IA Scholar Query: Solving a Linear Diophantine Equation with Lower and Upper Bounds on the Variables.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 24 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On the Diophantine equation ∑ _k=1^5F_n_k=2^a
https://scholar.archive.org/work/njqhnmv65rcphdbwpgubaeqtdq
Let (F_n)_n≥ 0 be the Fibonacci sequence given by F_0 = 0, F_1 = 1 and F_n+2 = F_n+1+F_n for n ≥ 0. In this paper, we have determined all the powers of 2 which are sums of five Fibonacci numbers with few exceptions that we characterize. We have also stated an open problem relating to the number of solutions of equations like those studied in this paper.Pagdame Tiebekabe, Ismaïla Dioufwork_njqhnmv65rcphdbwpgubaeqtdqSat, 24 Sep 2022 00:00:00 GMTOn smooth plane models for modular curves of Shimura type
https://scholar.archive.org/work/skje54gnzzflpix6ifp4hxhrke
In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves.Samuele Anni, Eran Assaf, Elisa Lorenzo Garcíawork_skje54gnzzflpix6ifp4hxhrkeFri, 23 Sep 2022 00:00:00 GMTTilings of an Isosceles Triangle
https://scholar.archive.org/work/ob4btokxsreele3zwnn2erp77u
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we study the case of isosceles (but not equilateral) ABC. We study three possible forms of the tile: right-angled, or with one angle double another, or with a 120 degree angle. In the case of a right-angled tile, we give a complete characterization of the tilings, for N even, and prove that N must be even. In the latter two cases we prove the ratios of the sides of the tile are rational, and give a necessary condition for the existence of an N-tiling. For the case when the tile has one angle double another, we prove N cannot be prime or even squarefree.Michael Beesonwork_ob4btokxsreele3zwnn2erp77uThu, 22 Sep 2022 00:00:00 GMTSeparation of periods of quartic surfaces
https://scholar.archive.org/work/limexnqigza4hb72wcuiufzpu4
We give a computable lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether--Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups.Pierre Lairez, Emre Can Sertözwork_limexnqigza4hb72wcuiufzpu4Thu, 22 Sep 2022 00:00:00 GMTA dynamical Thouless formula
https://scholar.archive.org/work/ybfbcxmupba7pf375qcglwzcua
In this paper we establish an abstract, dynamical Thouless-type formula for affine families of GL (2,ℝ) cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.Jamerson Bezerra and Ao Cai and Pedro Duarte and Catalina Freijo and Silvius Kleinwork_ybfbcxmupba7pf375qcglwzcuaMon, 19 Sep 2022 00:00:00 GMTBounded solutions of KdV: uniqueness and the loss of almost periodicity
https://scholar.archive.org/work/fqrrt44en5d23a33ixfomu5tqu
We address two pressing questions in the theory of the Korteweg--de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.Andreia Chapouto, Rowan Killip, Monica Vişanwork_fqrrt44en5d23a33ixfomu5tquThu, 15 Sep 2022 00:00:00 GMTThe Herman invariant tori conjecture
https://scholar.archive.org/work/mlw6xxrfdfh2jl72ra7epjf6ey
We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/Mauricio Garay, Duco van Stratenwork_mlw6xxrfdfh2jl72ra7epjf6eyMon, 12 Sep 2022 00:00:00 GMTSatisfiability Phase Transtion for Random Quantum 3XOR Games
https://scholar.archive.org/work/n7tv53idpjhi3flhiixhcygywi
Recent results showed it was possible to determine if a modest size 3XOR game has a perfect quantum strategy. We build on these and give an explicit polynomial time algorithm which constructs such a perfect strategy or refutes its existence. This new tool lets us numerically study the behavior of randomly generated 3XOR games with large numbers of questions. A key issue is: how common are pseudotelephathy games (games with perfect quantum strategies but no perfect classical strategies)? Our experiments strongly indicate that the probability of a randomly generated game being pseudotelpathic stays far from 1, indeed it is bounded below 0.15. We also find strong evidence that randomly generated 3XOR games undergo both a quantum and classical "phase transition", transitioning from almost certainly perfect to almost certainly imperfect as the ratio of number of clauses (m) to number of questions (n) increases. The locations of these two phase transitions appear to coincide at m/n ≈ 2.74.Adam Bene Watts, J. William Helton, Zehong Zhaowork_n7tv53idpjhi3flhiixhcygywiSat, 10 Sep 2022 00:00:00 GMTHausdorff dimension of Gauss–Cantor sets and two applications to classical Lagrange and Markov spectra
