Extremal norms of graphs and matrices

In recent years, the trace norm of graphs has been extensively studied under the name graph energy. In this paper, this research is extended to more general matrix norms, for example, the Schatten p-norms and the Ky Fan k-norms. Whenever possible, the results are given both for graphs and general matrices. In various contexts, a puzzling fact was observed: the Schatten p-norms are widely different for 1????p?<?2 and for p????2.     

Positive definite matrix approximation with condition number constraint

Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one when we use a unitary similarity invariant norm as a metric. We can especially convert it to a univariate piecewise convex optimization problem when we use the Ky Fan p-k norm. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.     

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the kp-approximation property (kp-AP) if for every Banach space Y, the space F(Y,X) of finite rank operators is dense in the space Kp(Y,X) of p-compact operators endowed with its natural ideal norm kp. In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the kp-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the kp-AP and Saphar's approximation property. As another application, the kp-AP is characterized in terms of a trace condition. Finally, we relate the kp-AP to the (p,p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of Lp(μ)-spaces.     

Polynomial algorithms for solving the vector sum problem

The problem is investigated of weighted sum maximization of a given finite set of vectors from the finite-dimensional vector space ? ^k . Polynomial algorithms solving it are presented and analyzed in the case when a finite polyhedral norm or the l ₂ norm is defined on ? ^k .     

Пространства Lp со сме шанной нормой на беск онечном декартовом произвед ении вероятностных п ространств

Letp=(p ₁,p₂,...) be a vector with an infinite number of coordinates, 1≦p _k≦,k=1,2,... On the set of random functions depending on infinite number of variables, a mixed norm ∥.∥ _p is introduced, and thus the spacesL _p with mixed norm are defined. Part 1 contains observations of general properties of those spaces (in particular, convergence properties depending on the behaviour of the exponentsp _k ask→ ∞). Part 2 contains the proof of infinite-dimensional version of S. L. Sobolev's theorem (in mixed norm) for potentials of Wiener semigroup on infinite dimensional torusT ^∞.     

On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems

For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the residual,whose norm is ξk,of the linear system obtained by the Lanczos method at outer iteration k+1.Based on them,we make a refned analysis and establish new local convergence results.It is proved that(i) the inexact RQI with Lanczos converges quadratically provided that ξk≤ξ with a constant ξ1 and (ii) the method converges linearly provided that ξk is bounded by some multiple of1/||rk|| with rkthe residual norm of the approximate eigenpair at outer iteration k.The results are fundamentally diferent from the existing ones that always require ξk<1,and they have implications on efective implementations of the method.Based on the new theory,we can design practical criteria to control ξkto achieve quadratic convergence and implement the method more efectively than ever before.Numerical experiments confrm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.     

Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation

Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., L_p) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k ? 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.     

A trace class operator inequality

In this note, we generalize the inequality about the trace of positive semidefinite matrix trk(AB)?k(trA)k(trB) to Hilbert space, and obtain a relevant inequality about positive trace class operator.     

Rank 72 high minimum norm lattices

Given a polarization of an even unimodular lattice and integer k?1, we define a family of unimodular lattices L(M,N,k). Of special interest are certain L(M,N,3) of rank 72. Their minimum norms lie in {4,6,8}. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a L(M,N,3). Our method constructs such L(M,N,k) in dimensions 96, 120 and 128 with minimum norms 8.     

Exact constants in generalized inequalities for intermediate derivatives

Consider the Sobolev space W ₂ ⁿ (?₊) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A _n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f ^(k)(0)| ≤ A _n,k ‖f‖. In the general case, the expression for the constants A _n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A _n,k in the case of the following norms: $$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$ In the case of the norm ‖ · ‖₁, formulas for the constants A _n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A _n,k is also studied in the case of the norm ‖ · ‖₂. In addition, we prove a symmetry property of the constants A _n,k in the general case.     

On operator norms of submatrices

Both of the following conditions are equivalent to the absoluteness of a norm ν in $\text{C}$ⁿ: (1) for all n×n diagonal matrices D=(d_k), the subordinate operator norm N_ν(D)=max_k|d_k|; (2) for all n×n matrices A, N_ν(A) ?N_ν(|A|). These conditions are modified for partitioned matrices by replacing absolute values with norms of blocks. A generalization of absoluteness is thus obtained.     

Maps leaving functional values of operator products invariant

Let A be a standard operator algebra on a complex Hilbert space H of dimension greater than 2. By invariants of certain functional values of operator products, we characterize some surjective maps on A. Furthermore, several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as, for example, k-numerical radius (k?1), operator norm, Ky Fan k-norm, Schatten p-norm (1?p<∞), and so on.     

A mean value theorem for discriminants of abelian extensions of a number field

Let k be an algebraic number field and let N(k,C?;m) denote the number of abelian extensions K of k with G(K/k)≅C?, the cyclic group of prime order ?, and the relative discriminant D(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for _∑m?XN(k,C?;m) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.     

Note on the number of integral ideals in Galois extensions

Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

Minimum L ∞ accelerations in Riemannian manifolds

Riemannian cubics are curves in Riemannian manifolds M that are critical points for the L ² norm of covariant acceleration, and are already rather well studied as elementary curves for interpolation problems in engineering. In the present paper the L ² norm is replaced by the L ^∞ norm, which may be more appropriate for some applications. However it is more difficult to derive the analogue of the Euler-Lagrange equation for the L ^∞ norm, requiring techniques from optimal control, and the resulting necessary conditions take a different form. These necessary conditions are examined when M is a sphere or a bi-invariant Lie group, and some examples are given.     

Variation of Grassman powers and spectra

The operator norm of the derivative of the map which takes a finite-dimensional linear operator to its kth Grassman power (the kth compound) is evaluated. This leads to a bound for the distance between the Grassman powers of two operators. As an important application, a bound for the distance between the eigenvalues of two operators is obtained.     

Representations of Anisotropic unitary groups. II

Let SU(f) be the special unitary group of an anisotropic hermitian form f over a field k. Assume f represents only one norm class in k. The representations α:SU(f)→SL(n, R) are characterized when R is a commutative local ring with 3 a unit and n=dimf?4.     

A noncommutative analogue of |D(Xk|=|kXk−1|

The paper concerns alternating powers of a Hilbert space. Let ∧^k be defined by ∧^k(A)(x₁∧?∧x_k)=Ax₁∧?∧Ax_k. It is proved that the norm of the linear map D∧^k(A) depends only upon |A| and is assumed at the identity.     

On solving the planar k-centrum problem with Euclidean distances

This paper presents a solution procedure based on a gradient descent method for the k-centrum problem in the plane. The particular framework of this problem for the Euclidean norm leads to bisector lines whose analytical expressions are easy to handle. This allows us to develop different solution procedures which are tested on different problems and compared with existing procedures in the literature of Location Analysis. The computational analysis reports that our procedures provide better results than the existing ones for the k-centrum problem.