0
votes
0answers
4 views

If the second moments are uniformly bounded, does $Y_n$ converge in $L^2$?

Let $\{X_n\}$ be a pairwise uncorrelated sequence of random variables such that there exists a fixed constant $c>0$ such that $E(X_n^2)\leq c$ for all $n\geq1$. Does it imply that for any ...
0
votes
0answers
19 views

How to insert Gothic letters in Word?

I do not manage to find in any of the options provided by Word the real Gothic letters used my German mathematicians in their original works. Noe even under the two "gothic" rubrics. just some sparse ...
0
votes
0answers
6 views

Show the following including number of divisors d(n)

I know how to show that $(d ∗ \mu)(n) = 1$ for all n ≥ 1.But.. I have two solutions. Firstly... result is trivial, because $d = 1 ∗ 1$ Secondly We know that both sides are multiplicative. Thus it ...
0
votes
1answer
4 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
0
votes
0answers
5 views

Maximum number of independent parameters for defining a subspace of a vector space

Consider a subspace $W$ in a vector space $V$. The basis of $W$ is a funciton of a set of parameters $\{\alpha_i\}$. What is the maximum number of independent parameters for fully defining the ...
0
votes
1answer
12 views

How to find complex coordinates of a square?

If one coordinate is given by: $z_{1}=\frac{3}{2}+\frac{3}{2}i$ and $Re(z_{2})=6,Re(z_{4})=1$. How to find $z_{2},z_{3},z_{4}$ so that $z_{1}z_{2}z_{3}z_{4}$ forms a square in the first quadrant? ...
0
votes
0answers
8 views

Are the following logical statements equal? Solution verification

We were requested to rewrite the following statement: $((\phi \rightarrow(\psi \lor \lnot X)) \land (\phi \rightarrow (\psi \land X))) $ Using $\exists, \land, \lnot $ only. My result: ...
2
votes
0answers
12 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
1answer
6 views

How to express outer sum in a matrix form?

So I have the following equation for a matrix $\mathbf{B}$ given $\mathbf{A}$: $$ b_{ij} = \sum_k \sum_l a_{ki} a_{jl} $$ The question is if there is anyway that I can write that one compactly in ...
1
vote
0answers
6 views

Can critical point that $f''$ has different sign in its every neighborhood be a local extreme point?

Suppose that $f$ is a second order derivable function on $[0,1)$, and $f'(0)=f''(0)=0$. If for any $\delta>0$ there exits $x_1,x_2\in [0,\delta)$ such that $f''(x_1)>0$ and $f''(x_2)<0$, is ...
1
vote
0answers
15 views

Find the matrix $P$

$A= \begin{bmatrix}1 & -2 & 3\\-2 & 6 &-9 \\3 & -9 & 4 \end{bmatrix}$ Find $P$ with non-negative integer entries and has determinant $2$. $P^TAP=\begin{bmatrix}a & 0 ...
1
vote
1answer
25 views

My proof that an n digit number, times an n digit number can be expressed as a 2n digit number

I am very proud to say this is the first time I've actually used maths to endeavour to prove something without it being related to a question from my course! Statement In a base $B$, an $n$ digit ...
1
vote
1answer
26 views

Can a function be differentiable at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a,b]$. Is it possible that $f$ is differentiable on the closed interval $[a,b]$, or must the maximal domain for $f'$ be $(a,b)$?
0
votes
1answer
15 views

If mutiplication of probabilities of two events is equal to their intersection,then are the events always independent?

Here is an example , Let a ball be drawn from an urn containing four balls, numbered $1, 2, 3, 4$. Let $E = \{1, 2\}$, $F = \{1, 3\}$ If all four outcomes are assumed equally likely,then we have ...
1
vote
2answers
12 views

Derivation of second moment of area of a circle, a small question

I hope this question will be allowed on math.stackexchange, as the question is a mathematical one even though the subject might be from engineering. I am trying to derive the formula for the second ...
1
vote
2answers
20 views

Factoring Gaussian integers

How do I factor the elements $2, 3$ and $5$ of the ring $\mathbb{Z}[i]$? Are they not primes, that is $ 2=2 \times 1$, etc? (an exercise from Vinberg's Algebra).
0
votes
0answers
19 views

What is the name of this geometric shape?

