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Determine whether this series converges or diverges: $$\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)!}$$
Thought about using the limit theorem or by comparison but am so stuck. any pointers would be appreciated guys
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professi... |
We first determine all the eigenvalues of the matrix $A$.The characteristic polynomial $p(t)$ of $A$ is given by\begin{align*}p(t)&=\det(A-tI)\\[6pt]&=\begin{vmatrix}3-t & -12 & 4 \\-1 & -t &-2 \\-1 & 5 & -1-t\end{vmatrix}.\end{align*}Using the first row cofactor expansion, we compute\begin{align*}p(t)&=(3-t)\begin{vma... |
I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over velocities.
But when I've tried to derive NS equations with viscosity and bulk coefficients, I've failed. Most textbooks contai... |
skills to develop
To understand what chi-square distributions are. To understand how to use a chi-square test to judge whether two factors are independent. Chi-Square Distributions
As you know, there is a whole family of \(t\)-distributions, each one specified by a parameter called the degrees of freedom, denoted \(df\... |
Is the Derivative Linear Transformation Diagonalizable?
Problem 690
Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by\[ T( ax^2 + bx + c ) = 2ax + b . \]
Is $T$ diagonalizable? If so, f... |
I have just learned about the Seifert-Van Kampen theorem and I find it hard to get my head around. The version of this theorem that I know is the following (given in Hatcher):
If $X$ is the union of path - connected open sets $A_\alpha$ each containing the basepoint $x_0 \in X$ and if each intersection $A_\alpha \cap A... |
I wish to compute $$ \int_C \frac{\sin z}{(z+1)^7}\ \text{d}z $$ where $C$ is the circle of radius $6$, centre $0$, positively oriented.
Now I know that $f(z) = \frac{\sin z}{(z+1)^7}$ is analytic on $\mathbb{C}\setminus\{-1\}$ since $f$ is the composition and quotient of analytic functions.
I tried to use the Cauchy-G... |
"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
As forest and natural resource managers, we must be aware of how our timber management practices impact the biological communities in which they occur. A silvicultural prescription is going to influence not only the timber we are growing but also the plant and wildlife communities that inhabit these stands. Landowners,... |
Let $G^n$ denote the OR product of a graph with itself $n$ times, i.e. the graph which has an edge between distinct vertices $(v_1,v_2,\ldots,v_n)$ and $(u_1,u_2,\ldots,u_n)$ if there exists some $i$ so that $(v_i,u_i)$ is an edge in $G.$
The book Fractional Graph Theory by Scheinerman and Ullman has a number of wonder... |
They make a virtual image on a plane behind the HUD. A good model system to understand this kind of thing is the
aplanatic sphere: Born and Wolf "Principles of Optics"gives a good discussion of this if you can get hold of it.
If not, here's a simple explanation: imagine a lens system that collimates a point source, so ... |
Let's say we have an object in 3D space located at
P(1,1,1) and if we decided we wanted to rotate this object within any of the planes
XY,
XZ,
YZ along any of the 3 axis where this is a basic 3D Cartesian system in a Right-Hand System the rotation matrices $R_n$ for
P will look like these:
$$ R_x \space|| \space R_y \s... |
I think I've figured this out. The point is that, the rigorous meaning one can draw from the formal covariance of $J^\mu$ is that the momentum-space coefficient functions of $J^\mu$ (i.e. the functions in front of monomials of $a_p$ and $a^\dagger_p$) transform covariantly under the change of variable $p\to \Lambda p$.... |
27 3 Summary Derive Q of R in parallel with tank circuit
I've been experimenting with an LC tank circuit in series with a resistance R, and I've noted that the Q seems to increase with R. I've tried to derive this result via phasor analysis, but I'm not sure if my expression is correct.
To make things clear, I'm talkin... |
Likelihood with |omega_K \ne 0
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1of 1 Posts:69 Joined:June 13 2007 Affiliation:Malaviya National Institute of Technology Jaipur
Hi,
I am trying to get the likelihood for the models with \Omega_k \ne 0. I am using CAMB to get the [tex]cls[/tex] and then using these [tex]cls[/tex] to get the like... |
I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand most simple proof which can be written on a forum are not "high-level" enough for MO and other places are more appropriat... |
I think I've figured this out. The point is that, the rigorous meaning one can draw from the formal covariance of $J^\mu$ is that the momentum-space coefficient functions of $J^\mu$ (i.e. the functions in front of monomials of $a_p$ and $a^\dagger_p$) transform covariantly under the change of variable $p\to \Lambda p$.... |
Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$ Problem 613
Let $m$ and $n$ be positive integers such that $m \mid n$.
