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”.     

Decomposition of Graphs into (k,r)‐Fans and Single Edges

Let $\phi (n,H)$ be the largest integer such that, for all graphs G on n vertices, the edge set $E(G)$ can be partitioned into at most $\phi (n,H)$ parts, of which every part either is a single edge or forms a graph isomorphic to H. Pikhurko and Sousa conjectured that $\phi (n,H)=\mathrm{ex}(n,H)$ for $\chi (H)⩾3$ and all sufficiently large n, where $\mathrm{ex}(n,H)$ denotes the maximum number of edges of graphs on n vertices that do not contain H as a subgraph. A $(k,r)$‐fan is a graph on $(r-1)k+1$ vertices consisting of k cliques of order r that intersect in exactly one common vertex. In this article, we verify Pikhurko and Sousa's conjecture for $(k,r)$‐fans. The result also generalizes a result of Liu and Sousa.     

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.     

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.     

Exceptions and characterization results for type‐1 λ‐designs

Let $X$ be a finite set with $v$ elements, called points and $\beta$ be a family of subsets of $X$, called blocks. A pair $\left(X,\beta \right)$ is called $\lambda$‐design whenever $\mid \beta \mid =\mid X\mid$ and

• 1. for all $B_i,B_j\in \beta ,i\ne j,\mid B_i\cap B_j\mid =\lambda$;
• 2. for all $B_j\in \beta ,\mid B_j\mid =k_j>\lambda$, and not all $k_j$ are equal.

The only known examples of $\lambda$‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$‐designs are type‐1. Let $r,r^{*}?(r>r^{*})$ be replication numbers of a $\lambda$‐design $D=\left(X,\beta \right)$ and $g=\text{gcd}\left(r?1,r^{*}?1\right),m=\text{gcd}\left(\left(r?r^{*}\right)∕g,\lambda \right)$, and $m\prime =m$, if $m$ is odd and $m\prime =m∕2$, otherwise. For distinct points $x$ and $y$ of $D$, let $\lambda \left(x,y\right)$ denote the number of blocks of $X$ containing $x$ and $y$. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if $\frac{r\left(r?1\right)}{\left(v?1\right)}?\frac{k\left(r?r^{*}\right)}{m\prime \left(v?1\right)}$ are not integers for $k=1,2,\text{…},m\prime ?1$, then $D$ is type‐1. As an application of these results, we show that for fixed positive integer $\theta$ there are finitely many nontype‐1 $\lambda$‐designs with $r=r^{*}+\theta$. If $r?r^{*}=27$ or $r?r^{*}=4p$ and $r^{*}\ne {\left(p?1\right)}^2$, or $v=7p+1$ such that $p?1,13\left(\mathrm{mod}21\right)$ and $p?4,9,19,24\left(\mathrm{mod}35\right)$, where $p$ is a positive prime, then $D$ is type‐1. We further obtain several inequalities involving $\lambda (x,y)$, where equality holds if and only if D is type‐1.     

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.     

A note on 3‐partite graphs without 4‐cycles

Let $C_4$ be a cycle of order 4. Write $ex\left(n,n,n,C_4\right)$ for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has $n$ vertices that have no subgraph isomorphic to $C_4$. In this paper, we show that $ex\left(n,n,n,C_4\right)\ge \frac{3}{2}n\left(p+1\right)$, where $n=\frac{p\left(p?1\right)}{2}$ and $p$ is a prime number. Note that $ex\left(n,n,n,C_4\right)\le \left(\frac{3\sqrt{2}}{2}+o\left(1\right)\right)n^{\frac{3}{2}}$ from Tait and Timmons's works. Since for every integer $m$, one can find a prime $p$ such that $m\le p\le \left(1+o\left(1\right)\right)m$, we obtain that $\underset{n\to \infty }{\lim }\frac{ex\left(n,n,n,C_4\right)}{\frac{3\sqrt{2}}{2}{n}^{\frac{3}{2}}}=1$.     

Nonsymmetric 2‐(v,k,λ) designs,with (r,λ) = 1, admitting a solvable flag‐transitive automorphism group of affine type

Nonsymmetric $2‐\left(v,k,\lambda \right)$ designs, with $\left(r,\lambda \right)=1$, admitting a solvable flag‐transitive automorphism group of affine type not contained in $A\mathrm{\Gamma }{L}_1\left(v\right)$ are classified.     

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.     

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.     

New lower bounds on the size of (n,r)‐arcs in PG(2,q)

An $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ is a set of $n$ points such that each line contains at most $r$ of the selected points. It is well known that $\left(n,r\right)$‐arcs in $\text{PG}\left(2,q\right)$ correspond to projective linear codes. Let $m_r\left(2,q\right)$ denote the maximal number $n$ of points for which an $\left(n,r\right)$‐arc in $\text{PG}\left(2,q\right)$ exists. In this paper we obtain improved lower bounds on $m_r\left(2,q\right)$ by explicitly constructing $\left(n,r\right)$‐arcs. Some of the constructed $\left(n,r\right)$‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.     

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

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.