y_t given theta and Y_{0:t-1} return stats.norm.logpdf(self.data[t], loc=theta['mu'], scale = theta['sigma']) In words, we are considering a model where the observations are $Y_t\sim N(\mu, \sigma^2)$. The parameter is $\theta=(\mu, \sigma)$. Class ToyModel contains information about the likelihood of the considered model, but not about its prior, or the considered data. First, let’s define those: [2]: T = 1000my_data = stats.norm.rvs(loc=3.14, size=T) # simulated datamy_prior = dists.StructDist({'mu': dists.Normal(scale=10.), 'sigma': dists.Gamma()}) For more details about to define prior distributions, see the documentation of module distributions, or the previous tutorial on Bayesian estimation of state-space models. Now that we have everything, let’s specify our static model: [3]: my_static_model = ToyModel(data=my_data, prior=my_prior) This time, object my_static_model has enough information to define the posterior distribution(s) of the model (given all data, or part of the data). In fact, it inherits from StaticModel method logpost, which evaluates (for a collection of $\theta$ values) the posterior log-density at any time $t$ (meaning given data $y_{0:t}$). [4]: thetas = my_prior.rvs(size=5)my_static_model.logpost(thetas, t=2) # if t is omitted, gives the full posterior [4]: array([ -103.81939366, -8.74342665, -1574.47064329, -33.24706434, -89.17306505]) The input of logpost (and output of myprior.rvs()) is a structured array, with the same keys as the prior distribution: [5]: thetas['mu'][0] [5]: 4.5636339239154635 Typically, you won’t need to call logpost yourself, this will be done by the SMC sampler for you. IBIS¶ The IBIS (iterated batch importance sampling) algorithm is a SMC sampler that samples iteratively from a sequence of posterior distributions, $p(\theta\|y_{0:t})$, for $t=0,1,\ldots$. Module smc_samplers defines IBIS as a subclass of FeynmanKac. [6]: my_ibis = ssp.IBIS(my_static_model)my_alg = particles.SMC(fk=my_ibis, N=1000, store_history=True)my_alg.run() Since we set store_history to True, the particles and their weights have been saved at every time (in attribute hist, see previous tutorials on smoothing). Let’s plot the posterior distributions of $\mu$ and $\sigma$ at various times. [7]: plt.style.use('ggplot')for i, p in enumerate(['mu', 'sigma']): plt.subplot(1, 2, i + 1) for t in [100, 300, 900]: plt.hist(my_alg.hist.X[t].theta[p], weights=my_alg.hist.wgt[t].W, label="t=%i" % t, alpha=0.5, density=True) plt.xlabel(p)plt.legend(); As expected, the posterior distribution concentrates progressively around the true values. As before, once the algorithm is run, my_smc.X contains the N final particles. However, object my_smc.X is no longer a simple (N,) or (N,d) numpy array. It is a ThetaParticles object, with attributes: theta: a structured array: as mentioned above, this is an array with fields; i.e. my_smc.X.theta['mu']is a (N,) array that contains the the $\mu-$component of the $N$ particles; lpost: a (N,) numpy array that contains the target (posterior) log-density of each of the N particles; acc_rates: a list of the acceptance rates of the resample-move steps. [8]: print(["%2.2f%%" % (100 * np.mean(r)) for r in my_alg.X.acc_rates])plt.hist(my_alg.X.lpost, 30); ['23.33%', '12.70%', '26.07%', '31.82%', '32.75%', '31.90%', '35.50%', '35.98%', '33.75%', '35.28%'] You do not need to know much more about class ThetaParticles for most practical purposes (see however the documention of module smc_samplers if you do want to know more, e.g. in order to implement other classes of SMC samplers). Regarding the Metropolis steps¶ As the text output of my_alg.run() suggests, the algorithm “resample-moves” whenever the ESS is below a certain threshold ($N/2$ by default). When this occurs, particles are resampled, and then moved through a certain number of Metropolis-Hastings steps. By default, the proposal is a Gaussian random walk, and both the number of steps and the covariance matrix of the random walk are chosen automatically as follows: the covariance matrix of the random walk is set to scaletimes the empirical (weighted) covariance matrix of the particles. The default value for scaleis $2.38 / \sqrt{d}$, where $d$ is the dimension of $\theta$. the algorithm performs Metropolis steps until the relative increase of the average distance between the starting point and the end point is below a certain threshold $\delta$. Class IBIS takes as an optional argument mh_options, a dictionary which may contain the following (key, values) pairs: 'type_prop': either 'random walk'or 'independent’; in the latter case, an independent Gaussian proposal is used. The mean of the Gaussian is set to the weighted mean of the particles. The variance is set to scaletimes the weighted variance of the particles. 'scale’: the scale of the proposal (as explained above). 'nsteps': number of steps. If set to 0, the adaptive strategy described above is used. Let’s illustrate all this by calling IBIS again: [9]: alt_ibis = ssp.IBIS(my_static_model, mh_options={'type_prop': 'independent', 'nsteps': 10})alt_alg = particles.SMC(fk=alt_ibis, N=1000, ESSrmin=0.2)alt_alg.run() Well, apparently the algorithm did what we asked. We have also changed the threshold of Let’s see how the ESS evolved: [10]: plt.plot(alt_alg.summaries.ESSs)plt.xlabel('t')plt.ylabel('ESS') [10]: Text(0, 0.5, 'ESS') As expected, the algorithm waits until the ESS is below 200 to trigger a resample-move step. SMC tempering¶ SMC tempering is a SMC sampler that samples iteratively from the following sequence of distributions: with $0=\gamma_0 < \ldots < \gamma_T = 1$. In words, this sequence is a geometric bridge, which interpolates between the prior and the posterior. SMC tempering implemented in the same was as IBIS: as a sub-class of FeynmanKac, whose __init__ function takes as argument a StaticModel object. [11]: fk_tempering = ssp.AdaptiveTempering(my_static_model)my_temp_alg = particles.SMC(fk=fk_tempering, N=1000, ESSrmin=1., verbose=True)my_temp_alg.run() t=0, ESS=500.00, tempering exponent=2.94e-05t=1, Metropolis acc. rate (over 6 steps): 0.275, ESS=500.00, tempering exponent=0.000325t=2, Metropolis acc. rate (over 6 steps): 0.261, ESS=500.00, tempering exponent=0.0018t=3, Metropolis acc. rate (over 6 steps): 0.253, ESS=500.00, tempering exponent=0.0061t=4, Metropolis acc. rate (over 6 steps): 0.287, ESS=500.00, tempering exponent=0.0193t=5, Metropolis acc. rate (over 6 steps): 0.338, ESS=500.00, tempering exponent=0.0636t=6, Metropolis acc. rate (over 6 steps): 0.347, ESS=500.00, tempering exponent=0.218t=7, Metropolis acc. rate (over 5 steps): 0.358, ESS=500.00, tempering exponent=0.765t=8, Metropolis acc. rate (over 5 steps): 0.366, ESS=941.43, tempering exponent=1 Note: Recall that SMC resamples every time the ESS drops below value N times option ESSrmin; here we set it to to 1, since we want to resample at every time. This makes sense: Adaptive SMC chooses adaptively the successive values of $\gamma_t$ so that the ESS drops to $N/2$ (by default). Note: we use option verbose=True in SMC in order to print some information on the intermediate distributions. We have not saved the intermediate results this time (option store_history was not set) since they are not particularly interesting. Let’s look at the final results: [12]: for i, p in enumerate(['mu', 'sigma']): plt.subplot(1, 2, i + 1) sb.distplot(my_temp_alg.X.theta[p]) plt.xlabel(p) This looks reasonable! You can see from the output that the algorithm automatically chooses the tempering exponents $\gamma_1, \gamma_2,\ldots$. In fact, at iteration $t$, the next value for $\gamma$ is set that the ESS drops at most to $N/2$. You can change this particular threshold by passing argument ESSrmin to TemperingSMC. (Warning: do not mistake this with the ESSrmin argument of class SMC): [13]: lazy_tempering = ssp.AdaptiveTempering(my_static_model, ESSrmin = 0.1)lazy_alg = particles.SMC(fk=lazy_tempering, N=1000, verbose=True)lazy_alg.run() t=0, ESS=100.00, tempering exponent=0.00097t=1, Metropolis acc. rate (over 5 steps): 0.233, ESS=100.00, tempering exponent=0.0217t=2, Metropolis acc. rate (over 6 steps): 0.323, ESS=100.00, tempering exponent=0.315t=3, Metropolis acc. rate (over 5 steps): 0.338, ESS=520.51, tempering exponent=1 The algorithm progresses faster this time, but the ESS drops more between each step. Another optional argument for Class TemperingSMC is options_mh, which works exactly as for IBIS, see above. That is, by default, the particles are moved according to a certain (adaptative) number of random walk steps, with a variance calibrated to the particle variance.
A 60 degrees Rhombus in a 60 degrees Isosceles Trapezoid What is this about? A Mathematical Droodle Created with GeoGebra, 24 November, 2016 Problem Proof WLOG, assume $AB=BC=AD=1\;$ and denote $AX=x.\;$ Since $\Delta ABC\;$ is isosceles with base angles of $30^{\circ},\;$ so is $\Delta AXM,\;$ implying $XM=AX=x\;$ and, by the construction, $BY=x.\;$ By the Law of Cosines, $\begin{align} XY^2 &= BX^2+BY^2-2BX\cdot BY\cos\angle XBY\\ &= (1-x)^2+x^2 + x(1-x)\\ &=x^2-x+1. \end{align}$ Let $T'\in AD\;$ be such that $DT=x.\;$ Then, as before, $T'X^2=x^2-x+1.\;$ in $\Delta DTZ,\;$ we have $\begin{align} T'Z^2 &= DT'^2+DZ^2-2DT'\cdot DZ\cos\angle TDZ\\ &= x^2+1^2 - x. \end{align}$ Similarly, $YZ=x^2-x+1,\;$ such that $T'Z=TX=XY=YZ,\;$ showing that $XYZT'\;$ is a rhombus. In particular, $T'X\parallel YZ\;$ and $T'Z\parallel XY,\;$ making $T'=T.\;$ Now, in $\Delta AXZ,\;$ $AX=x=DT,\;$ $AZ=DZ=1,\;$ and $\angle XAZ=60^{\circ}=\angle TDZ.\;$ It follows that $\Delta AXZ=\Delta TDZ\;$ and, therefore, $XZ=TZ.\;$ This shows that $\Delta TXZ\;$ is equilateral and $\angle XYZ=\angle XTZ=60^{\circ}.$ Acknowledgment Tran Quang Hung has kindly posted at the CutTheKnotMath facebook page a construction of the Golden Ratio in an isosceles trapezoid with an angle of $60^{\circ}.\;$ His construction, in part, reveals an inscribed rhombus with a $60^{\circ}\;$ angle. The above is a generalization that I think should be treated separately. The proof is a simple application of the Law of Cosines. 65461406
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient [1] R(M, x), is defined as: [2] [3] R(M,x) := {x^{} M x \over x^{} x}. For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x^{} to the usual transpose x'. Note that R(M, c x) = R(M,x) for any non-zero real scalar c. Recall that a Hermitian (or real symmetric) matrix has real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value \lambda_\min (the smallest eigenvalue of M) when x is v_\min (the corresponding eigenvector). Similarly, R(M, x) \leq \lambda_\max and R(M, v_\max) = \lambda_\max. The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration. The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range, (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, \lambda_\max is known as the spectral radius. In the context of C-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh-Ritz quotient R(M,x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra. Contents Bounds for Hermitian M 1 Special case of covariance matrices 2 Formulation using Lagrange multipliers 2.1 Use in Sturm–Liouville theory 3 Generalizations 4 See also 5 References 6 Further reading 7 Bounds for Hermitian M As stated in the introduction, it is R(M,x) \in \left[\lambda_\min, \lambda_\max \right]. This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: R(M,x) = {x^{} M x \over x^{} x} = \frac{\sum_{i=1}^n \lambda_i y_i^2}{\sum_{i=1}^n y_i^2} where (\lambda_i, v_i) is the ith eigenpair after orthonormalization and y_i = v_i^* x is the ith coordinate of x in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors v_\min, v_\max. The fact that the quotient is a weighted average of the eigenvalues can be used to identify the second, the third, ... largest eigenvalues. Let \lambda_{max} = \lambda_1 \ge \lambda_2 \ge ... \ge \lambda_n = \lambda_{min} be the eigenvalues in decreasing order. If x is constrained to be orthogonal to v_1, in which case y_1 = v_1^x = 0 , then R(M,x) has the maximum \lambda_2, which is achieved when x = v_2. Special case of covariance matrices An empirical covariance matrix M can be represented as the product A' A of the data matrix A pre-multiplied by its transpose A'. Being a positive semi-definite matrix, M has non-negative eigenvalues, and orthogonal (or othogonalisable) eigenvectors, which can be demonstrated as follows. Firstly, that the eigenvalues \lambda_i are non-negative: M v_i = A' A v_i = \lambda_i v_i \Rightarrow v_i' A' A v_i = v_i' \lambda_i v_i \Rightarrow \left\\| A v_i \right\\|^2 = \lambda_i \left\\| v_i \right\\|^2 \Rightarrow \lambda_i = \frac{\left\\| A v_i \right\\|^2}{\left\\| v_i \right\\|^2} \geq 0. Secondly, that the eigenvectors v are orthogonal to one another: i \begin{align} &\qquad \qquad M v_i = \lambda _i v_i \\ &\Rightarrow v_j' M v_i = \lambda _i v_j' v_i \\ &\Rightarrow \left (M v_j \right )' v_i = \lambda _j v_j' v_i \\ &\Rightarrow \lambda_j v_j ' v_i = \lambda _i v_j' v_i \\ &\Rightarrow \left (\lambda_j - \lambda_i \right ) v_j ' v_i = 0 \\ &\Rightarrow v_j ' v_i = 0 \end{align} If the eigenvalues are different – in the case of multiplicity, the basis can be orthogonalized. To now establish that the Rayleigh quotient is maximised by the eigenvector with the largest eigenvalue, consider decomposing an arbitrary vector x on the basis of the eigenvectors v: i x = \sum _{i=1} ^n \alpha _i v_i, where \alpha_i = \frac{x'v_i}{v_i'v_i} = \frac{\langle x,v_i\rangle}{\left\\| v_i \right\\| ^2} is the coordinate of x orthogonally projected onto v. Therefore we have: i R(M,x) = \frac{x' A' A x}{x' x} = \frac{ \left (\sum _{j=1} ^n \alpha _j v_j \right )' \left ( A' A \right ) \left (\sum _{i=1} ^n \alpha _i v_i \right )}{ \left (\sum _{j=1} ^n \alpha _j v_j \right )' \left (\sum _{i=1} ^n \alpha _i v_i \right )} which, by orthogonality of the eigenvectors, becomes: R(M,x) = \frac{\sum _{i=1} ^n \alpha_i^2 \lambda _i}{\sum _{i=1} ^n \alpha_i^2} = \sum_{i=1}^n \lambda_i \frac{(x'v_i)^2}{ (x'x)( v_i' v_i)} The last representation establishes that the Rayleigh quotient is the sum of the squared cosines of the angles formed by the vector x and each eigenvector v, weighted by corresponding eigenvalues. i If a vector x maximizes R(M,x), then any non-zero scalar multiple kx also maximizes R, so the problem can be reduced to the Lagrange problem of maximizing \sum _{i=1}^n \alpha_i^2 \lambda _i under the constraint that \sum _{i=1} ^n \alpha _i ^2 = 1. Define: β = i α2 i. This then becomes a linear program, which always attains its maximum at one of the corners of the domain. A maximum point will have \alpha_1 = \pm 1 and \alpha _i = 0 for all i > 1 (when the eigenvalues are ordered by decreasing magnitude). Thus, as advertised, the Rayleigh quotient is maximised by the eigenvector with the largest eigenvalue. Formulation using Lagrange multipliers Alternatively, this result can be arrived at by the method of Lagrange multipliers. The problem is to find the critical points of the function R(M,x) = x^T M x , subject to the constraint \\|x\\|^2 = x^Tx = 1. I.e. to find the critical points of \mathcal{L}(x) = x^T M x -\lambda \left (x^Tx - 1 \right), where λ is a Lagrange multiplier. The stationary points of \mathcal{L}(x) occur at \frac{d\mathcal{L}(x)}{dx} = 0 \therefore 2x^T M^T - 2\lambda x^T = 0 \therefore M x = \lambda x and R(M,x) = \frac{x^T M x}{x^T x} = \lambda \frac{x^Tx}{x^T x} = \lambda. Therefore, the eigenvectors x_1, \cdots, x_n of M are the critical points of the Rayleigh Quotient and their corresponding eigenvalues \lambda_1, \cdots, \lambda_n are the stationary values of R. This property is the basis for principal components analysis and canonical correlation. Use in Sturm–Liouville theory Sturm–Liouville theory concerns the action of the linear operator L(y) = \frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + q(x)y\right) on the inner product space defined by \langle{y_1,y_2}\rangle = \int_a^b w(x)y_1(x)y_2(x) \, dx of functions satisfying some specified boundary conditions at a and b. In this case the Rayleigh quotient is \frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle} = \frac{\int_a^b y(x)\left(-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + q(x)y(x)\right)dx}{\int_a^b{w(x)y(x)^2}dx}. This is sometimes presented in an equivalent form, obtained by separating the integral in the numerator and using integration by parts: \begin{align} \frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle} &= \frac{ \left \{ \int_a^b y(x)\left(-\frac{d}{dx}\left[p(x)y'(x)\right]\right) dx \right \}+ \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b{w(x)y(x)^2} \, dx} \\ &= \frac{ \left \{\left. -y(x)\left[p(x)y'(x)\right] \right \|_a^b \right \} + \left \{\int_a^b y'(x)\left[p(x)y'(x)\right] \, dx \right \} + \left \{\int_a^b{q(x)y(x)^2} \, dx \right \}}{\int_a^b w(x)y(x)^2 \, dx}\\ &= \frac{ \left \{ \left. -p(x)y(x)y'(x) \right \|_a^b \right \} + \left \{ \int_a^b \left [p(x)y'(x)^2 + q(x)y(x)^2 \right] \, dx \right \} } {\int_a^b{w(x)y(x)^2} \, dx}. \end{align} Generalizations For a given pair ( A, B) of matrices, and a given non-zero vector x, the generalized Rayleigh quotient is defined as: R(A,B; x) := \frac{x^ A x}{x^* B x}. The Generalized Rayleigh Quotient can be reduced to the Rayleigh Quotient R(D, C^x) through the transformation D = C^{-1} A {C^}^{-1} where CC^* is the Cholesky decomposition of the Hermitian positive-definite matrix B. For a given pair ( x, y) of non-zero vectors, and a given Hermitian matrix H, the generalized Rayleigh quotient can be defined as: R(H; x,y) := \frac{y^* H x}\sqrt{y^y \cdot x^x} which coincides with R(H,x) when x= y. See also References ^ Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh. ^ Horn, R. A. and C. A. Johnson. 1985. Matrix Analysis. Cambridge University Press. pp. 176–180. ^ Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998 Further reading Shi Yu, Léon-Charles Tranchevent, Bart Moor, Yves Moreau, Kernel-based Data Fusion for Machine Learning: Methods and Applications in Bioinformatics and Text Mining, Ch. 2, Springer, 2011. This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002. Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles. By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as$$ u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L$$and the quotient monoid $X^{\ast} / \sim_L$ is called the syntactic monoid of $L$. Now what monoids arise as syntactic monoids of languages? I found languages for symmetric groups and for the set of all mappings on some underlying finite set. But what about other, are there finite monoids that could not be written as the syntactic monoid of some language? For a given automaton, by considering the monoid generated by the mappings induced by the letters on the states (the so called transformation monoid) when function composition is read from left to right, it holds that the transformation monoid of the minimal automaton is precisely the syntactic monoid. This observation helped me in constructing the above mentioned examples. Let me also not that it is quite simple to realise any finite monoid $M$ as the transformation monoid of some automaton, simply take the elements of $M$ as the states, and consider every generator of $M$ as a letter of the alphabet and the transitions are given by $qx$ for some state $q$ and letter $x$, then the transformation monoid is isomorphic to $M$ itself (this is similar to the Cayley theorem about how groups embed into symmetric groups).
Tagged: matrix Problem 121 Let $A$ be an $m \times n$ real matrix. Then the null space $\calN(A)$ of $A$ is defined by \[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\] That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$. Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$. (Note that the null space is also called the kernel of $A$.) Read solution Problem 115 Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2= \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{v}_3= \begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}.\] ( The Ohio State University, Linear Algebra Exam) Problem 104 Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don’t lose a point if any of these is your exam problems. (These are actual exam problems at the Ohio State University.) You can take the quiz as many times as you like. The solutions will be given after completing all the 10 problems. Click the View question button to see the solutions. Problem 102 Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a)\[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$. (b)$A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix. (c)A homogeneous system of $3$ equations in $4$ unknowns. (d)$A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A\|\mathbf{b}]$ looks as follows: \[\begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 &1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.\] ( The Ohio State University, Linear Algebra Exam) Read solution Add to solve later Problem 98 Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a counterexample. Problem 85 Consider a polynomial \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Define the matrix \[A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}.\] Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$. The matrix is called the of the polynomial $p(x)$. companion matrix Add to solve later Problem 79 Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero. That is, \begin{equation} V:=\left\{ A=\begin{bmatrix} a_{11} & 0 & \dots & 0 \\ 0 &a_{22} & \dots & 0 \\ 0 & 0 & \ddots & \vdots \\ 0 & 0 & \dots & a_{nn} \end{bmatrix} \quad \middle\| \quad \begin{array}{l} a_{11}, \dots, a_{nn} \in \C,\\ \tr(A)=0 \\ \end{array} \right\} \end{equation} Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere. (a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.) (b) Show that matrices \[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\] are a basis for the vector space $V$. Add to solve later (c) Find the dimension of $V$. Read solution
Proving (-infinity, 0) is open Hi, I've done most of the proof(I think), but I'm lost anyone willing to finish it off for me? I've spent a while on it and i'm stuck. Let x be an element of (-infinity, 0) this implies x < 0 chose an r such that abs(r) < abs(x) let y be an element of (x -r, x+r). This implies x-r<y<x+r which implies 0 - x + x > x - r < y now what exactly do I do to show y < 0 which would be the end goal, right? Re: Proving (-infinity, 0) is open If $\displaystyle x\in (-\infty,0)$ then $\displaystyle x\in (2x,0)\subset (-\infty,0) $. $\displaystyle (2x,0)$ is an open set containing $\displaystyle x$. Quote: Originally Posted by glambeth Does that work? Re: Proving (-infinity, 0) is open Hm that makes sense, but how would I be able to get to that stage? Re: Proving (-infinity, 0) is open What you did was perfectly good- the "r-neighborhood" of x, (x-r, x+ r) is a subset of $\displaystyle (-\infty, 0)$ so x is an interior point of $\displaystyle (-\infty, 0)$. Since x could be any point in the set, the set is open. Re: Proving (-infinity, 0) is open Hm, I think my part is wrong though. Looking at it i say choose an r such that abs(r) < abs(x) but then i say x - r < y < x + r so i end up saying 0 < y < 2x but isn't this false? Given y should be less than 0? Re: Proving (-infinity, 0) is open If $\displaystyle \|r\|<\|x\|$, it is possible that $\displaystyle r<0$, which you don't want. You want $\displaystyle 0<r<\|x\|$. Now, $\displaystyle (x-r,x+r)$ is correct. This is because $\displaystyle x<0$, so $\displaystyle x-r<0$. Since $\displaystyle 0<r<\|x\|$, $\displaystyle x+r = -\|x\|+r<0$. So, you have both endpoints less than zero.
