Abstract: Given an arbitrary set \(\Omega \) , we call a triple \({\mathfrak {P}}=(U,F, \Lambda )\) , where U and \(\Lambda \) are two non-empty sets and F is a map from \(U\times \Omega \) into \(\Lambda \) , a pairing on \(\Omega \) . A pairing is an abstract mathematical generalization of the notion of information table, classically used in several scopes of granular computing and rough set theory. In this paper we undertake the study of pairings in relation to specific types of set operators, set systems and binary relations appearing in several branches of pure mathematics and information sciences. For example, an intersection-closed system \(MAXP({\mathfrak {P}})\) on \(\Omega \) can be canonically associated with any pairing \({\mathfrak {P}}\) on \(\Omega \) and we showed that for any intersection-closed system \(\mathfrak {S}\) on an arbitrary (even infinite) set \(\Omega \) there exists a pairing \({\mathfrak {P}}\) on \(\Omega \) such that \(MAXP({\mathfrak {P}})=\mathfrak {S}\) . Next, we introduce some classes of pairings whose properties have a close analogy with corresponding notions derived from topology and matroid theory. We describe such classifications by means of a binary relation \(\leftarrow _{{\mathfrak {P}}}\) on the power set \(\mathcal {P}(\Omega )\) canonically associated with any pairing \({\mathfrak {P}}\) . Using such a relation, we analyze new properties of intersection-closed systems and related operators, both within concrete models induced by metric spaces and also in connection with basic notions of common interest in several scopes of pure and applied mathematics and information sciences. PubDate: 2019-02-19 DOI: 10.1007/s11565-019-00314-5

Abstract: An old conjecture of Voisin describes how 0-cycles on a surface S should behave when pulled-back to the self-product \(S^m\) for \(m>p_g(S)\) . We show that Voisin’s conjecture is true for a 3-dimensional family of surfaces of general type with \(p_g=q=2\) and \(K^2=7\) constructed by Cancian and Frapporti, and revisited by Pignatelli–Polizzi. PubDate: 2019-01-29 DOI: 10.1007/s11565-019-00313-6

Abstract: We consider a system associated to Klein–Gordon equations with homogeneous time-dependent electric fields. The upper and lower boundaries of a time-evolution propagator for this system were proven by Veselić (J Oper Theory 25:319–330, 1991) for electric fields that are independent of time. We extend this result to time-dependent electric fields. PubDate: 2018-11-01 DOI: 10.1007/s11565-017-0294-y

Abstract: We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation. PubDate: 2018-11-01 DOI: 10.1007/s11565-018-0307-5

Abstract: On projective spaces of dimension \(d\ge 2\) defined over a field of positive characteristic we construct rank \(d+1\) uniform toric but non-homogeneous bundles, which do not exists in characteristic zero. These bundles are obtained by choosing suitable equivariant extensions of the Frobenius pullbacks of \(T_{\mathbb {P}^d}\) by a line bundle. PubDate: 2018-11-01 DOI: 10.1007/s11565-018-0305-7

Abstract: In his second notebook, Ramanujan recorded total of seven P–Q modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained for higher orders. As a consequence, several values of quotients of theta-function are evaluated. The cubic singular modulus is evaluated at \(q=\exp (-2\pi \sqrt{n/3})\) for \(n\in \{5k, 1/5k, 5/k, k/5\}\) , where \(k\in \{4,7,16\}\) . PubDate: 2018-11-01 DOI: 10.1007/s11565-017-0286-y

Abstract: This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier–Stokes equations in the half space when the initial data \( h\in {\dot{B}}_{q \sigma }^{\alpha -\frac{2}{q}}({{\mathbb {R}}}^n_+)\) and the boundary data \( g\in \dot{ B}_q^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}({{\mathbb {R}}}^{n-1}\times {\mathbb R}_+) \) with \(g_n\in \dot{B}^{\frac{1}{2} \alpha }_q ({{\mathbb {R}}}_+; \dot{B}^{-\frac{1}{q}}_q ({{\mathbb {R}}}^{n-1}))\cap L^q({\mathbb R}_+;{\dot{B}}^{\alpha -\frac{1}{q}}({{\mathbb {R}}}^{n-1}))\) , for any \(0<\alpha <2\) and \(q =\frac{n+2}{\alpha +1}\) . Compatibility condition (1.3) is required for h and g. PubDate: 2018-11-01 DOI: 10.1007/s11565-018-0312-8

