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MinRank is an NP-complete problem in linear algebra whose characteristics make it attractive to build post-quantum cryptographic primitives. Several MinRank-based digital signature schemes have been proposed. In particular, two of them, MIRA and MiRitH, have been submitted to the NIST Post-Quantum Cryptography Standardization Process. In this paper, we propose a key-generation algorithm for MinRank-based schemes that reduces the size of the public key to about 50% of the size of the public key generated by the previous best (in terms of public-key size) algorithm. Precisely, the size of the public key generated by our algorithm sits in the range of 328-676 bits for security levels of 128-256 bits. We also prove that our algorithm is as secure as the previous ones.

The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than $2w_{\min}$, where $w_{\min}$ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than $2w_{\min}$. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.

Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.

A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.

Classical multi-sorted equational theories and their free algebras have been fundamental in mathematics and computer science. In this paper, we present a generalization of multi-sorted equational theories from the classical ($Set$-enriched) context to the context of enrichment in a symmetric monoidal category $V$ that is topological over $Set$. Prominent examples of such categories include: various categories of topological and measurable spaces; the categories of models of relational Horn theories without equality, including the categories of preordered sets and (extended) pseudo-metric spaces; and the categories of quasispaces (a.k.a. concrete sheaves) on concrete sites, which have recently attracted interest in the study of programming language semantics. Given such a category $V$, we define a notion of $V$-enriched multi-sorted equational theory. We show that every $V$-enriched multi-sorted equational theory $T$ has an underlying classical multi-sorted equational theory $|T|$, and that free $T$-algebras may be obtained as suitable liftings of free $|T|$-algebras. We establish explicit and concrete descriptions of free $T$-algebras, which have a convenient inductive character when $V$ is cartesian closed. We provide several examples of $V$-enriched multi-sorted equational theories, and we also discuss the close connection between these theories and the presentations of $V$-enriched algebraic theories and monads studied in recent papers by the author and Lucyshyn-Wright.

An important element of the $S$-matrix bootstrap program is the relationship between the modulus of an $S$-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this integral equation cannot be solved analytically and numerical approaches are required. We apply modern machine learning techniques to studying the unitarity constraint. We find that for a given modulus, when a phase exists it can generally be reconstructed to good accuracy with machine learning. Moreover, the loss of the reconstruction algorithm provides a good proxy for whether a given modulus can be consistent with unitarity at all. In addition, we study the question of whether multiple phases can be consistent with a single modulus, finding novel phase-ambiguous solutions. In particular, we find a new phase-ambiguous solution which pushes the known limit on such solutions significantly beyond the previous bound.

Analysis-suitable $G^1$ (AS-$G^1$) multi-patch spline surfaces [4] are particular $G^1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C^1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties [16]. We present a novel local approach for the design of AS-$G^1$ multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-$G^1$ multi-patch spline surface by approximating a given $G^1$-smooth but non-AS-$G^1$ multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-$G^1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, over them.

We show that the VC-dimension of a graph can be computed in time $n^{\log d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that see a specific subset of vertices inside of a (small) query set. The construction of this data structure takes time $O(d2^dn)$, afterwards queries can be computed efficiently using fast M\"obius inversion. This data structure turns out to be useful for a range of tasks, especially for finding bipartite patterns in degenerate graphs, and we outline an efficient algorithms for counting the number of times specific patterns occur in a graph. The largest factor in the running time of this algorithm is $O(n^c)$, where $c$ is a parameter of the pattern we call its left covering number. Concrete applications of this algorithm include counting the number of (non-induced) bicliques in linear time, the number of co-matchings in quadratic time, as well as a constant-factor approximation of the ladder index in linear time. Finally, we supplement our theoretical results with several implementations and run experiments on more than 200 real-world datasets -- the largest of which has 8 million edges -- where we obtain interesting insights into the VC-dimension of real-world networks.

Given univariate random variables $Y_1, \ldots, Y_n$ with the $\text{Uniform}(\theta_0 - 1, \theta_0 + 1)$ distribution, the sample midrange $\frac{Y_{(n)}+Y_{(1)}}{2}$ is the MLE for $\theta_0$ and estimates $\theta_0$ with error of order $1/n$, which is much smaller compared with the $1/\sqrt{n}$ error rate of the usual sample mean estimator. However, the sample midrange performs poorly when the data has say the Gaussian $N(\theta_0, 1)$ distribution, with an error rate of $1/\sqrt{\log n}$. In this paper, we propose an estimator of the location $\theta_0$ with a rate of convergence that can, in many settings, adapt to the underlying distribution which we assume to be symmetric around $\theta_0$ but is otherwise unknown. When the underlying distribution is compactly supported, we show that our estimator attains a rate of convergence of $n^{-\frac{1}{\alpha}}$ up to polylog factors, where the rate parameter $\alpha$ can take on any value in $(0, 2]$ and depends on the moments of the underlying distribution. Our estimator is formed by the $\ell^\gamma$-center of the data, for a $\gamma\geq2$ chosen in a data-driven way -- by minimizing a criterion motivated by the asymptotic variance. Our approach can be directly applied to the regression setting where $\theta_0$ is a function of observed features and motivates the use of $\ell^\gamma$ loss function for $\gamma > 2$ in certain settings.

The languages of mathematical physics and modelling are endowed with a rich "grammar of dimensions" that common abstractions of programming languages fail to represent. We propose a dependently typed domain-specific language (embedded in Idris) that captures this grammar. We apply it to explain basic notions of dimensional analysis and Buckingham's Pi theorem. We argue that the language makes mathematical physics more accessible to computer scientists and functional programming more palatable to modelers and physicists.