https://scholar.archive.org/work/c7inaujxhbbq5lwdzyxr36jfxi
This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum L and Markov spectrum M. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value t_1 such that the portion of the Markov spectrum (-∞,t_1)∩ M has Hausdorff dimension 1. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference M∖ L. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss–Cantor sets of continued fraction expansions whose entries satisfy appropriate restrictions.Carlos Matheus, Carlos Gustavo Moreira, Mark Pollicott, Polina Vytnovawork_c7inaujxhbbq5lwdzyxr36jfxiTue, 30 Aug 2022 00:00:00 GMTExceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
https://scholar.archive.org/work/zh6burqdzne4tg5taxa5sfjrum
Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either X_K has infinitely many rational curves or X has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.Ananth N. Shankar, Arul Shankar, Yunqing Tang, Salim Tayouwork_zh6burqdzne4tg5taxa5sfjrumMon, 29 Aug 2022 00:00:00 GMTOn Classifying Continuous Constraint Satisfaction Problems
https://scholar.archive.org/work/de6jy7vvvjgxjce3qv22rmev3e
A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain U ⊂ℝ. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form ∃ x_1, ..., x_n ∈ℝ : Φ(x_1, ..., x_n), where Φ is a well-formed quantifier-free formula consisting of the symbols {0, 1, +, ·, ≥, >, ∧, ∨, }, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP ⊆ ER ⊆ PSPACE. We restrict our attention on CCSPs with addition constraints (x + y = z) and some other mild technical condition. Previously, it was shown that multiplication constraints (x · y = z), squaring constraints (x^2 = y), or inversion constraints (x· y = 1) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical condition) that have any one well-behaved curved equality constraint (f(x,y) = 0) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint (f(x,y) ≥ 0 and g(x,y) ≥ 0) imply ER-completeness on the class of such CCSPs.Tillmann Miltzow, Reinier F. Schmiermannwork_de6jy7vvvjgxjce3qv22rmev3eMon, 29 Aug 2022 00:00:00 GMTAn Efficient Identification Scheme Based on Bivariate Function Hard Problem
https://scholar.archive.org/work/2g2lc5phsvh2biukxdo6uddjze
Symmetric cryptography allows faster and more secure communication between two entities using the identical pre-established secret key. However, identifying the honest entity with the same secret key before initiating symmetric encryption is vital since the communication may be impersonated. Tea and Ariffin, in 2014, proposed a new identification (ID) scheme based on the Bivariate Function Hard Problem (BFHP) that proved secure against impersonation under passive, active and concurrent attacks via the BFHP-hardness assumption. In this paper, we upgrade the ID scheme and improve some of its settings. Next, we provide the security proof against impersonation under active and concurrent attacks in the random oracle model via the hardness assumption of the One-More BFHP. Finally, we include an additional discussion about the computational efficiency of the upgraded ID scheme based on BFHP and present its comparison with other selected ID schemes.Boon Chian Tea, Muhammad Rezal Kamel Ariffin, Amir Hamzah Abd Ghafar, Siti Hasana Sapar, Mohamat Aidil Mohamat Johariwork_2g2lc5phsvh2biukxdo6uddjzeSat, 27 Aug 2022 00:00:00 GMTThe Beurling-Selberg Box Minorant Problem via Linear Programming Bounds
https://scholar.archive.org/work/77f72fffybgzrjnrfu562j2vfi
In this paper we investigate a high dimensional version of Selberg's minorant problem for the indicator function of an interval. In particular, we study the corresponding problem of minorizing the indicator function of the box Q_N=[-1,1]^N by a function whose Fourier transform is supported in the same box Q_N. We show that when the dimension is sufficiently large there are no minorants with positive mass and we give an explicit lower bound for such dimension. On the other hand, we explicitly construct minorants for dimensions 1,2,3,4 and 5 and, as an application, we use them to produce an improved diophantine inequality for exponential sums.Jacob Carruth, Noam Elkies, Felipe Gonçalves, Michael Kellywork_77f72fffybgzrjnrfu562j2vfiWed, 24 Aug 2022 00:00:00 GMTDagstuhl Reports, Volume 12, Issue 1, January 2019, Complete Issue
