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What are the primitive notions and axioms in model theory?

I know every theory has its primitive notions and axioms. Now I am reading Basic Model Theory, and there is no term or sentence referred as to a primitive notion or an axiom. But, I think I know that ...
Wenchuan Zhao's user avatar
0 votes
0 answers
46 views

Elimination over $\mathbb F_p[x,y]$

Let $p$ be a prime. Consider the two independent modular equations: $$a_1x^2+b_1y^2+c_1xy\equiv d_1\bmod\mathbb F_p$$ $$a_2x^2+b_2y^2+c_2xy\equiv d_2\bmod\mathbb F_p$$ Is it possible to extract the ...
Turbo's user avatar
  • 13.6k
3 votes
0 answers
44 views

Is the Whitehead bracket $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$ injective?

Let $X$ and $Y$ be finite CW-complexes and $p,q\geq 2$. The Whitehead bracket induces a homomorphism $\pi_{p}(X)\otimes \pi_{q}(Y)\to \pi_{p+q-1}(X\vee Y)$, $\alpha\otimes \beta\mapsto [\alpha,\beta]$....
J.K.T.'s user avatar
  • 435
0 votes
0 answers
51 views

Hilbert scheme of points on an arithmetic surface

Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\mathrm{Hilb}^n(X)$ is regular. My primary question is: if $X$ is a smooth (quasi-projective) surface ...
Stephen McKean's user avatar
1 vote
0 answers
34 views

What is the proper name for this "tersest path" problem in Infinite Craft?

The web game Infinite Craft gives you a starting set of elements $V_0\subset V$ and a mapping $E$ of type $V\times V\rightarrow V$. In fact, $F$ is commutative: $E(v_a,v_b) = E(v_b,v_a)$. So another ...
Quuxplusone's user avatar
1 vote
0 answers
22 views

One parameter subgroups of reductive algebraic groups

If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
Ekta's user avatar
  • 43
3 votes
0 answers
35 views

Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?

(Originally on MSE.) Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial ...
RavenclawPrefect's user avatar
1 vote
0 answers
48 views

If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?

Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...
user479223's user avatar
  • 1,257
2 votes
0 answers
37 views

Some questions about induced subgraphs of the directed hypercube graph

Let $Q^n$ be the hypercube graph in $n$ dimensions. Hao Huang famously showed that any induced subgraph on more than $2^{n-1}$ must have maximum degree $ \geq \sqrt{n}$. It is also known that this ...
Agile_Eagle's user avatar
2 votes
0 answers
19 views

Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
0xbadf00d's user avatar
  • 171
1 vote
0 answers
101 views

Tame-Wild dichotomy; why cant tame algebras be wild?

I would like to understand the Tame-Wild dichotomy, and in particular why an algebra cannot be tame and (semi-)wild at the same time. I've looked in the papers by Drozd and Crawley-Boevey[D80, CB88]. ...
Jacob FG's user avatar
  • 425
2 votes
0 answers
21 views

Smoothness of the Fréchet Function

Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as $$ F(x) = \...
Yueqi's user avatar
  • 21
1 vote
0 answers
43 views

Affine Springer fibers for symmetric spaces

Springer fibers are defined to be the varieties of "isotropic" full flags which are fixed by a certain element in the symmetric space. In a similar manner, affine Springer fibers can be ...
211's user avatar
  • 93
5 votes
1 answer
206 views

Trying to understand the topology of the Weil group for the local Langlands conjecture

I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group. Let $F$ be a non ...
Mario's user avatar
  • 151
3 votes
0 answers
73 views

Intersection complex of genus-zero curves?

I would like to have a very explicit description of $\bar M_{0, n}$, especially its boundary divisors and how they intersect. All I can do in my construction is add divisors and blow up at strata, ...
Leo Herr's user avatar
  • 1,034
3 votes
0 answers
62 views

Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...
kindasorta's user avatar
  • 1,199
4 votes
0 answers
74 views

Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
asv's user avatar
  • 20.9k
1 vote
0 answers
34 views

Normality in a tower of cyclic extensions of global fields, as in Artin-Tate

Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field, and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
Mikhail Borovoi's user avatar
1 vote
0 answers
39 views

Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$). The sequence begins with $$ 8, 16, 32, 48, 64, ...
Notamathematician's user avatar
0 votes
1 answer
66 views

Closed unbounded sets and partitions

Let $\kappa$ be a regular, uncountable cardinal. Let $S\subseteq \kappa$ be a closed and unbounded set. Suppose that we partition $S$ into $<\kappa$ pieces. Does one of those pieces contain a ...
Pace Nielsen's user avatar
1 vote
0 answers
73 views

Discrete spectrum of $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
Hugo's user avatar
  • 11
0 votes
0 answers
96 views

Vanishing of first group cohomology

Let $\Gamma$ be a finitely presented group, acting by isometries on a normed vector space $(V,\|\cdot\|)$. The first group cohomology $H^{1}(\Gamma,V)$ is the space of crossed homomorphisms modulo ...
studiosus's user avatar
  • 265
0 votes
0 answers
51 views

Gradient estimates of linear elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain. Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ $$u(x)\Big|_{\...
mnmn1993's user avatar
1 vote
0 answers
112 views

Effective bound for odd numbers expressed as sums of three primes

I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
Anton Rechenauer's user avatar
4 votes
1 answer
132 views

Fppf or étale extension of group algebraic spaces

Let $S$ be a scheme and let $$0 \to A \to B \to C \to 0$$ be an exact sequence of abelian sheaves on $(\mathrm{Sch}/S)_\text{fppf}$. Assume that $A$ and $C$ are representable by flat algebraic spaces. ...
Joseph's user avatar
  • 41
-6 votes
0 answers
26 views

Similar Rectangles [closed]

Garden a with a length of 6ft and a width of 8feet is similar to garden b with a length of 15 feet. it takes you 45 minutes to plant garden a. how long does it take you to plant garden b?enter image ...
Holly Jean's user avatar
4 votes
0 answers
68 views

Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
Alexander Chervov's user avatar
2 votes
1 answer
57 views

Let $G$ be a graph of genus $g$. Is the number of (non necessarily disjoint) 5-clique subgraphs at most $f(g)$ for some function $f$?

