<Emphasis Type=Italic>K</Emphasis><Subscript><Emphasis Type=Italic>n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.      

Extreme values of zeta and <Emphasis Type=Italic>L</Emphasis>-functions

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions. The author is partially supported by the National Science Foundation (DMS 0500711) and the American Institute of Mathematics (AIM).     

On <Emphasis Type=Italic>K</Emphasis>-large and generalized <Emphasis Type=Italic>K</Emphasis>-large Abelian groups

The concepts of K-large and generalized K-large Abelian groups are introduced, and their main properties and interconnections are studied. We consider groups that are K-large (generalized K-large) with respect to the class of all groups and torsion-free groups. The main focus is on the case where K consists of bounded p-groups for an infinite set of prime numbers p. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 51–60, 2007.     

Characterizations of some rings with <Emphasis Type=Italic>C</Emphasis>-projective, <Emphasis Type=Italic>C</Emphasis>-(FP)-injective and <Emphasis Type=Italic>C</Emphasis>-flat modules

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.     

Weak interpolation in extensions of the logics <Emphasis Type=Italic>S</Emphasis><Subscript>4</Subscript> and <Emphasis Type=Italic>K</Emphasis><Subscript>4</Subscript>

Conditions are specified which are necessary and sufficient for a logic over K4 to possess the weak interpolation property. For this goal to be met, simple transitive modal algebras are described, and we establish a criterion for the class of such algebras to be amalgamable. For extensions of K4, the weak interpolation property is proved decidable. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 705–722, November–December, 2008.     

Expectation values of scaling fields in <Emphasis Type=Italic>Z</Emphasis><Subscript><Emphasis Type=Italic>N</Emphasis></Subscript> Ising models

We compute exact vacuum expectation values of physically relevant scaling fields in the perturbed Z_N parafermionic models. Our algebraic method is based on analyzing vertex operators in the corresponding off-critical lattice model. The results coincide with the expressions derived by perturbed conformal field theory methods. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 557–583, March, 2008     

<Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

<Emphasis Type=Italic>N</Emphasis>-homogeneous <Emphasis Type=Italic>C</Emphasis>*-algebras generated by idempotents

In this paper we consider n-homogeneous C*-algebras generated by idempotents. We prove that a finitely generated unital n-homogeneous (when n is greater than or equals 2) C*-algebra A can be generated by a finite set of idempotents if and only if the algebra A contains at least one nontrivial idempotent.     

On the torsion in <Emphasis Type=Italic>K</Emphasis><Subscript>2</Subscript> of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

Value-distribution of twisted automorphic <Emphasis Type=Italic>L</Emphasis>-functions

In this paper, we prove a limit theorem for twisted with character automorphic L-functions with an increasing modulus of the character.     

On <Emphasis Type=Italic>l</Emphasis><Subscript>2,<Emphasis Type=Italic>p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Spin L-functions for <Emphasis Type=Italic>GSO</Emphasis><Subscript>10</Subscript> and <Emphasis Type=Italic>GSO</Emphasis><Subscript>12</Subscript>

Two multi-variable Rankin-Selberg integrals are studied. They may be regarded as extending the theory begun in [G-H1]. Each is shown to be Eulerian with the unramified contribution given explicitly in terms of partial Langlands L-functions.     

On deformations of <Emphasis Type=Italic>F</Emphasis><Subscript><Emphasis Type=Italic>n</Emphasis></Subscript> in compact Lie groups

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular, we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(F _n ) on such variety when n ≥ 3. The author was partially supported by NSF grant DMS-0404557, BSF grant 2004010, and the ‘Finite Structures’ Marie Curie Host Fellowship, carried out at the Alfréd Rényi Institute of Mathematics in Budapest.     

<Emphasis Type=Italic>L</Emphasis><Subscript>1</Subscript>-Quantization and clustering in Banach spaces

LetX be a random variable with distribution μ taking values in a Banach spaceH. First, we establish the existence of an optimal quantization of μ with respect to the L ₁-distance. Second, we propose several estimators of the optimal quantizer in the potentially infinite-dimensional space H, with associated algorithms. Finally, we discuss practical results obtained from real-life data sets.     

Computing the values of <Emphasis Type=Italic>L</Emphasis>-functions,and the chebyshev polynomials

The Chebyshev polynomials and Chebyshev’s economization method are applied to speed up the computation of the values of L-functions. Bibliography: 3 titles.     

Completeness properties of the pseudocompact-open topology on <Emphasis Type=Italic>C</Emphasis>(<Emphasis Type=Italic>X</Emphasis>)

This is a study of the completeness properties of the space C _ps (X) of continuous real-valued functions on a Tychonoff space X, where the function space has the pseudocompact-open topology. The properties range from complete metrizability to the Baire space property. Dedicated to Professor Robert A. McCoy