3
votes
0answers
79 views
+50

A weak* dense subset intersected with norm ball contains no ball

I'm struggling with this problem in general. Represent $\ell^1$ as the space of all real functions $x$ on $S = \{(m,n): m\geq 1, n \geq 1\}$, such that $$ \|x\|_1 = \sum |x(m,n)| < \infty $$ ...
3
votes
1answer
108 views
+50

The path of the shock

Here I am using the shock speed to work out the path the shock takes. I don't understand why we cannot take the value of $u_{-}$ at $t=1/u_0$ i.e $u_{-}=u_0$. and calculate the speed of the ...
4
votes
3answers
162 views
+200

Method of solving no-homogeneous recurrence equation

I need to obtain a closed form of $M(t)$, satisfying the following recurrence equation: $$M(t+1)=a+bM(t)+\frac{c}{t+1}\sum_{t'=0}^tM(t')+df(t)$$ Where $f(t)$ is a known function and $a$, $b$, $c$ ...
9
votes
3answers
3k views
+50

Proving rigorously the supremum of a set

Suppose $A \subset \mathbb{R} \neq \emptyset$. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ This is my attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 ...
4
votes
1answer
91 views
+50

How to check some topological concepts in product and direct sum spaces

Given $a=(a_i)_{i=1}^\infty$ with $a_i \geq 0$ and $b=(b_i)_{i=1}^\infty$ with $b_i \in \mathbb{R}$, let $$E_i = \lbrace (x_n)_{n=1}^\infty : n^{b_i}|x_n|\leq a_i, \forall n\in \mathbb{N} \rbrace$$ ...
2
votes
0answers
86 views
+50

basic concept about edge graphs (line graphs)

I was learning about the edge graphs or line graphs $L(G)$ of a graph $G$. I read about the relation between degree of any two vertices $u$ and $v$ in $G$ and that of edge $uv$ in $L(G)$. I am just ...
1
vote
1answer
63 views
+50

vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
28
votes
0answers
998 views
+50

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$
3
votes
1answer
100 views
+50

Maximize the largest eigenvalue of a Hermitian matrix constrained by quadratic polynomials

I am looking for a method to maximize under $\mathbf{y}$ the largest eigenvalue of the following Hermitian matrix \begin{equation} S = \left [ \begin{array}{ccc} \mathbf{y}^{H}S_{11}\mathbf{y} ...
-3
votes
1answer
141 views
+50

Matrix Derivations-Research

$DT_3(R)$ is the upper triangular matrix with the diagonal being the same element, for example $\begin{bmatrix}a&b&c\\0&a&d\\0&0&a\end{bmatrix}$ where $a,b,c,d ∈R$. Based on ...
2
votes
1answer
130 views
+100

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
7
votes
1answer
161 views
+100

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
3
votes
1answer
67 views
+50

Algebraic number theory topics for undergrads

What are some interesting topics or problems in algebraic number theory which could be presented to students in a first undergraduate algebra course (which covers some elementary number theory, ...
3
votes
0answers
45 views
+50

What book or website has nice, colorful diagrams illustrating real quadratic integer rings?

I'm sure you all have seen diagrams, colorful or not, illustrating prime numbers in $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, with some of them helpfully pointing out the inert and splitting primes ...
1
vote
0answers
84 views
+50

Altering a Lease Calculation to take into account an upfront payment

I am trying to find the interest rate of a lease if we know the monthly payment amount but have an advance payment. I have found a site with part of the calculation we need (Scenario 2 on the link ...
2
votes
1answer
67 views
+50

Improper integral Riemann sum limit in the derivation of Fourier series to Fourier transform

To give background to my question, in all the books I've looked at to derive the inverse Fourier transform of a continuous function $f$ on $\mathbb{R}$, they seem to work as follows. Let $k$ be a ...
2
votes
0answers
97 views
+50

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
2
votes
0answers
37 views
+50

What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?

Warning: This problem requires a bit of setting. Fix a finite set $A \subset \Bbb{Z}$ and consider an infinite non (ultimately) periodic sequence $\mathbf{a}=(a_i)_{i \geq 1}$ of elements of $A$ such ...
1
vote
0answers
56 views
+50

Estimate $|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi \lambda_k x)}{\lambda_k}\right|$.

We known that there exists a constant $C\geq 0$ such that for all $n\geq 1$ : $$|S_n(x)|=\left|\sum _{k=1}^n \frac{\cos (2\pi kx)}{k}\right| \leq C -\log |\sin (\pi x)|, \quad \forall x\in (0,1]. $$ ...
3
votes
0answers
51 views
+50

Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
1
vote
1answer
81 views
+50

Linear Algebra: System of Equations

Consider a finite sequence $x_i \in (0,1)$ for $i=1,\ldots, n$ and define $y_i=\dfrac{\Pi_{j=1}^n x_j }{x_i}$. I solved this system for $x$ in terms of $y$ and got $$x_i=\dfrac{\left(\Pi_{j=1}^n y_j ...
3
votes
1answer
64 views
+50

Cardinality of a minimal open cover of the disc

Consider $D_1^2(0)=\{x\in\Bbb R^n: ||x||_2\leq 1\}$ and let $\epsilon>0$. Consider the open cover $\mathcal{O}=\{B_\epsilon^2(x):x\in D_1^2(0)\}$ of $D_1^2(0)$. What is the minimum cardinality ...
1
vote
2answers
134 views
+250

Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that ...
1
vote
2answers
62 views
+200

