Continuous Convex Research Articles

On the Lipschitz Regularity of Minimizers of Anisotropic Functionals

We prove a Lipschitz regularity result for minimizers of functionals of the calculus of variations of the form ∫Ωf(Du(x))dx, where f is a continuous convex function from Rn into [0,+∞), not necessarily depending on the modulus of Du.

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on X X has a continuous biconjugate function on X ∗ ∗ X^{**} , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

Flame Temperature of Laminar Premixed Flame with Wrinkling Flame Surface.

The flame temperature of a laminar premixed flame with continuous convex and concave curvatures was experimentally investigated. The flame temperature and the stretch rate along the flame surface were measured for the flame formed in the lean propane/air mixture (φ=0.72, Le=1.74). As a result, the flame temperature at the convex toward the unburned mixture was lower than the adiabatic flame temperature and that of the concave was higher, because of the flame stretch and the Lewis number effect. However, the measured flame temperature was different from the estimated flame temperature by the theory for the local flame structure proposed by C. J. Sun et al. We guess the reason why such difference was arisen is the existence of the heat transfer along the flame surface, so we suggest that the heat transfer along the flame surface should be considered in the case of the flame with continuous convex and concave curvatures.

On the solvability of a variational inequality problem and application to a problem of two membranes

The purpose of this work is to give a continuous convex function, for which we can characterize the subdifferential, in order to reformulate a variational inequality problem: find u = (u1, u2) ∈ K such that for all v = (v1, v2) ∈ K, ∫Ω∇u1∇(v1 − u1)+∫Ω∇u2∇(v2 − u2) + (f, v − u) ≥ 0 as a system of independent equations, where f belongs to L2(Ω) × L2(Ω) and .

ＴｉＮセラミック膜の伸びの形態

The elongated morphology of TiN films deposited on SUS430 stainless steel coil has been studied. TiN-coated stainless steel coil has been deposited using a new horizontal-type high current (1500 A) hollow cathode discharge (HCD) gun installed in a continuous ion plating apparatus of air-to-air type.In the scanning electron microscope (SEM) observation of TiN films after bulge press-forming and 180° bend-forming, the TiN films showed comparatively sharp, straight cracks characteristic of ceramics. However, very small areas across cracks of the TiN films showed the “bridgelike” elongation on TiN-TiN surface matrix. The formation of these bridges was associated with the formation of the continuous convex lines on the film surface.

Absolute Norms on [formula omitted

In this paper, we study the absolute normalized norms on Cn. We show that the set of all absolute normalized norms on Cn is in one-to-one correspondence with the set Ψn of all continuous convex functions on Δn with some suitable conditions, where Δn={(s1,…,sn−1)∈Rn−1; s1+s2+···+sn−1≤1, si≥0 (1≤i≤n−1)}. As some applications, we show that an absolute norm on Cn is strictly convex if and only if the corresponding function ψ on Δn is strictly convex. Further, we consider the von Neumann–Jordan constant of Cn with an absolute normalized norm.

Optimal partition of QoS requirements with discrete cost functions

The future Internet is expected to support applications with quality of service (QoS) requirements. To this end, several mechanisms are suggested in the IETF; the most promising among them is DiffServ. An important problem in this framework is how to partition the QoS requirements of an application along a selected path. The problem which is, in general, NP-complete, was solved for continuous convex cost functions by Lorenz and Orda (see IEEE/ACM Trans. Networking. vol.6, p.768-78, 1998 and Proc. IEEE INFOCOM'99, p.246-53, 1999). This paper concentrates on discrete cost functions, which better model the existing and upcoming mechanisms in the Internet. We present efficient exact and approximated solutions for various conditions of the problem. We also show that although the more complex problem of QoS sensitive routing with discrete cost functions is hard, it has a fully polynomial approximation scheme.

A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant

We give a generalization of Ekeland's ϵ-Variational Principle and of its Borwein–Preiss smooth variant, replacing the distance and the norm by a “gauge-type” lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz β-smooth “bump function,” then every continuous convex function on an open subset U of X is densely β-differentiable in U. This generalizes the Borwein–Preiss theorem on the differentiability of convex functions.

Generic Convergence of Descent Methods in Banach Spaces

Given a continuous convex function f on a Banach space X, we consider a complete metric space of vector fields V on X with the topology of uniform convergence on bounded subsets. With each such vector field we associate two iterative processes. We show that for a generic V the values of the function f tend to its infimum for both processes.

Von Neumann–Jordan Constant of Absolute Normalized Norms on [formula omitted

We determine and estimate the von Neumann–Jordan constant of absolute normalized norms on C2 by means of their corresponding continuous convex functions on [0,1]. This provides many new interesting examples including those of non-ℓp-type as well as some previous ones. It is also shown that all such norms are uniformly non-square except ℓ1- and ℓ∞-norms.

Stability in Semilinear Problems

In the paper, we derive results concerning a continuous dependence of solutions on the right-hand side for a semilinear operator equation Lu=∇g(u), by assuming that L:D(L)⊂H→H (H−a Hilbert space) is self-adjont, with a closed range, and g:H→R is continuous convex on H and Gâteaux differentiable on D(L). Using these results, we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order ü+au=DuF(t, u, ω), t∈[0, π] a.e., with the Dirichlet boundary conditions u(0)=u(π)=0, where a⩾1, and ω is a functional parameter.

