Indian Journal of Pure and Applied Mathematics最新文献

The crux of the present paper is approximation of bivariate Lipschitz continuous functions by hidden variable fractal functions. We propose the construction of hidden variable fractal perturbation associated with a given bivariate Lipschitz continuous function defined on a rectangle (mathcal {D}) in the Euclidean space. This procedure yields a fractal operator on the space of all (mathbb {R}^2)-valued Lipschitz continuous functions defined on a rectangle (mathcal {D}). Some basic and important properties of this fractal operator will be discussed. Subsequently, we extend this fractal operator to the norm preserving bounded linear operator on the the space of all (mathbb {R}^2)-valued continuous functions defined on a rectangle (mathcal {D}). We investigate the stability of hidden variable fractal functions with respect to a perturbation in the scaling factors. Finally, existence of optimal hidden variable fractal function which approximates the given bivariate Lipschitz continuous function is discussed.

This article provides a numerical study of two-dimensional Volterra integro-differential equations involving fractional derivatives in the Caputo sense of order ( alpha ,gamma ) ( (0< alpha ,gamma <1). ) First, we establish a sufficient condition for the existence and uniqueness of the solution using the Banach fixed point theorem. Due to the limitation of finding the exact analytical solution, we derive and analyze an efficient numerical scheme to approximate the solution. The proposed scheme uses the L1 technique to discretize the differential components, whereas a composite trapezoidal rule is used to approximate the double integral. The convergence analysis and error estimation are carried out. It is shown that the proposed scheme converges with an optimal convergence rate of ( min {2-alpha ,2-gamma } ) for sufficiently smooth initial data. In addition, we apply the proposed difference scheme to solve the semilinear problem. The well-known Newton’s linearization technique is used to deal with semilinearity. Finally, a couple of numerical experiments are conducted to support our theoretical findings and validate the proposed scheme.

Our purpose of this article is to study nonexistence of positive super solutions for Lane-Emden system involving inverse-square potentials

where (p,q>0), (mu _1,mu _2ge -N^2/4), (Omega ) is a bounded smooth domain in (mathbb {R}^N) with (Nge 3) such that (0in partial Omega ) and (B^+_2(0):={x=(x',x_N)in mathbb {R}^{N-1}times mathbb {R}: x_N>0,, |x|<2}subset Omega ). Sharp critical curves of (q, p) are derived for nonexistence of positive super solutions to system (0.1) in the case that (-N^2/4le mu _1,mu _2<1-N) and (-N^2/4le mu _1<1-Nle mu _2). Our method is to iterate an initial singularities at the origin to improve the blowing-up rate until the nonlinearities are not admissible in some weighted (L^1) space.

Recently, a general version of the Hoffmann-Jørgensen inequality was shown jointly with Rajaratnam [Ann. Probab. 2017], which (a) improved the result even for real-valued variables, but also (b) simultaneously unified and extended several versions in the Banach space literature, including that by Hitczenko–Montgomery-Smith [Ann. Probab. 2001], as well as special cases and variants of results by Johnson–Schechtman [Ann. Probab. 1989] and Klass–Nowicki [Ann. Probab. 2000], in addition to the original versions by Kahane and Hoffmann-Jørgensen. Moreover, our result with Rajaratnam was in a primitive framework: over all semigroups with a bi-invariant metric; this includes Banach spaces as well as compact and abelian Lie groups. In this note we show the result even more generally: over every semigroup ({mathscr {G}}) with a strongly left- (or right-)invariant metric. We also prove some applications of this inequality over such ({mathscr {G}}), extending Banach space-valued versions by Hitczenko and Montgomery-Smith [Ann. Probab. 2001] and by Hoffmann-Jørgensen [Studia Math. 1974]. Furthermore, we show several other stochastic inequalities – by Ottaviani–Skorohod, Mogul’skii, and Lévy–Ottaviani – as well as Lévy’s equivalence, again over ({mathscr {G}}) as above. This setting of generality for ({mathscr {G}}) subsumes not only semigroups with bi-invariant metric (thus extending the previously shown results), but it also means that these results now hold over all Lie groups (equipped with a left-invariant Riemannian metric). We also explain why this primitive setting of strongly left/right-invariant metric semigroups ({mathscr {G}}) is equivalent to that of left/right-invariant metric monoids ({mathscr {G}}_circ ): each such ({mathscr {G}}) embeds in some ({mathscr {G}}_circ ).

