0
votes
1answer
336 views

Existence of a weak solution due to Fredholm alternative in “Evans-Partial differential Equation”

I have a question about a proof in Evans "Partial Differential Equation, Theorem 5 (Third Existence Theorem for weak solution) in Chapter 6.2. They have shown that the equation $$Lu=\lambda u+f ...
1
vote
1answer
79 views

Ring thrown in space

A ring is thrown randomly in the space and let $A(t)$ be its position at moment $t$. Let us say that the moment $t_{0}$ is "twisted", if the ring $A(t_{0})$ is linked with the rings $A(t)$ for $t$ ...
2
votes
1answer
276 views

Are compacta in a complete infinite dimensional normed space nowhere dense?

Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense. I've been able to prove the statement for ...
2
votes
0answers
500 views

Difference between “substitution” and “replacement”

Is there a technical difference between "substitution" and "replacement"? For example, if I use another expression for x, am I replacing for it? Or substituting? What is I use another value of x?
3
votes
2answers
75 views

Are there diagonalisable endomorphisms which are not unitarily diagonalisable?

I know that normal endomorphisms are unitarily diagonalisable. Now I'm wondering, are there any diagonalisable endomorphisms which are not unitarily diagonalisable? If so, could you provide an ...
2
votes
1answer
289 views

There exists a finite set of natural numbers which isn't a proper subset of any other finite set of natural numbers?

Here's a proof: Let $A$ be a family of all finite sets of natural numbers partially ordered by set inclusion. Let $A' = \{A_k\}$ ($k \in K$) be any totally/simply ordered subset of $A$. Consider the ...
1
vote
1answer
102 views

Diophantine Equation $x^n+y^n=z^n$

Problem Using simple mathematical operators (+,- ,> etc.) can it be shown that (assuming $ x<y$) Fermat’s theorem is always true when $$ n\ge x$$ Request I am sure this approach has been ...
1
vote
1answer
65 views

What are the solutions of $x^n+a^n=0$?

For what values $a$, the equation $x^n+a^n=0$ has $n$ different solutions? what are the solutions? (the question refers to complex solutions).
8
votes
2answers
287 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
2
votes
2answers
910 views

Graph isomorphism as permutation matrix.

The automorphism group of a graph (lets us consider undirected) is the set of all permutation on vertices that preserve the adjacency. It is claimed: automorphism group of graph may be equivalently ...
1
vote
0answers
68 views

Universal coefficient formula

Let $X$ be a compact manifold (so that all (co)homologyies have finite rank). The universal coefficient formula ($\mathbb{Z}$-coefficient for simplicity) says that we have the following short exact ...
4
votes
1answer
104 views

Equivalent form of definition of manifolds.

I am studying topology on my own, and I am having trouble proving the following. For a Hausdorff, connected, locally euclidean paracompact space $X$, there exists a countable basis for $X$. I ...
0
votes
1answer
57 views

Determing if two k subsets are disjoint given the product of their elements

Consider the following problem (phrased with the use of a black box). You choose $n$ numbers $X = \{x_1,\ldots,x_n\}$ and pass it to a black box that returns a list $Y = \{y_1,\ldots,y_m\}$ where ...
2
votes
1answer
71 views

Approximate rational dependence

After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers ...
0
votes
2answers
157 views

Mordell equation implemented anywhere?

