Extended Taylor series and interpolation of physically meaningful functions

An extension of the concept of the Taylor series to arbitrary functions that are physically meaningful is presented. The series is obtained using a matrix to describe the behaviour of the function at any position of discontinuity while performing the Taylor series expansion in the rest of the considered domain. The matrix is derived from the continuity conditions that are inherent to a particular physical problem. This allows an introduction of function classes that are relevant from the physical problem point of view. Several illustrative examples of physically meaningful function classes are derived. It is then demonstrated that using the derived function classes and the concept of the extended Taylor series one can obtain the interpolation formulae fitted for a particular class of physically meaningful functions. The application of the interpolation formulae based on these novel concepts shows clear advantages over the standard approach. Finally, it is also shown that the developed concept of the extended Taylor series and classes of physically meaningful functions can be used for the extrapolation and an elegant derivation of finite difference approximations for physical problems.     

MicroRNA Delivery by Graphene-Based Complexes into Glioblastoma Cells

Glioblastoma (GBM) is the most common primary and aggressive tumour in brain cancer. Novel therapies, despite achievements in chemotherapy, radiation and surgical techniques, are needed to improve the treatment of GBM tumours and extend patients’ survival. Gene delivery therapy mostly uses the viral vector, which causes serious adverse events in gene therapy. Graphene-based complexes can reduce the potential side effect of viral carries, with high efficiency of microRNA (miRNA) or antisense miRNA delivery to GBM cells. The objective of this study was to use graphene-based complexes to induce deregulation of miRNA level in GBM cancer cells and to regulate the selected gene expression involved in apoptosis. The complexes were characterised by Fourier transform infrared spectroscopy (FTIR), scanning transmission electron microscopy and zeta potential. The efficiency of miRNA delivery to the cancer cells was analysed by flow cytometry. The effect of the anticancer activity of graphene-based complexes functionalised by the miRNA sequence was analysed using 2,3-bis-(2-methoxy-4-nitro-5-sulfophenyl)-2H-tetrazolium-5-carboxyanilide salt (XTT) assays at the gene expression level. The results partly explain the mechanisms of miRNA deregulation stress, which is affected by graphene-based complexes together with the forced transport of mimic miR-124, miR-137 and antisense miR-21, -221 and -222 as an anticancer supportive therapy.     

Asymptotically commuting families of operators

We study families of symmetric operators (Q _n) with domains given by the range of self-adjoint contraction semigroups (e ^−tHn ). Assuming the asymptotic commutativity, lim _n [Q _n, e^−tHn]=0, and certain other estimates, we establish the existence and properties of a limiting self-adjoint operatorQ=lim _n Q _n. We apply these results to the study of an elementary supersymmetry algebra. Supported in part by the National Science Foundation under Grant DMS/PHY 88-16214.     

Inductive Detection for Homotopy Equivalences of Manifolds

Let f:M N be a homotopy equivalence of CAT manifolds M and N (CAT := PL, TOP or DIFF) with finite fundamental groups. Each subgroup H 1(M) determines a homotopy equivalence fH:MH NH of the corresponding covering spaces. Suppose now that for each subgroup H in some particular class C (for example: elementary, hyperelementary or solvable) fH is homotopic to a CAT isomorphism. The general problem studied in this paper can be formulated as follows: If each map fH as above is homotopic to a CAT isomorphism, under what additional conditions on M, C and CAT is f itself (or f × idR) homotopic (properly homotopic) to a CATisomorphism?     

Synthesis of dinucleoside (N3'-->MeP5') methanephosphonamidates

[structure: see text] Three different approaches were used for the synthesis of dinucleoside methanephosphonamidates [3'-NH-P(O)(CH3)O-5'], starting from dichloromethylphosphine or dichloromethanephosphonate as the phosphorus-containing moiety. 5'-DMT-3'-amino-3'-deoxythymidine and N(4)-benzoyl-5'-DMT-3'-amino-2',3'-dideoxycytidine were used as the aminonucleoside precursors and the respective 3'-protected nucleosides (thymidine or N(4)-benzoyl-2'-deoxycytidine) as the 5'-hydroxyl reagents.     

Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. II. Basic gauge-invariant variables with applications to de sitter space-time

In a companion paper, a systematic treatment of linearized perturbations and a new geometric definition of gauge-invariant variables, based on the theory of vector bundles and applicable to the case of an arbitrary system of covariant field equations, were carefully presented. One of the purposes of the present paper is to specify a necessary and sufficient condition that a given, finite set of gaugeinvariant variables, denoted collectively by ω and referred to as the complete set of basic variables, can be used to extract the equivalence classes of perturbations from ω in a unique way. The above set is complete because it has the following property: a knowledge of ω is all one needs in the sense that ifx represents an arbitrary point of the “space-time” manifoldX andG denotes any gauge-invariant tensor field onX, then the value ofG atx∈X is uniquely specified by giving the germs of basic gauge-invariant variables atx∈X. Arguments are proposed that ω also has a stronger property which is more immediately useful: anyG is obtainable directly from the basic variables through purely algebraic and differential operations. These results are of practical interest, and one concrete setting where one is led to the explicit definition of ω occurs when considering the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time and obeying the linearized empty-space Einstein equations with nonnegative cosmological constant Λ; the case Λ=0 corresponds to linear perturbation theory in Minkowski space-time.     

Two ideals

We find two ideals on neither of which is whose quotient Boolean algebras are homogeneous but nonisomorphic. This solves a problem of Just and Krawczyk (1984).