Weak Polynomial Identities for M1,1(E)

* Partially supported by Universita` di Bari: progetto “Strutture algebriche, geometriche e descrizione degli invarianti ad esse associate”.

Compactness in the First Baire Class and Baire-1 Operators

For a polish space M and a Banach space E let B1 (M, E) be the space of first Baire class functions from M to E, endowed with the pointwise weak topology. We study the compact subsets of B1 (M, E) and show that the fundamental results proved by Rosenthal, Bourgain, Fremlin, Talagrand and Godefroy, in case E = R, also hold true in the general case. For instance: a subset of B1 (M, E) is compact iff it is sequentially (resp. countably) compact, the convex hull of a compact bounded subset of B1 (M, E) is relatively compact, etc. We also show that our class includes Gulko compact. In the second part of the paper we examine under which conditions a bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this case are connected with classical results of Choquet, Odell and Rosenthal.

Partial Quasi-Bilateral Generating Function Involving some Special Functions

A group-theoretic method of obtaining more general class of generating functions from a given class of partial quasi-bilateral generating functions involving Hermite, Laguerre and Gegenbaur polynomials are discussed.

Spline Subdivision Schemes for Compact Sets. A Survey

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel

Fragmentability of the Dual of a Banach Space with Smooth Bump

We prove that if a Banach space X admits a Lipschitz β-smooth bump function, then (X ∗ , weak ∗ ) is fragmented by a metric, generating a topology, which is stronger than the τβ -topology. We also use this to prove that if X ∗ admits a Lipschitz Gateaux-smooth bump function, then X is sigma-fragmentable.

On Typical Compact Convex Sets in Hilbert Spaces

Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

Speculating About Mountains

∗Partially supported by Grant MM 409/94 of the Mininstry of Education, Science and Technology, Bulgaria. ∗∗Partially supported by Grants MM 521/95, MM 442/94 of the Mininstry of Education, Science and Technology, Bulgaria.

On Uniformly Convex and Uniformly Kadec-Klee Renomings

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak* lower semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee Property for the weak*-topology (UKK*). Then we solve the UKK or UKK* renorming problems for Lp(X) spaces and C(K) spaces for K scattered compact space.

Strong-weak Stackelberg Problems in Finite Dimensional Spaces

We are concerned with two-level optimization problems called strongweak Stackelberg problems, generalizing the class of Stackelberg problems in the strong and weak sense. In order to handle the fact that the considered two-level optimization problems may fail to have a solution under mild assumptions, we consider a regularization involving ε-approximate optimal solutions in the lower level problems. We prove the existence of optimal solutions for such regularized problems and present some approximation results when the parameter ǫ goes to zero. Finally, as an example, we consider an optimization problem associated to a best bound given in [2] for a system of nondifferentiable convex inequalities.

Radon Measures on Banach Spaces with their Weak Topologies

The main concern of this paper is to present some improvements to results on the existence or non-existence of countably additive Borel measures that are not Radon measures on Banach spaces taken with their weak topologies, on the standard axioms (ZFC) of set-theory. However, to put the results in perspective we shall need to say something about consistency results concerning measurable cardinals.

Compact sensors based on cascaded single-mode-multimode-single-mode fiber structures

Concatenated single-mode-multimode-single-mode (SMS) structures are demonstrated as functional sensing platforms. The devices are fabricated by periodically inserting micrometric sections of multimode optical fiber (MMF) in a single-mode fiber (SMF). The periodic change of the core diameter produces a single strong resonant transmission notch, tunable in the wavelength range from 1200 to 1600 nm. It was found that the position of the notch changed with temperature and refractive index. The devices introduced here are highly compact (length less than 5 mm), simple to fabricate and robust; hence, they are adequate for diverse sensing applications. © 2013 The Japan Society of Applied Physics.

A compact, portable and low cost generic interrogation strain sensor system using an embedded VCSEL, detector and fibre Bragg grating

We present a compact, portable and low cost generic interrogation strain sensor system using a fibre Bragg grating configured in transmission mode with a vertical-cavity surface-emitting laser (VCSEL) light source and a GaAs photodetector embedded in a polymer skin. The photocurrent value is read and stored by a microcontroller. In addition, the photocurrent data is sent via Bluetooth to a computer or tablet device that can present the live data in a real time graph. With a matched grating and VCSEL, the system is able to automatically scan and lock the VCSEL to the most sensitive edge of the grating. Commercially available VCSEL and photodetector chips are thinned down to 20 µm and integrated in an ultra-thin flexible optical foil using several thin film deposition steps. A dedicated micro mirror plug is fabricated to couple the driving optoelectronics to the fibre sensors. The resulting optoelectronic package can be embedded in a thin, planar sensing sheet and the host material for this sheet is a flexible and stretchable polymer. The result is a fully embedded fibre sensing system - a photonic skin. Further investigations are currently being carried out to determine the stability and robustness of the embedded optoelectronic components. © 2012 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).

Generating More Boundary Elements of Subset Projections

Composition problem is considered for partition constrained vertex subsets of n dimensional unit cube E^n . Generating numerical characteristics of E^n subsets partitions is considered by means of the same characteristics in 1 − n dimensional unit cube, and construction of corresponding subsets is given for a special particular case. Using pairs of lower layer characteristic vectors for E^(1-n) more characteristic vectors for E^n are composed which are boundary from one side, and which take part in practical recognition of validness of a given candidate vector of partitions.

A Note on a Classical Generating Function for the Jacobi Polynomials

Mathematics Subject Classification: 33C45.

Fractional Derivatives and Fractional Powers as Tools in Understanding Wentzell Boundary Value Problems for Pseudo-Differential Operators Generating Markov Processes

Mathematics Subject Classification: 26A33, 31C25, 35S99, 47D07.