#1 I am trying to find the name for this when $d1 = d2$ What is the name of this object? #2 Assume d1 is different than d2. What is the name of this kind of object?
0
votes
2answers
22 views

Can someone help me prove these two limits? I need them for probability.

$$1.\lim_{n \to \ +\infty} \binom{n}{k} \frac{1}{n^k}\left(1- \frac{1}{n}\right)^{n-k}= \frac{e^{-1}}{k!}$$ $$2.\lim_{n \to \ +\infty} \binom{r}{k} \frac {(n-1)^{r-k}}{n^r}= \frac{2^{k}}{k!}e^{-2}$$ ...
0
votes
1answer
13 views

How to find solution to $y'=y_1(x)g(x)+y_2(x)f(x)$?

Asuume that function $y=y_1(x)$ is one of the solutions of differential equation $y'=f(x)$ as well as $y=y_2(x)$ of $y'=g(x)$. You need to find at least one solution of this equation: ...
0
votes
3answers
25 views

What are some simple examples illustrating the definition of “cover”

In my class the word "cover" is used very informally such as this set covers another set (this is for a class in PDE not topology by the way). Can someone provide a trivial example of cover to get ...
0
votes
1answer
18 views

Can a function be continuous at the end points of its (closed interval) domain?

Assume $f$ has a domain of $[a, b]$. Is it possible that $f$ is continuous at $x = a$ and $x = b$? If the definition of continuity is that the left and right limits are equal to the function at the ...
-1
votes
0answers
21 views

the formula for the volume

If you know that the volume of cube ($a^3$) represents the sum of the surface squares ($a^2$). Following the same logic: If we know the area of the circular segment is equal to the area of the ...
0
votes
1answer
20 views

Two dice thrown together.

Each face of a die is marked with a different number from 1 to 6. The number on the faces of the die are marked in such a way that the sum of the numbers on any pair of opposite faces is 7. Two such ...
2
votes
2answers
18 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
0
votes
1answer
21 views

General question on notations when dealing multiplicative and additive modulo

One of the property for the requirement for a set to be a group is associativity. Under ordinary multiplication: $\large{a(bc)=(ab)c}$ Under ordinary addition: $\large{a+(b+c)=(a+b)+c}$ What ...
1
vote
0answers
8 views

Estimation for the principal Matrix solution using eigenvalues: $\| R(t,t_0)\| \leq \text{exp}\left(\int_{t_0}^{t} v(u) \text{d}u \right)$

Suppose $R(t,t_0)$ is the principal Matrix solution at $t_0$ satisfying the matrix-valued Initial value Problem $$ R'(t,t_0)=A(t)R(t,t_0), \qquad R(t_0,t_0)=I_d. $$ I want to prove the inequality ...
3
votes
8answers
77 views

What is the mathematical distinction between closed and open sets?

If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would ...
0
votes
0answers
16 views

$ id + t T $ invertible for $t$ small enough?

I would be thankful if someone could help me. Let $T \in C_c^\infty(R^d; R^d)$. Then why is $ id + t T $ invertible for $t$ small enough? The statement for real-valued is clear, but how does it apply ...
0
votes
0answers
14 views

Hatcher 2.1.10…

Hatcher asked a question in chapter 2 (a) Show the quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with ...
0
votes
2answers
7 views

Is it possible to find Median of set contains string value as well as Null value?

I am doing one calculation on Median and AFSIK about Median, it can be calculated by a "set of Finite set of Numbers". Now if a set has following value as: A = {3,4.56, 2, 11, JAMES, , Joy, 54, 9} ...
0
votes
2answers
17 views

About chain rule

I have $z=f(x,y)$ known and I'd like to express $\partial z / \partial x$ as a function of $\partial y / \partial x$ and $\partial y / \partial z$. I know the solution is $$\frac{\partial z}{\partial ...
0
votes
1answer
11 views

Conditional expectation of random variable

I have this home assignment in Introduction to Probability, and I'm not comfortable with definitions and heuristics. I really need someone to check if I'm even in the right direction. The question: ...
0
votes
0answers
18 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
0
votes
1answer
15 views