The 0.1-100 keV spectrum and variability of Mrk 421 in a high state

ABSTRA C T The results of a BeppoSAX target of opportunity (TOO) observation of the BL Lac object Mrk 421 during a high-intensity state are reported and compared with monitoring X-ray data collected with the BeppoSAX Wide Field Cameras (WFC) and the RXTE All Sky Monitor (ASM). The 0.1‐100 keV spectrum of Mrk 421 shows continuous convex curvature that can be interpreted as the high-energy end of the synchrotron emission. The source shows significant short-term temporal and spectral variability, which can be interpreted in terms of synchrotron cooling. The comparison of our results with those of previous observations when the source was a factor 3‐5 fainter shows evidence for strong spectral variability, with the maximum of the synchrotron power shifting to higher energy during high states. This behaviour suggests an increase in the number of energetic electrons during high states.

Regularization and existence of solutions in the equilibrium problem for an elastoplastic plate

UDC 517.9 At present, there are many articles devoted to the solvability of the boundary value problems describing deformation of static and quasistatic ideally elastoplastic bodies. Namely, solvability of variational inequalities resulting from the original (exact) statements of boundary value problems is proven in the case of quasistatic deformation for the flow-type models and in the case of static deformation for the Hencky-type models [1-7]. A peculiarity of solutions to the variational inequalities is the fact that some boundary conditions are lost. The smoothness assumption on a solution to the variational inequality does not allow us to establish the presence of the lost boundary conditions, since the set of feasible stresses is not a vector space. It was demonstrated in [8-11] that in the case of one-dimensional models of elastoplastic deformation (beam, rod, etc.) a solution to the variational inequality does not loose the boundary conditions that are involved in the original statement of the boundary value problem; i.e., all boundary conditions hold. Until recently, there were no such results for three-dimensional (two-dimensional) models of elastoplasticity. In the present article, we prove an existence theorem for the equilibrium problem for an elastoplastic plate which guarantees validity of all boundary conditions. Moreover, we consider domains with smooth boundaries as well as domains with cuts. The proof is based on a special combination of the regularization method and the penalty method. Namely, we apply elliptic regularization for solving the penalized boundary value problem that approximates the original elastoplastic problem. We obtain an existence theorem for the original problem by passing to the limit over the regularization and penalty parameters. The proposed way of regularization turns out to be rather effective and useful for analysis of other elastoplastic problems. The essence of the method consists in the fact that the regularization of the penalized equations is accompanied by special regularization of the boundary conditions. 1. A domain with smooth boundary. Suppose that fl C R 2 is a bounded domain with smooth boundary F and �9 : R 3 ---* R is a continuous convex function. The equilibrium problem for an elastoplastic plate is stated as follows [12]: Find functions w, m = {mij}, and ~ii, i,j = 1, 2, in satisfying the following equations and inequalities: -rnii,ij = f,

Randomized online interval scheduling

Online interval scheduling problems arise in important application areas such as continuous media and call control. Woeginger gives an optimal 4-competitive deterministic algorithm for online interval scheduling. He leaves as an open problem whether randomization can be used to achieve a competitive ratio less than 4. A randomized (2+ 3 <3.73206) -competitive algorithm is presented for the case where a job of length ℓ has value φ(ℓ), where φ is any continuous convex non-negative function with φ(0)=0.

Some thoughts on generalized Weber location models

The Weber and inverse Weber location problem is defined for a continuous one-dimensional convex region in the plane and solved using constructive numerical techniques. It is conjectured that the Weber functional for a continuous one-dimensional convex region is concave. The equivalence between the one-dimensional inverse Weber model and a polar geometric optimization problem is demonstrated, and an alternative symbolic expression for the integral functional is described.

Lipschitz functions on Banach spaces which are actually on Asplund spaces

SINCE Namioka and ~ h e l ~ s " ] , starting with Asplund's pioneering work[21, introduced the notion of Asplund spaces (those are Banach spaces on which every continuous convex function is FrCchet differentiable on a dense Gd subset) and proved that the dual of an Asplund space has the Radon-Nikodym property (RNP) , the study of differentiability properties of functions on infinite dimensional spaces has continued widely and deeply (see, for instance, ~ h e l ~ s ' ~ ] and ~ i l e s ' ~ ] ) . The research attained a great achievment after Stegall's theoremE5]: If the dual space E * has the RNP, then E is an Asplund space. Because of the N-Ph-S theorem, we have discovered that a number of apparently disparate mathematical topics ( monotone operators, perturbed optimization of real-valued functions, differentiability of vector valued measures, geometry of Banach spaces) are in fact closely intertwined, with differentiability of convex functions forming a common thread. A more general result, the Preiss differentiability theorem[6-81 that every Lipschitz function on an Asplund space is densely FrCchet differentiable,

α-Covex Sets and Strong Quasiconvexity

A major property of continuous strong convex functions is that they are inf-compact; hence, their minima are reached. In the past, several attempts have been made to extend strong convexity to generalized convexity, but they fail in the sense that they do not ensure inf-compactness. In this paper, we introduce a concept of strong convexity on sets from which we deduce new concepts of strong quasiconvexity related to inf-compactness.

Generic Fréchet Differentiability of Convex Functions on Non-Asplund Spaces

Letfbe a continuous convex function on a Banach spaceE. This paper shows that every proper convex functiongonEwithg≤fis generically Fréchet differentiable if and only if the image of the subdifferential map ∂foffhas the Radon–Nikodým property, and in this case it is equivalent to showing that the image of ∂fis separable on each separable subspace ofE.

Core Equivalence Theorems for Infinite Convex Games

We show that the core of a continuous convex game on a measurable space of players is a von Neumann–Morgenstern stable set. We also extend the definition of the Mas–Colell bargaining set to games with a measurable space of players and show that for continuous convex games the core may be strictly included in the bargaining set but it coincides with the set of all countably additive payoff measures in the bargaining set. We provide examples which show that the continuity assumption is essential to our results.Journal of Economic LiteratureClassification Number: C71.

Convex functions on Banach spaces not containing ℓ1

AbstractThere is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of ℓ1. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.