Some further results on a semilinear equation with concave–convex nonlinearity

Following our previous work [J. Math. Anal. Appl. 295 (2004) 341], we give an exact growth order near zero for positive solutions of a class of elliptic equation and use it to give the existence of multiple solutions with negative energy, multiple positive solutions and sign changing solutions of the considered problem. Even in the particular case, our results extend a previous work by Ambrosetti–Garcia–Peral [J. Funct. Anal. 137 (1996) 219].     

Absolute value equation solution via concave minimization

The NP-hard absolute value equation (AVE) Ax − |x| = b where and is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization. A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations.     

Toward a generalized computability theory

Translated fromAlgebra i Logika, Vol. 32, No. 4, pp. 349–386, July–August, 1993.     

Toward a bipolar theory of risk

Every man's attitude toward risk is a priori characterized by both a speculative pole and a conservative pole. Most men like high (speculative) gains and simultaneously dislike losses associated with a given venture. Such an attitude generates a conflict, which must be resolved when dealing with decisions under uncertainty. The traditional theories of risk implicitly assume that this conflict is erased when fitting an everywhere-concave (or convex) utility function. Such unipolar theories should be compared with a bipolar theory and more generally with a multicriterion theory of risk. Our mean-bipolar risk vector analysis generalizes the traditional mean-risk analysis. In Portfolio Selection, the mean-variance approach can be extended to include the speculative pole as well as the conservative pole: the variance is viewed as both desired and undesired components of riskiness.     

Toward a formal theory of model integration

The aim of this paper is to provide the first steps toward a formal theory of model integration. This is supported at least by three arguments: (a) increasing the productivity of the modeling work; (b) decreasing errors; (c) saving time and money. Of course, any formal theory has to be based on a given framework; in our case, we consider only models which satisfy the core concepts of Structured Modeling. The outline of the paper is as follows. After the motivations are pointed out, some preliminary results are given in section 2. Section 3 defines the levels of integration, while in sections 4 and 5 some examples are presented. Remarks and future extensions conclude the paper.On leave from the University of Siena on a fellowship from CNR, Italy. Permanent address: Department of Quantitative Methods, Piazza S. Francesco 17, I-53100 Siena, Italy (contact author).     

Toward a general theory of fuzzy variables

A general framework for a theory is presented that encompasses both statistical uncertainty, which falls within the province of probability theory, and nonstatistical uncertainty, which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. Fuzzy Sets1 (1978), 3–28]. The concept of a fuzzy integral is used to define the expected value of a random variable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence is formulated, various convergence concepts are defined and their relationships are studied by using a Chebyshev-like inequality for the fuzzy integral. The possibility of using this theory in Bayesian estimation with fuzzy prior information is explored.     

Toward a theory of nonlinear stochastic systems

The inequalities of Bihari, Langenhop, and Chebyshev are applied to a system of stochastic nonlinear differential equations. As a result, a one-parametric family of determinate estimates from above and below are obtained for the norm of the solution of the system. Each of these estimates is valid with a probability not less than some quantity depending upon the parameter of the family.Translated from Matematicheskii Zametki, Vol. 5, No. 5, pp. 607–614, May, 1969.The author expresses his appreciation to G. Ya. Lyubarski for his interest in the study and his valuable comments.     

Toward a general theory of uniform spaces

This paper contains a historical and logical foundation for the general theory of uniform spaces, and also an account of the basic concepts of this theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 7–18, 1985.     

Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation. With the help of the Nehari manifold and the center mass function, we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole.     

A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

Fourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u₁ with and . We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains.     

Toward equivariant Iwasawa theory

 Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G _∞ , X _∞ the Galois groups of K _∞ /k and M _∞ /K _∞ , respectively, where K _∞ is the cyclotomic l-extension of K and M _∞ the maximal abelian S-ramified l-extension of K _∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G _∞ ]]-module X _∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when lł[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l. Received: 7 August 2001 / Revised version: 6 May 2002 We acknowledge financial support provided by NSERC. Mathematics Subject Classification (2000): 11R23, 11R27, 11R32, 11R33, 11R37, 11R42, 11S20, 11S23, 11S31, 11S40     

Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space (H,á·,·?){(\mathcal H,\langle\cdot,\cdot\rangle)}, and interval L ì (0,+￥){\Lambda\subset(0,+\infty)} and a map K ? C²(H,\mathbb R){K\in C^2(\mathcal H,\mathbb R)} whose gradient is a compact mapping, we consider the family of functionals of the type:

Dirac equation in a 5-dimensional Kaluza-Klein theory

Dirac equation is discussed in 5-dimensional space time having topology M ⁴ ×T ¹, whereM ⁴ and T ¹ both are curved. It is shown that 4-dimensional fermion can be obtained from 5-dimensional fermion, as a result of compactification of extra dimension. It is found that the realistic 4-dimensional fermions are possible in higher modes earlier than those in lower modes during the course of expansion of 4-dimensional universe. 4-dimensional Dirac equation, obtained from 5-dimensional Dirac equation after compactification, is solved for an arbitrary mode for superheavy as well as light (realistic) fermions. Time-dependence of polarization vector and magnetization density, as a result of Gordon-decomposition of the current vector for 4-dimensional spin-½ field (with arbitrary mode), is exhibited.     

On a functional equation arising in information theory

We derive the representation of generalized additive measures of information, depending on two probability distributions and having the sum property with a regular generating function. This family of measures includes the Shannon entropy, the entropy of degree (,), the generalized information energy, the sine entropy and the generalized directed divergence.