1
vote
0answers
56 views

Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms: $$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$ Assume $z=i$: $$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$ with ...
0
votes
0answers
109 views

excess intersection theory

Can the excess intersection theory be applied to the following problem: I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, ...
18
votes
2answers
293 views

Is fixed point property for posets preserved by products?

Recall that a Partially Ordered Set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point. Theorm : Suppose $P$ and $Q$ are posets ...
2
votes
0answers
65 views

What are the first non-maximal non-group-subgroup simple subfactors?

Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$ is normal if the biprojections ...
0
votes
1answer
50 views

Schur norm for self-adjoint operators

If $A$ is a $n \times n$ complex matrix then the Schur norm of $A$ is given by $$ || A||_S := \max_{||B||=1} ||A*B||,$$ where $||. ||$ is the operator norm and $*$ is the Hadamard (entry-wise) ...
1
vote
0answers
43 views

The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$), let $\mathcal{F}$ be the set of ...
9
votes
1answer
153 views

Finite subgroups of mapping class groups

Given a closed, oriented surface $\Sigma$ of genus greater than 1, let $Mod(\Sigma)$ denote the mapping class group of orientation preserving diffeomorphisms of $\Sigma$ up to isotopy. Given any ...
6
votes
1answer
69 views

$RUCar^{V}$-semiproperness implies properness

This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some ...
3
votes
1answer
118 views

Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$ (e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...
6
votes
0answers
111 views

A compactification of the non-negative rationals with the discrete topology

Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...
2
votes
1answer
266 views

Paper by Mumford

In the paper of "The spectrum of difference operators and algebraic curves", by P. van Moerbeke and D. Mumford, Acta Mathematics, vol.143, 1979, (link: ...
-5
votes
0answers
66 views

True lies logic [on hold]

Santa says lie on all days except one day... Once he made 3 statemnts on consequtive days : 1) I lie on monday and tuesday. 2) today is thursday, saturday or sunday. 3) I lie on wednesday and friday. ...
-2
votes
0answers
37 views

Direct sum of simple modules [on hold]

I would like to ask the simplest example of a simple module M in which M \oplus M has infinite sub-modules. Thanks a lot.
0
votes
0answers
98 views

Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp The ...
7
votes
1answer
324 views

A question on BSD conjecture

If $E$ is an elliptic curve over $\mathbb{Q}$, and $K$ is an imaginary quadratic field. If $rank E(K)\leq 1$ and both $E$ and the quadratic twist of $E$ by $K$ satisfy the full BSD conjecture, does ...
0
votes
0answers
24 views

reducing an n-order differential equation to a first order system of equations using either sagemath or sympy

I want to reduce a n-order ordinary differential equation into a first order system of equations. This is in preparation for numerical analysis. I use both Sympy and Sagemath for Computer Algebra, but ...
8
votes
1answer
271 views

Examples of continuous differential equations with no solution

Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem. ...
2
votes
1answer
24 views

Making a system of second-order ODEs chaotic

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have X'' ...
3
votes
2answers
141 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...
2
votes
1answer
135 views

Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form $ \left( ...
-1
votes
0answers
35 views

Machine learning approach [on hold]

I will illustrate my question my example let us say, I want to build a classifier to label each word in a huge set or words into either "GOOD" or "BAD" word. I have a set of GOOD words and I have ...
-4
votes
0answers
31 views

Lagrange error bound for the approximation of sinx≈x [on hold]

The problem is to bound the error of the approximation of sinx≈x on the interval [-1,1]. Here is what I tried/understand: -I know this is a n=1, x is a 1st degree Taylor polynomial of sin x -On the ...
3
votes
0answers
43 views

Different definitions of linkless graphs

Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows: An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...
1
vote
1answer
56 views

Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?

Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space? Moreover I would like to know if any ...
8
votes
1answer
173 views

A random variation on Polya's orchard problem

Polya's orchard problem is as follows: "How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?" See, ...
7
votes
1answer
78 views

Can the graph Laplacian be well approximated by a Laplace-Beltrami operator?

It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L_{G}$ for some $G$ (where $G$ is ...
2
votes
1answer
99 views

Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$. We can generalize the notion of indecomposable from groups to inclusion of groups as ...
1
vote
1answer
147 views

Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
2
votes
0answers
44 views

Uniqueness of the tensor product decomposition of subfactors

A subfactor $(N \subset M)$ is indecomposable if (for $N_i \subset M_i)$: $$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$ Then, a subfactor $(N ...
5
votes
1answer
212 views

On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if $x^{T}M x\geq 0$ holds for all non-negative real $x_1,x_2,\cdots,x_n$, where $x=(x_1,x_2,\cdots,x_n)^T$. ...
1
vote
0answers
25 views

Family of (Cumulative Distribution) Functions

I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties: For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing. $F$ is closed under products. ...
-2
votes
0answers
37 views

Integrating a differential form over a box [on hold]

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
0
votes
0answers
57 views

Smoothness in Ecalle's method for fractional iterates

Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
4
votes
1answer
221 views

Is there an analog of the Barratt-Eccles construction for E_∞-groups and E_∞-rings?

The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an E∞-operad; algebras in spaces over the Barratt-Eccles operad model E∞-spaces, i.e., homotopy ...
7
votes
1answer
329 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...
4
votes
0answers
44 views

The regularity of Dirichlet form in Besov space

Let $C_{0}(\mathbb{R}^n)$ denote the set of continuous functions in $\mathbb{R}^n$ with compact support, $C_0^\infty(\mathbb{R}^n)$ denote the set of infinite differentiable functions in ...
-1
votes
0answers
64 views

Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit?

[This question was cross-posted from MSE.] A positive integer $N$ is said to be a perfect number if $$\sigma(N) = 2N,$$ where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is ...
2
votes
0answers
52 views

Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167: Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...
0
votes
0answers
52 views

odd squarefree and squareful neighbors [migrated]

There are squarefree numbers $n=\prod_{i=1}^{k}p_i$ so that $n+2$ is not squarefree (e.g. $115+2=3^2.13$). Are there infinite many such $n$? Are there numbers n with arbitrarely many prime-factors? ...
4
votes
1answer
272 views

Differential Geometric Aspects of Rubber Bands

What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only ...
1
vote
0answers
77 views

Ring of endomorphisms as a criterion of a dimension 1 module

Let $R$ be a unital ring and $M$ be an $R$-module. I have some questions about relation between the ring $\operatorname{End}_R M$ of endomorphisms and the notion of “dimension” of a module. I’m not ...
7
votes
0answers
85 views

First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...
1
vote
0answers
92 views

Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...
1
vote
0answers
26 views

The topology of complete minimal surfaces of finite total Gaussian curvature [on hold]

Suppose that M is a complete minimal surface with finite total curvature. If M is embedded in $\mathbf{R}^3$, then we observe that M viewed from infinity looks like a plane passing through the ...
1
vote
3answers
281 views

Is there a general notion of semigroup action?

The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a ...
4
votes
1answer
33 views

What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
0
votes
0answers
66 views

Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
14
votes
1answer
257 views

Three old questions on the Sacks forcing

I came across the two following Qs in 1970. Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
1
vote
1answer
178 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
0
votes
0answers
58 views

A doubt on balaji meyn's ergodic theorem paper

I have a question regarding the classic paper by Balaji and Meyn: "Multiplicative ergodicity and Large Deviations for an Irreducible Markov Chain". Consider a recurrent apeiodic irreducible Markov ...

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