Piecewise linear approximations in nonconvex nonsmooth optimization

We present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables. The algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function. In particular, at each iteration, a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model. A proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness. This research has been partially supported by the Italian “Ministero dell’Istruzione, dell’Università e della Ricerca”, under PRIN project Ottimizzazione Non Lineare e Applicazioni (20079PLLN7_003).     

The orders of nonsingular derivations of modular Lie algebras

We extend the results of Shalev [Sh] on the orders of nonsingular derivations of finite-dimensional non-nilpotent modular Lie algebras. The author is grateful to Ministero dell’Università e della Ricerca Scientifica, Italy, for financial support to the project “Graded Lie algebras and pro-p-groups of finite width”.     

An Identification Problem for First-Order Degenerate Differential Equations

We study a first-order identification problem in a Banach space. We discuss the nondegenerate and mainly the degenerate case. As a first step, suitable hypotheses on the involved closed linear operators are made in order to obtain unique solvability after reduction to a nondegenerate case; the general case is then handled with the help of new results on convolutions. Some applications to partial differential equations motivate this abstract approach.Communicated by I. GalliganiWork partially supported by MIUR (Ministero dell’ Istruzione, dell’ Università e dalla Ricerca), Project PRIN 2004011204 “Analisi Matematica nei Problemi Inversi,” and by the University of Bologna Funds for Selected Research Topics.     

The orders of nonsingular derivations of Lie algebras of characteristic two

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional nonnilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k − 1 with n ⁴ > (2^k − n)³ belongs to . Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups. This work was partially supported by Ministero dell’Istruzione e dell’Università, Italy, through PRIN “Graded Lie algebras and pro-p-groups of finite width”.     

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Acceptable moves for the “worthwhile-to-move” incremental principle are such that “advantages-to-move” are higher than some fraction of “costs-to-move”. When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland’s ϵ-variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties.     

On the radical of Brauer algebras

The radical of the Brauer algebra is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of . The proof is by direct approach for x  =  0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley–Lieb algebra—realised inside —acting on it.

“Ahi quanto a dir che sia è cosa dura lo radical dell’algebra di Brauer pur se’l pensier già muove a congettura” N. Barbecue, “Scholia”

Partially supported by the European RTN “LieGrits”, contract no. MRTN-CT-2003-505078, and by the Italian PRIN 2005 “Moduli e teorie di Lie”.     

Rubber rolling over a sphere

“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G ₂). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S ² and base S ², that can be reduced to an almost Hamiltonian system in T*S ² with a non-closed 2-form ω_NH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I _j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω_NH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I _j are known.      

The <Emphasis Type="Italic">Q</Emphasis> Method for Symmetric Cone Programming

The Q method of semidefinite programming, developed by Alizadeh, Haeberly and Overton, is extended to optimization problems over symmetric cones. At each iteration of the Q method, eigenvalues and Jordan frames of decision variables are updated using Newton’s method. We give an interior point and a pure Newton’s method based on the Q method. In another paper, the authors have shown that the Q method for second-order cone programming is accurate. The Q method has also been used to develop a “warm-starting” approach for second-order cone programming. The machinery of Euclidean Jordan algebra, certain subgroups of the automorphism group of symmetric cones, and the exponential map is used in the development of the Newton method. Finally we prove that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum. Hence the Q method for symmetric cone programming is accurate and can be used to “warm-start” a slightly perturbed symmetric cone program.     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces

A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L _p (μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated. The first author was supported by NSF grant DMS-0555670. The second author was supported by the Russian Foundation for Basic Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1. The third author was supported in part by the Ministero dell’Università e della Ricerca of Italy.     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Graph factors

This exposition is concerned with the main theorems of graph-factor theory, Hall’s and Ore’s Theorems in the bipartite case, and in the general case Petersen’s Theorem, the 1-Factor Theorem and thef-Factor Theorem. Some published extensions of these theorems are discussed and are shown to be consequences rather than generalizations of thef-Factor Theorem. The bipartite case is dealt with in Section 2. For the proper presentation of the general case a preliminary theory of “G-triples” and “f-barriers” is needed, and this is set out in the next three Sections. Thef-Factor Theorem is then proved by an argument of T. Gallai in a generalized form. Gallai’s original proof derives the 1-Factor Theorem from Hall’s Theorem. The generalization proceeds analogously from Ore’s Theorem to thef-Factor Theorem.     

A note on some relations among special sums of reciprocals modulo <Emphasis Type="Italic">p</Emphasis>

In this note the sums s(k, N) of reciprocals are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers. Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan MSM 0021630518 “Simulation modeling of mechatronic systems”.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the B _π characters, which can be viewed as the “π-modular” characters in G, such that the B _p′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs’ work, Slattery has developed some Brauer’s ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer’s three main theorems to the π-blocks. In this paper, depending on Isaacs’ and Slattery’s work, we will extend the first main theorem for π-blocks.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of H?lder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.