Abstract We study arithmetic functions Φ(x;d,a), called prime running functions, whose value at x sums the gaps between primes pk≡a(mod d) below x and the next following prime pk+1, up to x. (The following prime pk+1 may be in any residue class (mod d).) We empirically observe systematic biases of order x/ log x in Φ(x;d,a)−Φ(x;d,b) for different a, b. We formulate modified Cramér models for primes

Liedtke has introduced group functors K and K˜, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work, we relate K and K˜ to a group functor τ arising in the construction of the non-abelian exterior square of a group. In contrast to K˜, there exist

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between

We present a fast algorithm that takes as input an elliptic curve defined over Q and an integer d and returns all the number fields K of degree d ′ dividing d such that E ( K ) tors contains E ( F ) tors as a proper subgroup, for all F ⊈ K . We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all d ≤ 23 and collected various interesting data

Abstract We used the MACH2 supercomputer to study coefficients in the q-series expansion of (1−q)(1−q2)…(1−qn), for all n≤75000. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd n. Remarkably, the observed period is 62,624.

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our earlier algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus Bg of the moduli space Mg of compact Riemann surfaces of genus g≥2, corresponding to group actions with orbit genus 0. That is, it works for actions on surfaces

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of primitive lattice surfaces, we show that the Teichmüller curve of the unfolded dodecahedron

Abstract We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai L-functions of Hilbert modular forms.

In this paper we discuss minimizing a generalized frame potential - F(p,m,n) - for vectors in CPn. We discuss the minimal frame potential as well as the space of solutions for the minimization problem. We investigate this question both numerically and analytically. We find that for some values of p - Equiangular Tight Frames (ETF) are unique solutions, in contrast to the case of p = 2 where they are

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. In the first part of this paper, we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is “which 3-manifolds embed smoothly in S4?” The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only

In this paper, we enumerate superspecial trigonal curves of genus 5 over Fpa for any natural number a if p≤7 and for odd a if p≤13. This paper provides an algorithm enumerating superspecial trigonal curves of genus 5 over Fq for arbitrary q. We implemented the algorithm over computer algebra system Magma. With help of computational results obtained by our implementation, we prove the nonexistence (resp

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

We present eigenvalue data and pictures of eigenfunctions of the classic and quadratic snowflake fractal and of quadratic filled Julia sets. Furthermore, we approximate the area and box-counting dimension of selected Julia sets to compare the eigenvalue counting function with the Weyl term.

Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions

Recently, Hirschhorn investigated vanishing coefficients of the arithmetic progressions in the following two q-series expansions ∑n=0∞a(n)qn:=∏n=1∞(1+q5n−4)(1+q5n−1)(1−q10n−9)3(1−q10n−1)3,∑n=0∞b(n)qn:=∏n=1∞(1+q5n−3)(1+q5n−2)(1−q10n−7)3(1−q10n−3)3. He proved that for n≥0,a(5n+2)=a(5n+4)=b(5n+1)=b(5n+4)=0. In this paper, we further study these two q-series expansions and obtain the generating functions

This paper presents some configuration theorems concerning rectangles inscribed in four lines.

The problem of determining DSn, the complex numbers that occur as an eigenvalue of an n-by-n doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, PMn, is contained in DSn and is known to be exactly DSn for n≤4, but strictly contained within DSn for n = 5. Here, we present a Boundary Conjecture that asserts that the boundary of DSn is achieved by eigenvalues

We relate Reiner, Tenner, and Yong’s coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave “as if” they had isomorphic comparability graphs, even though they do not. We further explore the

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4–manifolds. Lower bounds on trisection genus are

We construct a one (real) dimensional deformation of an acylindrical hyperbolic 3-manifold, compute some associated arithmetic invariants and exhibit the skinning map along the deformation locus. The key idea is to construct a one-dimensional family of hyperbolic polyhedra; reflections in some of the faces of the polyhedra then generate a family of Kleinian groups.

In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavor (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of discrete Dirac operators, the latter on discrete surfaces with polygonal faces and normals

In this note, given a pair (g,λ), where g is a complex semisimple Lie algebra and λ∈h* is a dominant integral weight of g, where h⊂g is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU(2) and Weyl equivariant smooth map f:X→(Pm(C))n, where X⊂h⊗R3 is the configuration space of regular triples in h, and m, n depend on the initial data (g,λ). We conjecture that, for any

A monic, irreducible polynomial in one variable having integer coefficients and all real roots deserves particular interest if its roots lie in an interval of length 4 whose end-points are not integers. This follows by some pioneering studies by R. Robinson. Thanks to the crucial support of computers, a number of contributions over the decades settled the existence question for such polynomials up

We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface x2+y2+z2=3xyz modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases

We present a property satisfied by a large variety of complex continued fraction algorithms (the “finite building property”) and use it to explore the structure of bijectivity domains for natural extensions of Gauss maps. Specifically, we show that these domains can each be given as a finite union of Cartesian products in C×C. In one complex coordinate, the sets come from explicit manipulation of the

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner leading to a more general notion of index. This allows us to produce explicit examples of destabilizing perturbations for the fundamental

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments

(2020). Correction. Experimental Mathematics: Vol. 29, No. 3, pp. 360-360.

Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m)≥valp(A(E))>0

We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval [0, 1]. We implement algorithms from semi-infinite programing and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree n. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in

We construct an explicit example of a genus 2 curve C over a number field K such that the adelic Galois representation arising from the action of Gal(K¯/K) on the Jacobian of C has image GSp4(Ẑ).

We define oldforms and newforms for Drinfeld cusp forms of level t and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix U associated to the action of the Atkin operator Ut on cusp forms of level t and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on the diagonalizability

Abstract We propose an approach for showing rationality of an algebraic variety X. We try to cover X by rational curves of certain type and count how many curves pass through a generic point. If the answer is 1, then we can sometimes reduce the question of rationality of X to the question of rationality of a closed subvariety of X. This approach is applied to the case of the so-called Ueno-Campana

Abstract In a convex mosaic in Rd we denote the average number of vertices of a cell by v¯ and the average number of cells meeting at a node by n¯. Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the [n¯,v¯] plane (except for the unphysical n¯,v¯

A Lie (super)algebra with a nondegenerate invariant symmetric bilinear form B is called a nis-(super)algebra. The double extension g of a nis-(super)algebra a is the result of simultaneous adding to a a central element and a derivation so that g is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations

We describe conjecturally the generalized Samuel multiplicities c0,…,cd−1 of a monomial ideal I⊂K[x1,…,xd] in terms of its Newton polyhedron NP(I). More precisely, we conjecture that ci equals the sum of the normalized (d−i)-volumes of pyramids over the projections of the (d−i−1)-dimensional compact faces of NP(I) along the infinite-directions of i-unbounded facets in which they are contained. For

Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was studied by Musiker and Reiner. We answer a question posed by the two authors regarding homotopy type of certain subcomplexes of the associated simplicial complex when the degree of the cyclotomic

The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmüller curves. We show that the stratum Prym(1, 2) contains no such Teichmüller curve and the stratum Prym(2) at most 92 such Teichmüller curves. This complements the recent progress establishing general – but non-effective – methods to prove finiteness results for Teichmüller

We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a “q-analogue of a real.” The construction is based on the notion of q-deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.

We continue research into the cyclically presented groups with length three positive relators. We study small cancelation conditions, SQ-universality, and hyperbolicity, we obtain the Betti numbers of the groups’ abelianisations, we calculate the orders of the abelianisations of some groups, and we study isomorphism classes of the groups. Through computational experiments we assess how effective the

A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this article is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov

Let Φn denotes the nth cyclotomic polynomial. In this paper, we show that if m and n are distinct positive integers and x is a nonzero real number with Φm(x)=Φn(x), then 12<|x|<2 except when {m,n}={2,6} and x = 2. We also observe that 2 appears to be the largest real limit point of the set of values of x for which Φm(x)=Φn(x) for some m≠n.

We investigate two truncated series derived recently by S. H. Chan, T. P. N. Ho, and R. Mao from the Watson quintuple product identity and experimentally discover two stronger results. In this context, for each S∈{1,2}, we obtain two infinite families of linear homogeneous inequalities for the number of partitions of n into parts congruent to ±Smod5.

We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-Émery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the Möbius ladders. As a consequence of the classification result we show that non-negatively curved cubic expanders do not exist. We also introduce the Graph Curvature Calculator

It is a long standing open problem whether the Thompson group F is an amenable group. In this article, we show that if A, B, C denote the standard generators of Thompson group T and D:=CBA−1 then 2+3 < 112||(I+C+C2)(I+D+D2+D3)|| ≤ 2+2.Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring CT is computed in B(ℓ2(T)) via the regular representation

Given a real elliptic curve E with non-empty real part and a real effective divisor D on E arising via pullback from P1 under the hyperelliptic structure map, we study the real inflection points of distinguished subseries of the complete real linear series |D| on E. We define inflection polynomials whose roots index the (x-coordinates of) inflection points of the linear series, away from the points

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite

How can we arrange n lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin, and Sloane (1996) produced line packings for n≤55 that they conjectured to be within numerical precision of optimal in this sense, but until now only the cases n≤7 have been solved. In this paper, we resolve the case n = 8. Drawing

We study the octahedral configuration O6 [Kuperberg] of six equal cylinders touching the unit sphere. We show that the configuration O6 is a local sharp maximum of the distance function. Thus, it is not unlockable and, moreover, rigid.

We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the “strong d-step Theorem” that allows to construct such large-diameter polytopes from “non-d-step” prismatoids still works at this combinatorial level. Then, using metaheuristic methods on the

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by Tao for dimensions 5 and higher by giving a counterexample in Z35. We present a computer program that determines that the Fuglede conjecture holds in Z35 by exhausting the search space. Recently Iosevich, Mayeli and Pakianathan showed that the Fuglede conjecture holds over prime

The covering number of a group G, denoted by σ(G), is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most 129 by introducing

We determine which non-crystallographic, almost-crystallographic groups of dimension-4 have the R∞-property. We then calculate the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups.

Abstract We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of π, log (2), or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in

We consider a semi-random walk on the space X of lattices in Euclidean n-space which attempts to maximize the sphere-packing density function Φ. A lattice (or its corresponding quadratic form) is called “sticky” if the set of directions in X emanating from it along which Φ is infinitesimally increasing has measure 0 in the set of all directions. Thus the random walk will tend to get “stuck” in the