2
votes
1answer
25 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
2
votes
0answers
14 views

Characteristics of a Character table and what it tells me.

I am trying to solve the character table and some related questions. The questions are below, and what I have done is below that. Any help on any pieces I am sure will enlightening. For parts c and ...
0
votes
0answers
27 views

Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x)

Here is the theorem in mathjax: A real value function $f$ is continuous at $x \in R$ iff whenever a sequence of real numbers $x_{n}$ converges to $x$ then the sequence $f(x_{n})$ $\rightarrow f(x)$. ...
2
votes
2answers
13 views

Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?

Let $X$ be a random variable defined on a measure space $(\Omega, F, P)$. Let $N$ be a sub sigma algebra of $F$. Then conditional expectation $E(X|N)$ is an a.e. equivalent class. Is the a.e. ...
0
votes
4answers
42 views

Why does the definiton of the Euler's number not violate the rule agaisnt division by zero? [duplicate]

e= appears to be defined as the sum of the series 1/n! as n goes from zero to infinity. But this implies that the first term is 1/0! which appears to violate the rule against division by zero
0
votes
0answers
16 views

Weierstrass transform on a Riemmanin manifold

As it has been written on this Wiki page, the Weierstrass transform can be defined on a Riemannian manifold. Even though, I couldn't find any references I guess this transform for a function $f: ...
1
vote
2answers
31 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
1
vote
0answers
18 views

Ternary expansion and Cantor set

If $x$ has a ternary expansion $\sum \limits_{k=1}^{\infty}\dfrac{c_k}{3^k}$ where each $c_k\in \{0,2\}$ then $x$ belongs to Cantor set. Proof: Suppose $x$ has a ternary expansion $\sum ...
3
votes
0answers
37 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
13
votes
1answer
184 views

Is there a name for this “mean”?

We all know these means: $$GM = \sqrt[3]{xyz} $$ $$AM = \frac{x + y + z}{3}$$ $$QM = \sqrt{\frac{x^2 + y^2 + z^2}{3}} $$ Of course: $$GM \le AM \le QM $$ What about this one: $$XM = ...
2
votes
0answers
36 views

What solution would you come up with for this problem?

So the question is: put numbers $1, 3, 5, 7, 9, 11, 13$ and $15$ into gaps in the following expression: $$\_\_ + \_\_ + \_\_ = 30$$ The most naive approach to use summation in the group of integers ...
0
votes
0answers
8 views

Name of the probability distribution

If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows: $$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$ ...
0
votes
0answers
13 views

$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ for all $a\leq x,y \leq b$. For which values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t\leq 1$, let $V_t$ be the set of all functions $f\in \mathbb{R}^{\mathbb{R}}$ satisfying the condition that if $a<b$ in $\mathbb{R}$ then there exists a real number ...
2
votes
2answers
22 views

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$

Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$ This is from Hodges' A Shorter Model Theory. My idea is to take ...
3
votes
0answers
20 views

How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $K\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
2
votes
3answers
25 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
1
vote
6answers
67 views

Proving that $\frac{1}{2}<\frac{2}{3}<\frac{3}{4}<$…$<\frac{n-1}{n}$

In an attempt to find a pattern, I did this: Let a,b,c,d be non-zero consecutive numbers. Then we have: $a=a$ $b=a+1$ $c=a+2$ $d=a+3$ This implies: $\frac{a}{b}=\frac{a}{a+1}$ ...
-4
votes
0answers
35 views

Entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is to be proven constant [on hold]

Show that an entire function satisfying $f(z + 1) = f(z)$ and $f(z + i) = f(z)$ is a constant.
3
votes
0answers
22 views

Is this a “locally surjective” function?

I quote the "locally surjective" part because I haven't found any reference of that concept, but it kind of fits what I mean. Let $f:\mathbb{R}^N \to \mathbb{R}^M, f \in C^1, x_0 \in \mathbb{R}^N : ...
2
votes
1answer
19 views

Find all functions with given property

Find all functions $f$ that are holomorphic on $B = \{z: -\pi/2 < \operatorname{Im}(z) < \pi/2 \}$ with $f(B) \subset B$ and such that $f(0) = 0$ and $f'(0) = i$. Thoughts so far: My first ...
1
vote
0answers
29 views

Is this a good generating function for Sum-of-divisors function?

