0
votes
1answer
24 views

Control theory: when does $G(s) = \frac{1}{P_\lambda(A)}$

In other words, under what condition is the system transfer function G(s) = Y(s)/U(s) equivalent to the reciprocal of the characteristic equation of the $A$ matrix in state space realization?
1
vote
0answers
42 views

Random word problem for unknown groups.

Suppose that I pick a group $G$ from a distribution $X$, unknown to you, and I give you a generating set $|S|$ for $G$, and a word $x$ in $S$ and their formal inverses, and a set $T$ of sentences of ...
6
votes
3answers
164 views

A closed form for the sum of $(e-(1+1/n)^n)$ over $n$

I have been having some trouble trying to find a closed form for this sum. It seems to converge really slowly
0
votes
3answers
123 views

Matrix diagonalization - eigenvalues on diagonal

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $\Delta$ such that $A=PD P^{-1}$ where $D$ is a diagonal matrix. What theorem tells us that $P$ is a matrix composed of the ...
0
votes
2answers
87 views

How to prove Existence and unicity

Prove or give a counterexample: Let $g$ be a continuous extension of a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$ to $\mathbb{R}$. $g$ exists $g$ is unique I am interested in how ...
1
vote
0answers
22 views

base of open neighborhood for dual group in k-topology

I wanted to ask the following: Suppose I have an abelian topological $G$, and $G^*$ is its dual group (all the continuous homomorphisms from $G$ to the circle group $T$). How can I show that the ...
1
vote
4answers
128 views

Integrating $\int_{0}^{2} (1-x)^2 dx$

I solved this integral $$\int_{0}^{2} (1-x)^2 dx$$ by operating the squared binomial, first. But, I found in some book, that it arrives at the same solution and I don't understand why it appears ...
3
votes
2answers
144 views

Let $S(n)$ be the sum of the digits of a number $n$. Solve $n+S(n)+S(S(n))=1993$

My first step is to just to try to understand the problem by considering specific values of $n$. In the simplest case, $n=1$, and then $S(n)=1$ and $S(S(n))=1$, and the sum of these values is 3, which ...
2
votes
2answers
298 views

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$. Now, I know that this involves the Sylow Theorems, of course. This is very new ...
2
votes
1answer
81 views

Lp spaces - example of function

Can you give me an example of function which: $$f \in L^{p}[a,b]$$ but $$f \not\in L^{\infty}[a,b]$$ $L^{\infty}[a,b]$ is space of essentially bounded function at interval $[a,b]$ $1 \le p < \...
5
votes
1answer
491 views

Standard Deviation. Why do we take the square root of the entire equation?

Please forgive my lack of maths knowledge, It is my understanding that: Standard Deviation is the average distance from the mean in a data set of numbers. Therefore it stands to reason that working ...
0
votes
2answers
83 views

Laurent Series Expansion computing terms

I need to compute the -5th term to the 5th term of the Laurent expansion of $(\cos(z))^2/\sin(z)$. I know that I can make this into $\csc(z)-\sin(z)$ but I wouldn't know what to do with the $\csc(z)$ ...
1
vote
0answers
84 views

Does the following limit exist? (involving harmonic numbers)

Let $H_m$ denote the $m$-th harmonic number (with the convention $H_0:=0$). Fix an integer $n$. Define for $k=0,1,\dots,n-1$ $$ d_{n,k}:={1\over{n^2}}\biggl\{\sum_{j=0}^k \bigl(H_n-H_k+H_{n-1}-H_{n-k+...
2
votes
2answers
696 views

Proving convergence of sum of two convergent series'.

Suppose $$\sum_{n=1}^\infty a_n~,~\sum_{n=1}^\infty b_n$$ are convergent series' how could I use this to prove that $$\sum_{n=1}^\infty \frac{a_n+b_n}{2}$$ is also convergent using the definition for ...
1
vote
1answer
95 views

Least squares approximation to a subspace.

