Abstract: In this paper we provide a criterion of essential self-adjointness for operators in the tensor product of a separable Hilbert space and a Fock space. The class of operators we consider may contain a self-adjoint part, a part that preserves the number of Fock space particles and a non-diagonal part that is at most quadratic with respect to the creation and annihilation operators. The hypotheses of the criterion are satisfied in several interesting applications.

Abstract: Unlike the real number field, a set of p−adic Gibbs measures of p−adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over p−adic fields. Recently, all translation-invariant p−adic Gibbs measures of the p−adic Potts model on the Cayley tree of order two were described by means of roots of a certain quadratic equation over some domain of the p−adic field. In this paper, we consider the same problem on the Cayley tree of order three. In this case, we show that all translation-invariant p−adic Gibbs measures of the p−adic Potts model can be described in terms of roots of some cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ . In own its turn, we also provide a solvability criterion of a general cubic equation over $\mathbb {Z}_{p}\setminus \mathbb {Z}_{p}^{*}$ for p > 3.

Abstract: The classical result of concentration of the Gaussian measure on the sphere in the limit of large dimension induces a natural duality between Gaussian and spherical models of spin glass. We analyse the Legendre variational structure linking the free energies of these two systems, in the spirit of the equivalence of ensembles of statistical mechanics. Our analysis, combined with the previous work (Barra et al., J. Phys. A: Math. Theor. 47, 155002, 2014), shows that such models are replica symmetric. Lastly, we briefly discuss an application of our result to the study of the Gaussian Hopfield model.

Abstract: We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.

Abstract: We study the irreducible decomposition under $Sp(2n,{\mathbb R})$ of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M. We rediscover Ph. Tondeur’s Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.

Abstract: We study resolvent estimates, spectral theory and large time dispersive properties of scalar and matrix Schrodinger-type operators on ℍ n+1 for n ⩾ 1.

Abstract: We consider in this work some class of strongly perturbed for the semilinear wave equation with conformal power nonlinearity. We obtain an optimal estimate for a radial blow-up solution and we have also obtained two less stronger estimates. These results are achieved in three-steps argument by the construction of a Lyapunov functional in similarity variables and the Pohozaev identity derived by multiplying (1.14) by y∂yw.

Abstract: In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyroscopic forces, these conditions, when expressed in terms of a multiplier matrix, reduce to those obtained in Mestdag et al. (Differential Geom. Appl. 29(1), 55–72, 2011). When the involved geometric structures are homogeneous with respect to the fibre coordinates, we show how one can further simplify the generalized Helmholtz conditions. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.

Abstract: We study the asymptotics of the difference of the ground-state energies of two non-interacting N-particle Fermi gases in a finite volume of length L in the thermodynamic limit up to order 1/L. We are particularly interested in subdominant terms proportional to 1/L, called finite-size energy. In the nineties (Affleck, Nuc. Phys. B 58, 35–41 1997; Zagoskin and Affleck, J. Phys. A 30, 5743–5765 1997) claimed that the finite-size energy is related to the decay exponent occurring in Anderson’s orthogonality. We prove that the finite-size energy depends on the details of the thermodynamic limit and is therefore non-universal. Typically, it includes an additional linear term in the scattering phase shift.

Abstract: We prove a variant of a theorem by Nekhoroshev on persistence of invariant tori in systems with symmetry. The new proof applies to reversible non Hamiltonian systems equivariant under the action of an Abelian group and is much simpler then the original one.

Abstract: We develop an algebraic formalism for topological $\mathbb {T}$ -duality. More precisely, we show that topological $\mathbb {T}$ -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra $\mathcal {D}$ . Then we show that there is a functorial topological K-theory symmetric spectrum construction $\mathbf {K}_{\Sigma }^{\textup {top}}(-)$ on the category of separable C ∗-algebras, such that $\mathbf {K}_{\Sigma }^{\text {top}}(\mathcal {D})$ is an algebra over this operad; moreover, $\mathbf {K}_{\Sigma }^{\text {top}}(A\hat {\otimes }\mathcal {D})$ is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that $\mathcal {O}_{\infty }$ -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+b-semigroup C ∗-algebras coming from number theory and that of $\mathcal {O}_{\infty }$ -stabilized noncommutative tori.

