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How to generate a random function with conditions?

The background is as follows: I consider the following differential equation $$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$ where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
61 views

Is there a category theoretic definition of a cryptographic commitment scheme?

I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...
• 123
15 views

From average degree to a highly connected subhypergraph

I'm looking for a result in $k$-uniform hypergraphs analogous to the following result for graphs, due to Mader: Every graph of average degree $4r$ has a $r$-connected subgraph.
• 123
1 vote
53 views

A question about cohomology with local coefficient

Let's consider the next theorem. Theorem [The cohomology Leray-Serre Spectral sequence] Let $R$ be a commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{% \rightarrow }B$, ...
• 1,001
117 views

Hilbert irreducibility and the inverse Galois problem?

A few years ago I wrote a note about Hilbert irreducibility and the Galois problem, which I recently re-uploaded here. It has not been peer-reviewed, so the following should be taken with a caution ...
• 2,104
1 vote
50 views

derived completion and flat base change

Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings. We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete. For a ...
• 347
27 views

Kernel perfection in some powers of cycles

Suppose I orient the edges of the power of cycle graph $G=C_n^k$ where $n=16$ and $k=4$ in such a way that all the generated cycles by the elements $1,2,3,4$ are given the standard lexical orientation....
• 1,841
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Three-dimensional analogues of Hirzebruch surfaces

There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
• 31
1 vote
43 views

Random walk with same directions and different step sizes

Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$. Consider the following two random ...
• 479
259 views

Groups $G$ with only non-trivial quotient isomorphic to $G$

If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we ...
45 views

Expansion with Laguerre polynomials

Let $L_{m}^{(\alpha)}\left(x^{2}\right)$ be the Laguerre polynomial of degree $m$ and order $\alpha$. Put $\varphi_{m}^{(\alpha)}(x)=e^{-\frac{x^{2}}{2}} L_{m}^{(\alpha)}\left(x^{2}\right)$. It's ...
28 views

Truncating the high degree part of a positive boolean function doesn't change the distance to positive functions too much

Given $n\in\mathbb{Z}^{+}$, suppose $f:\{-1,1\}^n\to[0,1],$ then $f$ has a Fourier expansion: $f(x)=\sum_{S\subseteq[n]} \tilde{f}(S)x^S,$ where $x^S=\Pi_{i\in S}x_i$ , $\tilde{f}(S)\in\mathbb{R}$. (...
• 91
1 vote
24 views

Understanding simple point processes

Background I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes. A simple point process is a random variable whose ...
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• 111
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• 11
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Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer

In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...
40 views

• 731
1 vote
75 views

Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms? Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
• 141
77 views

What are the internal adjunctions in the bicategory $\mathsf{Span}$?

Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...
108 views

Pontryagin product on the homology of cyclic groups

Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
• 83
1 vote
29 views

Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
• 10.3k
29 views

Define this set of power curves bounded above by a given geometric curve and below by the x-axis

Let $f(x) = 1/(1-x)$, with $x$ a real number in $[0, 1]$ and $f(x)$ a real number in $[1, \infty]$. This is clearly part of a geometric curve, as well as part of a branch of a hyperbola with ...
• 139
143 views

Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

Question: is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$? I'm convinced it must be true, but can't remember having seen ...
• 12.4k
1 vote
60 views

Adjunction correspondence for Blow up of double point

Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$. Why holds for ...
• 5,650
301 views

Countable chain condition in topology

A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
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Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...