Authors:Andrea Vaccaro; Matteo Viale Pages: 293 - 319 Abstract: Abstract It is common knowledge in the set theory community that there exists a duality relating the commutative \(C^*\) -algebras with the family of \(\mathsf {B}\) -names for complex numbers in a boolean valued model for set theory \(V^{\mathsf {B}}\) . Several aspects of this correlation have been considered in works of the late 1970s and early 1980s, for example by Takeuti (Two Applications of Logic to Mathematics. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, Kanô Memorial Lectures, vol 3. Publications of the Mathematical Society of Japan, No. 13, 1978) and Fourman et al. (eds.) (Applications of sheaves. In: Lecture Notes in Mathematics, vol 753. Springer, Berlin, 1979), and by Jech (Trans Am Math Soc 289(1):133–162, 1985). Generalizing Jech’s results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space Y (such as the complex numbers) in a boolean valued model for set theory \(V^\mathsf {B}\) as a space \(C^+(X,Y)\) consisting of functions f whose domain X is the Stone space of \(\mathsf {B}\) , and whose range is contained in Y modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of \(C^+(X,Y)\) . PubDate: 2017-09-01 DOI: 10.1007/s40574-017-0124-2 Issue No:Vol. 10, No. 3 (2017)

Authors:Margarida Melo Pages: 321 - 334 Abstract: Abstract In the present paper we consider the following question: does there exist a Néron model for families of Jacobians of curves with sections' By applying a construction of the author of universal compactified Jacobians over the moduli stack of reduced curves with markings and a result by J. Kass, we give a positive answer to the question holding for curves with planar singularities. PubDate: 2017-09-01 DOI: 10.1007/s40574-016-0103-z Issue No:Vol. 10, No. 3 (2017)

Authors:Ulisse Stefanelli Pages: 335 - 354 Abstract: Abstract Molecular Mechanics models molecules as configurations of particles interacting via classical potentials. The specific geometry of covalent bonding in carbon is described by the combination of an attractive-repulsive two-body interaction and a three-body bond-orientation part. We investigate the strict local minimality of specific carbon configurations under general assumptions on the interaction potentials. Carbyne, graphene, some fullerenes, and diamond are proved to be stable. PubDate: 2017-09-01 DOI: 10.1007/s40574-016-0102-0 Issue No:Vol. 10, No. 3 (2017)

Authors:Bhakti B. Manna; Angela Pistoia Pages: 355 - 368 Abstract: Abstract We consider the Neumann problem $$\begin{aligned} - \Delta v + v= v^{q-1} \quad \text { in } \; \mathcal {D}, \quad v > 0 \quad \text { in } \; \mathcal {D},\ \partial _\nu v = 0 \quad \text { on } \; \partial \mathcal {D}, \qquad \qquad (P) \end{aligned}$$ where \(\mathcal {D} \) is an open bounded domain in \( \mathbb {R}^N,\) \(\nu \) is the unit inner normal at the boundary and \(q>2\) . For any integer, \(1\le h\le N-3,\) we show that, in some suitable domains \(\mathcal D,\) problem (P) has a solution which blows-up along a \(h-\) dimensional minimal submanifold of the boundary \(\partial \mathcal D\) as q approaches from either below or above the higher critical Sobolev exponent \({2(N-h)\over N-h-2}\) . PubDate: 2017-09-01 DOI: 10.1007/s40574-016-0108-7 Issue No:Vol. 10, No. 3 (2017)

Authors:Luca Motto Ros Pages: 369 - 410 Abstract: Abstract We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research. PubDate: 2017-09-01 DOI: 10.1007/s40574-017-0125-1 Issue No:Vol. 10, No. 3 (2017)

Authors:Alessio Porretta Pages: 411 - 439 Abstract: Abstract Mean field games theory describes the strategic interactions in a large population of similar agents whereas the strategy of the individuals depend on the distribution law of the state. The equilibria solve a system of partial differential equations where a backward Hamilton-Jacobi-Bellman equation for the value function is coupled with a forward Fokker-Planck equation for the mass distribution. If the cost criteria depend on the density of the distribution law, a theory of weak solutions is needed to handle mean field games systems, including new results concerning the Fokker-Planck equation with \(L^2\) drift. Here we prove existence and uniqueness of weak solutions when the dynamics takes place in the whole space \(\mathbb R^N\) . We extend previous results obtained so far only for compact state space. PubDate: 2017-09-01 DOI: 10.1007/s40574-016-0105-x Issue No:Vol. 10, No. 3 (2017)

