Related papers: Coding is hard
Coding in the projective space has received recently a lot of attention due to its application in network coding. Reduced row echelon form of the linear subspaces and Ferrers diagram can play a key role for solving coding problems in the…
In continuation to earlier works where the problem of joint information embedding and lossless compression (of the composite signal) was studied in the absence \cite{MM03} and in the presence \cite{MM04} of attacks, here we consider the…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
We show that if we enrich first order logic by allowing quantification over isomorphisms between definable ordered fields the resulting logic, L(Q_{Of}), is fully compact. In this logic, we can give standard compactness proofs of various…
The problem of characterizing normed ordered spaces which admit a representation in the algebraic, order and norm sense as a subspace of $C(X)$, the space of all continuous functions on a compact Hausdorff space is a classical problem that…
We discuss the concepts of fine and coarse moduli spaces in the context of finite dimensional algebras over algebraically closed fields. In particular, our formulation of a moduli problem and its potential strong or weak solution is adapted…
An important endeavor in computer science is to understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. This investigation requires therefore a…
Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework…
This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's…
Traditional neural embeddings represent concepts as points, excelling at similarity but struggling with higher-level reasoning and asymmetric relationships. We introduce a novel paradigm: embedding concepts as linear subspaces. This…
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several…
Dependency distance minimization (DDm) is a well-established principle of word order. It has been predicted theoretically that DDm implies compression, namely the minimization of word lengths. This is a second order prediction because it…
Sorting algorithms are fundamental to computer science, and their correctness criteria are well understood as rearranging elements of a list according to a specified total order on the underlying set of elements. As mathematical functions,…
The list-decodable code has been an active topic in theoretical computer science.There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. In this paper we show that rates,…
Reverse Mathematics (RM for short) is a program in the foundations of mathematics with the aim of finding the minimal axioms required for proving theorems about countable and separable objects. RM usually takes place in second-order…
We investigate how sorting algorithms efficiently overcome the exponential size of the permutation space. Our main contribution is a new continuous-time formulation of sorting as a gradient flow on the permutohedron, yielding an independent…
The study of convex functions - in particular, of their optimization (really minimization) is one of the most important fields of applied mathematics. Convexity seems to be one of those incredibly well-chosen hypotheses which is just…
In the absence of the axiom of choice, new results concerning sequential, Fr\'echet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other…
Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear…
It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and well-studied. We prove precise structure…