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In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials. [ABSTRACT FROM AUTHOR]

We study the Nash and essential valuations of terminal threefolds with type c A / r. If r = 1 or the given threefold is ℚ -factorial, we can provide a complete description of all Nash and essential valuations. We also constructed counterexamples to the Nash problem for non-Gorenstein or non- ℚ -factorial threefolds. [ABSTRACT FROM AUTHOR]

The two-dimensional intermittency and its self-affine nature are investigated for p–p collisions at s = 1 3 TeV in the two-dimensional anisotropic (η − ϕ) space. The UrQMD model has been employed to generate and accumulate the p–p collisions data. Our investigation is made in the framework of scaled factorial moment (SFM) method. The concept of Hurst exponent H is incorporated to bring a qualitative comparison between the UrQMD generated minimum bias (MB) events and the events at a particular impact parameter b = 0. 4 fm. The variation of the fractal strength with the variation of H as well as with the variation of the order of the moment q has been analyzed. Also, the nonlinearity in the variation of SFM with that of H has been accompanied in this paper. It is observed that the fractal strength and the intermittent type of fluctuations are found to be much stronger in the region with H < 1 compared to the region with H > 1 and the self-affine nature in the fluctuations increases as H deviates from unity. [ABSTRACT FROM AUTHOR]

Local actions of ℙ ℕ , the group of finite permutations on ℕ , on quasi-local algebras are defined and proved to be ℙ ℕ -abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of C * -algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon ℙ ℕ in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here. [ABSTRACT FROM AUTHOR]

The aim of this paper is two-fold. First, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius–Euler polynomials. Second, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them. [ABSTRACT FROM AUTHOR]

This paper presents a study of bin–bin correlation of the produced shower particles in the pseudo-rapidity space by the method of factorial correlator in 1 6 O-AgBr and 3 2 S-AgBr interactions at 4.5 A GeV/ c. The correlated moments are found to increase with decreasing bin–bin separation D, following a power law. Strong bin–bin correlation is exhibited by the experimental data. Experimental data also supports the validity of log normal approximation. Experimental analysis has been compared with the results obtained from the analysis of events simulated by UrQMD model. [ABSTRACT FROM AUTHOR]

In this study, we have performed a detailed analysis of genuine pion correlations and fluctuations in terms of the normalized factorial comulant moments of second and third orders, K 2 and K 3 , in case of 1 6 O–AgBr interactions at 6 0 A and 2 0 0 A GeV/c. The experimental results are compared with the predictions of AMPT and UrQMD model simulated events. The UrQMD model reproduces the trends in experimental results but the strength of correlation is much smaller. However, the AMPT model does not also replicate all features of the experimental data. The genuine two-particle and three-particle correlations are found to become weaker with the increase in momentum of the projectile nucleus. [ABSTRACT FROM AUTHOR]

There is abundant research on the factors that influence backers' decision to support a crowdfunding project. Yet work validating existing results and offering in a single study a direct comparison of those factors among crowdfunding types is needed. We draw upon the theory of perceived values and examine empirically which perceived values influence backers' decision across equity, lending and reward-based crowdfunding. We employ a factorial survey that testes how changing the type of crowdfunding (ceteris paribus) alters the significance of the perceived values that a project must demonstrate in order to attract funding. A total of 337 participants evaluated one of the three variations of the same crowdfunding project with a manipulation of the type of reward that was offered (shares for equity, interest rate for lending and product for reward-based). Our findings validate and extend existing results, confirming the heterogeneity in backers' decision-to support a project. [ABSTRACT FROM AUTHOR]

The high-order cumulants and factorial cumulants of conserved charges are suggested to study the critical dynamics in heavy ion collisions. In this paper, using parametric representation of the three-dimensional Ising model, density plots on the phase diagram and temperature dependence of the order parameter cumulants and factorial cumulants are studied and compared. In the vicinity of the critical point, cumulants and factorial cumulants can not be distinguished. Far away from the critical point, sign changes occur in the factorial cumulants comparing with the same order cumulants. The cause of sign changes is analyzed. These features may be used to measure the distance to the critical point. [ABSTRACT FROM AUTHOR]

Let R ⊂ A be factorial domains containing ℚ. In this paper, we give a criterion, in terms of locally nilpotent derivations, for A to be R -isomorphic to R [ v , w ] / (c w − h (v)) , where c ∈ R is nonzero and h (v) ∈ R [ v ]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc.145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent R -derivations of the polynomial ring in two variables over R. [ABSTRACT FROM AUTHOR]

Let D be an integral domain, S be a nonzero torsionless commutative cancellative monoid with quotient group 〈 S 〉 , and D [ S ] be the semigroup ring of S over D. In this paper, among other things, we show that if char (D) = 0 (respectively, char (D) = p > 0) , then D [ S ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain, S is a weakly factorial GCD-semigroup, and 〈 S 〉 is of type (0 , 0 , 0 , ...) (respectively, (0 , 0 , 0 , ...) except p). [ABSTRACT FROM AUTHOR]

