Authors:Changxing Miao, Xingdong Tang, Guixiang Xu Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [math] and the critical parameters [math], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [math], [math] and the supercritical parameters [math], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [math], with [math], [math] or [math]. On the one hand, different with the scattering result for the [math]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [math] because of the existence of infinity many small solitary/traveling waves in [math] with [math], [math] or [math]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6). Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:31:06Z DOI: 10.1142/S0219199717500493
Authors:Luis Barreira, Claudia Valls Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. We give a complete characterization of the existence of Lyapunov coordinate changes bringing an invertible sequence of matrices to one in block form. In other words, we give a criterion for the block-trivialization of a nonautonomous dynamics with discrete time while preserving the asymptotic properties of the dynamics. We provide two nontrivial applications of this criterion: we show that any Lyapunov regular sequence of invertible matrices can be transformed by a Lyapunov coordinate change into a constant diagonal sequence; and we show that the family of all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices (with finite exponent) coincides with the family of Lyapunov coordinate changes. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:31:02Z DOI: 10.1142/S0219199717500274
Authors:Amal Attouchi, Mikko Parviainen Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. In this paper, we study an evolution equation involving the normalized [math]-Laplacian and a bounded continuous source term. The normalized [math]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [math] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:56Z DOI: 10.1142/S0219199717500353
Authors:Fashun Gao, Minbo Yang Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. In this paper, we are concerned with the following nonlinear Choquard equation −Δu + V (x)u = ∫ℝN G(y,u) x − y μdyg(x,u)in ℝN, where [math], [math] and [math]. If [math] lies in a gap of the spectrum of [math] and [math] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579]. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:52Z DOI: 10.1142/S0219199717500377
Authors:Wenhua Zhao Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. Let [math] be a commutative ring and [math] an [math]-algebra. An [math]-[math]-derivation of [math] is an [math]-linear map of the form [math] for some [math]-algebra endomorphism [math] of [math], where [math] denotes the identity map of [math]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [math]-derivation or [math]-[math]-derivation of [math] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [math], and whether or not a locally nilpotent (LN) [math]-derivation or [math]-[math]-derivation of [math] maps every ideal of [math] to a Mathieu subspace of [math]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [math] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [math]-[math]-derivations of integral domains of characteristic zero. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:48Z DOI: 10.1142/S0219199717500560
Authors:Arka Mallick, Cyril Tintarev Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain [math] in [math] containing the origin. supu∈W01,n(Ω),In[u,Ω,R]≤1∫Ωecn u(x) E2β x R n n−1dx < +∞, for some cn> 0 depending only on n. Here, [math], [math], [math] and [math], [math] for [math] This improves an earlier result by Psaradakis and Spector. Also, we prove that for any [math] in the place of [math], the above inequality is false if we take [math]. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:40Z DOI: 10.1142/S0219199717500341
Authors:Chiara Camere Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. We construct quasi-projective moduli spaces of [math]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:37Z DOI: 10.1142/S0219199717500444
Authors:Andrei Minchenko, Alexey Ovchinnikov Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:34Z DOI: 10.1142/S0219199717500389
Authors:Miaomiao Niu, Zhongwei Tang, Lushun Wang Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: Δ2u + (λV (x) − δ(x))u = u p−2uinℝN, (P λ) where [math], [math], [math] is a parameter, [math] is a nonnegative potential function with nonempty zero set [math], [math] is a positive function such that the operator [math] is indefinite and non-degenerate for [math] large. We show that both in subcritical and critical cases, equation [math] admits a least energy solution which for [math] large localized near the zero set [math]. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:31Z DOI: 10.1142/S021919971750047X
Authors:Jaume Llibre, Regilene Oliveira Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [math]: [math], and a Darboux invariant of the form [math] with [math]. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:27Z DOI: 10.1142/S021919971750033X
Authors:Huyuan Chen, Feng Zhou Abstract: Communications in Contemporary Mathematics, Volume 20, Issue 04, June 2018. Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [math] −Δu + u = Iα[up]uqinℝN ∖{0},lim x →+∞u(x) = 0, where [math] and [math] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [math]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation. Citation: Communications in Contemporary Mathematics PubDate: 2018-05-21T01:30:19Z DOI: 10.1142/S0219199717500407
Authors:Xuezhang Chen, Liming Sun Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [math]. We prove the existence of such conformal metrics in the cases of [math] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [math], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [math]. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:27:00Z DOI: 10.1142/S0219199718500219
Authors:Salvatore A. Marano, Sunra J. N. Mosconi Abstract: Communications in Contemporary Mathematics, Ahead of Print. The existence of optimizers [math] in the space [math], with differentiability order [math], for the Hardy–Sobolev inequality is established through concentration-compactness. The asymptotic behavior [math] as [math] and the summability information [math] for all [math] are then obtained. Such properties turn out to be optimal when [math], in which case optimizers are explicitly known. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:27:00Z DOI: 10.1142/S0219199718500281
