Parametric duality on minimax programming involving generalized convexity in complex space

In this paper we employ generalized convexity of complex functions to establish several sufficient optimality conditions for minimax programming in complex spaces. Using such criteria, we constitute a parametrical dual, and establish the weak, strong, and strict converse duality theorems in the framework.     

Splitting for rare event simulation: A large deviation approach to design and analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set B$B$ before another set A$A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.     

Optimal extended optical flow subject to a statistical constraint

This work is motivated by the necessity to improve heart image tracking. This technique is related to the ability of generating an apparent continuous motion, which is observable through the variation of intensity from a starting image to an ending one. Given two images ρ₀ and ρ₁, we calculate an evolution process ρ(t,⋅) which transports ρ₀ to ρ₁ by using the optimal extended optical flow. Such a strategy is found to be well suited for heart image tracking, provided the motion is controlled by a statistical model. In this paper we use viability theory to give sufficient conditions to handle the optimal extended optical flow subject to a point-wise statistical constraint by using Parzen’s approximation. The strategy is implemented in a 1D case and the numerical results which are presented show the efficiency of the proposed strategy.     

Quantum families of maps and quantum semigroups on finite quantum spaces

Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.     

Abelian gauge theories on homogeneous spaces

An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell-Chern-Simons theory on S ³ is calculated. A generalization of the Chern-Simons action is suggested and analyzed with the example of SU(3)/U(1) X U(1).     

Fuzzy measures and discreteness

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the σ-algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG 8543g13263t32/xxlarge8834.gif" alt="sub" align="MIDDLE" BORDER="0"> 8543g13263t32/xxlarge8477.gif" alt="Ropf" align="BASELINE" BORDER="0"> ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: G8543g13263t32/xxlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0">C(S ^n–1) and a kernelK: ((G × G 8543g13263t32/xxlarge8726.gif" alt="setmn" align="MIDDLE" BORDER="0"> }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with 8543g13263t32/xxlarge8466.gif" alt="Lscr" align="MIDDLE" BORDER="0">(E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Solutions and multiple solutions for problems with the <Emphasis Type="Italic">p</Emphasis>-Laplacian

In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ₂ = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory. Second author is Corresponding author.