2
votes
0answers
76 views

Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [on hold]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
4
votes
0answers
94 views

Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let's ...
2
votes
0answers
122 views

Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and ...
2
votes
1answer
90 views

(Partial) crepant resolutions

Consider de orbifold $\mathbb{C}^2$/$\mathbb{Z}_n$. In this case a full crepant resolution exists and it is unique. However, this orbifold admits partial resolutions. So my question is: Are all those ...
5
votes
0answers
119 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
2
votes
0answers
80 views

To determine if a 2 variable symmetric function is addition formula of one variable function or not?

Since $$f(x+y)=f(y+x)$$, So an addition formula must be symmetric. $$f(x+y)=U(f(x),f(y))=U(f(y),f(x))$$ $$f(f^{-1}(x)+f^{-1}(y))=U(x,y)=U(y,x)$$ An example: $$f(x+y)=f(x)f(y)(f(x)+f(y))$$ ...
3
votes
0answers
93 views

An inequality with a sum of integrals

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad ...
0
votes
1answer
186 views

If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with quasi-Euler ...
4
votes
1answer
90 views

Glueing together functions defined on the leaves of a foliation

Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying. Consider a Poisson ...
3
votes
0answers
49 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ ...
9
votes
1answer
185 views

Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
-1
votes
0answers
65 views

value of Riemann's zeta function at even negative numbers [on hold]

The zeta function has trivial zeros at -2,-4,......But direct substitution of say -2 makes the sum diverge as the negative exponent in the denominator makes the terms 2 squared,3 squared etc. please ...
8
votes
3answers
474 views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product ...
4
votes
0answers
66 views

Non-universally trivial Chow group of zero-cycles on Fano hypersurfaces

Let $X$ be a smooth projective variety over a field $k$. By (one) definition, the Chow group of zero-cycles $CH_0(X)$ is universally trivial if, for every field extension $k \subset K$, the degree map ...
0
votes
0answers
17 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
3
votes
2answers
164 views

How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...
2
votes
1answer
73 views

Discrepancy of elements in minimal members of a union-closed set

This question is motivated by Frankl's union-closet sets conjecture. Let $n\in\mathbb{N}$ and set $[n] = \{0,1,\ldots,n\}$. We say that a family ${\cal A} \subseteq {\cal P}([n])$ is union-closed if ...
4
votes
0answers
64 views

Understanding homotopy t-structure

The following question came up while reading Hoyois' From algebraic cobordism to motivic cohomology. Let $S$ be a Noetherian scheme of finite Krull dimension and let $SH(S)$ denote the homotopy ...
0
votes
3answers
154 views

Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which $$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...
0
votes
1answer
186 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
0
votes
1answer
49 views

Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$: $\require{AMScd}$ \begin{CD} a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...
3
votes
1answer
56 views

Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...
0
votes
0answers
94 views

Ramanujan graphs and stable real polynomials

I have a problem with the lemma 6.5 and the theorems 5.1, 6.6 in the paper of Adam Marcus, Daniel Spielman and Nikhil Srivastava. http://arxiv.org/pdf/1304.4132.pdf I do not understand how they use ...
1
vote
0answers
45 views

Can a semigroup be defined on a Banach algebra? [on hold]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...
0
votes
0answers
24 views

Why do we say a level 1 Menger Sponge has 5 holes? [on hold]

I've heard that a level 1 Menger Sponge has 5 holes, but what is the justification for this? I can understand starting with a hole down the center, and making 4 more to meet it from the sides, but ...
1
vote
0answers
32 views

Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix ...
0
votes
1answer
73 views

In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made: Let $u_m$ converges weakly to $u$ in ...
0
votes
0answers
22 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
12
votes
1answer
397 views

Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.
1
vote
1answer
140 views

Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...
2
votes
0answers
33 views

Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...
0
votes
0answers
35 views

Is preimage of closure equal to closure of preimage under continuous topological maps? [on hold]

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$ Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
9
votes
1answer
184 views

The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says: The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$. The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the ...
4
votes
1answer
77 views

Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...
0
votes
0answers
57 views

Prove in GL that no statement can be proven consistent with PA unless PA is inconsistent [on hold]

Edit: I moved this question to math.stackexchange.com here as sugested. I'm trying to do a exersie on page 16 of this paper. It says: Exercise. Show, using the rules of Godel-Lob modal logic ...
0
votes
1answer
604 views

Hard maths on viXra? [on hold]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...
1
vote
0answers
35 views

How to calculate the derivative of logarithm of a matrix? [migrated]

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(1-M)^k}{k}$$ The derivative of ...
12
votes
1answer
158 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...
2
votes
0answers
49 views

Zauner's conjecture [migrated]

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 ...
-4
votes
0answers
43 views

Distance Between Sets- Jaccard Coefficient [on hold]

How do I estimate whether a given distance between two sets obeys triangle law of inequality. lets say d(x,y)= |A-B|+|B-A| or d(x,y)=(|A-B|+|B-A|)/|A U B|
1
vote
0answers
93 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
1
vote
1answer
73 views

Vectors which average to zero over any graph neighborhood

Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum_{j \in N(i)} x_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set ...
1
vote
1answer
95 views

The space of loops as a Banach space [on hold]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...
0
votes
0answers
26 views

Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space. I'm intrested in informative examples and applications of such systems. I know about ...
1
vote
2answers
51 views

Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...
2
votes
1answer
69 views

Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
3
votes
2answers
84 views

References about the matrix generators of the finite subgroups of the orthogonal group O(4)

"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
3
votes
0answers
72 views

henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme? For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme. Let $Y_{Z}^{h}$ the ...
7
votes
0answers
50 views

Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
0
votes
0answers
18 views

Characterization of Dedekind complete Riesz spaces by strictly positive functionals

I was browsing throughout the literature and I found the following fact: Every $\sigma$-Dedekind complete Riesz space $E$ which admits a strictly positive functional $f$ is Dedekind complete. I ...

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