1
vote
1answer
90 views

Inequalities of summations lift to inequalities of summations of powers?

Let us assume: an at most countable set of indexes $I$; a set of reals $q_i$ with $i \in I$ such that $0\le q_i \le 1$ and $\sum_{i \in I} q_i \le 1$; a natural $n_i$ for each $i \in I$; a real $x ...
1
vote
1answer
23 views

Vector Field Problem

Why does $\frac{\mathbf{\hat{r}}}{ r^2} = \frac{\mathbf{\vec{r}}}{ r^3}$? I've never seen this before and I'm not sure what I should consult to understand this situation.
8
votes
2answers
260 views

How find this limit $\lim \limits_{x\to+\infty}e^{-x}\left(1+\frac{1}{x}\right)^{x^2}$

find this limit $$\lim \limits_{x\to+\infty}e^{-x}\left(1+\dfrac{1}{x}\right)^{x^2}$$ my idea: $$\lim \limits_{x\to+\infty}e^{-x}\left(1+\dfrac{1}{x}\right)^{x^2}=\lim ...
-2
votes
1answer
131 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
1
vote
1answer
76 views

Number of draws needed to get a positive element using a *weighted* sampling without replacement

Imagine we have $M$ elements, where some of them ($y$) are positive and the rest, $z=M-y$, are negative. The probability of drawing any of them is given by a distribution. Let's call $p_1$, $p_2$, ...
1
vote
0answers
167 views

Discrete to Continuous Representations of Functions via Laplace Transforms?

The Laplace transform can be thought of as the continuous analogue of a power series, as in this video. From this perspective, think of the function $ a : \mathbb{N} \rightarrow \mathbb{R}$ as a ...
2
votes
2answers
94 views

Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
3
votes
2answers
789 views

How to evaluate $\sqrt{5+2\sqrt{6}}$ + $\sqrt{8-2\sqrt{15}}$ ?

My exams are approaching fast and I found this question in one of the unsolved sample papers. I tried squaring the whole term but couldn't work out the answer . I am a ninth grader so please try to ...
0
votes
1answer
28 views

What is the relation between this subgroup and its conjugates?

Let $S$ be an infinite set, and let $A(S)$ denote the group of all bijections of $S$ onto itself. Let $M \subset A(S)$ be the set of all elements $f \in A(S)$ such that $f(s) \neq s$ for at most a ...
2
votes
2answers
371 views

Proving that $\left(1 + \frac{x}{n}\right)^n, n \in \mathbb N$ is bounded

Please can you help me with the following question $$E(x) = \left\{ \left(1 + \frac{x}{n}\right)^n : n \in \mathbb N \right\}$$ Let a(x) = sup E(x) (least upper bound) without finding the sup of ...
1
vote
0answers
92 views

Stationary distribution behavior - Markov chain

I have modeled a process with a Markov chain with K+1 states which is irreducible and apperiodic. The transition matrix is a centrosymmetric matrix where all it's entries has a positive probability. ...
1
vote
2answers
95 views

Predicting the graph of the given reaction

I have no idea on how to answer this question. Please help.
2
votes
1answer
77 views

I want to prove this result

Let $g$ be a complex analytic function that have infinitely many real zeros. Let $f$ be its restriction to $ℝ$. I want to verify and prove this result: Lemma: If $f$ is a real analytic function with ...
3
votes
1answer
69 views

Is there an automorphism of symmetric group of degree 6 sending a transposition to product of two transpositions?

$\operatorname{Aut}(S_6)\cong S_6\rtimes C_2$. there are several (720) automorphisms sending a transposition to product of three transpositions. Is there an automorphism sending a transposition to ...
1
vote
1answer
82 views

Laplace transformation problem

There is a timely unchanged continuous function : $$H(s)=\frac{s-1}{s+1}$$ At the entry of the system exists a $x(t)$ which Laplace's transformation is: $$X(s)=\frac{(5s^2 - 15s + ...
1
vote
1answer
57 views

What's the meaning of a formula in MatrixCookbook?

I'm learning the derivatives of matrices and vectors. In Matrix Cookbook Chapter 2(page 7), there is a formula as follows: $$\frac{\partial{X_{kl}}}{\partial{X_{ij}}}=\delta_{ik}\delta_{lj}$$ The ...
0
votes
1answer
57 views

Are $d_{\infty}$ and $d_{p}$ distance functions?

