Riesz potential operator in continual variable exponents Herz spaces

We find conditions on the variable parameters $p\left(x\right),q\left(t\right)$ and $\alpha \left(t\right)$, defining the Herz space $H^{p(·),q(·),\alpha (·)}\left({\mathbb{R}}^n\right)$, for the validity of Sobolev type theorem for the Riesz potential operator to be bounded within the frameworks of such variable exponents Herz spaces. We deal with a “continual” version of Herz spaces (which coincides with the “discrete” one when q is constant).     

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 $m=2?n$.     

Extensions of Laba‐Wang's condition for spectral pairs

Let ${\mu }_{M,D}$ be a self‐affine measure associated with an expanding matrix $M\in M_n\left(\mathbf{Z}\right)$ and a finite digit set $D?{\mathbf{Z}}^n$. We consider in this paper the spectrality of ${\mu }_{M,D}$. In the case when $(M^{?1}D,S)$ is a compatible pair for some $S?{\mathbf{Z}}^n$, a necessary condition is obtained for the spectral pair $({\mu }_{M,D},\Lambda (M,S))$. This condition is shown to be equivalent to the known necessary conditions for the same spectral pair. Moreover, we prove that all these necessary conditions are not sufficient in the higher dimensions, but they are sufficient in the dimension one. This extends Laba‐Wang's condition for spectral pairs.     

Minimal energy problems for strongly singular Riesz kernels

We study minimal energy problems for strongly singular Riesz kernels ${|\mathbf{x}-\mathbf{y}|}^{\alpha -n}$, where $n\ge 2$ and $\alpha \in (-1,1)$, considered for compact $(n-1)$‐dimensional $C^{\infty }$‐manifolds Γ immersed into ${\mathbb{R}}^n$. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such minimization problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order $\beta =1-\alpha$ on Γ. The measures with finite energy are shown to be elements from the Sobolev space $H^{\beta /2}(\mathrm{\Gamma })$, $0<\beta <2$, and the corresponding minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D. P. Hardin and E. B. Saff.     

Moufang sets of mixed type F4

Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so‐called mixed groups of type ${\mathsf{F}}_4$ in characteristic 2, obtained as the fixed point set under a suitable involution.     

Large‐time behavior of solutions to a 1D liquid crystal system

In this paper, we prove the large‐time behavior, as time tends to infinity, of solutions in $H^i\times H_0^i\times H^{i+1}(i=1,2)$ and $H^4\times H_0^4\times H^4$ for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier‐Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.     

Rigidity of ‐quasi Einstein manifolds

This paper deals with the study on ‐quasi Einstein manifolds. First, we give some characterizations of an ‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an ‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an ‐dimensional Riemannian manifold and q is a smooth function on I . Finally, a non‐trivial example of an ‐quasi Einstein manifold verifying our results in terms of the potential function is presented.     

Critical point equation on four‐dimensional compact manifolds

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M⁴. In fact, we shall show that for a nontrivial must be isometric to a sphere and f is some height function on      

On infinite‐dimensional Banach spaces and weak forms of the axiom of choice

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to $\mathsf{AC}$; “ $\mathbb{R}$ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$”, thus the latter statement is true in every Fraenkel‐Mostowski model of $\mathsf{ZFA}$; “No infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$” is not provable in $\mathsf{ZF}$; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality $<2^{{\aleph }_0}$” is provable in $\mathsf{ZF}$; ${\mathsf{AC}}_{\mathsf{fin}}^{{\aleph }_{\mathsf{0}}}$ (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X , and $\epsilon >0$, then there is a unit vector $x\in X$ such that $||y||\le (1+\epsilon )||y+\alpha x||$ for all $y\in Y$ and all scalars α”) is provable in $\mathsf{ZF}$; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball $B_X=\{x\in X:||x||\le 1\}$ is compact” is provable in $\mathsf{ZF}$; $\mathsf{DC}$ (Principle of Dependent Choices) + “ $\mathbb{R}$ can be well‐ordered” does not imply the Hahn‐Banach Theorem ( $\mathsf{HB}$) in $\mathsf{ZF}$; $\mathsf{HB}$ and “no infinite‐dimensional Banach space has a Hamel basis of cardinality $<2^{{\aleph }_0}$” are independent from each other in $\mathsf{ZF}$; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between ${\mathsf{AC}}^{{\aleph }_0}$ (the Axiom of Countable Choice) and ${\mathsf{AC}}_{\mathsf{fin}}^{{\aleph }_{\mathsf{0}}}$; $\mathsf{DC}$ implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ${\mathsf{AC}}^{{\aleph }_0}$; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of $\mathsf{ZF}+{\mathsf{AC}}^{{\aleph }_0}$, but not a theorem of $\mathsf{ZF}$; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality $\ge 2^{{\aleph }_0}$” implies “every Dedekind‐finite set is finite”.     

Maximal functions for multipliers on stratified groups

In this paper we obtain a refined $L^p$ bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi‐parameter multipliers on product spaces of stratified groups. As an application we find a refined $L^p$ bound for maximal functions of joint spectral multipliers on Heisenberg group.     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

Extremal results on feedback arc sets in digraphs

For an oriented graph $G$, let $\beta (G)$ denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any $m$-edge oriented graph $G$ satisfies $\beta (G)=m/2-\mathrm{\Omega }(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the bound $\beta (G)=m/2-\mathrm{\Omega }(m^{3/4})$ is sharp as a function of $m$ if $B$ is not bipartite, but the exponent $3/4$ in the lower order term can be improved if $B$ is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in $[3/4,1]$ is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.     

On steepest descent curves for quasi convex families in Rn

A connected, linearly ordered subset $\gamma ?{\mathbb{R}}^n$ satisfying $$x_1,x_2,x_3\in \gamma ,\text{and}{x}_1?x_2?x_3\to |x_2?x_1|\le |x_3?x_1|$$ is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these curves are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length bounds are proved for them.     

Group divisible covering designs with block size four

Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k‐GDCD of type $g^u$ is a covering number denoted by $C(k,g^u)$. When $k=3$, the values of $C(3,g^u)$ have been determined completely for all possible pairs $(g,u)$. When $k=4$, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type ${(g,h)}^u$. Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for $(g,u)$ whose status remains undetermined arise when $g\equiv 7mod12$ and $u\equiv 3mod6$, when $g\equiv 11,14,17,23mod24$ and $u\equiv 5mod6$, and in several small families for which one of g and u is fixed.     

Packing list-colorings

List coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colors to each vertex $v\in V(G)$, we study the existence of $k$ pairwise-disjoint proper colorings of $G$ using colors from these lists. We may refer to this as a list-packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$. (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalizations of the problem above in the same spirit.     

Random perfect matchings in regular graphs

We prove that in all regular robust expanders $G$, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.     

A Ramsey–Turán theory for tilings in graphs

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted ${\alpha }_r(G)$, is the largest size of a $K_r$-free set of vertices in $G$. In this article, we discuss Ramsey–Turán-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.