2018-09-11 04:29 Proprieties of FBK UFSDs after neutron and proton irradiation up to $6*10^{15}$ neq/cm$^2$ / Mazza, S.M. (UC, Santa Cruz, Inst. Part. Phys.) ; Estrada, E. (UC, Santa Cruz, Inst. Part. Phys.) ; Galloway, Z. (UC, Santa Cruz, Inst. Part. Phys.) ; Gee, C. (UC, Santa Cruz, Inst. Part. Phys.) ; Goto, A. (UC, Santa Cruz, Inst. Part. Phys.) ; Luce, Z. (UC, Santa Cruz, Inst. Part. Phys.) ; McKinney-Martinez, F. (UC, Santa Cruz, Inst. Part. Phys.) ; Rodriguez, R. (UC, Santa Cruz, Inst. Part. Phys.) ; Sadrozinski, H.F.-W. (UC, Santa Cruz, Inst. Part. Phys.) ; Seiden, A. (UC, Santa Cruz, Inst. Part. Phys.) et al. The properties of 60-{\mu}m thick Ultra-Fast Silicon Detectors (UFSD) detectors manufactured by Fondazione Bruno Kessler (FBK), Trento (Italy) were tested before and after irradiation with minimum ionizing particles (MIPs) from a 90Sr \b{eta}-source . [...] arXiv:1804.05449. - 13 p. Preprint - Full text Detaljnije - Slični zapisi 2018-08-25 06:58 Charge-collection efficiency of heavily irradiated silicon diodes operated with an increased free-carrier concentration and under forward bias / Mandić, I (Ljubljana U. ; Stefan Inst., Ljubljana) ; Cindro, V (Ljubljana U. ; Stefan Inst., Ljubljana) ; Kramberger, G (Ljubljana U. ; Stefan Inst., Ljubljana) ; Mikuž, M (Ljubljana U. ; Stefan Inst., Ljubljana) ; Zavrtanik, M (Ljubljana U. ; Stefan Inst., Ljubljana) The charge-collection efficiency of Si pad diodes irradiated with neutrons up to $8 \times 10^{15} \ \rm{n} \ cm^{-2}$ was measured using a $^{90}$Sr source at temperatures from -180 to -30°C. The measurements were made with diodes under forward and reverse bias. [...] 2004 - 12 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 533 (2004) 442-453 Detaljnije - Slični zapisi 2018-08-23 11:31 Detaljnije - Slični zapisi 2018-08-23 11:31 Effect of electron injection on defect reactions in irradiated silicon containing boron, carbon, and oxygen / Makarenko, L F (Belarus State U.) ; Lastovskii, S B (Minsk, Inst. Phys.) ; Yakushevich, H S (Minsk, Inst. Phys.) ; Moll, M (CERN) ; Pintilie, I (Bucharest, Nat. Inst. Mat. Sci.) Comparative studies employing Deep Level Transient Spectroscopy and C-V measurements have been performed on recombination-enhanced reactions between defects of interstitial type in boron doped silicon diodes irradiated with alpha-particles. It has been shown that self-interstitial related defects which are immobile even at room temperatures can be activated by very low forward currents at liquid nitrogen temperatures. [...] 2018 - 7 p. - Published in : J. Appl. Phys. 123 (2018) 161576 Detaljnije - Slični zapisi 2018-08-23 11:31 Detaljnije - Slični zapisi 2018-08-23 11:31 Characterization of magnetic Czochralski silicon radiation detectors / Pellegrini, G (Barcelona, Inst. Microelectron.) ; Rafí, J M (Barcelona, Inst. Microelectron.) ; Ullán, M (Barcelona, Inst. Microelectron.) ; Lozano, M (Barcelona, Inst. Microelectron.) ; Fleta, C (Barcelona, Inst. Microelectron.) ; Campabadal, F (Barcelona, Inst. Microelectron.) Silicon wafers grown by the Magnetic Czochralski (MCZ) method have been processed in form of pad diodes at Instituto de Microelectrònica de Barcelona (IMB-CNM) facilities. The n-type MCZ wafers were manufactured by Okmetic OYJ and they have a nominal resistivity of $1 \rm{k} \Omega cm$. [...] 2005 - 9 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 548 (2005) 355-363 Detaljnije - Slični zapisi 2018-08-23 11:31 Silicon detectors: From radiation hard devices operating beyond LHC conditions to characterization of primary fourfold coordinated vacancy defects / Lazanu, I (Bucharest U.) ; Lazanu, S (Bucharest, Nat. Inst. Mat. Sci.) The physics potential at future hadron colliders as LHC and its upgrades in energy and luminosity Super-LHC and Very-LHC respectively, as well as the requirements for detectors in the conditions of possible scenarios for radiation environments are discussed in this contribution.Silicon detectors will be used extensively in experiments at these new facilities where they will be exposed to high fluences of fast hadrons. The principal obstacle to long-time operation arises from bulk displacement damage in silicon, which acts as an irreversible process in the in the material and conduces to the increase of the leakage current of the detector, decreases the satisfactory Signal/Noise ratio, and increases the effective carrier concentration. [...] 2005 - 9 p. - Published in : Rom. Rep. Phys.: 57 (2005) , no. 3, pp. 342-348 External link: RORPE Detaljnije - Slični zapisi 2018-08-22 06:27 Numerical simulation of radiation damage effects in p-type and n-type FZ silicon detectors / Petasecca, M (Perugia U. ; INFN, Perugia) ; Moscatelli, F (Perugia U. ; INFN, Perugia ; IMM, Bologna) ; Passeri, D (Perugia U. ; INFN, Perugia) ; Pignatel, G U (Perugia U. ; INFN, Perugia) In the framework of the CERN-RD50 Collaboration, the adoption of p-type substrates has been proposed as a suitable mean to improve the radiation hardness of silicon detectors up to fluencies of $1 \times 10^{16} \rm{n}/cm^2$. In this work two numerical simulation models will be presented for p-type and n-type silicon detectors, respectively. [...] 2006 - 6 p. - Published in : IEEE Trans. Nucl. Sci. 53 (2006) 2971-2976 Detaljnije - Slični zapisi 2018-08-22 06:27 Technology development of p-type microstrip detectors with radiation hard p-spray isolation / Pellegrini, G (Barcelona, Inst. Microelectron.) ; Fleta, C (Barcelona, Inst. Microelectron.) ; Campabadal, F (Barcelona, Inst. Microelectron.) ; Díez, S (Barcelona, Inst. Microelectron.) ; Lozano, M (Barcelona, Inst. Microelectron.) ; Rafí, J M (Barcelona, Inst. Microelectron.) ; Ullán, M (Barcelona, Inst. Microelectron.) A technology for the fabrication of p-type microstrip silicon radiation detectors using p-spray implant isolation has been developed at CNM-IMB. The p-spray isolation has been optimized in order to withstand a gamma irradiation dose up to 50 Mrad (Si), which represents the ionization radiation dose expected in the middle region of the SCT-Atlas detector of the future Super-LHC during 10 years of operation. [...] 2006 - 6 p. - Published in : Nucl. Instrum. Methods Phys. Res., A 566 (2006) 360-365 Detaljnije - Slični zapisi 2018-08-22 06:27 Defect characterization in silicon particle detectors irradiated with Li ions / Scaringella, M (INFN, Florence ; U. Florence (main)) ; Menichelli, D (INFN, Florence ; U. Florence (main)) ; Candelori, A (INFN, Padua ; Padua U.) ; Rando, R (INFN, Padua ; Padua U.) ; Bruzzi, M (INFN, Florence ; U. Florence (main)) High Energy Physics experiments at future very high luminosity colliders will require ultra radiation-hard silicon detectors that can withstand fast hadron fluences up to $10^{16}$ cm$^{-2}$. In order to test the detectors radiation hardness in this fluence range, long irradiation times are required at the currently available proton irradiation facilities. [...] 2006 - 6 p. - Published in : IEEE Trans. Nucl. Sci. 53 (2006) 589-594 Detaljnije - Slični zapisi
№ 8 All Issues Volume 57, № 10, 2005 Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1315–1326 We suggest a method for obtaining a monotonically decreasing sequence of upper bounds of percolation threshold of the Bernoulli random field on $Z^2$. On the basis of this sequence, we obtain a method of constructing approximations with the guaranteed exactness estimate for a percolation probability. We compute the first term $c_2 = 0,74683$ of the considered sequence. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1327–1333 We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1334–1343 A general solution of the degenerate Nevanlinna-Pick problem is described in terms of fractional-linear transformations. A resolvent matrix of the problem is obtained in the form of a J-expanding matrix of full rank. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1344–1358 We consider the singular Cauchy problem $$txprime(t) = f(t,x(t),x(g(t)),xprime(t),xprime(h(t))), x(0) = 0,$$ where $x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t$, and $h(t) ≤ t, t ∈ (0, τ)$, for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set of continuously differentiable solutions $x: (0, ρ] → ℝ$ ($ρ$ is sufficiently small) with required asymptotic properties. On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1359–1384 For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution of the supremum, infimum, and value of the process. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1385–1394 We present conditions under which the subdifferential of a proper convex lower-semicontinuous functional in a Fréchet space is a bounded upper-semicontinuous mapping. The theorem on the boundedness of a subdifferential is also new for Banach spaces. We prove a generalized Weierstrass theorem in Fréchet spaces and study a variational inequality with a set-valued mapping. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1395–1408 Asymptotic equalities are established for upper bounds of approximants by Fourier partial sums in a metric of spaces $L_p,\quad 1 \leq p \leq \infty$ on classes of the Poisson integrals of periodic functions belonging to the unit ball of the space $L_1$. The results obtained are generalized to the classes of $(\psi, \overline{\beta})$-differentiable functions (in the Stepanets sense) that admit the analytical extension to a fixed strip of the complex plane. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1409–1417 As $n \rightarrow \infty$ the exact order of relative widths of classes $W^r_1$ of periodic functions in the space $L_1$ is found under restrictions on higher derivatives of approximating functions. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1418-1419 Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1420–1423 We prove the generalized convexity of domains satisfying the condition of acyclicity of their sections by a certain continuously parametrized family of two-dimensional planes. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1424–1429 For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1430–1434 Let $Γ$ be the set of all permutations of the natural series and let $α = \{α_j\}_{j ∈ ℕ},\; ν = \{ν_j\}_{j ∈ ℕ}$, and $η = {η_j}_{j ∈ ℕ}$ be nonnegative number sequences for which $$\left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1 : = \sum\limits_{j = 1}^\infty {v _j \alpha _{\gamma (_j )} } \eta _{\gamma (_j )}$$ is defined for all $γ:= \{γ(j)\}_{j ∈ ℕ} ∈ Γ$ and $η ∈ l_p$. We find $\sup _{\eta :\left\\| \eta \right\\|_p = 1} \inf _{\gamma \in \Gamma } \left\\| {\nu (\alpha \eta )_\gamma } \right\\|_1$ in the case where $1 < p < ∞$. Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1435–1440 We investigate the finite-dimensionality and growth of algebras specified by a system of polylinearly interrelated generators. The results obtained are formulated in terms of a function $\rho$.
The Wikipedia page for Monad says just that for a monad $(T,\eta,\mu)$ we can define the category of all adjunctions that define the monad: Let $\textbf{Adj}(C,T)$ be the category whose objects are the adjunctions $(F,G,e,\varepsilon)$ such that $(GF,e,G\varepsilon F)=(T,\eta,\mu)$ and whose arrows are the morphisms of adjunctions which are the identity on $C$. Then this category has an initial object $(F_T,G_T,\eta,\mu_T) : C\to C_T$, where $C_T$ is the Kleisli category, a terminal object $(F^T,G^T,\eta,\mu^T) : C\to C^T$, where $C^T$ is the Eilenberg-Moore category. What are the morphisms of adjunctions in $\textbf{Adj}(C,T)$ and what is meant by ... which are the identity on $C$?
A common question exists regarding the use of logarithm base 10 ($\log$ or $\log_{10}$) vs. logarithm base $e$ ($\ln$). The logarithm base $e$ is called the natural logarithm since it arises from the integral: \[ \ln (a) = \int_1^a \dfrac{dx}{x}\] Of course, one can convert from $\ln$ to $\log$ with a constant multiplier. \[ \ln (10^{\log a}) = \log (a) \ln(10) \approx 2.3025 \log (a)\] but $ 10^{\log a}=a$ so \[ \ln a \approx 2.4025 \log a\] The analysis of the reaction order and rate constant using the method of initial rates is performed using the $\log_{10}$ function. This could have been done using the $\ln$ function just as well. The initial rate is given by \[r_o=k'[A]_0^a\] The analysis can proceed by taking the logarithm base 10 of each side of the equation \[ \log r_o = \log k' + a\log [A]_0\] or the $\ln$ of each side of the equation \[ \ln r_o = \ln k' + a\ln [A]_0\] as long as one is consistent. Once can think of the $\log$ or the $\ln$ as a way to 'linearize data' that has some kind of power law dependence. The only difference between these two functions is a scaling factor ($\ln 10 \approx 2.3025$) in the slope. Obviously, if you multiply both sides of the equation by the same number the relative values of the constants remains the same on both sides.
Computational Aerodynamics Questions & Answers Very good question. We made this assumption only when deriving the GCI. So, within the $\rm GCI_{f}$ equation, you should change the term to $\left\vert\left(\frac{\triangle x_c}{\triangle x_f }\right)^p-1\right\vert$. That is, the GCI should always be positive. However, when determining order of accuracy $p$, we did not use the GCI and we did not assume that $p$ should be greater than 1. So you should not change any of the equations used to determine $p$. The user-defined constant $\epsilon$ is included to prevent a division by zero. Set it to a very small value. As for $\gamma_0$ and $\gamma_1$, they are fixed constants: don't change them. For the inflow BC, the stagnation pressure can be assumed equal to the one in the freestream because the flow along a streamline is isentropic. But such is not the case for the outflow BC. What if there is a shock somewhere within the domain? Then, the entropy rises and the stagnation pressure will go down and not be equal to the one in the freestream. However, for external flows around a body, the pressure will eventually become equal to the freestream pressure even if shocks are present (as long as the BC is far away from the body). Hence why it's better at the outflow BC to choose to fix pressure rather than stagnation pressure. OK, will look into this now. $\pi$
Suppose we have $3$ parties Alice, Bob and Charlie such that Alice can't talk with Bob. Suppose that Alice has some string $x\in\{0,1\}^n$ and Bob has a string $y\in\{0,1\}^n$. Suppose that Alice and Bob have access to a shared random string $r$ and a shared secret $s$. Suppose that Charlie knows both $x$ and $y$ and he doesn't have access to $r$. Describe a protocol in which Alice sends one message to Charlie (it may depend on $x,r,s$) and simultaneously Bob sends one message to Charlie (it may depend on $y,r,s$). Afterwards, if $x=y$ Charlie will know $s$ and if $x\neq y$ Charlie will not be able to know $s$. The general context of this problem is secret sharing, e.g. Shamir's secret sharing and so on. I have a proposition for solution to the inverse version of this question, i.e. if $x\neq y$ Charlie will know $s$ and if $x=y$ Charlie will not be able to know $s$: My proposition is to think of $x,y,r$ as members of the field $\mathbb{Z}_{2^n}$. Alice computes $s+rx$ and sends it to Charlie. Bob computes $s+ry$ and sends it to Charlie. If $x\neq y$ Charlie can interpolate to get $s$, otherwise $x=y$ and Charlie only saw some random element of $\mathbb{Z}_{2^n}$: $s+rx=s+ry$, so he can not recover $s$. But this is not the original question and I don't know what to do with the original. Can we generalize it to some boolean function $f:\{0,1\}^n\times\{0,1\}^n\to\{0,1\}$, i.e. only if $f(x,y)=1$ Charlie will be able to recover $s$?
Is it possible to approximate a monotone submodular function using a concrete coverage function, i.e. Given a ground set $U$ and a monotone submodular function, $f:2^U\to \mathbb{R}$, the goal is to find another ground set, $V$, and map each element from $U$ to a subset of $V$ ($m:U \to 2^V$) such that $f(S) \leq g(S) \leq \alpha f(S)$ for every $S \subseteq U$, where $\alpha$ is the approximation ratio (hopefully a constant) and $g(S) = c\|\bigcup_{e \in S}m(e)\|$ for some constant $c$. An approximation that uses a weighted ground set, $V$, is also very welcomed (I guess we can't avoid that). Thank you
Since $G$ is a $p$-group, its center is not trivial (see post 1 for a proof.) If the center $Z(G)=G$, then $G$ is abelian so assume that $Z(G)$ is a proper nontrivial subgroup. Then the center must have order $p$ and it follows that the order of the quotient $G/Z(G)$ is $p$, hence $G/Z(G)$ is a cyclic group. (b) The group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ Let $x \in G$ be any nontrivial element of $G$. If $x \in G$ has order $p^2$, then $G=\langle x \rangle \cong \Zmod{p^2}$. If the order of $x$ is $p$, then take $y \in G \setminus \langle x \rangle$. Then the order of $y$ is also $p$. Since the subgroup $\langle x, y \rangle$ generated by $x$ and $y$ is properly bigger than the subgroup $\langle x \rangle$, we must have $G=\langle x, y \rangle$.We claim that $\langle x, y \rangle \cong \langle x \rangle \times \langle y \rangle$. Define a map $f:\langle x \rangle \times \langle y \rangle \to \langle x, y \rangle$ by sending $(x^a, y^b)$ to $x^ay^b$. This is a group homomorphism because for any elements $(x^{a_1}, y^{b_1})$ and $(x^{a_2}, y^{b_2})$, we have\begin{align}f\left( (x^{a_1}, y^{b_1})(x^{a_2}, y^{b_2}) \right) &=f\left (x^{a_1+a_2}, y^{b_1+b_2})\right) =x^{a_1+a_2} y^{b_1+b_2}\\& =x^{a_1}y^{b_1}x^{a_2}y^{b_2} =f\left( (x^{a_1}, y^{b_1}) \right) f\left( (x^{a_2}, y^{b_2}) \right).\end{align} Here we used the result of part (a) that $G$ is abelian in the third equality. We claim that the homomorphism $f$ is injective.If $f\left (x^a, y^b) \right)=1$, we have $x^a=y^b$ but since $y \not \in \langle x \rangle$ we must have $a=b=0$. Thus the kernel is trivial, hence $f$ is injective. Since $\langle x \rangle \times \langle y \rangle \cong \Zmod{p} \times \Zmod{p}$ has order $p^2$ and $f$ is injective, the homomorphism must be surjective as well, hence it is an isomorphism. Abelian Normal subgroup, Quotient Group, and Automorphism GroupLet $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.Let $\Aut(N)$ be the group of automorphisms of $G$.Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.Then prove that $N$ is contained in the center of […] Group of Order $pq$ is Either Abelian or the Center is TrivialLet $G$ be a group of order $\|G\|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers.Then show that $G$ is either abelian group or the center $Z(G)=1$.Hint.Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […] If the Quotient by the Center is Cyclic, then the Group is AbelianLet $Z(G)$ be the center of a group $G$.Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian.Steps.Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$.Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$.Using […] The Center of the Symmetric group is Trivial if $n>2$Show that the center $Z(S_n)$ of the symmetric group with $n \geq 3$ is trivial.Steps/HintAssume $Z(S_n)$ has a non-identity element $\sigma$.Then there exist numbers $i$ and $j$, $i\neq j$, such that $\sigma(i)=j$Since $n\geq 3$ there exists another […] Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$Let $D_8$ be the dihedral group of order $8$.Using the generators and relations, we have\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\](a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.Prove that the centralizer […] Abelian Group and Direct Product of Its SubgroupsLet $G$ be a finite abelian group of order $mn$, where $m$ and $n$ are relatively prime positive integers.Then show that there exists unique subgroups $G_1$ of order $m$ and $G_2$ of order $n$ such that $G\cong G_1 \times G_2$.Hint.Consider […]
Matrices Satisfying $HF-FH=-2F$ Problem 69 Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation \[HF-FH=-2F.\] (a) Find the trace of the matrix $F$. (b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$. Contents Hint. For (a), take the trace of the both sides of the given relation. For (b), show that if $F^k\mathbf{v}\neq \mathbf{0}$ then there are infinitely many eigenvalues, hence a contradiction. Proof. (a) The trace of the matrix $F$ Using the given relation we compute the trace of $F$ as follows. By taking the trace of both sides we have \[\tr(-2F)=\tr(HF-FH).\] The right hand side is $-2\tr(F)$ and the left hand side is \begin{align} \tr(HF-FH)&=\tr(HF)-\tr(FH)\\ &=\tr(HF)-\tr(HF)=0. \end{align} Therefore we have $\tr(F)=0$. (b) There exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$. Since $\mathbf{v}$ is an eigenvector corresponding to the eigenvalue $\lambda$ of $H$, we have $H\mathbf{v}=\lambda \mathbf{v}$ or equivalently $(H-\lambda I)\mathbf{v}=\mathbf{0}$. Now we compute \begin{align} & F(H-\lambda I)=FH-\lambda F\\ &=(HF+2F)-\lambda F=(H-(\lambda-2)I)F. \end{align} Therefore we have \[F(H-\lambda I)=(H-(\lambda-2)I)F.\] Evaluating at $\mathbf{v}$, we obtain \[\mathbf{0}=F(H-\lambda I)\mathbf{v}=(H-(\lambda-2)I)F\mathbf{v}.\] If $F\mathbf{v} \neq \mathbf{0}$, then this equality implies that $F\mathbf{v}$ is an eigenvector corresponding to the eigenvalue $\lambda-2$ of $H$. In this case we further calculate \begin{align} \mathbf{0}&=F(H-(\lambda-2)I)F\mathbf{v} \\ &=(FH-(\lambda-2)F)F=(HF+2F-(\lambda-2)F)F\mathbf{v}\\ &=(H-(\lambda-4))F^2\mathbf{v}. \end{align} If the vector $F^2\mathbf{v}\neq \mathbf{0}$, then this equality implies that $F^2\mathbf{v}$ is an eigenvector corresponding to the eigenvalue $\lambda-4$ of $H$. Repeating this procedure, we see that \[\mathbf{0}=(H-(\lambda-2k))F^k\mathbf{v}\] for all $k$. Therefore, if $F^k\mathbf{v}$ is nonzero vector for all $k$, then there are infinitely many eigenvalues $\lambda-2k$ but this is impossible since $H$ is an $n \times n$ matrix and hence $H$ has at most $n$ eigenvalues. Therefore there exists $N$ such that $F^N\mathbf{v}=\mathbf{0}$. Related Question. See the problem “Matrices satisfying the relation HE-EH=2E” for similar questions. As noted there, the relation $HF-FH=-2F$ comes from the Lie algebra $\mathfrak{sl}(2)$. Add to solve later
Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$.The (field) norm $N$ of an element $a+b\sqrt{5}$ is defined by\[N(a+b\sqrt{5})=(a+b\sqrt{5})(a-b\sqrt{5})=a^2-5b^2.\] Consider the case when $a=3, b=1$.Then we have\[(3+\sqrt{5})(3-\sqrt{5})=4=2\cdot 2. \tag{}\] We prove that elements $2, 3\pm \sqrt{5}$ are irreducible in $\Z[\sqrt{5}]$.Note that the norms of these elements are $4$. We claim that each element $\alpha \in \Z[\sqrt{5}]$ of norm $4$ is irreducible. Suppose that $\alpha=\beta \gamma$ for some $\beta, \gamma \in \Z[\sqrt{5}]$.Our objective is to show that either $\beta$ or $\gamma$ is a unit. Since we have\[4=N(\alpha)=N(\beta \gamma)=N(\beta) N(\gamma)\]and the norms are integers, the value of $N(\beta)$ is one of $\pm 1, \pm 2, \pm 4$. If $N(\beta)=\pm 1$, then $\beta$ is a unit.If $N(\beta)=\pm 4$, then $N(\gamma)=\pm 1$ and hence $\gamma$ is a unit. Let us consider the case $N(\beta)=\pm 2$.We show that this case does not happen.Write $\beta=a+b\sqrt{5}$ for some integers $a, b$.Then we have\[\pm 2 =N(\beta)=a^2-5b^2.\]Considering the above equality modulo $5$ yields that\[\pm 2 \equiv a^2 \pmod{5}.\]However note that any square of an integer modulo $5$ is one of $0, 1, 4$.So this shows that there is no such $a$. Therefore, we have proved that either $\beta$ or $\gamma$ is a unit, hence $\alpha$ is irreducible.The claim is proved. It follows from () that the element $4 \in \Z[\sqrt{5}]$ has two different decompositions into irreducible elements.Thus the ring $\Z[\sqrt{5}]$ is not a UFD. Related Question. Problem.Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). This problem only differs from the current problem by the sign.($-5$ is used instead of $5$.) Ring of Gaussian Integers and Determine its Unit ElementsDenote by $i$ the square root of $-1$.Let\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]be the ring of Gaussian integers.We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]Here $\bar{\alpha}$ is the complex conjugate of […] A ring is Local if and only if the set of Non-Units is an IdealA ring is called local if it has a unique maximal ideal.(a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.(b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.Prove that if every […] 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$In the ring\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\]show that $5$ is a prime element but $7$ is not a prime element.Hint.An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […] The Ring $\Z[\sqrt{2}]$ is a Euclidean DomainProve that the ring of integers\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]of the field $\Q(\sqrt{2})$ is a Euclidean Domain.Proof.First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$.We use the […] Three Equivalent Conditions for an Ideal is Prime in a PIDLet $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.(1) The ideal $(a)$ generated by $a$ is maximal.(2) The ideal $(a)$ is prime.(3) The element $a$ is irreducible.Proof.(1) $\implies$ […]
