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Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.

Let $\{\Lambda_n=\{\lambda_{1,n},\ldots,\lambda_{d_n,n}\}\}_n$ be a sequence of finite multisets of real numbers such that $d_n\to\infty$ as $n\to\infty$, and let $f:\Omega\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function defined on a domain $\Omega$ with $0<\mu_d(\Omega)<\infty$, where $\mu_d$ is the Lebesgue measure in $\mathbb R^d$. We say that $\{\Lambda_n\}_n$ has an asymptotic distribution described by $f$, and we write $\{\Lambda_n\}_n\sim f$, if \[ \lim_{n\to\infty}\frac1{d_n}\sum_{i=1}^{d_n}F(\lambda_{i,n})=\frac1{\mu_d(\Omega)}\int_\Omega F(f({\boldsymbol x})){\rm d}{\boldsymbol x}\qquad\qquad(*) \] for every continuous function $F$ with bounded support. If $\Lambda_n$ is the spectrum of a matrix $A_n$, we say that $\{A_n\}_n$ has an asymptotic spectral distribution described by $f$ and we write $\{A_n\}_n\sim_\lambda f$. In the case where $d=1$, $\Omega$~is a bounded interval, $\Lambda_n\subseteq f(\Omega)$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, B\"ottcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[\lambda_{1,n},\ldots,\lambda_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\ldots,f(x_{d_n,n})]$, i.e., \[ \lim_{n\to\infty}\,\max_{i=1,\ldots,d_n}|f(x_{i,n})-\lambda_{\tau_n(i),n}|=0, \qquad\qquad(**) \] where $x_{1,n},\ldots,x_{d_n,n}$ is a uniform grid in $\Omega$ and $\tau_n$ is the sorting permutation. We extend this result to the case where $d\ge1$ and $\Omega$ is a Peano--Jordan measurable set (i.e., a bounded set with $\mu_d(\partial\Omega)=0$). See the rest of the abstract in the manuscript.

We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha,\beta\ge 0$, $\alpha\neq \beta$, $$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=\psi_{\alpha,\beta}(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by $\psi_{\alpha,\beta}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020}, is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $\psi_{1,0}$. We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries ones.

We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$

A pair of linear codes whose intersection is of dimension $\ell$, where $\ell$ is a non-negetive integer, is called an $\ell$-intersection pair of codes. This paper focuses on studying $\ell$-intersection pairs of $\lambda_i$-constacyclic, $i=1,2,$ and conjucyclic codes. We first characterize an $\ell$-intersection pair of $\lambda_i$-constacyclic codes. A formula for $\ell$ has been established in terms of the degrees of the generator polynomials of $\lambda_i$-constacyclic codes. This allows obtaining a condition for $\ell$-linear complementary pairs (LPC) of constacyclic codes. Later, we introduce and characterize the $\ell$-intersection pair of conjucyclic codes over $\mathbb{F}_{q^2}$. The first observation in the process is that there are no non-trivial linear conjucyclic codes over finite fields. So focus on the characterization of additive conjucyclic (ACC) codes. We show that the largest $\mathbb{F}_q$-subcode of an ACC code over $\mathbb{F}_{q^2}$ is cyclic and obtain its generating polynomial. This enables us to find the size of an ACC code. Furthermore, we discuss the trace code of an ACC code and show that it is cyclic. Finally, we determine $\ell$-intersection pairs of trace codes of ACC codes over $\mathbb{F}_4$.

Challenges with data in the big-data era include (i) the dimension $p$ is often larger than the sample size $n$ (ii) outliers or contaminated points are frequently hidden and more difficult to detect. Challenge (i) renders most conventional methods inapplicable. Thus, it attracts tremendous attention from statistics, computer science, and bio-medical communities. Numerous penalized regression methods have been introduced as modern methods for analyzing high-dimensional data. Disproportionate attention has been paid to the challenge (ii) though. Penalized regression methods can do their job very well and are expected to handle the challenge (ii) simultaneously. Most of them, however, can break down by a single outlier (or single adversary contaminated point) as revealed in this article. The latter systematically examines leading penalized regression methods in the literature in terms of their robustness, provides quantitative assessment, and reveals that most of them can break down by a single outlier. Consequently, a novel robust penalized regression method based on the least sum of squares of depth trimmed residuals is proposed and studied carefully. Experiments with simulated and real data reveal that the newly proposed method can outperform some leading competitors in estimation and prediction accuracy in the cases considered.

A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of $K_{1, 3}$. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. A {\em butterfly} is a graph obtained from two traingles by sharing exactly one vertex. A graph $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick {\em et al}). As an application, we show that $\chi(G)\le\binom{\omega(G)+1}{2}$ if $G$ is (fork, gem)-free or (fork, butterfly)-free.

Given a matrix-valued function $\mathcal{F}(\lambda)=\sum_{i=1}^d f_i(\lambda) A_i$, with complex matrices $A_i$ and $f_i(\lambda)$ analytic functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity for $\mathcal{F}(\lambda)$. The closest matrix-valued function $\widetilde {\mathcal{F}}(\lambda)$ with respect to the Frobenius norm is approximated using an iterative method. The condition of singularity on the matrix-valued function is translated into a numerical constraint for a suitable minimization problem. Unlike the case of matrix polynomials, in the general setting of matrix-valued functions the main issue is that the function $\det ( \widetilde{\mathcal{F}}(\lambda) )$ may have an infinite number of roots. The main feature of the numerical method consists in the possibility of extending it to different structures, such as sparsity patterns induced by the matrix coefficients.

We consider the minimal thermodynamic cost of an individual computation, where a single input $x$ is mapped to a single output $y$. In prior work, Zurek proposed that this cost was given by $K(x\vert y)$, the conditional Kolmogorov complexity of $x$ given $y$ (up to an additive constant which does not depend on $x$ or $y$). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that $K(x\vert y)$ is a minimal cost of mapping $x$ to $y$ that must be paid using some combination of heat, noise, and protocol complexity, implying a tradeoff between these three resources. Our result is a kind of "algorithmic fluctuation theorem" with implications for the relationship between the Second Law and the Physical Church-Turing thesis.

For the Euler scheme of the stochastic linear evolution equations, discrete stochastic maximal $ L^p $-regularity estimate is established, and a sharp error estimate in the norm $ \|\cdot\|_{L^p((0,T)\times\Omega;L^q(\mathcal O))} $, $ p,q \in [2,\infty) $, is derived via a duality argument.