Authors:Laiachi El Kaoutit, Paolo Saracco Abstract: Communications in Contemporary Mathematics, Ahead of Print. Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:59Z DOI: 10.1142/S0219199718500153
Authors:Hoai-Minh Nguyen, Marco Squassina Abstract: Communications in Contemporary Mathematics, Ahead of Print. We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain–Brezis–Mironescu formula, including the magnetic case both for Sobolev and BV functions. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:58Z DOI: 10.1142/S0219199718500177
Authors:Van Hoang Nguyen Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study the existence and nonexistence of maximizers for variational problem concerning the Moser–Trudinger inequality of Adimurthi–Druet type in [math] MT(N,β,α) =supu∈W1,N(ℝN),∥∇u∥NN+∥u∥NN≤1∫ℝNΦN(β(1+α∥u∥NN) 1 N−1 u N N−1)dx, where [math], [math] both in the subcritical case [math] and critical case [math] with [math] and [math] denotes the surface area of the unit sphere in [math]. We will show that MT[math] is attained in the subcritical case if [math] or [math] and [math] with [math] being the best constant in a Gagliardo–Nirenberg inequality in [math]. We also show that MT[math] is not attained for [math] small which is different from the context of bounded domains. In the critical case, we prove that MT[math] is attained for [math] small enough. To prove our results, we first establish a lower bound for MT[math] which excludes the concentrating or vanishing behaviors of their maximizer sequences. This implies the attainability of MT[math] in the subcritical case. The proof in the critical case is based on the blow-up analysis method. Finally, by using the Moser sequence together with the scaling argument, we show that MT[math]. Our results settle the questions left open in [J. M. do Ó and M. de Souza, A sharp inequality of Trudinger–Moser type and extremal functions in [math], J. Differential Equations 258 (2015) 4062–4101; Trudinger–Moser inequality on the whole plane and extremal functions, Commun. Contemp. Math. 18 (2016) 32 pp.]. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:58Z DOI: 10.1142/S0219199718500232
Authors:Francesca Colasuonno, Eugenio Vecchi Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem: infρ∈Pinfu∈ð’²∖{0}∫Ω(Δu)2 ∫Ωρu2 , where [math] is a class of admissible densities, [math] for Dirichlet boundary conditions and [math] for Navier boundary conditions. The associated Euler–Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator [math]. In the spirit of [S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys. 214 (2000) 315–337], we study qualitative properties of the optimal pairs [math]. In particular, we prove existence and regularity and we find the explicit expression of [math]. When [math] is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of [math] and radial symmetry of both [math] and [math]. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:57Z DOI: 10.1142/S0219199718500190
Authors:Nabil Kahouadji, Niky Kamran, Keti Tenenblat Abstract: Communications in Contemporary Mathematics, Ahead of Print. We consider the class of evolution equations of the form [math], [math], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [math]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [math] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [math]' We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [math]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [math] are universal, i.e. they are independent of [math]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:57Z DOI: 10.1142/S0219199718500256
Authors:X. Fang, E. Feigin, G. Fourier, I. Makhlin Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type [math]. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:56Z DOI: 10.1142/S0219199718500165
Authors:Valentine Roos Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:55Z DOI: 10.1142/S0219199718500189
Authors:Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu Abstract: Communications in Contemporary Mathematics, Ahead of Print. We consider a variant of Gamow’s liquid drop model, with a general repulsive Riesz kernel and a long-range attractive background potential with weight [math]. The addition of the background potential acts as a regularization for the liquid drop model in that it restores the existence of minimizers for arbitrary mass. We consider the regime of small [math] and characterize the structure of minimizers in the limit [math] by means of a sharp asymptotic expansion of the energy. In the process of studying this limit we characterize all minimizing sequences for the Gamow model in terms of “generalized minimizers”. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:55Z DOI: 10.1142/S0219199718500220
Authors:Mikyoung Lee Abstract: Communications in Contemporary Mathematics, Ahead of Print. We prove interior Hessian estimates in the setting of weighted Orlicz spaces for viscosity solutions of fully nonlinear, uniformly elliptic equations F(D2u,x) = f(x)in B 1 under asymptotic assumptions on the nonlinear operator [math] The results are further extended to fully nonlinear, asymptotically elliptic equations. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:54Z DOI: 10.1142/S0219199718500244
Authors:Minbo Yang, Carlos Alberto Santos, Jiazheng Zhou Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we consider the existence of least action nodal solutions for the quasilinear defocusing Schrödinger equation in [math]: −Δu + k 2uΔu2 + V (x)u = g(u) + λ u p−2u, where [math] is a positive continuous potential, [math] is of subcritical growth, [math] and [math] are two non-negative parameters. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of least action nodal solution via deformation flow arguments and [math]-estimates. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-10T07:26:53Z DOI: 10.1142/S0219199718500268
Authors:Joontae Kim, Myeonggi Kwon, Junyoung Lee Abstract: Communications in Contemporary Mathematics, Ahead of Print. For a Liouville domain [math] whose boundary admits a periodic Reeb flow, we can consider the connected component [math] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [math] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [math] has infinite order in [math] if there is an admissible Lagrangian [math] in [math] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [math]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences. Citation: Communications in Contemporary Mathematics PubDate: 2018-04-06T02:40:33Z DOI: 10.1142/S0219199718500141