维基教科书:格式手册 The Wikibooks Manual of Style serves to illustrate good structural and stylistic practices to help editors produce higher quality wikibooks. Wikibook titles编辑 See also Wikibooks:Naming policy for information on how to name books and their chapters. Wikibooks should be titled based on their aspect. This is a combination of the target audience, scope, and depth of the book. Several books on one topic but with differing aspects can exist (see Wikibooks:Forking policy for details). For example, a book on woodwork aimed at the general public may be entitled simply "Woodworking", and a mathematics text for commerce students may be called "Mathematics for Commerce" or "Commercial Mathematics" instead of simply "Mathematics". Some people prefer to use title casing like books often do, while other people prefer to use sentence casing like Wikipedia does. Title casing can reduce the potential for conflict with categories. For this reason title casing is often preferred more by Wikibookians. Whichever scheme you use please be consistent and follow the existing style for books you are editing. Structure编辑 See also book design for some helpful tips and ideas. Main page编辑 The main page is generally the first page a new reader sees. It should give a quick overview of the scope, intended audience and layout of the book. Splash pages should be avoided. Often the layout is simply the full table of contents. Collections, printable versions and PDF files should be easily accessible from this page. Links to the book from the card catalog office and external websites, should point to a book's main page. The subject index on the card catalog office requires the {{subject}} template to be placed on the main page. The book's main page or category should also be placed in any other categories that it belongs to. If you still require help with categorizing a book, please request assistance in the reading room. Interlingual links should be placed on the book's main page. Books across language projects may be dissimilar even when about the same subject. Be wary about placing interlingual links on any other pages. Table of contents编辑 In general, the table of contents should reside on the main page, giving readers an overview of the available material. In cases where this is impractical, a special page can be created for it. Common names for such pages are , Contents or similar. Table of contents Introduction编辑 An introduction page is the first page where learning begins. A book's introduction page commonly delves into purpose and goals of the book; what the book aims to teach, who the audience is, what the book's scope is, what topics the book covers, history of the subject, reasons to learn the subject, any conventions used in the book, or any other information that might make sense in an introductory chapter. Common names for such pages are or Introduction . The later is more commonly used when information about the book is split from the introduction of the subject matter. About Navigation aids are links included on pages to make navigating between pages easier. Navigation aids can help make reading books easier for readers, but can also make maintaining and contributing to books harder. Most web browsers can back track through pages already visited, and the wiki software also adds links for navigating back to the table of contents if pages use the slash convention in their name and a book's main page is the table of contents as suggested. Using a template on pages can help reduce some of the maintenance issues, since only one page must be edited if things change. There is no standard for navigation aids and are optional due to their potential drawbacks. Bibliography编辑 A bibliography is useful for collecting resources that were cited in the book, for linking to other useful resources on the subject that go beyond the scope of the book, and for including works that can aid in verifying the factual accuracy of what is written in the book. When used, such pages are commonly named , Further Reading , or similar. References Glossary编辑 A glossary lists terms in a particular domain of knowledge with the definitions for those terms. A glossary is completely optional and is most useful in books aimed at people new to a field of study. should always be used for such a page. Glossary Appendices编辑 An appendix includes important information that does not quiet fit into the flow of a book. For example a list of math formulas might be used as an appendix in a math book. Appendices are optional and books may have more than one appendix. Examples of common ways to name appendices are and Appendix/Keywords . Appendix:Forumlas Cover pages编辑 Cover pages are useful for print versions. These should be separated from the main page (remember: Wikibooks is not paper) but can be used to make print versions. Print versions编辑 See Wikibooks:Print versions for more on print versions. Examples编辑 Style编辑 Where appropriate the first use a word or phrase akin to the title of the page should be marked in bold and when a new term is introduced, it is italicised. Headers编辑 Headers should be used to divide page sections and provide content layout. Primary information should be marked with a "==", and information secondary to that should be marked with a "===" header, and so on, for example: == Animals == There are many kinds of animals. === Cats === Cats are animals. === Dogs === Dogs are animals. There is no need to provide systematised navigation between headers as the Linking section describes; only provide navigation between pages. A list of sections is automatically provided on the page when more then 2 headers are used. Linking编辑 See also: Help:Links Books with a deep sub-page hierarchy should include links on module pages to help navigate within its branch of the hierarchy. Templates can help maintain a navigation aid by including it on the necessary pages. Footnotes and references编辑 See also Help:References. Wikibooks has a couple of really simple implementations for footnotes and references. One uses {{note}} to place notes at the bottom of the page and {{ref}} to reference these at appropriate places in the text. The other places the references between <ref> and </ref> tags rigth in the text, and then uses the {{reflist}} to automatically generate a list of these at the bottom of the page. Mathematics编辑 Format using HTML or in-line Mediawiki markup when using variables or other simple mathematical notation within a sentence. Use <math></math> tags for more complicated forms. Italicise variables: a+ b, etc. Do not italicise Greek letter variables, function names or their brackets. To introduce mathematics notation in a display format, move to the start of the new line, add a single colon, and then the notation within the <math></math> tags. If punctuation follows the notation, do so inside the tags. For example: Markup Display : <math>\int_0^\infty {1 \over {x^2 + 1}}\,dx = \frac{\pi}{2},</math> is correctly used for "display" guidelines. If a notation does not render as a PNG, you may force it to do so by adding a " \," at the end of the formula. 结构编辑教科书一般可以分为“主页面”、“目录”和“内容”三部分。 “目录”和“内容”必须是“主页面”的子页面；例如创建一本“示例教科书”，其目录的标题应为“示例教科书/目录”。 “主页”和“目录”可以合并。主页用于简要介绍该教科书和其他的一些基本信息。主页必须包含的内容有——该教科书的分类、指向目录或内容的超链接等；若教科书参考了某些资料，也应在主页中说明。主页中可以包含的内容有——教科书的封面、此教科书的描述等。目录用于陈列教科书内容的链接。教科书中的内容应保持格式统一。格式的内容可以由教科书的创建者建立；若已有多人参与了该教科书的编写时，统一格式的建立应在该教科书主页面的讨论页征得共识之后。对现有统一格式的大幅更改亦需在主页面的讨论页征得共识。若存在统一格式，应在主页面说明或加入指向说明统一格式的页面的超链接。若教科书的结构设计确实美观合理的，不必拘泥于以上指引。
X Search Filters Format Subjects Library Location Language Publication Date Click on a bar to filter by decade Slide to change publication date range 1. Color connections of the four-quark Q Q Q ' Q ' system and doubly heavy baryon production in e + e - annihilation Physics Letters B, ISSN 0370-2693, 12/2013, Volume 727, Issue 4-5, p. 468 The hadronization effects induced by various color connections of the four-quark system in e.sup.+e.sup.- annihilation are briefly reviewed. A special color... Phenomenology \| High Energy Physics - Phenomenology \| High Energy Physics \| Nuclear Experiment \| Nuclear Theory \| Experiment \| Computer Science \| Three-jet event \| Color connection \| Doubly heavy baryon \| String effect \| Digital Libraries Phenomenology \| High Energy Physics - Phenomenology \| High Energy Physics \| Nuclear Experiment \| Nuclear Theory \| Experiment \| Computer Science \| Three-jet event \| Color connection \| Doubly heavy baryon \| String effect \| Digital Libraries Journal Article 2. Search for resonant diboson production in the $$\mathrm {\ell \ell }q\bar{q}$$ ℓ ℓ q q ¯ final state in $$pp$$ p p collisions at $$\sqrt{s} = 8$$ s = 8 TeV with the ATLAS detector The European Physical Journal C, ISSN 1434-6044, 2/2015, Volume 75, Issue 2, pp. 1 - 20 This paper reports on a search for narrow resonances in diboson production in the $$\mathrm {\ell \ell }q\bar{q}$$ ℓ ℓ q q ¯ final state using $$pp$$ p p... Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Journal Article EURASIP Journal on Advances in Signal Processing, ISSN 1687-6172, 12/2013, Volume 2013, Issue 1, pp. 1 - 17 Compressed sensing (CS) states that a sparse signal can exactly be recovered from very few linear measurements. While in many applications, real-world signals... Engineering \| IRLS \| Signal, Image and Speech Processing \| Block-sparse recovery \| Block-RIP \| l 2 / l q minimization \| Compressed sensing \| minimization Engineering \| IRLS \| Signal, Image and Speech Processing \| Block-sparse recovery \| Block-RIP \| l 2 / l q minimization \| Compressed sensing \| minimization Journal Article Physics Letters B, ISSN 0370-2693, 02/2006, Volume 633, Issue 4-5, pp. 664 - 670 We propose that the Baxter -operator for the spin-1/2 XXZ quantum spin chain is given by the limit of the transfer matrix with spin- (i.e., -dimensional)... Bethe ansatz \| Spin chain \| Fusion hierarchy \| Reflection equation \| Q-operator \| XXZ SPIN CHAIN \| PHYSICS, MULTIDISCIPLINARY \| fusion hierarchy \| reflection equation \| STATE \| bethe ansatz \| MODEL \| FUNCTIONAL RELATIONS \| CONFORMAL FIELD-THEORY \| 8-VERTEX \| spin chain \| BAXTER EQUATION \| NONDIAGONAL BOUNDARY TERMS \| BETHE-ANSATZ SOLUTION \| INTEGRABLE STRUCTURE \| Anisotropy \| Analysis \| Physics - High Energy Physics - Theory Bethe ansatz \| Spin chain \| Fusion hierarchy \| Reflection equation \| Q-operator \| XXZ SPIN CHAIN \| PHYSICS, MULTIDISCIPLINARY \| fusion hierarchy \| reflection equation \| STATE \| bethe ansatz \| MODEL \| FUNCTIONAL RELATIONS \| CONFORMAL FIELD-THEORY \| 8-VERTEX \| spin chain \| BAXTER EQUATION \| NONDIAGONAL BOUNDARY TERMS \| BETHE-ANSATZ SOLUTION \| INTEGRABLE STRUCTURE \| Anisotropy \| Analysis \| Physics - High Energy Physics - Theory Journal Article 5. Search for resonant diboson production in the $\mathrm {\ell \ell }q\bar{q}$ final state in $pp$ collisions at $\sqrt{s} = 8$ TeV with the ATLAS detector European Physical Journal. C, Particles and Fields, ISSN 1434-6044, 02/2015, Volume 75, Issue 2 This study reports on a search for narrow resonances in diboson production in the ℓℓqq¯ final state using pp collision data corresponding to an integrated... PHYSICS OF ELEMENTARY PARTICLES AND FIELDS PHYSICS OF ELEMENTARY PARTICLES AND FIELDS Journal Article 6. Search for $$W' \rightarrow tb \rightarrow qqbb$$ W ′ → t b → q q b b decays in $$pp$$ p p collisions at $$\sqrt{s}$$ s = 8 TeV with the ATLAS detector The European Physical Journal C, ISSN 1434-6044, 4/2015, Volume 75, Issue 4, pp. 1 - 23 A search for a massive $$W'$$ W ′ gauge boson decaying to a top quark and a bottom quark is performed with the ATLAS detector in $$pp$$ p p collisions at the... Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Journal Article International Journal of Approximate Reasoning, ISSN 0888-613X, 03/2019, Volume 106, pp. 146 - 154 Let be two comonotonic functions on the unit interval. The classical Chebyshev inequality states that the product of the integrals of and is a lower bound of... Capacity \| Q-integrals \| Q-cointegrals \| Chebyshev inequality \| CHOQUET \| OPERATORS \| AGGREGATION \| COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE \| Equality Capacity \| Q-integrals \| Q-cointegrals \| Chebyshev inequality \| CHOQUET \| OPERATORS \| AGGREGATION \| COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE \| Equality Journal Article 8. Some modular relations for Ramanujan’s function $$k(q)=r(q)r^2(q^2)$$ k ( q ) = r ( q ) r 2 ( q 2 ) The Ramanujan Journal, ISSN 1382-4090, 11/2014, Volume 35, Issue 2, pp. 243 - 251 In both his second and lost notebooks, Ramanujan introduced and studied a function $$k(q)=r(q)r^2(q^2)$$ k ( q ) = r ( q ) r 2 ( q 2 ) , where $$r(q)$$ r ( q )... 14K25 \| 05A30 \| Functions of a Complex Variable \| Rogers–Ramanujan continued fraction \| Field Theory and Polynomials \| 11A55 \| 11B65 \| Mathematics \| Modular equation \| Fourier Analysis \| Theta function \| Number Theory \| Combinatorics 14K25 \| 05A30 \| Functions of a Complex Variable \| Rogers–Ramanujan continued fraction \| Field Theory and Polynomials \| 11A55 \| 11B65 \| Mathematics \| Modular equation \| Fourier Analysis \| Theta function \| Number Theory \| Combinatorics Journal Article 9. Color connections of the four-quark Q (Q)over-bar Q' (Q)over-bar' system and doubly heavy baryon production in e(+)e(-) annihilation PHYSICS LETTERS B, ISSN 0370-2693, 12/2013, Volume 727, Issue 4-5, pp. 468 - 473 The hadronization effects induced by various color connections of the four-quark system in e(+)e(-) annihilation are briefly reviewed. A special color... STATES \| EVENTS \| PHYSICS, MULTIDISCIPLINARY \| Three-jet event \| SOFT GLUON INTERFERENCE \| REARRANGEMENT \| JET FRAGMENTATION \| QUARKS \| STRING MODEL \| Color connection \| Doubly heavy baryon \| E&E-ANNIHILATION \| PAIR PRODUCTION \| String effect STATES \| EVENTS \| PHYSICS, MULTIDISCIPLINARY \| Three-jet event \| SOFT GLUON INTERFERENCE \| REARRANGEMENT \| JET FRAGMENTATION \| QUARKS \| STRING MODEL \| Color connection \| Doubly heavy baryon \| E&E-ANNIHILATION \| PAIR PRODUCTION \| String effect Journal Article 10. Search for heavy resonances decaying to a $W$ or $Z$ boson and a Higgs boson in the $q\bar{q}^{(\prime)}b\bar{b}$ final state in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector ISSN 0370-2693, 07/2017 Phys. Lett. B 774 (2017) 494 A search for heavy resonances decaying to a $W$ or $Z$ boson and a Higgs boson in the $q\bar{q}^{(\prime)}b\bar{b}$ final state is... Physics - High Energy Physics - Experiment \| scattering [p p] \| triplet \| Subatomic Physics \| ((n)jet) [final state] \| Settore ING-INF/07 - Misure Elettriche e Elettroniche \| p p: colliding beams ; resonance: heavy ; resonance: decay ; W: associated production ; Z0: associated production ; W: hadronic decay ; Z0: hadronic decay ; p p: scattering ; resonance: mass ; Higgs particle: associated production ; Higgs particle: hadronic decay ; bottom: pair production ; bottom: particle identification ; final state: ((n)jet) ; ATLAS ; channel cross section: branching ratio: upper limit ; CERN LHC Coll ; CERN Lab ; triplet ; experimental results ; 13000 GeV-cms \| High Energy Physics - Experiment \| Settore FIS/04 - Fisica Nucleare e Subnucleare \| Science & Technology \| Phenomenology \| associated production [W] \| High Energy Physics \| mass [resonance] \| pair production [bottom] \| heavy [resonance] \| Subatomär fysik \| 13000 GeV-cms \| experimental results \| decay [resonance] \| Settore FIS/01 - Fisica Sperimentale \| hadronic decay [W] \| Nuclear and High Energy Physics \| Experiment \| particle identification [bottom] \| CERN LHC Coll \| Physics \| associated production [Z0] \| CERN Lab \| associated production [Higgs particle] \| branching ratio: upper limit [channel cross section] \| hadronic decay [Higgs particle] \| colliding beams [p p] \| hadronic decay [Z0] Physics - High Energy Physics - Experiment \| scattering [p p] \| triplet \| Subatomic Physics \| ((n)jet) [final state] \| Settore ING-INF/07 - Misure Elettriche e Elettroniche \| p p: colliding beams ; resonance: heavy ; resonance: decay ; W: associated production ; Z0: associated production ; W: hadronic decay ; Z0: hadronic decay ; p p: scattering ; resonance: mass ; Higgs particle: associated production ; Higgs particle: hadronic decay ; bottom: pair production ; bottom: particle identification ; final state: ((n)jet) ; ATLAS ; channel cross section: branching ratio: upper limit ; CERN LHC Coll ; CERN Lab ; triplet ; experimental results ; 13000 GeV-cms \| High Energy Physics - Experiment \| Settore FIS/04 - Fisica Nucleare e Subnucleare \| Science & Technology \| Phenomenology \| associated production [W] \| High Energy Physics \| mass [resonance] \| pair production [bottom] \| heavy [resonance] \| Subatomär fysik \| 13000 GeV-cms \| experimental results \| decay [resonance] \| Settore FIS/01 - Fisica Sperimentale \| hadronic decay [W] \| Nuclear and High Energy Physics \| Experiment \| particle identification [bottom] \| CERN LHC Coll \| Physics \| associated production [Z0] \| CERN Lab \| associated production [Higgs particle] \| branching ratio: upper limit [channel cross section] \| hadronic decay [Higgs particle] \| colliding beams [p p] \| hadronic decay [Z0] Journal Article OPTICS EXPRESS, ISSN 1094-4087, 10/2018, Volume 26, Issue 21, pp. 26916 - 26924 Journal Article 12. Measurement of $$WW/WZ \rightarrow \ell \nu q q^{\prime }$$ W W / W Z → ℓ ν q q ′ production with the hadronically decaying boson reconstructed as one or two jets in pp collisions at $$\sqrt{s} =8~\text {TeV}$$ s = 8 TeV with ATLAS, and constraints on anomalous gauge couplings The European Physical Journal C, ISSN 1434-6044, 8/2017, Volume 77, Issue 8, pp. 1 - 31 This paper presents a study of the production of WW or WZ boson pairs, with one W boson decaying to $$e\nu $$ e ν or $$\mu \nu $$ μ ν and one W or Z boson... Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Journal Article 13. Assembly of novel organic-decorated quaternary TM-Hg-Sb-Q compounds (TM = Mn, Fe, Co; Q = S, Se) by the combination of three types of metal coordination geometries Dalton Transactions, ISSN 1477-9226, 06/2012, Volume 41, Issue 22, pp. 6689 - 6695 Four organic-decorated quaternary TM-Hg-Sb-Q compounds, namely, [Mn(phen)](2)HgSb2S6 (phen = 1,10-phenanthroline) (1) and isomorphic [TM(tren)] HgSb2Se5 (TM =... SOLVOTHERMAL SYNTHESIS \| ANIONS \| ELECTRONIC-PROPERTIES \| HYDROGEN GENERATION \| ANTIMONY SULFIDES \| TRANSITION-METAL \| OPTICAL-PROPERTIES \| CRYSTAL-STRUCTURES \| ION \| THIOANTIMONATE(III) NETWORK \| CHEMISTRY, INORGANIC & NUCLEAR SOLVOTHERMAL SYNTHESIS \| ANIONS \| ELECTRONIC-PROPERTIES \| HYDROGEN GENERATION \| ANTIMONY SULFIDES \| TRANSITION-METAL \| OPTICAL-PROPERTIES \| CRYSTAL-STRUCTURES \| ION \| THIOANTIMONATE(III) NETWORK \| CHEMISTRY, INORGANIC & NUCLEAR Journal Article Neuron, ISSN 0896-6273, 12/2017, Volume 96, Issue 5, pp. 1070 - 1083.e5 Protein kinase A (PKA) integrates inputs from G-protein-coupled neuromodulator receptors to modulate synaptic and cellular function. Gαs signaling stimulates... G protein-coupled receptor \| neuromodulation \| muscarinic receptors \| hippocampus \| Gαq signaling \| optical reporters \| fluorescence lifetime imaging microscopy \| acetylcholine \| protein kinase A \| designer receptors \| GENETIC APPROACH \| CYCLIC-AMP \| SUBCELLULAR DYNAMICS \| ADENYLATE-CYCLASE ACTIVITY \| HAMSTER OVARY CELLS \| MUSCARINIC RECEPTORS \| RAT OLFACTORY-BULB \| IN-VIVO \| C ACTIVATION \| SYNAPTIC PLASTICITY \| NEUROSCIENCES \| Cyclic AMP-Dependent Protein Kinases - metabolism \| GTP-Binding Protein alpha Subunits, Gq-G11 - genetics \| Phosphorylation \| Receptors, G-Protein-Coupled - metabolism \| Calcium Signaling - physiology \| Humans \| Mice, Inbred C57BL \| Receptors, Muscarinic - metabolism \| Signal Transduction - genetics \| Hippocampus - cytology \| Pregnancy \| Hippocampus - metabolism \| Animals \| Protein Kinase C - metabolism \| HEK293 Cells \| Female \| Mice \| Receptors, Neurotransmitter - metabolism \| Calcium Signaling - genetics \| Enzyme Activation - genetics \| Receptors, Muscarinic - genetics \| Protein kinase A \| Protein kinase C \| Animal models \| G protein-coupled receptors \| Acetylcholine receptors (muscarinic) \| Kinases \| Neuromodulation \| Phospholipase C \| Calcium signalling \| Proteins \| Phospholipase \| Receptors \| Microscopy \| Rodents \| Physiology \| Acetylcholine \| Receptor mechanisms \| Index Medicus G protein-coupled receptor \| neuromodulation \| muscarinic receptors \| hippocampus \| Gαq signaling \| optical reporters \| fluorescence lifetime imaging microscopy \| acetylcholine \| protein kinase A \| designer receptors \| GENETIC APPROACH \| CYCLIC-AMP \| SUBCELLULAR DYNAMICS \| ADENYLATE-CYCLASE ACTIVITY \| HAMSTER OVARY CELLS \| MUSCARINIC RECEPTORS \| RAT OLFACTORY-BULB \| IN-VIVO \| C ACTIVATION \| SYNAPTIC PLASTICITY \| NEUROSCIENCES \| Cyclic AMP-Dependent Protein Kinases - metabolism \| GTP-Binding Protein alpha Subunits, Gq-G11 - genetics \| Phosphorylation \| Receptors, G-Protein-Coupled - metabolism \| Calcium Signaling - physiology \| Humans \| Mice, Inbred C57BL \| Receptors, Muscarinic - metabolism \| Signal Transduction - genetics \| Hippocampus - cytology \| Pregnancy \| Hippocampus - metabolism \| Animals \| Protein Kinase C - metabolism \| HEK293 Cells \| Female \| Mice \| Receptors, Neurotransmitter - metabolism \| Calcium Signaling - genetics \| Enzyme Activation - genetics \| Receptors, Muscarinic - genetics \| Protein kinase A \| Protein kinase C \| Animal models \| G protein-coupled receptors \| Acetylcholine receptors (muscarinic) \| Kinases \| Neuromodulation \| Phospholipase C \| Calcium signalling \| Proteins \| Phospholipase \| Receptors \| Microscopy \| Rodents \| Physiology \| Acetylcholine \| Receptor mechanisms \| Index Medicus Journal Article 15. New quaternary rare-earth chalcogenides Ba Ln Sn2Q6 (Ln = Ce, Pr, Nd, Q = S; Ln = Ce, Q = Se): Synthesis, structure, and magnetic properties Inorganic Chemistry, ISSN 0020-1669, 02/2014, Volume 53, Issue 4, pp. 2248 - 2253 Journal Article
Example of an Infinite Algebraic Extension Problem 499 Find an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$. Contents Definition (Algebraic Element, Algebraic Extension). Let $F$ be a field and let $E$ be an extension of $F$. The element $\alpha \in E$ is said to be algebraicover $F$ is $\alpha$ is a root of some nonzero polynomial with coefficients in $F$. The extension $E/F$ is said to be algebraicif every element of $E$ is algebraic over $F$. Proof. Consider the field \[K=\Q(\sqrt[3]{2}, \sqrt[5]{2}, \dots, \sqrt[2n+1]{2}, \dots).\] That is, $K$ is the field extension obtained by adjoining all numbers of the form $\sqrt[2n+1]{2}$ for any positive integers $n$. Note that $\sqrt[2n+1]{2}$ is a root of the monic polynomial $x^{2n+1}-2$, hence $\sqrt[2n+1]{2}$ is algebraic over $\Q$. By Eisenstein’s criterion with prime $2$, we know that the polynomial $x^{2n+1}-2$ is irreducible over $\Q$. Thus the extension degree is $[\Q(\sqrt[2n+1]{2}):\Q]=2n+1$. Since the field $K$ contains the subfield $\Q(\sqrt[2n+1]{2})$, we have \[2n+1=[\Q(\sqrt[2n+1]{2}):\Q] \leq [K:\Q]\] for any positive integer $n$. Therefore, the extension degree of $K$ over $\Q$ is infinite. Observe that any element $\alpha$ of $K$ belongs to a subfield $\Q(\sqrt[3]{2}, \sqrt[5]{2}, \dots, \sqrt[2n+1]{2})$ for some $n \in \Z$. Since each number $\sqrt[2k+1]{2}$ is algebraic over $\Q$, we know that this subfield is algebraic, hence $\alpha$ is algebraic. Thus, the field $K$ is algebraic over $\Q$. Is $K$ different from $\bar{\Q}$? It remains to show that $K\neq \bar{\Q}$. Consider $\sqrt{2}$. Since $\sqrt{2}$ is a root of $x^2-2$, it is algebraic, hence $\sqrt{2}\in \bar{\Q}$. We claim that $\sqrt{2}\not \in K$. Assume on the contrary that $\sqrt{2} \in K$. Then $\sqrt{2} \in F:=\Q(\sqrt[3]{2}, \sqrt[5]{2}, \dots, \sqrt[2n+1]{2}) \subset K$ for some $n \in \Z$. Note that the extension degree of this subfield $F$ is odd since each extension degree of $\Q(\sqrt[2k+1]{2})/\Q$ is odd. Since $\sqrt{2}\in F$, we must have \begin{align} [F:\Q]=[F:\Q(\sqrt{2})][\Q(\sqrt{2}):\Q]=2[F:\Q(\sqrt{2})], \end{align} which is even. This is a contradiction, and hence $\sqrt{2}\not \in K$. Thus, $K\neq \bar{\Q}$. Comment. With the same argument, we can prove that the field \[K=\Q(\sqrt[2]{2}, \sqrt[3]{2}, \dots, \sqrt[n]{2}, \dots)\] is infinite algebraic extension over $\Q$. However, it is not trivial to show that this field is different from $\bar{\Q}$. That’s why we used only $\sqrt[2k+1]{2}$ in $K$. We can also use $\sqrt[p]{2}$ for odd prime $p$. Add to solve later
Asymptotic for the perturbed heavy ball system with vanishing damping term 1. Institut Préparatoire aux Etude Scientifiques et Techniques, Université de Carthage, Bp 51 La Marsa, Tunisia 2. Faculté des Sciences de Tunis, Laboratoire EDP-LR03ES04, Université de Tunis El Manar Tunis, Tunisia 3. College of Sciences, Department of Mathematics and Statistics, King Faisal University, P.O. 400 Al Ahsaa 31982, Kingdom of Saudi Arabia $\ddot{x}(t)+\frac{c}{{{\left( 1+t \right)}^{\alpha }}}\dot{x}(t)+\nabla \Phi \left( x(t) \right)=g(t),~t\ge 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ $c$ $α∈\lbrack0,1[,Φ \ \ {\rm{is \ \ a}}\ \ C^{1}$ $\mathcal{H}$ $g∈ L^{1}(0,+∞;\mathcal{H}).$ $g(t)$ $x(t)$ $t\to ∞$ $Φ$ Keywords:Differential equation, asymptotically small dissipation, asymptotic behavior, energy function, convex function, rate of convergence, weak convergence. Mathematics Subject Classification:Primary: 34A34, 34A40; Secondary: 34D05, 34E10. Citation:Mounir Balti, Ramzi May. Asymptotic for the perturbed heavy ball system with vanishing damping term. Evolution Equations & Control Theory, 2017, 6 (2) : 177-186. doi: 10.3934/eect.2017010 References: [1] [2] H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, [3] H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method, Ⅰ: The continuous dynamical system: Global exploration of the the local minima of a real valued function by asymptotic analysis of a dissipative dynamical system, [4] M. Balti and R. May, Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source, Submitted, arXiv: 1608. 08760v1.Google Scholar [5] A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, [6] A. Haraux and M. A. Jendoubi, On a second order dissipative ODE in Hilbert space with an integrable source term, [7] M. A. Jendoubi and R. May, Asymptotics for a second-order differential equation with non-autonomous damping and an integrable source term, [8] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, [9] [10] show all references References: [1] [2] H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, [3] H. Attouch, X. Goudou and P. Redont, The heavy ball with friction method, Ⅰ: The continuous dynamical system: Global exploration of the the local minima of a real valued function by asymptotic analysis of a dissipative dynamical system, [4] M. Balti and R. May, Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source, Submitted, arXiv: 1608. 08760v1.Google Scholar [5] A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, [6] A. Haraux and M. A. Jendoubi, On a second order dissipative ODE in Hilbert space with an integrable source term, [7] M. A. Jendoubi and R. May, Asymptotics for a second-order differential equation with non-autonomous damping and an integrable source term, [8] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, [9] [10] [1] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. [2] Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. [3] Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. [4] Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. [5] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. [6] Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. [7] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. [8] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. [9] Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. [10] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. [11] Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. [12] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. [13] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. [14] [15] Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. [16] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. [17] Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. [18] [19] Matteo Focardi, Paolo Maria Mariano. Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence. [20] Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. 2018 Impact Factor: 1.048 Tools Metrics Other articles by authors [Back to Top]
Subspaces of Symmetric, Skew-Symmetric Matrices Problem 143 Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices. (c) The set $U$ consisting of all $n\times n$ nonsingular matrices. Contents Hint. Recall that a matrix $A$ is symmetric if $A^{\trans}=A$. a matrix $A$ is skew-symmetric if $A^{\trans}=-A$. Proof. To show that a subset $W$ of a vector space $V$ is a subspace, we need to check that (a) The set $S$ consisting of all $n\times n$ symmetric matrices. We will prove that $S$ is a subspace of $V$. The zero vector $O$ in $V$ is the $n \times n$ zero matrix and it is symmetric. Thus the zero vector $O\in S$ and the condition 1 is met. To check the second condition, take any $A, B \in S$, that is, $A, B$ are symmetric matrices. To show that $A+B \in S$, we need to check that the matrix $A+B$ is symmetric. We have \begin{align} (A+B)^{\trans}=A^{\trans}+B^{\trans}=A+B \end{align} since $A, B$ are symmetric. Thus $A+B$ is also symmetric, and $A+B \in S$. Condition 2 is also satisfied. Finally, to check condition 3, let $A \in S$ and let $r\in R$. We show that $rA \in S$, namely, we show that $rA$ is symmetric. We have \begin{align} (rA)^{\trans}=rA^{\trans}=rA \end{align} since $A$ is symmetric. Thus $rA$ is symmetric and hence $rA \in S$. Thus condition 3 is met. By the subspace criteria, the subset $S$ is a subspace of the vector space $V$. (b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices. We will prove that $T$ is a subspace of $V$. The zero vector $O$ in $V$ is the $n \times n$ matrix, and it is skew-symmetric because \[O^{\trans}=O=-O.\] Thus condition 1 is met. For condition 2, take arbitrary elements $A, B \in T$. The matrices $A, B$ are skew-symmetric, namely, we have \[A^{\trans}=-A \text{ and } B^{\trans}=-B \tag{}.\] We show that $A+B \in T$, or equivalently we show that the matrix $A+B$ is skew-symmetric. We have \begin{align} (A+B)^{\trans}=A^{\trans}+B^{\trans} \stackrel{}{=} -A+(-B)=-(A+B). \end{align} Therefore the matrix $A+B$ is skew-symmetric and condition 2 is met. To prove the last condition, consider any $A \in T$ and $r \in \R$. We show that $rA$ is skew-symmetric, and hence $rA \in T$. Using the fact that $A$ is skew-symmetric ($A^{\trans}=-A$), we have \[(rA)^{\trans}=rA^{\trans}=r(-A)=-rA.\] Hence $rA$ is skew-symmetric and condition 3 is satisfied. By the subspace criteria, the subset $T$ is a subspace of the vector space $V$. (c) The set $U$ consisting of all $n\times n$ nonsingular matrices. We claim that $U$ is not a subspace of $V$. As the zero vector of $V$ is the $n \times n$ matrix and the zero matrix is singular, the zero vector is not in $U$. Hence condition 1 is not met, and thus $U$ is not a subspace. Another reason that $U$ is not a subspace is that the addition is not closed. For example, if $A$ is a nonsingular matrix (say, $A$ is $n\times n$ identity matrix), then $-A$ is also nonsingular matrix but their addition $A+(-A)=O$ is nonsingular, hence it is not in $U$. Add to solve later
Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space Department of Mathematics, Northwest Normal University, Lanzhou 730070, China $\left\{ \begin{array}{l} -\text{div}\big(\frac{\nabla u}{\sqrt{1-\|\nabla u\|^2}}\big) = λ f(x,u,\nabla u)\ \ \ \ \ \ & \text{in}\ Ω,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{on}\ \partial Ω.\\\end{array} \right.$ $Ω$ $\mathbb{R}^N$ $f$ $f$ $u$ $0$ Keywords:Prescribed mean curvature equations, Lorentz-Minkowski space, positive solution, bifurcation, connected component. Mathematics Subject Classification:Primary: 35J25; Secondary: 47H11, 47J10, 34B18. Citation:Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 References: [1] [2] C. Bereanu, P. Jebelean and J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski space-A variational approach, [3] C. Bereanu, P. Jebelean and P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, [4] C. Bereanu, P. Jebelean and P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, [5] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, [6] G. Chen, [7] [8] S. N. Chow and J. K. Hale, [9] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, [10] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, [11] C. Corsato, F. Obersnel and P. Omari, The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space, [12] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, [13] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, [14] [15] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, [16] H. Kielhöfer, [17] [18] R. Ma and Y. An, Global structure of positive solutions for superlinear second order $m$-point boundary value problems, [19] R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, [20] R. Ma, H. Gao and Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, [21] [22] [23] G. T. Whyburn, [24] E. Zeidler, show all references References: [1] [2] C. Bereanu, P. Jebelean and J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski space-A variational approach, [3] C. Bereanu, P. Jebelean and P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, [4] C. Bereanu, P. Jebelean and P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, [5] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, [6] G. Chen, [7] [8] S. N. Chow and J. K. Hale, [9] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, [10] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, [11] C. Corsato, F. Obersnel and P. Omari, The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space, [12] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, [13] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, [14] [15] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, [16] H. Kielhöfer, [17] [18] R. Ma and Y. An, Global structure of positive solutions for superlinear second order $m$-point boundary value problems, [19] R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, [20] R. Ma, H. Gao and Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, [21] [22] [23] G. T. Whyburn, [24] E. Zeidler, [1] Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. [2] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. [3] Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. [4] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. [5] Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. [6] Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. [7] Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. [8] Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. [9] Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. [10] Franco Obersnel, Pierpaolo Omari. On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation. [11] Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. [12] Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. [13] Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. [14] Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. [15] [16] [17] [18] [19] [20] Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. 2018 Impact Factor: 1.008 Tools Metrics Other articles by authors [Back to Top]
Answer $\text{Set Builder Notation: } \left\{ y\|y\le-13 \right\} \\\text{Interval Notation: } \left( -\infty,-13 \right]$ Work Step by Step $\bf{\text{Solution Outline:}}$ Use the properties of inequality to solve the given inequality, $ -5y\ge65 .$ Write the answer in both set-builder notation and interval notation. $\bf{\text{Solution Details:}}$ Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} -5y\ge65 \\\\ y\le\dfrac{65}{-5} \\\\ y\le-13 .\end{array} Hence, the solution set is \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \left\{ y\|y\le-13 \right\} \\\text{Interval Notation: } \left( -\infty,-13 \right] .\end{array}
Answer $\text{Set Builder Notation: } \left\{ t\|t\ge-1 \right\} \\\text{Interval Notation: } \left[ -1,\infty \right)$ Work Step by Step $\bf{\text{Solution Outline:}}$ To solve the given inequality, $ \dfrac{1}{2}t-\dfrac{1}{4}\le\dfrac{3}{4}t ,$ remove first the fraction by multiplying both sides by the $LCD.$ Then use the properties of inequality to isolate the variable. $\bf{\text{Solution Details:}}$ The $LCD$ of the denominators, $\{ 2,4,4 \},$ is $ 4 $ since this is the least number that can be evenly divided (no remainder) by all the denominators. Multiplying both sides by the $LCD,$ the given inequality is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{2}t-\dfrac{1}{4}\le\dfrac{3}{4}t \\\\ 4\left( \dfrac{1}{2}t-\dfrac{1}{4} \right) \le4\left( \dfrac{3}{4}t \right) \\\\ 2t-1\le3t .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} 2t-1\le3t \\\\ 2t-3t\le1 \\\\ -t\le1 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} -t\le1 \\\\ \dfrac{-t}{-1}\le\dfrac{1}{-1} \\\\ t\ge-1 .\end{array} Hence, the solution set is \begin{array}{l}\require{cancel} \text{Set Builder Notation: } \left\{ t\|t\ge-1 \right\} \\\text{Interval Notation: } \left[ -1,\infty \right) .\end{array}
Helo, every one. May I ask for help about how to solve this problem? \[\begin{align} max_{x_i} \quad \|\sum_{i=1}^{4} a_i x_i \| \\ s.t. \quad \sum_{i=1}^4 x_i^2=1\end{align}\] The goal is to find the optimal $x_i$, the $a_i$ is known. Thank you very much. Intuitively, the optimal $x$ will be proportional to $a$. A hint to get you started, in case this is homework: first ignore the absolute value and use the method of Lagrange multipliers to show that $x_i = a_i / \sqrt(\sum_j a_j^2)$. By the way, there is nothing special about 4 here; you can replace it with arbitrary $n$. answered Rob Pratt This looks like a homework problem, so I'll just give a hint: there is an "obvious" optimal solution that does not require the use of Lagrange multipliers. It relies on the inequality \[\|\sum_{i} z_{i}\| \le \sum_{i} \|z_{i}\|.\] answered Paul Rubin ♦♦ Here is yet another way to look at it. Assume we have an optimal solution. If sum ai xi is negative, then we can multiply x by -1 and get another optimal solution. It shows that we can get rid of the absolute value in the objective. In that case the objective is the inner product of vectors a and x, given a is fixed and the length of x is fixed. That inner product is maximal when the two vectors are collinear. answered jfpuget
Each element of $G/N$ is a coset $aN$ for some $a\in G$.Let $aN, bN$ be arbitrary elements of $G/N$, where $a, b\in G$. Then we have\begin{align}(aN)(bN)&=(ab)N \\&=(ba)N && \text{since $G$ is abelian}\\&=(bN)(aN).\end{align}Here the first and the third equality is the definition of the group operation of $G/N$. Remark Since $N$ is a normal subgroup of $G$, the set of left cosets $G/H$ becomes a group with group operation\[(aN)(bN)=(ab)N\]for any $a, b\in G$. Related Question. As an application, try the following problem. Problem.Let $H$ and $K$ be normal subgroups of a group $G$. Suppose that $H < K$ and the quotient group $G/H$ is abelian. Then prove that $G/K$ is also an abelian group. Commutator Subgroup and Abelian Quotient GroupLet $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.Let $N$ be a subgroup of $G$.Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.Definitions.Recall that for any $a, b \in G$, the […] Normal Subgroups, Isomorphic Quotients, But Not IsomorphicLet $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$.Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.Proof.We give a […] Any Subgroup of Index 2 in a Finite Group is NormalShow that any subgroup of index $2$ in a group is a normal subgroup.Hint.Left (right) cosets partition the group into disjoint sets.Consider both left and right cosets.Proof.Let $H$ be a subgroup of index $2$ in a group $G$.Let $e \in G$ be the identity […] Group of Order 18 is SolvableLet $G$ be a finite group of order $18$.Show that the group $G$ is solvable.DefinitionRecall that a group $G$ is said to be solvable if $G$ has a subnormal series\[\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G\]such […]
Suppose $G$ is a finite group. We will say, that it force solvability if any finite group $H$, such that $G$ is isomorphic to its maximal proper subgroup, is solvable. Does there exist some sort of classification of such groups? On one hand, all such groups have to be solvable. On the other hand, there are several large classes of such groups known. One of them is yielded by a theorem from “A condition for the solvability of a finite group” by W. E. Deskins: All groups of nilpotency class 2 force solvability The other comes from a theorem by Thomas Browning: If a finite group is nilpotent and all its $2$-subgroups are normal, then it forces solvability. His proof is here: Let $G$ be minimal such that $G$ is not solvable and such that $G$ contains a maximal subgroup $M$ that is nilpotent and whose 2-subgroups are normal. If $M$ contains a nontrivial normal subgroup $N$ of $G$ then $G/N$ contradicts the minimality of $G$. Thus, $M$ does not contain nontrivial normal subgroups of $G$. In particular, $N_G(P)=M$ for all Sylow $p$-subgroups $P$ of $M$. Then $P$ is a Sylow $p$-subgroup of $N_G(P)$ so $P$ is a Sylow $p$-subgroup of $G$. This shows that $M$ is a Hall subgroup of $G$. If $P$ is a Sylow $p$-subgroup of $M$ and if $Q$ is a nontrivial normal subgroup of $P$ then $N_G(Q)=M$ which has a normal $p$-complement. For $p=2$, Frobenius' normal $p$-complement theorem gives that $G$ has a normal $p$-complement. For $p\geq3$, Thompson's normal $p$-complement theorem or Glauberman's normal $p$-complement theorem gives that $G$ has a normal $p$-complement (since you only have to consider characteristic $p$-subgroups). Thus, for each prime $p$ dividing the order of $M$, $G$ has a normal $p$-complement. Then $M$ has a normal complement $N$ in $G$. Since $M$ is solvable but $G$ is not solvable, $N$ is not solvable. In particular, $N$ does not admit a fixed-point-free automorphism of prime order. If $m\in Z(M)$ has prime order then $C_N(m)$ is nontrivial. Then $C_N(m)M$ is a subgroup of $G$ that properly contains $M$ so $C_N(m)M=G$ by the maximality of $M$. Comparing cardinalities shows that $C_N(m)=N$ so $m\in Z(G)$. Then $\langle m\rangle$ is a nontrivial normal subgroup of $G$ contained in $M$ which is a contradiction. It is also known, that if $H$ and $K$ force solvability, then $H \times K$ also does. However, I do not know, whether there is anything else here...
Rikka with Number Time Limit: 8000/4000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others) Description As we know, Rikka is poor at math. Yuta is worrying about this situation, so he gives Rikka some math tasks to practice. There is one of them: In radix $d$, a number $K=(A_1A_2...A_m)_d(A_i \in [0,d),A_1 \neq 0)$ is good if and only $A_1-A_m$ is a permutation of numbers from $0$ to $d-1$. A number $K$ is good if and only if there exists at least one $d \geq 2$ and $K$ is good under radix $d$. Now, Yuta wants to calculate the number of good numbers in interval $[L,R]$ It is too difficult for Rikka. Can you help her? Input The first line contains a number $t(1 \leq t \leq 20)$, the number of the testcases. For each testcase, the first line contains two decimal numbers $L,R(1 \leq L \leq R \leq 10^{5000})$. Output For each testcase, print a single line with a single number -- the answer modulo $998244353$. Source 2017 Multi-University Training Contest - Te
Viscous Flow Questions & Answers This is a good question. The conservative form is not necessary to solve most viscous flow problems in this course, so I prefer to derive only the non-conservative form. In other courses (like Intro to CFD for instance), then we need the conservative form because it's easier to discretize. I fixed the mistakes, thank you for pointing these out. It must have been quite late when I wrote this. As for Assign. 4, let's look into this again when we get there. Don't go too fast ;) Expand the terms $\partial \rho v/\partial t$ as $\rho \partial v/\partial t+...$ and $\partial \rho v^2/\partial y$ as $ \rho v \partial v/\partial y+...$ and so on. Then regroup terms so that the mass conservation equation appears and set those terms to zero. Hm, I don't understand perfectly. Are you referring to the sign of $\tau$? Then please rewrite your question below and make this more clear. $\pi$
№ 8 All Issues Türkmen E. ↓ Abstract Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 400-411 Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V \| Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a $\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper class $\scr{RS}$ . Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1140-1146 Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a no-local Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.
Finance theory suggests that capital markets reflect all available information about firms in the firms' stock prices. Given this basic premise, one can study how a particular event changes a firm's prospects by quantifying the impact of the event on the firm's stock. Finance scholars have developed the 'event study methodology' to perform this type of analysis - in its most common form, with a focus on stock returns, in less used forms, with a focus on trading volumes and volatilities. Return event studies quantify an events economic impact in so-called abnormal returns. Abnormal returns are calculated by deducting the returns that would have been realized if the analyzed event would not have taken place (normal returns) from the actual returns of the stocks. While the actual returns can be empirically observed, the normal returns need to be estimated. For this, the event study methodology makes use of expected return models, which are also common to other areas of Finance research. The 'market model' is the most frequently used expected return model. It builds on the actual returns of a reference market and the correlation of the firm's stock with the reference market. Equation (1) describes the model formally. The abnormal return on a distinct day within the event window represents the difference between the actual stock return ($R_{i,t}$) on that day and the normal return, which is predicted based on two inputs; the typical relationship between the firm's stock and its reference index (expressed by the $\alpha$ and $\beta$ parameters), and the actual reference market's return ($R_{m,t}$). $$AR_{i,t}=R_{i,t}-(\alpha_i+\beta_i R_{m,t}) (1)$$ Such an analysis performed for multiple events of the same event type (i.e., a sample study) may yield typical stock market response patterns, which have been at the center of prior academic research. Typical abnormal returns associated with a distinct point of time before or after the event day are defined as follows. $$AAR= \frac{1}{N} \sum\limits_{i=1}^{N}AR_{i,t} (2)$$ To measure the total impact of an event over a particular period of time (termed the 'event window'), one can add up individual abnormal returns to create a 'cumulative abnormal return'. Equation (2) formally shows this practice. The most common event window found in studies is a three-day event window starting at $t_1=-1$ and ending at $t_2=1$. $$CAR(t_1,t_2)=\sum\limits_{t=t_1}^{t_2} AR_{i,t} (3)$$ Figure 1 plots the CAR values of two different corporate event types, 'FDA approvals' and the issuance of 'special dividends' as they change when the event window is gradually extended. The figure suggests that capital market perceive both event types as good news. Figure 1: Cumulative Abnormal Returns Over Expanding Event Windows Lengths Adapted from Neuhierl et al. (2011: 48) In a 'sample event study' that holds multiple observations of individual event types (e.g., acquisitions), one can further calculate 'cumulative average abnormal returns (CAARs)', which represent the mean values of identical events. Equation 3 shows the formal equation for CAARs and Figure 2 illustrates CAARs and their standard deviations at the example of a ten-year study in the global insurance industry (Schimmer, 2012). The presented CAARs represent the average stock market responses (in percent) to press releases describing different types of corporate decisions. $$CAAR= \frac{1}{n} \sum\limits_{i=1}^{n}CAR(t_1,t_2) (4)$$ Figure 2: Cumulative Average Abnormal Returns (-1,+1) in a Sample Event Study Own Figure For a further introduction to the methodology, you may also view this third-party video from youtube: References and additional links Neuhierl, A., Scherbina, A. and Schlusche, B. 2011. 'Market reaction to corporate press releases'. Available at SSRN: http://ssrn.com/abstract=1556532. Schimmer, M. 2012. Competitive dynamics in the global insurance industry: Strategic groups, competitive moves, and firm performance. Wiesbaden: SpringerGabler.
Many inductive arguments work by projecting an observed pattern onto as-yet unobserved instances. All the ravens we’ve observed have been black, so all ravens are. All the emeralds we’ve seen have been green, so all emeralds are. The assumption that the unobserved will resemble the observed seems to be central to induction. Philosophers call this assumption the Principle of Induction.11 See Section 2.5 and Appendix C for previous discussions of the Principle of Induction. But what justfies this assumption? Do we have any reason to think the parts of reality we’ve observed so far are a good representative of the parts we haven’t seen yet? Actually there are strong reasons to doubt whether this assumption can be justified. It may be impossible to give any good argument for expecting the unobserved to resemble the observed. We observed previously that there are two kinds of argument, inductive and deductive. Some arguments establish their conclusions necessarily, others only support them with high probability. If there is an argument for the Principle of Induction, it must be one of these two kinds. Let’s consider each in turn. Could we give an inductive argument for the Principle of Induction? At first it seems we could. Scientists have been using inductive reasoning for millenia, often with great success. Indeed, it seems humans, and other creatures too, have relied on it for much longer, and could not have survived without it. So the Principle of Induction has a very strong track record. Isn’t that a good argument for believing it’s correct? Figure D.1: David Hume (1711–1776) raised the problem of induction in $1739$. Our presentation of it here is somewhat modernized from his original argument. No, because the argument is circular. It uses the Principle of Induction to justify believing in the Principle of Induction. Consider that the argument we are attempting looks like this: The principle has worked well when we’ve used it in the past. Therefore it will work well in future instances. This is an inductive argument, an argument from observed instances to ones as yet unobserved. So, under the hood, it appeals to the Principle of Induction. But that’s exactly the conclusion we’re trying to establish. And one can’t use a principle to justify itself. What about our second option: could a deductive argument establish the Principle of Induction? Well, by definition, a deductive argument establishes its conclusion with necessity. Is it necessary that the unobserved will be like the observed? It doesn’t look like it. It seems perfectly possible that tomorrow the world will go haywire, randomly switching from pattern to pattern, or even to no pattern at all. Maybe tomorrow the sun will fail to rise. Maybe gravity will push apart instead of pull together, and all the other laws of physics will reverse too. And just as soon as we get used to those patterns and start expecting them to continue, another pattern will arise. And then another. And then, just as we give up and come to have no expectation at all about what will come next, everything will return to normal. Until we get comfortable and everything changes again. Thankfully, our universe hasn’t been so mischievous. We get surprised now and again, but for the most part inductive reasoning is pretty reliable, when we do it carefully. But we’re lucky in this respect, is the point. Nature could have been mischievous, totally unpredictable. It is not a necessary truth that the unobserved must resemble the observed. And so it seems there cannot be a deductive argument for the Principle of Induction. Because such an argument would establish the principle as a necessary truth. The two problems are quite different, but it’s easy to get them confused. The problem we’re discussing here is about justifying the Principle of Induction. Is there any reason to believe it’s true? Whereas the grue paradox points out that we don’t even really know what the principle says, in a way. It says that what we’ve observed is a good indicator of what we haven’t yet obsered. But in what respects? Will unobserved emeralds be green, or will they be grue? So the challenge posed by grue is to spell out, precisely, what the Principle of Induction says. But even if we can meet that challenge, this challenge will remain. Why should we believe the principle, once it’s been spelled out? Neither a deductive argument nor an inductive argument seems possible. The Problem of Induction is centuries old. Isn’t it out of date? Hasn’t the modern, mathematical theory of probability solved the problem for us? Not at all, unfortunately. One thing we learn in this book is that the laws of probability are very weak in a way. They don’t tell us much, without us first telling them what the prior probabilities are. And as we’ve seen over and again throughout Part III, the problem of priors is very much unsolved. For example, suppose we’re going to flip a mystery coin five times. We don’t know whether the coin is fair or biased, but we hope to have some idea after a few flips. Now suppose we get through the first four flips and they’ve all been tails. The Principle of Induction says we should expect the next flip to be tails too. At least, that outcome should now be more probable. Do the laws of probability agree? Well, we need to calculate the quantity: \[ \p(T_5 \given T_1 \wedge T_2 \wedge T_3 \wedge T_4).\] The definition of conditional probability tell us: \[ \begin{aligned} \p(T_5 \given T_1 \wedge T_2 \wedge T_3 \wedge T_4) &= \frac{\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4 \wedge T_5)} {\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4)}. \end{aligned} \] But the laws of probability don’t tell us what numbers go in the numerator and the denominator. The numbers have to be between $0$ and $1$. And we have to be sure mutually exclusive propositions have probabilities that add up, according to the Additivity rule. But that still leaves things wide open. For example, we could finish the calculation this way: \[ \begin{aligned} \p(T_5 \given T_1 \wedge T_2 \wedge T_3 \wedge T_4) &= \frac{\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4 \wedge T_5)} {\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4)}\\ &= \frac{1/32}{1/16}\\ &= 1/2. \end{aligned} \] Or we could finish it this way: \[ \begin{aligned} \p(T_5 \given T_1 \wedge T_2 \wedge T_3 \wedge T_4) &= \frac{\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4 \wedge T_5)} {\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4)}\\ &= \frac{5/20}{6/20}\\ &= 5/6. \end{aligned} \] We could even do this: \[ \begin{aligned} \p(T_5 \given T_1 \wedge T_2 \wedge T_3 \wedge T_4) &= \frac{\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4 \wedge T_5)} {\p(T_1 \wedge T_2 \wedge T_3 \wedge T_4)}\\ &= \frac{0}{1}\\ &= 0. \end{aligned} \] All these options result from different choices of prior probabilities. And the laws of probability don’t tell us what prior probabilities we must choose, as we learned in Part III. So the laws of probability don’t by themselves tell us what to expect. It could be undecided, with heads and tails equally probable on the final toss ($1/2$). Or the pattern of tails could continue into the future with high probability ($5/6$). There could even be no chance of the pattern continuing ($0$). The laws of probability only tell us what to expect once we’ve specified the necessary prior probabilities. The problem of induction challenges us to justify one choice of prior probabilities over the alternatives. In the $280$ years since this challenge was first raised by David Hume, no answer has gained general acceptance.
Current browse context: cond-mat.stat-mech Change to browse by: References & Citations Bookmark(what is this?) Condensed Matter > Statistical Mechanics Title: Correlation Function and Simplified TBA Equations for XXZ Chain (Submitted on 30 Dec 2010 (v1), last revised 8 Jan 2011 (this version, v2)) Abstract: The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by $\log 2$ and Riemann's zeta functions with odd integer argument $\zeta(3),\zeta(5),\zeta(7),...$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions. Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$. Submission historyFrom: Minoru Takahashi [view email] [v1]Thu, 30 Dec 2010 03:40:44 GMT (93kb) [v2]Sat, 8 Jan 2011 09:20:53 GMT (64kb)
№ 8 All Issues Volume 62, № 3, 2010 Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 291–300 We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment $[–1, 1]$ with respect to a measure, or, more precisely, with respect to the functional $μ(f) = \text{mes}\left\{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1 \right\}$. We also discuss an analogous problem with respect to the integral functionals $∫_{–1}^1 φ (∣f (x)∣) dx$ for functions $φ$ that are defined, nonnegative, and nondecreasing on the semiaxis $[0, +∞)$. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 301–314 Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ \|x(t)\|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(\|t_j\|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 315–326 We obtain explicit formulas that express the Hankel determinants of functions given by their expansions in continued $P$-fractions in terms of the parameters of the fraction. As a corollary, we obtain a lower bound for the capacity of the set of singular points of these functions, an analog of the van Vleck theorem for $P$-fractions with limit-periodic coefficients, another proof of the Gonchar theorem on the Leighton conjecture, and an upper bound for the radius of the disk of meromorphy of a function given by a $С$-fraction. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 327–331 A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 332–368 In a linear space of dimension $n$ over the field $\mathbb{F}_2$, we construct a set $A$ of given density such that the Fourier transform of $A$ is large on a large set, and the intersection of $A$ with any subspace of small dimension is small. The results obtained show, in a certain sense, the sharpness of one theorem of J. Bourgain. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 369–386 In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 387–395 We prove the following statement, which is a quantitative form of the Luzin theorem on $C$-property: Let $(X, d, μ)$ be a bounded metric space with metric $d$ and regular Borel measure $μ$ that are related to one another by the doubling condition. Then, for any function $f$ measurable on $X$, there exist a positive increasing function $η ∈ Ω\; \left(η(+0) = 0\right.$ and $η(t)t^{−a}$ decreases for a certain $\left. a > 0\right)$, a nonnegative function $g$ measurable on $X$, and $a$ set $E ⊂ X, μE = 0$, for which $$\|f(x)−f(y)\| ⩽ [g(x)+g(y)]η(d(x,y)),\;x,y ∈ X \setminus E.$$ If $f ∈ L^p(X),\; p >0$, then it is possible to choose $g$ belonging to $L^p (X)$. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 396–408 We study the approximation of the classes of functions by the manifold $R_n$ formed by all possible linear combinations of $n$ ridge functions of the form $r(a · x))$. It is proved that, for any $1 ≤ q ≤ p ≤ ∞$, the deviation of the Sobolev class $W^r_p$ from the set $R_n$ of ridge functions in the space $L_q (B^d)$ satisfies the sharp order $n^{-r/(d-1)}$. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 409–422 An asymptotically sharp estimate is obtained for the best one-sided approximation of a step by algebraic polynomials in the space $L_1$. Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 423–431 We obtain an upper bound for the least value of the factor $М$ for which the Kolmogorov widths $d_n (W_C^r, C)$ are equal to the relative widths $K_n (W^C_r, MW^C_j, C)$ of the class of functions $W_C^r$ with respect to the class $MW^C_j$, provided that $j > r$. This estimate is also true in the case where the space $L$ is considered instead of $C$.
