Arbitrary Norm Research Articles

There is No Lattice Preserving Natural Tensor Norm

We prove that for every natural tensor norm α, one can find a Banach lattice X such that the tensor product X ⊗ X endowed with the norm α does not have the Gordon-Lewis property and, therefore, cannot be isomorphic to any Banach lattice. We also discuss the situation for arbitrary tensor norms.

Robust linear optimization under general norms

We explicitly characterize the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm. Our approach encompasses several approaches from the literature and provides guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.

Calderón--Lozanovskii; construction for mixed norm spaces

We show that the Calderon--Lozanovskii; construction φ(.) commutes with arbitrary mixed norm spaces, that is, φ(E0[F0], E1[F1]) = φ(E0, E1) [φ(F0, F1)] if and only if φ is equivalent to a power function. This result we obtain by giving characterizations of the corresponding embeddings of φ(E0[F0], E1[F1]) into φ0 (E0, E1)[φ1 (F0, F1)] and vice versa in terms of the functions φ, φ0, φ1. As a particular case, we get embeddings of an Orlicz space with mixed norms into an Orlicz space on a product of measure spaces. Applications to classical operators between mixed norm Orlicz spaces are also discussed.

Duality with BF-type I functions for a class of nondifferentiable multiobjective variational problems

A class of BF-type I functions and its extensions are introduced in the continuous case, an example is presented in support. Utilizing these new concepts, sufficient optimality conditions and duality results are presented for multiobjective variational problems involving arbitrary norms.

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of C n with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f( z, t) and the transition mapping v( z, s, t) associated to f( z, t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈ B. Moreover, we prove that a mapping f∈ H( B) has parametric representation if and only if there exists a Loewner chain f( z, t) such that the family { e − t f( z, t)} t⩾0 is a normal family on B and f( z)= f( z,0) for z∈ B. Also we show that univalent solutions f( z, t) of the generalized Loewner differential equation in higher dimensions are unique when { e − t f( z, t)} t⩾0 is a normal family on B. Finally we show that the set S 0( B) of mappings which have parametric representation on B is compact.

A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities

We show that mass transportation methods provide an elementary and powerful approach to the study of certain functional inequalities with a geometric content, like sharp Sobolev or Gagliardo–Nirenberg inequalities. The Euclidean structure of R n plays no role in our approach: we establish these inequalities, together with cases of equality, for an arbitrary norm.

On Existence Theorems for Solutions of Non‐Linear Systems

AbstractAn equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton‐Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G → ℝn, G ⊆ ℝn, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton‐Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized.

Semiparametric proper efficiency principles and duality models for constrained multiobjective fractional optimal control problems containing arbitrary norms

Semiparametric necessary and sufficient proper efficiency conditions are established for a class of constrained multiobjective fractional optimal control problems with linear dynamics, containing arbitrary norms. Moreover, utilizing these proper efficiency results, eight semiparametric duality models are formulated and appropriate duality theorems are proved. These proper efficiency and duality criteria contain, as special cases, similar results for several classes of unorthodox optimal control problems with multiple, fractional, and conventional objective functions, which are particular cases of the main problem considered in this paper.

Duality models for some nonclassical problems in the calculus ofvariations

Parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonconvex variational problems with generalized fractional objective functions and nonlinear inequality constraints containing arbitrary norms. Based on these optimality criteria, ten parametric and parameter‐free dual problems are constructed and appropriate duality theorems are proved. These optimality and duality results contain, as special cases, similar results for minmax fractional variational problems involving square roots of positive semidefinite quadratic forms as well as for variational problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here subsume various existing duality formulations for variational problems and include variational generalizations of a great variety of cognate dual problems investigated previously in the area of finite‐dimensional nonlinear programming by an assortment of ad hoc methods.

Generalized Goal Programming: polynomial methods and applications

In this paper we address a general Goal Programming problem with linear objectives, convex constraints, and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets. In particular, classical Linear Goal Programming problems, as well as several models in Location and Regression Analysis are modeled within this framework. In spite of its generality, this problem can be analyzed from a geometrical and a computational viewpoint, and a unified solution methodology can be given. Indeed, a dual is derived, enabling us to describe the set of optimal solutions geometrically. Moreover, Interior-Point methods are described which yield an $\varepsilon$-optimal solution in polynomial time.

Pareto-optimality of compromise decisions

The questions of the existence of Pareto-optimal decisions and the existence of compromise decisions for decision problems in finite dimensional spaces are investigated. Several conditions for a compromise decision to be Pareto-optimal are derived. The concept of the compromise decision and some of the results are then generalized by allowing in the definition of the compromise solution arbitrary triangular norms instead of the triangular norm defined by the minimum operation only.

