Conjecture

Summary

In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all i

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https://en.wikipedia.org/wiki/Conjecture

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Combinatorics For General Kinetically Constrained Spin Models

Laure Irène Marêché

We study the set of possible configurations for a general kinetically constrained model (KCM), a nonmonotone version of the U-bootstrap percolation cellular automata. We solve a combinatorial question that, is a generalization of a problem addressed by Chung, Diaconis, and Graham [Adv. in Appl. Math., 27 (2001), pp. 192-206] for a specific one-dimensional KCM, the East model. Since the general models we consider are in any dimension and lack the oriented character of the East dynamics, we have to follow a completely different route than the one taken by Chung, Diaconis, and Graham. Our combinatorial result is used by Mareche, Martinelli, and Toninelli to complete the proof of a conjecture put forward by Morris.

SIAM PUBLICATIONS2020

A new REM conjecture

Véronique Gayrard

We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit, as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the P-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down.

Birkhauser Boston, C/O Springer-Verlag, Service Center, 44 Hartz Way, Secaucus, Nj 07096-2491 Usa2008

Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over Q and by the subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over Q that are not (Q) over bar -isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.

SPRINGER2022