11
votes
4answers
472 views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
1
vote
0answers
90 views

Criterion for normality of a schematic image

Consider a projective flat morphism $$ f\colon X\to Y $$ between normal varieties. Let's say over the complex numbers. The geometric fibers of $f$ are all irreducible. I would like a criterion to ...
4
votes
1answer
470 views

textbooks on modern algebraic geometry for 21st-century starters

As for learners in algebraic geometry in 21st century, is there a textbook, lecture note or anything like that to introduce algebraic geometry utilizing the language of derived categories and stacks? ...
0
votes
0answers
20 views

Combining Pearson correlations [on hold]

I have variables a, b, c, x I know sample values A, B, C, but not X I know Pearson correlations pairwise: ax, bx, cx (and ab, ac, bc too if it helps) Now what is the most likely value for X? ...
-3
votes
0answers
33 views

Odd-cycle inequality [on hold]

Consider the stable set problem. An odd hole is a cycle with an odd number if nodes and no edges between nonadjacent nodes of the cycle. Show that if H is the node set of an odd hole, the following ...
1
vote
1answer
88 views

Rademacher type of a Banach space is always less than or equal to 2

Before I ask my question I will provide a brief introduction. I came across the notion of Rademacher type while reading Assaf Naor's article An introduction to the Ribe program, which can be found ...
2
votes
1answer
84 views

A surface on which all regular curves have nowhere vanishing curvature

Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that ...
1
vote
1answer
46 views

Groups arising as direct limits of a stationary system of primitive matrices over the integers

I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...
-8
votes
0answers
126 views

Maths to take a user chosen number to a predictable number [on hold]

As part of simple card trick, I want to allow a user to choose a number between 1 and 100 and then ask them to do various maths to lead them to the same number so their choice becomes irrelevant. One ...
4
votes
0answers
76 views

Contractibility of a poset-indexed colimit

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...
0
votes
0answers
74 views

Reference request: Superconformal algebra

I am looking for references that deal with the basics of $\mathcal{N=1,2}$ superconformal algebras in 2D, their representations and applications to physics, in particular I want to read about spectral ...
-6
votes
0answers
30 views

Boolean algebra 1´=0 ; 0´=1 ; x+1=1 [on hold]

Hi I have a problem to solve, in Boolean algebra. I have to prove that 1´=0 ; 0´=1 ; x+1=1 I solve the first problem x*0=0 -> x*0=x*0+0=x*0+x* x´=x*(0+x´)=x*x´=0 previous 3 problems ...
5
votes
1answer
188 views

Is there a left orderable profinite group?

Is there a profinite group $G$ with a binary transitive relation $<$ such that for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < ...
-1
votes
0answers
35 views

Metrics mappings which are metrics [on hold]

A function f: Z x Z => R is a metric iff forall a,b in Z. f(a,b) >= 0. forall a,b in Z. f(a,b) = f(b,a). forall a,b in Z. f(a,b) = 0 iff a = b. forall a,b,c in Z. f(a,b) + f(b,c) >= f(a,c). Given ...
2
votes
1answer
58 views

Universal and left-factoring order-preserving maps

Trying to get a different angle for the question Fixed points and universal maps for posets, I want to compare universal maps to a different kind of functions. First recall that for posets $P,Q$ an ...
4
votes
0answers
113 views

The cohomology of an $S_{3}$ cover of an elliptic curve ramified in one point

Let $E/\mathbb{C}$ be an elliptic curve. Let $C \to E$ be a Galois cover with group $G = S_{3}$ (symmetric group on $3$ elements), ramified in one point. (To clarify: there is a unique point in $E$ ...
7
votes
1answer
122 views

Intersection Cohomology and $L^2$ cohomology

In the study of singular spaces, topological methods like intersection cohomology have played an important role. They have led to the development of technology like perverse sheaves and these find ...
2
votes
1answer
217 views

A perfect domain that is not integrally closed?

Does there exist an integral domain $R$ of characteristic $p > 0$ that is perfect (i.e., $x \mapsto x^p$ is bijective on $R$) but not integrally closed in its field of fractions?
2
votes
0answers
55 views

Topological/numerical constraints for the existence of more than one pencil

A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective ...
4
votes
1answer
168 views

Existence of internal toposes/inner models in a topos

It has been known for some time that one can define a topos as a model of a (finitary) essentially algebraic theory (or in other words, can be defined internal to any category with finite limits). In ...
2
votes
1answer
123 views

A Category-ish Structure with Morphism Domains containing Multiple Objects?

I am working on formalizing software design using category theory. However the most natural way for me to express what I want is with a Category where multiple morphisms can join into a single ...
27
votes
1answer
596 views

Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations, where we can use forcing to prove the existence of finite objects with some ...
-3
votes
0answers
76 views

What is a discrete shape [on hold]

I saw this term in a paper on tiling using shapes. Can you give the definition for discrete shape? I googled on the web, and could not find any explanation for this concept...
0
votes
0answers
26 views

What is the relation between linear subgraph and matching polynomial? [on hold]

I am confused about these following three concepts, An edge-cycle subgraph of a graph $G$ (also called a linear subgraph of $G$) is a subgraph of $G$ whose components are cycles and edges. A set of ...
26
votes
2answers
509 views

Why do sporadic simple groups have so few conjugacy classes?

