Related papers: Two Lorentzian Lattices
We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].
The point of this note is to prove that the secrecy function attains its maximum at y=1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Sol\'e. Further, we will give a very simple…
In this paper we prove a generalization of famous Larchr's theorem concerning good lattice points.
Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.
The following is a concise exposition of the conjecture and three of its proofs for the case of positive entropy by D. Rudolph [22] , B. Host [14] and W. Parry [21]. A simpler theorem of R. Lyons [19] - preceding them - is also presented…
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…
This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties…
This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of thetheory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M.…
It is well-known that global hyperbolicity implies that the Lorentzian distance is finite and continuous. By carefully analysing the causes of discontinuity of the Lorentzian distance, we show that in most other respects the finiteness and…
We prove that every lattice with more than one element has a proper congruence-preserving extension.
This note provides a self-contained exposition of the proof of the artinian conjecture, following closely Djament's Bourbaki lecture. The original proof is due to Putman, Sam, and Snowden.
We revisit a subexponential bound for the $abc$ conjecture due to the first author, and we establish a variation of it using linear forms in logarithms. As an application, we prove an unconditional subexponential bound towards the $4$-terms…
We prove a conjecture of Frenkel-Gaitsgory-Kazhdan-Vilonen on some exponential sums related to the geometric Langlands correspondence. Our main ingredients are the resolution of Lusztig scheme of lattices introduced by Laumon and the…
We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.
We give a proof of Willcocks's Conjecture, stating that if $p - q$ and $p + q$ are relatively prime, then there exists a Hamiltonian tour of a $(p, q)$-leaper on a square chessboard of side $2(p + q)$. The conjecture was formulated by T. H.…
These are lecture notes (by the first author) from a course (by the second author) given over two extended semesters at the University of Sydney. The first part provides an introduction to the Langlands correspondence from an arithmetical…
This paper explores special conditions on the starting value of a Collatz sequence which imply that the Collatz conjecture is true. This is the result of the collaboration of a retired mathematics professor (Koelzer) and a retired physics…
New cases of the multiplicity conjecture are considered.
We give an informal introduction to the authors' work on some conjectures of Kazhdan and Lusztig, building on work of Soergel and de Cataldo-Migliorini. This article is an expanded version of a lecture given by the second author at the…
In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.