(a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined. (b) Prove that $\phi$ is a group homomorphism. (c) Prove that $\phi$ ... |
One more post before I return to core algebra. We need to look a bit more at division and fractions.
Now, as you’ve seen, something like 8 ÷ 2 indicates division, but another way to show exactly the same thing is \[\frac{8}{2}\]. In other words, fractions are just another way to show division. Now before I expand on th... |
How did the early sailors determine their latitude position without GPS? That is the topic of today’s post.
Now first, a little background. The earth’s axis is tilted with respect to its orbit about the sun. The angle of this tilt is approximately 23.5°. This causes the northern and southern hemispheres to get more sun... |
The problem is this-
Evaluate $\int\int_R \sqrt{xy - y^2}dxdy$ where R is a triangle with vertices (0,0), (10,1) and (1,1).
I have done up to this- $\int_{y=0}^{1}\int_{x=y}^{10y} (x^2 - xy)^{\frac12}dxdy = \int_{y=0}^{1}dy\int_{x=y}^{10y}(x^2 - xy)^{\frac12}dx$
I was stuck here, and opened the book to check up the sol... |
In the last 10 years there has been more and more crossover of techniques from high energy physics being used in AMO and condensed matter scenarios, in particular diagrammatic techniques and related perturbative calculations. Unlike the relativistic case, a simple Schrodinger field theory will conserve the number of pa... |
Put into one sentence, Noether's first Theorem states that a continuous, global, off-shell symmetry of an action $S$ implies a local on-shell conservation law. By the words
on-shell and off-shell are meant whether Euler-Lagrange equations of motion are satisfied or not.
Now the question asks if
continuous can be replac... |
For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
gmc_nxtman Posts: 1147 Joined: May 26th, 2015, 7:20 pm Kazyan wrote:
Component found in a CatForce result:
Code: Select all
x = 42, y = 67, rule = LifeHistory
A$.2A$2A5$7.A$8.2A$7.2A19$24.2A$24.2A2$39.A$37.A3.A$3... |
The idea, as I understand it, is that the term between parenthesis in Big O bounds the rest of the series asymptotically (as $x$ goes to $0$ or $\infty$) giving and elegant way of writing the series (if you don't like "+ ..." or "etc") while also providing an error approximation. Taking the example of $e^x$:
$$ e^x=1+x... |
Presented by:
Jeremie Unterberger Université de Lorraine
Date:
Thursday 25th October 2018 - 10:00 to 11:00
Venue:
INI Seminar Room 1
Abstract:
We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta(t,x), \qquad (t,x)\in{\math... |
I was recently considering how to justify the formula relating to the Doppler effect for sound waves to a group of eleventh grade students who are likely encountering it for the first time. The formula in question is
$$f'=f \frac{v-v_o}{v+v_s}, $$
where $v$ is the speed of the sound wave, $f$ is the frequency of the wa... |
ISSN:
1547-5816
eISSN:
1553-166X
All Issues
Journal of Industrial & Management Optimization
October 2013 , Volume 9 , Issue 4
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Export/Reference:
Abstract:
In this paper, we propose a primal-dual approach for solving the generalized fractional programming problem. The outer iteration of the algorithm ... |
Let me first recall the definition and some properties of the exponential function. The exponential function in basis $a$ is defined as follows:$ \exp_a(x) = a^x$. If $a>1$, such a function
grows faster than any polynomial $p(x)$.More formally, this property is interpreted in the two following ways.
1) For any polynomi... |
The sequence $\{a_n\}_{n=1}^\infty$ is cauchy if for every $\epsilon>0$, there is a corresponding natural number $N$ such that
$$ m,n\geq N\Rightarrow |a_m-a_n|<\epsilon $$
I am doing a particular problem where the problem talks about cauchy sequences of rational numbers and I am not sure how that is different from a n... |
I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iterati... |
A Simple Abelian Group if and only if the Order is a Prime NumberLet $G$ be a group. (Do not assume that $G$ is a finite group.)Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.Definition.A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of... |
When using a hypothesis test for matched or paired samples, the following characteristics should be present:
Simple random sampling is used. Sample sizes are often small. Two measurements (samples) are drawn from the same pair of individuals or objects. Differences are calculated from the matched or paired samples. The... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
Suppose that we have a twice-differentiable function $f$ on $x\in [0,\infty)$ such that
$f(x)>0$ on $x\in [0,\infty)$ (i.e. strictly positive) $f'(x)<0$ on $x\in [0,\infty)$ (i.e. strictly decreasing) $f^{''}(x) >0$ on $x\in [0,\infty)$ (i.e. strictly convex) $\lim_{x \to \infty} f(x)=0$
and we have a twice-differentia... |
I've written the following code for a quantum circuit using the qcircuit package and everything looks how I want it to look except that the curly brackets enclosing the bottom 2 lines are too far away.
