scholarly journals The Sprague-Grundy Functions of Saturations of Misère Nim

10.37236/8916 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Yuki Irie
Keyword(s):  
Explicit Formula ◽  
Explicit Formulas ◽  
Terminal Position ◽  
Play Game

We consider misère Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of misère Nim. All three of these games can be considered as position restrictions of Nim. What are the differences between them? We point out that Nim and Welter's game are saturated, but misère Nim is not. Moreover, we present explicit formulas for the Sprague-Grundy functions of saturations of misère Nim, which are obtained from misère Nim by adjoining some moves.  

scholarly journals Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

2021 ◽  
Vol 14 (1) ◽  
pp. 65-81
Author(s):  
Roberto Bagsarsa Corcino ◽  
Jay Ontolan ◽  
Maria Rowena Lobrigas
Keyword(s):  
Recurrence Relation ◽  
Explicit Formula ◽  
Taylor Series ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Difference Operator ◽  
Interpolation Formula ◽  
Explicit Formulas ◽  
Fundamental Properties ◽  
Whitney Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.

scholarly journals Generalized Shifted Chebyshev Koornwinder’s Type Polynomials: Basis Transformations

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 692 ◽  
Author(s):  
Mohammad AlQudah ◽  
Maalee AlMheidat
Keyword(s):  
Explicit Formula ◽  
Bernstein Polynomial ◽  
Continuous Functions ◽  
The Other ◽  
Numerical Techniques ◽  
Bernstein Basis ◽  
Fixed Degree ◽  
Explicit Formulas ◽  
Degree Elevation ◽  
Polynomial Bases

Approximating continuous functions by polynomials is vital to scientific computing and numerous numerical techniques. On the other hand, polynomials can be characterized in several ways using different bases, where every form of basis has its advantages and power. By a proper choice of basis, several problems will be removed; for instance, stability and efficiency can be improved, and numerous complications can be resolved. In this paper, we provide an explicit formula of the generalized shifted Chebyshev Koornwinder’s type polynomial of the first kind, T r * ( K 0 , K 1 ) ( x ) , using the Bernstein basis of fixed degree. Moreover, a Bézier’s degree elevation was used to rewrite T r * ( K 0 , K 1 ) ( x ) in terms of a higher degree Bernstein basis without altering the shapes. In addition, explicit formulas of conversion matrices between generalized shifted Chebyshev Koornwinder’s type polynomials and Bernstein polynomial bases were given.

scholarly journals DISCRETE INTERPOLATION BETWEEN MONOTONE PROBABILITY AND FREE PROBABILITY

2006 ◽  
Vol 09 (01) ◽  
pp. 77-106 ◽  
Author(s):  
ROMUALD LENCZEWSKI ◽  
RAFAŁ SAŁAPATA
Keyword(s):  
Explicit Formula ◽  
Free Product ◽  
Central Limit ◽  
Limit Theorems ◽  
Free Probability ◽  
Explicit Formulas ◽  
Cauchy Transforms ◽  
New Type ◽  
Monotone Probability

We construct a sequence of states called m-monotone product states which give a discrete interpolation between the monotone product of states of Muraki in monotone probability and the free product of states of Avitzour and Voiculescu in free probability. We derive the associated basic limit theorems and develop the combinatorics based on non-crossing ordered partitions with monotone order starting from depth m. We deduce an explicit formula for the Cauchy transforms of the m-monotone central limit measures and for the associated Jacobi coefficients. A new type of combinatorics of inner blocks in non-crossing partitions leads to explicit formulas for the mixed moments of m-monotone Gaussian operators, which are new even in the case of monotone independent Gaussian operators with arcsine distributions.

scholarly journals Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 319-327 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Integral Representations ◽  
Logarithmic Function ◽  
Explicit Formulas

In the paper, by establishing a new and explicit formula for computing the n-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial n! are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.

scholarly journals ON DIFFERENTIAL CHARACTERISTIC CLASSES

2014 ◽  
Vol 99 (1) ◽  
pp. 30-47 ◽  
Author(s):  
MAN-HO HO
Keyword(s):  
Explicit Formula ◽  
Chern Class ◽  
Natural Transformation ◽  
Euler Class ◽  
Characteristic Classes ◽  
Characteristic Class ◽  
Chern Classes ◽  
Explicit Formulas ◽  
Differential Characteristic ◽  
Differential Characters

In this paper we give explicit formulas for differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and the differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons–Sullivan) differential $K$-theory to (Cheeger–Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern class on Freed–Lott differential $K$-theory. Finally, we discuss the odd differential Chern classes.

scholarly journals The Typical Cell of a Voronoi Tessellation on the Sphere

2021 ◽  
Author(s):  
Zakhar Kabluchko ◽  
Christoph Thäle
Keyword(s):  
Explicit Formula ◽  
Unit Sphere ◽  
Voronoi Tessellation ◽  
Explicit Formulas ◽  
Dimensional Unit ◽  
Typical Cell ◽  
Low Dimensional

AbstractThe typical cell of a Voronoi tessellation generated by $$n+1$$ n + 1 uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ S d is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in $$\mathbb {R}^d$$ R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals Closed formulas for special bell polynomials by Stirling numbers and associate Stirling numbers

2020 ◽  
Vol 108 (122) ◽  
pp. 131-136
Author(s):  
Feng Qi ◽  
Dongkyu Lim
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Special Values

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

scholarly journals A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers

2015 ◽  
Vol 3 (1) ◽  
pp. 33 ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Genocchi Numbers

<p>In the paper, the authors concisely review some explicit formulas and establish a new explicit formula for the Bernoulli and Genocchi numbers in terms of theStirlingnumbers of the second kind.</p>

ON THE NUMBER OF SOLUTIONS TO CERTAIN DIAGONAL EQUATIONS OVER FINITE FIELDS

2010 ◽  
Vol 06 (01) ◽  
pp. 1-14 ◽  
Author(s):  
IOULIA BAOULINA
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Explicit Formula ◽  
Finite Fields ◽  
Characteristic P ◽  
Explicit Formulas ◽  
Number Of Solutions

We consider a diagonal equation, which can be reduced to the form [Formula: see text] over a finite field of characteristic p > 2. In 1997, Sun obtained the explicit formula for the number of solutions to an equation of this type when n is even. In this paper, we find explicit formulas for the number of solutions when n is odd, k = 2rh, and there exists a positive integer ℓ such that pℓ ≡ 2m-1h + 1 ( mod 2mh), m = 3 or 4, r ≥ m, h = 1 or 3.

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