-4
votes
0answers
41 views

Arun Bhandari,Master of philosophy in applied mathematicsm, Kathmandu University ,Nepal [on hold]

Greetings from Arun Bhandari, I am doing research in Numerical methods for nonlinear differential equations. Currently, I am working on He's Variational Iteration Method for this I need following ...
1
vote
0answers
5 views

tangent space of line bundles over projective space

Let a line bundle $$ \eta:\mathbb{R}\to E(\eta)\to \mathbb{R}P^m.$$ I want to study the tangent bundle $TE(\eta)$. Question 1. When $n$ is even, $\mathbb{R}P^m$ is non-orientable. Does this imply ...
0
votes
0answers
4 views

About irreducible representation of symmetric group [migrated]

Consider the tensor space $$\mathbb{C}^m\otimes \mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n$$ with $k$ factors. The symmetric group $S_k$ on $k$ letters acts on this space (on ...
-2
votes
0answers
26 views

What would be the impact - to the foundation of First Order Logic - of a sentence whose truth value is impossible to verify or know?

Suppose there's a sentence F written in L(PA) that is undecidable in PA, and whose truth value is impossible to verify (know), then face value it seems both the formal systems T1 = PA + {F} and T2 = ...
0
votes
0answers
16 views

Gauss Curvature Equation for hypersurface in Semi-Riemannian manifold

We have the Gauss curvature equation: $$\langle R(V,W)X,Y\rangle = \langle R'(V,W)X,Y\rangle - \langle II(V,X),II(W,Y)\rangle + \langle II(V,Y),II(W,X)\rangle$$ Here $M$ is an immersion in $N$. ...
0
votes
1answer
41 views

Is the locus of points which have irreducible fibers constructible?

Suppose $X \rightarrow Y$ is a map of projective schemes over a field $k$. Is $\{y \in Y: \pi^{-1}(y) \text{ is irreducible}\}$ a constructible subset of $Y$? Note: One cannot hope to do "better" ...
0
votes
1answer
48 views

Solving Shroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system

Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$ $r_1$ is the distance between the proton $1$ and the electron. $r_2$ is the distance between the proton $2$ and the electron. $R$ is the ...
24
votes
5answers
1k views

Which polynomial's roots are its coefficients?

Start with any polynomial of degree $n$ with complex coefficients, e.g., $$z^3+z^2+2 z+3 \;.$$ Find its $n$ roots, and list them in order of their modulus: $$-1.28, (0.14\pm 1.53 i)$$ Now form a new ...
1
vote
1answer
124 views

Covering space theory, category theory

Requiring covering spaces of a well-behaved connected topological space $X$ to be connected, let $\mathcal{Cov}(X)$ be the category of covering spaces of $X$ and maps over $X$ and maps over $X$. Can ...
0
votes
0answers
256 views

Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...
-3
votes
0answers
20 views

How to compute the direction of flattest ascent for a convex function

Consider an infinitely differentiable convex function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ at the point $x_0$. So long as $x_0$ is not the minimum, it is well known that we can compute a unit vector ...
3
votes
0answers
45 views

Adjacency matrix, quivers

Let $Q$ be a quiver with finitely many edges and such that the underlying graph is connected. Let $I = \{1, \dots, n\}$ be the vertex set of $Q$, so we have $\mathbb{R}\{I\} \cong \mathbb{R}^n$. For ...
5
votes
2answers
303 views

The sum of a series, continued

In this question the OP asks whether the sum $$ f(q, \alpha) = \sum _{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)_k} $$ is ever zero. An experiment with Mathematica indicates, to any ...
0
votes
1answer
58 views

Why does optimization of a sum of two terms result in “neat” answers? [on hold]

This is a somewhat vague and philosophical question. Consider the following three problems: Problem 1: Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$ ...
3
votes
1answer
132 views

characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space $$ B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2 $$ where $$ \Delta=\{(m,m)\mid ...
0
votes
0answers
21 views

Problem regarding sum of a recursive sequence

Problem of the recursive sum is as follows. Find the sum $$\sum_{r=1}^n U_r$$ where $$U_r = \frac{U_{r-1}M_r}{M_{r-1}(a+b M_r)}$$ and $$U_1 = \frac{M_1}{a+b M_1} , \ \ \sum_{r=1}^{n} M_r = 1.$$ Here ...
2
votes
2answers
38 views

Algorithm to determine isomorphism of 2 maximal planar graphs

I read on wikipedia that there are efficient algorithms to answer the question whether 2 (maximal) planar graphs F and G are isomorphic. However, after some (IMHO) substantial searching I don't seem ...
7
votes
1answer
108 views

Physical interpretation of the mellin transform variable?

I shall keep this to the point: Given a time domain signal say microphone recording of a conversation: Laplace tranfrom of x is some function X(s) say defined in the complex plane. I like to think ...
0
votes
0answers
22 views

Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof. Paper 1, page 1, line 10 says : Consider the topological image G of a ...
1
vote
1answer
88 views

Complete regularity in C*-algebras

It is clear that commutative C*-algebras correspond to locally compact Hausdorff spaces. And locally compact Hausdorff spaces are completely regular. Now, does the complete regularity statement have ...
1
vote
0answers
27 views

Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...
3
votes
0answers
58 views

How can I include irreducibility in a Groebner basis calculation?