I'm trying to get a conclusive numerical value for Mean Squared Error (MSE) as the performance metric of a few CS sparse recovery algorithms. To do this, I vary the number of measurements ($M$) taken from an $N$-dimensional vector with $M\ll N$ using the sensing matrix $\mathbf A \in \mathbb K^{M\times N}$. The goal is to find the MSE at each $M$. At the moment, randomly generated sparse signals are sensed using random sensing matrices ($\mathbf A$), leading to $M$ measurements from which the original sparse vectors are then recovered. For each $M$, I generate 30 sensing matrices and 30 signals for each $\mathbf A$. This totals 900 trials for a single $M$. I first find the difference between the original and reconstructed signal. Then I find the $l_2$ norm of the difference (error) vector. This value is found for all trials for a single value of $M$, then squared and averaged to find the MSE at that value of $M$. I am curious if there is a lower limit to the number of trials I should perform for each MM so that I get a conclusive value of MSE? How do I defend against a person who says I should do more? Kindly point me to the appropriate forum if the question is not relevant to this domain.
A Concentration Phenomenon for Semilinear Elliptic Equations Article First Online: 529 Downloads Citations Abstract For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$. Here ${V \geqq 0}$ and $$-\Delta{u} + V(x)u = Q_n(x)\|{u}\|^{p-2}u$$ Q are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets { n Q > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$. We show that if n u is a nontrivial solution corresponding to n Q , then the sequence ( n u ) concentrates at n x 0with respect to the H 1and certain L -norms. We also show that if the sets { q Q > 0} shrink to two points and n u are ground state solutions, then they concentrate at one of these points. n KeywordsSoliton Nontrivial Solution Dielectric Response Kerr Nonlinearity Ground State Solution These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. Preview Unable to display preview. Download preview PDF. References 1.Ambrosetti, A., Arcoya, D., Gámez, J.L.: Asymmetric bound states of differential equations in nonlinear optics. Rend. Sem. Math. Univ. Padova 100, 231–247 (1998). http://www.numdam.org/item?id=RSMUP_1998__100__231_0 2.Bandle, C., Marcus, M.: “Large” solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour. J. Anal. Math. 58, 9–24 (1992). doi: 10.1007/BF02790355 (Festschrift on the occasion of the 70th birthday of Shmuel Agmon) 3. 4. 5. 6. 7. 8. 9.Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, Vol. 224, 2nd edn. Springer, Berlin, 1983Google Scholar 10. 11. 12. 13. 14. 15. 16.Strauss, W.A.: The nonlinear Schrödinger equation. Contemporary developments in continuum mechanics and partial differential equations. Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. North-Holland Math. Stud., Vol. 30. North-Holland, Amsterdam, 452–465, 1978Google Scholar 17. 18. 19. 20. 21. Copyright information © Springer-Verlag Berlin Heidelberg 2012
Computational Aerodynamics Questions & Answers You don't necessarily need to express $\phi$ as a function of $U$ to determine $\partial \phi/\partial U$ in the same way as you don't need to express $F$ as a function of $U$ to obtain $\partial F/\partial U$. Use a 1D extrapolation polynomial. 2D is too time consuming to compute. No, $\|A\|$ is a $2 \times 2$ matrix determined from an average state function of $Z_L$ and $Z_R$. I updated the answers to make them more clear. Hm, I see a problem in your approach. You shouldn't be finding a polynomial when determining the flux with a TVD minmod2 limiter. $\pi$
I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial representation of the compact circle group $K$ induces a discrete series representation. I have found a description of the discrete series in the book of Knapp: $$ \{ f: \text{analytic for Im}~z > 0 ~:~ \\| f \\|^2 = \int \|f(z)^2\| y^{n-2} dx dy < \infty \}. $$ with action $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} f(z) = \left(-bz + d\right)^{-n} f \left( \frac{az -c}{-bz +d } \right). $$ I would like to show explicit how to realize a discrete series representation as the kernel of a Dirac operator, as it discussed in (Atiyah-Schmid). I think that the idea is that the kernel of the Dirac operator ensures that $f$ is analytic, and that the $y^{n-2}$ comes from the $G$-invariant metric on the vector bundle $S \otimes V$ over $G/K$, where $S$ is the spinor bundle and $V$ an irreducible representation of $K$. I don’t know how to compute the metric on the twisted vector bundle $S \otimes V$, and how to show that the sections of this vector bundle give a discrete series representation. Is there any reference where they show how to construct the discrete series of SL(2,R) as the kernel of twisted Dirac operators?
Problem 4 from the IMO 2013 What Might This Be About? Problem Let $ABC$ be an acute triangle with orthocenter $H,$ and let $W$ be a point on the side $BC,$ lying strictly between $B$ and $C.$ The points $M and N$ are the feet of the altitudes from $B$ and $C,$ respectively. Denote by $\omega_1$ is the circumcircle of $BWN,$ and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1.$ Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM,$ and let $Y$ be the point such that $WY$ is a diameter of $\omega_2.$ Prove that $X,$ $Y$ and $H$ are collinear. Generalization The solution below is by Leonard Giugiuc, Romania. As a matter of fact, Leonard proves a more general statement: Let $ABC$ be a triangle with orthocenter $H;$ assume neither of the angles at $B$ or $C$ is right, and let $W$ be a point on the side line $BC$ different from either $B$ or $C.$ The points $M and N$ are the feet of the altitudes from $B$ and $C,$ respectively. Denote by $\omega_1$ is the circumcircle of $BWN,$ and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1.$ Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM,$ and let $Y$ be the point such that $WY$ is a diameter of $\omega_2.$ Prove that $X,$ $Y$ and $H$ are collinear. Solution Choose Cartesian coordinate system in which $A(0,2),$ $B(-2b,0),$ and $C(2c,0),$ $(b\ne 0\ne c).$ Then, $a=\cot B,$ $c=\cot C,$ and $H(0, 2bc).$ Let $W(2a,0),$ with $-b\ne a\ne c,$ and let $D$ and $E$ be the centers of $\omega_1$ and $\omega_2,$ respectively. Then $D(a-b,y)$ and $E(a+c,z),$ for some $y$ and $z.$ $D$ is the midpoint of $WX$ and $E$ is the midpoint of $WY.$ Therefore, $X(-2b,2y)$ and $Y(2a,2z).$ The points $X,$ $Y,$ $H$ are collinear iff $\left\| \begin{array}{ccc} 0 & 2bc & 1 \\ -2b & 2y & 1 \\ 2c & 2z & 1 \end{array} \right\|=0,$ i.e., when $bz+cy=bc(b+c).$ Let $N(2n,2q)$ be the feet the feet of the altitude from $C.$ Then, since the equation of $AB$ is $-x+by=2b,$ the equation of $CN$ is $\displaystyle\frac{x-2c}{-1}=\frac{y}{b}.$ Solving for $x$ and $y$ leads to $\displaystyle n=\frac{b(bc-1)}{1+b^2}$ and $\displaystyle q=\frac{b(b+c)}{1+b^2}.$ Further, $DN=DW$ implies $(a+b)^{2}+y^{2}=(a-b-2n)^{2}+(y-2q)^{2}$ from which $\displaystyle y=\frac{n^2+q^2+nb}{q}-\frac{a(n+b}{q}.$ Substituting the values of $n$ and $q$ gives $y=bc-ab.$ Similarly, $z=bc+ab$ such that, indeed, $bz+cy=bc(b+c).$ Note: there is another, synthetic solution. 65461492
A friend of mine and me have decided to try the brute-force method and compute some values of $t$ for small values of $n$ and $d$. This is totally impossible without employing pruning, and we hope that the tricks we have found will give some insight in the rest of the problem. So far, we have not managed to get the doubly-exponential running time of the brute force method down significantly (roughly $3^{2^n}$ is our best bound so far) and hence we have not yet reached our original goal of somehow predicting the function behind $f$ from its first few values. We have also not studied all the comments of the previous threads in detail, so some of this may already be known - we basically had fun making our code fast and wanted to post our results somewhere, if I had a functioning LaTeX environment I would have put this on the ArXiV. Code (it's not exactly production code...): http://pastebin.com/bSetW8JS. Values: f(d=2, n)=2n-1 for n <= 6 f(d=3, n=3) = 6 {} {0} {01} {012} {12} {2} f(d=4, n=4) = 8 f(d=3, n=4) = 8 {} {0} {01} {1,02,03} {2,13} {123} {23} {3} {} {0} {01} {2,013} {1,02,03} {023} {23} {3} f(d=5, n=5) = 11 f(d=4, n=5) = 11 f(d=3, n=5) = 11 {} {0} {01} {1,02} {2,13,04} {12,03,14} {3,124} {23,24} {234} {34} {4} {} {0} {01} {1,02} {2,13,04} {12,03,14} {3,124} {23,24} {234} {34} {4} {} {0} {01} {012,3} {02,12,013,014} {13,023,04,124} {123,024} {23,24} {234} {34} {4} {} {0} {01} {012,13} {02,12,013} {03,123,014,024} {023,124} {23,24} {234} {34} {4} We say that the sequence ${\cal F}_1, ..., {\cal F}_t$ is convex if (*) holds. Our approach constructs convex sequences by appending families to shorter convex sequences, essentially using that if ${\cal F}_1, ..., {\cal F}_t$ is convex, then ${\cal F}_1, ..., {\cal F}_{t-1}$ is convex. We note that ${\cal F}_1, ..., {\cal F}_t$ is convex if and only if for all $A \in {\cal F}_t$ we have that ${\cal F}_1, ..., {\cal F}_{t-1}, \{ A \}$ is convex. We say that $A$ is compatible with ${\cal F}_1, ..., {\cal F}_{t-1}$ if ${\cal F}_1, ..., {\cal F}_{t-1}, \{ A \}$ is convex - we save computation time by computing the sets that are compatible with a sequence and then taking the elements of their powerset as the new ${\cal F}_t$, rather than determining if ${\cal F}_1, ..., {\cal F}_t$ is convex directly. Our next speedup is essentially dynamic programming. We try to find an equivalence relation $\sim$ on convex sequences with the following two properties. First, if ${\cal F}_1, ..., {\cal F}_t \sim {\cal F}'_1, ..., {\cal F}'_t$ for two convex sequences, then $A$ is compatible with ${\cal F}_1, ..., {\cal F}_t$ if and only if it is compatible with ${\cal F}'_1, ..., {\cal F}'_t$. Second, if ${\cal F}_1, ..., {\cal F}_t \sim {\cal F}'_1, ..., {\cal F}'_t$ and ${\cal F}_1, ..., {\cal F}_t, {\cal F}_{t+1}$ is convex, then ${\cal F}_1, ..., {\cal F}_t, {\cal F}_{t+1} \sim {\cal F}'_1, ..., {\cal F}'_t, {\cal F}_{t+1}$. Furthermore, we would like it if we can determine whether a set is compatible with elements from an equivalence class, and determine a representative of the equivalence class of ${\cal F}_1, ..., {\cal F}_t, {\cal F}_{t+1}$ given ${\cal F}_{t+1}$ and a representative of the equivalence class of ${\cal F}_1, ..., {\cal F}_t$. The ensuing dynamic programming algorithm is then obvious. The number of equivalence classes (along with the time taken by the above two operations) then gives a bound on the running time of the obvious dynamic programming algorithm. For the equivalence we use to get our bound, we use a characterization of convexity that is based on `intervals' as follows. Given a subset $A$ of $\{ 1, \dots, n \}$, we say $A$ is contiguous with respect to a (not necessarily convex) sequence ${\cal F}_1, ..., {\cal F}_t$ if $\{ k \mid \exists B \in {\cal F}_k : A \subseteq B \} = \{ i, \dots, j \}$ for some integers $1 \leq i \leq j \leq n$. We say that $(i, j)$ is the interval for $A$ w.r.t. this sequence. It is easily seen that ${\cal F}_1, ..., {\cal F}_t$ is convex if and only if all subsets of $\{ 1, \dots, n \}$ are contiguous with respect to the sequence. Now, given a convex sequence ${\cal F}_1, ..., {\cal F}_t$, we mark all subsets of $\{ 1, \dots, n \}$ as not touched, disallowed or active as follows: all elements of ${\cal F}_t$ are active, all elements of ${\cal F}_1, ..., {\cal F}_{t-1}$ are disallowed and all supersets $B$ of sets $A$ whose interval with respect to ${\cal F}_1, ..., {\cal F}_{t-1}$ is $(i, j)$ with $j < t - 1$ are also disallowed. It is immediate that a set $A$ is compatible with the sequence if and only if it is marked as not touched. We define two sequences as equivalent under $\sim$ if their marking is equal. It is easily seen that this equivalence relation satisfies our two properties. For computing whether a set $B$ should be disallowed by the interval condition, we can use the equivalent condition 'there exists a set $C \in {\cal F}_t$ such that for no set $D \in {\cal F}_{t+1}$, $B \cap C \subseteq D$'. $3^{2^n}$ is an immediate bound on the number of equivalence classes. We also use various extra prunings. We only consider antichains for ${\cal F}_{t+1}$ and we require that the elements of its elements come from ${1, \dots, i}$. Lastly, we use the optimization that ${\cal F}_1 = \{ \{ 1 \} \}, {\cal F}_2 = \{ \{ 1, 2 \} \}$ for optimally long sequences (and similar for ${\cal F}_{t-1}$ and ${\cal F}_t$). We imagine that investigating the behaviour of ${\cal F}_3$ may result in more drastic savings.
As a matter of fact, it isn't hard to construct a multiplicative sequence $a_n$ such that $f(z)$ is an entire function without zeroes. Unfortunately, it is completely useless for the questions that you brought up as "motivation". Here is the construction. Claim 1: Let $\lambda_j\in [0,1]$ ($j=0,\dots,M$). Assume that $\|a_j\|\le 1$ and $\sum_ja_j\lambda_j^p=0$ for all $0\le p\le P$. Then, if $P>2eT$, we have $\left\|\sum_j a_je^{\lambda_j z}\right\|\le (M+1)(eT/P)^P$ Proof: Taylor decomposition and a straightforward tail estimate. Claim 2: Let $P$ be large enough. Let $\Delta>0$, $M>P^3$ and $(M+1)\Delta<1$. Let $I_j$ ($0\le j\le M$) be $M+1$ adjacent intervals of length about $\Delta$ each arranged in the increasing order such that $I_0$ contains $0=\lambda_0$. Suppose that we choose one $\lambda_j$ in each interval $I_j$ with $j\ge 1$. Then for every $\|a_0\|\le 1$, there exist $a_j\in\mathbb C$ such that $\|a_j\|\le 1$ and $\sum_{j\ge 0} a_j\lambda_j^p=0$ for $0\le p\le P$. Proof: By duality, we can restate it as the claim that $\sum_{j\ge 1}\|Q(\lambda_j)\|\ge \|Q(0)\|$ for every polynomial $Q$ of degree $P$. Now, let $I$ be the union of $I_j$. It is an interval of length about $M\Delta$, so, by Markov's inequality, $\|Q'\|\le CP^2(M\Delta)^{-1} A\le CP^{-1}\Delta^{-1}A$ where $A=\max_I \|Q\|\ge \|Q(0)\|$. But then on the 5 intervals $I_j$ closest to the point where the maximum is attained, we have $\|Q\|\ge A-5\Delta CP^{-1}\Delta^{-1}A\ge A/2$. The rest is obvious. Claim 3: Suppose that $a_0$ is fixed and $a_j$ ($j\ge 1$) satisfy $\sum_{j\ge 0} a_j\lambda_j^p=0$ for $0\le p\le P$. Then we can change $a_j$ with $j\ge 1$ so that all but $P+1$ of them are exactly $1$ in absolute value and the identities still hold. Proof: As long as we have more than $P+1$ small $a_j$, we have a line of solutions of our set of $P+1$ linear equations. Moving along this line we can make one of small $a_j$ large. Repeating this as long as possible, we get the claim. Now it is time to recall that the logarithm of the function $f(z)$ is given by $$L(z)=\sum_{n\in\Lambda}a_ne^{-z\log n}$$where $\Lambda$ is the set of primes and prime powers and $a_n=m^{-1}a_p^m$ if $n=p^m$. We are free to choose $a_p$ for prime $p$ in any way we want but the rest $a_n$ will be uniquely determined then. The key point is that we have much more primes than prime powers for unit length. So, split big positive numbers into intervals from $u$ to $e^\Delta u$ where $\Delta$ is a slowly decaying function of $u$ (we'll specify it later). Formally we define the sequence $u_k$ by $u_0=$something large, $u_{k+1}=e^{\Delta(u_k)}u_k$ but to put all those backward apostrophes around formulae is too big headache, so I'll drop all indices. Choose also some slowly growing functions $M=M(u)$ and $P=P(u)$ to be specified later as well. We need a few things: 1) Each interval should contain many primes. Since the classical prime number theorem has the error term $ue^{-c\sqrt{\log u}}$, this calls for $\Delta=\exp\{-\log^{\frac 13} u\}$Then we still have at least $u^{4/5}$ primes in each interval (all we need is to beat $u^{1/2}$ with some margin). 2) We should have $M\Delta\ll 1$, $M>P^3$, and $u\left(\frac{eT}P\right)^P\le (2u)^{-T-3}$ for any fixed $T>0$ and all sufficiently large $u$. This can be easily achieved by choosing $P=\log^2 u$ and $M=\log^6 u$. 3) At last, we'll need $M(P+\sqrt u)\ll u^{4/5}$, which is true for our choice. Now it is time to run the main inductive construction. Suppose that $a_n$ are already chosen for all $n$ in the intervals up to $(u,e^{\Delta}u)$ and we still have almost all primes in the intervals following the current interval free (we'll check this condition in the end). We want to assign $a_p$ for all $p$ in our interval for which the choice hasn't been made yet or was made badly. We start with looking at all $a_p$ that are not assigned yet or assigned in a lame way, i.e., less than one in absolute value. Claim 3 (actually a small modification of it) allows us to upgrade all of them but $P+1$ to good ones (having absolute value $1$) at the expense of adding an entire function that in the disk of radius $T$ is bounded by $(2u)^{T} u\left(\frac{eT}P\right)^P\le u^{-3}$ to $L(z)$. Now we are left with at most $\sqrt u$ powers of primes and $P+1$ lame primes to take care of. We need the prime powers participate in small sums as they are and we need the small coefficients to be complemented by something participating in small sums too. For each of them, we choose $M$ still free primes in the next $M$ intervals (one in each) and apply Claim 2 to make a (lame) assignment so that the corresponding sum is again bounded by $u^{-3}$ in the disk of radius $T$. We have at most $u$ such sums, so the total addition will be at most $u^{-2}$. This will finish the interval off. Now it remains to notice that we used only about $\sqrt u+P$ free primes in each next interval and went only $M$ intervals ahead. This means that in each interval only $M(\sqrt u+P)$ free primes will ever be used for compensating the previous intervals, so we'll never run out of free primes. Also, the sum of the blocks we constructed will converge to an entire function. At last, when $\Re z>1$, we can change the order of summation and exponentiate finally getting the Dirichlet series representation that we need. The end.
