Ground state solutions for a modified fractional Schrödinger equation with critical exponent

In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations $${(-\mathrm{\Delta })}^{\alpha }u+\mu u+\kappa [{(-\mathrm{\Delta })}^{\alpha }{u}^2]u=\sigma |u{|}^{p-1}u+|u{|}^{q-2}u,x\in R^N,$$ where $N\ge 2$, $\alpha \in (0,1)$, $\mu$, $\sigma$ and $\kappa$ are positive parameters, $2<p+1<q\le 2_{\alpha }^{\ast }:=\frac{2N}{N-2\alpha }$, ${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian of order $\alpha$. For the case $2<p+1<q<2_{\alpha }^{\ast }$ and the case $q=2_{\alpha }^{\ast }$, the existence results of ground state solutions are given, respectively.     

Variational formulation for nonlinear impulsive fractional differential equations with (p,q)-Laplacian operator

In this paper, we focus on a class of generalized (p, q)-Laplacian-type impulsive fractional differential equation for 1 < p≤q < ∞, with a nonlinearity f containing fractional derivatives ${}_0{D}_t^{\alpha }u$ and ${}_0{D}_t^{\beta }u$ simultaneously and being imposed on mild assumptions contrasting with previous works. The underlying idea is based on the Mountain pass theorem and the iterative technique to obtain the existence of nontrivial solutions for our equation. Our work extends and supplements some existing results.     

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

In this paper, we mainly study the local energy equation of the weak solutions of the compressible isentropic MHD equation defined on ${\mathrm{??}}^3$. We prove that the regularity of the solution is sufficient to guarantee the balance of the total energy in the $B_3^{\alpha ,\infty }((0,T)\times {\mathrm{??}}^3)$ space. We adopt a variant of the method of Feireisl et al.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

Degrees of categoricity of trees and the isomorphism problem

In this paper, we show that for any computable ordinal α, there exists a computable tree of rank $\alpha +1$ with strong degree of categoricity ${\mathbf{0}}^{(2\alpha )}$ if α is finite, and with strong degree of categoricity ${\mathbf{0}}^{(2\alpha +1)}$ if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with strong degree of categoricity ${\mathbf{0}}^{(\alpha )}$ (which equals ${\mathbf{0}}^{(2\alpha )}$). It follows from our proofs that, for every computable ordinal $\alpha >0$, the isomorphism problem for trees of rank α is ${\mathrm{\Pi }}_{2\alpha }^0$‐complete.     

The extremal function for K9= minors

We prove the extremal function for $K_9^{=}$ minors, where $K_9^{=}$ denotes the complete graph $K_9$ with two edges removed. In particular, we show that any graph with $n\ge 8$ vertices and at least $6n-20$ edges either contains a $K_9^{=}$ minor or is isomorphic to a graph obtained from disjoint copies of $K_8$ and $K_{2,2,2,2,2}$ by identifying cliques of size 5. We utilize computer assistance to prove one of our lemmas.     

Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface $X\subset {\mathbb{P}}^{n+1}$ of degree $d⩾2$, we study the cones $V_p^h\subset {\mathbb{P}}^{n+1}$ swept out by lines having contact order $h⩾2$ at a point $p\in X$. In particular, we prove that if X is general, then for any $p\in X$ and $2⩽h⩽\min \{n+1,d\}$, the cone $V_p^h$ has dimension exactly $n+2-h$. Moreover, when X is a very general hypersurface of degree $d⩾2n+2$, we describe the relation between the cones $V_p^h$ and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies $d-⌊\frac{\sqrt{16n+25}-3}{2}⌋⩽conn.gon(X)⩽d-⌊\frac{\sqrt{8n+1}+1}{2}⌋$.     

Backward error analysis of specified eigenpairs for sparse matrix polynomials

This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include $T$-symmetric, $T$-skew-symmetric, Hermitian, skew Hermitian, $T$-even, $T$-odd, $H$-even, $H$-odd, $T$-palindromic, $T$-anti-palindromic, $H$-palindromic, and $H$-anti-palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real-life applications.     

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters $t_1,\dots ,t_k$. A formula in this language defines a parametric set $S_{\mathbf{t}}?{\mathbb{Z}}^d$ as $\mathbf{t}$ varies in ${\mathbb{Z}}^k$, and we examine the counting function $|S_{\mathbf{t}}|$ as a function of t . For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming $\mathbf{P}\ne \mathbf{NP}$) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1,t_2}|$ is not even polynomial‐time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_{\mathbf{t}}?{\mathbb{Z}}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_{\mathbf{t}}|$ is always polynomial‐time computable in the size of t , and in fact can be represented using the gcd and similar functions.     

A novel result on regularity of time-periodic Lagrangians

The regularity of the Peierls barrier $h_c^{\infty }$ for time-periodic Lagrangian systems is equivalent to the convergence of the associated Lax-Oleinik semigroup. In this paper, we established a novel result on the regularity, which has been applied to the convergence and nonconvergence of the Lax-Oleinik semigroup for time-periodic Lagrangians.     

Legendre G‐array pairs and the theoretical unification of several G‐array families

We investigate how Legendre $G$‐array pairs are related to several different perfect binary $G$‐array families. In particular we study the relations between Legendre $G$‐array pairs, Sidelnikov‐Lempel‐Cohn‐Eastman ${\mathbb{Z}}_{q?1}$‐arrays, Yamada‐Pott $G$‐array pairs, Ding‐Helleseth‐Martinsen ${\mathbb{Z}}_2\times {\mathbb{Z}}_p^m$‐arrays, Yamada ${\mathbb{Z}}_{\left(q?1\right)∕2}$‐arrays, Szekeres ${\mathbb{Z}}_p^m$‐array pairs, Paley ${\mathbb{Z}}_p^m$‐array pairs, and Baumert ${\mathbb{Z}}_{p_1}^{m_1}\times {\mathbb{Z}}_{p_2}^{m_2}$‐array pairs. Our work also solves one of the two open problems posed by Ding. Moreover, we provide several computer search‐based existence and nonexistence results regarding Legendre ${\mathbb{Z}}_n$‐array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre ${\mathbb{Z}}_{57}$‐array pair.     

Noncompact quasi‐Einstein manifolds conformal to a Euclidean space

The goal of this article is to investigate nontrivial m‐quasi‐Einstein manifolds globally conformal to an n‐dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n?1)$‐dimensional translation group, we provide a complete classification when $\lambda =0$ and $m\ge 1$ or $m=2?n$.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.