0
votes
0answers
8 views

Counterexample for the Generalized Associativity Equation

The Generalized Associativity Equation is given by $$ F(G(x,y),z)=K(x,H(y,z)),$$ where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional ...
0
votes
0answers
14 views

Gromov hausdorff convergence

Is the Gromov-Hausdorff limit of a sequence of compact submanifolds of an Euclidean space also in the Euclidean space ?
0
votes
0answers
7 views

Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional approach to finding a good simultaneous diophantine approximation is the following: given a set of real numbers $\alpha=(g_1,\ldots,g_d)$, an integer $N$, and a real $\gamma>0$ we ...
0
votes
0answers
7 views

Equivalence classes of (2,3)-pairs in PSL(2,q)

This may not be a research-level question, which is why I submitted it to math.stackexchange first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it ...
1
vote
0answers
10 views

Number of cyclic difference sets

A subset $D=\{a_1,\ldots,a_k\}$ of $\mathbb{Z}/v\mathbb{Z}$ is said to be a $(v,k,\lambda)$-cycic difference set if for each nonzero $b\in\mathbb{Z}/v\mathbb{Z}$, there are exactly $\lambda$ ordered ...
0
votes
0answers
10 views

Rational mapping related to cubic surfaces

A theorem in Manin's book on cubic forms leads to the conclusion that the least degree of a rational mapping from $\mathbb{P}^2$ to general cubic surfaces(e.g. Picard rank=1) over $\mathbb{Q}$ is 6. ...
0
votes
0answers
10 views

First order pde with characteristics

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point). Is it still possible to apply in ...
0
votes
0answers
8 views

Complexity of an algorithm to solve linear diophantine equations

A friend of mine ask me yesterday a problem, however, the interesting part for me it is not his problem, but what I will ask here. I want to know the optimal complexity of an algorithm (I mean the ...
1
vote
0answers
35 views

If matrices describe simplices, what do matrix operations describe?

Suppose we are given a $d \times d$ matrix $M$ with rows $m_1, \dots, m_d$. This matrix describes a simplex, namely the convex closure of the origin with the vectors $m_1, \dots, m_d$. Now, scaling ...
1
vote
0answers
6 views

Wide cylinders on half-translation surfaces

Take a collection of pairwise disjoint, simple polygons in the plane. Identify pairs of sides between polygons by either a translation in the plane, or, a translation composed with a rotation by ...
0
votes
0answers
23 views

Models of BL$\forall$

What results are known about the construction of models for a theory $T$ of the logic BL$\forall$ for languages of higher cardinality? The construction for the countable case relies on 1) The fact ...
0
votes
0answers
21 views

Extension of Radon measure

I will mention this specific case, Let $m_1$ be a (homogeneous) Radon probability measure on $X=\{0,1\}^\kappa$ of Maharam type $\kappa$. As $X$ is zero-dimensional space, so it has an algebra ...
0
votes
2answers
49 views

quotient of planar groups

If G is an infinite planar group (it means that it has a generating subset C such that Cay (S, C) is a planar graph) and H is a normal subgroup of it, I would be very grateful if somebody helps me and ...
0
votes
0answers
35 views

Which morphisms of varieties and motives yield isomorphisms of their lower Chow groups?

This question is a contination of Varieties with Chow groups supported in positive codimension: examples and properties? What examples are known of morphisms of varieties and Chow motives (say, over ...
-6
votes
0answers
19 views

Borel function and random variable [on hold]

Positive part of a function X+ is borel function and will be a random variable if X is random variable. I know this things, but how should i prove it mathematically ?
4
votes
0answers
39 views

Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit ...
-3
votes
0answers
21 views

Breadth First Search and Depth First Search on Graphs [on hold]

What i would like to know is if it's possible to use these two algorithms on a directed or on a not directed graph and visit every node.From what i've seen it seems impossible to me to visit every ...
0
votes
0answers
58 views

Surjectivity of the algebraic K-functors

Let $R \to S$ be a surjective morphism of commutative rings. For a fixed integer $q$, is there any known condition under which the resulting morphism of the K-groups, $K_q(R) \to K_q(S)$ is ...
1
vote
0answers
35 views

Substitutions and Sturmian sequences

We know that any substitution can generate sequence, for example the Fibonacci substitution: $\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of ...
2
votes
2answers
122 views

