Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

Mittag–Leffler Functions and the Truncated $${\mathcal {}}$$-fractional Derivative

In this paper, we introduce a new type of fractional derivative, which we called truncated \({\mathcal {V}}\)-fractional derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated \({\mathcal {V}}\)-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the \({\mathcal {V}}\)-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-fractional derivative and the truncated \({\mathcal {V}}\)-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0<\alpha <1\)).     

John Playfair on British decline in mathematics

This article considers the role John Playfair (1748–1819) played in creating and popularizing the myth that mathematical development halted in Great Britain in the eighteenth century due to mathematicians' irrational attachment to a geometrical approach to the calculus. By the turn of the nineteenth century, Playfair had established his reputation as an energetic teacher, gifted expositor, and skilled natural philosopher. He served as joint professor of mathematics at the University of Edinburgh and as general secretary of the Royal Society of Edinburgh, editing the Society's Transactions, while his written accomplishments included Elements of geometry (1795) and Illustrations of the Huttonian theory of the earth (1802). He then contributed his talents to the opinionated journal, Edinburgh Review, where his assessments of the contemporary state of mathematics reached a wide audience of intellectuals, gentlemen, and merchants, albeit anonymously. The article expands upon a section of a talk delivered to the Fourth Joint Meeting of the BSHM and CSHPM in Montreal, 27–29 July 2007 (see also Ackerberg-Hastings 2007 Ackerberg-Hastings, A. 2007. “‘Euler and the enlightenment mathematicians: a Scottish perspective’”. In Proceedings of the Canadian society for history and philosophy of mathematics Edited by: Cupillari, A. Vol. 20, 12–23.  [Google Scholar]).     

Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let ?? be a bounded open subset of ${\mathbb{G}}$ , where ${\mathbb{G}}$ is a Carnot group, and let ${u: \Omega \rightarrow \mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(\Omega,\mathbb{R}^d)}$ , with p?>?1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.     

Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane \({\prod}\) are considered. By using a special Möbius transform, an equivalent relation between \({\widehat{H}^\mu}\) class functions over \({\prod}\) and \({H^\mu}\) class functions over the unit sphere is shown. For \({\widehat{H}^\mu}\) class functions over \({\prod}\) , we prove the existence of Cauchy type integral and singular integral over \({\prod}\) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over \({\prod}\) and unit sphere is built.     

Homogeneous $$$$-difference euations and generating functions for $$$$-hypergeometric polynomials

In this paper we show how to deduce several types of generating functions for \(q\)-hypergeometric polynomials by the method of homogeneous \(q\)-difference equations. In addition, we build relations between transformation formulas and homogeneous \(q\)-difference equations. Moreover, we generalize the Andrews–Askey integral from the perspective of \(q\)-integrals by the method of homogeneous \(q\)-difference equations.     

Operational Calculus Approach to Nonlocal Cauchy Problems

Let Φ be a linear functional of the space ${\mathcal{C} =\mathcal{C}(\Delta)}$ of continuous functions on an interval Δ. The nonlocal boundary problem for an arbitrary linear differential equation $$ P\left(\frac{d}{d t}\right)y = F(t) $$ with constant coefficients and boundary value conditions of the form $$ \Phi\{\,y^{(k)}\} =\alpha_k,\,\,\,k = 0,\,1,\,2,\, \ldots,\,{\rm deg} P-1 $$ is said to be a nonlocal Cauchy boundary value problem. For solution of such problems an operational calculus of Mikusiński’s type, based on the convolution $$ (f*g)(t) = \Phi_\tau\, \left\{{\int\limits_\tau^t} f(t+\tau - \sigma)\,g(\sigma)\, d \sigma\, \right\}, $$ is developed. In the frames of this operational calculus the classical Heaviside algorithm is extended to nonlocal Cauchy problems. The obtaining of periodic, antiperiodic and mean-periodic solutions of linear ordinary differential equations with constant coefficients both in the non-resonance and in the resonance cases reduces to such problems. Here only the non-resonance case is considered. Extensions of the Duhamel principle are proposed.     

Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System

In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data \({\{(x_n,y_n):n\geq 2\}}\) with countable iterated function system, where \({x_n}\) is a monotone and bounded sequence, \({y_n}\) is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.     

Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities

We show that if ${{\mathcal A} \subset \mathbb{R}^N}$ is an annulus or a ball centered at zero, the homogeneous Neumann problem on ${{\mathcal A}}$ for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and ${\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}$ The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.     

Two Meromorphic Functions Share Some Pairs of Small Functions

In this article, we prove that two meromorphic functions $f$ and $g$ must be linked by a quasi-Möbius transformation if they share two pairs of small functions ignoring multiplicities and share other four small functions with multiplicities truncated by 2. We also show that two meromorphic functions which share seven pairs of small functions ignoring multiplicities are linked by a quasi-Möbius transformation.     

Rate of convergence of Fourier series on the classes of \bar \Psi -integrals-integrals

We introduce the notion of $\bar \Psi $ -integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\bar \Psi } $ . We obtain integral representations of deviations of the trigonometric polynomials U _n(f;x;Λ) generated by a given Λ-method for summing the Fourier series of functions $f{\text{ }}\varepsilon {\text{ }}L^{\bar \Psi } $ . On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\bar \Psi } $ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\bar \Psi } $ , which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.     