Authors:Elisabetta Colombo, Paola Frediani, Alessandro Ghigi, Matteo Penegini Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study Shimura curves of PEL type in [math] generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases, we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is one-dimensional and the quotient of the base curve by the group is [math]. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples obtained in this way up to genus 28. We obtain 43 Shimura curves contained in the unramified Prym locus and 9 families contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:44Z DOI: 10.1142/S0219199718500098
Authors:Izzet Coskun, Eric Riedl Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be a general Fano complete intersection of type [math]. If at least one [math] is greater than [math], we show that [math] contains rational curves of degree [math] with balanced normal bundle. If all [math] are [math] and [math], we show that [math] contains rational curves of degree [math] with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that [math] is separably rationally connected by exhibiting very free rational curves in [math] of optimal degrees. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:44Z DOI: 10.1142/S0219199718500116
Authors:Fathi Ben Aribi Abstract: Communications in Contemporary Mathematics, Ahead of Print. We prove a Torres-like formula for the [math]-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the [math]-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:44Z DOI: 10.1142/S021919971850013X
Authors:Dražen Adamović, Gordan Radobolja Abstract: Communications in Contemporary Mathematics, Ahead of Print. This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [math] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [math] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [math], Adv. Math. 289 (2016) 438–479, arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [math]-modules. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:43Z DOI: 10.1142/S0219199718500086
Authors:Henri Berestycki, Alessandro Zilio Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study a mathematical model of environments populated by both preys and predators, with the possibility for predators to actively compete for the territory. For this model we study existence and uniqueness of solutions, and their asymptotic properties in time, showing that the solutions have different behavior depending on the choice of the parameters. We also construct heterogeneous stationary solutions and study the limits of strong competition and abundant resources. We then use these information to study some properties such as the existence of solutions that maximize the total population of predators. We prove that in some regimes the optimal solution for the size of the total population contains two or more groups of competing predators. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:43Z DOI: 10.1142/S0219199718500104
Authors:Kang Lu, Evgeny Mukhin Abstract: Communications in Contemporary Mathematics, Ahead of Print. We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory. Citation: Communications in Contemporary Mathematics PubDate: 2018-02-13T08:17:43Z DOI: 10.1142/S0219199718500128
Authors:Alberto Boscaggin, Maurizio Garrione Abstract: Communications in Contemporary Mathematics, Ahead of Print. By using a shooting technique, we prove that the quasilinear boundary value problem div ∇u 1− ∇u 2 + λq( x ) u p−1u = 0,x ∈ℬ,u = 0, x ∈ ∂ℬ, where [math] is a ball and [math], has more and more pairs of nodal solutions on growing of the parameter [math]. The radial Neumann problem and the periodic problem for the corresponding one-dimensional equation are considered, as well. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:58Z DOI: 10.1142/S0219199718500062
Authors:Byungdo Park, Corbett Redden Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be a compact Lie group acting on a smooth manifold [math]. In this paper, we consider Meinrenken’s [math]-equivariant bundle gerbe connections on [math] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [math], and isomorphism classes of [math]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:57Z DOI: 10.1142/S0219199718500013
Authors:Xuemei Zhang, Meiqiang Feng Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of − u′ 1−u′2′ = λf(u),u(x)> 0,−L < x < L,u(−L) = u(L) = 0, where [math] is a bifurcation parameter, [math], the radius of the one-dimensional ball [math], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:57Z DOI: 10.1142/S0219199718500037
Authors:Dorin Bucur, Ilaria Fragalà Abstract: Communications in Contemporary Mathematics, Ahead of Print. We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:57Z DOI: 10.1142/S0219199718500074
Authors:Xiang Mingqi, Vicenţiu D. Rădulescu, Binlin Zhang Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity M(∥u∥s,A2)[(−Δ) Asu + u] = λ∫ℝN F( u 2) x − y αdyf( u 2)u + u 2s∗−2uin ℝN, ∥u∥s,A = ∬ℝ2N u(x) − ei(x−y)⋅A(x+y 2 )u(y) 2 x − y N+2s dxdy + ∫ℝN u 2dx1/2, where [math] with [math], [math] is the Kirchhoff function, [math] is the magnetic potential, [math] is the fractional magnetic operator, [math] is a continuous function, [math], [math] is a parameter, [math] and [math] is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:56Z DOI: 10.1142/S0219199718500049
Authors:Ran Zhuo, Yan Li Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [math] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [math] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:56Z DOI: 10.1142/S0219199718500050
Authors:Nicola Abatangelo, Sven Jarohs, Alberto Saldaña Abstract: Communications in Contemporary Mathematics, Ahead of Print. We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power [math] of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if [math]. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of [math]-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions. Citation: Communications in Contemporary Mathematics PubDate: 2018-01-25T10:29:55Z DOI: 10.1142/S0219199718500025
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