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In previous notes, we have discussed a method of obtaining equilibrium distributions by balancing reaction probabilities, i.e. taking the ln (log) of a balance equation and linking it ( up to constants) to conservation of energy e1+e2=e3+e4. We have also linked such a scenario to factorial schemes M!/ [m1! M2! …] ((1)) subject to energy and number constraints. In a recent note (1), we argued that reaction balance for the Maxwell-Boltzmann case is based on complete uncertainty of reaction products for each reaction e1+e2=E. In this note, we try to link this idea of uncertainty of reaction products to the factorial scheme ((1)). For more general cases, f(ei) is replaced by g(f(ei)) which obeys Maxwell-Boltzmann statistics and yields results for non-Maxwell-Boltzmann f(ei) such as Fermi-Dirac and Bose-Einstein cases.

n regular statistical mechanics, the idea of “maximizing” the number of physical arrangements of particles with different energies leads to both Shannon’s entropy and the canonical distribution (1). This “maximization of arrangements”, it seems, is treated as a fundamental idea of nature. In a series of notes, we argued that the equilibrium distribution for general cases as well as for the Maxwell-Boltzmann case may be obtained from a time reversal balance of elastic two body collisions. In a previous note (2), we argued one may map collisions (e.g. e1+e2=e3+e4) into M!/ (m1! m2! …) where m1 is the number of reactions of particles with energy m1. (For the Maxwell-Boltzmann case, then number of reactions is proportional to the number of particles, but this is not the general case.) If e1 and e2 appear next to each other in the factorial arrangement, they may react. In (2), however, we did not argue why one needs to “maximize” the scattering arrangements. Instead, we showed maximization leads to the same result as reaction balance. In this note, we try to investigate why maximization is the same reaction balance.

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities, including an adapted version of the Pascal's rule. We then investigate the associated generating functions, for which we establish a recursive, combinatorial and integral formulation. From this, we derive an asymptotic version of the Binomial Theorem. A combinatorial and asymptotic analysis of some finite sums completes the paper.

Since G.E.P. Box introduced central composite designs in early fifties of 20th century, the classic design of experiments (DoE) utilizes response surface models (RSM), however usually limited to the simple form of low-degree polynomials. In the case of small size datasets, the conformity with the normal distribution has very weak reliability and it leads to very uncertain assessment of a parameter statistical significance. The bootstrap approach appears to be better solution than - theoretically proved but only asymptotically equal - t distribution based evaluation. The authors presents the comparison of the RSM model evaluated by a classic method and bootstrap approach.

We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis., Comment: 8 pages, to appear in Proceedings of the AMS

We give an overview of the use of asymptotic expansions of gamma and related functions --- ratio of gamma functions, powers, digamma and polygamma functions. The aim is to introduce a general theory which can unify various particular formulas for factorial functions and binomial coefficients. The connection with inequalities for gamma function is established. Also, a systematic approach to asymptotic expansion of various integral means, bivariate classical and parameter means is given, with applications to comparison of means.

In this paper, we show under the abc conjecture that the Diophantine equation f(x)=u!+v! has only finitely many integer solutions (x,u,v) whenever f(X) Q[X] is a polynomial of degree at least three.

Problem: The prediction of cutting forces is very important in designing mechanical micromachining to ensure geometrical accuracy of the machined feature and avoid tool breakage. These predictions can be done either via analytical models or finite-element models. The finite-element models are precise but usually time consuming to run. Analytical models, on the other hand, are less accurate but computationally much cheaper. The problem here is to integrate these two types of physics-based models and obtain an easy-to-evaluate statistical model that can approximate the machining forces. Approach: We propose performing a sensitivity analysis using the computationally cheap analytical models prior to conducting the computationally intensive finite-element simulations. With the elicited prior knowledge from the sensitivity analysis, a two-stage strategy is presented for designing the finite-element simulations in which customized number of levels can be assigned for each input factor. The finite-element simulation data can then be integrated with the analytical models in developing the final metamodel. Results: We show that the initial sensitivity analysis can reveal critical information about the underlying system and guide us to more efficiently extract information from the finite-element models. The proposed design for the finite-element simulations is comprised of two subarrays and overall can achieve desirable orthogonality and space-filling properties. Compared with using n different levels for all input factors, as in the traditional space-filling design, the new design is more capable for estimating factor interactions while still maintaining the ability to capture necessary nonlinear effects. By using fewer levels, it can also improve the efficiency of estimating the effects when the simulation is subject to stochastic noise. In model validation, our numerical results indicate that the fitted integrated metamodel can more precisely approximate the machining forces than either using the analytical models or the traditional metamodel based on the finite-element simulations alone.

We study the solutions of the equation ϕ ( C m ) / ϕ ( C n ) = r , where r is a fixed rational number, C k is the k th Catalan number and ϕ is the Euler function. We note that the number r = 4 is special for this problem and for it we construct solutions ( m , n ) to the above equation which are related to primes p such that 2 p − 1 or 4 p − 3 is also prime.

Consider the multiplicities ep1(n),ep2(n),…,epk(n) in which the primes p1,p2,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m1,m2,…,mk) for any fixed integers m1,m2,…,mk, thus improving a result of Luca and Stănică [F. Luca, P. Stănică, On the prime power factorization of n!, J. Number Theory 102 (2003) 298–305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.

Let p1,p2,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given εi∈{0,1}(i=1,2,…,k), there exist infinitely many positive integers n withe1(n!)≡ε1(mod2),e2(n!)≡ε2(mod2),…,ek(n!)≡εk(mod2),where ei(n!) denotes the exponent of the prime pi in the standard factorization of positive integer n!. In 1997 Berend proved a conjecture of Erdős and Graham, that is, the conclusion with all εi=0.

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials., Comment: 16 pages, to be published in the Journal of Number Theory

WOS: 000314309400001 Binomial coefficients are used in many fields such as computational and applied mathematics, statistics and probability, theoretical physics and chemistry. For accurate numerical results, the correct calculation of these coefficients is very important. We present some new recurrence relationships and numerical methods for the evaluation of binomial coefficients for negative integers. For this purpose, we give some comparisons of the outputs for different computer programming languages in case of negative integers, and also we wrote two new algorithms for computations.

In this paper, we solve a few diophantine equations linked to balancing numbers and factorials. The basic problem (B_y=x!) has only the one nontrivial solution (B_2=6=3!), but it is a direct consequence of a theorem of F. Luca. The more difficult problem (B_y=x_2!/x_1!) is still open, but we solve it under different conditions. Two related problems are also studied.