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An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, and has garnered significant interest among researchers. An $(r,\delta)$-LRC is called an optimal code if its parameters achieve the Singleton-like bound. In this paper, we construct three classes of $q$-ary optimal cyclic $(r,\delta)$-LRCs with new parameters by investigating the defining sets of cyclic codes. Our results generalize the related work of \cite{Chen2022,Qian2020}, and the obtained optimal cyclic $(r, \delta)$-LRCs have flexible parameters. A lot of numerical examples of optimal cyclic $(r, \delta)$-LRCs are given to show that our constructions are capable of generating new optimal cyclic $(r, \delta)$-LRCs.

A fully discrete semi-convex-splitting finite-element scheme with stabilization for a degenerate Cahn-Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and the solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFEM illustrate the phase segregation and pattern formation.

In this paper we generalize the notion of $n$-equivalence relation introduced by Chen et al. in \cite{Chen2014} to classify constacyclic codes of length $n$ over a finite field $\mathbb{F}_q$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call this relation $(n,\sigma)$-equivalence relation, where $n$ is the length of the code and $ \sigma$ is an automorphism of the finite field. We compute the number of $(n,\sigma)$-equivalence classes, and we give conditions on $ \lambda$ and $\mu$ for which $(\sigma, \lambda)$-constacyclic codes and $(\sigma, \lambda)$-constacyclic codes are equivalent with respect to our $(n,\sigma)$-equivalence relation. Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are equivalent to cyclic codes. We also prove that when $q$ is even and $\sigma$ is the Frobenius autmorphism, skew constacyclic codes of length $n$ are equivalent to cyclic codes when $\gcd(n,r)=1$. Finally we give some examples as applications of the theory developed here.

Currently, the best known tradeoff between approximation ratio and complexity for the Sparsest Cut problem is achieved by the algorithm in [Sherman, FOCS 2009]: it computes $O(\sqrt{(\log n)/\varepsilon})$-approximation using $O(n^\varepsilon\log^{O(1)}n)$ maxflows for any $\varepsilon\in[\Theta(1/\log n),\Theta(1)]$. It works by solving the SDP relaxation of [Arora-Rao-Vazirani, STOC 2004] using the Multiplicative Weights Update algorithm (MW) of [Arora-Kale, JACM 2016]. To implement one MW step, Sherman approximately solves a multicommodity flow problem using another application of MW. Nested MW steps are solved via a certain ``chaining'' algorithm that combines results of multiple calls to the maxflow algorithm. We present an alternative approach that avoids solving the multicommodity flow problem and instead computes ``violating paths''. This simplifies Sherman's algorithm by removing a need for a nested application of MW, and also allows parallelization: we show how to compute $O(\sqrt{(\log n)/\varepsilon})$-approximation via $O(\log^{O(1)}n)$ maxflows using $O(n^\varepsilon)$ processors. We also revisit Sherman's chaining algorithm, and present a simpler version together with a new analysis.

In this paper, we consider a problem which we call LTL$_f$ model checking on paths: given a DFA $\mathcal{A}$ and a formula $\phi$ in LTL on finite traces, does there exist a word $w$ such that every path starting in a state of $\mathcal{A}$ and labeled by $w$ satisfies $\phi$? The original motivation for this problem comes from the constrained parts orienting problem, introduced in [Petra Wolf, "Synchronization Under Dynamic Constraints", FSTTCS 2020], where the input constraints restrict the order in which certain states are visited for the first or the last time while reading a word $w$ which is also required to synchronize $\mathcal{A}$. We identify very general conditions under which LTL$_f$ model checking on paths is solvable in polynomial space. For the particular constraints in the parts orienting problem, we consider PSPACE-complete cases and one NP-complete case. The former provide very strong lower bound for LTL$_f$ model checking on paths. The latter is related to (classical) LTL$_f$ model checking for formulas with the until modality only and with no nesting of operators. We also consider LTL$_f$ model checking of the power-set automaton of a given DFA, and get similar results for this setting. For all our problems, we consider the case where the required word must also be synchronizing, and prove that if the problem does not become trivial, then this additional constraint does not change the complexity.

Tensor operations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we establish a few basic properties of the range and null space of a tensor using block circulant matrices and the discrete Fourier matrix. We then discuss the outer inverse of tensors based on $t$-product with a prescribed range and kernel of third-order tensors. We address the relation of this outer inverse with other generalized inverses, such as the Moore-Penrose inverse, group inverse, and Drazin inverse. In addition, we present a few algorithms for computing the outer inverses of the tensors. In particular, a $t$-QR decomposition based algorithm is developed for computing the outer inverses.

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called $k$-times bin packing ($k$BP), where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of fair electricity distribution. In many developing countries, the total electricity demand is higher than the supply capacity. We show that $k$-times bin packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit-Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem.

In the analysis of the $h$-version of the finite-element method (FEM), with fixed polynomial degree $p$, applied to the Helmholtz equation with wavenumber $k\gg 1$, the $\textit{asymptotic regime}$ is when $(hk)^p C_{\rm sol}$ is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here $C_{\rm sol}$ is the norm of the Helmholtz solution operator, normalised so that $C_{\rm sol} \sim k$ for nontrapping problems. In the $\textit{preasymptotic regime}$, one expects that if $(hk)^{2p}C_{\rm sol}$ is sufficiently small, then (for physical data) the relative error of the Galerkin solution is controllably small. In this paper, we prove the natural error bounds in the preasymptotic regime for the variable-coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition $\textit{either}$ realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball $\textit{or}$ approximated either by a radial perfectly-matched layer (PML) or an impedance boundary condition. Previously, such bounds for $p>1$ were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition. Our result is obtained via a novel generalisation of the "elliptic-projection" argument (the argument used to obtain the result for $p=1$) which can be applied to a wide variety of abstract Helmholtz-type problems.

The paper studies nonstationary high-dimensional vector autoregressions of order $k$, VAR($k$). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, $T$, and the number of coordinates, $N$, are assumed to be large and of the same order. Under this regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai-Bartlett, and Hotelling-Lawley tests for cointegration are derived: the test statistics converge to nonrandom integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy$_1$ point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond those considered in our theorems. The paper presents empirical implementations of the approach for the analysis of S$\&$P$100$ stocks and of cryptocurrencies. The latter example has a strong presence of multiple cointegrating relationships, while the results for the former are consistent with the null of no cointegration.

In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.