Abstract: We study the regularity of Gevrey vectors for Hörmander operators $$\begin{aligned} P = \sum _{j=1}^m X_j^2 + X_0 + c \end{aligned}$$ where the \(X_j\) are real vector fields and c(x) is a smooth function, all in Gevrey class \(G^{s}.\) The principal hypothesis is that P satisfies the subelliptic estimate: for some \(\varepsilon >0, \; \exists \,C\) such that $$\begin{aligned} \Vert v\Vert _{\varepsilon }^2 \le C\left( (Pv, v) + \Vert v\Vert _0^2\right) \qquad \forall v\in C_0^\infty . \end{aligned}$$ We prove directly (without the now familiar use of adding a variable t and proving suitable hypoellipticity for \(Q=-D_t^2-P\) and then, using the hypothesis on the iterates of P on u, constructing a homogeneous solution U for Q whose trace on \(t=0\) is just u) that for \(s\ge 1,\) \(G^s(P,\Omega _0) \subset G^{s/\varepsilon }(\Omega _0);\) that is, $$\begin{aligned}&\forall K\Subset \Omega _0, \;\exists C_K: \Vert P^j u\Vert _{L^2(K)}\le C_K^{j+1} (2j)!^s, \;\forall j\\&\quad \implies \forall K'\Subset \Omega _0, \;\exists \tilde{C}_{K'}:\,\Vert D^\ell u\Vert _{L^2(K')} \le \tilde{C}_{K'}^{\ell +1} \ell !^{s/\varepsilon }, \;\forall \ell . \end{aligned}$$ In other words, Gevrey growth of derivatives of u as measured by iterates of P yields Gevrey regularity for u in a larger Gevrey class. When \(\varepsilon =1,\) P is elliptic and so we recover the original Kotake–Narasimhan theorem (Kotake and Narasimhan in Bull Soc Math Fr 90(12):449–471, 1962), which has been studied in many other classes, including ultradifferentiable functions (Boiti and Journet in J Pseudo-Differ Oper Appl 8(2):297–317, 2017). We are indebted to M. Derridj for multiple conversations over the years. PubDate: 2018-11-01 DOI: 10.1007/s11565-017-0293-z

Abstract: We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on \((-\infty ,\infty )\) . Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too. PubDate: 2018-11-01 DOI: 10.1007/s11565-018-0303-9

Abstract: We define and study the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators defined on the half plane \(]0,+\infty [\times \mathbb {R}\) . We prove a Plancherel theorem and an inversion formula that we use to establish the classical Heisenberg uncertainty principle. Next, we study this transform on subsets of \(([0,+\infty [\times \mathbb {R})^2\) with finite measures, in particular we establish a well generalized Heisenberg–Pauli–Weyl uncertainty principle for this transform (with general magnitude). Also, we check a local uncertainty principle and we give nice applications. PubDate: 2018-10-29 DOI: 10.1007/s11565-018-0311-9

Abstract: In this paper, we prove an analog of Titchmarsh’s theorem for the q-Bessel Fourier transform using a generalized q-translation operator. Using the q-Bessel operator we define the Sobolev-type spaces, K-functionals and we give the proof of the equivalence theorem for a K-functional and a modulus of smoothness. PubDate: 2018-10-04 DOI: 10.1007/s11565-018-0309-3

Abstract: In this article we study the boundary behaviour of surface potentials on the plane \(\mathbb {R}^{n-1}\) ( \(n\ge 2\) ) with hypersingular kernels. We show the existence of pointwise and “weak” one-sided limits for such hypersingular integrals. The pointwise result will be also expressed in terms of Hadamard finite-part integrals. Moreover, we determine a two-sided jump relation which yields necessary and sufficient conditions for the validity of a Lyapunov–Tauber property. The paper ends with some applications to harmonic, elastic and Stokes potentials. PubDate: 2018-10-01 DOI: 10.1007/s11565-018-0310-x