T-Sets and some algebraic properties in β S and in ℓ ∈ fty(S) *

For a large class of infinite discrete semigroups, we prove that right cancellative points in βS can have arbitrary norms or sizes. More precisely, if for x ∈ βS, we let \(\left\| x \right\| = \min \left\{ {\left| A \right|:x \in \bar A} \right\}\) and for each infinite cardinal κ, we let Pκ(S) = {x ∈ βS : ∥x ∥ = κ} then the set of points in Pκ(S) which are right cancellative in βS has an interior which is dense in Pκ(S). The method to prove this result enables us also to calculate the already known cardinal of the pairwise disjoint left ideals in \(2^{2^{\left| S \right|} } \). We give an application to the Banach algebra l∞(S)*, by showing that the vector space dimension of any non-zero right ideal in this algebra is at least \(2^{2^{\left| S \right|} } \).

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.

Parametric Representation of Univalent Mappings in Several Complex Variables

AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

Some new applications of the pointwise relations for the relative rearrangement

Let $V=V(\Omega)$ be a set satisfying the Poincaré-Sobolev pointwise inequalities for the relative rearrangement and $\rho$ an arbitrary norm on the set of measurable functions on the interval $\Omega_*=(0,measure(\Omega))$. Then using the associate norm of $\rho$, we give some sufficient conditions to ensure that a function $u$ of $V$ has to be bounded or integrable in an Orlicz space. We show similar results for a solution of quasilinear equation under some conditions between the data and $\rho$. We then give an unified approach for many kinds of estimates. Using a main relation involving some pointwise relations for the relative rearrangement, we prove a regularity result for the time derivative of a parabolic system well posed in a Grand Sobolev space.

From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities

We develop several applications of the Brunn—Minkowski inequality in the Prekopa—Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Prekopa—Leindler theorem the Brascamp—Lieb inequality for strictly convex potentials. We deduce similarly the logarithmic Sobolev inequality for uniformly convex potentials for which we deal more generally with arbitrary norms and obtain some new results in this context. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.

Uniform estimate of a compact convex set by a ball in an arbitrary norm

The problem of the best uniform approximation of a compact convex set by a ball with respect to an arbitrary norm in the Hausdorff metric corresponding to that norm is considered. The question is reduced to a convex programming problem, which can be studied by means of convex analysis. Necessary and sufficient conditions for the solubility of this problem are obtained and several properties of its solution are described. It is proved, in particular, that the centre of at least one ball of best approximation lies in the compact set under consideration; in addition, conditions ensuring that the centres of all balls of best approximation lie in this compact set and a condition for unique solubility are obtained.

Evolving Standards For Academic Publishing: A q-r Theory

This paper develops a model of evolving standards for academic publishing. It is motivated by the increasing tendency of academic journals to require multiple revisions of articles and by changes in the content of articles. Papers are modeled as varying along two quality dimensions: q and r. The former represents the clarity and importance of a paper's main ideas and the latter its craftsmanship and polish. Observed trends are regarded as increases in r-quality. A static equilibrium model in which an arbitrary social norm determines how q and r are weighted is developed and used to discuss comparative statics explanations for increases in r. The paper then analyzes a learning model in which referees (who have a biased view of the importance of their own work) try to learn the social norm from observing how their own papers are treated and the decisions editors make on papers they referee. The model predicts that social norms will gradually but steadily evolve to increasingly emphasize r-quality.

Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system

In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero dynamics as the gain parameters are increased to infinity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary L2 norm. For the controlled Burgers' equation forced by a disturbance we prove that, for all initial data in L2(0, 1), the closed-loop trajectories converge in L2(0, 1), uniformly in t∈[0, T] and in H1(0, 1), uniformly in t∈[t0, T] for any t0>0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional boundary controls are included in the closed-loop system. This provides a proof of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous Dirichlet boundary controlled Burgers' equation. As an application of our convergence of trajectories results, we demonstrate that our boundary feedback control scheme provides a semiglobal exponential stabilizing feedback law in L2, H1 and L∞ for the open-loop system consisting of Burgers' equation with Neumann boundary conditions and zero forcing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is ‘practically semiglobally stabilizable’ in L2-norm. Copyright © 1999 John Wiley & Sons, Ltd.

Global continuation of Lyapunov centre orbits in Hamiltonian systems

One-parameter families of periodic solutions emanating from equilibrium points of a Hamiltonian system are investigated. A class of families that cannot merge on continuation is indicated; as a result, a lower bound for the number of families which are continuable to an arbitrary large norm or period is obtained. Via these findings, some generic multiplicity results for the prescribed period and prescribed energy problems are established. In particular, it is proved that a starshaped energy surface carries at least n distinct periodic solutions, and bilateral bounds for their periods are found.