On permutations of $$\{1,\ldots,n\}$$ { 1 , … , n } and related topics

Journal of Algebraic Combinatorics - In this paper, we study combinatorial aspects of permutations of $$\{1,\ldots ,n\}$$ and related topics. In particular, we prove that there is a unique...     

�˻�ͼ\mathbb{Z}^2\times\{0,1,\ldots,l-1\}�ij����Եij���֤��

It is well known that the two dimensional integer lattice \mathbb{Z}^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\} is recurrent. The proof approach only utilizes the elementary methods in probability theory (without the words of electric networks).     

圈的$L(d_1,d_2,\ldots,d_t)$-数$\lambda(C_n;d_1,d_2,\ldots,d_t)$

An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）     

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

For a given finite index subgroup \(H\subseteq \mathrm {SL}(2,\mathbb {Z})\), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) found by Athreya and Cheung to the finite cover \(\mathrm {SL}(2,\mathbb {R})/H\) of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd?s, Szüsz, and Turán.     

Estimates for the Entropy Numbers of the Classes B p , θ Ω $$ {B}_{p,\theta}^{\varOmega } $$ of Periodic Multivariable Functions in the Uniform Metric

Ukrainian Mathematical Journal - We establish order estimates for the entropy numbers of the classes $$ {B}_{p,\theta}^{\varOmega } $$ of periodic multivariable functions in the uniform metric. For...     

Optimal portfolio selections via $$\ell _{1, 2}$$ ℓ 1 , 2 -norm regularization

Computational Optimization and Applications - There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ norm and/or $$\ell _2$$ -norm squared. The common...     

Lattices from codes over $$\mathbb {Z}_q$$: generalization of constructions D, $$ D'$$ D′ and $$\overline{ D}$$ D¯

In this paper, we extend the lattice Constructions D, \(D'\) and \(\overline{D}\) (this latter is also known as Forney’s code formula) from codes over \(\mathbb {F}_p\) to linear codes over \(\mathbb {Z}_q\), where \(q \in \mathbb {N}\). We define an operation in \(\mathbb {Z}_q^n\) called zero-one addition, which coincides with the Schur product when restricted to \(\mathbb {Z}_2^n\) and show that the extended Construction \(\overline{D}\) produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction \(A'\) is also derived and we show that this construction produces a lattice if and only if the corresponding code over \(\mathbb {Z}_q[X]/X^a\) is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of q-ary lattices in cryptography.     

$${\overline{1}}$$ 1 ¯ -Evaluating linear functionals on function spaces

Positivity - Let C(X) denotes the Riesz algebra of all real valued continuous functions on a Tychonoff space X, and let L be a vector subspace of C(X). A linear functional $$H:L\rightarrow {\mathbb...     

Approximating Characteristics of the Nikol’skii–Besov Classes S1,θrBℝd$$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$

Ukrainian Mathematical Journal - We establish the exact-order estimates for the approximation of the classes $$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$ by entire functions of...     

Existence of Multi-bump Solutions for a Class of Quasilinear Schrödinger Equations in $${\mathbb{R}^{N}}$$ Involving Critical Growth

In this paper we prove the existence of multi-bump solutions for a class of quasilinear Schrödinger equations of the form \({-\Delta{u} + (\lambda{V} (x) + Z(x))u - \Delta(u^{2})u = \beta{h}(u) + u^{22*-1}}\) in the whole space, where h is a continuous function, \({V, Z : \mathbb{R}^{N} \rightarrow \mathbb{R}}\) are continuous functions. We assume that V(x) is nonnegative and has a potential well \({\Omega : = {\rm int} V^{-1}(0)}\) consisting of k components \({\Omega_{1}, \ldots , \Omega{k}}\) such that the interior of Ω _i is not empty and \({\partial\Omega_{i}}\) is smooth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that for any given non-empty subset. \({\Gamma \subset \{1, \ldots ,k\}}\), a bump solution is trapped in a neighborhood of \({\cup_{{j}\in\Gamma}\Omega_{j}}\) for\({\lambda > 0}\) large enough.     

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Using infinite time Turing machines we define two successive extensions of Kleene’s O{\mathcal{O}} and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call O⁺{\mathcal{O}^{+}}—has height equal to the supremum of the writable ordinals, and that the other extension—which we will call O⁺⁺{\mathcal{O}}^{++}—has height equal to the supremum of the eventually writable ordinals. Next we prove that O⁺{\mathcal{O}^+} is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that O⁺⁺{\mathcal{O}^{++}} is Turing computably isomorphic to the halting problem of eventual computability.     

The Teichmüller distance between finite index subgroups of $${PSL_2(\mathbb{Z})}$$

For a given , we show that there exist two finite index subgroups of which are -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any there are two finite regular covers of the Modular once punctured torus T ₀ (or just the Modular torus) and a -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S ^p ) of the punctured solenoid S ^p under the action of the corresponding Modular group (which is the mapping class group of S ^p [6], [7]) has the closure in T(S ^p ) strictly larger than the orbit and that the closure is necessarily uncountable.      

Projective Compactification of $$\mathbb {R}^{1,1}$$ and Its Möbius Geometry

We examine the semi-Riemannian manifold \(\mathbb {R}^{1,1}\), which is realized as the split complex plane, and its conformal compactification as an analogue of the complex plane and the Riemann sphere. We also consider conformal maps on the compactification and study some of their basic properties.     

Short-time behavior of the heat kernel and Weyl’s law on $${{\mathrm{RCD}}}^*(K,N)$$ spaces

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of \({{\mathrm{RCD}}}^{*}(K,N)\)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in \({{\mathrm{RCD}}}^*(K,N)\)-spaces. We use then these results to initiate the study of Weyl’s law in the \({{\mathrm{RCD}}}\) setting.     

On Cohen–Macaulay modules over the plane curve singularity of type $$\varvec{T_{44}}$$

For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type \(T_{44}\), we explicitly describe the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

A Kähler structure for the $$\text {PU}(2, 1)$$ PU ( 2 , 1 ) configuration space of four points in $$S^3$$ S 3

Geometriae Dedicata - We show that an open subset $${\mathfrak {F}}_4''$$ of the $$\mathrm{PU}(2,1)$$ configuration space of four points in $$S^3$$ is in bijection with an open subset of...     

The Intrinsic π-Operator on Domain Manifolds in $${\mathbb{C}^{n+1}}$$

The main aim of this article is to study the hypercomplex π-operator over \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} via real, compact, n + 1-dimensional manifolds called domain manifolds. We introduce an intrinsic Dirac operator for such types of domain manifolds and define an intrinsic π-operator, study its mapping properties and introduce a Clifford–Beltrami equation in this context.