Multitype pair correlation function (cross-type) Calculates an estimate of the cross-type pair correlation function for a multitype point pattern. Usage pcfcross(X, i, j, ..., r = NULL, kernel = "epanechnikov", bw = NULL, stoyan = 0.15, correction = c("isotropic", "Ripley", "translate"), divisor = c("r", "d")) Arguments X The observed point pattern, from which an estimate of the cross-type pair correlation function $g_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). i The type (mark value) of the points in Xfrom which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of marks(X). j The type (mark value) of the points in Xto which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of marks(X). … Ignored. r Vector of values for the argument $r$ at which $g(r)$ should be evaluated. There is a sensible default. kernel Choice of smoothing kernel, passed to density.default. bw Bandwidth for smoothing kernel, passed to density.default. stoyan Coefficient for default bandwidth rule; see Details. correction Choice of edge correction. divisor Choice of divisor in the estimation formula: either "r"(the default) or "d". See Details. Details The cross-type pair correlation function is a generalisation of the pair correlation function pcf to multitype point patterns. For two locations $x$ and $y$ separated by a distance $r$, the probability $p(r)$ of finding a point of type $i$ at location $x$ and a point of type $j$ at location $y$ is $$ p(r) = \lambda_i \lambda_j g_{i,j}(r) \,{\rm d}x \, {\rm d}y $$ where $\lambda_i$ is the intensity of the points of type $i$. For a completely random Poisson marked point process, $p(r) = \lambda_i \lambda_j$ so $g_{i,j}(r) = 1$. Indeed for any marked point pattern in which the points of type i are independent of the points of type j, the theoretical value of the cross-type pair correlation is $g_{i,j}(r) = 1$. For a stationary multitype point process, the cross-type pair correlation function between marks $i$ and $j$ is formally defined as $$ g_{i,j}(r) = \frac{K_{i,j}^\prime(r)}{2\pi r} $$ where $K_{i,j}^\prime$ is the derivative of the cross-type $K$ function $K_{i,j}(r)$. of the point process. See Kest for information about $K(r)$. The command pcfcross computes a kernel estimate of the cross-type pair correlation function between marks $i$ and $j$. If divisor="r"(the default), then the multitype counterpart of the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284--285) is used. By default, the recommendations of Stoyan and Stoyan (1994) are followed exactly. If divisor="d"then a modified estimator is used: the contribution from an interpoint distance $d_{ij}$ to the estimate of $g(r)$ is divided by $d_{ij}$ instead of dividing by $r$. This usually improves the bias of the estimator when $r$ is close to zero. There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects associated with the boundary of the spatial window): correction="translate" is the Ohser-Stoyan translation correction, and correction="isotropic" or "Ripley" is Ripley's isotropic correction. The choice of smoothing kernel is controlled by the argument kernel which is passed to density. The default is the Epanechnikov kernel. The bandwidth of the smoothing kernel can be controlled by the argument bw. Its precise interpretation is explained in the documentation for density.default. For the Epanechnikov kernel with support $[-h,h]$, the argument bw is equivalent to $h/\sqrt{5}$. If bw is not specified, the default bandwidth is determined by Stoyan's rule of thumb (Stoyan and Stoyan, 1994, page 285) applied to the points of type j. That is, $h = c/\sqrt{\lambda}$, where $\lambda$ is the (estimated) intensity of the point process of type j, and $c$ is a constant in the range from 0.1 to 0.2. The argument stoyan determines the value of $c$. Value Essentially a data frame containing columns the vector of values of the argument $r$ at which the function $g_{i,j}$ has been estimated the theoretical value $g_{i,j}(r) = 1$ for independent marks. See Also Mark connection function markconnect. Aliases pcfcross Examples # NOT RUN { data(amacrine) p <- pcfcross(amacrine, "off", "on") p <- pcfcross(amacrine, "off", "on", stoyan=0.1) plot(p)# } Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)
[Click here for a PDF of this post with nicer formatting] This is my first set of notes for the UofT course ECE1229, Advanced Antenna Theory, taught by Prof. Eleftheriades, covering ch. 2 [1] content. Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides that match the textbook so closely, there is little value to me taking notes that just replicate the text. Instead, I am annotating my copy of textbook with little details instead. My usual notes collection for the class will contain musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book.) Poynting vector The Poynting vector was written in an unfamiliar form \begin{equation}\label{eqn:chapter2Notes:560} \boldsymbol{\mathcal{W}} = \boldsymbol{\mathcal{E}} \cross \boldsymbol{\mathcal{H}}. \end{equation} I can roll with the use of a different symbol (i.e. not $\BS$) for the Poynting vector, but I’m used to seeing a $ \frac{c}{4\pi} $ factor ([6] and [5]). I remembered something like that in SI units too, so was slightly confused not to see it here. Per [3] that something is a $ \mu_0 $, as in \begin{equation}\label{eqn:chapter2Notes:580} \boldsymbol{\mathcal{W}} = \inv{\mu_0} \boldsymbol{\mathcal{E}} \cross \boldsymbol{\mathcal{B}}. \end{equation} Note that the use of $ \boldsymbol{\mathcal{H}} $ instead of $ \boldsymbol{\mathcal{B}} $ is what wipes out the requirement for the $ \frac{1}{\mu_0} $ term since $ \boldsymbol{\mathcal{H}} = \boldsymbol{\mathcal{B}}/\mu_0 $, assuming linear media, and no magnetization. Typical far-field radiation intensity It was mentioned that \begin{equation}\label{eqn:advancedantennaL1:20} U(\theta, \phi) = \frac{r^2}{2 \eta_0} \Abs{ \BE( r, \theta, \phi) }^2 = \frac{1}{2 \eta_0} \lr{ \Abs{ E_\theta(\theta, \phi) }^2 + \Abs{ E_\phi(\theta, \phi) }^2}, \end{equation} where the intrinsic impedance of free space is \begin{equation}\label{eqn:advancedantennaL1:480} \eta_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} = 377 \Omega. \end{equation} (this is also eq. 2-19 in the text.) To get an understanding where this comes from, consider the far field radial solutions to the electric and magnetic dipole problems, which have the respective forms (from [3]) of \begin{equation}\label{eqn:chapter2Notes:740} \begin{aligned} \boldsymbol{\mathcal{E}} &= -\frac{\mu_0 p_0 \omega^2 }{4 \pi } \frac{\sin\theta}{r} \cos\lr{w t – k r} \thetacap \\ \boldsymbol{\mathcal{B}} &= -\frac{\mu_0 p_0 \omega^2 }{4 \pi c} \frac{\sin\theta}{r} \cos\lr{w t – k r} \phicap \\ \end{aligned} \end{equation} \begin{equation}\label{eqn:chapter2Notes:760} \begin{aligned} \boldsymbol{\mathcal{E}} &= \frac{\mu_0 m_0 \omega^2 }{4 \pi c} \frac{\sin\theta}{r} \cos\lr{w t – k r} \phicap \\ \boldsymbol{\mathcal{B}} &= -\frac{\mu_0 m_0 \omega^2 }{4 \pi c^2} \frac{\sin\theta}{r} \cos\lr{w t – k r} \thetacap \\ \end{aligned} \end{equation} In neither case is there a component in the direction of propagation, and in both cases (using $ \mu_0 \epsilon_0 = 1/c^2$) \begin{equation}\label{eqn:chapter2Notes:780} \Abs{\boldsymbol{\mathcal{H}}} = \frac{\Abs{\boldsymbol{\mathcal{E}}}}{\mu_0 c} = \Abs{\boldsymbol{\mathcal{E}}} \sqrt{\frac{\epsilon_0}{\mu_0}} = \inv{\eta_0}\Abs{\boldsymbol{\mathcal{E}}} . \end{equation} A superposition of the phasors for such dipole fields, in the far field, will have the form \begin{equation}\label{eqn:chapter2Notes:800} \begin{aligned} \BE &= \inv{r} \lr{ E_\theta(\theta, \phi) \thetacap + E_\phi(\theta, \phi) \phicap } \\ \BB &= \inv{r c} \lr{ E_\theta(\theta, \phi) \thetacap – E_\phi(\theta, \phi) \phicap }, \end{aligned} \end{equation} with a corresponding time averaged Poynting vector \begin{equation}\label{eqn:chapter2Notes:820} \begin{aligned} \BW_{\textrm{av}} &= \inv{2 \mu_0} \BE \cross \BB^\conj \\ &= \inv{2 \mu_0 c r^2} \lr{ E_\theta \thetacap + E_\phi \phicap } \cross \lr{ E_\theta^\conj \thetacap – E_\phi^\conj \phicap } \\ &= \frac{\thetacap \cross \phicap}{2 \mu_0 c r^2} \lr{ \Abs{E_\theta}^2 + \Abs{E_\phi}^2 } \\ &= \frac{\rcap}{2 \eta_0 r^2} \lr{ \Abs{E_\theta}^2 + \Abs{E_\phi}^2 }, \end{aligned} \end{equation} verifying \ref{eqn:advancedantennaL1:20} for a superposition of electric and magnetic dipole fields. This can likely be shown for more general fields too. Field plots We can plot the fields, or intensity (or log plots in dB of these). It is pointed out in [3] that when there is $ r $ dependence these plots are done by considering the values of at fixed $ r $. The field plots are conceptually the simplest, since that vector parameterizes a surface. Any such radial field with magnitude $ f(r, \theta, \phi) $ can be plotted in Mathematica in the $ \phi = 0 $ plane at $ r = r_0 $, or in 3D (respectively, but also at $ r = r_0$) with code like that of the following listing Intensity plots can use the same code, with the only difference being the interpretation. The surface doesn’t represent the value of a vector valued radial function, but is the magnitude of a scalar valued function evaluated at $ f( r_0, \theta, \phi) $. The surfaces for $ U = \sin\theta, \sin^2\theta $ in the plane are parametrically plotted in fig. 2, and for cosines in fig. 1 to compare with textbook figures. fig 1. Cosinusoidal radiation intensities fig 2. Sinusoidal radiation intensities Visualizations of $ U = \sin^2 \theta$ and $ U = \cos^2 \theta$ can be found in fig. 3 and fig. 4 respectively. Even for such simple functions these look pretty cool. fig 3. Square sinusoidal radiation intensity fig 4. Square cosinusoidal radiation intensity dB vs dBi Note that dBi is used to indicate that the gain is with respect to an “isotropic” radiator. This is detailed more in [2]. Trig integrals Tables 1.1 and 1.2 produced with tableOfTrigIntegrals.nb have some of the sine and cosine integrals that are pervasive in this chapter. Polarization vectors The text introduces polarization vectors $ \rhocap $ , but doesn’t spell out their form. Consider a plane wave field of the form \begin{equation}\label{eqn:chapter2Notes:840} \BE = E_x e^{j \phi_x} e^{j \lr{ \omega t – k z }} \xcap + E_y e^{j \phi_y} e^{j \lr{ \omega t – k z }} \ycap. \end{equation} The $ x, y $ plane directionality of this phasor can be written \begin{equation}\label{eqn:chapter2Notes:860} \Brho = E_x e^{j \phi_x} \xcap + E_y e^{j \phi_y} \ycap, \end{equation} so that \begin{equation}\label{eqn:chapter2Notes:880} \BE = \Brho e^{j \lr{ \omega t – k z }}. \end{equation} Separating this direction and magnitude into factors \begin{equation}\label{eqn:chapter2Notes:900} \Brho = \Abs{\BE} \rhocap, \end{equation} allows the phasor to be expressed as \begin{equation}\label{eqn:chapter2Notes:920} \BE = \rhocap \Abs{\BE} e^{j \lr{ \omega t – k z }}. \end{equation} As an example, suppose that $ E_x = E_y $, and set $ \phi_x = 0 $. Then \begin{equation}\label{eqn:chapter2Notes:940} \rhocap = \xcap + \ycap e^{j \phi_y}. \end{equation} Phasor power In section 2.13 the phasor power is written as \begin{equation}\label{eqn:chapter2Notes:620} I^2 R/2, \end{equation} where $ I, R $ are the magnitudes of phasors in the circuit. I vaguely recall this relation, but had to refer back to [4] for the details. This relation expresses average power over a period associated with the frequency of the phasor \begin{equation}\label{eqn:chapter2Notes:640} \begin{aligned} P &= \inv{T} \int_{t_0}^{t_0 + T} p(t) dt \\ &= \inv{T} \int_{t_0}^{t_0 + T} \Abs{\BV} \cos\lr{ \omega t + \phi_V } \Abs{\BI} \cos\lr{ \omega t + \phi_I} dt \\ &= \inv{T} \int_{t_0}^{t_0 + T} \Abs{\BV} \Abs{\BI} \lr{ \cos\lr{ \phi_V – \phi_I } + \cos\lr{ 2 \omega t + \phi_V + \phi_I} } dt \\ &= \inv{2} \Abs{\BV} \Abs{\BI} \cos\lr{ \phi_V – \phi_I }. \end{aligned} \end{equation} Introducing the impedance for this circuit element \begin{equation}\label{eqn:chapter2Notes:660} \BZ = \frac{ \Abs{\BV} e^{j\phi_V} }{ \Abs{\BI} e^{j\phi_I} } = \frac{\Abs{\BV}}{\Abs{\BI}} e^{j\lr{\phi_V – \phi_I}}, \end{equation} this average power can be written in phasor form \begin{equation}\label{eqn:chapter2Notes:680} \BP = \inv{2} \Abs{\BI}^2 \BZ, \end{equation} with \begin{equation}\label{eqn:chapter2Notes:700} P = \textrm{Re} \BP. \end{equation} Observe that we have to be careful to use the absolute value of the current phasor $ \BI $, since $ \BI^2 $ differs in phase from $ \Abs{\BI}^2 $. This explains the conjugation in the [4] definition of complex power, which had the form \begin{equation}\label{eqn:chapter2Notes:720} \BS = \BV_{\textrm{rms}} \BI^\conj_{\textrm{rms}}. \end{equation} Radar cross section examples Flat plate. \begin{equation}\label{eqn:chapter2Notes:960} \sigma_{\textrm{max}} = \frac{4 \pi \lr{L W}^2}{\lambda^2} \end{equation} fig. 6. Square geometry for RCS example. Sphere. In the optical limit the radar cross section for a sphere fig. 7. Sphere geometry for RCS example. \begin{equation}\label{eqn:chapter2Notes:980} \sigma_{\textrm{max}} = \pi r^2 \end{equation} Note that this is smaller than the physical area $ 4 \pi r^2 $. Cylinder. fig. 8. Cylinder geometry for RCS example. \begin{equation}\label{eqn:chapter2Notes:1000} \sigma_{\textrm{max}} = \frac{ 2 \pi r h^2}{\lambda} \end{equation} Tridedral corner reflector fig. 9. Trihedral corner reflector geometry for RCS example. \begin{equation}\label{eqn:chapter2Notes:1020} \sigma_{\textrm{max}} = \frac{ 4 \pi L^4}{3 \lambda^2} \end{equation} Scattering from a sphere vs frequency Frequency dependence of spherical scattering is sketched in fig. 10. Low frequency (or small particles): Rayleigh\begin{equation}\label{eqn:chapter2Notes:1040} \sigma = \lr{\pi r^2} 7.11 \lr{\kappa r}^4, \qquad \kappa = 2 \pi/\lambda. \end{equation} Mie scattering (resonance),\begin{equation}\label{eqn:chapter2Notes:1060} \sigma_{\textrm{max}}(A) = 4 \pi r^2 \end{equation} \begin{equation}\label{eqn:chapter2Notes:1080} \sigma_{\textrm{max}}(B) = 0.26 \pi r^2. \end{equation} optical limit ( $r \gg \lambda$ )\begin{equation}\label{eqn:chapter2Notes:1100} \sigma = \pi r^2. \end{equation} fig 10. Scattering from a sphere vs frequency (from Prof. Eleftheriades’ class notes). FIXME: Do I have a derivation of this in my optics notes? Notation Time average. Both Prof. Eleftheriades and the text [1] use square brackets $ [\cdots] $ for time averages, not $ <\cdots> $. Was that an engineering convention? Prof. Eleftheriades writes $\Omega$ as a circle floating above a face up square bracket, as in fig. 1, and $ \sigma $ like a number 6, as in fig. 1. Bold vectors are usually phasors, with (bold) calligraphic script used for the time domain fields. Example: $ \BE(x,y,z,t) = \ecap E(x,y) e^{j \lr{\omega t – k z}}, \boldsymbol{\mathcal{E}}(x, y, z, t) = \textrm{Re} \BE $. fig. 11. Prof. handwriting decoder ring: Omega fig 12. Prof. handwriting decoder ring: sigma References [1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005. [2] digi.com. Antenna Gain: dBi vs. dBd Decibel Detail, 2015. URL http://www.digi.com/support/kbase/kbaseresultdetl?id=2146. [Online; accessed 15-Jan-2015]. [3] David Jeffrey Griffiths and Reed College. Introduction to electrodynamics. Prentice hall Upper Saddle River, NJ, 3rd edition, 1999. [4] J.D. Irwin. Basic Engineering Circuit Analysis. MacMillian, 1993. [5] JD Jackson. Classical Electrodynamics. John Wiley and Sons, 2nd edition, 1975. [6] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980. ISBN 0750627689.
AliPhysics 5403132 (5403132) Scripts used in the analysis Algorithms Corrections Monte-carlo code Mid-rapidity tracklet code for dN/deta Tasks Topical file AliAODForwardMult.h Per-event $ N_{ch}$ per $(\eta,\varphi)$ bin. file AliFMDEventPlaneFinder.h file AliForwardUtil.h Various utilities used in PWGLF/FORWARD. file AliLandauGaus.h Declaration and implementation of Landau-Gauss distributions. file AliLandauGausFitter.h Declaration and implementation of fitter of Landau-Gauss distributions to energy loss spectra. class AliForwardCreateResponseMatrices class AliForwardTriggerBiasCorrection class AliForwardUtil struct AliForwardUtil::RingHistos struct AliForwardUtil::Histos class AliLandauGaus class AliLandauGausFitter Code to do the multiplicity analysis in the forward psuedo-rapidity regions The code in this section defines methods to measure the charged-particle pseudorapidity density over $\|\eta\|<2$ using SPD tracklets. The code in this module constitutes tools for analysing SPD tracklet data for the charged-particle pseudorapidity density. It is based on Ruben's original code (see Ruben's SPD tracklet dNch/deta analysis (deprecated)), but differs in some important aspects. This code also requires a pass of real-data (AliTrackletAODTask) and simulated (AliTrackletAODMCTask) ESDs, but the output is not near-final histograms but an array of data structures (AliAODTracklet) stored on the output AOD. The data structure contains basic information on each tracklet Tracklet structures from simulations contain in addition During the AOD production, no cuts, except those defined for the re-reconstruction are imposed. In this way, the AOD contains the minimum bias information on tracklets for all events, And for AODs corresponding to simulated data, we can also In this way, we do a single pass of ESDs for real and simulated data, and we can then process the generated AODs with various cuts imposed. The AODs are generally small enough that they can be processed locally and quickly (for example using ProofLite). This scheme allows for fast turn-around with the largest possible flexibility. The final charged-particle pseudorapidity density is produced by an external class (AliTrackletdNdeta2). Other differences to Ruben's code is that the output files are far more structured, allowing for fast browsing of the data and quality assurance. The analysis requires real data and simulated data, anchored to the real data runs being processed. For both real and simulated data, the analysis progresses through two steps: A pass over ESD plus clusters to generate an AOD branch containing a TClonesArray of AliAODTracklet objects. In this pass, there are no selections imposed on the events. In this pass, the SPD clusters are reprocesed and the tracklets are re-reconstructed. In this pass, we also form so-called injection events. In these events, a real cluster is removed and a new cluster put in at some other location in the detector. The tracklets of the event is then reconstructed and stored. This procedure is repeated as many times as possible. The injection events are therefore superpositions of many events - each with a real cluster removed and replaced by a fake cluster. The injection events are used later for background estimates. When processing simulated data, the tracklets are also inspected for their origin. A tracklet can have three distinct classes of origins: The last class is the background from combinations of clusters that does not correspond to true particles. This background must be removed from the measurements. The second class, tracklets from secondaries, also form a background, but these tracklets are suppressed by cuts on the sum-of-square residuals \[ \Delta = \left[\frac{\Delta_{\theta}^2\sin^2\theta}{\sigma_{\theta}^2}+ \frac{(\Delta_{\phi}-\delta_{\phi})^2}{\sigma_{\phi}^2}\right] \] \[ \frac{d^2N_X}{d\eta d\mathrm{IP}_z}\quad, \] where $ X$ is $ M$ or $ I$ for real data, or $ M',I',P',S',C'$ or $ G'$ for simulated data. For each of these tracklet samples, except $ G'$, we also form the 3-dimensional differential $\Delta$ distributions \[ \frac{d^3N_X}{d\eta d\mathrm{IP}_z d\Delta}\quad, \] which are later used to estimate the background due to wrong combinations of clusters into tracks. Once both the real and simulated data has passed these two steps, we combine the to data sets into the final measurement. The final measurement is given by \[ R = \frac{G'}{(1-\beta')M'}(1-\beta)M, \] where \[ \beta' = \frac{C'}{M'}\quad\mathrm{and}\quad \beta = k\beta'\quad. \] Here, $ k$ is some scaling derived from the 3-dimensional differential $\Delta_M$ and $\Delta_{M'}$ distributions . There are classes for containing data, classes that represent analysis tasks, and classes that perform calculations, as well as specialized classes for analysis of simulation (MC) output. The classes AliAODTracklet and AliAODMCTracklet stores individual tracklet parameters. The difference between the two are that AliAODMCTracklet also stores the PDG code(s) and transverse momentum (momenta) of the mother primary particle (which may be the particle it self). The pass over the ESD is done by the classes AliTrackletAODTask and AliTrackletAODMCTask. These tasks generated the array of tracklets in the AOD events. The difference between the two is that AliTrackletAODMCTask inspects and groups each tracklet according to it's origin, and create pseudo-tracklets corresponding to the generated primary, charged particles. \[ \frac{d^2N_X}{d\eta d\mathrm{IP}_z}\quad, \] and \[ \frac{d^3N_X}{d\eta d\mathrm{IP}_z d\Delta}\quad. \] The first task does this for $ X=M$ and $ I$, while the second and thhird tasks does this for $ X=M',I',P',S',C'$ and $ G'$. The third task reweighs all tracklets according to the particle specie(s) and transverse momentum (momenta) of the mother primary particle(s). AliTrackletWeights defines the interface used for reqeighing the data. To produce the AODs with the tracklet information in, one needs to run a train with a task of the class AliTrackletAODTask (or AliTrackletAODMCTask for simulated data) and a task of the class AliSimpleHeaderTask in it. This is most easily done using the TrainSetup (Using the TrainSetup facility) derived class TrackletAODTrain. For example for real data from run 245064 of LHC15o using the first physics pass runTrain --name=LHC15o_245064_fp_AOD \ --class=TrackletAODTrain.C \ --url="alien:///alice/data/2015/LHC15o?run=245064&pattern=pass_lowint_firstphys//AliESDs.root&aliphysics=last,regular#esdTree" or for simulated data from the LHC15k1a1 production anchored to run 245064 runTrain --name=LHC15k1a1_245064_fp_AOD \ --class=TrackletAODTrain.C \ --url=alien:///alice/sim/2015/LHC15k1a1?run=245064&pattern=/AliESDs.root&aliphysics=last,regular&mc#esdTree (note the addition of the option "&mc" to the URL argument) In both cases a sub-directory - named of the name argument - of the current directory is created. In that sub-directory there are scripts for merging the output, downloading results, and downloading the generated AODs. It is highly recommended to download the generated AODs to your local work station to allow fast second step analysis. To download the AODs, go to the generated sub-directory an run the DownloadAOD.C script. For example, for the real data analysis of run 245064 of LHC15o, one would do (cd LHC15o_245064_fp_AOD && root -l -b -q DownloadAOD.C) and similar for the analysis of the simulated data. To produce the histograms for the final charged-particle pseudorapidity density , one needs to run a train with a task of the class AliTrackletAODdNdeta (or AliTrackletAODMCdNdeta for simulated data) in it. This is most easily done using the TrainSetup derived class TrackletAODdNdeta. For example for real data from run 245064 of LHC15o where we store the AODs generated above on the grid runTrain --name=LHC15o_245064_fp_dNdeta \ --class=TrackletAODdNdeta.C \ --url="alien:///alice/cern.ch/user/a/auser/LHC15o_245064_fp_dNdeta/output?run=245064&pattern=* /AliAOD.root&aliphysics=last,regular#aodTree" or for simulated data from the LHC15k1a1 production anchored to run 245064 runTrain --name=LHC15k1a1_245064_fp_dNdeta \ --class=TrackletAODdNdeta.C \ --url=alien:///alice/cern.ch/user/a/auser/LHC15k1a1_245064_fp_AOD/output?run=245064&pattern=* /AliAOD.root&aliphysics=last,regular&mc#esdTree (note the addition of the option "&mc" to the URL argument) If we had downloaded the AODs, we can use ProofLite to do this step runTrain --name=LHC15o_245064_fp_dNdeta \ --class=TrackletAODdNdeta.C \ --url="lite:///${PWD}/LHC15o_245064_fp_dNdeta?pattern=AliAOD_.root#aodTree" and similar for simulated data runTrain --name=LHC15k1a1_245064_fp_dNdeta \ --class=TrackletAODdNdeta.C \ --url="lite:///${PWD}/LHC15k1a1_245064_fp_dNdeta?pattern=AliAOD_.root&mc#aodTree" (note the addition of the option "&mc" to the URL argument) The final result is obtained by runnin the class AliTrackletdNdeta2 over the histograms from both real data and simulations. As an example, suppose we ran or histogram production on the Grid and have downloaded the merged results into LHC15o_245064_fp_dNdeta/root_archive_000245064/AnalysisResult.root (using LHC15o_245064_fp_dNdeta/Download.C) and LHC15k1a1_245064_fp_dNdeta/root_archive_245064/AnalysisResult.root (using LHC15k1a1_245064_fp_dNdeta/Download.C). Then we should do (see AliTrackletdNdeta2::Run for more information on arguments) Alternatively one can use the script Post.C to do this. The script is used like. where simFile is the simulation data input file (or directory) realFile is the real data input file (or directory) outDir is the output directory (created) process are the processing options visualize are the visualisation options nCentralities is the maximum number of centrality bins Processing options: 0x0001 Do scaling by unity 0x0002 Do scaling by full average 0x0004 Do scaling by eta differential 0x0008 Do scaling by fully differential 0x0010 Correct for decay of strange to secondary 0x1000 MC closure test Visualization options: 0x0001 Draw general information 0x0002 Draw parameters 0x0004 Draw weights 0x0008 Draw dNch/deta 0x0010 Draw alphas 0x0020 Draw delta information 0x0040 Draw backgrounds 0x0100 Whether to make a PDF 0x0200 Whether to pause after each plot 0x0400 Draw in landscape 0x0800 Alternative markers By default, each plot will be made and the process paused. To advance, simple press the space-bar. If we had made the histograms using ProofLite, we should do
I have a problem involving a time-indexed variable $x_{it}$ representing the amount of a resource allotted to a task $i$ in time $t$. The quantity of the (renewable) resource is constrained at a value $R$ per period: $$ \sum_{i=1}^{I} x_{it}\leq R\,\,\,\forall t \in T $$ and I have to allott a given quantity $A_i$ of $R$ to each task over the time horizon $1...T$: $$ \sum_{t=1}^{T} x_{it}\geq A_{i}\,\,\,\forall i \in I $$ My question is: is it possible with this formulation to get the highest and lowest $t$s such as $x_{it}$ is not zero, keeping the problem linear? $$ t_{i}^{MIN}=\{ \tau \;\|\; x_{i\tau}>0 \wedge x_{it}=0\;\forall t<\tau\} $$ $$ t_{i}^{MAX}=\{ \tau \;\|\; x_{i\tau}>0 \wedge x_{it}=0\;\forall t>\tau\} $$ What I am trying to get is the starting time ($t_{i}^{MIN}$) and the completion time ($t_{i}^{MAX}$) of each task. The objective function could be to minimize the maximum of the $t_{i}^{MAX}$, or to minimize the difference between the maximum $t_{i}^{MAX}$ and the minimum $t_{i}^{MIN}$ (that is the makespan). I think this formulation is not the ideal one, but I can't figure out a suitable, simple alternative. The fact is that the duration of the task is dependent upon the amount of resources allotted in each period, so a formulation using variables representing the starting time and the completion time seems not ok either. Technically, \[ \left. \begin{align} x_{it} & \leq R \cdot y_{it} & \scriptstyle{// ~[1]}\\ t \cdot y_{it} & \leq t_i^{MAX} & \\ T + ( -T + t ) \cdot y_{it} & \geq t_i^{MIN} & \scriptstyle{// ~[2]}\\ y_{it} & \in \{0,1\} & \\ t_{i}^{\{MIN, MAX\}} & \in \mathrm{Z} & \\ \end{align} \quad \right\} \quad \forall ~i = 1 \ldots I, ~t = 1 \ldots T \] should suffice your needs ...but, of course, at the cost of introducing $I \times (T + 2)$ integer/binary variables and even $3 \times I \times T$ additional constraints. While I'm not sure whether integer variables can be completely avoided, I'm almost certain that "we" have a more compact [re]formulation for this. I. e.: come forward, schedulers! Corrections: I think I can reduce the number of binaries in @Florian's approach by noting that you don't really care about individual end times, just the cumulative end time (which equals makespan if you assume that operations start at time 0 -- nothing in what you described indicates any reason for a "dead time" at the outset). Let $Z_t\in\{0,1\}$ be 1 if production terminates at the end of time epoch $t\in T$ (making $t$ the makespan), 0 otherwise. That introduces just $\|T\|$ new binary variables. The additional constraints are that $x_{it} \le \min(A_i,R)(1 - \sum_{\tau \lt t}Z_\tau)$ for all $i\in I,t \in T$, along with an SOS1 constraint $\sum_{t\in T}Z_t = 1$. To minimize end time, the objective is $\min \sum_{t \in T}t Z_t$.
