Euclidean embeddings of finite metric spaces

This is a brief survey on Euclidean embeddings of finite metric spaces, focusing on the power transform metric with many examples. Some old results are presented in slightly improved forms, and the last section contains a few new results. Proofs are given if they are elementary and not too long. Several problems and conjectures are also given.     

Linear operators in finite dimensional probabilistic normed spaces

In this paper, we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces. Then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete.     

On the probability that finite spaces with random distances are metric spaces

For a set of 3 or 4 points we compute the exact probability that, after assigning the distances between these points uniformly at random from the set 1,…,n , the space obtained is metric. The corresponding results for random real distances follow easily. We also prove estimates for the general case of a finite set of points with uniformly random real distances.     

Generalized degree for classes of finite dimensional upper hemi-continuous mappings in separable Banach spaces

Let E be a real separable Banach space, E^∗ the dual space of E, and Ω⊂E an open bounded subset, and let T:D(T)⊆E→2^E∗ be a finite dimensional upper hemi-continuous mapping with . A generalized degree theory is constructed for such a mapping. This degree is then applied to study the existence of approximate weak solutions to the equation 0∈Tx.     

Inner product spaces and minimal values of functionals

We consider a real function which depends on the distances between a variable point and the points of a finite subset A of a linear normed space X. We show that X is an inner product space if this function attains its local minimum on a barycenter of points of A with well-chosen weights. Our result generalizes classical results about characterization of inner product spaces and answers a question of R. Durier, which was posed in his article [J. Math. Anal. Appl. 207 (1997) 220-239].     

Constructing frames for finite dimensional Hilbert spaces

In this paper, we use the characterization of frames, introduced in [W. Rolli, A characterization of frames for finite dimensional Hilbert spaces, preprint], to construct frames for finite dimensional Hilbert spaces. Furthermore this approach allows for the construction of frames with prescribed properties.     

Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Integrability of the general product Hardy operators on the product Hardy spaces

In this paper,we prove that the general product Hardy operators are bounded from the product Hardy space H1/n ( Rm1 ×…× Rmn ) to L 1 ( RΣni=1 mi).     

Decomposition of finite pseudometric spaces

Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting ofn points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric is a linear combination of basic pseudometrics with nonnegative coefficients. Forn ≤ 7, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish some other conditions of decomposability and indecomposability. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 225–234, February, 1998.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

Preservers of generalized orthogonality on finite dimensional real vector and projective spaces

We describe the general form of bijective orthogonality preserving maps on ? ⁿ equipped with a pair of generalized indefinite inner products. The relations between the projective space and vector space versions of this result are examined and an example is given showing that the hypotheses of our main theorem are essential.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Riesz product spaces and representation theory

Let {E _i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ _i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ _i∈I E_i is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX _i such thatE _i is Riesz isomorphic toC(X _i). Supported by the National Natural Science Foundation of China     

D-boundedness andD-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces.     

Matrix product codes with rosenbloom-tsfasman metric

In this article, the Rosenbloom-Tsfasman metric of matrix product codes over finite commutative rings is studied and the lower bounds for the minimal Rosenbloom-Tsfasman distances of the matrix product codes are obtained. The lower bounds of the dual codes of matrix product codes over finite commutative Frobenius rings are also given.     

On Segre's product of partial line spaces and spaces of pencils

In this paper we prove that every collineation of the Segre product of strongly connected partial line spaces is (up to permutation of indices) the product of collineations of its components (Thm. 1.10). Spaces of pencils are strongly connected, so the claim holds for Segre products of them (Thm. 1.14). In the second part we study the extendability of collineations of Segre products of spaces of pencils under some natural embeddings. Received 4 Mai 1998.     

Classes of plus matrices in finite dimensional indefinite scalar product spaces

We describe in terms of canonical forms various classes of plus matrices in real and complex finite dimensional spaces with indefinite scalar product and study their topological structures (such as closure and interior).Dedicated to M.G. Kreîn, one of the founding fathers of the theory of plus-operators.The work of this author was performed under the auspices of C.N.R.-G.N.F.M. and partially supported by the research project, Nonlinear problems in analysis and its physical, chemical and biological applications: Analytical, modelling and computational aspects, of the Italian Ministry of Higher Education and Research (M.U.R.S.T.)The work of this author partially supported by the NSF grant DMS 9500924.The work of all authors is partially supported by the NATO grant CGR 960700.