I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...
0
votes
0answers
18 views

Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation $$dg_t = dB_t \circ g_t$$ where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
0
votes
1answer
43 views

Probability of having no cycles of fixed length in $d$-regular graphs

According to this paper, the probability that a random $d$-regular graph of order $n$ has no cycles of length $c_1,c_2,\ldots,c_t$ is $$P=\exp\left(-\sum_{i=1}^t\mu_i+o(1)\right)$$ as ...
4
votes
0answers
147 views

Is a determinant 2x2 positive and increasing?

Let $X_1,X_2,X_3$ be a three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and ...
0
votes
1answer
57 views

Intersection of compact sets in the compact-open topology

Let $(X,\tau)$ be a topological space. We topologize $\tau$ itself in the following way. For $K\subseteq X$ compact, we set $${\cal V}_K=\{U\in \tau: U \supseteq K\}.$$ The compact-open topology on ...
3
votes
0answers
71 views

Do very general hypersurfaces contain smooth surfaces with $c_1^2>2c_2?$

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n.$ Does $X$ contain a smooth surface $S$ with $c_1(T_S)^2>2c_2(T_S)$? For $d<<\sqrt{n}$ the answer is yes, as $X$ will ...
0
votes
0answers
19 views

graph reconstruction via canonical labeling [on hold]

Graph edge deleted reconstruction: Given an edge deleted deck find the canonical lexicographically ordered largest representation for each card. Find the card with largest canonical label this is the ...
2
votes
0answers
23 views

How much larger than the relaxation time can the mixing time be?

The notation is mostly taken from the book "Markov chains and mixing times" by Levin, Peres, and Wilmer. Consider an irreducible, aperiodic, time-reversible, discrete-time Markov chain on a finite ...
2
votes
0answers
72 views

A conjecture like Cayley–Bacharach theorem

Let five conics and 12 points on a plane, one conic through six points, and one point lie on two conics. Then every cubic that passes through any eleven of the points also passes through the 12th ...
3
votes
0answers
45 views

Wavelet-like Schauder basis for standard spaces of test functions?

The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of real numbers with faster than power-like decay. Likewise, the space ...
0
votes
1answer
40 views

When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space $$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := ...
3
votes
1answer
221 views

characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal $$ ...
2
votes
0answers
63 views

Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$. I read ...
3
votes
1answer
67 views

Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...
10
votes
0answers
135 views

Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
0
votes
1answer
93 views

Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification. Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...
0
votes
1answer
40 views

Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ...
2
votes
0answers
62 views

Polynomial constraints triggered by irreducibility [on hold]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic: $$af^2 + bf + c = 0$$ If we're working in a ring, ...
-2
votes
0answers
80 views

almost complex embedding of $S^2$ and $S^6$ into $\mathbb{C}^N$ [migrated]

In Which Spheres are Complex Manifolds? , I find that $S^2=\mathbb{C}P^1$ is a complex manifold and $S^6$ is an almost complex manifold. Are there references about: What is the smallest integer $N$ ...
6
votes
1answer
242 views

The sum of a series

Let $0< \alpha <1$ and $q>1.$ Consider the (alternating) series: $$ \sum_{k=1}^\infty (-1)^k \frac{q^k (q^k-1)^\alpha}{(q^k-1)\dots (q-1)}.$$ Denote its sum by $f(q,\alpha).$ Prove (or ...
2
votes
0answers
89 views

Effective Realization of GCD of middle binomials?

So, it is well-known that $$ \gcd \left(\binom{m}{k}\mid 1\leq k\lt m \right) = e^{\Lambda(m)}$$ which can incidentally be sparsified for prime $p$ $$ \gcd ...
5
votes
0answers
88 views

Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...
5
votes
0answers
79 views

$E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral ...
0
votes
1answer
39 views

General Markov Chains on same Probability Space?

A Markov chain $(X_i)_{i\in \mathbb{N}}$ on a measurable space $(E,\Sigma)$ is (see e.g. Revuz or Meyn/Tweedie) constructed on the following probabilty space. $$ \Omega = \{ (x_l)_{l \in \mathbb{N}} ...
5
votes
1answer
102 views

Are all transversely oriented, transversely measured foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...
1
vote
0answers
58 views

Normal Subgroups of $U_n(q)$

What is known about normal subgroups of $U_n(q)$, the group of upper triangular matrices with entries in the finite field $\mathbb{F}_q$ and ones on the diagonal?
0
votes
0answers
16 views

Estimate $\int_0^1 g_k(x) d\mu(x)$, where $\{g_k(x)\}_{k=0}^\infty$ is a generic Parseval frame for $L^2(\mu)$

I came across the following integral in some applied problem: $$\int_0^1 g_k(x) d\mu(x)$$ where $\mu$ is a Borel probability measure on $[0, 1)$, and $\{g_k(x)\}_{k=0}^\infty$ is a generic Parseval ...
2
votes
0answers
51 views

The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following. Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...
1
vote
1answer
60 views

Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...

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