Abstract: We study almost real spectral triples on quantum lens spaces, as orbit spaces of free actions of cyclic groups on the spectral geometry on the quantum group S U q (2). These spectral triples are given by weakening some of the conditions of a real spectral triple. We classify the irreducible almost real spectral triples on quantum lens spaces and we study unitary equivalences of such quantum lens spaces. Applying a useful characterization of principal U(1)-fibrations in noncommutative geometry, we show that all such quantum lens spaces are principal U(1)-fibrations over quantum teardrops.

Abstract: In this paper, we study the three-dimensional inhomogeneous incompressible Navier-Stokes equations, and establish several regularity criteria in terms of only velocity which allow the initial density to contain vacuum. Therefore, our results can be considered as further improvement to the previous results.

Abstract: We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. The main result is that, if all the deformations of the satellite dissipate some energy, then under a suitable nondegeneracy condition there are only three possible outcomes for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the satellite falls on the planet, (iii) the satellite is captured in synchronous resonance i.e. its orbit is asymptotic to a motion in which the barycenter moves on a circular orbit, and the satellite moves rigidly, always showing the same face to the planet. The result is obtained by making use of LaSalle’s invariance principle and by a careful kinematic analysis showing that energy stops dissipating only on synchronous orbits. We also use in quite an extensive way the fact that conservative elastodynamics is a Hamiltonian system invariant under the action of the rotation group.

Abstract: In this paper, two methods of approximate symmetries for partial differential equations with a small parameter are applied to a perturbed nonlinear Ostrovsky equation. To compute the first-order approximate symmetry, we have applied two methods which one of them was proposed by Baikov et al. in which the infinitesimal generator is expanded in a perturbation series; whereas the other method by Fushchich and Shtelen [3] is based on the expansion of the dependent variables in perturbation series. Especially, an optimal system of one dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.

Abstract: Holomorphic vector bundles corresponding to the static soliton solution of the Ward equation were explicitly presented by Ward in terms of a meromorphic framing. Bundles (for simplicity, “bundle” is to be taken throughout to mean “holomorphic vector bundle”) corresponding to all Ward k-soliton solutions whose extended solutions have only simple poles, and some Ward 2-soliton solutions whose extended solutions have only a second-order pole, were explicitly described by us in a previous paper. In this paper, we go on to present some bundles corresponding to soliton-antisoliton solutions of the Ward equation, and Ward 3-soliton solutions whose extended solutions have a simple pole and a double pole. To give some more interpretation of the bundles, we study the second Chern number of the corresponded bundles and find that it can be obtained directly from the patching matrices. We also point out some information about bundles corresponding to Ward soliton solutions whose extended solutions have general pole data at the end of the paper.

Abstract: We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S 2 × S 2. Its principal order Hamiltonian H 0 was known already to Pauli in 1926. H 0 defines an isochronous system with a linear flow on S 2 × S 2 and with frequencies depending on the external fields. Small perturbations of H 0 due to higher order terms can be studied by further normalization, either resonant or nonresonant. We study the question, raised previously, of how to decide for given parameters of the fields what normalization should be used and with regard to which resonances. We base this analysis on the Nekhoroshev theory—a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. Our answer depends on the a priori choice of the maximal order N of resonances that are going to be taken into account (a cutoff). For any given N, there is a decomposition of the parameter space into resonant and nonresonant zones, and a normal form with a remainder of order $\exp (-N)$ may be consistently constructed in each of such zones.

Abstract: In this paper, we are interested in regularity problem for the incompressible Navier-Stokes equations in a bounded domain of \(\mathbb {R}^{N}\) with smooth boundary. Some new sufficient conditions which guarantee the regularity of Leray-Hopf weak solutions concerning the combination of the unknowns are established.

