Generalized bi-circular projections

Recall that a projection P on a complex Banach space X is a generalized bi-circular projection if P+λ(I−P) is a (surjective) isometry for some λ such that |λ|=1 and λ≠1. It is easy to see that every hermitian projection is generalized bi-circular. A generalized bi-circular projection is said to be nontrivial if it is not hermitian. Botelho and Jamison showed that a projection P on C([0,1]) is a nontrivial generalized bi-circular projection if and only if P−(I−P) is a surjective isometry. In this article, we prove that if P is a projection such that P+λ(I−P) is a (surjective) isometry for some λ, then either P is hermitian or λ is an nth unit root of unity. We also show that for any nth unit root λ of unity, there are a complex Banach space X and a nontrivial generalized bi-circular projection P on X such that P+λ(I−P) is an isometry.     

Simultaneous rayleigh-quotient minimization methods for Ax=λBx

New simultaneous iteration techniques are developed for solving the generalized eigenproblem Ax=λBx, where A and B are real symmetric matrices and B is positive definite. The approach is to minimize the generalized Rayleigh quotient in some sense over several independent vectors simultaneously. In particular, each new vector iterate is formed from a linear combination of current iterates and correction vectors that are derived from either gradient or conjugate-gradient techniques. A Ritz projection or simultaneous iteration process is used to accelerate convergence. For one of the gradient versions, convergence and asymptotic rates of convergence are established. Also, some numerical experiments are reported that demonstrate the convergence behavior of these methods.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

On robust classification using projection depth

This article uses projection depth (PD) for robust classification of multivariate data. Here we consider two types of classifiers, namely, the maximum depth classifier and the modified depth-based classifier. The latter involves kernel density estimation, where one needs to choose the associated scale of smoothing. We consider both the single scale and the multi-scale versions of kernel density estimation, and investigate the large sample properties of the resulting classifiers under appropriate regularity conditions. Some simulated and real data sets are analyzed to evaluate the finite sample performance of these classification tools.     

On operations induced by hereditary classes on generalized topological spaces

We introduce the operator $\gamma_{\mu}^{*}$ given by a hereditary class and a generalized topology?μ, and investigate its properties. With the help of? $\gamma_{\mu}^{*}$ , we define $\gamma_{\mu}^{*}$ -semi-open sets which are generalized open sets on generalized topologies and study the relation between such sets and some generalized open sets (e.g. μ-semi-open sets, μ-β-open sets) on generalized topologies.     

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.     

Some refined bounds for the perturbation of the orthogonal projection and the generalized inverse

In this paper, we consider the perturbation of the orthogonal projection and the generalized inverse for an n × n matrix A and present some perturbation bounds for the orthogonal projections on the rang spaces of A and A^?, respectively. A combined bound for the orthogonal projection on the rang spaces of A and A^? is also given. The proposed bounds are sharper than the existing ones. From the combined bounds of the orthogonal projection on the rang spaces of A and A^?, we derived new perturbation bounds for the generalized inverse, which always improve the existing ones. The combined perturbation bound for the orthogonal projection and the generalized inverse is also given. Some numerical examples are given to show the advantage of the new bounds.     

On amoebas of algebraic sets of higher codimension

The amoeba of a complex algebraic set is its image under the projection onto the real subspace in the logarithmic scale. We study the homological properties of the complements of amoebas for sets of codimension higher than 1. In particular, we refine A. Henriques’ result saying that the complement of the amoeba of a codimension k set is (k ? 1)-convex. We also describe the relationship between the critical points of the logarithmic projection and the logarithmic Gauss map of algebraic sets.     

Behavior of different ansätze in the generalized projection operator method

Using the recently reported generalized projection operator method for the nonlinear Schrödinger equation, we derive the generalized pulse parameters equations for ansätze like hyperbolic secant and raised cosine functions. In general, each choice of the phase factor θ in the projection operator gives a different set of ordinary differential equations. For θ = 0 or θ = π/2, the corresponding projection operator scheme is equivalent to the Lagrangian variation method or the bare approximation of the collective variable theory. We prove that because of the inherent symmetric property between the pulse parameters of a Gaussian ansätz results the same set of pulse parameters equations for any value of the generalized projection operator parameter θ. Finally we prove that after the substitution of the ansätze function, the Lagrange function simplifies to the same functional form irrespective of the ansätze used because of a special property shared by all the anätze chosen in this work.     

Metric projection onto finite-dimensional subspaces of C and L

In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or x(t)≡0 for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space L[a, b] of real integrable functions is not uniformly continuous.     

On intutionistic fuzzy gpr-closed sets

In this paper, a new class of intuitionistic fuzzy closed sets called intuitionistic fuzzy generalized preregular closed sets (briefly intuitionistic fuzzy gpr-closed sets) and intuitionistic fuzzy generalized preregular open sets (briefly intuitionistic fuzzy gpr-open sets) are introduced and their properties are studied. Further the notion of intuitionistic fuzzy preregular T _1/2-spaces and intuitionistic fuzzy generalized preregular continuity (briefly intuitionistic fuzzy gpr-continuity) are introduced and studied.     

Supervised box clustering

In this work we address a technique for effectively clustering points in specific convex sets, called homogeneous boxes, having sides aligned with the coordinate axes (isothetic condition). The proposed clustering approach is based on homogeneity conditions, not according to some distance measure, and, even if it was originally developed in the context of the logical analysis of data, it is now placed inside the framework of Supervised clustering. First, we introduce the basic concepts in box geometry; then, we consider a generalized clustering algorithm based on a class of graphs, called incompatibility graphs. For supervised classification problems, we consider classifiers based on box sets, and compare the overall performances to the accuracy levels of competing methods for a wide range of real data sets. The results show that the proposed method performs comparably with other supervised learning methods in terms of accuracy.     

Modified hybrid projection methods for finding common solutions to variational inequality problems

In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.     

On generalized cluster sets of functions and multifunctions

This paper introduces and studies generalized cluster sets (g-cluster sets) of functions and multifunctions on GTS, which unifies the existing notions of cluster sets, θ-cluster sets, δ-cluster sets, S-cluster sets, s-cluster sets, p-cluster sets and many more. Several properties of the functions and multifunctions as well as their range and domain spaces are observed via degeneracies of their g-cluster sets. Characterizations of g-cluster sets through filterbases and grills on a typical class of GTS’s are also obtained. Moreover, μ-compactness of a GTS is characterized through g-cluster sets of multifunctions.     

Error estimates for kernel and projection methods of recovering the orientation distribution function on <Emphasis Type="Italic">SO</Emphasis>(3)

The orientation density function is recovered from a sample of orientations on the rotation group SO(3) of the three-dimensional Euclidean space. Sufficient conditions for the consistency of kernel and projection estimates in L ₂, L ₁, and C are considered. Numerical results concerning the error estimation of projection methods over the basis of generalized spherical functions are given for normal distributions on SO(3).     

Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s₁+s₂,s_i∈S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g?3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Kolountzakis’ idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo, Ruzsa and Trujillo, Jia, and Habsieger and Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.     

Extensions of barrier sets to nonzero roots of the matching polynomial

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to their connection to maximum matchings in a graph. For a root θ of the matching polynomial, we define θ-barrier and θ-extreme sets. We prove a generalized Berge-Tutte formula and give a characterization for the set of all θ-special vertices in a graph.     

On convex bodies of constant width

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.