M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

Algebraic Independence of Certain Generalized Mahler Type Numbers

In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.     

The Weight Distribution of C 5(1, n)

In [2] the codes C _q (r,n) over were introduced. These linear codes have parameters , can be viewed as analogues of the binary Reed-Muller codes and share several features in common with them. In [2], the weight distribution of C ₃(1,n) is completely determined.In this paper we compute the weight distribution of C ₅(1,n). To this end we transform a sum of a product of two binomial coefficients into an expression involving only exponentials an Lucas numbers. We prove an effective result on the set of Lucas numbers which are powers of two to arrive to the complete determination of the weight distribution of C ₅(1,n). The final result is stated as Theorem 2.     

Bass Numbers of Local Cohomology Modules with Respect to an Ideal

Let be a commutative Noetherian local ring and let be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module are finite if one of the following holds:

A divergent Khintchine theorem in the real,complex, and <Emphasis Type="Italic">p</Emphasis>-adic fields

In this paper, we show that if the sum ∑_r=1^∞ Ψ(r) diverges, then the set of points (x, z, w) ∈ ℝ × ℂ × ℚ_p satisfying the inequalities , and for infinitely many integer polynomials P has full measure. With a special choice of parameters v _i and λ _i , i = 1, 2, 3, we can obtain all the theorems in the metric theory of transcendental numbers which were known in the real, complex, or p-adic fields separately.     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ? _p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ? _p . By using this symmetry of fermionic p-adic invariant integral on ? _p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ? _p .     

Decomposition Numbers and Canonical Bases

We obtain some simple relations between decomposition numbers of quantized Schur algebras at an nth root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space representation of . It follows that these coefficients can all be expressed in terms of those of the global crystal basis of the irreducible subrepresentation generated by the vacuum vector. As a consequence, using the works of Ariki and Varagnolo and Vasserot, it is possible to give a new proof of Lusztig"s character formula for the simple U _v (sl _r )-modules at roots of unity, which does not involve representations of of negative level.     

The odd-number sequence: squares and sums

Direct study of various characteristics of integers and their interactions is readily accessible to undergraduate students. Integers obviously fall in different classes of modular rings and thus have features unique to that class which can result in a variety of formations, particularly with sums of squares. The sum of the first n odd numbers is itself the square of n within the odd number sequence, from which testing for primality within the Fibonacci sequence is investigated in this note.     

Some new Fibonacci and Lucas identities by matrix methods

The aim of this article is to characterize the 2 × 2 matrices X satisfying X ² = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers.     

A New Approach to Polynomial Identities

Using the formal derivative idea, we give a generalization for the Cauchys Theorem relating to the factors of (x + y)ⁿ–xⁿ– yⁿ. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial

Hua＇s Theorem on Five Almost Equal Prime Squares

In this paper we sharpen Hua＇s result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N=p1^2＋p2^2＋p3^2＋p4^2＋p5^2,with │pj-√N/5│≤U=N^1/2-1/28＋ε,where pj are primes.     

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.      

The strong law for the integral of P-L estimate under left truncation and right censoring

For left truncated and right censored model, letF _n be the product-limit estimate and φ a nonnegative measurable function. The almost sure limits of the cumulative hazard function based onF _n pd the integral ∫ ϕdF _n are given. The results are useful in establishing strong consistent results of various estimates. For left truncated data, similar results were obtained in literature.     

The Ramsey number for a cycle of length five vs. a complete graph of order six

It has been conjectured that r(C_m, K_n) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 4. This has been proved recently for n = 4 and n = 5. In this paper, we prove that r(C₅, K₆) = 21. This raises the possibility that r(C_m, K₆) = 5m − 4 for all m ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 99–108, 2000     

Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit

Manufacturing of steel involves thermal energy intensive processes with coal as the major input. Energy generated is a direct function of ash content of coal and as such it weighs very high as regards the choice of coal. In this paper, we study a multiobjective transportation problem to introduce a new type of coal in a steel manufacturing unit in India. The use of new type of coal serves three non-prioritized objectives, viz. minimization of the total freight cost, the transportation time and the ratio of ash content to the production of hot metal. It has been observed from the past data that the supply and demand points have shown fluctuations around their estimated values because of changing economic conditions. To deal with uncertainties of supply and demand parameters, we transform the past data pertaining to the amount of supply of the ith supply point and the amount of demand of the jth demand point using level (λ,ρ) interval-valued fuzzy numbers. We use a linear ranking function to defuzzify the fuzzy transportation problem. A transportation algorithm is developed to find the non-dominated solutions for the defuzzified problem. The application of the algorithm is illustrated by numerical examples constructed from the data provided by the manufacturing unit.      

Rationally smooth algebraic monoids

A reductive monoid M is called rationally smooth if it has sufficiently mild singularities as a topological space. We characterize this class of monoids in combinatorial terms. We then use our results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the “monoid BB-decomposition”.