Abstract: It is well known that the subgradient mapping associated with a lower semicontinuous function is maximal monotone if and only if the function is convex, but what characterization can be given for the case in which a subgradient mapping is only maximal monotone locally instead of globally' That question is answered here in terms of a condition more subtle than local convexity. Applications are made to the tilt stability of a local minimum and to the local execution of the proximal point algorithm in optimization. PubDate: 2019-03-21 DOI: 10.1007/s10013-019-00339-5
Abstract: We present a criterion for local surjectivity of mappings between graded Fréchet spaces in the spirit of a well-known criterion in Banach spaces. As applications, we get “hard inverse mapping theorem” in the flavor of Nash–Moser. The technology of proofs was strongly influenced by a recent paper of Ekeland. PubDate: 2019-03-20 DOI: 10.1007/s10013-019-00342-w
Abstract: The purpose of this paper is to obtain the existence of common fixed points of family of multivalued mappings satisfying generalized F-contraction conditions in ordered metric spaces. Some examples are presented to support the results proved herein. Our results generalize and extend various comparable results in the existing literature. PubDate: 2019-03-19 DOI: 10.1007/s10013-019-00341-x
Abstract: The present paper introduces approximate analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations (CSTFPDEs) by using a generalized conformable fractional partial differential transform (GCFPDT). The GCFPDT is a modified version of the conformable fractional partial differential transform introduced in our recent work and it can be used as an efficient alternative transform to find analytical solutions for nonlinear systems of CSTFPDEs available in the literature. The convergence and error estimation of the proposed GCFPDT are also considered. Moreover, approximate analytical solutions to nonlinear system of gas dynamic equations, nonlinear system of KdV equations, and nonlinear system of approximate long water wave equations in the sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the effectiveness and efficiency of the proposed GCFPDT. PubDate: 2019-03-16 DOI: 10.1007/s10013-019-00340-y
Abstract: In this paper, recent results of Khatibzadeh and Ranjbar (Vietnam J. Math. 44, 307–313, 2016) on △-convergence are discussed. In particular, two supplementary strong convergence theorems are derived. Moreover, an example is given to show that their results are not necessarily analogous to that of Maingé (J. Comput. Appl. Math. 219, 223–236, 2008) in the setting of CAT(0) spaces. PubDate: 2019-03-11 DOI: 10.1007/s10013-019-00338-6
Abstract: In 1969, Boyd and Wong proved a fixed point result for single-valued contractions, that was extended by Reich in 1972, to multifunctions with compact values. Reich then asked whether the result holds for multifunctions with closed, bounded values, which is known as Reich’s conjecture. Using a simple variational approach, we give in this note a partial positive answer to the conjecture, which contains the previous related contributions. PubDate: 2019-03-09 DOI: 10.1007/s10013-019-00334-w
Abstract: Let \(T=\left (\begin {array}{cc} R & V\\ W & S \end {array}\right )\) be the ring of a Morita context such that VW = R and WV = S. In this paper, we give necessary and sufficient conditions for T to be a G-Dedekind prime ring. PubDate: 2019-03-08 DOI: 10.1007/s10013-019-00335-9
Abstract: Let Ω be a pseudoconvex domain in \(\mathbb C^{n}\) satisfying an f -property for some function f. We show that the Bergman metric associated with Ω has the lower bound \(\tilde {g}(\delta _{\Omega }(z)^{-1})\) where δΩ(z) is the distance from z to the boundary ∂Ω and \(\tilde g\) is a specific function defined by f. This refines Khanh–Zampieri’s work in Khanh and Zampieri (Invent. Math. 188, 729–750, 2012) with weaker smoothness assumption of the boundary. PubDate: 2019-03-06 DOI: 10.1007/s10013-019-00337-7
Abstract: It is shown that if R is a right automorphism-invariant ring and satisfies ACC on right annihilators, then R is a semiprimary ring. By this useful fact, we study finiteness conditions which ensure an automorphism-invariant ring is quasi-Frobenius (QF). Thus, we prove, among other results, that: (1) R is QF if and only if R is right automorphism-invariant, right min-CS and satisfies ACC on right annihilators; (2) R is QF if and only if R is left Noetherian, right automorphism-invariant and every complement right ideal of R is a right annihilator; (3) If R is right CPA, right automorphism-invariant and every complement right ideal of R is a right annihilator, then R is QF. PubDate: 2019-03-02 DOI: 10.1007/s10013-019-00336-8
Abstract: Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is noncompact, constitute a prototype for the formation of singularities, the so-called bubbles, in geometric analysis. In theoretical physics, they arise from the nonlinear \(\sigma \) -model of quantum field theory. That model possesses a supersymmetric extension, coupling a harmonic map like field with a nonlinear spinor field. In the physical model, that spinor field is anticommuting. In this contribution, we analyze both a mathematical version with a commuting spinor field and the original supersymmetric version. Moreover, this model gives rise to a further field, a gravitino, that can be seen as the supersymmetric partner of a Riemann surface metric. Altogether, this leads to a beautiful combination of concepts from quantum field theory, structures from Riemannian geometry and Riemann surface theory, and methods of nonlinear geometric analysis. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0298-7
Abstract: We examine the local geometry of affine surfaces which are locally symmetric. There are six non-isomorphic local geometries. We realize these examples as type \(\mathcal {A}\) , type \(\mathcal {B}\) , and type \(\mathcal {C}\) geometries using a result of Opozda and classify the relevant geometries up to linear isomorphism. We examine the geodesic structures in this context. Particular attention is paid to the Lorentzian analogue of the hyperbolic plane and to the pseudosphere. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0280-4