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows ...
6
votes
0answers
39 views

Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent). Consider an element ...
0
votes
2answers
26 views

Metric properties of a quadratic differential at an essential singularity

Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| ...
1
vote
0answers
10 views

singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation $$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$ where $\gamma\in (1, 2]$ is some ...
-6
votes
0answers
20 views

Graph Theory - Adjacent certices in a simple graph [on hold]

Let u and v be adjacent vertices in a simple graph G. Prove that uv belongs to at least d(u) + d(v) - n(G) triangles in G.
-4
votes
0answers
21 views

Graph Theory - Question on hypercubes and cycles [on hold]

Prove that every cycle of length 2r in a hypercube is contained in a subcube of dimension at most r. Can a cycle of length 2r be contained in a subcube of dimension less than r?
4
votes
2answers
68 views

Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
0
votes
0answers
16 views

Bounds on Product of CDF or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
24
votes
8answers
2k views

Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
-4
votes
0answers
38 views

Normal line, regular surface [on hold]

Let $\mathcal A$ be a regular surface in $\mathbb R^3$ and $P$ a point in $\mathbb R^3\setminus\mathcal A$. Suppose that $C$ is a point at minimum distance from $P$. Show that $P$ belongs to the ...
1
vote
1answer
67 views

Ricci flow and conformal classes

Is it true that the conformal class of the metric is preserved under Ricci flow? I have seen it mentioned in an answer on this site. Is there an easy argument? (This question was asked on MSE but it ...
0
votes
0answers
22 views

computational question concerning singular integral theory

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
0
votes
2answers
36 views

Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions: (1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$ (2) $f(n,M) ...
1
vote
0answers
13 views

IVP accuracy - scheme accuracy Vs. derivative accuracy?

General Question: If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ...
1
vote
0answers
38 views

Reference needed for Hilbert-Schmidt result regarding basis of $V \subset H$

I am seeking a reference that says: If $V \subset H \subset V^*$ is a Gelfand triple with all spaces Hilbert spaces and if $V \subset H$ is a compact embedding, then there is a basis of $V$ which ...
2
votes
1answer
34 views

Bound the degree of the generator of polynomial ring

Suppose we are given two polynomial rings $R_1$ and $R_2$ by presenting their generators, $S_1$ and $S_2$, where $S_i$ are finite set of $m_i$ variables, $i.e.$, $S_i\subset ...
4
votes
0answers
109 views

analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...
3
votes
1answer
214 views

Why $( \infty , n)$-categories are useful for?

I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this ...
0
votes
0answers
50 views

Vanishing of the module of differentials of a extension of perfect fields

Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...
6
votes
1answer
84 views

Limiting probabilities for two-player game drawing random uniform numbers

Consider this simple 2-person game I just made up: Player A goes gets to draw a uniform U[0,1] number up to X times. At any time, he may either keep his number, or draw a brand new uniform number. ...
0
votes
1answer
125 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
-4
votes
0answers
54 views

2d crank-nicolson [on hold]

I require your assistance in creating the system of equations for a 2-dimensional heat diffution equation, using a finite differences crank nicolson scheme. Can you direct me to a solved example, or ...
9
votes
1answer
112 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
38
votes
2answers
762 views

Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?
2
votes
0answers
55 views

What is the (mixed strategies) equilibrium of this game?

Given a weight vector $w\in [0,1]^d$ such that $\sum w_i=1$, the game goes as follows: Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x_i = \sum y_i = 1$. The utility (profit) ...
0
votes
0answers
32 views

Is $[u_1,u_2]$ an edge of the polytope $conv(F)$? [on hold]

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
5
votes
1answer
94 views

Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
1
vote
0answers
98 views

On monotonicity of the roots of polynomials

I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied. Here is my ...
-1
votes
0answers
53 views

Colon ideal and Artin-Rees lemma [on hold]

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$. (homework of nagata's local rings, page ...
0
votes
0answers
16 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
-1
votes
0answers
134 views

Two Questions on $\pi(x)$ [on hold]

I have recently came to know about this conjecture. The questions that naturally came to my mind are, $\forall$ $x,y \geq2$ and $x,y \in \mathbb{N}$ prove that $\pi(x) \pi(y) \leq \pi(xy)$ ...

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