I admit I have no idea how to tag this post, but I'm looking for a CAS/number theory software package that would implement a decent algorithm for computing the integral solutions to $x^2 = y^3 - k$, ...
3
votes
3answers
103 views

A basic question related with the positive definite matrix

I have a one doubt related with positive definite matrices. Suppose that we have an arbitrary non zero matrix $A$ . Can we find such matrix $B$ which may depend on $A $such that product $AB$ is ...
-2
votes
2answers
259 views

Primes of the form $a^2+qb^2$

Now I came with some very interesting results. Take $p = a^2 + qb^2$ with p is some odd prime and a, b are some integers. Then, (1) Fixing q = 10, p = m (mod 40) for m belongs to the set of 1, 9, ...
0
votes
1answer
104 views

Is there a solution to the definite integral, $\int\limits_{0}^{\infty} \frac{1}{x^{\frac{1}{n}}}\frac{1}{1+x^2}\mathrm{d}x$ where, $n \in \mathbb{N}$ [duplicate]

Possible Duplicate: Evaluating this integral for different values of a constant Is there a solution to the definite integral, $$\int\limits_{0}^{\infty} ...
10
votes
2answers
354 views

Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$

Consider the series $$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$ Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is: Prove that ...
1
vote
2answers
160 views

Product of two elements is identity implies they are mutual inverses..

Let $A$ be an associative unital n-dimensional algebra over field $F$. Show that if $ab=1$ for some $a,b \in A$ then $a=b^{-1}$
2
votes
2answers
151 views

Generating random string

I'm working on a small web service that will require users to upload files. The file type is P12. When the file is uploaded and I need to save it on my server, I need to generate a random name for it. ...
1
vote
0answers
97 views

How to solve these exponential equations for D?

I'm curious if this is even possible to solve for D. D is the only variable, x, y, z, and w are all constants, and e is the mathematical constant e. $$ (\frac{x+yD^{2}}{zD})^{\sqrt{2D}} = ...
6
votes
4answers
230 views

Solving $|a| < |b|$

I apologize if this question in general, but I've been having trouble finding solutions as Google discards absolute value signs and inequality symbols. I am looking for a way to eliminate absolute ...
3
votes
1answer
121 views

The stucture of invariant polynomials on matrix

Let field $\mathbb F$ be either $\mathbb R$ or $\mathbb C$ and $M_n(\mathbb F)$ all $n \times n$ matrixes. We denote by $I_n(\mathbb F)$ the space of all functions: $P : M_n(\mathbb F) \rightarrow ...
5
votes
1answer
178 views

'Quantum' approach to classical probability

Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached ...
2
votes
1answer
65 views

Equivalence of properties defining unramified morphisms

SGA I.3 claims the following three properties of a finite type morphism $X \rightarrow Y$ are equivalent (Let $x \in X$, $y = f(x)$: Let $A$ be the stalk of $X$ at $x$, and $B$ be the stalk of $Y$ at ...
5
votes
4answers
776 views

Why isn't this a regular language?

I'm stuck as to figuring out why $L_1$={$n^p$ | $p$ = a prime number} is not a regular language but $L_2$={$n^p$ | $p$ = a prime number bounded by some fixed number f} is. I can see that $L_2$ is a ...
0
votes
0answers
58 views

What are Rough Eigenvalues?

I am unable to find any definition of what rough eigenvalues are. My intuition tells me that this definition only makes sense when we specify some space, say $H$, and suppose we have an operator $O$, ...
2
votes
1answer
200 views

Contragradient representation of a finite group

I am reading Serre's Linear Representations of Finite Groups and in an exercise in there he asks to show if $\rho$ is a representation of a finite group on $\textrm{GL}(V)$ with $V$ a finite ...
36
votes
4answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
0
votes
1answer
555 views

Derivatives of vectors involving the expectation operator - Part I

So, I am trying to take the derivative of the following equation, because it is needed in an optimization problem. I want to make sure I am on the right track. The equation is: $$ -3 \mathbb ...
1
vote
1answer
97 views

Computing sign table for $2^k$ factorial experiment design

all! I need to compute the sign table for a generic $2^k$ factorial design. For $k$ factors we compute $2^k$ experiments and need to compute a $2^k \times 2^k$ matrix, as the following example for ...
1
vote
3answers
69 views

Describing the effect on $ax^2$ by manipulating $a$

Please take, for example, $y = x^2$ and $y = 2x^2$. Graphs: Wolfram Alpha What is the most appropriate way to describe the effect of $a$? "$a$ causes the parabola to open at $1/a$ the rate of $y = ...
31
votes
4answers
2k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
3
votes
1answer
239 views

I'm searching for some books with guidance into mathematical study.

