Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

The Growth Rate of Symplectic Homology and Affine Varieties

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.     

Natural density distribution of Hermite normal forms of integer matrices

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present an explicit computation of this distribution together with some applications. More precisely, for integer matrices whose entries are upper bounded in absolute value by a large bound, we compute the asymptotic number of such matrices whose HNF has a prescribed diagonal structure. We apply these results to the analysis of some procedures and algorithms whose dynamics depend on the HNF of randomly looking integer matrices.     

On maximal immediate extensions of valued fields

We study the maximal immediate extensions of valued fields whose residue fields are perfect and whose value groups are divisible by the residue characteristic if it is positive. In the case where there is such an extension which has finite transcendence degree we derive strong properties of the field and the extension and show that the maximal immediate extension is unique up to isomorphism, although these fields need not be Kaplansky fields. If the maximal immediate extension is an algebraic extension, we show that it is equal to the perfect hull and the completion of the field.     

The polynomial analogue of a theorem of Rényi

Rényi's result on the density of integers whose prime factorizations have excess multiplicity has an analogue for polynomials over a finite field.

     

线性时变系统二次最优控制问题的保辛近似求解

状态空间的最优控制体系是保守的,其近似算法应当保辛．提出了基于分段常值精细积分方法的保辛摄动近似方法,在同一框架下求解了线性时变LQ最优控制中的计算问题,即变系数矩阵Riccati方程和状态反馈方程．该算法是保辛的,具有很好的数值稳定性和精度．算例验证了算法的有效性．     

Adjacency posets of planar graphs

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.     

具有一个T．I．Sylow 2-子群的有限群的类保持Coleman自同构

设G是一个有限群,它的Sylow 2-子群是T.I.集,证明了如果G的2的方幂阶类保持自同构在G任意的Sylow子群上的限制等于G的某个内自同构的限制,则它一定是一个内自同构.对这样的自同构的研究是由整群环的同构问题所引起的.     

Convex drawings of graphs with non-convex boundary constraints

In this paper, we study a new problem of convex drawing of planar graphs with non-convex boundary constraints, and call a drawing in which every inner-facial cycle is drawn as a convex polygon an inner-convex drawing. It is proved that every triconnected plane graph with the boundary fixed with a star-shaped polygon whose kernel has a positive area admits an inner-convex drawing. We also prove that every four-connected plane graph whose boundary is fixed with a crown-shaped polygon admits an inner-convex drawing. We present linear time algorithms to construct inner-convex drawings for both cases.     

On biquadratic fields that admit unit power integral basis

We consider biquadratic number fields whose maximal orders have power integral bases consisting of units. We prove an effective and efficient criteria to decide whether the maximal order of a biquadratic field has a unit power integral basis or not. In particular we can determine all trivial biquadratic fields whose maximal orders have a unit power integral basis.     

Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups

We give three equivalent conditions for weak convergence of almost orbits of an asymptotically nonexpansive commutative semigroup acting on a nonempty bounded closed convex subset of a uniformly convex Banach space whose dual has the Kadec property.     

A network of congruences on an ample semigroup

Semigroup Forum - For an admissible congruence $$\gamma $$ on an ample semigroup S, the minimum admissible congruence $$\gamma _t$$ whose trace is $$\mathrm {tr}\gamma $$ has been determined by the...     

Subsets of Euclidean Space with Nearly Maximal Gowers Norms

A set E ⊂ ℝd whose indicator function 1E has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If 1E has nearly maximal Gowers norm then E...     

Projectivity,prime ideals,and chain conditions of Stone algebras

Some properties of projective stone algebras are exhibited, which are connected with the ordered set of prime ideals. From this we derive a simple characterization of finite projective Stone algebras, and of those projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective distributive lattice. Finally, we give some conditions under which a Stone algebra has no chains of type λ, where λ is an infinite regular cardinal. The results of this paper are part of the author's Ph.D. Thesis written under the direction of S. Koppelberg. The author wishes to express his gratitude to Prof. Koppelberg for her guidance and her patience. Presented by K. A. Baker.     

Extreme Values of Artin L-Functions and Class Numbers

Assuming the generalized Riemann hypothesis (GRH) and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (>1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin L-functions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin L-functions with arbitrarily large conductor whose value at s=1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary nontrivial finite irreducible subgroup of GL(n, ) with property Gal _T .     

Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms

We give an example of an incompressible diffusion equation whose solution is nonunique. It is shown that this equation has an approximation solution as well as another solution that cannot be obtained by approximation. We give sufficient conditions for the uniqueness of a solution as well as for the uniqueness of an approximation solution.     

Numerical solution of a miscible displacement problem with dispersion term using a two-grid mixed finite element approach

Numerical Algorithms - Seriation is an important ordering problem which consists of finding the best ordering of a set of units whose interrelationship is defined by a bipartite graph. It has...     

Superlarge deviation probabilities for sums of independent lattice random variables with exponential decreasing tails

In the note we study large and superlarge deviation probabilities of sum of i.i.d. lattice random variables, whose distribution function has an exponentially decreasing tail at infinity.     

Characterizing subgroups of compact abelian groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.