Laplace transforms were used to solve the MNA equations for time dependent systems, and to find the moments used to in MOR. For the record, the Laplace transform is defined as: \begin{equation}\label{eqn:laplaceTransformVec:20} \boxed{ \LL( f(t) ) = \int_0^\infty e^{-s t} f(t) dt. } \end{equation} The only Laplace transform pair used in the lectures is that of the first derivative \begin{equation}\label{eqn:laplaceTransformVec:40} \begin{aligned} \LL(f'(t)) &= \int_0^\infty e^{-s t} \ddt{f(t)} dt \\ &= \evalrange{e^{-s t} f(t)}{0}{\infty} – (-s) \int_0^\infty e^{-s t} f(t) dt \\ &= -f(0) + s \LL(f(t)). \end{aligned} \end{equation} Here it is loosely assumed that the real part of $ s $ is positive, and that $ f(t) $ is “well defined” enough that $ e^{-s \infty } f(\infty) \rightarrow 0 $. Where used in the lectures, the laplace transforms were of vectors such as the matrix vector product $ \LL(\BG \Bx(t)) $. Because such a product is linear, observe that it can be expressed as the original matrix times a vector of Laplace transforms \begin{equation}\label{eqn:laplaceTransformVec:60} \begin{aligned} \LL( \BG \Bx(t) ) &= \LL {\begin{bmatrix} G_{i k} x_k(t) \end{bmatrix}}_i \\ &= {\begin{bmatrix} G_{i k} \LL x_k(t) \end{bmatrix}}_i \\ &= \BG {\begin{bmatrix} \LL x_i(t) \end{bmatrix}}_i. \end{aligned} \end{equation} The following notation was used in the lectures for such a vector of Laplace transforms \begin{equation}\label{eqn:laplaceTransformVec:80} \BX(s) = \LL \Bx(t) = {\begin{bmatrix} \LL x_i(t) \end{bmatrix}}_i. \end{equation}
Answer (a) The electron moved to a lower potential. (b) $\Delta V = -190~J/C$ Work Step by Step (a) By conservation of energy, the decrease in kinetic energy is equal in magnitude to the increase in electric potential energy. Since the charge on an electron is negative, the electron must have moved to a lower potential. (b) We can use conservation of energy to find the potential difference: $K_2+U_2 = K_1+U_1$ $U_2-U_1 = K_1-K_2$ $(\Delta V)~q = \frac{1}{2}m~v_1^2-\frac{1}{2}m~v_2^2$ $\Delta V = \frac{\frac{1}{2}m~(v_1^2-v_2^2)}{q}$ $\Delta V = \frac{\frac{1}{2}(9.1\times 10^{-31}~kg)[(8.50\times 10^6~m/s)^2-(2.50\times 10^6~m/s)^2]}{-1.6\times 10^{-19}~C}$ $\Delta V = -190~J/C$
Sorry I cannot find a clearer title for this question. I have a constraint like the following: $$ \sum_{t \in T} w_{t} \cdot y_{t} \geq A $$ where $w_t$ are given, positive weights, and $y_t \in \{0,1\} $. I want to add (if possible) the following constraint: $$ y_{t} = \dots 0,0,1,1,1,0,0\dots $$ while this is not: $$ y_{t} = \dots 0,1,0,1,1,0,0\dots $$ The problem I find is that I don't know \[ T \cdot (1 - x_i + x_{i+1}) ~\geq \sum_{j = i+1}^T x_j \quad, ~\forall ~i = 1 \ldots T-1 \] \[ \begin{array}[t]{cccc} x_i & x_{i+1} & & LHS & & RHS\\ 0 & 0 & \rightarrow & T & \geq & [0, T-1]\\ 0 & 1 & \rightarrow & 2T & \geq & [1, T-1]\\ 1 & 0 & \rightarrow & 0 & \geq & 0\\ 1 & 1 & \rightarrow & T & \geq & [1, T-1]\\ \end{array} \] doesn't introduce aux. variables (just like the approach @Austin suggested), and requires only $T-1$ constraints. In fact [and contrary to my initial estimate], this approach is both more memory efficient *P.S.: Here, "faster" means: the solvers I've tried (CPLEX, Gurobi) solve [AMPL] model Model A: Model B: Model C: performs best among all three discussed approaches [by far!]. You could just use a set of (binary) variables, say $z_t$, to indicate the first non-zero $y_t$ variable. You can do this with something like $y_t\ge z_t\ge y_t-y_{t-1}$. Then, bound $\sum z_t$ to be (smaller or) equal to 1. There's no need to introduce new variables. You want to avoid situations where two variables are set to one, but a variable between them is set to zero. So, for each triple $ (i,j,k)$ with $1 \le i < j < k \le n:= \|T\|$, include the inequality $ y_j \ge y_i + y_k - 1$. If we have $ y_i = y_k = 1$ then all variables $x_j$ in between (i.e., with $ i < j < k$) will be forced to one as well. The number of such inequalities is $ n \choose 3$ which is $ O(n^3)$. A couple things. Assuming we ignore the weight A constraint:
If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal Problem 424 Let $A$ and $B$ be $n\times n$ matrices.Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$. Since $A$ and $B$ have $n$ linearly independent eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$, they are diagonalizable.Specifically, if we put $S=[\mathbf{x}_1, \dots, \mathbf{x}_n]$. Then $S$ is invertible (as column vectors of $S$ are linearly independent) and we have\[S^{-1}AS=D \text{ and } S^{-1}BS=D,\]where $D$ is the diagonal matrix whose diagonal entries are eigenvalues:\[D=\begin{bmatrix}\lambda_1 & 0 & \cdots & 0 \\0 & \lambda_2 & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \lambda_n\end{bmatrix}.\]It follows that we have\[S^{-1}AS=D=S^{-1}BS,\]and hence $A=B$. This completes the proof. Diagonalize the 3 by 3 Matrix if it is DiagonalizableDetermine whether the matrix\[A=\begin{bmatrix}0 & 1 & 0 \\-1 &0 &0 \\0 & 0 & 2\end{bmatrix}\]is diagonalizable.If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.How to […] Diagonalize a 2 by 2 Symmetric MatrixDiagonalize the $2\times 2$ matrix $A=\begin{bmatrix}2 & -1\\-1& 2\end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.Solution.The characteristic polynomial $p(t)$ of the matrix $A$ […] How to Diagonalize a Matrix. Step by Step Explanation.In this post, we explain how to diagonalize a matrix if it is diagonalizable.As an example, we solve the following problem.Diagonalize the matrix\[A=\begin{bmatrix}4 & -3 & -3 \\3 &-2 &-3 \\-1 & 1 & 2\end{bmatrix}\]by finding a nonsingular […] Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary MatrixConsider the Hermitian matrix\[A=\begin{bmatrix}1 & i\\-i& 1\end{bmatrix}.\](a) Find the eigenvalues of $A$.(b) For each eigenvalue of $A$, find the eigenvectors.(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […] True or False. Every Diagonalizable Matrix is InvertibleIs every diagonalizable matrix invertible?Solution.The answer is No.CounterexampleWe give a counterexample. Consider the $2\times 2$ zero matrix.The zero matrix is a diagonal matrix, and thus it is diagonalizable.However, the zero matrix is not […]
Churn is a really big deal in SaaS... but which is a "bigger" deal? Customer churn, or revenue churn? Today, I'm looking at the differences between these two essential SaaS metrics, and helping you to use them to monitor and improve the growth of your own SaaS startup. Customer Churn Customer churn (also known as "Logo Churn") measures the rate at which your customers cancel their subscription to your service. Almost all SaaS companies are subscription-based business. As a result, your growth depends on both maximising new customers, and minimising the churn of existing customers. The customer churn formula below allows you to monitor the rate at which you're losing customers each month: $$\text{% Customer churn rate}=\frac{\text{Customers that churned in period t}}{\text{Total customers at the start of period t}}$$ For example, if we ended January with 20 paying customers, and a month later, we'd lost 2 of those customers to churn, we'd have a customer churn rate of 10%. In other words, during the month of February, we lost 10% of our customer base to churn: $$\text{January: 20 customers}$$ $$\text{February: 18 customers}$$ $$\text{% Customer churn rate}=\frac{(20-18)}{20}=\frac{2}{20}=10\%$$ Revenue Churn Revenue churn (also known as "MRR churn rate") is used to look at the rate at which monthly recurring revenue (MRR) is lost, as a result of churned customers and downgraded subscriptions. The formula for revenue churn is similar to customer churn: $$\text{% MRR churn rate}=\frac{\text{Churned MRR}}{\text{Previous month's MRR}}$$ If we go back to our earlier example, and add in MRR figures, we end up with a revenue churn rate of 10%. In other words, as a result of those 2 churned customers, February saw a loss of 10% of our monthly recurring revenue: $$\text{January: 20 customers}\times$200=$4,000\text{MRR}$$ $$\text{February: 18 customers}\times$200=$3,600\text{MRR}$$ $$\text{Revenue churn}=\frac{(4,000-3,600)}{4,000}=\frac{400}{4,000}=10\%$$ For a few alternative ways to calculate both customer churn and revenue churn, check out the Essential SaaS Metrics Glossary. Customer Churn vs Revenue Churn When it comes to churn metrics, it might be tempting to dial-in on the one "ultimate" metric and disregard all others. In this instance though, we need to keep a close eye on both customer churn and revenue churn. Customer churn tells you how good you are at retaining customers. Revenue churn tells you how good you are at retaining customer revenue. These two concepts are similar, but they aren't the same. The examples we calculated above were incredibly simple: 10% customer churn resulted in 10% revenue churn. In reality though, things like different pricing packages, and additional seats, users and storage, can make churn calculations much more complicated. And as we're about to see, customer churn doesn't always do a great job at predicting revenue churn (and vice versa). Negative Revenue Churn Time for another example: imagine we have 3 customers, each paying $200 per month. In January, our MRR looks like this: $$\text{January:}$200+$200+$200=$600\text{MRR}$$ When it comes time to renew in February, we lose a customer to churn... $$\text{February:}$200+$200+$0=$400\text{MRR}$$ ...but successfully upsell the remaining two customers, switching them to a higher-priced package for an additional $150 MRR per customer. $$\text{February:}$350+$350+$0=$700\text{MRR}$$ Here's what customer churn and revenue churn look like over that period: $$\text{Customer churn}=\frac{(3-2)}{3}=\frac{1}{3}\approx33\%$$ $$\text{Revenue churn}=\frac{(600-700)}{600}=\frac{-100}{600}\approx-17\%$$ In isolation, these metrics would tell very conflicting stories. A churn rate of 33% is hugely problematic in SaaS, and you'd be inclined to assume that your MRR had fallen alongside. On the other hand, we have negative revenue churn over the same time period: thanks to the power of upselling, we've actually increased our MRR, despite losing a customer. Two Crucial SaaS Metrics Both of these metrics look at different aspects of your company's health, and in terms of deciding which is "best", or which takes precedence, the simple answer is neither. To get a balanced view of your growth, you need to analyse both of these metrics on a regular basis, for two simple reasons: Retention matters. If left unchecked, even low customer churn can become a serious problem, so it's vital to monitor and improve customer churn over time. Upselling matters. If you're able to upsell your customers onto higher-value packages, and sell additional recurring seats/users/storage, you can combat the effects of customer churn. Revenue churn will help you understand how efficient you are at retaining (and growing) your MRR.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by $\mathbf{0}(x) = 0$. This shows that $\mathbf{0} \in W_a$ if and only if $a=0$. We have shown that if $a \neq 0$, then $W_a$ is not a subspace as every subspace contains the zero vector. Now we consider the case $a=0$ and prove that $W_0$ is a subspace. We verify the subspace criteria: the zero vector of $C(\R)$ is in $W_0$, and $W_0$ is closed under addition and scalar multiplication. As mentioned before, $\mathbf{0} \in W_0$. Now suppose $f, g \in W_0$. Then $f(0) = g(0) = 0$, and so\[(f+g)(0) = f(0) + g(0) = 0.\]Thus $f+g \in W_0$. Finally, if $c \in \mathbb{R}$ is a scalar and $f \in W_0$, then\[(cf)(0) = c f(0) = c \cdot 0 = 0.\]Thus $cf \in W_0$, and $W_0$ is a vector subspace. The Centralizer of a Matrix is a SubspaceLet $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define\[W = \{ A \in V \mid AM = MA \}.\]The set $W$ here is called the centralizer of $M$ in $V$.Prove that $W$ is a subspace of $V$.Proof.First we check that the zero […] The Sum of Subspaces is a Subspace of a Vector SpaceLet $V$ be a vector space over a field $K$.If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset\[W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}\]is a subspace of the vector space $V$.Proof.We prove the […] Prove that the Center of Matrices is a SubspaceLet $V$ be the vector space of $n \times n$ matrices with real coefficients, and define\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]The set $W$ is called the center of $V$.Prove that $W$ is a subspace […] Sequences Satisfying Linear Recurrence Relation Form a SubspaceLet $V$ be a real vector space of all real sequences\[(a_i)_{i=1}^{\infty}=(a_1, a_2, \cdots).\]Let $U$ be the subset of $V$ defined by\[U=\{ (a_i)_{i=1}^{\infty} \in V \mid a_{k+2}-5a_{k+1}+3a_{k}=0, k=1, 2, \dots \}.\]Prove that $U$ is a subspace of […]
Sample Quantiles The generic function quantile produces sample quantiles corresponding to the given probabilities. The smallest observation corresponds to a probability of 0 and the largest to a probability of 1. Keywords univar Usage quantile(x, …) # S3 method for defaultquantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE, names = TRUE, type = 7, …) Arguments x numeric vector whose sample quantiles are wanted, or an object of a class for which a method has been defined (see also ‘details’). NAand NaNvalues are not allowed in numeric vectors unless na.rmis TRUE. probs numeric vector of probabilities with values in $[0,1]$. (Values up to 2e-14outside that range are accepted and moved to the nearby endpoint.) na.rm logical; if true, any NAand NaN's are removed from xbefore the quantiles are computed. names logical; if true, the result has a namesattribute. Set to FALSEfor speedup with many probs. type an integer between 1 and 9 selecting one of the nine quantile algorithms detailed below to be used. … further arguments passed to or from other methods. Details A vector of length length(probs) is returned; if names = TRUE, it has a names attribute. The default method works with classed objects sufficiently like numeric vectors that sort and (not needed by types 1 and 3) addition of elements and multiplication by a number work correctly. Note that as this is in a namespace, the copy of sort in base will be used, not some S4 generic of that name. Also note that that is no check on the ‘correctly’, and so e.g. quantile can be applied to complex vectors which (apart from ties) will be ordered on their real parts. Types quantile returns estimates of underlying distribution quantiles based on one or two order statistics from the supplied elements in x at probabilities in probs. One of the nine quantile algorithms discussed in Hyndman and Fan (1996), selected by type, is employed. All sample quantiles are defined as weighted averages of consecutive order statistics. Sample quantiles of type $i$ are defined by: $$Q_{i}(p) = (1 - \gamma)x_{j} + \gamma x_{j+1}$$ where $1 \le i \le 9$, $\frac{j - m}{n} \le p < \frac{j - m + 1}{n}$, $x_{j}$ is the $j$th order statistic, $n$ is the sample size, the value of $\gamma$ is a function of $j = \lfloor np + m\rfloor$ and $g = np + m - j$, and $m$ is a constant determined by the sample quantile type. Discontinuous sample quantile types 1, 2, and 3 For types 1, 2 and 3, $Q_i(p)$ is a discontinuous function of $p$, with $m = 0$ when $i = 1$ and $i = 2$, and $m = -1/2$ when $i = 3$. Type 1 Inverse of empirical distribution function. $\gamma = 0$ if $g = 0$, and 1 otherwise. Type 2 Similar to type 1 but with averaging at discontinuities. $\gamma = 0.5$ if $g = 0$, and 1 otherwise. Type 3 SAS definition: nearest even order statistic. $\gamma = 0$ if $g = 0$ and $j$ is even, and 1 otherwise. Continuous sample quantile types 4 through 9 For types 4 through 9, $Q_i(p)$ is a continuous function of $p$, with $\gamma = g$ and $m$ given below. The sample quantiles can be obtained equivalently by linear interpolation between the points $(p_k,x_k)$ where $x_k$ is the $k$th order statistic. Specific expressions for $p_k$ are given below. Type 4 $m = 0$. $p_k = \frac{k}{n}$. That is, linear interpolation of the empirical cdf. Type 5 $m = 1/2$. $p_k = \frac{k - 0.5}{n}$. That is a piecewise linear function where the knots are the values midway through the steps of the empirical cdf. This is popular amongst hydrologists. Type 6 $m = p$. $p_k = \frac{k}{n + 1}$. Thus $p_k = \mbox{E}[F(x_{k})]$. This is used by Minitab and by SPSS. Type 7 $m = 1-p$. $p_k = \frac{k - 1}{n - 1}$. In this case, $p_k = \mbox{mode}[F(x_{k})]$. This is used by S. Type 8 $m = (p+1)/3$. $p_k = \frac{k - 1/3}{n + 1/3}$. Then $p_k \approx \mbox{median}[F(x_{k})]$. The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x. Type 9 $m = p/4 + 3/8$. $p_k = \frac{k - 3/8}{n + 1/4}$. The resulting quantile estimates are approximately unbiased for the expected order statistics if xis normally distributed. Further details are provided in Hyndman and Fan (1996) who recommended type 8. The default method is type 7, as used by S and by R < 2.0.0. References Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician 50, 361--365. 10.2307/2684934. See Also Aliases quantile quantile.default Examples library(stats) # NOT RUN {quantile(x <- rnorm(1001)) # Extremes & Quartiles by defaultquantile(x, probs = c(0.1, 0.5, 1, 2, 5, 10, 50, NA)/100)### Compare different typesquantAll <- function(x, prob, ...) t(vapply(1:9, function(typ) quantile(x, prob=prob, type = typ, ...), quantile(x, prob, type=1)))p <- c(0.1, 0.5, 1, 2, 5, 10, 50)/100signif(quantAll(x, p), 4)## for complex numbers:z <- complex(re=x, im = -10*x)signif(quantAll(z, p), 4)# } Documentation reproduced from package stats, version 3.6.1, License: Part of R 3.6.1
Schönbrodt's and Perugini's method defines a point of stability (POS), a sample size beyond which one is reasonably confident that an estimate is within a specified range (labeled the corridor of stability, or COS) of its population value. For more details on how the point of stability is estimated, you can read either my previous post or Schönbrodt's and Perugini's paper. By adapting Schönbrodt's and Perugini's freely available source code, I found that, in two-group, three-group, and interaction designs, statistical stability generally requires sample sizes around 150-250. In this post, I will apply this same method to simple mediation designs. Simple mediation designs Simple mediation designs involve three variables: a predictor variable $X$ (which is usually manipulated), an outcome variable $Y$, and a hypothesized mechanism (mediator) for the relationship between $X$ and $Y$, $M$. Assuming that there is a causal relationship between $X$ and $Y$, $M$ a simple mediation design aims to establish whether $M$ is part of the reason for this causal relationship, as diagrammed below. In this diagram, $X$ is connected to $Y$ in two ways -- once via path $c$, and a second time through the two paths $a$ and $b$. The indirect effect of $X$ on $Y$ through $M$ is therefore the product of paths $a$ and $b$ ($ab$), which is an estimator of the population indirect effect $\alpha\beta$. If path $c$ is non-zero, $X$ may also have a direct effect on $Y$, which is the effect of $X$ on $Y$ that is not mediated through $M$. The total effect of $X$ on $Y$ is merely the sum of the indirect and direct effects, $ab + c$. The paths $a$, $b$, and $c$ can be estimated using the following two statistical models: The paths $a$, $b$, and $c$ can be estimated using the following two statistical models: $$M = aX + e_1$$ $$Y = bM + cX + e_2$$ Here, coefficient $c$ estimates the direct effect of $X$ on $Y$, and the product $ab$ estimates the indirect effect of $X$ on $Y$ through $M$. Thus, the population indirect effect $\alpha\beta$ of $X$ on $Y$ through $M$ can be estimated using the above two statistical models by taking the product $ab$. Effect sizes for the indirect effect What I have described so far is standard and well-known about simple mediation designs. What is less obvious is the proper estimator one should use to measure the size of the indirect effect. One of the more obvious options is to use standardized estimates of $\alpha$ and $\beta$ and simply calculate their product. Thus, for $\alpha$, one could estimate the correlation between $X$ and $M$, and, for $\beta$, one could estimate the semipartial correlation between $M$ and $Y$ (controlling for $X$). Their product, which I will call $ab_{\text{standard}}$, yields a standardized estimate of the indirect effect. Other commonly used alternatives express the magnitude of $ab$ relative to other quantities in a simple mediation model. The most common approach is to estimate the ratio between the indirect and total effects. Mathematically, this reduces to estimating the proportion $P_M$ as follows: One of the more obvious options is to use standardized estimates of $\alpha$ and $\beta$ and simply calculate their product. Thus, for $\alpha$, one could estimate the correlation between $X$ and $M$, and, for $\beta$, one could estimate the semipartial correlation between $M$ and $Y$ (controlling for $X$). Their product, which I will call $ab_{\text{standard}}$, yields a standardized estimate of the indirect effect. Other commonly used alternatives express the magnitude of $ab$ relative to other quantities in a simple mediation model. The most common approach is to estimate the ratio between the indirect and total effects. Mathematically, this reduces to estimating the proportion $P_M$ as follows: $$P_M=\frac{ab}{ab + c}$$ $P_M$ is not a true proportion in that it can take on values that are negative or greater than one, which hinders its interpretability. However, $P_M$ can be easily estimated using the same two models used to estimate $\alpha\beta$. $P_M$ also has some intuitive appeal as a comparison between two of the important quantities in mediation analysis, the indirect and total effects. Overall, there is little consensus about the best way to measure the size of the indirect effect -- in a recent paper, Preacher and Kelley (2011) describe fully 16 options. As the estimators most commonly used, I will focus on $ab_{\text{standard}}$ and $P_M$. Estimating the stability of an indirect effect estimator To apply Schönbrodt's and Perugini's method to a simple mediation model, I created populations with variables $X$, $M$, and $Y$ in which I had systematically varied $\alpha$ and $\beta$ while keeping the total effect constant to $\rho = .4$. I then drew 10,000 effect size trajectories, drawing an initial small sample and successively recalculating the target effect size metric as additional cases were added to the sample until the size of the sample reached 1000. Defining a corridor of stability (COS) and its half-width $w$ was a little tricky. Because there is no broad consensus as to what constitutes a "small", "medium", or "large" value of the indirect effect, there is therefore little guidance as to what might represent a small, medium, or large deviation of an estimate of the indirect effect from its population value. I used the following logic to choose values for $w$. Schönbrodt and Perugini argued that values of .1, .15, and .2 represent small, medium, and large deviations from a population $\rho$. $ab_{\text{standard}}$ is the product of two correlations; thus, $.1^2 = .01$, $.15^2 = .025$ and $.20^2 = .04$ seem like reasonable choices to represent represent small, medium and large deviations from the population indirect effect. Similarly, $P_M$ is just $ab_{\text{standard}}$ divided by the total effect, which across my simulations I am setting to .4. Thus, some reasonable values for the half-width of the COS for $P_M$ are $.01/.4= .025$, $.025/.4= .0625$, and $.04/.4=.1$. For each of $ab_{\text{standard}}$ and $P_M$, I investigated a case where $X$ is quantitative and a case where $X$ is categorical. The results for a quantitative and categorical $X$ were relatively similar, so I will only be presenting the results for a quantitative $X$. You can find both my results for a categorical $X$ and my source code here. Defining a corridor of stability (COS) and its half-width $w$ was a little tricky. Because there is no broad consensus as to what constitutes a "small", "medium", or "large" value of the indirect effect, there is therefore little guidance as to what might represent a small, medium, or large deviation of an estimate of the indirect effect from its population value. I used the following logic to choose values for $w$. Schönbrodt and Perugini argued that values of .1, .15, and .2 represent small, medium, and large deviations from a population $\rho$. $ab_{\text{standard}}$ is the product of two correlations; thus, $.1^2 = .01$, $.15^2 = .025$ and $.20^2 = .04$ seem like reasonable choices to represent represent small, medium and large deviations from the population indirect effect. Similarly, $P_M$ is just $ab_{\text{standard}}$ divided by the total effect, which across my simulations I am setting to .4. Thus, some reasonable values for the half-width of the COS for $P_M$ are $.01/.4= .025$, $.025/.4= .0625$, and $.04/.4=.1$. For each of $ab_{\text{standard}}$ and $P_M$, I investigated a case where $X$ is quantitative and a case where $X$ is categorical. The results for a quantitative and categorical $X$ were relatively similar, so I will only be presenting the results for a quantitative $X$. You can find both my results for a categorical $X$ and my source code here. The stability of $ab_{\text{standard}}$ Below are the points of stability for $ab_{\text{standard}}$, given varying values of $\alpha$, $\beta$, the half-width of the corridor of stability $w$, and the confidence of the point of stability. For reference, I have also included a column giving the value of $P_M$. $$ \begin{array}{cccc\|ccc\|ccc\|ccc} &&&&&80\%&&&90\% &&&95\%&&\\ \alpha&\beta&\alpha\beta&P_M&w = .01&w = .0225 & w = .04 & w = .01 & w = .0225 & w = .04 & w = .01 & w = .0225 & w = .04\\ \hline 0.1&0.1&0.01&0.025&541&133&57&808&204&83&986&296&112\\ 0.1&0.2&0.02&0.05&992&289&98&1000&434&146&1000&600&197\\ 0.1&0.3&0.03&0.075&1000&572&186&1000&805&272&1000&967&366\\ 0.1&0.4&0.04&0.1&1000&872&313&1000&1000&453&1000&1000&604\\ 0.2&0.1&0.02&0.05&959&260&90&1000&391&137&1000&533&186\\ 0.2&0.2&0.04&0.1&1000&411&125&1000&614&192&1000&820&271\\ 0.2&0.3&0.06&0.15&1000&641&204&1000&906&308&1000&1000&423\\ 0.2&0.4&0.08&0.2&1000&902&320&1000&1000&468&1000&1000&631\\ 0.3&0.1&0.03&0.075&1000&492&160&1000&710&236&1000&896&315\\ 0.3&0.2&0.06&0.15&1000&576&185&1000&827&276&1000&984&368\\ 0.3&0.3&0.09&0.225&1000&751&244&1000&976&359&1000&1000&488\\ 0.3&0.4&0.12&0.3&1000&916&337&1000&1000&489&1000&1000&658\\ 0.4&0.1&0.04&0.1&1000&766&267&1000&968&390&1000&1000&510\\ 0.4&0.2&0.08&0.2&1000&802&283&1000&988&404&1000&1000&521\\ 0.4&0.3&0.12&0.3&1000&847&304&1000&1000&432&1000&1000&563\\ 0.4&0.4&0.16&0.4&1000&896&343&1000&1000&482&1000&1000&626\\ \end{array} $$ According to my simulations, the points of stability for $ab_{\text{standard}}$ are extremely high, and these points of stability increaseas the population value of the indirect effect increases. In fact, in many of the cells of the POS table, the point of stability was beyond the maximum N that I sampled in the bootstrapped trajectories (1000). To see visually what's going on, I have plotted 100 trajectories with two different values of $\alpha\beta$, .01 and .16, along with a COS half-width of $w=.04$. 100 simulated trajectories, $\alpha\beta = .01$ 100 simulated trajectories, $\alpha\beta = .16$ From these graphs, we can see that the trajectories are much more unstable with a larger population value of $\alpha\beta$. This makes some intuitive sense because fluctuations in either $a$ or $b$ will ramify into fluctuations of $ab_{\text{standard}}$, and the fluctuations will be especially severe if the population values of either $\alpha$ or $\beta$ are large. Supporting this interpretation, the sampling distribution of $\alpha\beta$ is known to be non-normal due to skewness and excess kurtosis, and the departure from normality depends in part on the values of $\alpha$ or $\beta$ (MacKinnon, Fritz, Williams, & Lockwood, 2007). It is also instructive to compare the points of stability for the following values of $\alpha$ and $\beta$: $\alpha=.1; \beta=.4$, $\alpha=.2; \beta=.2$, and $\alpha=.4; \beta=.1$. In these cases, the magnitude of the indirect effect is equivalent. Nonetheless, the points of stability are consistently higher when either $\alpha$ or $\beta$ is equal to .4 than when they are both equal to .2. The stability of $P_M$ Based on the above discussion, one might expect the stability of $P_M$ to be even worse than the stability of $ab_{\text{standard}}$, because the value of $P_M$ depends on the estimated value of the total effect, which is yet another parameter that can take on an extreme value. Indeed, the table of the points of stability, below, supports this expectation. $$ \begin{array}{cccc\|ccc\|ccc\|ccc} &&&&&80\%&&&90\% &&&95\%&&\\ a&b&ab&P_M&w = .025&w = .05625 & w = .1 & w = .025 & w = .05625 & w = .1 & w = .025 & w = .05625 & w = .1\\ \hline 0.1&0.1&0.01&0.025&562&146&65&824&223&95&982&314&132\\ 0.1&0.2&0.02&0.05&985&286&103&1000&423&156&1000&577&207\\ 0.1&0.3&0.03&0.075&1000&546&182&1000&753&262&1000&930&349\\ 0.1&0.4&0.04&0.1&1000&808&286&1000&995&414&1000&1000&544\\ 0.2&0.1&0.02&0.05&975&280&100&1000&422&155&1000&565&214\\ 0.2&0.2&0.04&0.1&1000&423&133&1000&628&208&1000&835&291\\ 0.2&0.3&0.06&0.15&1000&630&205&1000&880&299&1000&1000&407\\ 0.2&0.4&0.08&0.2&1000&844&294&1000&1000&432&1000&1000&582\\ 0.3&0.1&0.03&0.075&1000&522&179&1000&745&265&1000&931&350\\ 0.3&0.2&0.06&0.15&1000&647&210&1000&888&315&1000&1000&439\\ 0.3&0.3&0.09&0.225&1000&763&270&1000&981&401&1000&1000&534\\ 0.3&0.4&0.12&0.3&1000&898&328&1000&1000&497&1000&1000&672\\ 0.4&0.1&0.04&0.1&1000&797&290&1000&986&421&1000&1000&547\\ 0.4&0.2&0.08&0.2&1000&854&326&1000&1000&466&1000&1000&619\\ 0.4&0.3&0.12&0.3&1000&905&360&1000&1000&521&1000&1000&677\\ 0.4&0.4&0.16&0.4&1000&957&404&1000&1000&569&1000&1000&764\\ \end{array} $$ If we just focus on the two columns where $w = .05625$ and where $w=.1$ (the two larger values of the half-width of the COS), the points of stability are on average 14.52 cases larger for $P_M$ than they are for $ab_{\text{standard}}$, and the problem is particularly acute as $\alpha\beta$ increases. You can see this pattern in the two sets of trajectories plotted below. 100 simulated trajectories, $P_M = .01$ 100 simulated trajectories, $P_M = .4$ Conclusions There are some obvious limitations to my approach. First, as I pointed out, I do not have a good basis to say what constitutes a "small" or "large" departure from the population indirect effect, so the values that I used to construct the corridors of stability were somewhat arbitrary and could very well have been too restrictive. Second, I only investigated one value for the total effect -- a generous $\rho = .4$ -- so I can't say with confidence how stability is affected if one systematically varies this value. However, based on the simulations that I conducted, I think it's reasonable to conclude that obtaining an accurate estimate of the indirect effect is quite difficult. Although it is possible to obtain reasonable power to detect a non-zero indirect effect at somewhat smaller sample sizes (Fritz & MacKinnon, 2007), obtaining a good or accurate estimate of the indirect effect probably requires a sample size that is extremely large, perhaps even in excess of 1000. References Fritz, M. S., & MacKinnon, D. P. (2007). Required sample size to detect the mediated effect. Psychological Science, 18, 233–239. http://doi.org/10.1111/j.1467-9280.2007.01882.x MacKinnon, D. P., Fritz, M. S., Williams, J., & Lockwood, C. M. (2007). Distribution of the product confidence limits for the indirect effect: Program PRODCLIN. Behavior Research Methods, 39, 384–389. http://doi.org/10.3758/BF03193007 Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative strategies for communicating indirect effects. Psychological Methods, 16, 93–115. http://doi.org/10.1037/a0022658 Schönbrodt, F. D., & Perugini, M. (2013). At what sample size do correlations stabilize? Journal of Research in Personality, 47, 609–612. http://doi.org/10.1016/j.jrp.2013.05.009
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
Search Now showing items 1-9 of 9 Production of $K(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV (Springer, 2012-10) The production of K(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 N/dydp_T$ at midrapidity \|y\|<0.5 in ... Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV (Springer, 2012-09) Measurements of the sphericity of primary charged particles in minimum bias proton--proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV with the ALICE detector at the LHC are presented. The observable is linearized to be ... Pion, Kaon, and Proton Production in Central Pb--Pb Collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (American Physical Society, 2012-12) In this Letter we report the first results on $\pi^\pm$, K$^\pm$, p and pbar production at mid-rapidity (\|y\|<0.5) in central Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, measured by the ALICE experiment at the LHC. The ... Measurement of prompt J/psi and beauty hadron production cross sections at mid-rapidity in pp collisions at root s=7 TeV (Springer-verlag, 2012-11) The ALICE experiment at the LHC has studied J/ψ production at mid-rapidity in pp collisions at s√=7 TeV through its electron pair decay on a data sample corresponding to an integrated luminosity Lint = 5.6 nb−1. The fraction ... Suppression of high transverse momentum D mesons in central Pb--Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV (Springer, 2012-09) The production of the prompt charm mesons $D^0$, $D^+$, $D^{*+}$, and their antiparticles, was measured with the ALICE detector in Pb-Pb collisions at the LHC, at a centre-of-mass energy $\sqrt{s_{NN}}=2.76$ TeV per ... J/$\psi$ suppression at forward rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (American Physical Society, 2012) The ALICE experiment has measured the inclusive J/ψ production in Pb-Pb collisions at √sNN = 2.76 TeV down to pt = 0 in the rapidity range 2.5 < y < 4. A suppression of the inclusive J/ψ yield in Pb-Pb is observed with ... Production of muons from heavy flavour decays at forward rapidity in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV (American Physical Society, 2012) The ALICE Collaboration has measured the inclusive production of muons from heavy flavour decays at forward rapidity, 2.5 < y < 4, in pp and Pb-Pb collisions at $\sqrt {s_{NN}}$ = 2.76 TeV. The pt-differential inclusive ... Particle-yield modification in jet-like azimuthal dihadron correlations in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (American Physical Society, 2012-03) The yield of charged particles associated with high-pT trigger particles (8 < pT < 15 GeV/c) is measured with the ALICE detector in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV relative to proton-proton collisions at the ... Measurement of the Cross Section for Electromagnetic Dissociation with Neutron Emission in Pb-Pb Collisions at √sNN = 2.76 TeV (American Physical Society, 2012-12) The first measurement of neutron emission in electromagnetic dissociation of 208Pb nuclei at the LHC is presented. The measurement is performed using the neutron Zero Degree Calorimeters of the ALICE experiment, which ...
