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Register a new userOn a generalized $k$-FL sequence and its applications
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We introduce a generalized $k$-FL sequence and special kind of pairs of real numbers that are related to it, and give an application on the integral solutions of a certain equation using those pairs. Also, we associate skew circulant and circulant matrices to each generalized $k$-FL sequence, and study the determinantal variety of those matrices as an application.
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Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
Sarnaks Density Conjecture is an explicit bound on the multiplicities of non-tempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue. The goal of this work is to discuss similar hypotheses, their interrelation and applications. We mainly focus on two properties - the Spectral Spherical Density Hypothesis and the Geometric Weak Injective Radius Property. Our results are strongest in the p-adic case, where we show that the two properties are equivalent, and both imply Sarnaks General Density Hypothesis. One possible application is that either the limit multiplicity property or the weak injective radius property imply Sarnaks Optimal Lifting Property. Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.
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The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the recurrence relation S_{k,n} = kS_{k,n-1} +S{k,n-2} for n >= 2, where a, b, k are real numbers. Using the sequence S_{k,n}, we introduce and prove some special identities. Also, we deal with the circulant and skew circulant matrices for the sequence S_{k,n}.
Demanding that charged Nariai black holes in (quasi-)de Sitter space decay without becoming super-extremal implies a lower bound on the masses of charged particles, known as the Festina Lente (FL) bound. In this paper we fix the $mathcal{O}(1)$ constant in the bound and elucidate various aspects of it, as well as extensions to $d>4$ and to situations with scalar potentials and dilatonic couplings. We also discuss phenomenological implications of FL including an explanation of why the Higgs potential cannot have a local minimum at the origin, thus explaining why the weak force must be broken. For constructions of meta-stable dS involving anti-brane uplift scenarios, even though the throat region is consistent with FL, the bound implies that we cannot have any light charged matter fields coming from any far away region in the compactified geometry, contrary to the fact that they are typically expected to arise in these scenarios. This strongly suggests that introduction of warped anti-branes in the throat cannot be decoupled from the bulk dynamics as is commonly assumed. Finally, we provide some evidence that in certain situations the FL bound can have implications even with gravity decoupled and illustrate this in the context of non-compact throats.
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