Let $X$ be a set equipped with a metric $d_x$, denoted by $\langle X,d_x\rangle$, and $Y$ equipped with a metric $d_y$, denoted by $\langle X,d_y\rangle$. Let $Z=X\times Y$. Let ...
1
vote
2answers
4k views

Finding the inverse of a matrix by elementary transformations.

While using the elementary transformation method to find the inverse of a matrix, our goal is to convert the given matrix into an identity matrix. We can use three transformations:- 1) Multiplying ...
0
votes
3answers
50 views

Finding the point on the real axis collinear with $i$ and $-1+2i$

Let $A = i$ and $B = -1+2i$ be two points on the complex plane. Find the point $D$ on the $x$-axis such that $A$, $B$ and $D$ are collinear. I don't know from where to start, and what does "$D$ ...
1
vote
4answers
559 views

Binomial coefficient question?

I'm unsure how to do these types of questions, so any help would be great: Find the coefficient of $x^2$ in the expansion of $(x+1/x)^3(x-1/x)^5$ Thanks
3
votes
3answers
162 views

What's the smallest exponent to give the identity in $S_n$?

Let $S_n$ denote the symmetric group on $n$ letters. We know that $\tau^{n!} = e$ for any element $\tau \in S_n,$ where $e$ denotes the identity element. Can we find a smaller positive integer $m$ ...
4
votes
2answers
108 views

Show that all the roots of $\frac{dx}{dt}=A(t)x$ are bounded in $[t_0, \infty)$.

For real system of equations$$\frac{dx}{dt}=A(t)x,(1)$$ where $A(t) \in C[t_0, +\infty)$. Prove that if $\int_{t_0}^{\infty} \|A(t_1)+A^T(t_1)\|< +\infty$ then all the roots of (1) are bounded in ...
0
votes
1answer
210 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
1
vote
1answer
219 views

Values of the Christoffel symbols

Are the values of the christoffel symbols the same for all coordinate systems on a surface/manifold? I would love to see an example for the cone in two different parametrizations.
2
votes
1answer
84 views

How to determine shape of powers of curves?

I'm trying to understand how to figure out the shape of a given polynomial curve quickly. $ax_1 + bx_2 = const$ gives a line. $ax^2_1 + bx_2 = const$ gives a parabola. $ax^2_1 + bx^2_2 = const$ ...
1
vote
2answers
78 views

Is there always a bijection mapping one element of an infinite set onto another?

Let $S$ be an infinite set, and let $s_1$, $s_2$ be any two distinct elements of $S$. Then how to determine whether or not there is always a bijection of $S$ onto itself that maps $s_1$ onto $s_2$? ...
0
votes
1answer
48 views

$\mathbb{R}$ equiped with topology generated by $(a,b)$ and $(a,b)\cap \mathbb{Q}$

$\mathbb{R}$ equiped with topology generated by $(a,b)$ and $(a,b)\cap \mathbb{Q}$ then which of the following statements is correct? It is normal It is regular $\mathbb{R}\setminus \mathbb{Q}$ is ...
2
votes
1answer
254 views

a question about the character table of dihedral group

As we know,quaternion group $Q_8$ and dihedral group $D_8$ have the same character table,but they are not isomorphic.I have proven that if the order of a group G is not divided by $8$,and $G$ has the ...
1
vote
2answers
30 views

Combinations Question?

In how many ways can 6 different books be distributed between 2 students, provided that both students receive at least one book? Thanks for helping
0
votes
1answer
138 views

Does integration over one complete cycle equals to 4 times integration over quarter-cyle?

From the article pendulum(mathmetics) from wikipedia. There is a demonstration that this equation: $$\dfrac{dt}{d\theta } = ...
5
votes
2answers
136 views

How Prove that $B$ is nilpotent.

Let $A$ and $B$ be complex matrices with $AB^2-B^2A=B$, Prove that $B$ is nilpotent. By the way:This problem is from $American Mathematical Monthly Problem 10339,and this solution post 1996 ...
8
votes
1answer
162 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
1
vote
4answers
312 views

arrangement in a circle with a condition [closed]

How many ways can we arrange 4 managers and 3 employees around a round table so that no 3 of the managers sit together...meaning 2 can sit together. I know it involves the PIE principle and that if I ...
2
votes
2answers
85 views

Onto and one-one

Let $f \colon A \to B$ be a surjection and $g \colon B \to C$ be such that $g\circ f$ is an injection. Prove that both $f$ and $g$ are injections. Since $f$ is onto then there exists a $f(a)$ ...
5
votes
2answers
452 views

How to Compute $\zeta (0)$?