Abstract: In order to treat high-dimensional problems, one has to find data-sparse representations. Starting with a six-dimensional problem, we first introduce the low-rank approximation of matrices. One purpose is the reduction of memory requirements, another advantage is that now vector operations instead of matrix operations can be applied. In the considered problem, the vectors correspond to grid functions defined on a three-dimensional grid. This leads to the next separation: these grid functions are tensors in \(\mathbb {R}^{n}\otimes \mathbb {R}^{n}\otimes \mathbb {R}^{n}\) and can be represented by the hierarchical tensor format. Typical operations as the Hadamard product and the convolution are now reduced to operations between \(\mathbb {R}^{n}\) vectors. Standard algorithms for operations with vectors from \(\mathbb {R}^{n}\) are of order \(\mathcal {O}(n)\) or larger. The tensorisation method is a representation method introducing additional data-sparsity. In many cases, the data size can be reduced from \(\mathcal {O}(n)\) to \(\mathcal {O}(\log n)\) . Even more important, operations as the convolution can be performed with a cost corresponding to these data sizes. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0300-4
Abstract: We prove estimates for the Green’s function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea is to transfer estimates for the continuous bilaplacian using a new discrete compactness argument and a discrete version of the Caccioppoli (or reverse Poincaré) inequality. One application that we have in mind is the study of entropic repulsion for the membrane model from statistical physics. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0325-8
Abstract: The “Monstrous Moonshine” conjecture (now a theorem) of Conway and Nortan has given rise to a large body of new mathematics. This theorem has been extended to other groups revealing unexpected relations to conformal field theory, quantum gravity, black holes, and string theory in physics and to Ramanujan’s mock theta functions and its extensions in mathematics. We call these results which have been discovered in the last few years “Mock Moonshine.” In this survey article, we will discuss some recent developments connecting diverse areas of mathematics and theoretical physics and indicate directions for future research. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0310-2
Abstract: Let Cb(X) be the C∗-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space X. Moreover, let \(\mathfrak {A}_{0}\) be some ideal and \(\mathfrak {A}_{1}\) be some unital C∗-subalgebra of Cb(X). For \(\mathfrak {A}_{0}\) and \(\mathfrak {A}_{1}\) having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate beforehand. Within the whole framework, e.g., the one-point compactification of X and the spectrum of the algebra of asymptotically almost periodic functions can be described. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0326-7
Abstract: Over many years, we developed the construction of the ϕ4-model on four-dimensional Moyal space. The solution of the related matrix model \(\mathcal {Z}[E,J]=\int d{\Phi } \exp (\text {tr}(J{\Phi }- E{\Phi }^{2} -\frac {\lambda }{4} {\Phi }^{4}))\) is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. Locality is fulfilled. The Schwinger 2-point function is reflection positive in special cases. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0302-2
Abstract: This paper continues ongoing investigations into questions initiated by seventeenth-century experiments of Mariotte on mutual attractions and repulsions of floating objects, later reformulated mathematically by Laplace in the context of idealized surface tension theory. The characteristic nonlinearity in the governing equations imposes restrictions on behavior of solutions as graphs in the Laplace model, to the effect that a priori relations connecting height and inclination of a presumed surface interface occur, severely restricting the heights that can be achieved by solution curves. We develop below some relevant consequences of that phenomenon. The initial section provides general (perhaps inadequate) background, characterizes global qualitative behavior of all solutions of the underlying equations, and elucidates the significant distinctions in behavior that occur for solutions over an interval that extends unboundedly in one base direction. The section next following presents a non-existence result ensuing from the limitations imposed on such solutions (this result is elaborated in the Appendix). Next follow individual properties of general solutions. The remainder of the paper addresses the forces arising between partially submerged bodies. Theorem 6 offers an alternative derivation of a discovery due to Aspley, He, and McCuan, which displays a remarkable identification of net horizontal force with a first integral of the basic equations. These relations simplify significantly some earlier literature; additionally, as indicated in Section 5, they open the way to consideration of a much larger class of partially immersed objects than appears in the Laplace formulation. We complete Section 5 with an illustrative example in a particular case, indicative of a general study currently underway. PubDate: 2019-03-01 DOI: 10.1007/s10013-018-0307-x
Abstract: We define and study spaces of formal Fourier–Laplace series both in \(\mathbb {R}^{n}\) and in the cylinder \(\mathbb {S}^{n-1}\times \mathbb {R}\) . We show how such spaces of formal series appear in the study of the Radon transform. PubDate: 2019-01-30 DOI: 10.1007/s10013-018-00329-z
Authors:Felix Finster Abstract: Considering second variations about a given minimizer of a causal variational principle, we derive positive functionals in space-time. It is shown that the strict positivity of these functionals ensures that the minimizer is nonlinearly stable within the class of compactly supported variations with local fragmentation. As applications, we endow the space of jets in space-time with Hilbert space structures and derive a positive surface layer integral on solutions of the linearized field equations. PubDate: 2018-04-25 DOI: 10.1007/s10013-018-0295-x