Abstract: We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

Abstract: In a recent article (De Leo, R., Ann. Glob. Anal. Geom., 39, 3, 231–248 2011), we studied the global solvability of the so-called cohomological equation Lξf = g in \(C^{\infty }(\mathbb {R}^{2})\), where ξ is a regular vector field on the plane and Lξ the corresponding Lie derivative operator. In a joint article with T. Gramchev and A. Kirilov (2011), we studied the existence of global \(L^{1}_{loc}\) weak solutions of the cohomological equation for planar vector fields depending only on one coordinate. Here we generalize the results of both articles by providing explicit conditions for the existence of global weak solutions to the cohomological equation when ξ is intrinsically Hamiltonian or of finite type.

Abstract: In this paper nonlinear Hodge theory and Banach algebra estimates are employed to construct a convergent series expansion which solves the prescribed mean curvature equation \(\pm abla \cdot \left ( abla u/\sqrt {1\pm | abla u|^{2}}\right ) = nH\) for n-dimensional hypersurfaces in \(\mathbb {R}^{n+1}\) (+ sign) and \(\mathbb {R}^{1,n}\) (− sign) which are graphs \(\{(x,u(x)):x\in \mathbb {R}^{n}\}\) of a smooth function \(u:\mathbb {R}^{n}\to \mathbb {R}\), and whose mean curvature function H is α-Holder continuous and integrable, with small norm. The radius of convergence is estimated explicitly from below. Our approach is inspired by, and applied to, the Maxwell–Born–Infeld theory of electromagnetism in \(\mathbb {R}^{1,3}\), for which our method yields the first systematic way of explicitly computing the electrostatic potential \(\phi \propto u\) for regular charge densities \(\rho \propto H\) and small Born parameter, with explicit error estimates at any order of truncation of the series. In particular, our results level the ground for a controlled computation of Born–Infeld effects on the Hydrogen spectrum.

Abstract: We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group SUq(2). In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of SUq(2). We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of SUq(2). Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.

Abstract: We consider a semilinear heat equation with exponential nonlinearity in ℝ2. We prove that local solutions do not exist for certain data in the Orlicz space exp L 2(ℝ2), even though a small data global existence result holds in the same space exp L 2(ℝ2). Moreover, some suitable subclass of exp L 2(ℝ2) for local existence and uniqueness is proposed.

Abstract: The notion of polarity between sets, well-known from convex geometry, is a geometric version of the Fourier transform. We exploit this analogy to propose a new simple definition of quantum indeterminacy, using what we call “\(\hbar \)-polar quantum pairs”, which can be viewed as pairs of position-momentum indeterminacy with minimum spread. The existence of such pairs is consistent with the usual uncertainty principle, but is at the same time more general. We show that this quantum indeterminacy can be measured using the notion of symplectic capacity, which reduces to the notion of area in the phase plane.

Abstract: The purpose of this paper is threefold. Firstly, using Howe duality for \((\mathfrak {gl}(m_{1}|n_{1}),\mathfrak {gl}(m_{2}|n_{2}))\), we obtain integral formulas of the super Schur functions with respect to the super standard Gaussian distributions. Secondly, we give explicit expressions of the super Szego kernels and the weighted super Bergman kernels for the Cartan superdomains of type I. Thirdly, combining these results, we obtain duality relations of integrals over the unitary groups and the Cartan superdomains, and the marginal distributions of the weighted measure.

Abstract: We formulate a natural model of loops and isolated vertices for arbitrary planar graphs, which we call the monopole-dimer model. We show that the partition function of this model can be expressed as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of a polynomial with positive integer coefficients when the grid lengths are even. Finally, we analyse this formula in the infinite volume limit and show that the local monopole density, free energy and entropy can be expressed in terms of well-known elliptic functions. Our technique is a novel determinantal formula for the partition function of a model of isolated vertices and loops for arbitrary graphs.

Abstract: We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander.

Abstract: Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.

Abstract: We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass m, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for m larger than a critical value m ∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H 0 can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for m ∈ (m ∗,m ∗∗), where m ∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H 0,β , β ∈ ℝ, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.