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For which unary predicates $P_1, \ldots, P_m$ is the MSO theory of the structure $\langle \mathbb{N}; <, P_1, \ldots, P_m \rangle$ decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each $P_i$ can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable.

In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices $\mathcal{S}(Y)=YY^*$, where $Y=(y_{ij})$ is an $M\times N$ matrix with iid mean $0$ variance $N^{-1}$ entries. We prove a phase transition for its distribution, induced by the fatness of the tail of $y_{ij}$'s. More specifically, we assume that $y_{ij}$ is symmetrically distributed with tail probability $\mathbb{P}(|\sqrt{N}y_{ij}|\geq x)\sim x^{-\alpha}$ when $x\to \infty$, for some $\alpha\in (2,4)$. We show the following conclusions: (i). When $\alpha>\frac83$, the smallest eigenvalue follows the Tracy-Widom law on scale $N^{-\frac23}$; (ii). When $2<\alpha<\frac83$, the smallest eigenvalue follows the Gaussian law on scale $N^{-\frac{\alpha}{4}}$; (iii). When $\alpha=\frac83$, the distribution is given by an interpolation between Tracy-Widom and Gaussian; (iv). In case $\alpha\leq \frac{10}{3}$, in addition to the left edge of the MP law, a deterministic shift of order $N^{1-\frac{\alpha}{2}}$ shall be subtracted from the smallest eigenvalue, in both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by \cite{ALY} which is originally done for the bulk regime of the L\'{e}vy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of nonzero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. If $\mathcal{C}$ is a one-generator quasi-cyclic code, a tighter upper bound on the number of nonzero weights of $\mathcal{C}$ is obtained by considering a larger automorphism subgroup which is generated by the multiplier, $\rho^l$ and the scalar multiplications of $\mathcal{C}$. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}.

Are asymptotic confidence sequences and anytime $p$-values uniformly valid for a nontrivial class of distributions $\mathcal{P}$? We give a positive answer to this question by deriving distribution-uniform anytime-valid inference procedures. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests that enable inference at stopping times -- have been justified nonasymptotically. Nevertheless, asymptotic procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and weak assumptions. While recent work has derived asymptotic analogues of anytime-valid methods with the aforementioned benefits, these were not shown to be $\mathcal{P}$-uniform, meaning that their asymptotics are not uniformly valid in a class of distributions $\mathcal{P}$. Indeed, the anytime-valid inference literature currently has no central limit theory to draw from that is both uniform in $\mathcal{P}$ and in the sample size $n$. This paper fills that gap by deriving a novel $\mathcal{P}$-uniform strong Gaussian approximation theorem, enabling $\mathcal{P}$-uniform anytime-valid inference for the first time. Along the way, our Gaussian approximation also yields a $\mathcal{P}$-uniform law of the iterated logarithm.

We study time integration schemes for $\dot H^1$-solutions to the energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show first-order convergence in $L^2$ for the Lie splitting and convergence order $3/2$ for a corrected Lie splitting. To our knowledge this includes the first error analysis performed for scaling-critical dispersive problems. Our approach is based on discrete-time Strichartz estimates, including one (with a logarithmic correction) for the case of the forbidden endpoint. Our schemes and the Strichartz estimates contain frequency cut-offs.

We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$ admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in [T. Chen, A. Greenbaum, C. Musco, C. Musco, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 787--811] (arXiv:2106.09806) for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.

We define and study the model of patterned non-determinism in bipartite communication complexity, denoted by $PNP^{X\leftrightarrow Y}$. It generalises the known models $UP^{X\leftrightarrow Y}$ and $FewP^{X\leftrightarrow Y}$ through relaxing the constraints on the witnessing structure of the underlying $NP^{X\leftrightarrow Y}$-protocol. It is shown that for the case of total functions $PNP^{X\leftrightarrow Y}$ equals $P^{X\leftrightarrow Y}$ (similarly to $UP^{X\leftrightarrow Y}$ and $FewP^{X\leftrightarrow Y}$). Moreover, the corresponding exhaustive witness-searching problem -- determining the full set of witnesses that lead to the acceptance of a given input pair -- also has an efficient deterministic protocol. The possibility of efficient exhaustive $PNP^{X\leftrightarrow Y}$-search is used to analyse certain three-party communication regime (under the "number in hand" input partition): The corresponding three-party model is shown to be as strong qualitatively as the weakest among its two-party amplifications obtained by allowing free communication between a pair of players.

Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t = \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+\delta}$ deformation retracts to $S^f_t$ for all sufficiently small $\delta > 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.

A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I|/k$ for each $i$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $\alpha_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $\chi_b(H)$ in terms of the maximum degree $\Delta$. Our results match those of recent work of Chakraborti for $k = 2$.

When the unknown regression function of a single variable is known to have derivatives up to the $(\gamma+1)$th order bounded in absolute values by a common constant everywhere or a.e. (i.e., $(\gamma+1)$th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ in the literature. This paper shows that: (i) if $n\leq\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\frac{\log n}{n\log(\log n)}$ and the optimal degree of smoothness to exploit is roughly $\max\left\{ \left\lfloor \frac{\log n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\} $; (ii) if $n>\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ and the optimal degree of smoothness to exploit is $\gamma+1$. The fundamental contribution of this paper is a set of metric entropy bounds we develop for smooth function classes. Some of our bounds are original, and some of them improve and/or generalize the ones in the literature (e.g., Kolmogorov and Tikhomirov, 1959). Our metric entropy bounds allow us to show phase transitions in the minimax optimal MISE rates associated with some commonly seen smoothness classes as well as non-standard smoothness classes, and can also be of independent interest outside the nonparametric regression problems.