Stability analysis of higher order nonlinear differential equations in β–normed spaces

In this paper, we interrogate different Ulam type stabilities, ie, β–Ulam–Hyers stability, generalized β–Ulam–Hyers stability, β–Ulam–Hyers–Rassias stability, and generalized β–Ulam–Hyers–Rassias stability, for nth order nonlinear differential equations with integrable impulses of fractional type. The existence and uniqueness of solutions are investigated by using the Banach contraction principle. In the end, we give an example to support our main result.     

Poincaré–Birkhoff–Witt Deformation of Koszul Calabi–Yau Algebras

The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained by He et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.     

The differential analytic index in Simons–Sullivan differential K-theory

We define the Simons–Sullivan differential analytic index by translating the Freed–Lott differential analytic index via explicit ring isomorphisms between Freed–Lott differential K-theory and Simons–Sullivan differential K-theory. We prove the differential Grothendieck–Riemann–Roch theorem in Simons–Sullivan differential K-theory using a theorem of Bismut.     

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class \({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\), we recover isomorphism classes of geometric string structures on Spin _n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class \({\frac{1}{2} p_1}\) together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.     

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.     

Longest increasing subsequences,Plancherel-type measure and the Hecke insertion algorithm

We define and study the Plancherel–Hecke probability measure on Young diagrams; the Hecke algorithm of Buch–Kresch–Shimozono–Tamvakis–Yong is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of Thomas–Yong on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of Schensted and of Knuth, respectively, on the Schensted and Robinson–Schensted–Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of Vershik–Kerov, Logan–Shepp and others on the “longest increasing subsequence problem” for permutations. We also include a related extension of Aldous–Diaconis on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K-theoretic Schubert calculus.     

Kähler–Einstein metrics along the smooth continuity method

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kähler–Einstein manifolds. We also give analogous results for twisted Kähler–Einstein metrics and Kahler–Ricci solitons.     

Weighted matrix means and symmetrization procedures

Here we prove the convergence of the Ando–Li–Mathias and Bini–Meini–Poloni procedures for matrix means. Actually it is proved here that for a two-variable function which maps pairs of positive definite matrices to a positive definite matrix and is not greater than the square mean of two positive definite matrices, the Ando–Li–Mathias and Bini–Meini–Poloni procedure converges. In order to be able to set up the Bini–Meini–Poloni procedure, a weighted two-variable matrix mean is also needed. Therefore a definition of a two-variable weighted matrix mean corresponding to every symmetric matrix mean is also given. It is also shown here that most of the properties considered by Ando, Li and Mathias for the n-variable geometric mean hold for all of these n-variable maps that we obtain by this two limiting process for all two-variable matrix means. As a consequence it also follows that the Bini–Meini–Poloni procedure converges cubically for every matrix mean.     

Li–Yorke sensitive and weak mixing dynamical systems

Akin and Kolyada in 2003 [E. Akin, S. Kolyada, Li–Yorke sensitivity, Nonlinearity 16 (2003), pp. 1421–1433] introduced the notion of Li–Yorke sensitivity. They proved that every weak mixing system (X, T), where X is a compact metric space and T a continuous map of X is Li–Yorke sensitive. An example of Li–Yorke sensitive system without weak mixing factors was given in [M. ?iklová, Li–Yorke sensitive minimal maps, Nonlinearity 19 (2006), pp. 517–529] (see also [M. ?iklová-Mlíchová, Li–Yorke sensitive minimal maps II, Nonlinearity 22 (2009), pp. 1569–1573]). In their paper, Akin and Kolyada conjectured that every minimal system with a weak mixing factor, is Li–Yorke sensitive. We provide arguments supporting this conjecture though the proof seems to be difficult.     

Fibrations with few rational points

We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning–Dietmann, Bright–Browning–Loughran, Graber–Harris–Mazur–Starr, et al.     

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.     

Representations and T*-Extensions of Hom–Jordan–Lie Algebras

The purpose of this article is to study representations and T*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties of T*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application.     

Monomial Gorenstein algebras and the stably Calabi–Yau property

A celebrated result by Keller–Reiten says that 2-Calabi–Yau tilted algebras are Gorenstein and stably 3-Calabi–Yau. This note shows that the converse holds in the monomial case: a 1-Gorenstein monomial algebra and stably 3-Calabi–Yau is 2-Calabi–Yau tilted, moreover is Jacobian. We study the case of other stably Calabi–Yau Gorenstein monomial algebras.

     

Smooth perturbations the self–adjoint operators defined by the ℋ︁–2–construction

The existence of the wave operators for the self–adjoint operator constructed by ??_–2–construction is discussed in the abstract theory by using Kato's smooth perturbation method. Moreover, the s–matrix is calculated explicitly.     

Robin spectral rigidity of nearly circular domains with a reflectional symmetry

This is a note on a paper of De Simoi–Kaloshin–Wei. We show that by combining their techniques with the wave trace invariants of Guillemin–Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of De Simoi–Kaloshin–Wei to the case of Robin boundary conditions. We will consider the same generic subset as did by De Simoi–Kaloshin–Wei of smooth strictly convex ?₂-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary ?₂-symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.     

Canonical reduction of self-dual Yang–Mills equations to Fitzhugh–Nagumo equation and exact solutions

The (constrained) canonical reduction of four-dimensional self-dual Yang–Mills theory to two-dimensional Fitzhugh–Nagumo and the real Newell–Whitehead equations are considered. On the other hand, other methods and transformations are developed to obtain exact solutions for the original two-dimensional Fitzhugh–Nagumo and Newell–Whitehead equations. The corresponding gauge potential A_μ and the gauge field strengths F_μν are also obtained. New explicit and exact traveling wave and solitary solutions (for Fitzhugh–Nagumo and Newell–Whitehead equations) are obtained by using an improved sine-cosine method and the Wu’s elimination method with the aid of Mathematica.     

On the Nonexistence of Global Weak Solutions to the Navier–Stokes–Poisson Equations in ℝ N

In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ? ^N , N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ? ^N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.     

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

In this short note, we generalized an energy estimate due to Malchiodi–Martinazzi (J Eur Math Soc 16:893–908, 2014) and Mancini–Martinazzi (Calc Var 56:94, 2017). As an application, we used it to reprove existence of extremals for Trudinger–Moser inequalities of Adimurthi–Druet type on the unit disc. Such existence problems in general cases had been considered by Yang  (Trans Am Math Soc 359:5761–5776, 2007; J Differ Equ 258:3161–3193, 2015) and Lu–Yang (Discrete Contin Dyn Syst 25:963–979, 2009) by using another method.