I understand that my code for getting the labels on the dotted lines is probably not the best way of achieving this, b... |
ISSN:
1930-5311
eISSN:
1930-532X
All Issues
Journal of Modern Dynamics
October 2013 , Volume 7 , Issue 4
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Export/Reference:
Abstract:
Let ${\cal Q}$ be a connected component of a stratum in the moduli space of abelian or quadratic differentials for a nonexceptional Riemann surface $S$ of finite type.... |
Latest results from NA62
Pre-published on: 2019 July 05
Published on: 2019 October 04
Abstract
The ultra rare kaon decay $K\rightarrow \pi \nu \bar{\nu}$, being one of the theoretically cleanest meson decays, is very sensitive to the effects of New Physics at high mass scales. NA62 is a fixed-target experiment at the C... |
Not quite an answer, but a heuristics from the point of view of Weil philosophy about why the rationality mod $p$ is much easier.
Weil reduced the rationality of zeta-function to the existence of a good cohomology theory. The crucial property of a cohomology which satisifies the Weil axioms is that it has zero-characte... |
Let $x\in X$. We have $P_n(x)=\sum_{j=1}^n\langle x,e_j\rangle e_j$ so , by Bessel-Parseval equality $$\lVert P_nx-x\rVert^2=\sum_{j\geq n+1}|\langle x,e_j\rangle|^2.$$As the latest series is convergent we have the result.
Now, let $K\subset H$ compact. Fix $\varepsilon >0$. Then we can find an integer $N$ and $x_1,\do... |
Set up
I'm reading a text where formulae are numbered in the following way. First there is a numbering function $\nu$ such that all logical particles (truth-functional operators and quantifiers, are congruent to some fixed $i$ mod 5). In the author's choice, $\nu(\neg)=0, \nu(\rightarrow)=5$ and so on with some arbitra... |
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to every "space" $X$ a triangulated category $\newcommand{\D}{\mathsf D} \D (X)$, its derived category, and to each morphism ... |
Gregory from Magnus C of E School inNottingham and Luke from St Patrick's School reasoned correctly.Here is Luke's explanation:
For this question I found a pattern in it:
For example $C_1$ has length of 1 and $C_2$ has length of$\frac{2}{3}$
and there is a pattern in this because 1 $\times$2 = 2, sothere is your numera... |
Reference documentation for deal.II version 8.4.1
#include <deal.II/dofs/function_map.h>
typedef std::map< types::boundary_id, const Function< dim, Number > * > type
This class declares a local typedef that denotes a mapping between a boundary indicator (see GlossBoundaryIndicator) that is used to describe what kind of... |
S. G. Dani and C. R. E. Raja Asymptotics of measures under group automorphisms and an application to factor sets Lie groups and ergodic theory (Mumbai, 1996), 59-73, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.
S. G. Dani and C. R. E. Raja A note on Tortrat Groups Journal of Theoretical P... |
I always thought that both “proof by reductio ad absurdum” and “proof by contradiction” mean the same, but now my professor asked this question on my homework and I don't know.
I believe that in both cases you assume the negation of the conclusion and develop a contradiction through the premises. This will imply the co... |
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1. The graphic work of Felicien Rops. Notes on the life of Rops / by Lee Revens. "Instrumentum Diaboli" / by J.-K. Huysmans
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2012, 10th ed., ISBN 9780071761... |
You can design that filter manually without problems. Matlab just uses a very simplistic approach to comb filtering with a delay line.
In order to keeps things as simple as possible I would recommend you use a series of notch filters to remove each partial of your harmonic noise separately. That also gives you more con... |
Haskell's answer does a great job of outlining conditions that a derivative $f'$
must satisfy, which then limits us in our search for an example. From there we see the key question: can we provide a concrete example of an everywhere differentiable function whose derivative is discontinuous on a dense, full-measure set ... |
It is unknown whether $P\subseteq CSL$ or $P\not\subseteq CSL$, where
$P$ is the set of all languages decidable in polynomial time on a deterministic Turing machine, and $CSL$ is the class of context-sensitive languages, known to be equivalent to $NSPACE(O(n))$, the languages decided by linear-bounded automata.