Ultimately, I am interested in analytically continuing the function $$ \eta _a(s):=\sum _{n=1}^\infty \frac{1}{(n^2+a^2)^s}, $$ where $a$ is a non-negative real number, and calculating $\eta _a$ and ...
1
vote
1answer
37 views

if $P\Rightarrow Q$ Then both are banach space?

$X,Y$ are norm linear space and $T_n$ be a sequence of bounded linear operators from $X\to Y$ consider the two statements below $P:\{\|T_n(x)\|\}$ is bounded for ever $n$ $Q:\{\|T_n\|\}$ is bounded ...
0
votes
3answers
802 views

Selecting a committe with two women refusing to sit together.

In how many ways can a committee of $3$ women and $4$ men be chosen from $8$ women and $7$ men if two particular women refuse to serve on the committee together? I've approached this question by ...
2
votes
1answer
32 views

A Pasting lemma for measurable functions

I have the following setting: Let $(\Omega,\Sigma)$, $(\Gamma, \mathcal{C})$, $(X_{1},\mathcal{B}_{1})$, and $(X_{2},\mathcal{B}_{2})$ be measurable spaces such that $\Omega = X_{1}\cup X_{2}$, ...
3
votes
3answers
81 views

$[E:\mathbb{Q}]$, the degree of $E$ over $\mathbb{Q}$

Algebraic Extension $E=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$, I need to find the $[E:\mathbb{Q}]$, the degree of $E$ over $\mathbb{Q}$ I know degree of $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$ is ...
1
vote
2answers
68 views

Basic Calculus Question On Trig substitution III

$$\int\sqrt{6x-x^2}\mbox{d}x$$ What is the answer to this tricky little problem??
1
vote
2answers
128 views

Permutations and Combinations Question?

Mr and Mrs Jones and 6 guests sit around the dinner table. In how many ways can they be arranged if the two host are separated? The answer says 3600, but I could never get that. My working out was ...
1
vote
1answer
40 views

Bernoulli Trial variables?

You are given n=10,000 light bulbs. Each has a reliability of p=99.99% Suppose you select a batch of r=190 light bulbs to light your warehouse. Can you use the equation to predict the ...
-1
votes
3answers
119 views

Any other method for finding Inverse of a matrix

Is there any method for finding inverse of a matrix other than Gauss-Jordan and (1/detA)(adjA methods?
2
votes
1answer
67 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $\|e\| = 1$ where ...
4
votes
1answer
200 views

Properties of special rectangle (measure)

Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent: a) $\lambda(I)=0$ b) $I^{\circ}=\emptyset$ (i.e., ...
1
vote
1answer
101 views

Smallest topology containing all topologies [duplicate]

Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $T_\alpha$, and a unique largest topology contained in all ...
5
votes
1answer
242 views

Find sufficient and necessary conditions to guarantees this property

Let $f$ be a real non polynomial analytic function. Suppose that the function $f$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $f(a)<−K$ and ...
1
vote
1answer
233 views

In triangle abc,angle BAC is 22 degrees. A circle with centre O has AB produced,AC produced and BC tangents. Find the number of degrees in angle BOC.

In triangle abc,angle BAC is 22 degrees. A circle with centre O has AB produced,AC produced and BC tangents. Find the number of degrees in angle BOC. I am not able to interpret it. Plz help me in ...
0
votes
2answers
78 views

for what values of $k$ $f(x,y)$ is continuous at $(0,0)$

$$f(x,y)=\begin{cases}{xy\over(x^2+y^2)^{5\over 2}}[1-\cos (x^2+y^2)] & (x,y)\ne(0,0)\\ k & (x,y)=(0,0) \end{cases}$$ could anyone give me hints how to solve this one? I tried to find $\lim ...
0
votes
1answer
2k views

Why the expected value of the error when doing regression by OLS is 0?

What I know to begin with is that the sum will be 0 if there is a y-intercept b0 , why is that? my book doesnt say and can't figure it out. I also know that an importantant assumption for the OLS ...

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