It is well known that sequences such as the Fibonacci numbers and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by P (d) = log 1 0 (1 + 1 / d) , d = 1 , 2 , ... , 9. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order k if, roughly speaking, k -tuples of consecutive leading digits behave like k independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford's Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a "best-possible" local Benford Law; that is, we determine the maximal value k such that the sequence is locally Benford distributed of order k. The result applies, in particular, to sequences of the form { a n } , { a n d } , and { n β a n α } , as well as the sequence of factorials { n ! } and similar iterated product sequences. [ABSTRACT FROM AUTHOR]

Let S be a multiplicative set in an integral domain D. A nonzero ideal I of D is said to be S - υ -principal if there exist an s ∈ S and a ∈ D such that s I ⊆ a D ⊆ I υ . Call D an S -GCD domain if each finitely generated nonzero ideal of D is S - υ -principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and S -class group of formal power series rings, J. Pure Appl. Algebra221 (2017) 2869–2879]. One aim of this paper is to characterize S -GCD domains, giving several equivalent conditions and showing that if D is an S -GCD domain then D S is a GCD domain but not conversely. Also we prove that if D is an S -GCD S -Noetherian domain such that every prime w -ideal disjoint from S is a t -ideal, then D is S -factorial and we give an example of an S -GCD S -Noetherian domain which is not S -factorial. We also consider polynomial and power series extensions of S -GCD domains. We call D a sublocally s -GCD domain if D is a { s n | n ∈ N } -GCD domain for every non-unit s ∈ D \ { 0 } and show, among other things, that a non-quasilocal sublocally s -GCD domain is a generalized GCD domain (i.e. for all a , b ∈ D \ { 0 } , a D ∩ b D is invertible). [ABSTRACT FROM AUTHOR]

In this paper, we initiate a study on Gauss factorials of polynomials over finite fields, which are the analogues of Gauss factorials of positive integers. [ABSTRACT FROM AUTHOR]

The theorems of Gauss and Jacobi that give modulo evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo . In this paper, we further extend these results to congruences modulo . In the process, we prove congruences to arbitrarily high powers of for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's -adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients. [ABSTRACT FROM AUTHOR]

The aim of this note is to improve an approximation formula of Ramanujan-type discussed by Hirschhorn and Villarino. [ABSTRACT FROM AUTHOR]

Assuming Dickson's conjecture, we give explicit formulas for Bhargava's factorials associated to the set ℙ2 of twin primes. We also characterize the polynomials of degree n which take integral values on ℙ2 by means of their values on {1, ..., 2n + 1}. Doing this, we give a negative answer to a conjecture of Mingarelli. Similar results are also obtained when replacing ℤ by 픽q[T]. [ABSTRACT FROM AUTHOR]

We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor. [ABSTRACT FROM AUTHOR]

We prove that the Diophantine equation $p_1^{r_1} \cdots p_k^{r_k} = \left( \begin{array}{cc}r_1 + \cdots + r_k\\ r_1, \ldots, r_k \end{array} \right)$ has only finitely many positive integer solutions k, p1, ..., pk, r1, ..., rk, where p1, ..., pk are distinct primes. If a positive integer n has prime factorization $p_1^{r_1}\cdots p_k^{r_k}$, then $ \left(\begin{array}{cc}{r_1} \+ \cdots \+{r_k}} \\ {{r_1}, \ldots \, {r_k}} \end{array} \right)$ represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers. [ABSTRACT FROM AUTHOR]

A theorem of Gauss extending Wilson's theorem states the congruence (n - 1)n! ≡ -1 (mod n) whenever n has a primitive root, and ≡ 1 (mod n) otherwise, where Nn! denotes the product of all integers up to N that are relatively prime to n. In the spirit of this theorem, we study the multiplicative orders of $(\frac{n-1}{M})_n! (mod n) for odd prime powers pα. We prove a general result about the connection between the order for pα and for pα+1 and study exceptions to the general rule. Particular emphasis is given to the cases M = 3, M = 4 and M = 6, while the case M = 2 is already known. [ABSTRACT FROM AUTHOR]

We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities. [ABSTRACT FROM AUTHOR]

Different fractal methods (Factorial moment, Takagi moment) have been used for the analysis of fractal behavior of multiparticle production in π--AgBr interaction at 350 GeV/c of compound multiplicity distribution in emission angle (cos θ) space and azimuthal angle (ϕ) space. The generalized fractal dimensions Dq are determined according to two methods. Results indicate multifractal behavior in compound multiplicity distribution. From the knowledge of Dq, the multifractal specific heat is calculated. Results are compared with the corresponding results of shower multiplicity. [ABSTRACT FROM AUTHOR]

We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition. Then we prove that for each factorial language, a canonical decomposition exists and is unique. [ABSTRACT FROM AUTHOR]