For a graph of genus $g$, it holds that it cannot have too many disjoint 5-cliques, as each clique requires a new handle. It feels that given a graph of genus $g$, it cannot have an unbounded number ...
master bob's user avatar
2 votes
0 answers
50 views

Cohomological dimension of functors from fields to vector spaces

Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are pairs $F,j$ consisting of a field $F$ and the embedding $...
Galois group's user avatar
4 votes
1 answer
146 views

Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$

Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
David Fernandez-Breton's user avatar
3 votes
1 answer
119 views

Solution of SDE at finite time, continuity of pdf

I'm looking at the Langevin dynamics described by the following SDE $$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$ where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...
Simone256's user avatar
0 votes
0 answers
72 views

A Combinatorial Geometric Problem [closed]

It's a math (geometry, combinatorial geometry) problem. Ten cowboys are on the ground (which is totally plane), when the clock strikes at 12 o'clock,they shoot at the one who is nearest to them. How ...
GoGeo Li's user avatar
5 votes
0 answers
68 views

Spherical functions in the space of functions on real Grassmannians

Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$. Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
asv's user avatar
  • 20.9k
0 votes
0 answers
31 views

Product of two circles and holomorphic functions

Assume that $f$ and $g$ are holomorphic functions in the unit disk having boundary values on the unit circle $T$ almost everywhere. Assume further that $$\int_0^{2\pi}\int_0^{2\pi}|f(e^{it})+g(e^{is})|...
AlphaHarmonic's user avatar
0 votes
0 answers
145 views

Degree 6 Galois extension over $\mathbb{Q} $

Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
Sky's user avatar
  • 913
3 votes
0 answers
102 views

A question about Gromov's proof of a "more effective version of the main theorem"

In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem" For any positive integers $d$ and $k$, there ...
A Name's user avatar
  • 31
1 vote
0 answers
16 views

Riemannian gradient flow over a convex set

Given a function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ that is Lipschitz continuous, and $A(x)$ is positive definite for all $x \in \text{int} C$ and $A(x) = 0$ for all $x \in \text{bd}...
dkyopt's user avatar
  • 11
1 vote
0 answers
82 views

A cell complex constructed from singular knots

Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...
Eric Ley's user avatar
  • 111
1 vote
1 answer
117 views

How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?

In the proof of Lemma 4.1, pp. 962–963 in "Endpoint Strichartz Estimates" by Tao and Keel (1997) (see MR1646048 or Zbl 0922.35028), the authors first proved the statements hold for some ...
Elvis's user avatar
  • 11
1 vote
0 answers
60 views

Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?

Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
Bram Cohen's user avatar
6 votes
0 answers
175 views

Do acyclic amenable groups exist?

Is there an example of a nontrivial discrete amenable group with vanishing integral homology? To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
Denis T's user avatar
  • 4,369
1 vote
0 answers
61 views

Lower bounding a partition-related sum

We say the $\mathbb{N}$-valued, non-increasing, eventually zero sequence $\lambda=(\lambda_1\geq\lambda_2,\cdots)$ is a partition of $N$ if $|\lambda|:=\sum_{k\geq 1}\lambda_k=N$, and denote $m_k(\...
MikeG's user avatar
  • 613
3 votes
0 answers
51 views

Subspaces of $\mathbb{F}_2^N$ containing many pairs of far apart vectors

Let $S$ be a subset of vectors in $\mathbb{F}_2^{3n}$ having Hamming weight $n$. Suppose that $S$ contains $m$ pairs of vectors having disjoint supports (that is, they are at Hamming distance $2n$ ...
Kevin's user avatar
  • 510
2 votes
1 answer
160 views

Construction of the Mayer-Vietoris spectral sequence

Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here ...
Francis's user avatar
  • 21
0 votes
1 answer
131 views

Does weak convergence in $L^2$ imply convergence a.e. of a subsequence?

The title pretty much explains it all. Let $u_n\in L^2(\mathbb{R}^n)$ be a sequence converging weakly in $L^2$ to some $u\in L^2(\mathbb{R}^n)$, that is $\int u_k v \to \int u v$ for all $v\in L^2(\...
No-one's user avatar
  • 1,037
0 votes
0 answers
67 views

Number of solutions $x$ of equation $a_1 b_1^x + \dotsb + a_n b_n^x=0$ over a finite field

Let $F$ be a finite field and let $a_1, b_1, \dotsc, a_n, b_n \in F$ be field elements. I am interested in the number of solutions $0\leq x \leq |F|-1$ such that \begin{equation}\label{e:1} a_1 b_1^x +...
Albert Garreta's user avatar
5 votes
2 answers
286 views

A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
Mohammad Ghomi's user avatar
1 vote
1 answer
86 views

Defect between modulus and conductor of ray class field

I have following question about a remark in J. Neukirch's Algebraic Number Theory around page 397. The context: We consider ideal theoretic formulation of global class field theory of a number field $...
user267839's user avatar
  • 5,806
2 votes
0 answers
104 views

A relative cycle class map

Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...
Asvin's user avatar
  • 7,578
3 votes
0 answers
50 views

Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$

I have a (fairly large) system of polynomial equations, of the form $$ c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots $$ (In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
Pace Nielsen's user avatar

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