0
votes
0answers
2 views

quastion in character theory of group

let X be an irreducible character of G and C be a conjugacy class of G. for any g of C, if (o(g) , |C|) = 1 then X(g)=0 or |X(g)|= X(1).
0
votes
0answers
9 views

Integral of floor function.

How can I solve this integral? $$\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$$
0
votes
0answers
5 views

How to show that SVM is convex problem

It's well-known fact that SVM is convex problem $min \frac{1}{2} \left \| w \right \|^2$ s.t. $(wx_i+b)y_i \geq 1$ I don't understand how given the LP formulation of SVM I can coclude that it's ...
0
votes
1answer
7 views

Prime numbers making constant : 1.2527

Reading "Excursion in calculus" (Robert M. Young, 1992), exercice 13 on page 71 ask the reader to show there is a constant $c\approx 1.25$ such that $a_0=2^c$ $a_{n+1}=2^{a_n}$ $\forall n\; \lfloor ...
1
vote
0answers
5 views

Henkin-Konstruktion:Goedel completeness THM

I am trying to understand better the Henkin konstruktion, which consist first in an extension of the signature and then of the theory. Here are my question about this topic: -we extend the ...
0
votes
0answers
14 views

How do I verify that $\int_0^1 (1-t) \, f''(t) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$

How do I verify that: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
0
votes
0answers
7 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
0
votes
1answer
7 views

Does first isomorphism theorem work both sides?

The theorem says that if I have a group homomorphism, then the kernel is normal and the image is isomorphic to the domain group modulo the kernel. Now, suppose I have $G/K \cong{H}$ where $G$ and ...
0
votes
1answer
8 views

Additive function and continuity at a point

Does continuity at a point and Additive function imply continuity at all other points in a normed linear space. Is there some result like there exist a in field such that f(x) = ax for all x in normed ...
0
votes
1answer
7 views

A space homotopy equivalent to its subspace implies the inclusion map is a homotopy equivalence?

I find it not easy to understand the proof of corollary 0.21 of Hatcher's algebraic toplology. If the question of the title is true, I can understand it. But I don't know how to prove it. Can somebody ...
0
votes
0answers
8 views

Basic definition of continuity

Ltf(c+h) = f(c)(h goes to 0) if and only if Ltf(x) = f(c)(x goes to c). I am able to prove this fact using sequential criterion of continuity. But sequential criterion is dependent on Axiom of ...
1
vote
0answers
12 views

Solve these simple simultaneous equations?

Assuming $x_1, x_2 \geq 0, \lambda \neq 0, w_1,w_2 > 0$ We have the equalities: $$w_1 - \lambda x_2 = 0 ... (1)$$ $$w_2 - \lambda x_1 = 0 ... (2)$$ $$\bar y - x_1x_2= 0 ... (3)$$ My solutions ...
0
votes
0answers
9 views

Vectors components that are not contra or covariant?

I know that a vector can have contravariant and covariant components, but is it possible to have components that are neither contravarient or covariant? I suspect that the answer is yes, and that most ...
-1
votes
0answers
9 views

conditional probability proof 3 varables

Suppose that a,b and c are dependent variables. P(a|b)=sum(P(a|b,c)+P(c|b)) can anyone explain it how we get it?
1
vote
1answer
4 views

Determining a constant to a Bessel function solution to an ODE

I am using the generalized Bessel function to obtain a solution to an ODE. The derivative of the general solution is something like \begin{align} \frac{dT_{1}}{dx} = \frac{3}{4}Ax^{\frac{1}{4}} ...
1
vote
1answer
20 views

Is there something wrong with this proof of the cancellation rule?

I was helping a friend of mine with math homework. We were dealing with real numbers, and had to prove the cancellation rule; that is $$a,b,c, \in \mathbb{R} \land a+c=b+c \implies a=c$$ He had a ...
0
votes
2answers
12 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of 3 rolls of a die being less than or equal to 9. In order to solve the problem I try first to find the number of ...
-1
votes
0answers
5 views

Mapping from a point inside a disk to a point inside an annulus

How do I map any point inside a disk to a point inside an annulus ? Disk and annulus are concentric (at the origin).
2
votes
1answer
6 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
0
votes
0answers
13 views

Can you recommend a book with techniques for solving hard algebra/arithmetic problems?

I'm a university student who never really studied maths in high school (I did the basic courses, but because Im dyslexic I was to embarrassed to try the harder courses) now I'm getting back into it, ...
0
votes
0answers
7 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
0
votes
2answers
32 views

If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$ .

Having trouble solving this. If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$. .
1
vote
2answers
28 views

Prove that $(a,b)$ is not compact based on the definition of compactness

I am trying to show that $(a,b)$ is not compact while $[a,b]$ is compact, purely based on definition of compactness. Here are examples, which I write it better: $[a,b]$ is covered by the intervals ...
0
votes
1answer
16 views

triple integral over aregion

I have an assignment given on a triple integral. I tried to evaluate it but was unable to get the answer. The question is as follows: Evaluate the following: $$\iiint_D{6xy\space dV},$$ ...
0
votes
0answers
7 views

How many cubic inches of lead are in a one-pond sample of lead?

