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We construct certain entire function $\lambda(s)$ which for integer s coincides with the well-known Keiper-Li coefficients, i.e. $\lambda(n)={\lambda}_{n}$. This is an even function ${\lambda}(s)={\lambda}(-s)$ and has an infinitude of complex zeros exhibiting interesting distribution. Extensive tables of more than 3500 complex zeros of ${\lambda}(s)$ with precision of 14 significant digits are included. A detailed analysis of the distribution of these zeros may shed some light on the Riemann hypothesis. It turns out that possible violation of the Riemann hypothesis (if such is the case) would be clearly reflected in the specific distribution of these zeros. More specifically, they form complex quadruplets if the Riemann hypothesis is true, and aligned real doublets if it is false., Comment: 41 pages, 13 figures

We present a new asymptotic formula for the Stieltjes constants which is both simpler and more accurate than several others published in the literature (see e.g. \cite{Fekih-Ahmed}, \cite{Knessl Coffey}, \cite{Paris}). More importantly, it is also a good starting point for a detailed analysis of some surprising regularities in these important constants., Comment: 21 pages 11 figures

We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 \cite{Maslanka 1}. The crucial ingredient in this method is a sequence of high-precision numerical values of the Riemann zeta function computed in equally spaced real arguments, i.e. $\zeta(1+\varepsilon),\zeta(1+2\varepsilon),\zeta(1+3\varepsilon),...$ where $\varepsilon$ is some real parameter. (Practical choice of $\varepsilon$ is described in the main text.) Such values of zeta may be readily obtained using the PARI/GP program, which is especially suitable for this., Comment: 24 pages, 11 figures

The nongeneric six- and eightdimensional orbits of SO(4,2) are described in explicitly covariant way. The relevant Hamiltonian dynamical systems are constructed and canonically quantized. It is shown that the resulting unitary representations of conformal group fit into the classification described by Mack (G. Mack, Comm. Math. Phys. 55 (1977),1)., Comment: 20 pages, no figures

Khnichin's theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss--Kuzmin theorem to several thousand high precision (up to more than 53000 significant digits) initial Stieltjes constants $\gamma _{n}$, $n=0,1,...,5000$ in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We study also the normality of these important constants., Comment: 9 figures, 1 table

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group., Comment: 30 pages, no figures, references added

We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process., Comment: version 2: 54 pages

We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes., Comment: version 3. 42 pages. Final version. New: Section 2.7 (conjecture about the rate of convergence)

We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the \emph{second class particle} converges (as the size of the system tends to infinity) to a random arc of an ellipse. In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux., Comment: Version 3: 92 pages. ----------- Version 2: Published as a 12 page extended abstract in the proceedings of FPSAC 2020 conference. "Limit shapes of evacuation and jeu de taquin paths in random square tableaux" Seminaire Lotharingien de Combinatoire, 84B.8 (2020), 12 pp https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2020/8.html

The explicit form of conformal generators is found which provides the extension of Poincare symmetry for massless particles of arbitrary helicity. The helicity 1/2 particles are considered as the particular example. The realization of conformal symmetry in the semiclassical regime of Weyl equation is obtained., Comment: 15 pages, no figures; final version to appear in Phys. Lett. B

In 2013 Balka and M\'ath\'e showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS, in which they considered families of mappings defined on finite Cartesian product $X^m$ with values in $X$. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ endowed with supremum metric. In the paper we show that in Euclidean spaces a typical compact set is an attractor in sense of Secelean and that in general in the polish spaces it can be perceived as selfsimilar in such sense., Comment: 21 pages

Miculescu and Mihail in 2008 introduced the concept of a \emph{generalized iterated function system} (GIFS in~short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space~$X$, GIFSs consist of maps defined on a finite Cartesian {$m$-th power} $X^m$ with values in $X$ (in such a case we say that a GIFS is \emph{of order} $m$). It turned out that a great part of the classical {Hutchinson theory} has natural counterpart in this GIFSs' framework. On the other hand, there are known only few examples of~fractal sets which are generated by GIFSs, but which are not IFSs' {attractors}. In the paper we study $0$-dimensional compact metrizable spaces from the perspective of GIFSs' theory. We prove that each such space $X$ (in particular, countable with limit {scattered} height) is homeomorphic to the~attractor of some GIFS on the real line. Moreover, we prove that $X$ can be embedded into the real line $\R$ as {the attractor of some} GIFS of order $m$ and (in the same time) {a nonattractor} of any GIFS of order $m-1$, as well as it can be embedded as {a nonattractor of any GIFS}. {Then} we show that a relatively simple modifications of $X$ deliver spaces whose each connected component is "big" and which are GIFS's { attractors} not homeomorphic with IFS's {attractors}. Finally, we use obtained results to show that a generic compact subset of a Hilbert space is not {the} attractor of any Banach GIFS.

