A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter λ. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases λ=1 and λ=−1 were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials.

Divisibility of Fibonomial coefficients in terms of their digital representations and applications

<abstract><p>We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.</p></abstract>

Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction. Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions , is a non-negative integer

Sums Over Primes II

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

The second moment of Sn(t) on the Riemann hypothesis

Let [Formula: see text] be the argument of the Riemann zeta-function at the point [Formula: see text]. For [Formula: see text] and [Formula: see text] define its antiderivatives as [Formula: see text] where [Formula: see text] is a specific constant depending on [Formula: see text] and [Formula: see text]. In 1925, Littlewood proved, under the Riemann Hypothesis (RH), that [Formula: see text] for [Formula: see text]. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of [Formula: see text] and [Formula: see text]. This was extended by Fujii for [Formula: see text], when [Formula: see text]. Assuming the RH, we give the explicit asymptotic formula for the second moment of [Formula: see text] up to the second-order term, for [Formula: see text]. Our result conditionally refines Selberg’s and Fujii’s formulas and extends previous work by Goldston in [Formula: see text], where the case [Formula: see text] was considered.

Proof of the Bessenrodt–Ono Inequality by Induction

In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

In this paper, we consider the sum Snξ = ξ1 + ... + ξn of possibly dependent and nonidentically distributed real-valued random variables ξ1, ... , ξn with consistently varying distributions. By assuming that collection {ξ1, ... , ξn} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((Snξ)α1(Snξ > x)) and E((Snξ – x)+)α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.

The Rayleigh wave scattering on a rectangular lattice of the solid roughness discontinuities

The problem of the surface acoustic Rayleigh wave scattering on a deterministic three-dimensional roughness, occupying a finite size rectangular region of an isotropic solid free surface, is solved in the Rayleigh-Born approximation of the perturbation theory in a roughness amplitude. Formula for the displacement field in the scattered Rayleigh wave at a big distance from the roughness, as compared to rough region sizes L1,2 along the x1,2- axes respectively, and asymptotic formulas for this displacement field in the Bragg, i.e. short-wavelength λ≪ L1,2 limit, where λ is the wavelength, are derived. The new laws of scattering are obtained. They are caused by a strong modulation of scattering by the roughness form. They exceed the fundamental physical conception, that a wave scattering in the short-wavelength limit takes place on a medium discontinuities, by the statement, that a wave strongly senses the structure of a medium in the near vicinity of discontinuities as well as the form-factor of the discontinuities lattice. This form-factor is a dependence of the discontinuity amplitude, i.e. of a difference of the left and right limit values of a roughness non-zero derivative, including one of zero order, in coordinate at a point of discontinuity, on a number of this discontinuity in a lattice. This exceeded physical conception violates the classical Laue-Bragg-Wulff laws of scattering.