Showing papers in "Annali Dell'universita' Di Ferrara in 2018"

Estimates of anisotropic Sobolev spaces with mixed norms for the Stokes system in a half-space

Abstract: We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in $${\mathbb {R}}^n_+ \times (0,T)$$ . We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier–Stokes equations in a half-space in dimension three.

Pairings and related symmetry notions

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.

Microscopic modeling and analysis of collective decision making: equality bias leads suboptimal solutions

Abstract: We discuss a novel microscopic model for collective decision-making interacting multi-agent systems. In particular we are interested in modeling a well known phenomena in the experimental literature called equality bias, where agents tend to behave in the same way as if they were as good, or as bad, as their partner. We analyze the introduced problem and we prove the suboptimality of the collective decision-making in the presence of equality bias. Numerical experiments are addressed in the last section.

Direct and inverse problems for degenerate differential equations

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.

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

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 Hs 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.

On local Gevrey regularity for Gevrey vectors of subelliptic sums of squares: an elementary proof of a sharp Gevrey Kotake–Narasimhan theorem

Abstract: We study the regularity of Gevrey vectors for Hormander 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.

New P – Q mixed modular equations and their applications

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 $$k\in \{4,7,16\}$$

Effective adjunction theory

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 $$H^0(X, K_X+tL) e 0$$ .

q- Lupas Kantorovich operators based on Polya distribution

Abstract: The purpose of the present paper is to introduce a Kantorovich modification of the q-analogue of the Stancu operators defined by Nowak (J Math Anal Appl 350:50–55, 2009). We study a local and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Further A-statistical convergence properties of these operators are investigated. Next, a bivariate generalization of these operators is introduced and its rate of convergence is discussed with the aid of the partial and complete modulus of continuity and the Peetre‘s K-functional.

On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Abstract: Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\), the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\), and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\). We show that any dominating component of \({\mathscr {W}}^1_{d}(|L'|)\) corresponds to \(\mu _{L'}\)-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in |L|\) and a base-point free \(g^1_d\) on C, say (A, V), we study the \(\mu _L\)-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the \(g^1_d\), we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\)-semistable.

Multilinear dyadic operators and their commutators

Abstract: We obtain a generalized paraproduct decomposition of the pointwise product of two or more functions that naturally gives rise to multilinear dyadic paraproducts and Haar multipliers. We then study the boundedness properties of these multilinear operators and their commutators with dyadic BMO functions. We also characterize the dyadic BMO functions via the boundedness of (a) certain paraproducts, and (b) the commutators of multilinear Haar multipliers and paraproduct operators.

Comparing predator–prey models with hidden and explicit resources

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.

Characterization of some derivations on von Neumann algebras via left centralizers

Abstract: Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.

Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation

Abstract: We consider the initial value problem for the fractional nonlinear Schrodinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.

Some results on a special case of a general continued fraction of Ramanujan

Abstract: We derive a new special case C(q) of a general continued fraction recorded by Ramanujan in his Lost Notebook. We give a representation of the continued fraction C(q) as a quotient of Dedekind eta-function and then use it to prove modular identities connecting C(q) with each of the continued fractions $$C(-q)$$ , $$C(q^{2})$$ , $$C(q^{3})$$ , $$C(q^{5})$$ , $$C(q^{7})$$ , $$C(q^{11})$$ , $$C(q^{13})$$ and $$C(q^{17})$$ . We also prove general theorems for the explicit evaluation of the continued fraction C(q) by using Ramanujan’s class invariants.

Klein–Gordon equations with homogeneous time-dependent electric fields

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 Veselic (J Oper Theory 25:319–330, 1991) for electric fields that are independent of time. We extend this result to time-dependent electric fields.

Remarks on uniform bundles on projective spaces

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.

Existence of positive periodic solutions of nonlinear neutral differential systems with variable delays

Abstract: In this work we use some mixed techniques of the Mawhin coincidence degree theory and fixed point theorem to prove the existence of positive periodic solutions of delay systems. As a consequence, we offer existence criteria and sufficient conditions for existence of periodic solutions to the systems with feedback control. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved.

On the existence of the biharmonic Green kernels and the adjoint biharmonic functions

Abstract: We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.

On $$\mathcal {SSH}$$ SSH -subgroups of finite groups

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.

Homological equivalence relations on an algebraic curve

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.

Stable hypersurfaces via the first eigenvalue of the anisotropic Laplacian operator

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.

A note on regularity of the pressure in the Navier–Stokes system

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.