We will study the Henstock–Kurzweil delta and nabla integrals, which generalize the Henstock–Kurzweil integral. Many properties of these integrals will be obtained. These results will enable time scale researchers to study more general dynamic equations. The Hensock–Kurzweil delta (nabla) integral contains the Riemann delta (nabla) and Lebesque delta (nabla) integrals as special cases.

the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  wea...

In this paper we derive the analogue of the Lebesque-Radon-Nikodym theorem with respect to fermionic p-adic invariant measure on Zp. 2010 Mathematics Subject Classification : 11S80, 48B22, 28B99

In this paper we study the multiplicative, tensor, Sobolev’s and convolution inequalities in certain Banach spaces, the so-called Bide Side Grand Lebesque Spaces, and give examples to show their sharpness. 2000 Mathematics Subject Classification. Primary 37B30, 33K55; Secondary 34A34, 65M20, 42B25.

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise-interaction. Specifically, we focus on systems with non-negative total energy. We show that on the zero energy level set, most of the orbits lead to binary escape configurations and the set of initial conditions leading to escape configurations where al...

In this paper, we introduce the notions of [V,λ ]-summability and λ−statistical convergence of function by taking a nonnegative real-valued Lebesque measurable function in the interval (1,∞) and introduce new notions, namely, [V,λ ] (I )− summability, and Iλ−statistical convergence of function. We mainly examine the relation between these two new methods.

Given a one dimensional SOM with a monotonically decreasing neighbourhood and an input distribution which is not Lebesque continuous, a set of su cient conditions and a Theorem are stated which ensure probability one organisation of the neuron weigh ts. This leads to a rule for choosing the number of neurons and width of the neighbourhood to improve the c hances of reaching an organised state i...

In the following, 5 will denote the boundary of the unit ball in En, and u a variable point of S (so re is a unit vector, or "direction"). The polar equation of the boundary of DK is given by p = /»(«), uES, so piu) is the radius of DK in the direction u. Then p(w) is the maximum length of a chord of K having direction u—the length of a "diameter" of K having direction u. Let p. denote w-dimens...

We (i.e. I) present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is the ideal generated by a Suslin ccc forcing (e.g. Lebesque– null or meager), and P is a Suslin forcing, and P is I–preserving (i.e. it doesn’t make any positive Borel–set small), then P preserves generics over candidates and therefore is strongly I–preserving (i.e. doesn’t make any positi...