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4
votes
1answer
135 views

How to show that $H(x)=\int_0^1 f(y-x)g(y)~dy$ is bounded and continuous on $\mathbf{R}$

Given that $f$ and $g$ belong to $L^2(\mathbf{R})$, how can I show that $$ H(x)=\int_0^1 f(y-x)g(y)~dy$$ is a bounded and continuous function on $\mathbf{R}$. My attempt for the boundedness part: $$\...
2
votes
1answer
517 views

The fundamental group of the product of a 3-sphere and circle

I know that a torus the product of circles. But what the fundamental group of the product of a 3-sphere and a circle? ie $\pi_1((S^3 \times S^1), (1,1))$? Thanks!
6
votes
1answer
135 views

Help with convergence in distribution

$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$ Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity? I let $Z = \frac{Y-E(Y)}{\...
3
votes
1answer
259 views

Showing a $\alpha\times\beta$ is well-ordered for $\alpha,\beta$ ordinals

I am concerned with showing the least element existence in every subset of $\alpha\times\beta$. Below is my attempt. My teacher has used similar argument to show $\mathbb{N}\times\mathbb{N}$ is a well-...
0
votes
1answer
122 views

Exterior product of Modules, problem wih tensor product

Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$. Then the exterior product $M \boxtimes N $ is defined ...
5
votes
4answers
5k views

Example of a real-life graph with a “hole”?

Anyone ever come across a real non-textbook example of a graph with a hole in it? In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled ...
5
votes
1answer
721 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...
4
votes
1answer
213 views

Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$

I encountered this notation in a paper by Carron: When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ... $\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...
3
votes
1answer
74 views

Substitution to get separable equation

I need help with converting $$y' = \sin(x-y)$$ to separable form. What I've done so far is to apply the difference formula: $$y' = \sin(x)\cos(y) - \cos(x)\sin(y)$$
1
vote
2answers
237 views

Regarding a notation related to divisors & elliptic curves

Section 5.8 of the book An Introduction to Mathematical Cryptography defines the divisor of a rational function $f(X,Y)$ defined on an elliptic curve $E: Y^2 = X^3 + AX + B$ as the formal sum: $\text{...
5
votes
1answer
201 views

Modulus of infinite product of complex functions

We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus. What about if we have an infinite product? Is it true that $$\bigg| \prod_{k=1}...
1
vote
2answers
3k views

bird traveling to a nest wants to save energy

This is a multiple choice question in one of tests I just wrote and I did not know the answer to it. I was just stuck on this during the test. It is a very weird question, one I find to be impossible. ...
4
votes
1answer
296 views

Is the diagonal morphism always a monomorphism?

What properties of a category have to be fulfilled such that every diagonal $\Delta:X\to X\times X$ is a monomorphism? Is this true for ''reasonable'' categories?
2
votes
3answers
120 views

Finding a matrix

How might I find matrix $M\in M_2(\mathbb C)$ such that $M^t A =M^{-1}$ where $A=\left[ \begin{array}{cc} a & a \\ ...
1
vote
2answers
343 views

Upper and lower bounds in regards to 0.(9) [duplicate]

Possible Duplicate: Does .99999… = 1? I'm only doing this at GSCE and I'm really only asking here because of an interesting email conversation between my Grandfather and I regarding the ...
3
votes
2answers
3k views

Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1 $ is true. we need to ...
3
votes
1answer
250 views

Non compact embedding

Could someone please explain to me, why the embedding $\iota \colon C^0( \overline{\Omega}) \to L^2(\Omega)$ is not compact? $\Omega=(0,1)$ and $ \overline{\Omega}$ denotes the closure of $\Omega$. ...
4
votes
1answer
333 views

Dual space $E^*$ metrizable iff E has a countable basis

I have trouble proving the following theorem: If $E$ is a locally convex, Hausdorff topological vector space, then $E^*$ is metrizable if and only if $E$ has an (at most) countable basis. I've ...
0
votes
2answers
64 views

Finding a untransitive relation

I have tried without luck for a few hours now... given $$A=\{1,2,3\}$$ find $R \subseteq A \times A$ such that $R \cup R^2$ is not transitive.
0
votes
1answer
55 views