I have an expression for the sum-of-divisors function defined as $$\sigma(n)=\sum_{d\mid n}d.$$ However I do not know how nontrivial or practical it actually is. Let us define ...
-6
votes
1answer
57 views

I'm taking a statistics class right now, and I get stuck on these problems.

For a random variable $W$ where $P(W = 0) = 0.1$ and $P(W = 1) = 0.2$ and the density of $W$ for values between $0$ and $1$ is $f(w) = 1.4w$, draw a graph of the CDF. Is this a valid probability ...
2
votes
0answers
6 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
0
votes
1answer
24 views

Russian Roulette consecutive bullets

You are playing a game of Russian Roulette. If instead of one bullet, two bullets are randomly put in the chamber. Your opponent played the first and he was alive after the first trigger pull. You ...
0
votes
0answers
21 views

Topology generated by basis equals intersection of all topologies that contain A

Here is my proof I was wondering any critiques to my proof. If A is a basis;The topology generated by A equals collections of all unions of elements of A that is $\tau = \bigcup_{i \in I: B_i \in ...
0
votes
1answer
16 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
4
votes
1answer
31 views

Proof of stars and bars formula

I am trying to prove a formula (for ways of distributing n identical balls among r persons when each person may get any number of balls) C(n+r-1, r-1). But I am not able to prove it. I may be doing ...
-1
votes
0answers
26 views

Conversion of exponential terms

How do I convert $$ e^{\pi} + e^{2\pi} + e^{3\pi} +1 $$ into $$\frac{e^{4\pi} -1}{e^{\pi} -1}$$ I have no idea how to proceed further.
1
vote
1answer
31 views

how to show that $f_n \uparrow f$

How to show that $f_n \uparrow f$ where $$f_n(x)=\min\left(\frac{\lfloor 2^nf(x)\rfloor}{2^n},n\right)$$ It is clear to me that $f_n(x) \leq f(x)$ But how do I show that the limit is indeed $f$ ? ...
2
votes
0answers
41 views

“Race” of the primes modulo $1,3,7,9\ \pmod {10}$

The "race" starts with the prime $11$. The number of primes $1, 3, 7, 9 \pmod {10}$ is denoted after every occurring prime. Does the lead change infinitely often? And does every "runner" have ...
-1
votes
0answers
38 views

Three line segments made by intersection in harmonic progression [on hold]

I'm learning coordinate geometry in high school and have this question as a doubt. The equations of three lines are $7x + y = 16$ , $5x - y - 8 = 0$ and $x - 5y + 8 = 0$. A variable line through ...
0
votes
1answer
21 views

Parametric Equation explanation

Explain how the expression $tX + (1-t)Y$, $0\le t\le 1$, produces a segment that connects point $X (x_1, y_1)$ with point $Y (x_2,y_2)$. So I rearranged the problem such that $t(X - Y) + Y$ which I ...
0
votes
3answers
55 views

What we exactly do when we take derivative of any function? [on hold]

When we take differentiation of any function then what actually we do with that function? Ex.d/dx of x^2 is 2x. So what we have actually done with x^2.
-2
votes
1answer
20 views

Find parametric equations?

Find parametric equations A.) Part of line that goes through points $(2,5)$ and $(3,2)$ and $y∈[1,2]$. $\mathbf{r}(t)=\mathbf{a}+t(\mathbf{b}-\mathbf{a}),\;\; t\in\mathbb{R}$ B.) Intersection ...
-1
votes
1answer
41 views

Full stop as mutiplication sign

I have often seen, that users of this forum use "$.$" (full-stop) as a multiplication sign, e.g. $4.5=20$ I have thought, that this notation is the american style. But a user of this forum told me ...
1
vote
0answers
21 views

How do you expand a matrix to a power?