Authors:Jasmin Raissy Pages: 441 - 450 Abstract: Abstract In this short note we give an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two in a neighbourhood of an invariant fiber, dividing our discussion according to the different possible kinds of invariant fibers. PubDate: 2017-09-01 DOI: 10.1007/s40574-016-0101-1 Issue No:Vol. 10, No. 3 (2017)

Authors:Luca Migliorini Pages: 467 - 485 Abstract: Abstract We give an introduction to non-abelian Hodge theory for curves with the aim of stating the \(P = W\) conjecture both in its original cohomological version and in the more recent geometric one, and proposing a strategy to relate the two conjectures. PubDate: 2017-09-01 DOI: 10.1007/s40574-017-0122-4 Issue No:Vol. 10, No. 3 (2017)

Authors:Timoteo Carletti; Duccio Fanelli Abstract: Abstract We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the \((3x+1)\) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8 (mod8), obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates in the original, infinite-dimensional, space of positive integers. Then, we consider a Markov chain which runs on the reduced space of mod8 congruence classes of integers. The transition probabilities of the Markov chain are computed from the deterministic map, by employing a measure that is invariant for the map itself. Working in this setting, we demonstrate that the stationary distribution sampled by the stochastic system induces a contracting behaviour for the orbits of the deterministic map on the original space of the positive integers. Sampling the equilibrium distribution on the congruence classes mod \(8^m\) for any m, which amounts to arbitrarily reducing the degree of imposed coarse graining, returns an identical conclusion. PubDate: 2017-10-03 DOI: 10.1007/s40574-017-0145-x

Authors:N. Chernyavskaya; L. Shuster Abstract: Abstract We consider the equation 1 $$\begin{aligned} - (r(x)y'(x))'+q(x)y(x)=f(x),\quad x\in {\mathbb {R}} \end{aligned}$$ where \(f\in L_1({\mathbb {R}}) \) and 2 $$\begin{aligned}&r >0,\quad q\ge 0,\quad 1/r\in L_1^{\mathrm{loc}}({\mathbb {R}}),\quad q\in L_1^{\mathrm{loc}}({\mathbb {R}}),\end{aligned}$$ 3 $$\begin{aligned}&\lim _{ d \rightarrow \infty }\int _{x-d}^x\frac{dt}{r(t)}\cdot \int _{x-d}^x q(t)dt=\infty . \end{aligned}$$ By a solution of (1), we mean any function y, absolutely continuous in \({\mathbb {R}}\) together with \(ry'\) , which satisfies (1) almost everywhere in \({\mathbb {R}}.\) Under conditions (2) and (3), we give a criterion for correct solvability of (1) in the space \(L_1({\mathbb {R}})\) . PubDate: 2017-09-25 DOI: 10.1007/s40574-017-0144-y

Authors:Ebrahim Ghashghaei; Mehrdad Namdari Abstract: Abstract In this paper, a generalization of max modules is introduced by applying the concept of corational submodules. PubDate: 2017-09-13 DOI: 10.1007/s40574-017-0143-z