Generators for Rings of Compactly Supported Distributions

Let \({{\tt C}}\) denote a closed convex cone in \({\mathbb R^d}\) with apex at 0. We denote by \({\mathcal E'({\tt C})}\) the set of distributions on \({\mathbb R^d}\) having compact support contained in \({{\tt C}}\). Then \({\mathcal E'({\tt C})}\) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on \({\widehat{f}_1,\dots, \widehat{f}_n}\) for \({f_1,\dots ,f_n \in \mathcal E'({\tt C})}\) to generate the ring \({\mathcal E'({\tt C})}\). (Here \({\widehat{\;\cdot\;}}\) denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.     

A New Characterization of Gundersen’s Example of Two Meromorphic Functions Sharing Four Values

Let ? and g be nonconstant meromophic functions sharing four values IM and satisfying ?^?1({a}) ? ^?1({b}) for two values a, b not shared by ? and g. Then either ? = T o g with a Möbius transformation T or \(f=L\ {\rm o}\ \hat{f}\ {\rm o}\ h\) and \(g=L\ {\rm o}\ \hat{g}\ {\rm o}\ h\) , where \(\hat{f}(z)=({\rm exp}\ z+1)/({\rm exp}\ z-1)^2\) and \(\hat{g}(z)=({\rm exp}\ z+1)^2/(8({\rm exp}\ z-1))\) are the functions in Gundersen’s example [1], L is a Möbius transformation and h is an entire function.     

Arithmetic of pall orders and representation of integers by some ternary quadratic forms

The discrete ergodic method (Yu. V. Linnik, Izv. Akad. Nauk SSSR, Ser. Mat.,4, 363–402 (1940); A. V. Malyshev, Tr. Mat. Inst. Akad. Nauk SSSR,65) is applied to the study of properties of integral points on the ellipsoids $$\sum\nolimits_{g,m} : g(x) = m,x = (x_1 ,x_2 ,x_3 ), g(x) = \bar f(Cx),$$ where \(\bar f\) is the adjoint of one of the 39 quadratic forms of Pall (Trans. Am. Math. Soc,59, 280–332 (1946);C is an integral matrix,¦detC¦?1. We construct a flow of integral points on the genus surface of the ellipsoidsg(x)=m. The ergodicity of this flow and a mixing theorem are proved. We obtain an asymptotic formula for the number of representations ofm belonging to a given domain on the ellipsoid and lying in a given residue class.     

The analogue of the law of large numbers for additive functions on sparse sets

An analog of the Turan'n-Kubilyus inequality is proved for a sufficiently wide class of sequences which contains, in particular,a _n=f (n) and a_n=f (p_n), wheref (n) is a polynomial with integral coefficients. This result helps us to obtain integral limit theorems for additive functions on the class of sequences under investigation.     

On a Stochastic Fourier Coefficient: Case of Noncausal Functions

Given a random function $f(t,\omega )$ and an orthonormal basis $\{\varphi _n \}$ in $L^2(0,1),$ we are concerned with the basic question whether the function can be reconstructed from the complete set of its stochastic Fourier coefficients $\{{\hat{f}}_n(\omega )\}$ which are defined by the following stochastic integral with respect to the Brownian motion $W.$ : ${\hat{f}}_n(\omega ):=\int _0^1 f(t,\omega ) \overline{\varphi _n(t)}{\text{ d}}_*W_t$ , where the symbol $\int {\text{ d}}_*W_t$ stands for the stochastic integral of noncausal type. In an earlier article (Stochastics, doi: 10.1080/17442508.2011.651621, 2012), Ogawa studied the question in the limited framework of homogeneous chaos and gave some affirmative answers when the random functions are causal and square integrable Wiener functionals for which the Itô integral is used for the definition of the stochastic Fourier coefficient. In this note, we aim to extend those results to the more general case where the functions are free from the causality restriction and the Skorokhod integral is employed instead of the Itô integral.     

On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift

Second order elliptic operators and their parabolic counterparts are studied in the case of complete boundary degeneration of the leading order coefficients in the presence of a strong outward pointing drift. It is shown that the problem generates a positive analytic \(C_0\)-semigroup with maximal \(L_p\)-regularity in \(L_q\)-spaces. This result is based on hard analysis estimates for integral operators in combination with modern functional analytic tools like \({\mathcal R}\)-boundedness and the operator-valued functional calculus.     

<Emphasis Type="Italic">k</Emphasis>-Monogenic Functions Over 6-Dimensional Euclidean Space

Recently, physicists are interested in 6-dimensional physics including the massless field operators on Lorentzian space \(\mathbb R^{5,1}\). The elliptic version \(\mathcal {D}_{k}\) of these operators coincides with the higher spin massless field operators on \(\mathbb R^{6}\) introduced by Sou?ek earlier. The embedding of \(\mathbb R^{6}\) into the space of complex antisymmetric matrices allows us to use two-component notation, generating the Penrose two-spinor notation for dimension 4, which makes the spinor calculus on \(\mathbb R^6\) more concrete and explicit. A function annihilated by \(\mathcal {D}_{k}\) is called k-monogenic. Applying the Penrose integral formula, which can be checked by direct differentiation, we give infinite number of such k-monogenic polynomials for fixed k. So the function theory of k-monogenic functions is abundant and interesting.