Demonstration of a problem using combinatorics

I got an assigment for homework and haven't got a clue about how to solve it, it goes as follows: "Given a set of 12 integers, demonsrate that the subtraction of 2 integers of said set is divisible ...
3
votes
1answer
274 views

Maximize sum of reciprocals vs Minimize sums

Will the returned result of the function $$\max\{\tfrac{1}{a}+\tfrac{1}{f}, \tfrac{1}{b}+\tfrac{1}{e}, \tfrac{1}{c}+\tfrac{1}{d}\}$$ return the same set $\{a,f\}$, $\{b,e\}$ or $\{c,d\}$ as the ...
5
votes
1answer
879 views

Joint distribution of non homogeneous Poisson event times?

I am trying to calculate the density of $(T_1,T_2)$ where $T_1$ is the time of the first event and $T_2$ is the time of the second event. I have been looking at the Wiki article on Poisson process and ...
2
votes
2answers
112 views

Determine the set of real numbers x for which the following series diverges

Determine the set of real numbers x for which the following series diverges $$\sum_{n=1}^{\infty}\left(\frac{1}{n}\csc\frac{1}{n}-1\right)^x$$
4
votes
1answer
936 views

Finding some missing subfields of a splitting field of $x^4-7$

I was looking at the Galois group of the splitting field of $x^4-7$ over $\mathbb{Q}$. I found it to be $\mathbb{Q}(\sqrt[4]{7},i)$, and the Galois group to be the dihedral group of order $8$. Now $...
0
votes
2answers
156 views

Calculating dB output from this example

This is my extra credit assignment so don't tell me answers, but please guide me how I should do this. I need to learn. Question states: Determine the power output of the receiver in watts and in the ...
2
votes
1answer
112 views

Rewriting a matrix as a polynomial

If a finite dimensional matrix $A$ is positive semidefinite, how can we write $A$ as a polynomial in $A^{2}$? Thanks.
1
vote
1answer
230 views

Identifying quotient rings of product rings

Given two rings $R,S$ and a principal ideal $((a,b))=I\in R\times S$ where $(a,b)\in R \times S$; is it true in general that $(R\times S) / ((a,b))\cong R/(a)\times S/(b)$?
4
votes
2answers
4k views

Definition of Compact Mapping

I was reading around the other day and came across the term "compact mapping". After googling, I saw the following two definitions: Let $X$ be a topological space. Then a mapping $f:X \to X$ is ...
2
votes
4answers
172 views

Calculating a Taylor Polynomial of a mystery function

I need to calculate a taylor polynomial for a function $f:\mathbb{R} \to \mathbb{R}$ where we know the following $$f\text{ }''(x)+f(x)=e^{-x} \text{ } \forall x$$ $$f(0)=0$$ $$f\text{ }'(0)...
0
votes
1answer
695 views

Finding Highest Order of Contact

Determine the highest order of contact at $x_0 = 0$ $f(x) = x^2$ $g(x) = \sin x$ My definition of contact is fuzzy. So I took the derivative at each step, and inspected when $f^{(k)}=g^{(k)}=0$ It'...
4
votes
1answer
414 views

Polynomials that are orthogonal over curves in the complex plane

Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other ...
0
votes
1answer
232 views

Factoring $x^4+1$ into irreducibles over $\mathbb{Q}$ and over $\mathbb{F}_2$

I've been struggling with factoring polynomials into irreducibles. I found an example and would like to know how to tackle it. If our polynomial is $x^4+1$, what are the irreducible polynomial factors ...
1
vote
2answers
478 views

How to do Lebesgue Integration

How can you prove that the following is Lebesgue Integrable on $(0,1)$? $$ f(x)= \frac{1}{\sqrt[3]{1-x}} $$ I don't know how to approach it with Lebesgue. Any help would be appreciated.
13
votes
4answers
3k views

What is the difference between real and complex manifolds?