Suppose I have an nxn matrix A, where t is a natural number >0. Is A^t=A^(t-1)A or A^t=AA^(t-1) I would think that the operation of splitting them up into these two should work. However, A and ...
0
votes
1answer
12 views

What does “Normal” mean in the context of linear equations?

My summer packet has the question: "Write equations of the line through the given point a)parallel and b) normal to the given line: $(−6, 2)$, $5x + 2y = 7$" I had no problem with finding the ...
0
votes
0answers
22 views

Conditional Probability Error Question

Let $P_0 = 0.25,$ $P_1 = 0.35$, $P_2 = 0.25,$ and $P_3 = 0.15.$ What is the probability of more than one error? I thought to sum $P_1, P_2, P_3$ together but that doesn't seem to work. What formula ...
0
votes
1answer
44 views

Can I solve this integral with a squared sum in it?

Title says it all. By now I have tried by hand and I think that it is indeed solvable, but I can't handle the very long terms. I tried to run the thing through SAGEs integrator: ...
0
votes
1answer
12 views

Query about non-singular transformation of vectors

Suppose we are given a probability function, P (x^T (Y-z)≥0) , where ‘x’ is a vector, ‘Y’ is a random variable and ‘z’ is a known value. Now, suppose, we make a non-singular transformation w=Ax, ...
1
vote
0answers
21 views

let $n \in \mathbb{N}$, $V=\{i\in \mathbb{Z}:0\leq i \leq 2^n \}$. Define operations of addition and scalar multiplication to make a vector space.

let $n \in \mathbb{N}$ and let $V=\{i\in \mathbb{Z}:0\leq i < 2^n \}$. Define operations of vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the ...
1
vote
0answers
17 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
0
votes
1answer
16 views

Logic formula translation

I'm studying for a logic exam I've come across the following L-formula $$ \forall i (\operatorname{in}(i,xs) \Rightarrow \exists j (i = 2 \cdot j)) $$ Where the $\operatorname{in}(x,xs)$ predicate ...
0
votes
0answers
8 views

On Weisner Method

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
0
votes
0answers
14 views

If $L(v_1)=L(v_2)=L(v_3)=w_1$ then what is the $rank(L)$?

it is an elementary question. $L:V\to W$ is a linear transformation and $S=\{v_1,v_2,v_3\}$ is an ordered base of V, and $T=\{w_1,w_2\}$ an ordered base of W. If $L(v_1)=L(v_2)=L(v_3)=w_1$ what is ...
0
votes
1answer
28 views

Does this matrix operation hold?

Suppose A is an nxn matrix and b is a constant scalar. t is some natural number >0 Can i apply binomial expansion on (A-Ib)^t?
1
vote
2answers
30 views

determinant of a vector times vector transpose

I have a vector $x$ of dimension $N \times 1$ and let's say I create a matrix $S = x x'$ which a matrix of dimension $N \times N$. If I calculate the determinant of $S$, I get it as $0$. Is this a ...
0
votes
1answer
29 views

Show that a point lies on the diagonal of quadrilateral

In a quadrilateral ABCD we choose a point E on the side AD and a point F on the side CD. Then we choose a point G on the line EF. Let H be the second point of the intersection between the circles that ...
0
votes
0answers
23 views

Find the maximum value of $\log_{10}(\frac {c_2}{x})\log_{10}(x+1-c_1)$, where $c_1 ,c_2$ are real constants and x is a real number,$x\in [c_1,c_2]$

What is the maximum value of: $$\log_{10}\left(\frac {c_2}{x}\right)\log_{10}(x+1-c_1)$$ where $c_1$ & $c_2$ are real constants and $x$ is a real number, $x\in [c_1,c_2]$. For which $x$ is this ...
0
votes
1answer
11 views

Show that this construction preserves connectedness

Let $G_1$ and $G_2$ be $k$-connected graphs and let $v_1\in V(G_1)$ and $v_2\in V(G_2)$ be such that $\deg v_1=\deg v_2=k$. Form a new graph, $H$, by putting an $M$-matching of size $k$---conneect ...

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