Authors:Sanja Friganović Abstract: In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\) , the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\) , in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \) . As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous. PubDate: 2018-05-11 DOI: 10.1007/s11565-018-0302-x

Authors:Jonatan Floriano da Silva; Henrique Fernandes de Lima; Marco Antonio Lázaro Velásquez Abstract: In this paper we provide a characterization for stable hypersurfaces with constant anisotropic mean curvature immersed in the Euclidean space \(\mathbb {R}^{n+1}\) through the analysis of the first eigenvalue of the anisotropic Laplacian operator. PubDate: 2018-04-11 DOI: 10.1007/s11565-018-0301-y

Authors:Marco Andreatta; Claudio Fontanari Abstract: Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: A projective variety X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have \(H^0(X, m_0K_X+H)=0\) for some \(m_0=m_0(H)>0\) . Let X be a projective 4-fold, L an ample divisor and t an integer with \(t \ge 3\) . If \(K_X+tL\) is pseudo-effective, then \(H^0(X, K_X+tL) \ne 0\) . PubDate: 2018-02-09 DOI: 10.1007/s11565-018-0300-z

Authors:T. M. Al-Gafri; S. K. Nauman Abstract: Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an \(\mathcal {SSH}\) -subgroup in G if G has an s-permutable subgroup K such that \(H^{sG} = HK\) and \(H^g \cap N_K (H) \leqslant H\) , for all \(g \in G\) , where \(H^{sG}\) is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are \(\mathcal {SSH}\) -subgroups in G. Several recent results from the literature are improved and generalized. PubDate: 2018-02-05 DOI: 10.1007/s11565-018-0299-1

Authors:Luciana Mafalda Elias de Assis; Malay Banerjee; Ezio Venturino Abstract: In this paper two mathematical models are proposed and analyzed to elucidate the influence on a generalist predator of its hidden and explicit resources. Boundedness of the system’s trajectories, feasibility, local and global stability of the equilibria for both models are established, as well as possible local bifurcations. The findings indicate that the relevant behaviour of the system, including switching of stability, extinction and persistence of the involved populations, depends mainly on the reproduction rate of the favorite prey. To achieve full ecosystem survival some balance between the respective grazing pressures exerted by the predator on the prey populations needs to be maintained, while higher grazing pressure just on one species always leads to its extinction. PubDate: 2018-01-22 DOI: 10.1007/s11565-018-0298-2

Authors:G. Chiaselotti; T. Gentile; F. Infusino Abstract: In this paper, we give a purely mathematical generalization of an information table. We call pairing on a given set \(\Omega \) a triple \(\mathfrak {P}=(U, F, \Lambda )\) , where U and \(\Lambda \) are non-empty sets and \(F:U\times \Omega \rightarrow \Lambda \) is a map. We provide several examples of pairings: graphs, digraphs, metric spaces, group actions and vector spaces endowed with a bilinear form. Moreover, we reinterpret the usual notion of indiscernibility (with respect to a fixed attribute subset of an information table) in terms of local symmetry on U and, then, we study a global version of symmetry, that we called indistinguishability. In particular, we interpret the latter relation as the symmetrization of a pre-order \(\le _{\mathfrak {P}}\) , that describes the symmetry transmission between subsets of \(\Omega \) . Hence, we introduce a global average of symmetry transmission and studied it for some basic digraph families. Finally, we prove that the partial order of any finite lattice can be described in terms of the above pre-order. PubDate: 2018-01-15 DOI: 10.1007/s11565-018-0297-3

Authors:Emmanuel Franck; Laurent Gosse Abstract: By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax–Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the \(H^s\) norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former “shape functions” and “symmetric potential schemes” are highlighted. PubDate: 2017-12-06 DOI: 10.1007/s11565-017-0296-9

Authors:Lucio Guerra Abstract: We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve. PubDate: 2017-11-23 DOI: 10.1007/s11565-017-0295-x