The notion of position operator for massless spinning particles is discussed in some detail. The noncommutativity of coordinates is related to the gauge symmetry following from the freedom in definition of standard state in Wigner's construction of induced representations of Poincare group. The counterpart of Newton-Wigner operator is discussed. It is explained why the Newton-Wigner construction works only for helicity $ \mid\lambda\mid\leq1/2$., Comment: 19 pages, no figures, few references added

The application of the chiral decomposition procedure to hybrid graphene h-BN systems revealed rules for the partition of the system into effective subsystems being bilayers plus monolayer in case the number of layers is odd. Three types of subsystems have been detected namely purely graphene bilayers and monolayers, mixed bilayers and pure h-BN monolayers depending on the hybrid composition. The effective parameters characterizing these chiral subsystems consist of the interlayer couplings and on-site potentials which shows the mechanism of compensation of the asymmetry introduced into the system by h-BN layers. For illustration, we provide a pedagogical overview about chiral tunneling in graphene subsystems (MLG, BLG) present in hybrid with one h-BN layer. We have established the parameter ranges for which the characteristic features in the spectrum are observed, such as Fabry-P\'erot resonances in the case of MLG and "magic angles" in the case of effective BLG. We also consider different hybrid stacking in order to indicate effective systems with the desired properties required in the electronic and spintronic applications.

The quasiclassical theory of massless chiral fermions is considered. The effective action is derived using time-dependent variational principle which provides a clear interpretation of relevant canonical variables. As a result their transformation properties under the action of Lorentz group are derived from first principles., Comment: 11 pages, no figures; final version to appear in JHEP

We discuss the chiral decomposition of non-symmetric stacking structures. It is shown that the low-energy electronic structure of Bernal stacked graphene multilayer deposited on h-BN consists of chiral pseudospin doublets. N-layer graphene stocks on h-BN layer have N/2 effective bilayer graphene systems and one effective h-BN layer if N is even or (N-1)/2 effective graphene bilayers plus one graphene monolayer modified by h-BN layer if N is odd. We present the decomposition procedure and we derive the recurrence relations for the effective parameters characterizing the chiral subsystems. The effective parameters consist in this case of the interlayer couplings and on-site potentials in contrast to pure graphene multilayer systems where only interlayer couplings are modified. We apply this procedure to discuss the Klein tunneling phenomena and compare quantitatively the results with pure graphene multilayer systems.

We consider the branching random walk drifting to $-\infty$ and we investigate large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem., Comment: 19 pages

The aim of this paper is to prove a counterpart of the Banach fixed point principle for mappings $f: \ell_\infty(X) \to X$, where $X$ is a metric space and $\ell_\infty(X)$ is the space of all bounded sequences of elements from~$X$. Our result generalizes the theorem obtained by Miculescu and Mihail in 2008, who proved a~counterpart of the Banach principle for mappings $f:X^m\to X$, where $X^m$ is the Cartesian product of $m$ copies of $X$. We also compare our result with a recent one due to Secelean, who obtained a weaker assertion under less restrictive assumptions. We illustrate our result with several examples and give an application.

Miculescu and Mihail in 2008 introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS. Instead of families of selfmaps of a metric space $X$, they considered families of mappings defined on finite Cartesian product $X^m$. It turned out that a great part of the classical Hutchinson--Barnsley theory has natural counterpart in this GIFSs' case. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ and obtained versions of the Hutchinson--Barnsley theorem for appropriate families of such functions. In the paper we study some further aspects of Secelean's setting. In particular, we introduce and investigate a bit more restrictive framework and we show that some problems of the theory have more natural solutions within such a case. Finally, we present an example which shows that this extended theory of GIFSs gives us fractal sets that cannot be obtained by any IFSs or even by any GIFSs.%In a recent paper of the authors and Jachymski we itroduced fixed point theory for mappings defined on $\ell_\infty(X)$ product of a metric space $X$. In this note we use this approach to study a version of Hutchinson--Barnsley theory of fractals for function systems consisting of such mappings. In particular, we generalize the earlier concept of generalized generated function systems introduced by Mihail and Miculescu in 2008, in which mappings defined on finite Cartesian product are considered, and compare our approach with the one introduced recently by Secelean.

Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode to some extent the well-known algorithm of the sieve of Eratosthenes. Most part of the text is written in pedagogical style, however some formulas are new., Comment: 16 pages, 5 figures

We consider solutions to the maximum recursion on weighted branching trees given by$$X\,{\buildrel d\over=}\,\bigvee_{i=1}^{N}{A_iX_i}\vee B,$$where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in\mathbb{N}}$ are random positive numbers and $X_i$ are independent copies of $X$, also independent of $N$, $B$, $\{A_i\}_{i\in\mathbb{N}}$. Properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E}\big[\sum_{i=1}^NA_i^s\big]$. Recently, Jelenkovi\'c and Olvera-Cravioto proved, assuming e.g. $m(s)<1$ for some $s$, that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e.$$\mathbb{P}[R>t]\sim Ct^{-\alpha}$$for some $\alpha>0$ and $C>0$. In this paper we assume $m(s)\ge 1$ for all $s$ and prove analogous results., Comment: 16 pages

Recently (Phys. Rev. Lett. 114 (2015), 210402) the influence of the so called "Wigner translations" (more generally-Lorentz trans- formations) on circularly polarized Gaussian packets ( providing the solution to Maxwell equations in paraxial approximation) has been studied. It appears that, within this approximation, the Wigner trans- lations have an efect of shifting the wave packet trajectory parallel to itself by an amount proportional to the photon helicity. It has been suggested that this shift may result from specific properties of the algebra of Poincare generators for massless particles. In the present letter we describe the general relation between transformation proper- ties of electromagnetic field on quantum and classical levels. It allows for a straightforward derivation of the helicity-dependent transforma- tion rules. We present also an elementary derivation of the formula for sideways shift based on classical Maxwell theory. Some comments are made concerning the generalization to higher helicities and the relation to the coordinate operator defined long time ago by Pryce., Comment: 13 pages; no figures; section "Conclusions" added; missing normalization factor in eq.(20) inserted; final form to appear in Physics Letters B

Let $S_n$ be partial sums of an i.i.d. sequence $\{X_i\}$. We assume that $\mathbb{E} X_1 <0$ and $\mathbb{P}[X_1>0]>0$. In this paper we study the first passage time $$ \tau_u = \inf\{n:\; S_n > u\}. $$ The classical Cram\'er's estimate of the ruin probability says that $$ \mathbb{P}[\tau_u<\infty] \sim C e^{-\alpha_0 u}\quad \text{as } u\to \infty, $$ for some parameter $\alpha_0$. The aim of the paper is to describe precise large deviations of the first crossing by $S_n$ a linear boundary, more precisely for a fixed parameter $\rho$ we study asymptotic behavior of $\mathbb{P}\left[\tau_u = \lfloor u/\rho\rfloor \right]$ as $u$ tends to infinity., Comment: 10 pages

The paper is devoted to searching algorithms which will allow to generate images of attractors of \emph{generalized iterated function systems} (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space $X$, we consider mappings form the Cartesian product $X\times...\times X$ to $X$). Two presented algorithms are counterparts of classical \emph{deterministic algorithm} and so-called \emph{chaos game}. The third and fourth one one is fitted to special kind of GIFSs - to \emph{affine} GIFS, which are, in turn, also investigated., Comment: 22 pages, 6 figures

The coadjoint orbit method is applied to the construction of Hamiltonian dynamics of massless particles of arbitrary helicity. The unusual transformation properties of canonical variables are interpreted in terms of nonlinear realizations of Poincare group. The action principle is formulated in terms of new space-time variables with standard transformation properties., Comment: 26 opages; no figures; few references added

The generalization of Bertrand's theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analyzed for the case of Kepler and harmonic oscillator potentials., Comment: 15 pages, 3 figures; some references added, few misprints corrected

We consider the dynamics invariant under the action of l-conformal Galilei group using the method of nonlinear realizations. We find that by an appropriate choice of the coset space parametrization one can achieve the complete decoupling of the equations of motion. The Lagrangian and Hamiltonian are constructed. The results are compared with those obtained by Galajinsky and Masterov [Nucl. Phys. B860, (2013), 212]., Comment: 16 pages, no figures; substantial improvements of the text; several typos corrected; accepted for publication in Nucl. Phys. B

It is well-known that the dynamics of low energy electron in graphene honeycomb lattice near the K/K' points can be described, in tight-binding approximation, by 2+1 massless Dirac equation. Graphene's spin equivalent, "pseudospin", arises from the degeneracy introduced by the honeycomb lattice's two inequivalent atomic sites per unit cell. Mecklenburg and Regan (Phys. Rev. Lett. 106 (2011), 116803) have shown that, contrary to the common view, the pseudospin has all attributes of real angular momentum. In some circumstances, the internal symmetries can produce an important contribution to angular momentum. This phenomenon has been known for many years in particle physics and called "spin from isospin". We show that similar mechanism works in the case of lattice pseudospin.

Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the "mass" parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N+1)-th order equation of motion. As a special case, the Schroedinger group and the standard Newton equations are obtained for N=1 (l=1/2)., Comment: 18 pages, no figures; few references added

It is well-known that the tight-binding Hamiltonian of graphene describes the low-energy excitations that appear to be massless chiral Dirac fermions. Thus, in the continuum limit one can analyze the crystal properties using the formalism of quantum electrodynamics in 2+1 dimensions (QED2+1) which provides the opportunity to verify the high energy physics phenomena in the condensed matter system. We study the symmetry properties of 2+1-dimensional Dirac equation, both in the non-interacting case and in the case with constant uniform magnetic field included in the model. The maximal symmetry group of the massless Dirac equation is considered by putting it in the Jordan block form and determining the algebra of operators leaving invariant the subspace of solutions. It is shown that the resulting symmetry operators expressed in terms of Dirac matrices cannot be described exclusively in terms of gamma matrices (and their products) entering the corresponding Dirac equation. It is a consequence of the reducibility of the considered representation in contrast to the 3+1-dimensional case. Symmetry algebra is demonstrated to be a direct sum of two gl(2,C) algebras plus an eight-dimensional abelian ideal. Since the matrix structure which determines the rotational symmetry has all required properties of the spin algebra, the pseudospin related to the sublattices (M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. 106, 116803 (2011)) gains the character of the real angular momentum, although the degrees of freedom connected with the electron's spin are not included in the model. This seems to be graphene's analogue of the phenomenon called "spin from isospin" in high energy physics.

Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of O(2,1){\oplus}SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistors components. For arbitrary Hermitian quantum-mechanical Galilean twistor realizations we obtain the result that the representations of GCA with positive-definite Hamiltonian do not exist. For non-positive H we construct for N{\geq}2 the Hermitian Galilean N-twistor realizations of GCA; for N=2 such realization is provided explicitly., Comment: v3: 1+12 pages, with typos corrected in PLB proofs included

The alternative version of Hamiltonian formalism for higher-derivative theories is proposed. As compared with the standard Ostrogradski approach it has the following advantages: (i) the Lagrangian, when expressed in terms of new variables yields proper equations of motion; no additional Lagrange multipliers are necessary (ii) the Legendre transformation can be performed in a straightforward way provided the Lagrangian is nonsingular in Ostrogradski sense. The generalization to singular Lagrangians as well as field theory are presented., Comment: 22 pages; no figures; some comments added; accepted for publication in Phys. Rev. D

We consider the Euclidean path integral approach to higher-derivative theories proposed by Hawking and Hertog (Phys. Rev. D65 (2002), 103515). The Pais-Uhlenbeck oscillator is studied in some detail. The operator algebra is reconstructed and the structure of the space of states revealed. It is shown that the quantum theory results from quantizing the classical complex dynamics in which the original dynamics is consistently immersed. The field-theoretical counterpart of Pais-Uhlenbeck oscillator is also considered., Comment: 14 pages; no figures;the paper considerably extended; field-theoretical part added

An alternative version of Hamiltonian formalism for higher-derivative theories is presented. It is related to the standard Ostrogradski approach by a canonical transformation. The advantage of the approach presented is that the Lagrangian is nonsingular and the Legendre transformation is performed in a straightforward way., Comment: 8 pages, no figures

We consider an elementary discrete process which starts from purely random configuration and leads to well-ordered and stable state. Complete analytical solution to this problem is presented., Comment: LaTeX, 13 pages, 6 figures (included)

The theories defined by Lagrangians containing second time derivative are considered. It is shown that if the second derivatives enter only the terms multiplied by coupling constant one can consistently define the perturbative sector via Dirac procedure. The possibility of introducing standard canonical variables is analysed in detail. The ambiguities in quantization procedure are pointed out., Comment: 16 pages, no figures

It is argued that the twisted gauge theory is consistent provided it exhibits also the standard noncommutative gauge symmetry., Comment: 7 pages, no figures;two references added

Criterion for the Riemann hypothesis found by B\'{a}ez-Duarte involves certain real coefficients $c_{k\text{}}$defined as alternating binomial sums. These coefficients can be effectively investigated using N\"{o}% rlund-Rice's integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of $c_{k\text{}}$for large values of $k$, at least to $k=4\cdot 10^{8}$. We give explicit expressions both for the trend and for the oscillations. The first tends to zero and is therefore, in view of the criterion, irrelevant for the Riemann hypothesis. The oscillations can be further decomposed into a series of harmonics with amplitudes diminishing quickly. Possible violation of the Riemann hypothesis would indicate that the amplitude of some high harmonic increases., Comment: 10 pages, 6 figures

Simple argument in favour of unitarity, to all orders, of space-like noncommutative theory is given., Comment: 8 pages, no figures;some changes in the main text, and the list of references modified

The global counterpart of infinitesimal symmetries of noncommutative space-time is discussed., Comment: 7 pages, no figures; minor changes in the bibliography; final version accepted for publication in Phys. Rev.D

We consider the deformed Poincare group describing the space-time symmetry of noncommutative field theory. It is shown how the deformed symmetry is related to the explicit symmetry breaking., Comment: 7 pages, no figures, final version, accepted for publication in Phys. Lett. B