Superintegrability of (2n + 1)-body choreographies, n = 1,2,3,…,∞ on the algebraic lemniscate by Bernoulli (inverse problem of classical mechanics)

For one 3-body and two 5-body planar choreographies on the same algebraic lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies [Formula: see text] moving choreographically (without collisions) along given algebraic lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit [Formula: see text] is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic lemniscate and it is characterized by infinitely many constants of motion.

Mobile Robots with Uncertain Visibility Sensors: Possibility Results and Lower Bounds

We consider mobile robotic entities that have to cooperate to solve assigned tasks. In the literature, two models have been used to model their visibility sensors: the full visibility model, where all robots can see all other robots, and the limited visibility model, where there exists a limit [Formula: see text] such that all robots closer than [Formula: see text] are seen and all robots further than [Formula: see text] are not seen. We introduce the uncertain visibility model, which generalizes both models by considering that a subset of the robots further than [Formula: see text] cannot be seen. An empty subset corresponds to the full visibility model, and a subset containing every such robot corresponds to the limited visibility model. Then, we explore the impact of this new visibility model on the feasibility of benchmarking tasks in mobile robots computing: gathering, uniform circle formation, luminous rendezvous, and leader election. For each task, we determine the weakest visibility adversary that prevents task solvability, and the strongest adversary that allows task solvability. Our work sheds new light on the impact of visibility sensors in the context of mobile robot computing, and paves the way for more realistic algorithms that can cope with uncertain visibility sensors.

Nonlocal characterizations of variable exponent Sobolev spaces

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

Chiral gauge leptoquark mass limits and branching ratios of KL0,B0,Bs → li+lj− decays with account of the general fermion mixing in leptoquark currents

The contributions of the chiral gauge leptoquarks [Formula: see text] induced by the chiral four-color quark–lepton symmetry to the branching ratios of [Formula: see text] decays are calculated and analyzed using the general parametrizations of the fermion mixing matrices in the leptoquark currents. From the current experimental data on these decays under assumption [Formula: see text], the lower mass limit [Formula: see text] is found, which in particular case of equal gauge coupling constants gives [Formula: see text]. The branching ratios of the decays under consideration predicted by the chiral gauge leptoquarks are calculated and analyzed in dependence on the leptoquark masses and the mixing parameters. It has shown that in consistency with the current experimental data, these branching ratios for [Formula: see text] decays can be close to their experimental limits and those for [Formula: see text] decays can be of order of [Formula: see text]. The calculated branching ratios will be useful in the further experimental searches for these decays.

How Do Campaign Spending Limits Affect Elections? Evidence from the United Kingdom 1885–2019

In more than half of the democratic countries in the world, candidates face legal constraints on how much money they can spend on their electoral campaigns, yet we know little about the consequences of these restrictions. I study how spending limits affect UK House of Commons elections. I contribute new data on the more than 70,000 candidates who ran for a parliamentary seat from 1885 to 2019, and I document how much money each candidate spent, how they allocated their resources across different spending categories, and the spending limit they faced. To identify the effect on elections, I exploit variation in spending caps induced by reforms of the spending-limit formula that affected some but not all constituencies. The results indicate that when the level of permitted spending is increased, the cost of electoral campaigns increases, which is primarily driven by expenses related to advertisement and mainly to the disadvantage of Labour candidates; the pool of candidates shrinks and elections become less competitive; and the financial and electoral advantages enjoyed by incumbents are amplified.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

On Petersson’s partition limit formula

For each prime [Formula: see text] consider the Legendre character [Formula: see text]. Let [Formula: see text] be the number of partitions of [Formula: see text] into parts [Formula: see text] such that [Formula: see text]. Petersson proved a beautiful limit formula for the ratio of [Formula: see text] to [Formula: see text] as [Formula: see text] expressed in terms of important invariants of the real quadratic field [Formula: see text]. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald’s conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman–Erdős.

Anisotropic Effect on Local Magnetic Properties of 3D Extended Ising Model

The magnetic properties of anisotropic 3D Ising model on a cubic lattice are studied by Monte Carlo simulation. In particular, we have considered an extended 3D Ising model with spatially uniaxial anisotropic bond randomness on the simple cubic lattice parameterized by exchange interaction parameter [Formula: see text], anisotropy parameter [Formula: see text] and external longitudinal magnetic field [Formula: see text]. The obtained numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model at critical temperature [Formula: see text] that is strongly correlated to [Formula: see text] and [Formula: see text]. Especially, in the limit, [Formula: see text], the spin ½ cubic lattice becomes a collection of noncorrelated Ising chains, whereas in the other limit, [Formula: see text], the system becomes a stack of noncorrelated Ising square lattice.