https://scholar.archive.org/work/sz47bglqcjeoha4rv2m7lcgyba
Dagstuhl Reports, Volume 12, Issue 1, January 2019, Complete Issuework_sz47bglqcjeoha4rv2m7lcgybaTue, 23 Aug 2022 00:00:00 GMTOn quadratic Waring's problem in totally real number fields
https://scholar.archive.org/work/eey2dhb5qfdn5b2qui7u6khnfu
We improve the bound of the g-invariant of the ring of integers of a totally real number field, where the g-invariant g(r) is the smallest number of squares of linear forms in r variables that is required to represent all the quadratic forms of rank r that are representable by the sum of squares. Specifically, we prove that the g_𝒪_K(r) of the ring of integers 𝒪_K of a totally real number field K is at most g_ℤ([K:ℚ]r). Moreover, it can also be bounded by g_𝒪_F([K:F]r+1) for any subfield F of K. This yields a sub-exponential upper bound for g(r) of each ring of integers (even if the class number is not 1). Further, we obtain a more general inequality for the lattice version G(r) of the invariant and apply it to determine the value of G(2) for all but one real quadratic field.Jakub Krásenský, Pavlo Yatsynawork_eey2dhb5qfdn5b2qui7u6khnfuSun, 21 Aug 2022 00:00:00 GMTBeyond the spherical sup-norm problem
https://scholar.archive.org/work/2bu3ob5n3jgw5akh7klwiyveyq
We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of G=SL_2(C) with K=SU_2(C). Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.Valentin Blomer, Gergely Harcos, Péter Maga, Djordje Milićevićwork_2bu3ob5n3jgw5akh7klwiyveyqSat, 20 Aug 2022 00:00:00 GMTLarge hyperbolic circles
https://scholar.archive.org/work/tbylvxz6q5gzxgv4biuo5fow24
We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of SL_2(ℝ). Our proof relies on a spectral method pioneered by Ratner and subsequently developed by Burger in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss applications to the classical circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen.Emilio Corso, Davide Ravottiwork_tbylvxz6q5gzxgv4biuo5fow24Tue, 16 Aug 2022 00:00:00 GMTOn solutions of the Diophantine equation 𝒫_m-L_n=c
https://scholar.archive.org/work/eu7rq7jisvg2rfqt3zhdzhjtzu
In this article, we determine all the integers c having at least two representations as difference between two linear recurrent sequences. This is a variant of the Pillai's equation. This equation is an exponential Diophantine equation. The proof of our main theorem uses lower bounds for linear forms of logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.Pagdame Tiebekabe, Serge Adonsou, Ismaïla Dioufwork_eu7rq7jisvg2rfqt3zhdzhjtzuThu, 11 Aug 2022 00:00:00 GMTSIC-POVMs from Stark units: Prime dimensions n^2+3
https://scholar.archive.org/work/7sjnvijmwrdqpllo355uqz4hyy
We propose a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form d=n^2+3, focussing on prime dimensions d=p. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being p=19603. The real quadratic base field K (in the standard SIC terminology) attached to such dimensions has fundamental units u_K of norm -1. Let ℤ_K denote the ring of integers of K, then pℤ_K splits into two ideals 𝔭 and 𝔭'. The initial entry of the fiducial is the square ξ^2 of a geometric scaling factor ξ, which lies in one of the fields K(√(u_K)). Strikingly, the other p-1 entries of the fiducial vector are each the product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s=0 of the first derivatives of partial L functions attached to the characters of the ray class group of ℤ_K with modulus 𝔭∞_1, where ∞_1 is one of the real places of K.Marcus Appleby, Ingemar Bengtsson, Markus Grassl, Michael Harrison, Gary McConnellwork_7sjnvijmwrdqpllo355uqz4hyyWed, 10 Aug 2022 00:00:00 GMTInduction and Skolemization in saturation theorem proving
https://scholar.archive.org/work/zahui23nbfhqfe2mcasckacspa
We consider a typical integration of induction in saturation-based theorem provers and investigate the effects of Skolem symbols occurring in the induction formulas. In a practically relevant setting we establish a Skolem-free characterization of refutation in saturation-based proof systems with induction. Finally, we use this characterization to obtain unprovability results for a concrete saturation-based induction prover.Stefan Hetzl, Jannik Vierlingwork_zahui23nbfhqfe2mcasckacspaMon, 08 Aug 2022 00:00:00 GMT