-3
votes
0answers
34 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements [on hold]

I am preparing to my abstract algebra exam and I try to find an example of such ring. Does it even exist? Thank you in advance.
9
votes
1answer
153 views

What is the reverse mathematical strength of the fundamental theorem of algebra?

Reverse mathemtics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the fom ...
0
votes
0answers
21 views

Solve complex exponencial equation [on hold]

I need to solve an expression of this kind(solve for x): e^(pi*i*x) -e^(-pi*i*x) = y*2i Both x and y are real numbers, y is given. I have no clue on how to solve it analytically. All I know is that ...
0
votes
0answers
26 views

Functor of order $n$ in Mumford's abelian variety

Let $T$ be a contravariant functor on the category of complete varieties into the Category $\underline{\mathrm{Ab}}$ of abelian groups. Let $X_0,\ldots,X_n$ be any system of complete varieties, ...
0
votes
0answers
13 views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ...
0
votes
0answers
18 views

Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
0
votes
0answers
32 views

tessellation of an arbitrary shape

Is there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate by some shapes?
2
votes
0answers
26 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
-3
votes
0answers
28 views

Cayley graph of dihedral group is isomorphic to which kind of graphs? [on hold]

Let D_{2n}= be dihedral group of order 2n. Also let D_{2n}= in which 1\ notin S=S^{-1}. In this case Cay(D_{2n}, S) is isomorphic to which kind of graphs? This is my conjecture that this graph is ...
-3
votes
0answers
30 views

Find the joint density function?

Assume that $X_t$ is the OU process , i.e, $dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$. Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$. I want ...
-1
votes
0answers
13 views

Branching process and process stochastic

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
0
votes
0answers
23 views

Necessary and sufficient condition for order relations which are realizable as subsets of real numbers [duplicate]

Is there any simple necessary and sufficient condition for a totally ordered set $S$ to be realizable as a subset of $\mathbb{R}$?
-2
votes
0answers
32 views

What is the formula to calculate the total sum of all radiuses of circles that can be contained in a square [on hold]

What is the formula to calculate the total (maximum possible) sum of all radiuses of circles that can be contained in a square considering that the square dimensions are fixed and that the circles are ...
-4
votes
0answers
19 views

Loss of Kinetic Energy [on hold]

can somebody help me with how we walked from ((m1u12)/2)+(m2u22)/2) to ((m12u12 + m22u22 + m1m2(u12+u22))/2(m1+m2) . probably a walk through of how the loss of kinetic energy was gotten to be ...
0
votes
0answers
57 views

Independent Generic Curves in the Projective Plane

I'm trying to read M. Nagata's paper On the Fourteenth Problem of Hilbert and I've run into some trouble understanding the definitions he's using. The setup is as follows: let $\pi$ be a prime field ...
0
votes
0answers
21 views

Weight shrinking in linear regression by L2 regularization [on hold]

Quoting Prof. Bengio from his Deep Learning text (http://www.iro.umontreal.ca/~bengioy/dlbook/regularization.html), $ w = (X^{T}X + \alpha I)^{-1}X^{T}y $ We can see L2 regularization causes ...
-1
votes
0answers
45 views

Clear estimate is not so clear [on hold]

In a paper I found the estimate (there it is said that the estimate is clear) for $U \subset \mathbb{R}^n$ and $u \in W_0^{2,2}(U)$ saying that for all $\varepsilon >0 $ we have $$\int_{U} ...
1
vote
0answers
54 views

Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...
-1
votes
0answers
18 views

Find the expectation of function of binomial random variable [on hold]

$\mathbb{E}\left[x^{\frac{1}{n}}\right]=?$ where $n\sim Bi(N,p)$ Thanks in advance
4
votes
0answers
89 views

Is it possible to evaluate Connect 4 positions with Combinatorial Game Theory?

The surreal numbers in Combinatorial Game Theory only work for certain classes of games (e.g. they must satisfy normal play convention). This rules out even reasonable games with fairly ...
0
votes
0answers
98 views

Morse theory in zero dimensions? [on hold]

Are there any known results for Morse theory of a compact 0-dimenionsal manifold (i.e. set of points)? In particular, can one define the analogue of a gradient flow for a finite set of points and ...
0
votes
1answer
38 views

Convergence of Fixed-Point Iteration of a dependent map

Suppose that we have two mappings $T_1(\cdot): Y \mapsto Y$ and $T_2(\cdot,\cdot) : X \times Y \mapsto X$ where both $X$ and $Y$ are compact and convex subsets of the same Euclidean space. Furthermore ...
2
votes
0answers
61 views

Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...
2
votes
0answers
74 views

Linear sections of $\mathbb{G}(1,4)$

Let $G = \mathbb{G}(1,4)\subset\mathbb{P}^9$ be the Grassmannian of lines in $\mathbb{P}^4$. Let us take two general hyperplanes $H_1,H_2$ in $\mathbb{P}^9$, and let $X = H_1\cap H_2\cap G$. Now, let ...
7
votes
2answers
212 views

When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that $$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$ for every positive integer $n$? ...
0
votes
0answers
11 views

How to solve a special linear system with noise?