Given an even-dimensional (smooth) manifold, what is the difference between its (real) smooth structure and its complex structure? I realize that in the real case, the overlap functions of charts need ...
1
vote
2answers
90 views

Lebesgue Integration Help

Let $E= \cup _{k=1}^{\infty} E_k$ where all the sets $E_k$'s are measurable and $m(E)< \infty$. Suppose $f$ is bounded and integrable one every $E_k$. Is $f$ integrable on $E$?
6
votes
3answers
779 views

Probability of rolling a die

I roll a die until it comes up $6$ and add up the numbers of spots I see. For example, I roll $4,1,3,5,6$ and record the number $4+1+3+5+6=19$. Call this sum $S$. Find the standard deviation of $S$...
9
votes
1answer
911 views

The Ring of Cauchy Sequences

Let $S$ be the ring of Cauchy sequences of $\mathbb{Q}$, i.e. $S=\{(a_n)\in\mathbb{Q}^{\mathbb{N}}|(a_n)\, \text{is a Cauchy rational sequence in the ordinary distance} \}$, $S$ is a subring of $\...
1
vote
1answer
78 views

finding interval where inequality holds

For which $x$ the following inequality is true: $$(1-x)^n\leq 1-\frac{nx}{2},$$ where $n$ is natural number? This is inequality opposite to Bernoulli inequality. For sure $x$ should be between zero ...
1
vote
3answers
166 views

Is $f:\mathbb{Q}\to\mathbb{Q}$ defined by $f(x)=\frac{x}{x^2+1}$ one-to-one? Onto?

I've been having trouble with determining if a function is one-to-one or onto. I found an example and would like to see how to go about this problem. If we have $f(x)=\frac{x}{x^2+1}$ where $f:\mathbb{...
2
votes
2answers
172 views

Solving a Second Order Linear Equation with Power Series

$$ y''+y'+xy=0 $$ I can't seem to get $y_1$ or $y_2$ to have any sort of pattern. I understand the technique to it but have no idea what the general solution is. The equation I have ...
3
votes
2answers
12k views

Finding second derivative of integral

Here is the problem I'm looking at: Given $f: \mathbb{R} \to \mathbb{R}$ is differentiable, define the function $$ H(x) = \int_{-x}^x [f(t)+f(-t)] dt \text{ } \text{ } \text{ for all x}$$ Find $H&#...
14
votes
2answers
2k views

What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?

Suppose $k$ is a field, and $k(t)$ is the rational function field. If $f(t)=P(t)/Q(t)$ for some polynomials $P(t)$ and $Q(t)\neq 0$, then the map $t\mapsto t+1$ sends $f(t)$ to $f(t+1)$. So the ...
4
votes
2answers
816 views

Can the concept of a jump discontinuity be extended to functions of the form $f\colon \mathbb{R}^2 \to \mathbb{R}$

We know that for a function $f \colon \mathbb{R} \to \mathbb{R}$, a jump discontinuity at a point $P$ is defined as the left and right limits exist but not equal. I'd like to know if this concept can ...
2
votes
2answers
84 views

Singular values and positive semi-definiteness

I have been working on the following question for a couple of days, and I am not getting anywhere. Suppose $A\in M_{n}$ with singular values $\sigma_{1}\geq \sigma_{2} \geq \cdots \sigma_n$. How can I ...
2
votes
0answers
177 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
2
votes
1answer
234 views

Questions about power sets and their ordering

Okay, so I'm stuck on a question and I'm not sure how to solve it, so here it is: In the following questions, $B_n = \mathcal{P}(\{1, ... , n\})$ is ordered by containment, the set $\{0,1\}$ is ...
2
votes
1answer
229 views

Independence between random variables — does it make sense?

A random variable is defined as a function from a probability space $\Omega$ to $\mathbb{R}$ (with certain properties). I think I understand the notion of independence, but my question is: If two ...
5
votes
1answer
197 views

Homotopy of singular $n$-simplices

I was wondering if there's any way to fit in homotopy into the definition of singular homology. Assume that two singular $n$-simplices are homotopic as maps, does this relate them in any way as ...
2
votes
0answers
391 views

Implicit function theorem and consistency of a semi-explicit DAE

This may be a trivial question, but here goes: Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows: $$ \begin{align} &\dot x = f(x,z,\theta),\qquad x(0) = ...
6
votes
3answers
265 views

One-to-one mapping from $\mathbb{R}^4$ to $\mathbb{R}^3$

I'm trying to define a mapping from $\mathbb{R}^4$ into $\mathbb{R}^3$ that takes the flat torus to a torus of revolution. Where the flat torus is defined by $x(u,v) = (\cos u, \sin u, \cos v, \sin ...

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