For man... |
Actually, on a dark night, the fraction of the sky that is light is pretty negligible. That's what it means to be a dark night ;-)
It's actually not hard to get an estimate of the density of light in the universe. Let's say that "light" includes photons of all wavelengths (not just visible light) for simplicity. A stra... |
Let $A=\begin{bmatrix}a& b \\c& d\end{bmatrix}$.Then as $A$ is a symmetric matrix, we have $A^{\trans}=A$.This implies that\[\begin{bmatrix}a& c \\b& d\end{bmatrix}=\begin{bmatrix}a& b \\c& d\end{bmatrix}.\]Hence we have $b=c$ by comparing entries.
Now, we find the characteristic polynomial $p(t)$ of $A$.We have\begin{... |
Talk:Celestial mechanics
The deleted discussions concern minor details on the two-body problem without importance for the understanding of the subject of this article. (Celestial Mechanics) I prefer not to overcrowd the article with many algebraic demonstrations.
Corrections
Using heliocentric vectors, the
angular mome... |
While the choice of activation functions for the hidden layer is quite clear (mostly sigmoid or tanh), I wonder how to decide on the activation function for the output layer. Common choices are linear functions, sigmoid functions and softmax functions. However, when should I use which one?
Regression: linear (because v... |
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Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut... |
Solutions Colligative Properties Colligative properties :- (1) The properties of dilute solution those depend on the number of solute particles irrespective to their nature. (2) Colligative properties are classified into four types a. Relative lowering of vapour pressure b. Elevation of boiling point c. Depression of f... |
Given regular expressions $R_1, \dots, R_n$, are there any non-trivial bounds on the size of the smallest context-free grammar for $R_1 \cap \cdots \cap R_n$?
This is a great question and it really lies within my interests. I'm glad that you asked it Max.
Let $n$ DFA's with at most $O(n)$ states each be given. It would... |
I have a bunch of points in $\mathbb{R}^3$ that I would like to translate and rotate so that their center is at the origin and the variance along the $x$ and $y$ axes are maximal (greedy, and in that order). To accomplish this I am trying to use python's principal components analysis algorithm. It is not behaving as I ... |
Skills to Develop
Compute probability in a situation where there are equally-likely outcomes Apply concepts to cards and dice Compute the probability of two independent events both occurring Compute the probability of either of two independent events occurring Do problems that involve conditional probabilities Compute ... |
It is well known that the generating function of a regular language $L$, i.e. $\sum n_kz^k$ where $n_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials $P(z)/Q(z)$. Suppose that $L$ is the language accepted by some finite automaton $\mathcal{A}$. How to find the polynomials ... |
Submodule Consists of Elements Annihilated by Some Power of an Ideal
Problem 417
Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$.Let $M’$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$.Prove that $... |
Abbreviation:
CRng$_1$
A
is a rings with identity $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ such that $\cdot$ is commutative: $x\cdot y=y\cdot x$ commutative ring with identity
Let $\mathbf{R}$ and $\mathbf{S}$ be commutative rings with identity. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\rightar... |
Images are essential elements in most of the scientific documents. LaTeX provides several options to handle images and make them look exactly what you need. In this article is explained how to include images in the most common formats, how to shrink, enlarge and rotate them, and how to reference them within your docume... |
It is well known that the generating function of a regular language $L$, i.e. $\sum n_kz^k$ where $n_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials $P(z)/Q(z)$. Suppose that $L$ is the language accepted by some finite automaton $\mathcal{A}$. How to find the polynomials ... |
I am currently working on a project where a two-phase flow is considered. The phases are described using a level set approach and a signed distance function from the interface between the phases where positive values are liquid and negative values are gas.
I consider a Stoke's flow which gives me a velocity field $v$ a... |
Recent Publications
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Mechanical power limitations emerge from the physical trade-off between force and velocity. Many biological systems incorporate power-enhancing mechanisms enabling extraordinary accelerations at small sizes. We establish how power enhancement emerges through the dynamic coupling of motors, springs... |
May be this question is really silly and obvious but I am missing something subtle. I am reading on Sensitivity and Block sensitivity.
Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a Boolean function.
Let $[n]=\{1,2,\dots,n\}$.
If $i\in[n]$, let $\Bbb 1_i$ be length $n$ vector with all $0$s except $1$ at $i$th position.