Special linear group as a submanifold of $M(n, \mathbb R)$

I have that $SL(n,\mathbb{R})$ is an embedded submanifold of dimension $n^2-1$ in $GL(n,\mathbb{R})$, and I know that $T_XGL(n,\mathbb{R})$ is isomorphic to $M(n,\mathbb{R})$ for all $X \in GL(n , ...
-1
votes
0answers
15 views

What is my error in this command? [on hold]

!(x[t_] := Sin[t]*(([ ExponentialE]\^Cos[t] - 2*Cos[4 t] - ((Sin\^5)([)(t/12))))
0
votes
0answers
34 views

Proof of limit of $x^3\ln x$ as $x$ goes to 0

I am trying to find an $\epsilon$-$\delta$ proof of \begin{equation*} \lim_{x \to 0^{+}} x^3\ln x=0 \end{equation*} Is there a way to construct such a $\delta$ and not find it by educated guessing?
-1
votes
0answers
13 views

Is $\ker(nat_{H})=H$ a true statement? [on hold]

Is $\ker(nat_{H})=H$? $nat_{H}$ defines as $nat_{H}(a)=a*H$ I know what $ker$ and $nat_{H}$ are but I am not familiar with $\ker(nat_{H})$.
1
vote
3answers
36 views

How is “p implies q” same as “q unless not p”?

I want to know how is "p implies q" same as "q unless not p"? ie how is "$p\Rightarrow q$" same as "$q$ unless $\neg p$" ?
1
vote
1answer
13 views

If a Bilinear Form is Non-Degenerte on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
0
votes
1answer
22 views

Subgroups of automorphisms of Finite fields

Let $G$ denote the group of all the automorphisms of the Field $F_{3^{100}}$.Then,what is the number of distinct subgroups of $G$? First of all I have to compute $G$. Now \begin{equation*} ...
0
votes
1answer
23 views

Find the vectors $x$ such that $T(x) =x$

I'm provided with a matrix $T$ which is $[2 -3; -1 4]$ and as the title says I'm supposed to find a vector $x$ such that when I multiply $T$ by it, $x$ is the result. The problem seems simple enough ...
0
votes
4answers
31 views

>Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$ I know that \begin{equation*} (1 + x)^n = 1 + nx +\frac {n(n-1)}2!\cdot x^2 +\frac {n(n-1)(n-2)}3! \cdot x^3 +... \end{equation*} ...
0
votes
0answers
26 views

Confusion regarding dF/dx=0, F=constant

I thought i found a theorem "Given a curve in the (y,x) plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of F along this curve satisfies relation $g = ...
2
votes
0answers
20 views

Suppose entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that such matrix exists

A matrix $A$ is interesting if entries of $A$ are either $-1$ or $1$, and $\det(A)=n!$. Find all positive numbers $n$ such that there exists an interesting matrix of size $n \times n$. Claim: If ...
2
votes
0answers
13 views

determinant of general linear group

I know that for the general linear group, the coordinate derivatives of the determinant function $\det:GL(n,\mathbb{R})\to \mathbb{R}$ are \begin{equation*} \frac{\partial}{\partial X^i_j}\det X=(\det ...
0
votes
1answer
23 views

Taking the partial derivative of an integral

Can I simply take the integral of this function with respect to $t$ by bringing the differential operator under the summation? $$u(x,t)=\int_{-\infty}^{\infty} ...
0
votes
1answer
25 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
1
vote
1answer
13 views

How to use Itō in this very simple case

I want to apply Ito for the following process: \begin{equation*} X_t = tW_t + \int_0^t W_u du, \end{equation*} where $W$ is a Brownian motion. I have no trouble with the part $tW_t$ This can be ...
0
votes
1answer
16 views

linear algebra-norm of matrix

Why $ \|A\| = \|A^*\| $ in matrix ? Suppose that A is a normal matrix. I know $ A^* = A^{-1} \det(A) $ and so $\|A^*\| = \|\det(A) A^{-1} \| \rightarrow \|A^*\|=\det(A) \|A^{-1}\|$ but I can't prove ...
0
votes
0answers
12 views

$H^1$ of some vector bundle on a cubic 3-fold

This question is a sequel to the following one Dimension of moduli space of some stable vector bundles on a cubic 3-fold. Let $E$ be a stable rank 2 vector bundle on a cubic 3-fold, say $X$, with ...

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