Consider the inner product space $C[0,1]$ with inner product $$\langle f,g\rangle =\int_0^1f(x)g(x)\,dx$$ Let $S$ be the subspace spanned by $1$ and $2x-1$ Find the best least squares approximation ...
2
votes
1answer
56 views

differentiable on $\Bbb R^{n}× \Bbb R^{n}$

Let $f : \Bbb R^{n} × \Bbb R^{n} → \Bbb R$ be defined by $f(x, y) = x·y$ , Show that $f$ is differentiable on $\Bbb R^{n}× \Bbb R^{n}$ and that $Df(a, b)(x, y) =b · x + a · y$ Here . denotes the dot ...
0
votes
1answer
125 views

Showing something is a vector space under pointwise addition and scalar multiplication

I'm asked to show that something forms a vector space under pointwise addition and scalar multiplication. I'm not sure what the term "pointwise addition and scalar multiplication" means in this ...
1
vote
1answer
203 views

Algebraically Deriving a function from a Table of Values

Is there an algebraic solution to deriving a function from a table of values, for example: \begin{array} {|r|r|} \hline x &f(x) \\ \hline 1 &2 \\ \hline 2 &4 \\ \hline \end{array} which ...
0
votes
1answer
49 views

Question on proof of elementary ordinary differential equation theorem

I am self-studying differential equations using MIT's publicly available materials. I'm afraid I'm not very far along--indeed, I'm just now looking at separable equations. The published lecture ...
1
vote
1answer
238 views

The membership table for (A U B) - C = (A - C) U (B - C)

I have attached the image of my m. table. I have supposedly mistook three values I marked with green question marks. Can someone please tell me why they are wrong? the correct values are ones ...
3
votes
0answers
85 views

Stochastic Integral and Ito Isometry

I am right now studying stochastic integral, and facing the following dilemma! I just want to check whether my understanding is right! The stochastic integral is defined by following: $I(t) =\int_0^...
3
votes
1answer
154 views

Hoffman's book question: the transpose of a linear transformation

Please, I need a hint to solve this question. Let $V$ be the vector space over the field $F$ and let $T$ be a linear operator on $V$.Let $c$ be a scalar and suppose there is a non-zero vector $\alpha$...
1
vote
1answer
61 views

A second-order non-constant coefficient differential equation

I encountered the following second-order differential equation in my research: $$x^2(1-x)^2y''+Ax(1-x)y'+By=0$$ where $y$ is a function of $x$ we are looking for, and $A$ and $B$ are constants. It ...
5
votes
2answers
147 views

Proving divergence of a sequence by proving the sequence is increasing.

We define a sequence recursively by $$a_{n+1}=\frac{1}{4}({a_n}^2+a_n+2)~~~~~~~(a_1=3)$$ By showing $a_n$ is increasing prove that $a_n$ does not converge. Not sure how to do this one. I tried ...
4
votes
3answers
82 views

Determining convergence of $\sum_{n=1}^{+\infty}(e^{\frac{1}{n}}-(1+\frac{1}{n}+\frac{1}{2n^2}))^{\frac{1}{2}}$

I have the following infinite series: $$\sum_{n=1}^{+\infty}(e^{\frac{1}{n}}-(1+\frac{1}{n}+\frac{1}{2n^2}))^{\frac{1}{2}}$$ I want to examine its convergence. First thing that came to my mind was "...
0
votes
1answer
43 views

Matrix can be orthogonally diagonalized iff its eigenvectors are linearly independent

Could anyone make the proof more obvious for me? The author should prove it in both directions, i.e. if a matrix can be orthogonally diagonalized, then its eigenvectors are linearly independent. And ...
1
vote
1answer
80 views

What's the name of the sequence of differences?

Let $a = (a_0,a_1,a_2,\ldots)$ be an infinite sequence of real numbers. What is the name of the sequence \begin{equation*} Da = (a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots) \qquad ? \end{equation*} ...
4
votes
1answer
107 views

base for finite dimensional vector space is not infinite dimensional vector space?

Let $V$ be a infinite dimensional vector space over a field $K$, and $\{v_i\}_{i\in I}$ be a basis of $V$. For each $i\in I$, let $f_i: V\to K$ be defined by $f_i(v_j)=\delta_{ij}$. Prove that $\{f_i\}...
3
votes
1answer
82 views

Definition of “deterministic coupling” [Villani]

I'm currently reading through "Optimal transport, old and new" by Cédric Villani. In the first chapter, he defines a coupling of two probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ as a ...
5
votes
5answers
97 views

Limit with number $e$ and complex number

This is my first question here. I hope that I spend here a lot of fantastic time. How to proof that fact? $$\lim_{n\to\infty} \left(1+\frac{z}{n}\right)^{n}=e^{z}$$ where $z \in \mathbb{C}$ and $e^...
-1
votes
1answer
254 views

Expectation of minimum and maximum of sum of iid random variables?