In this work, a high accurate method is given for solving the nonlinear fractional delay integro-differential equations, numerically. By considering the equation before and after delay time, we first apply the delay function in the equation and propose an equivalent system. By discretization in the Jacobi-Gauss collocation points, an algebraic nonlinear system is then proposed to approximate the solution of main equation. The convergence of method is fully given in spaces (L^{infty }_{omega ^{alpha ,beta }}(I)) and (L^{2}_{omega ^{alpha ,beta }}(I)), and the error bounds are specified for obtained approximations. Finally, some numerical examples are provided to show the capability and efficiency of method.

We use the crystallised (C^*)-algebra (C(SU_{q}(2))) at (q=0) to obtain a unitary that gives an approximate equivalence involving the GNS representation on the (L^{2}) space of the Haar state of the quantum SU(2) group and the direct integral of all the infinite dimensional irreducible representations of the (C^{*})-algebra (C(SU_{q}(2))) for nonzero values of the parameter q. This approximate equivalence gives a KK class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group (widehat{SU_q(2)}) with coefficients in a (C^*)-algebra in the sense of Mishchenko.

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting d- tuples of homogeneous normal operators. The Hahn–Hellinger theorem gives a canonical decomposition of a (*)- algebra representation (rho ) of (C_0({mathbb {S}})) (where ({mathbb {S}}) is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on ({mathbb {S}}) and is adapted to the (*)- representation (rho ) via a unitary representation U of the group G, in other words, if there is an imprimitivity, then the Hahn–Hellinger decomposition reduces to just one component, and the group representation U becomes an induced representation, which is Mackey’s imprimitivity theorem. We consider the case where a compact topological space (Ssubset {mathbb {C}}^d) decomposes into finitely many G- orbits. In such cases, the imprimitivity based on S admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of G- orbits.

This work is concerned with the properties of the ridge regression where the number of predictors p is proportional to the sample size n. Asymptotic properties of the means square error (MSE) of the estimated mean vector using ridge regression is investigated when the design matrix X may be non-random or random. Approximate asymptotic expression of the MSE is derived under fairly general conditions on the decay rate of the eigenvalues of (X^{T}X) when the design matrix is nonrandom. The value of the optimal MSE provides conditions under which the ridge regression is a suitable method for estimating the mean vector. In the random design case, similar results are obtained when the eigenvalues of (E[X^{T}X]) satisfy a similar decay condition as in the non-random case.

Marc Rieffel had introduced the notion of the quantum Gromov–Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on 2-sphere in this distance. One finds applications of similar approximations in many places in the theoretical physics literature. In this paper, we have defined a compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman spaces and have proved that the sequence converges to the space of continuous functions on odd spheres in the quantum Gromov–Hausdorff distance.

Let f, p, and q be Laurent polynomials with integer coefficients in one or several variables, and suppose that f divides (p+q). We establish sufficient conditions to guarantee that f individually divides p and q. These conditions involve a bound on coefficients, a separation between the supports of p and q, and, surprisingly, a requirement on the complex variety of f called atorality satisfied by many but not all polynomials. Our proof involves a related dynamical system and the fundamental dynamical notion of homoclinic point. Without the atorality assumption our methods fail, and it is unknown whether our results hold without this assumption. We use this to establish exponential recurrence of the related dynamical system, and conclude with some remarks and open problems.