If $... |
Recent Publications
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Baryogenesis through neutrino oscillations is an elegant mechanism that has found several realizations in the literature corresponding to different parts of the model parameter space. Its appeal stems from its minimality and dependence only on physics below the weak scale. In this paper we show th... |
01/14/16
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This is the third post in the “Machinery behind Machine Learning” series and after all the “academic” discussions it is time to show meaningful results for some of the most prominent Machine Learning algorithms – there have been quite a few requests from interested readers in this direction. Today ... |
Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 \le p_1, \ldots, p_n \le 1$
(ii) For $1\le j \le k$, $|(\sum_{i=1}^n p_i^j) - \alpha_j|\le \epsilon$.
Here $k \ll n$. I am ... |
T.R. Jain and V.K. Ohri Solutions for Class 11 Statistics for Economics Chapter 6 – Diagrammatic Presentation of Data- Bar Diagrams and Pies Diagrams is regarded as an important concept to be studied thoroughly by the students. Here, we have provided T.R. Jain and V.K. Ohri Solutions for Class 11.
Board CBSE Class Clas... |
Is it known or unknown whether hypergraph minimal covers are P-enumerable? I would be most happy with lower bounds. I'd also like to hear about conditional results, which assume some conjecture is true. Of course, I'd also want to know about closely related problems (such as independent sets and cliques).
Motivation. I... |
I have a finite automaton by the standard model Hopcroft & Ullman define: $$ M = (Q, \Sigma, \delta, q_0, F) $$
Where $\delta$ is the transition function mapping $Q \times \Sigma \mapsto Q$, such that $\delta(q, a)$ is a state for each state $q \in Q$, the set of all states, and input symbol $a \in \Sigma$, the alphabe... |
ISSN: 0538-8066
Keywords: Chemistry ; Physical Chemistry
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Chemistry and Pharmacology
Notes: The high-temperature reaction between sulfur dioxide and acetylene in an excess of argon was studied in a 1-in. i.d. single-pulse shock tube. Mixtures ranging from ... |
For (b), the statement is false. Try to find a counter example.A typical nilpotent matrix is an upper triangular matrix whose diagonal entries are all zero.
Proof.
(a) Show that $AB$ is nilpotent
Since $A$ is nilpotent, there exists a positive integer $k$ such that $A^k=O$. Then we have\[(AB)^k=(AB)(AB)\cdots (AB)=A^kB... |
I'm working with a simple local level model in a textbook \begin{align} y_t &= \alpha_t + \epsilon_t, \qquad \epsilon_t \sim N(0, \sigma_\epsilon^2) \\ \alpha_{t+1} &= \alpha_t + \eta_t, \qquad \eta_t \sim N(0, \sigma_\eta^2) \end{align}
The conditional distribution of $\alpha_t$ given $Y_{t-1}$ (the set of all observa... |
You compare SFCs at different speeds. That is like comparing payloads for differently sized aircraft. SFC goes up with speed and, therefore, must be compared at the same speed. The work performed by an engine is thrust times distance, and higher speed means that the same thrust will perform more work per unit of time w... |
Let $p$ be a prime number. Denote by $P$ the set of all primes which are not greater than $p$.
Is there a well known estimation of the product of all prime numbers in $P$ (i.e. $\prod_{q\in P}q$)?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
Associations to the word «Euclidean» Noun Geometry Manifold Rn Space Curvature Axiom Hilbert Subset Topology Subspace Vector Polynomial Theorem Integer Vertex Plane Algebra Pseudo Tensor Algorithm Norm Polygon Relativity Generalization Steiner Graph Coordinate Matrice Triangle Calculus Dimension Optimization Symmetry S... |
I want to show that
$$\int_0^{\pi} \log(2 - 2 \cos x) = 0$$
However, I cannot do this. I tried splitting the integral into $\int_0^{\pi/3} \log(2 - 2 \cos x)\,dx + \int_{\pi/3}^{\pi} \log(2 - 2 \cos x) \,dx$ and showing that the two parts were negatives of one another. Wolframalpha does not give very a simple antideriv... |
Since $AB=I$, we have\begin{align*}\det(A)\det(B)=\det(AB)=\det(I)=1.\end{align*}This implies that the determinants $\det(A)$ and $\det(B)$ are not zero.Hence $A, B$ are invertible matrices: $A^{-1}, B^{-1}$ exist.
