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Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

We define game semantics for the constructive $\mu$-calculus and prove its equivalence to bi-relational semantics. We use the game semantics to prove that the $\mu$-calculus collapses to modal logic over constructive variants of $\mathsf{S5}$. Finally, we use the collapse to prove the completeness of constructive variants of $\mathsf{S5}$ with fixed-point operators.

This paper presents an algorithm for the simulation of Hawkes-type processes where the intensity is expressed in terms of a continuous-time autoregressive moving average model. We identify upper bounds for both the univariate and the multivariate intensity functions that are used to develop simulation algorithms based on the thinning technique.

The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the $\ell_1$-norm, showing that a makespan of at most $5r$ can always be achieved, where $r$ is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most $5r$ can be computed in optimal time $O(n)$. Both bounds, the time and the makespan are optimal. This implies a new upper bound of $5\sqrt{2}r \approx 7.07r$ on the makespan in the $\ell_2$-norm, improving the best known bound so far $(5+2\sqrt{2}+\sqrt{5})r \approx 10.06r$.

Handling multiplicity without losing much power has been a persistent challenge in various fields that often face the necessity of managing numerous statistical tests simultaneously. Recently, $p$-value combination methods based on heavy-tailed distributions, such as a Cauchy distribution, have received much attention for their ability to handle multiplicity without the prescribed knowledge of the dependence structure. This paper delves into these types of $p$-value combinations through the lens of extreme value theory. Distributions with regularly varying tails, a subclass of heavy tail distributions, are found to be useful in constructing such $p$-value combinations. Three $p$-value combination statistics (sum, max cumulative sum, and max) are introduced, of which left tail probabilities are shown to be approximately uniform when the global null is true. The primary objective of this paper is to bridge the gap between current developments in $p$-value combination methods and the literature on extreme value theory, while also offering guidance on selecting the calibrator and its associated parameters.

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,L^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

The classical work of (Arora et al., 1999) provides a scheme that gives, for any $\epsilon>0$, a polynomial time $1-\epsilon$ approximation algorithm for dense instances of a family of $\mathcal{NP}$-hard problems, such as Max-CUT and Max-$k$-SAT. In this paper we extend and speed up this scheme using a logarithmic number of one-bit predictions. We propose a learning augmented framework which aims at finding fast algorithms which guarantees approximation consistency, smoothness and robustness with respect to the prediction error. We provide such algorithms, which moreover use predictions parsimoniously, for dense instances of various optimization problems.

This paper addresses the computational problem of deciding invertibility (or one to one-ness) of a Boolean map $F$ in $n$-Boolean variables. This problem has a special case of deciding invertibilty of a map $F:\mathbb{F}_{2}^n\rightarrow\mathbb{F}_{2}^n$ over the binary field $\mathbb{F}_2$. Further the problem can be extended and stated over a finite field $\mathbb{F}$ instead of $\mathbb{F}_2$. Algebraic condition for invertibility of $F$ in this special case over a finite field is well known to be equivalent to invertibility of the Koopman operator of $F$ as shown in \cite{RamSule}. In this paper a condition for invertibility is derived in the special case of Boolean maps $F:B_0^n\rightarrow B_0^n$ where $B_0$ is the two element Boolean algebra in terms of \emph{implicants} of Boolean equations. This condition is then extended to the case of general maps in $n$ variables. Hence this condition answers the special case of invertibility of the map $F$ defined over the binary field $\mathbb{F}_2$ alternatively, in terms of implicants instead of the Koopman operator. The problem of deciding invertibility of a map $F$ (or that of finding its $GOE$) over finite fields appears to be distinct from the satisfiability problem (SAT) or the problem of deciding consistency of polynomial equations over finite fields. Hence the well known algorithms for deciding SAT or of solvability using Grobner basis for checking membership in an ideal generated by polynomials is not known to answer the question of invertibility of a map. Similarly it appears that algorithms for satisfiability or polynomial solvability are not useful for computation of $GOE(F)$ even for maps over the binary field $\mathbb{F}_2$.

We study the dependent type theory CaTT, introduced by Finster and Mimram, which presents the theory of weak $\omega$-categories, following the idea that type theories can be considered as presentations of generalized algebraic theories. Our main contribution is a formal proof that the models of this type theory correspond precisely to weak $\omega$-categories, as defined by Maltsiniotis, by generalizing a definition proposed by Grothendieck for weak $\omega$-groupoids: Those are defined as suitable presheaves over a cat-coherator, which is a category encoding structure expected to be found in an $\omega$-category. This comparison is established by proving the initiality conjecture for the type theory CaTT, in a way which suggests the possible generalization to a nerve theorem for a certain class of dependent type theories

We give an alternative derivation of $(N,N)$-isogenies between fastKummer surfaces which complements existing works based on the theory oftheta functions. We use this framework to produce explicit formulae for thecase of $N = 3$, and show that the resulting algorithms are more efficient thanall prior $(3, 3)$-isogeny algorithms.