Search Now showing items 1-1 of 1 Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (Elsevier, 2014-09) Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, \|y\| $\le$ 0.8, in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV have been measured using the ALICE detector ...
Search Now showing items 31-40 of 165 Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV (Springer, 2012-09) Measurements of the sphericity of primary charged particles in minimum bias proton--proton collisions at $\sqrt{s}$=0.9, 2.76 and 7 TeV with the ALICE detector at the LHC are presented. The observable is linearized to be ... Measurement of inelastic, single- and double-diffraction cross sections in proton-proton collisions at the LHC with ALICE (Springer, 2013-06) Measurements of cross sections of inelastic and diffractive processes in proton--proton collisions at LHC energies were carried out with the ALICE detector. The fractions of diffractive processes in inelastic collisions ... Measurement of electrons from heavy-flavour hadron decays in p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV (Elsevier, 2016-03) The production of electrons from heavy-flavour hadron decays was measured as a function of transverse momentum ($p_{\rm T}$) in minimum-bias p–Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with ALICE at the LHC for $0.5 ... Production of $\Sigma(1385)^{\pm}$ and $\Xi(1530)^{0}$ in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV (Springer, 2017-06) The transverse momentum distributions of the strange and double-strange hyperon resonances ($\Sigma(1385)^{\pm}$, $\Xi(1530)^{0}$) produced in p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV were measured in the rapidity ... Direct photon production in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV (Elsevier, 2016-03) Direct photon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV was studied in the transverse momentum range $0.9 < p_{\rm T} < 14$ GeV/$c$. Photons were detected via conversions in the ALICE ... Multi-strange baryon production in p-Pb collisions at $\sqrt{s_\mathbf{NN}}=5.02$ TeV (Elsevier, 2016-07) The multi-strange baryon yields in Pb--Pb collisions have been shown to exhibit an enhancement relative to pp reactions. In this work, $\Xi$ and $\Omega$ production rates have been measured with the ALICE experiment as a ... Charged–particle multiplicities in proton–proton collisions at $\sqrt{s}=$ 0.9 to 8 TeV, with ALICE at the LHC (Springer, 2017-01) The ALICE Collaboration has carried out a detailed study of pseudorapidity densities and multiplicity distributions of primary charged particles produced in proton-proton collisions, at $\sqrt{s} =$ 0.9, 2.36, 2.76, 7 and ... Energy dependence of forward-rapidity J/$\psi$ and $\psi(2S)$ production in pp collisions at the LHC (Springer, 2017-06) We present ALICE results on transverse momentum ($p_{\rm T}$) and rapidity ($y$) differential production cross sections, mean transverse momentum and mean transverse momentum square of inclusive J/$\psi$ and $\psi(2S)$ at ... Transverse Momentum Distribution and Nuclear Modification Factor of Charged Particles in p-Pb Collisions at $\sqrt{s_{NN}}$ = 5.02 TeV (American Physical Society, 2013-02) The transverse momentum ($p_T$) distribution of primary charged particles is measured in non single-diffractive p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV with the ALICE detector at the LHC. The $p_T$ spectra measured ... Measurement of azimuthal correlations of D mesons and charged particles in pp collisions at $\sqrt{s}=7$ TeV and p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV (Springer, 2017-04) The azimuthal correlations of D mesons and charged particles were measured with the ALICE detector in pp collisions at $\sqrt{s}=7$ TeV and p-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV at the Large Hadron Collider. ...
July 11th, 2015, 05:37 AM # 1 Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry New Spherical Trig Function I have recently begun sorting through some of our old writings and I came across what looks to me like a new spherical trigonometric function. It goes like this: The relationship of the angle α, formed between the tangent of a point along a 45° small circle laid on the surface of a sphere and tilted to 45° (such that the small circle intersects both the pole and the equator) and a line of longitude passing through that same point, and the elevation angle E of this point (defined as the angle between the equator and tangent point as measured from the center of the sphere) is: α = cot ((1 – sin E) / sin E) = cot (1/sin E – 1) At this time, there are no hits at all on google for this equation: https://www.facebook.com/photo.php?f...3424896&type=1 https://www.facebook.com/photo.php?f...3424896&type=1 This was posted at Wolfram a couple days ago and there has been no discussion over there. Geometry Forum—Wolfram Community It would be ok if anyone has any questions. July 28th, 2015, 09:24 AM # 2 Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions Could you draw a diagram please? July 29th, 2015, 01:29 AM # 4 Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 Have you a derivation of this formula? It seems strange since you are working in degress yet you have an angle allegedly equal to the cotangent of an angle that can only be measured in radians. I can confirm that the small circle has to be the 45 degree parallel of altitude to fit the between the equator and pole as shown. It has to be translated as well as rotated to fit into the assigned position. Quote: Last edited by studiot; July 29th, 2015 at 01:57 AM. July 29th, 2015, 08:15 AM # 5 Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions I actually managed to obtain the following: $\displaystyle \tan E = \frac{1-\sin\alpha}{1+\sin\alpha}$ which makes sense to me. When $\displaystyle \alpha$ = 90 degrees, E = 0 (equator). When $\displaystyle \alpha$ = 0 degrees, E = 45 degrees. Here's the derivation (sorry, it's a bit wordy... I should draw a diagram and add points to it). Draw a line from the centre of the small, tilted circle (C) to the point where the circle touches the equator (A). This has length r (radius of circle). Now subtend an angle E upwards from the equatorial plane from the centre of the sphere and draw a plane. The plane will cut the radius of the circle at a given position X. Let's assign a length x to the line XA. Let's also name the point (actually two points, but we only care about one...) where the plane intersects the circumference of the circle B. So, there is a triangle BCX in the plane of the circle. The angle $\displaystyle \alpha$ is the angle between the South longitude line and the tangent of the circle at the point on the circle's circumference where the plane intersects the circle. This angle is the same as angle CBX. Therefore: $\displaystyle \sin \alpha = \frac{r-x}{r} = 1 - \frac{x}{r}$ $\displaystyle x = r(1-\sin\alpha)$ Now let's draw a line vertically downwards (from the persepective of someone sitting at the North pole) from X. The line hits the equatorial plane somewhere within the sphere at a point D. Let's assign the length h = DX. If the sphere has radius R, then $\displaystyle \tan E = \frac{h}{R-h}$ (since the small circle is tilted at 45 degrees) So far, we should have everything in accordance with the diagram posted by SteveUpson, albeit with names to some of the points and some algebraic letters assigned to various lengths. We also know $\displaystyle h = \sqrt{2} x$ because of the 45 degree isosceles triangle with hypoteneuse AX. Therefore: $\displaystyle \tan E = \frac{\frac{x}{\sqrt{2}}}{R - \frac{x}{\sqrt{2}}} = \frac{x}{\sqrt{2}R - x}$ We also know that $\displaystyle r = \frac{diameter}{2} = \frac{\sqrt{2} R}{2} = \frac{R}{\sqrt{2}}$ So $\displaystyle \tan E = \frac{x}{\sqrt{2}R-x} = \frac{x}{2r - x}$ Substituting for $\displaystyle x$: $\displaystyle \tan E = \frac{r(1-\sin\alpha)}{2r - r(1-\sin\alpha)} = \frac{1-\sin\alpha}{1 + \sin\alpha}$ I don't know if it's equivalent to your result or not. July 30th, 2015, 03:58 AM # 6 Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions I did a bit more investigating... The relation in my last post can be rearranged to form $\displaystyle \sin \alpha = \frac{1-\tan E}{1 + \tan E}$ I attached a plot of the function and this seems like a sensible curve. I think this is the correct result. Therefore, to answer your question, I think you cannot find your result in the literature because it is incorrect. July 30th, 2015, 05:06 AM # 7 Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions Attachment. The y-axis limits are 90 and -90 degrees, but for some reason the numbers didn't get displayed by my plotting tool. Also, here's an error correction from my first post: The line "We also know $\displaystyle h=\sqrt{2}x$ because of the 45 degree isosceles triangle with hypoteneuse AX. Therefore:" should read "We also know $\displaystyle h=\frac{x}{\sqrt{2}}$ because of the 45 degree isosceles triangle with hypoteneuse AX. Therefore:" The rest of the derivation is unaffected by this error. Last edited by Benit13; July 30th, 2015 at 05:11 AM. August 2nd, 2015, 07:29 PM # 8 Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry Quote: We KNOW that when your point B is at the North pole, then the tangent of this point is congruent with a line of longitude passing through this point. E=90° gives α=0° You started off correct with E=0° gives α=90° Quote: These are not the same. There is a method to use three discrete rotations about the x, y, and z in order to get a straight-on look at the angle α. also, if we graph the new function, we get this: E=0° α=90° E=30° α=45° E=45° α=22.5° E=60° α=8.79° E=90° α=0° August 2nd, 2015, 07:45 PM # 9 Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry transform for viewing angle in a plane tangent to the point transform for viewing angle in a plane tangent to the point (shear plane) August 2nd, 2015, 08:58 PM # 10 Senior Member Joined: Jul 2015 From: Florida Posts: 154 Thanks: 3 Math Focus: non-euclidean geometry My notes claim that we found a proof for this by "solving a rather awkward right spherical triangle (Napier)" but I haven't tried to go back and figure out what we were talking about. Edited to add> My best guess is that it's the right triangle formed by the equator, longitude line, and tangent. Edited again to add> It's awkward because the right angle is on the opposite side of the sphere. Nope, it's got to be the great circle formed by the plane that passes through the center of small circle. It's been years. Last edited by steveupson; August 2nd, 2015 at 09:25 PM. Tags function, non-euclidean, spherical, trig Thread Tools Display Modes Similar Threads Thread Thread Starter Forum Replies Last Post Spherical mirrors, spherical? proglote Physics 9 December 10th, 2018 06:20 AM Trig Function to a+bi WWRtelescoping Complex Analysis 4 February 26th, 2014 11:48 AM making transition from plane trig to spherical trig cr1pt0 Trigonometry 2 September 5th, 2013 06:11 PM Spherical Bessel function derivaties rkaminski Calculus 2 August 13th, 2012 04:26 AM function of a trig function integral jimooboo Calculus 3 February 26th, 2012 12:07 PM
Laustsen, Niels J. (2002) Maximal ideals in the algebra of operators on certain Banach spaces. Proceedings of the Edinburgh Mathematical Society, 45 (3). pp. 523-546. ISSN 0013-0915 PDF (download.pdf) download.pdf Download (296kB) Abstract For a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 < p < \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following. (i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$. (ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces. Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem. Item Type: Journal Article Journal or Publication Title: Proceedings of the Edinburgh Mathematical Society Additional Information: The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 45 (3), pp 523-546 2002, © 2002 Cambridge University Press. RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics Uncontrolled Keywords: /dk/atira/pure/researchoutput/libraryofcongress/qa Subjects: Departments: Faculty of Science and Technology > Mathematics and Statistics ID Code: 2379 Deposited By: ep_importer Deposited On: 01 Apr 2008 10:14 Refereed?: Yes Published?: Published Last Modified: 20 Sep 2019 23:52 URI: https://eprints.lancs.ac.uk/id/eprint/2379 Actions (login required) View Item
October 16th, 2014, 09:17 PM # 11 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 October 16th, 2014, 09:22 PM # 12 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond That is what's happening! The statement of the limit says "we can make f(x) as close to 1 as we like by choosing x sufficiently large". There is no restriction on the size of x. If there were, we'd have a constant, not a limit. Last edited by greg1313; October 16th, 2014 at 09:40 PM. October 16th, 2014, 09:39 PM # 13 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 Quote: October 16th, 2014, 09:48 PM # 14 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 October 16th, 2014, 10:02 PM # 15 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 Quote: The problem is at a much more basic level. If something is endless or limitless then by definition it cannot have an end point. The concept of a completed infinity implies an end. If something is truly endless then it can never be completed. And so saying something has no limit does not mean it is infinite. Furthermore, consider the expression 'as x approaches infinity'. Assuming we can use infinity in this context, then it does not matter how much we increase x because we will still be an infinite distance away from infinity. The same applies if we decrease x or leave x unchanged. Since increasing or decreasing or not changing x all result in being the same distance from infinity, it follows that we cannot 'approach' it. Last edited by Karma Peny; October 16th, 2014 at 10:43 PM. October 16th, 2014, 10:40 PM # 16 Senior Member Joined: Aug 2012 Posts: 2,395 Thanks: 749 One can employ infinity for its usefulness, even while denying it any ontological status. That is, one can if one likes adopt the following position: "I know that infinite numbers do not exist; and if they did exist I'd stamp them out; for I dislike them. Yet, they are curiously useful and make many computations simpler in physics, mathematics, biology, and economics. So I will freely use them, even though they do not exist." In fact that's a familiar sounding position. Some people felt that way about the idea that the earth revolves around the sun. Of course it doesn't, everybody knows that God put the earth at the center of the universe. But the model -- false though it is -- with the sun at the center of the solar system sure does simplify the calculations. So let's just use that system in astronomy class, even when we know it's not literally true. And that "imaginary" number i, with the property that if you square it you get -1. That doesn't exist either. Of course it does allow us to solve polynomial equations, and it makes the calculations of electromagnetic theory far simpler ... so let's just take it as a useful fiction. OP, would that satisfy you? From a philosophical point of view, what does "true" mean, anyway? Isn't truth just what we all find convenient to believe? You could in fact work out all the math with the earth as the center of the universe. The universe wouldn't change, only the math would. I don't care if the engineers building the bridge I'm driving on believe in infinity. I just expect that they've studied calculus. October 16th, 2014, 11:20 PM # 17 Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7 Quote: I think you are struggling with exactly the same problems with infinity as me. I looked at your articles, and I think the problem lies in what you have written under the topic "Removing infinity from repeating decimals", you write that: 0.999… = 1 this is incorrect (yet many mathematicians accept it) lim(0.999…) = 1 this is correct You also write that " we can say that the limit of 0.333… as the number of decimal places increases, equates to one third. This is completely different to saying that 0.333… equals one third, which would be wrong, as it never does." If you look at my last thread on this forum "What lies beyond infinity", I presented similar ideas there, but they were not accepted as valid. I did not suggest abandoning infinity. Instead, I was even more radical, I suggested abandoning the decimal number system altogether if we can't accept infinitesimals. In other words, I can deny that $\displaystyle $$\frac{1}{3}$ has a decimal number representation if the infinitesimals don't exist. $\displaystyle $$\frac{1}{3} = 0.333.....$ only if the infinitesimals exist, infinitesimals which are $\displaystyle \neq 0$. Otherwise, we are forced to admit that $\displaystyle $$\frac{1}{3} \approx 0.333.....$ Somehow I get the feeling that abandoning the decimal number system corresponds to your idea of abandoning infinity. We should abandon the infinite amount of decimals of $\displaystyle $$\frac{1}{3}$ if we cannot count them correctly. I see no other way out of this problem. Either we can count them all or we can't count them all. October 16th, 2014, 11:40 PM # 18 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 Quote: There are many processes that appear to use infinity, such as calculus, but the word is often used because people think of infinity as being an unimaginably large number. Calculus was not devised using infinity, the proof of the fundamental theorem does not require infinity, the use of differential and integral calculus does not require infinity. But still, such processes are claimed to justify the value of infinity. Gauss’s views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us to talk about limits. The notion of a completed infinity doesn’t belong in mathematics'. Quote: In order to solve previously unsolvable quadratic equations, we assume that the square root of -1 can exist. This appears wrong as it breaks the rule that something times itself must be positive (but maybe this rule was wrong all the time). It soon becomes apparent that we can form a logically consistent set of mathematical rules that include the square root of -1. In a similar fashion, it is perfectly acceptable to assume a completed collection of an endless sequence can exist. But in this case it is not possible to form a logically consistent set of rules. Many paradoxes arise and it appears to be possible to prove that such an object cannot exist. Last edited by Karma Peny; October 16th, 2014 at 11:53 PM. October 17th, 2014, 12:16 AM # 19 Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2 Quote: I've just looked through that thread you mentioned, it's a long one! The real number line supposedly stretches from -infinity to +infinity and any section of it, however small, contains an infinite number of numbers. As you might have guessed, I reject the whole basis of a continuum as it is based on the concept of infinity, which is nonsensical. I prefer to address the problem that the continuum is trying to provide a solution for. The problem, as I see it, is how can we fully express irrational numbers in a framework that we can work with? The solution is to use symbols, such as π and √2 rather than their decimal expansions. We can then work with irrational numbers in an abstract framework. If we want to see a numeric result rather than a set of expressions containing symbols then we have to expand the expressions using the constraints of the real or abstract world to which we are applying it. This problem has already been solved to some extent for the encoding of vector graphics in computer software. Vector images are made up from multiple objects. Each object consists of mathematical instructions that define shapes. And each shape is defined in terms of points and paths. This makes the image fully scalable without loss of quality. It only gets converted to discrete pixels when the image is rendered onto a real world object like a small screen or the large side of a building. Last edited by Karma Peny; October 17th, 2014 at 12:21 AM. October 17th, 2014, 01:34 AM # 20 Math Team Joined: May 2013 From: The Astral plane Posts: 2,274 Thanks: 943 Math Focus: Wibbly wobbly timey-wimey stuff. Tags infinity, mathematics, removed Search tags for this page Click on a term to search for related topics. Thread Tools Display Modes Similar Threads Thread Thread Starter Forum Replies Last Post Infinity ryanninjasheep Number Theory 11 September 30th, 2013 12:43 PM is 1/infinity = lim x->infinity 1/x sivela Calculus 1 June 25th, 2012 10:04 AM Infinity deanmullen10 Calculus 6 July 4th, 2010 01:24 AM Solve X where -infinity>x>infinity -DQ- Algebra 5 September 14th, 2009 05:13 AM Infinity... nerd9 Number Theory 1 December 31st, 1969 04:00 PM
In Absolute Value as Belief, Steven Daskal aims to save anti-Humeanism against Lewis’s attacks in the Desire as Belief papers by changing the connection between credences and values. I like the idea he’s trying to develop – trying to use the difference in value between $A$ and $\neg A$ to state the theory more carefully. But the particular way he does it isn’t quite working, and I don’t really know how to fix it. Here is the equation he ends up wanting to defend. $$\sum_y C(g(A) = y) \cdot y = \sum_w C(w) \cdot (V(w \bullet A) – V(w \bullet \neg A))$$ The sum on the left is over possible values. The sum on the right is over possible worlds. And the $\bullet$ is an imaging operator; so $w \bullet A$ is the nearest world to $w$ where $A$ is true. (The general form of this allows ties, but we won’t need that level of specificity.) I don’t think this can be right in general as it stands. Here is a puzzle case for the view. Assume there are three equiprobable worlds, $w_1, w_2, w_3$, and the first two have goodness 1, the third has goodness 0. Assume also that these goodness facts are known. Let $A$ be the proposition that $w_1$ obtains. So we have the following for the LHS of the equation. $$\sum_y C(g(A) = y) \cdot y = C(g(A) = 1) \cdot 1 = 1$$ Assuming that strong centring obtains for the ‘nearness’ function, we get the following. $w_1 \bullet A = w_1$ $w_2 \bullet A = w_1$ $w_2 \bullet \neg A = w_2$ $w_3 \bullet A = w_1$ $w_3 \bullet \neg A = w_3$ It isn’t clear what $w_1 \bullet \neg A$ should be; let’s call it $w_x$. Substituting all these into the RHS of the equation we get: $$\frac{V(w_1) – V(w_x)}{3} + \frac{V(w_1) – V(w_2)}{3} + \frac{V(w_1) – V(w_3)}{3}$$ The second term equals 0, and the third term equals 1/3. The value of the first term is unknown, but it is either 0 or 1/3. So the sum equals either 1/3 or 2/3. So we have LHS equals 1, and RHS equals either 1/3 or 2/3. So the equation doesn’t work. As I said, I like the idea of using differences between values of propositions and their negations in the theory of motivation. But I don’t think this particular way of doing it is quite right.

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.