Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

Spectral properties of λ-matrices

One investigates the spectral problem for polynomial -matrices of the general form (regular and singular). One establishes a relation between the elements of the complete spectral structure of the -matrix (i.e., its elementary divisors, its Jordan chains of vectors, its minimal indices and polynomial solutions) and of the matrices which occur in its transformation to the Smith canonical form. One establishes a correspondence between the complete spectral structures of a -matrix and of three linear accompanying matrix pencils. One notes the possibility of reducing the solution of the spectral problem for a -matrix to the solution of a similar problem for its accompanying pencil. One gives a factorization of a -matrix which allows to represent it by any of its considered accompanying pencils or by its Kronecker canonical form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 180–194, 1981.     

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.     

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.     

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.     

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.     

The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation

We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.     

Bounds for Determinants of Quaternion Matrices

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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.     

Zeros of Non-Baxter Paraorthogonal Polynomials on?the Unit Circle

We provide leading-order asymptotics for the size of the gap in the zeros around 1 of paraorthogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (−1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.     

An Algorithm for Finding All Isolated Zeros of Polynomial Systems

Finding all zeros of polynomial systems is very interesting and it is also useul for many applied science problems.In this paper,based on Wu‘s method,we give an algorithm to find all isolated zeros of polynomial systems (or polynomial equations).By solving Lorenz equations,it is shown that our algo-rithm is efficient and powerful.     

Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .     

A characterization of λ-central BMO space

We give a characterization of the λ-central BMO space via the boundedness of commutators of n-dimensional Hardy operators.     

Determinants of Cauchy-Riemann operators as τ-functions

The -functions introduced by Sato, Miwa, and Jimbo for the deformation theory associated with the Riemann-Hilbert problem on P ¹ is shown to be a determinant for a singular Cauchy-Riemann operator whose domain incorporates functions with prescribed branching behavior. The analysis relies heavily on previous work of Malgrange on monodromy preserving deformation theory.     

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.