0
votes
0answers
32 views

formula for sequence 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, [migrated]

There is a sequence with the values 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... (basically there are always four 0s followed by a 1, then it repeats). Is there a function for this sequence? Here are two ...
3
votes
0answers
127 views

A conjecture like Cayley–Bacharach theorem

Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...
4
votes
3answers
141 views

Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...
-3
votes
0answers
60 views

Maximal ideals of R [x] [on hold]

Let $R$ be a commutative ring with identity. Is there any relation between maximal ideas of $R[x]$ and maximal ideas of $R$?
2
votes
1answer
202 views

like prime for a ring

Let $R$ be a ring with identity. Is there any characterization for a ring $R$ such that has a decomposition?
1
vote
0answers
10 views

Stiefel-Whitney class of unordered configuration space

Let $S^n$ be the $n$-sphere. Then the unordered configuration space $B(S^m,2)=F(M,2)/\Sigma_2$ is the total space of a line bundle over $\mathbb{R}P^m$, i.e. we have a fibre bundle $$ \mathbb{R}\to ...
56
votes
8answers
5k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
0
votes
0answers
19 views

A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a 2x2 couterexample by Vinograd to the system $y'=A(t)y$ where $A(t)$= \begin{matrix} -1 -9 \cos^2 6t + ...
3
votes
0answers
46 views

Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from $$ xq+x^2q^4+x^3q^9+...+x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n}; $$ that is, $$ \sum_{d|n}x^{\frac ...
2
votes
3answers
44 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 ...
8
votes
1answer
119 views

Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
-3
votes
0answers
12 views

2's complement subtraction conversion to decimal for checking [on hold]

I was having some problem when trying to perform a 2's complement subtraction. So the question is: 01110101 - 11010110 ---------- Then I perform the following ...
1
vote
0answers
35 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 ...
2
votes
1answer
24 views

Avoiding the range of a bivariate function or Diophantine function

I have a bivariate integer function where x,y are positive integers in the function $f(x,y)=5+23x+7y+30xy$. The lattice points of this function, or its range, contain a large number of values. I'm ...
1
vote
0answers
50 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 ...
1
vote
0answers
32 views

Solution to $(A+x^2)e^x=B$ with Lambert W function

Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?
5
votes
1answer
114 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 ...
0
votes
1answer
46 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}} ...
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 ...
3
votes
1answer
141 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 ...
5
votes
1answer
127 views

Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in ...
-4
votes
0answers
54 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 ...
-3
votes
0answers
23 views

How to compute the direction of flattest ascent for a convex function [on hold]

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 ...
25
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 ...
11
votes
1answer
726 views

The maximum of the preimage of [1,x] through Euler's totient function

A friend of mine and I have shown the following: "For each $x \geq 1$ let $m := m(x)$ be the greatest positive integer such that $\varphi(m) \leq x$, where $\varphi$ is the Euler's totient function. ...
0
votes
1answer
67 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" ...
10
votes
5answers
2k views

Simple/efficient representation of Stirling numbers of the first kind

Stirling numbers of the second kind can be expressed by means of a simple hypergeometric (considering $n$ fixed) sum $$S_2(n,k) = \frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n. \qquad (1)$$ ...
-3
votes
0answers
52 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? [on hold]

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 = ...
1
vote
1answer
776 views

Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as ...
0
votes
0answers
19 views

Gauss Curvature Equation for hypersurface in Semi-Riemannian manifold [migrated]

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
57 views

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

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 ...
2
votes
1answer
72 views

Generating function of alternating even terms in the Vandermonde Convolution

I have \begin{equation} G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + ...
5
votes
1answer
92 views

Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing

The following result is on page 26 of this paper by Ferenczi [PDF]. Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) ...
2
votes
1answer
170 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
4
votes
0answers
471 views

Almost-Kahler Einstein four manifolds

Are the odd Betti numbers of a compact Almost-Kahler Einstein four manifold necessarily even ?
21
votes
6answers
2k views

undecidable sentences of first-order arithmetic whose truth values are unknown

Godel's undecidable sentences in first-order arithmetic were guaranteed to be true, by construction. But are there examples of specific sentences known to be undecidable in first-order arithmetic ...
1
vote
0answers
275 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 ...
0
votes
1answer
66 views

Ergodic automorphisms of a compact metric abelian group are Bernoulli

In the literature, such as in this article, it is proved that every ergodic automorphism of a compact metric abelian group is Bernoulli. A rotation is not isomorphic to a Bernoulli shift because it ...
42
votes
1answer
452 views

Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
0
votes
0answers
31 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 ...
5
votes
2answers
336 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 ...
1
vote
1answer
131 views

Covering space theory, category theory [on hold]

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 ...
10
votes
4answers
1k views

Elements of infinite order in a profinite group

Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general? A start for (A): we can ask the same question ...
0
votes
1answer
78 views

A question on theorem 1.1 of Fritz John ultrahyperbolic pde

I have the following paper: Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, pages 300-322 doi:10.1215/S0012-7094-38-00423-5 Now ...
2
votes
1answer
60 views

assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...
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, ...
3
votes
0answers
62 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 ...
2
votes
0answers
59 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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
62 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 $a,b>0.$ ...

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