0
votes
2answers
16 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
6 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
16 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
25 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 ...
2
votes
0answers
25 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 ...
1
vote
1answer
89 views

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

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
22 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 $\phi_F$). ...
2
votes
0answers
17 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
2answers
72 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 ...
-3
votes
0answers
44 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 ...
7
votes
1answer
57 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 ($C=\bf 0$) ...
21
votes
1answer
309 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?
0
votes
0answers
34 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
27 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
76 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" ...
0
votes
0answers
56 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
45 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
13 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
113 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 $\pi(x) \pi(y) \leq \pi(xy)$ ...
1
vote
0answers
86 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
0
votes
0answers
44 views

When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
2
votes
3answers
200 views

Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and $$G=\{P,2P,3P,4P,\cdots\}.$$ Is there an integral point $Q\in G?$
2
votes
1answer
71 views

Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research. Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...
0
votes
0answers
89 views

extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...
0
votes
0answers
38 views

Standard form for partitions of $Z_n$

Let $A$ and $B$ be partitions of $Z_n$. Suppose we say that $A$ and $B$ are equivalent if $A=Bx+y$ for some $x\in\{1, -1\}$ and $y\in Z_n$. In other words, two partitions are equivalent if one can be ...
0
votes
1answer
24 views

Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$. Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
0
votes
0answers
21 views

Strongly regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $(v,c k, c \lambda, c \mu)$ (simple) strongly regular graph that can be given an edge coloring with $c$ ...
0
votes
0answers
41 views

Probability of that the sum is even [on hold]

There are $2n$ letters which are randomly placed into $2n$ envelopes (each envelope can have only one letter). The letters and the envelopes are numbered from $1$ to $2n$. What is the probability that ...
3
votes
0answers
114 views

Not too classical group characters

When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation ...
2
votes
1answer
74 views

Characterization of a subset of [0,1] $III$

I have a question related to the previous one. Characterization of a subset of [0,1] $II$ Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e. $t_n$ is said to converge to ...
0
votes
0answers
35 views

Number of possible wall positions in the game Quoridor [on hold]

I'm trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I'm trying ...
2
votes
0answers
66 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
1
vote
0answers
40 views

Lifting a quadratic system to a non vanishing vector field on $S^{3}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
1
vote
0answers
46 views

Ordered statistics CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
2
votes
0answers
29 views

Removable sets for simply connectedness of a differentiable manifold

I am sorry that my question might be stupid for experts, but I really do not know the answer. Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...
-1
votes
1answer
47 views

Maximum size of set of points with distance bounded from below

I am interesting in finding a reference for a result of the following type: Suppose $D \subset \Bbb{R}^n$ is a bounded open set and $\delta>0$. Then the size $M$ of a family of points $F = ...
3
votes
0answers
63 views

Explicit non observability example for the wave equation?

Is there a simple (one dimensional, radial, by separation of variable...) example of non observability for the linear wave for a non constant in space velocity, in a simple domain? By this I mean: ...
3
votes
1answer
172 views

Existence of partitions

Good morning everybody. I would like to know if anybody is aware of nontrivial results of the following form : if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, ...
6
votes
0answers
53 views

Can one detect smoothness of $k$-forms with $k$-dimensional manifolds

Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le ...
0
votes
0answers
31 views

Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data: A $ C^{*} $-algebra $ A $. A locally compact Hausdorff group $ G $. A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...
1
vote
0answers
14 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
4
votes
0answers
266 views

What is the *smallest* number used in a mathematical paper? [on hold]

Numbers such as Skewe's Number and Graham's Number and a few others are relatively well-known in mathematical folklore as being among the largest, if not the largest, numbers encountered in proper ...
0
votes
0answers
87 views

A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions". Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...
3
votes
2answers
191 views

Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
3
votes
1answer
112 views

A question about some notation involving the exclamation mark

What does the symbol ‘!’ signify? Is it $ \text{argmin} $? For example, $ \| A x - y \| = \min! $.
-1
votes
0answers
38 views

Do the Hypersurfaces satisfy mean curvature $H$ is nonegative and $\langle F(p)-q_0,\nu(p)\rangle \geq 0$ are graphs? [on hold]

Let $F:M^n\to\mathbb{R}^{n+1}$ be the noncompact immersed hypersurface. Is the following ture? If the mean curvature $H$ is nonegative and there exists an fixed vector $q_0$ such that $\langle ...
0
votes
0answers
26 views

Help in finding the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
10
votes
3answers
980 views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...
6
votes
2answers
85 views

Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$. If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...
0
votes
1answer
58 views

Ways to order an algebraic extension

In following post, I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways. More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ ...

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