Some spectral characteristics of λ-matrices

The investigation of spectral properties and characteristics of -matrices of general form is continued. The concept of a matrix solution is extended to the case of singular -matrices. The concept of reducing subspaces and the concepts of a block eigenvalue and a block eigenvector are generalized. Existence theorems are proved; spectral properties of matrix quantities associated with the concepts indicated and their generalizations are established.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 111–124, 1984.     

Differential properties and Fourier coefficients of functions of λ-bounded variation

пУстьλ={λ _i} _i=1 ^∞ —пОслЕ ДОВАтЕльНОсть ВЕЩЕс тВЕННых ЧИсЕл сλ _i↑∞ Иλ ^m={λ_т+ i} _i=0 ^∞ . РАссМАтРМВАУтсь 2π-пЕ РИОДИЧЕскИЕ ФУНкцИИ, Дль кОтОРых $$V_\Lambda (f) = \mathop {\sup }\limits_x \mathop {\mathop {\sup }\limits_{(a_i ,b_i ) \cap (a_j ,b_j ) = \emptyset } }\limits_{(a_i ,b_i ) \subset (x,x + 2\pi ]} \mathop \sum \limits_{\iota = 1}^\infty \frac{{\left| {f(b_i ) - f(a_i )} \right|}}{{\lambda _i }}< \infty ,$$ И Дль кОтОРых $$\mathop {\lim }\limits_{m \to \infty } V_{\Lambda ^m } (f) = 0.$$ ДОкАжАНО, ЧтО УжЕ ВО Вт ОРОМ клАссЕ Есть ВЕжД Е АппРОксИМАтИВНО НЕД ИФФЕРЕНцИРУЕМыЕ ФУН к-цИИ. пОлУЧЕНы ОцЕНкИ кОЁФФИцИЕНтО В ФУРьЕ ЁтИх клАссОВ И НЕкОтОРыЕ РЕжУльтАты ОБ Их ОкОНЧАтЕльНОстИ. кАк слЕДстВИЕ ДАНО ДОстА тОЧНОЕ УслОВИЕ Дль Их НЕсОВп АДЕНИь.     

On Joint Realization of (0,1) -matrices

Let R=(r_1,…,r_m),S=(s_1,…,s_n),R’=(r_1',…,r_m') and S'=(S_1',…,S_m')be non-negative integral vectors.Denote by (R,S)the class of(0,1)-matrices withrow sum vector R and column sum vector S. The three classes (R,S), (R’,S’)and (R R’,S S’)are called jointly realizable if there exist a matrix A in (R,S)and a matrix B in (R’,S’)such that A B∈ (R R’,S S’). In 1980,R.A.Brualdi and R.P.Anstee posed the following conjecture inde-pendently(see [1]).     

Spectral properties of a limit-periodic Schrödinger operator in dimension two

We study the Schrödinger operator H = ?Δ + V(x) in dimension two, V(x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis, and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves exp( $i\left\langle {\overrightarrow k ,\overrightarrow x } \right\rangle $ ) at the high energy region. Second, the isoenergetic curves in the space of momenta $\overrightarrow k $ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.     

Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.     

On some conjectures on the spectra of τ-matrices

We will consider three conjectures of Schneider and Varga concerning the location of eigenvalues of ω- and τ-matices in the complex plane, and extend the known results to n ? 4. We will further show that in the neighborhood of certain cyclic M-matices the conjectures hold for all n.     

Spectral properties of k p schrödinger operators in one space dimension

In the physics of layered semiconductor devices the k · p method in combination with the envelope function approach is a well established tool for band structure calculations. We perform a rigorous mathematical analysis of spectral properties for the corresponding spatially one dimensional k · p Schrödinger operators;thereby encompassing a wide class of such operators. This class covers many of the k · p operators prevalent in solid state physics. It includes k · p Schrödinger operators with piecewise constant coefficients, a prerequisite for dealing with the important case of semiconductor hetero-structures. In particular, we address the question of persistence of a spectral gap over the wave vector range. We also introduce a regularization of the problem which gives rise to a consistent discretization of k · p operators with jumping coefficients and describe design patterns for the numerical treatment of k · p operators.     

Cardinal Invariants of λ-Topologies

We study the basic cardinal-valued invariants of C (X) such as weight, density, network weight, i-weight, and tightness, where C (X) is the space of all continuous real functions on X in the -topology.     

Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces

We investigate spectral properties of integral operators of the form

Nested Optimal λ-Packings and λ-Coverings of Pairs with Triples

We say that an optimal -packing ( -covering) of pairs with triples can be nested if it is possible to add one point to each block such that an optimal 2 -packing (2 -covering) of pairs with quadruples is obtained. A packing (covering) of order v can exist only if v 4 and v / 3 (v - 1) / 2 = v / 4 2(v - 1) / 3 ( v / 3 (v - 1) / 2 = v / 4 2 (v - 1) / 3 ). It is shown in this paper that this condition is also sufficient, except possibly for v = 20.     

Spectral Capacity of Decomposable Operator

In this paper,the notion of a spectral capacity for closed linear operator is introduced.We provethat T is closed decomposable operator if and only if T possesses a spectral capacity.     

Spectral Characterizations of Cliford Minimal Hypersurfaces

ＳｐｅｃｔｒａｌＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｓｏｆＣｌｉｆｏｒｄＭｉｎｉｍａｌＨｙｐｅｒｓｕｒｆａｃｅｓＺｈｏｕＺｈｅｎｒｏｎｇ（周振荣）（ＷｕｈａｎＦｏｏｄＩｎｄｕｓｔｒｙＣｏｌｅｇｅ，Ｗｕｈａｎ，４３００２２）ＡｂｓｔｒａｃｔＩｎｔｈｉｓｐａ...     

Distribution of Primitive λ-Roots of Composite Moduli II

We improve estimates for the distribution of primitiveλ-roots of a composite modulus q yielding an asymptotic formula for the number of primitiveλ-roots in any interval I of length |I| >> q1/2 c. Similar results are obtained for the distribution of ordered pairs (x,x-1) with x a primitive A-root, and for the number of primitiveλ-roots satisfying inequalities such as |x - x-1| < B.     

Enclosings of λ-fold 4-cycle systems

In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K _v . This supplementary solution builds on a result of Raines and Staniszlo.