The average weight for cast lead is 708 pounds per cubic foot. Consider a one-pound sample of cast lead. How many cubic inches of lead are in the sample? Choose the closest answer. One cubic foot is ...
1
vote
0answers
4 views

homotopy between continuous functions to an absolute retract

I have the following statement to prove as one of the "fundamental" questions our topology professor wants us to know for his final: Let $X$ be a topological space, and let $A$ be an absolute ...
0
votes
2answers
28 views

Re-arranging an equation help

How do I re-arrange the equation $$ -150 = (-9.8)(t) + 0.5(-9.8)(t)^2 $$ and solve for $t$? I collected the like terms firstly, so $$-150 = -48.02 \cdot t^3$$ then I knew I was doing something ...
0
votes
0answers
14 views

Find the area of the convex quadrilateral when you have the value of one diagonal and it's intersection point

ABCD is a convex quadrilateral and E is the intersection point of their diagonals if $DE=3$ and $BE=12$ find $\frac{ADC}{ABCD}$ I know the length of one diagonal so that's $BD=15$ and their now two ...
1
vote
1answer
9 views

Limit of translates of characteristic function

This might be silly, but what is a simple way of showing that given a characteristic function of a lebesgue measurable set in $\mathbb{R}$ then we have $\lim_{t \rightarrow 0} \chi (x-t) \rightarrow ...
0
votes
0answers
7 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
0
votes
1answer
21 views

Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
0
votes
1answer
25 views

Little o(h) limit about h=0

I understand that generally if a function $f(h)$ is described as $o(h)$ that $f(h)$ has a smaller rate of growth than $h$ (like it would have to be $\sqrt{h}$). i.e. $\sqrt{h} = o(h)$, just like (for ...
1
vote
4answers
20 views

How to caluclate the integral of $\int \frac{1}{\sqrt{4x^{2}+1}}dx$ using a trig substitution?

I am trying to determine the following integral: $\int \frac{1}{\sqrt{4x^{2}+1}} dx$ using a suitable substitution. My progress: let $x = \frac{1}{2} \tan \theta$ $dx = \frac{1}{2}\sec^{2} \theta ...
1
vote
0answers
3 views

How to find the long range transition matrix L of P

P is the transition matrix of a regular Markov chain. Find the long range transition matrix L of P. $$ P = \begin{bmatrix} 1/2 & 1/4 & 1/4\\1/2&1/2 &1/4\\0 &1/4 & ...
0
votes
0answers
16 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
1
vote
1answer
15 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
0
votes
0answers
11 views

Proof of Gelfand formula for spectral radius

STATEMENT: Let $A$ be a Banach algebra, then for every $x\in A$ we have $$\lim_{n\rightarrow\infty}||x^n||^{1/n}=r(x)$$ Proof: We know that $r(x)\leq \lim \inf_n||x^n||^{1/n}$, so it suffices to ...
0
votes
2answers
25 views

Property of SO(3)

Suppose $A\in SO(3).$ Show that there exists a vector $v\in \mathbb{R}^3$ such that $Av=v$. $ SO(3)={{A\in O(3)|detA=1}} $ and $ O(3)={A:\mathbb{R}^3\rightarrow ...
0
votes
0answers
4 views

How to convert this equation to close form (matrix form)

I have read that the matrix form for the following summation $$ Error(w) = \sum_{i=0}^{m} W^{T}x_i - y_i $$ $W^T$ is the transpose of weights vector in linear regression $x_i$ is the ith input in ...
0
votes
0answers
12 views

Law of Iterated Expectation Proof

I have a proof that needs to be done: $$ \mathbb{E}(XY) = \mathbb{E}[\mathbb{E}(Y|X)\,X] $$ So I start with the following \begin{align} \mathbb{E}(XY) &= \mathbb{E}(X)\cdot\mathbb{E}(Y)\\ ...
3
votes
1answer
66 views

If $N=a^2+b^2=c^2+d^2$ then $N$ cannot be a prime number.

The problem says that if $N$ can be expressed in two ways as the sum of two squares then $N$ is not prime. Clearly the first idea is to try and express $N$ as a product of two expressions containing ...
0
votes
0answers
12 views

Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
-1
votes
0answers
7 views

dividing a chebyshev polynomial by another polynomial

If I took a Chebyshev polynomial, is it possible to divide it completely by something that isn't a chebyshev polynomial?
0
votes
0answers
13 views

convex function divided by convave function is quasiconvex

$p(x) \geq 0$ is convex, and $q(x) > 0$ is concave. How to prove $f(x) = \frac{p(x)}{q(x)}$ is quasiconvex? My proof is using t-sublevel set: $\{x | \frac{p(x)}{q(x)} \leq t\}$ is ...
1
vote
1answer
39 views

Is $[a,\infty )$ closed

In standard topology, of course $[a,\infty )$ is closed since its complement is open. But I don't know how to prove closeness of $[a,\infty )$ in Real Analysis using just the definition of closeness, ...
2
votes
2answers
27 views

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once.

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once. This question is for a precalculus class so setting the derivative of the positive ...
1
vote
0answers
28 views

Trouble understanding some basic concepts of measure theory [on hold]

I am currently undergoing a course in Measure Theory. The book is "Principles of Real Analysis" by Charalambos D. Aliprantis and Owen Burkinshaw. The approach is little difficult for me to grasp and I ...
1
vote
2answers
14 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
3
votes
0answers
22 views

Basic question on appication of Sunflower lemma

A sunflower or $\Delta$-system is a collection of sets $\mathscr{F}$ whose pairwise intersections are all the same set $S$, possibly empty. Elements of the collection of sets $\mathscr{F}$ are called ...
4
votes
1answer
29 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...

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