Sorry the title may be confusing. I'm not so sure how to categorize this problem. Anyway, We have a real-valued vector $\overrightarrow a=(a_0,a_1,...,a_{2^n-1})$. Each $a_i$ comes from the inner ...
-2
votes
0answers
57 views

On cyclic decomposition of element in $S_n$? [on hold]

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
4
votes
0answers
57 views

Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
0
votes
0answers
18 views

Spectral densities of stationary Feller processes with no diffusion, constant positive drift and negative jumps

For a (real valued, finite variance, centered) stationary process $X_t$ on $\mathbb R$, the auto-correlation function $k(\tau) = \mathbb E(X_{t+\tau}-X_t)^2$ and its inverse Fourier transform $\rho$, ...
-6
votes
0answers
23 views

Identify a curve from bunch of numerical data [on hold]

I am trying to identify/compare a similar curve with the data I have. Data format: X, Y: (1, 0.01), (2, 0.02), (3, 0.03), (2, 0.04), (n, k), Lets say I have a plot or values which will generate a ...
1
vote
0answers
25 views

Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems. Are there any papers or books that ...
2
votes
1answer
49 views

Shared maximum eigenvector

Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$: Are there ...
8
votes
0answers
209 views

One-to-one correspondance between zeta zeros and the prime powers?

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here. I have noticed an interesting property related to the ...
1
vote
1answer
108 views

Counting matrices of special types

How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well). If only every ...
-4
votes
0answers
20 views

Relation between angle of rotation and change in coordinates in 2D plane [on hold]

I have 2 lines of different parallel to each other on the 2D plane. Now I want to convert it into coordinates of the 3D system where the lines are on the same plane and are of the same length. (When ...
0
votes
0answers
13 views

reference help on regular singular points of differential equations

I'm looking for books on systems of holomorphic partial differential equations with regular singular points. I know the book 'Équations différentielles à points singuliers réguliers' by Deligne, but I ...
-3
votes
0answers
26 views

How to find positive-definite matrices $Q$ and $W$ satisfying $x^T Q x \leq \xi^T W \xi<1$? [on hold]

Recently, I encounter a problem, that is, how to find positive-definite matrices $Q\in R^{n\times n}$ and $W\in R^{n\times n}$ satisfying $x^T Q x \leq \xi^T W \xi<1$? where $x\in R^{n\times 1}$ ...
0
votes
0answers
37 views

solve nonlinear congruence modulus prime [on hold]

I would like to solve the following congruence equation in positive integers $a$ and $b$. I would be grateful if anyone can give some hints and references. $$ 4\equiv (a+b)/(ab) (\mod p) $$ where ...
1
vote
0answers
37 views

Basic result in semi-infinite linear programming

Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
3
votes
0answers
277 views

Fermat's Last Theorem in $\mathbb{Z}/n\mathbb{Z}$

Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R\subseteq \mathbb{N}^3$ by $$ R = \{(x,y,z) \in \mathbb{N}^3: \exists n\in \mathbb{N}: 1< n \leq \max\{x,y,z\} \land ...
1
vote
0answers
38 views

Definition of Ito Integral

In Karatzas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable and adapted processes ($f(t,\omega)$), the authors say that there ...
0
votes
0answers
20 views

Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
0
votes
0answers
34 views

When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with ...
0
votes
0answers
24 views

Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...
10
votes
0answers
203 views

Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular? For example, for $n=3$ and $k=2$, the first ...
4
votes
1answer
131 views

Four Dimensional Rook Domination

Let $\gamma(G)$ denote the domination number of a graph, and $G\,\square\,H$ denote the cartesian product of two graphs. Then $K_8\,\square\, K_8$ is the rook graph, whose vertices are the squares of ...
11
votes
0answers
400 views

“To operate the machine, it is not necessary to raise the bonnet.”

The quotation in the title is attributed to Frank Adams and appears in several places: In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not ...
0
votes
0answers
136 views

Different proof's of Marten's theorem

I am referring to Marten's theorem on the dimension of $W_d^r $ as in ACGH p. 192 . It seems to me that an even shorter proof can be given using Hopf's Theorem that if $\nu : A \otimes B \to C $ is ...
7
votes
1answer
231 views

Generalization of Popoviciu's inequality

Popoviciu's inequality states that for convex $f$ and numbers $x_1,x_2,x_3$, we have $f(x_1)+f(x_2) + f(x_3) + 3\cdot f(\frac{x_1+x_2+x_3}3) \geq 2\cdot f(\frac{x_1+x_2}2)+2\cdot ...
0
votes
0answers
78 views

Topic in functional analysis [on hold]

i'm a graduate student and i like an analysis. What are current research topics in the functional analysis especially in geometry of Banach spaces? I would like to read about them.

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