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...
8
votes
2answers
257 views

Fano variety of lines on the Segre and the Grassmannian

Does every $\mathbb{P}^{19}\subset \mathbb{P}(\mathbb{C}^5\otimes\mathbb{C}^5)$ intersect the Segre variety of rank one matrices in at least a $\mathbb{P}^1$? A naive dimension count suggests this is ...
0
votes
2answers
75 views

Planar curves identical to their inverses

Is the right strophoid the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin) is identical to $C$?               ...
0
votes
1answer
89 views

Approximating an integral [on hold]

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
7
votes
2answers
288 views

Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...
0
votes
1answer
107 views

Short arithmetic progressions in quadratic residues

Let p be a prime number of form 4k+1. I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending ...
-3
votes
0answers
48 views

Chebyshev polynomials and uniformly convergence [on hold]

Let $P_n(x)=\csc(n\alpha)T_{n-1}(x)-\cot(n\alpha)x$ be a sequence of polynomials. Where $T_n(x)$ is the $n-th$ Chebyshev polynomial of the first kind, and $\alpha\in[0,2\pi]$. Is there uniformly ...
5
votes
1answer
148 views

Purely noncommutative algebra-Morita equivalence

Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all ...
0
votes
0answers
29 views

Perturbation method of a boundary value problem

Let $u(x, \epsilon, \theta)$ be the solution of $$u''+(\epsilon \cos(x)+\theta-u)u=0$$ with boundary conditions $u'(0)=0$ and $u'(\pi)=0$. Here $\theta\in [0, 1]$. I tried to put the solution in ...
1
vote
1answer
65 views

Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$ consider the (energy) functional ...
1
vote
0answers
57 views

Better version of “Monotonicity methods in Hilbert spaces and some applications to nonlinear PDEs..”

I am asking whether any one knows of a better source for the text Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations by H. Brezis which I ...
10
votes
1answer
151 views

Free subgroups in algebras of polynomial growth

What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are ...
4
votes
1answer
224 views

Geometrically connected components of an algebraic group

Suppose that $G$ is an algebraic group over a field $k$. Let $G^o$be the connected component of the identity. Since $G^0$ contains a $k$-rational point (the identity) therefore it is geometrically ...
2
votes
1answer
66 views

Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)? What is known about ...
10
votes
2answers
1k views

Existential statement without witness

Are there existential theorems of ZFC, or PA say, with no witnesses? Ie does there exist a formula $\phi$ such that ZFC $\vdash\exists x \phi(x)$, but for all numerals $\underline{n}$, ZFC $\nvdash ...
1
vote
1answer
99 views

what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
4
votes
1answer
227 views

Endomorphisms and almost all graphs

Is it known what fraction (almost all?) of graphs have a trivial endomorphism monoid? I can't seem to find any reference to the question. Maybe it's related to the question: what fraction of graphs ...
-1
votes
0answers
49 views

Length of Paths in Graph [on hold]

There is a very large directed graph with n number of vertices, in order of millions. We are given a number p much smaller than n, and two vertices v1 and v2. What is the efficient way of finding a ...
1
vote
0answers
35 views

Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous $f$ function such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is interpolation polynomial on ...
5
votes
0answers
136 views

Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups? My wish is something like this: Let $G$ be a compact Lie group. There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...
1
vote
0answers
31 views

Reason for the Choice of Line Parameters in the Radon Transform

Why are the lines, over which the integrals in a Radon Transform are calculated, apparently always parameterized as $L(t,\phi,\alpha) := ...
7
votes
0answers
94 views

Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?

Question: Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}_g$ for some $g\geq 1$? Here $\mathcal{A}_g$ is the ...
0
votes
0answers
51 views

Extension of Lebesgue measure [on hold]

For $n \in \Bbb N$, can we extend the Lebesgue measure to some measure (or complete measure) on $\Bbb R^n$? Can we extend the Lebesgue measure to some measure on the power set of $\Bbb R^n$? (The ...
0
votes
1answer
92 views

A conditional expectation question about consecutive inner products

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each $v_i \in \{-1,1\}$ independently and with equal probability. Each $w_j \in \{-1,0,1\}$ independently with equal ...
-1
votes
0answers
24 views

what is the best algorithm complexity known for finding a p.s.d symmetric matrix eigenvalues? and eigenvectors? [on hold]

I dont know if I can ask this question here or not, if I should delete it tell me:( what is the best algorithm complexity known for finding a p.s.d symmetric matrix eigenvalues? and eigenvectors?
4
votes
1answer
160 views

To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.] I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...

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