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We propose a new variant of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method (2018) for the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound of the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (i) equivalent to Roos's original method (2018) and superior to Louren\c{c}o et al.'s method (2019) when the symmetric cone is the nonnegative orthant, (ii) superior to Louren\c{c}o et al.'s method (2019) when the symmetric cone is a Cartesian product of second-order cones, and (iii) equivalent to Louren\c{c}o et al.'s method (2019) when the symmetric cone is the simple positive semidefinite cone, under the assumption that the costs of computing the spectral decomposition and the minimum eigenvalue are of the same order for any given symmetric matrix. We also conduct numerical experiments that compare the performance of our method with existing methods by generating instance in three types: (i) strongly (but ill-conditioned) feasible instances, (ii) weakly feasible instances, and (iii) infeasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

In this paper motivated from subspace coding we introduce subspace-metric codes and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes. The half-Singleton upper bounds for linear subspace-metric codes and linear subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. Our subspace-metric codes or subset-metric codes can be used to construct explicit well-structured insertion-deletion codes directly. $k$-deletion correcting codes with rate approaching $1$ can be constructed from subspace codes. By analysing the subset distances of folded codes from evaluation codes of linear mappings, we prove that they have high subset distances and then are explicit good insertion-deletion codes.

The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. List-decodable codes are also considered in rank-metric, subspace metric, cover-metric, pair metric and insdel metric settings. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes of finite metric spaces. The general covering code upper bounds can apply to the case when the volumes of the balls depend on the centers, not only on the radius case. Then any good upper bound on the covering radius or the size of covering code imply a good upper bound on the size of list-decodable codes.Our results give exponential improvements on the recent generalized Singleton upper bound in STOC 2020 for Hamming metric list-decodable codes, when the code lengths are large. Even for the list size $L=1$ case our covering code upper bounds give highly non-trivial upper bounds on the sizes of codes with the given minimum distance.The generalized Singleton upper bound for average-radius list-decodable codes is given. The asymptotic forms of covering code bounds can partially recover the Blinovsky bound and the combinatorial bound of Guruswami-H{\aa}stad-Sudan-Zuckerman in Hamming metric setting. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes. We apply our general covering code upper bounds for list-decodable rank-metric codes, list-decodable subspace codes, list-decodable insertion codes and list-decodable deletion codes. Some new better results about non-list-decodability of rank-metric codes and subspace codes are obtained.

This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.

A \textit{functional $k$-batch} code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^s-1$ servers. This conjecture is verified for $s\leq 5$ but previous work could only show that codes with $n=2^s-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left\lfloor \frac{ 2^{s/2}}{\sqrt{24}} \right\rfloor$ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac{5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result.These constructions are mainly based on Hadamard codes and equivalently provide constructions for \textit{parallel Random I/O (RIO)} codes.

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning , deep learning and more recently in kernel methods. In this article, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path. The general question of rates of convergence for these objects has been studied in depth in the recent monograph of Bobkov and Ledoux. By using these results, we generalise the original result of Hambly and Lyons from $C^3$ curves to a broad class of $C^2$ ones. We conclude by providing an explicit way to compute the limit in terms of a second-order differential equation.

Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to computationally approach general non-linear OT problems within the framework of Riemannian manifold optimization. The basis of this is the manifold of doubly stochastic matrices (and their generalization). Even though the manifold geometry is not new, surprisingly, its usefulness for solving general non-linear OT problems has not been popular. To this end, we specifically discuss optimization-related ingredients that allow modeling the OT problem on smooth Riemannian manifolds by exploiting the geometry of the search space. We also discuss extensions where we reuse the developed optimization ingredients. We make available the Manifold optimization-based Optimal Transport, or MOT, repository with codes useful in solving OT problems in Python and Matlab. The codes are available at \url{//github.com/SatyadevNtv/MOT}.

We show that it is provable in PA that there is an arithmetically definable sequence $\{\phi_{n}:n \in \omega\}$ of $\Pi^{0}_{2}$-sentences, such that - PRA+$\{\phi_{n}:n \in \omega\}$ is $\Pi^{0}_{2}$-sound and $\Pi^{0}_{1}$-complete - the length of $\phi_{n}$ is bounded above by a polynomial function of $n$ with positive leading coefficient - PRA+$\phi_{n+1}$ always proves 1-consistency of PRA+$\phi_{n}$. One has that the growth in logical strength is in some sense "as fast as possible", manifested in the fact that the total general recursive functions whose totality is asserted by the true $\Pi^{0}_{2}$-sentences in the sequence are cofinal growth-rate-wise in the set of all total general recursive functions. We then develop an argument which makes use of a sequence of sentences constructed by an application of the diagonal lemma, which are generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth in Mathematics". The argument establishes the result that it is provable in PA that $P \neq NP$. We indicate how to pull the argument all the way down into EFA.

For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results are illustrated by numerical experiments.

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.