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In Linear Hashing ($\mathsf{LH}$) with $\beta$ bins on a size $u$ universe ${\mathcal{U}=\{0,1,\ldots, u-1\}}$, items $\{x_1,x_2,\ldots, x_n\}\subset \mathcal{U}$ are placed in bins by the hash function $$x_i\mapsto (ax_i+b)\mod p \mod \beta$$ for some prime $p\in [u,2u]$ and randomly chosen integers $a,b \in [1,p]$. The "maxload" of $\mathsf{LH}$ is the number of items assigned to the fullest bin. Expected maxload for a worst-case set of items is a natural measure of how well $\mathsf{LH}$ distributes items amongst the bins. Fix $\beta=n$. Despite $\mathsf{LH}$'s simplicity, bounding $\mathsf{LH}$'s worst-case maxload is extremely challenging. It is well-known that on random inputs $\mathsf{LH}$ achieves maxload $\Omega\left(\frac{\log n}{\log\log n}\right)$; this is currently the best lower bound for $\mathsf{LH}$'s expected maxload. Recently Knudsen established an upper bound of $\widetilde{O}(n^{1 / 3})$. The question "Is the worst-case expected maxload of $\mathsf{LH}$ $n^{o(1)}$?" is one of the most basic open problems in discrete math. In this paper we propose a set of intermediate open questions to help researchers make progress on this problem. We establish the relationship between these intermediate open questions and make some partial progress on them.

This paper concerns an expansion of first-order Belnap-Dunn logic which is called $\mathrm{BD}^{\supset,\mathsf{F}}$. Its connectives and quantifiers are all familiar from classical logic and its logical consequence relation is very closely connected to the one of classical logic. Results that convey this close connection are established. Fifteen classical laws of logical equivalence are used to distinguish $\mathrm{BD}^{\supset,\mathsf{F}}$ from all other four-valued logics with the same connectives and quantifiers whose logical consequence relation is as closely connected to the logical consequence relation of classical logic. It is shown that several interesting non-classical connectives added to Belnap-Dunn logic in its expansions that have been studied earlier are definable in $\mathrm{BD}^{\supset,\mathsf{F}}$. It is also established that $\mathrm{BD}^{\supset,\mathsf{F}}$ is both paraconsistent and paracomplete. Moreover, a sequent calculus proof system that is sound and complete with respect to the logical consequence relation of $\mathrm{BD}^{\supset,\mathsf{F}}$ is presented.

We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\mathbf{Diff}$ or $\mathbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $\mathbb{R}^n$ decidable? As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $\mathbb{R}^n$ is undecidable when $n-m$ is even and $11m \geq 10n+1$.

For a large class of random constraint satisfaction problems (CSP), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular $k$-NAE-SAT, random $k$-SAT, and random regular $k$-SAT for large enough $k\geq k_0$ where $k_0$ is a large non-explicit constant. Establishing the same for small values of $k\geq 3$ remains an important open problem in the study of random CSPs. In this work, we study two closely related models of random CSPs, namely the $2$-coloring on random $d$-regular $k$-uniform hypergraphs and the random $d$-regular $k$-NAE-SAT model. For every $k\geq 3$, we prove that there is an explicit $d_{\ast}(k)$ which gives a satisfiability upper bound for both of the models. Our upper bound $d_{\ast}(k)$ for $k\geq 3$ matches the prediction from statistical physics for the hypergraph $2$-coloring by Dall'Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, $d_{\ast}(k)$ coincides with the satisfiability threshold of random regular $k$-NAE-SAT for large enough $k\geq k_0$ by Ding, Sly, Sun (2014).

We study the problem of maximizing a non-negative monotone $k$-submodular function $f$ under a knapsack constraint, where a $k$-submodular function is a natural generalization of a submodular function to $k$ dimensions. We present a deterministic $(\frac12-\frac{1}{2e})\approx 0.316$-approximation algorithm that evaluates $f$ $O(n^4k^3)$ times, based on the result of Sviridenko (2004) on submodular knapsack maximization.

We prove lower bounds for the randomized approximation of the embedding $\ell_1^m \rightarrow \ell_\infty^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N \in \mathbb{R}^{n \times m}$. These lower bounds reflect the increasing difficulty of the problem for $m \to \infty$, namely, a term $\sqrt{\log m}$ in the complexity $n$. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(\log\log m)$-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1/2} ( \log n)^{-1/2}$.

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are fourth order accurate schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme.

We first introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a distance is defined via a generalized Benamou-Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimization problem. Some recently proposed models, including the Kantorovich-Bures distance and the Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a complete characterization of the minimizer and explore the topological and geometrical properties of the space $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$. In particular, we show that $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ is a complete geodesic space and exhibits a conic structure. We further investigate the convergence property of the associated discrete transport problem. We present a convergence framework for abstract discretization and then propose a concrete convergent discretization scheme.

Let $F_q$ be the finite field with $q$ elements and $F_q[x_1,\ldots, x_n]$ the ring of polynomials in $n$ variables over $F_q$. In this paper we consider permutation polynomials and local permutation polynomials over $F_q[x_1,\ldots, x_n]$, which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-1)-1$ and local permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-2)$ when $q>3$, extending previous results.

The paper aims to study the performance of the amplitude-based model \newline $\widehat{\mathbf x} \in argmin{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$, where $b_j:=|\langle {\mathbf a}_j,{\mathbf x}_0\rangle|+\eta_j$ and ${\mathbf x}_0\in \mathbb{C}^d$ is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. {However, there are very few results available regarding the estimation performance in the complex case under noisy conditions.} In this paper, {we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem}. Specifically, we show that $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf \eta}\|_2}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m,$ are {i.i.d.} complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\mathbf \eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal ${\mathbf x}_0\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained $\ell_1$ minimization model. { To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.}