相关论文: Orthogonal matrices with rational components in co…
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…
Graded modalities have been proposed in recent work on programming languages as a general framework for refining type systems with intensional properties. In particular, continuous endomaps of the discrete time scale, or time warps, can be…
This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In…
We present a list of open questions in mathematical physics. After a historical introduction, a number of problems in a variety of different fields are discussed, with the intention of giving an overall impression of the current status of…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$,…
A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A…
The Pythagorean Theorem has been proved in hundreds of ways, yet it inspires fresh insights through geometry and trigonometry. In this paper, we offer a new proof based on three circles that circumscribe the sides of a right triangle.…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…
The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…
Whereas for a substantial part, Finite Geometry during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, several important…
The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…
We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a…
The unique and beautiful character of certain mathematical results and proofs is often considered one of the most gratifying aspects of engaging with mathematics. We study whether this perception of mathematical arguments having an…
Let $A\to B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond's Theorem 2.1 from his…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
Math is widely considered as a powerful tool and its strong appeal depends on the high level of abstraction it allows in modelling a huge number of heterogeneous phenomena and problems, spanning from the static of buildings to the flight of…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…