Spectrum restrictions in the signature consisting of just a single binary operation

In the signature {*}, where * is an operator of arity 2, is there a theory whose spectrum is the set of prime powers?
13
votes
0answers
121 views
+50

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discuss olympiad problem none of student solve it,therefore, meaning this problem is so hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
3
votes
0answers
40 views
+50

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
1
vote
0answers
37 views
+100

How to prove $\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $ around a phase singularity/over a cut

How would you prove that $$\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $$ We know that $\theta\in(-\pi,\pi)$, suppose that $\theta$ is continuous in the region bounded by and along $\Gamma$ apart ...
3
votes
0answers
57 views
+50

Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
-1
votes
0answers
83 views
+100

Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on f,$(x_0,y_0)$ is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$. In other ...
4
votes
1answer
81 views
+50

Values of the Herbrand quotient

For a finite cyclic group $G$, there is the Herbrand quotient in the theory of group cohomology. I calculated some of those quotients and I always came up with an Integer as solution. I failed at ...
2
votes
1answer
72 views
+150

Where to find an algorithm for decomposing rational functions into elementary fractions?

Specifically I need to decompose $\frac1{(1-x)(1-x^n)^2}$ into $\frac{f(x)}{(1-x)^3}+\frac{g(x)}{1-x^n\vphantom{()^2}}+\frac{h(x)}{(1-x^n)^2}$ where $f(x)$, $g(x)$, $h(x)$ are polynomials. Surely ...
1
vote
0answers
48 views
+50

Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
1
vote
0answers
61 views
+50

Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...
5
votes
2answers
272 views
+150

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
3
votes
0answers
94 views
+100

Complex submanifold has the minimal volume

I know that the following theorem is true: If $W$ is a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$, $sngW$ is the set of its singular points, $V \subset W$ is open, ...
2
votes
0answers
79 views
+50

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
1
vote
1answer
39 views
+50

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
6
votes
2answers
196 views
+100

diophantine equation $x^3+x^2-16=2^y$

Solve in integers: $x^3+x^2-16=2^y$. my attempt: of course $y\ge 0$, then $2^y\ge 1$, so $x\ge 1$. for $y=0,1,2,3$ there is no good $x$. so $y\ge 4$ and we have equation $x^2(x+1)=16(2^z+1)$, ...
3
votes
1answer
52 views
+50

Bifurcations in the Duffing oscillator

I'm trying to describe all the bifurcations in the two parameter Duffing oscillator: $$\ddot{x} + ax + bx^3 = 0$$ In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ ...
1
vote
2answers
126 views
+50

Example $x, y$ & $z$ values for, $(z = x ↑^{-3} y,\ z = G(-1, x, y))$ and $(z = x ↑^{-0.5} y,\ z = G(1.5, x, y))$

$\uparrow^n$ and $G(n,\cdot,\cdot)$ are notations for hyperoperation. http://en.m.wikipedia.org/wiki/Hyperoperation $n$ is the hyperoperations rank. Can example $x$, $y$ and $z$ values be ...
3
votes
0answers
68 views
+50

Asymptotic behavior a recursion involving min/max

Usually when I face solving recursions I use generating functions but I'm not aware of any "tools" to use when min/max expressions are involved. For example, I have the following recursive term: ...
3
votes
1answer
93 views
+50

pullabck of rational normal curve under Segre map

Let $\nu:P^1 \rightarrow P^2$ be the veronese map of degree $2$, i.e. $[Y_0 : Y_1] \mapsto [Y_0^2 : Y_0 Y_1 : Y_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Consider the ...
4
votes
0answers
87 views
+50

Diophantine equation $x^2 + 6(y+1)^2 = (y+2)^3$

I'm revising for exams and I've got stuck on an algebraic number theory question. The equation I'm trying to solve is $$ x^2 -2 = y^3, $$ and I was told to rewrite it as $$ x^2 + 6(y+1)^2 = (y+2)^3. ...
4
votes
1answer
86 views
+50

Prove (or disprove): $a\times b=c\times d$, the solutions for $x$ in the equation $\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$ is only $\pm 1$.

Prove (or disprove): If $a,b,c,d$ are positive real numbers with $a\times b=c\times d$, then the only solutions for $x$ in the equation $$\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$$ are $x = \pm 1$. ...
7
votes
1answer
166 views
+50

Rigorous separation of variables.

Let $I \subseteq \mathbb{R}$ denote an open, non-empty subinterval of the real line. We're given functions: $$f : I \rightarrow \mathbb{R}, \;\;g : \mathbb{R} \rightarrow \mathbb{R}.$$ Now suppose ...
15
votes
0answers
2k views
+200

What are some strong algebraic number theory PhD programs?

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
0
votes
0answers
38 views
+50

CDF of sum of 3 independent discrete uniform random variables on {1,2,…,n}

What is an approximate closed formula for this probability, with a derivation: p(k,n) is the probability, that among $n$ PC discs and $k$ errors in sum on them, there will be at least $1$ disc ...
1
vote
1answer
73 views
+50

Separable ODE and singular solutions

In most introductory ODE textbooks we can find the following definition: A separable first-order ODE is the one of the form $$y'=g(x)h(y)$$ and if $h(y)\neq0$, then the general solution is found by ...
47
votes
6answers
5k views
+500

Mathematically, why was the Enigma machine so hard to crack?

Mathematically, why was the Enigma machine so hard to crack? In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the ...
3
votes
0answers
112 views
+50

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in Polar coordinate $ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial u} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2}$ with ...

15 30 50 per page