Scattering and Bound States in Euclidean Lattice Quantum Field Theories

In this paper we sudy the property of asymptotic completeness in (massive) Euclidean lattice quantum field theories. We use the methods of Spencer and Zirilli [2] to prove, under suitable hypothesis, two-body asymptotic completeness, i.e., for the energy range just above the two-particle threshold.     

Some Results on the Lattice Parameters of Quaternionic Gabor Frames

Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case.     

Growth Conditions on a Function and the Error Bound Condition

Mathematical Notes - An Erratum to this paper has been published: https://doi.org/10.1134/S0001434618010261     

A New Bound on the Number of Designs with Classical Affine Parameters

The hyperplanes in the affine geometry AG(d, q) yield an affineresolvable design with parameters $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ . Jungnickel [3]proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parametersfor d ≥ 3. The bound of Jungnickel was improved recently [5] by a factor of $q^{\frac{{d^2 + d - 6}}{2}} (q - 1)^{d - 2}$ for any d ≥ 4. In this paper, a construction of $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ designs based on group divisible designs is given that yieldsat least $$\frac{{\left( {q^{d - 1} + q^{d - 2} + \cdots + 1} \right)!\left( {q - 1} \right)}}{{\left| {{\text{P}}\Gamma {\text{L(}}d,q{\text{)}}} \right|\left| {{\text{A}}\Gamma {\text{L(}}d,q{\text{)}}} \right|}}$$ non-isomorphic designs for any d ≥ 3. This new bound improves the bound of[5] by a factor of $$\frac{1}{{q^d }}\mathop \Pi \limits_{i = 1}^{(q^{d - 1} - q)/(q - 1)} (q^{d - 1} + i).$$ For any given q and d, It was previously known [2,11] that there are at least 8071non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is atleast 245,100,000.     

Validity and Failure of the Cauchy-Born Hypothesis in a Two-Dimensional Mass-Spring Lattice

Summary. The Cauchy-Born rule postulates that when a monatomic crystal is subjected to a small linear displacement of its boundary, then all atoms will follow this displacement. In the absence of previous mathematical results, we study the validity of this rule in the model case of a 2D cubic lattice interacting via harmonic springs between nearest and diagonal neighbours. Our main result is that for favourable values of the spring constants and spring equilibrium lengths, the CB rule is actually a theorem. Simple counterexamples show that for unfavourable spring parameters or large displacements the CB rule fails. Moreover the resulting overestimation of the lattice energy per unit volume by the CB rule cannot be cured by convexification (let alone quasiconvexification) of the CB energy. The main tool in our proof is a novel notion of lattice polyconvexity which allows us to overcome the difficulty that the elastic energy as a function of atomic positions can never be convex, due to frame-indifference.     

一类二元叠加码的构作及其d界

首先介绍了一种具有参数d,r的二元叠加(d,n,r)-码及偶特征正交空间上子空间的一些包含性质,然后利用这些性质及相关知识构作了二元叠加(d,n,r)-码并给出了其参数d的界.     

分配格上的全序幂格

研究了分配格上的幂格的偏序关系,给出了分配格上的幂格是链格的充要条件.     

Multifunction Optimization Problems Involving Parameters: Necessary Optimality Conditions

We prove the Fritz John and Kuhn-Tucker necessary optimality conditions for vector optimization problems involving multifunctions and parameters under relaxed assumptions.     

剩余格及有界psBCK-代数成为布尔代数的充要条件

研究了一般剩余格(未必可换)与布尔代数的关系,给出剩余格成为布尔代数的一系列充要条件.同时,进一步将这些结果推广到只含有蕴涵运算的有界psBCK-代数中,证明了在一定条件下由psBCK-代数可诱导出有界格且构成布尔代数.     

格上的T-幂等矩阵

利用完备的分配格L上三角模定义L上的矩阵运算,给出这些运算的一些基本性质,并且讨论L上的T-幂等矩阵。     

Lattice bimorphisms on f-algebras

Let A, B and C be lattice-ordered algebras and let be a bilinear map. We call a lattice bimorphism if for each and each the partial maps and are lattice homomorphisms of B and A into C, respectively; and we say that is multiplicative if holds in C for all and . In this paper, we study the connection between lattice bimorphisms and multiplicative bilinear maps on f-algebras in great detail. Our central result in this direction is the following: if A, B and C are Archimedean f-algebras with unit elements e _A , e _B and e _C respectively and is a Markov bilinear map (i.e., is positive and (e _A , e _B ) = e _C ) then is a lattice bimorphism if and only if is multiplicative. The main application of this result we present in this work is the Cauchy-Shwarz inequality in Archimedean (not necessarily commutative) d-algebras, which is an improvement of the result of Buskes and van Rooij, who established this inequality in the commutative case. Received April 26, 2001; accepted in final form August 9, 2001.     

Lattice uniformities on pseudo-D-lattices

Let L be a pseudo-D-lattice. We prove that the lattice uniformities on L which make uniformly continuous the operations of L are uniquely determined by their system of neighbourhoods of 0 and form a distributive lattice. Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.     

森谱的界

设λ_k(F)是树或者森的第k大特征值,[x]是不超过x的最大整数,q是F的边独立数.本文证明了:对于1≤k≤[(q 1)/2]有λ_k(F)≥1,并且这个下界是最好可能的;对于1≤i≤[q/2],若q为偶数,则有λ[(q 1)/2] i(F)≥2cos((2iπ)/(4i 1)),若q为奇数,则有λ_([(q 1)/2] i)(F)≥2cos(((2i 1)π)/(4i 3)),     

Optimization of Source Parameters in Multipoint Nonseparated Conditions for Linear Dynamical Systems

Computational Mathematics and Mathematical Physics - The problem of optimizing the right-hand sides of linear nonlocal multipoint conditions for a linear system of differential equations is...     

完备Brouwerian格上Fuzzy关系方程有极小解的条件

本文在有限论域上对完备Brouwerian格上Fuzzy关系方程极小的存在问题作了探讨，首先构造了Fuzzy关系方程有解但无极小解的一个例子，然后在解集非空时给出了对Fuzzy关系方程的每一个解都存在一个小于等于它的极小解的一个充分条件及一个充要条件，特别地，在充分条件下给出了一类Fuzzy关系方程所有极小解的个数的公式。     

依托面积为载体的几个不等式的直观证明及思考

依托面积为载体,在给出Young不等式的几何直观证明的基础上,继续讨论几何直观在几个相关不等式证明中的运用.探讨了数学教学中如何发挥几何直观的作用.     

Lattice orders on matrix algebras

We construct all the lattice orders (up to isomorphism) on a full matrix algebra over a subfield of the field of real numbers so that it becomes a lattice-ordered algebra. Received June 26, 2001; accepted in final form February 9, 2002.