Authors:Amin Hosseini; Ajda Fošner Abstract: Abstract In this paper, identities related to \((\sigma , \tau )\) -Lie derivations and \((\sigma , \tau \) )-derivations are considered. Let \(m,n \ge 1\) be integers and \(\mathcal {R}\) be an M!-torsion free unital ring, where \(M = \max \{m, n\}\) . Suppose that \(\sigma , \tau : \mathcal {R} \rightarrow \mathcal {R}\) are two endomorphisms such that \(\sigma (\mathbf e )\) , \(\tau (\mathbf e ) \in Z(\mathcal {R})\) , where \(\mathbf e \) denotes the unit element of \(\mathcal {R}\) . If \(D:\mathcal {R} \rightarrow \mathcal {R}\) is an additive map satisfying \(D[x^n, y^m] = [D(x^n), \sigma (y^m)] + [\tau (x^n), D(y^m)]\) for all \(x, y \in \mathcal {R}\) , then D is a \((\sigma , \tau \) )-Lie derivation. Moreover, we offer a characterization of ( \(\sigma , \tau \) )-derivations from a \(C^{*}\) -algebra \(\mathcal {A}\) into a Banach \(\mathcal {A}\) -bimodule \(\mathcal {M}\) which reads as follows. Let \(\mathcal {A}\) be a unital \(C^{*}\) -algebra, \(\mathcal {M}\) be a unital Banach \(\mathcal {A}\) -bimodule, and let \(\sigma , \tau :\mathcal {A} \rightarrow \mathcal {A}\) be continuous endomorphisms such that \(\sigma (\mathbf e ) = \mathbf e = \tau (\mathbf e )\) , where \(\mathbf e \) denotes the unit element of \(\mathcal {A}\) . Suppose that \(n > 1\) is an integer and \(d:\mathcal {A} \rightarrow \mathcal {M}\) is a linear map satisfying \(d(a^{n}) = \sum _{j = 1}^{n}\tau (a)^{n - j}d(a) \sigma (a)^{j - 1}\) for all PubDate: 2017-08-19 DOI: 10.1007/s40574-017-0141-1

Authors:Meera H. Chudasama; B. I. Dave Abstract: Abstract We defined an extended Bessel function as q- \(\ell \) - \(\varPsi \) Bessel function in (Boll. Unione. Mat. Ital. 8(4):239–256, 2016). Here we obtain as the main results, the infinite order difference equation, the generating function relation, and Contour integral representations. With the aid of these, some other properties are also deduced. PubDate: 2017-08-01 DOI: 10.1007/s40574-017-0139-8

Authors:Henrique F. de Lima; Fábio R. dos Santos; Jogli G. Araújo; Marco Antonio L. Velásquez Abstract: Abstract In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type \(I\times _fM^n\) , where \(M^{n}\) is a connected n-dimensional oriented Riemannian manifold and \(f:I\rightarrow \mathbb {R}\) is a positive smooth function. As applications, we obtain several uniqueness results concerning these hypersurfaces with constant mean curvature, under standard constraints on the Ricci curvature of \(M^n\) and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity. PubDate: 2017-07-25 DOI: 10.1007/s40574-017-0138-9

Authors:Sayed Saber Abstract: Abstract Let X be a Stein manifold of dimension \(n \ge 3\) . Let \(\Omega _{1}\) be a weakly q-convex and \(\Omega _{2}\) be a weakly \((n-q-1)\) -convex in X with smooth boundaries such that \(\overline{\Omega }_{2}\Subset \Omega _{1}\Subset X\) . Assume that \(\Omega =\Omega _{1}\backslash \overline{\Omega }_{2}\) . In this paper, we establish sufficient conditions for the closed range of \(\overline{\partial }\) on \(\Omega \) . Moreover, we study the global boundary regularity of the \(\overline{\partial }\) -problem on \(\Omega \) . PubDate: 2017-07-20 DOI: 10.1007/s40574-017-0135-z

Authors:Vincenzo Dimonte Abstract: Abstract This is a survey about I0 and rank-into-rank axioms, with some previously unpublished proofs. PubDate: 2017-07-15 DOI: 10.1007/s40574-017-0136-y

Authors:Pietro Caputo; Fabio Martinelli; Fabio Lucio Toninelli Abstract: Abstract We consider the \((2+1)\) -dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form \( \nabla \phi ^{p}\) , where \(p\in [1,+\infty ]\) . We show that at low temperature, for a square region \(\Lambda \) with side L, both under the infinite volume measure and under the measure with zero boundary conditions around \(\Lambda \) , the probability that the surface is nonnegative in \(\Lambda \) behaves like \(\exp (-4\beta \tau _{p,\beta } L H_p(L) )\) , where \(\beta \) is the inverse temperature, \(\tau _{p,\beta }\) is the surface tension at zero tilt, or step free energy, and \(H_p(L)\) is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in [8] for the standard SOS model ( \(p=1\) ). PubDate: 2017-07-15 DOI: 10.1007/s40574-017-0137-x

Authors:M. Di Francesco; S. Fagioli; M. D. Rosini Abstract: Abstract In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. The result is complemented with some numerical simulations. PubDate: 2017-06-20 DOI: 10.1007/s40574-017-0132-2