Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Two-dimensional regularity and exactness

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits [12] introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel–quotient factorisation, extending earlier work of Street and others  and .     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

Moduli of smoothness and approximation on the unit sphere and the unit ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a K-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding K-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

Dependence of the acoustic and mechanical properties of polyethylene on temperature and cyclic cooling and heating

The low-temperature stability of polyethylene is confirmed by acoustic and mechanical investigations. Data are given on the dependence of the acoustic and mechanical properties of polyethylene on temperature and repeated cyclic cooling to –50° and heating to +60°C, together with information on the change in Poisson's ratio and volume during deformation.Mekhanika Polimerov, Vol. 1, No. 3, pp. 145–150, 1965     

Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195-198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131-153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality.For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2^k function and a read 2^k formula for F can be obtained in polynomial time.     

Special relativity and theory of gravity via maximum symmetry and localization

Like Euclid,Riemann and Lobachevski geometries are on an almost equal footing,based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R,the de Sitter/anti-de Sitter(dS/AdS)special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein's special relativity on the Minkowski-space in the case of R→∞. Thus the dS-space is coin-like:a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side,and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S~3 fitting the cosmological principle on another side. If our universe is asymptotic to the Robertson-Walker-like dS-space of R(?)(3/Λ)~(1/2),it should be slightly closed in O(A)with entropy bound S(?)3πc~3k_B/ΛGh.Contrarily,via its asymptotic behavior, it can fix on Beltrami inertial frames without‘an argument in a circle’and acts as the origin of inertia. There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space,a null cone modulo projective equivalence[N](?)_p(AdS~5). Thus there should be a dS-space on the boundary of S~5×AdS~5 as a vacuum of supergravity. In the light of Einstein's‘Galilean regions’,gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics.Thus,this may lead to the theory of gravity of corresponding local symmetry.A simple model of dS-gravity characterized by a dimensionless constant g(?)(AGh/3c~3)~(1/2)～10~(-61)shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter. The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.     

Classical and Exceptional Root Systems and Quantizations

We present a method for quantizing semisimple Lie algebras. In Huru (Russ. Math. [2007]) we defined quantizations of the braided Lie algebra structure on a finite dimensional graded vector space V by quantizations of braided derivations on the exterior algebra of V ^* . We find quantizations of semisimple Lie algebras in this setting using the grading by their roots and shall go through all root systems, classical and exceptional.      

The splitting game and applications

First we define the splitting operator, which is related to the Shapley operator of the splitting game introduced by Sorin (2002). It depends on two compact convex sets C and D and associates to a function defined on C×D a saddle function, extending the usual convexification or concavification operators. We first prove general properties on its domain and its range. Then we give conditions on C and D allowing to preserve continuity or Lipschitz properties, extending the results in Laraki (2001a) obtained for the convexification operator. These results are finally used, through the analysis of the asymptotic behavior of the splitting game, to prove the existence of a continuous solution for the Mertens-Zamir system of functional equations (Mertens and Zamir (1971–72) and (1977)) in a quite general framework. Revised November 2001     

Cycles and Spectra

Some recent work on spaces of algebraic cycles is surveyed. The main focus is on spaces of real and quaternionic cycles and their relation to equivariant Eilenberg- MacLane spaces.Dedicated to IMPA on the occasion of its 50^th anniversary     

Technology and xx

The revolution of technology in education has heralded a new wave of learning styles and in some cases total dependence on technology. Yet, statistics reveal no significant improvement in student performance in Mathematics and a downward trend in basic algebra skills. The major impediment in the learning process is lack of understanding of the concepts and a virtual total dependence on technology. We expose the dangers of total dependence on technology by using the much-ignored function f(x) = x^x as a case study and conclude that technology can be a valuable tool provided the student has complete understanding of related concepts.     

Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity. In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.     

Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations

General second-order parabolic and hyperbolic equations on a bounded domain are considered. The input is applied in the Neumann or mixed boundary condition and is expressed as a finite-dimensional feedback. In the parabolic case, the feedback acts, in particular, on the Dirichlet trace of the solution: here it is shown that the resulting closed loop system defines a (feedback) C₀-semigroup on $\text{L}{}_2\text{(Ω)}$ (in fact, on $\text{H}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; 2?</mtext>}}\text{(Ω), ρ > 0}$), that is both analytic and compact for positive times, and whose generator has compact resolvent. In the hyperbolic case, the feedback acts on the position vector only, or on its Dirichlet trace in a special case: here a similar result is established regarding the existence of a feedback C₀-cosine operator. Moreover, an example is given, which hints that the class of prescribed feedbacks acting on the Dirichlet trace cannot be substantially enlarged. Functional analytic techniques are employed, in particular perturbation theory. However, perturbation theory for the original variable fails on $\text{L}{}_2\text{(Ω)}$, the space in which the final result is sought. Therefore, our approach employs perturbation theory, after a suitable continuous extension, on the larger space $\text{[H}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>\; +\; 2?</mtext>}}\text{(Ω)]′}$.     

Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

Length and Stable Length

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive ϵ there is a positive δ depending only on n and on ϵ such that an element of π₁(M) with stable commutator length less than δ is represented by a geodesic with length less than ϵ. Moreover, for any such M, the first accumulation point for stable commutator length on conjugacy classes is at least 1/12. Conversely, “most” short geodesics in hyperbolic 3-manifolds have arbitrarily small stable commutator length. Thus stable commutator length is typically good at detecting the thick-thin decomposition of M, and 1/12 can be thought of as a kind of homological Margulis constant. Received: June 2006 Revision: May 2007 Accepted: June 2007     

Maximum Orbit Weights in the σ-game and Lit-only σ-game on Grids and Graphs

Let G be a graph in which each vertex can be in one of two states: on or off. In the σ-game, when you “push” a vertex v you change the state of all of its neighbors, while in the σ⁺-game you change the state of v as well. Given a starting configuration of on vertices, the object of both games is to reduce it, by a sequence of pushes, to the smallest possible number of on vertices. We show that any starting configuration in a graph with no isolated vertices can, by a sequence of pushes, be reduced to at most half on, and we characterize those graphs for which you cannot do better. The proofs use techniques from coding theory. In the lit-only versions of these two games, you can only push vertices which are on. We obtain some results on the minimum number of on vertices one can obtain in grid graphs in the regular and lit-only versions of both games.     

Raney and Catalan

Raney’s lemma is often used in a counting argument to prove the formula for (generalized) Catalan numbers. It ensures the existence of “good” cyclic shifts of certain sequences, i.e. cyclic shifts for which all partial sums are positive.We introduce a simple algorithm that finds these cyclic shifts and also those with a slightly weaker property. Moreover it provides simple proofs of lemma’s of Raney type.A similar clustering procedure is also used in a simple proof of a theorem on probabilities of which many well-known results (e.g. on lattice paths and on generalized Catalan numbers) can be derived as corollaries. The theorem generalizes generalized Catalan numbers. In the end it turns out to be equivalent to a formula of Raney.     

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.