On the geometrized Skyrme and Faddeev models

The higher power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to _σ2${\sigma }_2$-energy, introduced by Eells and Sampson (1964) [1]. This paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined.     

An algorithmic approach to Ramanujan’s congruences

In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?_d|M?_n=1^￥(1-q^dn)^r_d=?_n=0^￥a(n)qⁿ\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).     

Density, Overcompleteness, and Localization of Frames. I. Theory

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames and ( a discrete abelian group), relating the decay of the expansion of the elements of in terms of the elements of via a map . A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of , the relative measure of — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set in . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.     

On extension results for n-cyclically monotone operators in reflexive Banach spaces

In this paper we provide some extension results for n-cyclically monotone operators in reflexive Banach spaces by making use of the Fenchel duality. In this way we give a positive answer to a question posed by Bauschke and Wang (2007) [4].     

An almost periodic noncommutative Wiener's Lemma

We develop a theory of almost periodic elements in Banach algebras and present an abstract version of a noncommutative Wiener's Lemma. The theory can be used, for example, to derive some of the recently obtained results in time-frequency analysis such as the spectral properties of the finite linear combinations of time-frequency shifts.     

On totally Fenchel unstable functions in finite dimensional spaces

We give an answer to the Problem 11.6 posed by Stephen Simons in his book “From Hahn-Banach to Monotonicity”: Do there exist a nonzero finite dimensional Banach space and a pair of extended real-valued, proper and convex functions which is totally Fenchel unstable? The answer is negative.     

General Voronovskaya and Asymptotic Theorems in Simultaneous Approximation

In the present note we prove general asymptotic and Voronovskaya theorems for simultaneous approximation. These generalize the Voronovskaya type theorems obtained recently by Floater for the Bernstein operators, and previously by Heilmann and Müller for the Durrmeyer operators.     

Newton method for reactive solute transport with equilibrium sorption in porous media

We present a mass conservative numerical scheme for reactive solute transport in porous media. The transport is modeled by a convection-diffusion-reaction equation, including equilibrium sorption. The scheme is based on the mixed finite element method (MFEM), more precisely the lowest-order Raviart-Thomas elements and one-step Euler implicit. The underlying fluid flow is described by the Richards equation, a possibly degenerate parabolic equation, which is also discretized by MFEM. This work is a continuation of Radu et al. (2008) and Radu et al. (2009) [1] and [2] where the algorithmic aspects of the scheme and the analysis of the discretization method are presented, respectively. Here we consider the Newton method for solving the fully discrete nonlinear systems arising on each time step after discretization. The convergence of the scheme is analyzed. In the case when the solute undergoes equilibrium sorption (of Freundlich type), the problem becomes degenerate and a regularization step is necessary. We derive sufficient conditions for the quadratic convergence of the Newton scheme.     

Analysis of an Euler implicit‐mixed finite element scheme for reactive solute transport in porous media

In this article, we analyze an Euler implicit‐mixed finite element scheme for a porous media solute transport model. The transporting flux is not assumed given, but obtained by solving numerically the Richards equation, a model for subsurface fluid flow. We prove the convergence of the scheme by estimating the error in terms of the discretization parameters. In doing so we take into account the numerical error occurring in the approximation of the fluid flow. The article, is concluded by numerical experiments, which are in good agreement with the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The canonical projection associated with certain possibly infinite generalized iterated function systems as a fixed point

In this paper, influenced by the ideas from Mihail (Fixed Point Theory Appl 2015:15, 2015), we associate to every generalized iterated function system \(\mathcal {F}\) (of order m) an operator \(H_{\mathcal {F}}:\mathcal {C} ^{m}\rightarrow \mathcal {C}\), where \(\mathcal {C}\) stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of \(\mathcal {F}\)) for the operator \(H_{\mathcal {F}}\) to be continuous, contraction, \(\varphi \)-contraction, Meir–Keeler or contractive. We also give sufficient condition under which \(H_{\mathcal {F}}\) has a unique fixed point \(\pi _{0}\). Moreover, we prove that, under these circumstances, the closure of the imagine of \(\pi _{0}\) is the attractor of \(\mathcal {F}\) and that \(\pi _{0}\) is the canonical projection associated with \(\mathcal {F}\). In this way we give a partial answer to the open problem raised on the last paragraph of the above-mentioned Mihail’s paper.