Looking for $\mathrm{E}[\min(\sum{X}) ]$ and $\mathrm{E}[\max(\sum{X})]$. Paper references much appreciated. Model: let's say we have 3 connected devices in a signal processing pipeline: $$ \...
0
votes
1answer
47 views

Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
1
vote
2answers
47 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} {...
0
votes
0answers
45 views

Estimating the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$

Estimate the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$ It seems too easy so I just want to make sure: Since $f(x)-p(x)\le R(x)$ and $R_5(x)=\cos (c) \frac {x^5} {5!}$ So we ...
1
vote
2answers
62 views

Continued Fraction Algorithm for 113/50

The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me ...
1
vote
0answers
31 views

Differentiability of Function of Two Variables

Define $g(x, y) := (|x| + |y|)^{1/2}$. Find those points in $\Bbb R^{2}$ at which $g$ is differentiable. My Idea of a Solution: In the $1^{st}$ Quadrant, $g(x,y)= (x + y)^{1/2}$, which is a ...
3
votes
1answer
53 views

A geometry homework question

Let $A,B,C$ and $D$ be $4$ points in the plane such that any combination of three or more of them are non-collinear. Let $[AB]$ and $[CD]$ intersect at $M$. Suppose that: $AM = 6$, $MB = 4$ and $MD = ...
2
votes
2answers
67 views

How can I express the ration of double factorials $\frac{(2n+1)!!}{(2n)!!}$ as a single factorial?

How can I change the double factorial of $$\frac{(2n+1)!!}{(2n)!!}$$ to single factorial?
1
vote
0answers
27 views

$T(n,k)$ is the number of k-element subsets of $\left\{1, 2, \cdots, n\right \}$ having pairwise coprime elements.

Based on https://oeis.org/A186974 This is based on the irregular triangle. However the formula of the sum $S(n,k)$ is not given(A186972), how to generate this for any $n$ and $k$ $$T(n,k)= \sum_{i=1}...
1
vote
1answer
142 views

Strongly Connected Tournament

I need to prove that every strongly connected tournament is Hamiltonian. My professor suggested proving a stronger result(A strongly connected tournament on $n$ vertices contains cycles of length $3, ...
1
vote
3answers
147 views

If p is prime and k is the smallest positive integer such that a^k=1(modp), then prove that k divides p-1

I know you need to use the Division Algorithm but I don't know where to start.
1
vote
1answer
66 views

an strange set $ \Xi_A =$ {$ n \in N | \exists k^2 \in A $ s.t $ k^2 \leq n$} is decidable ?, an Interview questions?

We are some student that had an Interview for M.sc Entrance Exam. This interview has two part and one multiple choice question. We see 1 strange question that some definition is so strange for us, we ...
0
votes
1answer
44 views

Combinatorics Chess Spot Problem

Very tough problem, I must say. NOT CONSIDERING the squares both can go in from one of the black square not considering the squares both can go to. The horse can go to is: $$4 + 4 = 8 \space \...
1
vote
2answers
51 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
0
votes
2answers
65 views

Prove H is a normal subgroup of G.

Let $G$ be a group and $H$ a subgroup of $G$. If for all $a, b \in G, ab \in H$ implies $ba \in H$, then prove that $H$ is a normal subgroup of $G$. How do I proceed on this? I tried to prove for all $...
5
votes
1answer
62 views

Intuitive explanation of the term manifold

I am reading Christopher Bishop's "Pattern Recognition and Machine Learning" and in the first chapter, where he talks about the curse of dimensionality, he gives the following example: Consider, ...
2
votes
1answer
64 views

Decide whether there exists an inversion that transforms one onto the other.

Question: For the following pairs of curves, Decide whether there exists an inversion that transforms one onto the other. Identify the inversion if it exists. The circles $x^2+y^2=16$ and $(...
34
votes
3answers
2k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
9
votes
1answer
295 views

When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
0
votes
0answers
36 views

Enumeration of Trees

Find the number of trees of $n$ vertices in which a given vertex is a leaf. I am having difficulty understanding what this question wants me to find. We know the total amount of trees with $n$ ...

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