Now we compute\begin{align*}I&=BB^{-1}=BIB^{-1}\\&=B(AB)B^{-1} &&\text{since $AB=I$}\\&=BAI=BA.\end{align... |
Answer
$$a=4,\:a=\frac{2}{7}$$
Work Step by Step
Solving the equation using the quadratic formula, we obtain: $$a_{1,\:2}=\frac{-\left(-30\right)\pm \sqrt{\left(-30\right)^2-4\cdot \:7\cdot \:8}}{2\cdot \:7} \\ a=4,\:a=\frac{2}{7}$$
You can help us out by revising, improving and updating this answer.Update this answer
... |
(a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If... |
https://www.youtube.com/watch?v=Z47IRtuF0eI
...
An excited atom of mass m and initial speed v emits a photon in its direction of motion. If v << c, show that the frequency of the photon is higher by \frac{\Delta f}{f} \approx \frac vc , than it would have been if the
...
q) Given k1 = 1500 N/m, k2 = 500 N/m , m1 = 2kg ... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
I am looking for single-number evaluation method that can be used in multi-class classification tasks that take into account imbalanced data-sets. For instance,
ROC-AUC defined by binary classifiers, is a single-number and takes into account imbalanced data-sets. On the other hand, accuracy is single-number, defined fo... |
Here is a interesting picture with two arrangements of four shapes.
How can they make a different area with the same shapes?
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This is a famous physical pu... |
Lattice problems are a good source of candidates. Given a basis for a lattice $L$ in $R^n$, one can look for a nonzero lattice vector whose ($\ell_2$) norm is smallest possible; this is the 'Shortest Vector Problem' (SVP). Also, given a basis for $L$ and a point $t \in R^n$, one can ask for a lattice vector as close as... |
Let $A_m$ and $B_{ij}$ be matrices of size $M\times N$ where $m\in\{1,2\}$ and $N \times N$ where $i,j\in\{1,2\}$, respectively. Note that $m,i,j$ can be any number in general, but here they are specified so that the question can be clearer and easy to see.
I am wondering whether there is an alternative way to write/de... |
ISSN:
1935-9179
eISSN:
1935-9179
All Issues
Electronic Research Announcements
2013 , Volume 20
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Abstract:
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mi... |
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Problem 10. (16 points) If
∑a=1∞∑b=1∞1a4+4b4=πpq,\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{1}{a^4+4b^4}=\frac{\pi^p}{q},a=1∑∞b=1∑∞a4+4b41=qπp,
then evaluate the integer value of p+qp+qp+q.
Announcement. This is the last CMC problem of the season. I expect to return... |
Differential and Integral Equations Differential Integral Equations Volume 13, Number 7-9 (2000), 1025-1038. Stable transition layers in a balanced bistable equation Abstract
This paper is concerned with the existence of steady-state solutions for $$ \left\{ \begin{array}{ll} u_t = \epsilon^2 u_{xx} - (u-a(x))(u-b(x))(... |
We model the expansion history of the Universe as a Gaussian process and find constraints on the dark energy density and its low-redshift evolution using distances inferred from the Luminous Red Galaxy (LRG) and Lyman-alpha (Ly$\alpha$) datasets of the Baryon Oscillation Spectroscopic Survey, supernova data from the Jo... |
ISSN:
2156-8472
eISSN:
2156-8499 Mathematical Control & Related Fields
March 2014 , Volume 4 , Issue 1
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We address in this work the null controllability problem for a linear heat equation with delay parameters. The control is exerted on a subdomain and we show how the glob... |
This week¶
Several paper caught my eye this week, but I'll be discussing only Efficient Exploration with Self-Imitation Learning via Trajectory-Conditioned Policy in more depth. I'm choosing this paper because, as happens sometimes, I had this idea myself a few weeks ago. It's especially exciting to see something you s... |
ISSN:
1556-1801
eISSN:
1556-181X
All Issues
Networks & Heterogeneous Media
June 2016 , Volume 11 , Issue 2
Special issue on contemporary topics in conservation laws
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Abstract:
During last 20 years the theory of Conservation Laws underwent a dramatic development.
Networks and Heterog... |
Use that $X$ is connected if and only if the only continuous functions $f:X\to\{0,1\}$ are constant, where $\{0,1\}$ is endowed with the discrete topology.
Now, you know each $F$ in $\mathscr F$ is connected. Consider $f:\bigcup \mathscr F\to\{0,1\}$, $f$ continuous.
Take $\alpha \in\bigcap\mathscr F$. Look at $f(\alph... |
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