Yesterday, I've found this. It's a PDF file with this purpose, from Oxford. Some weeks ago I've also found two books tha seems to fill this purpose: Prelude to Mathematics; I Want to Be a ...
0
votes
2answers
617 views

Subgame Perfect Nash Equilibrium

My homework question is summarized below: There are 7 players (say P1,P2,...,P7) trying to split 100 dollars. The game starts with P1 proposing an allocation of the 100 dollars to each ...
2
votes
1answer
67 views

Is this operator monotone?

Consider a convex optimization problem. $$\min_{u\in\Re^k} f(u)$$ s.t. $g_i(u)\leq0,\ i=1,\ldots,m$ Let ...
2
votes
3answers
168 views

Why does he need to first prove the sum of the $n$ integers and then proceed to prove the concept of arithmetical progression?

I'm reading What is Mathematics, on page 12 (arithmetic progression), he gives one example of mathematical induction while trying to prove the concept of arithmetic progression. There's something ...
4
votes
1answer
237 views

A type of local minimum

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
2
votes
4answers
416 views

Halting problem on finite set of programs

As I understand the halting problem, it imply the fact that there doesn't exist one program which can answer the halting problem for every computable program and it rely on Cantor diagonalization to ...
1
vote
1answer
218 views

What is a differentiable functional?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
3
votes
1answer
231 views

Concerning the lower incomplete gamma function

$\gamma$ is the lower incomplete gamma function. Is $\gamma(1, x) \ge \gamma(k, kx)$ when $k \in Z^+$, $x \in (0,1)$?
1
vote
1answer
140 views

Composition of Poisson arrivals and Bernoulli RVs

I was wondering if there is a known characterization of an arrival process defined as follows: a "potential" arrival occurs according to a Poisson process of rate L, then a Bernoulli RV with P(1)=B ...
8
votes
1answer
341 views

What is the history of “only if” in mathematics?

A quick search on the use of "only if" returns questions asking about its use and meaning in mathematics, such as here, here and here, revealing confusion in its interpretation and use for some ...
1
vote
1answer
87 views

Dimensionality effect on Fisher's Score

Consider Fisher's Linear Discriminant Analysis (LDA). Let $\mu_0 \in \mathcal{R}^D$ and $\mu_1 \in \mathcal{R}^D$ be means of the two classes. Similarly, let $\Sigma_0 \in \mathcal{R}^{D\times D}$ ...
0
votes
1answer
41 views

Quadrilateral Top Point

Suppose you are given a quadrilateral - orient so that one point is the "bottom" (ie. like a diamond). Given three points: the bottom point, the left point and right point, I want to solve for the top ...
4
votes
4answers
770 views

Let $M$ be a maximal ideal in $R$ such that for all $x\in M$, $x+1$ is a unit. Show that $R$ is a local ring with maximal ideal $M$

So I'm trying to prove that if $M$ is a maximal ideal in $R$ such that for all $x\in M$, $x+1$ is a unit, then $R$ is a local ring with maximal ideal $M$, that is to say $R$ has a unique maximal ...
3
votes
2answers
201 views

Given the pairwise distances between $n$ points, how can I find plausible coordinates for the points?

If I have three points $A, B, C$, and I know the distances between $A$ and $B$, $B$ and $C$, and $A$ and $C$, (1) How can I find (one possible value for) the coordinates of $A$, $B$, and $C$? (2) If ...
15
votes
4answers
6k views

What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
3
votes
1answer
233 views

Least quadratic non residue algorithm

I am trying to implement Tonelli-Shanks algorithm and at one of the steps I have to find the least quadratic non residue. I've searched the web for a while for some kind of algorithm but so far I've ...

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