We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For ...

S. Chowla conjectured that every prime p has the property that there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. Gauss’ genus theory guarantees the existence of infinitely many such fields when p = 2, and the work of Davenport and Heilbronn [D-H] suffices for the prime p = 3. In addition, the DavenportHeilbronn result demonstrates that a positive pr...

“FIBONACCI” directly implements the recurrence relation of the Fibonacci sequence. Each number in the sequence is the sum of the two previous numbers in the sequence. The running time is clearly O(n). The subproblem graph consists of n+ 1 vertices, v0, v1, ..., vn. For i = 2, 3, ..., n, vertex i has two leaving edges: to vertex vi−1 and to vertex vi−2. No edges leave vertices v0 or v1. Thus, th...

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation r t xΔ t γ Δ p t x g t 0 on an arbitrary time scale T with sup T ∞, where g t ≥ t and ∫∞ to Δs/ r1/γ s < ∞. Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify th...

In this paper, we analyse the dual pivot Quicksort, a variant of the standard Quicksort algorithm, in which two pivots are used for the partitioning of the array. We are solving recurrences of the expected number of key comparisons and exchanges performed by the algorithm, obtaining the exact and asymptotic total average values contributing to its time complexity. Further, we compute the averag...

background : meningiomas are common benign tumors of the central nervous system. patients with meningiomas achieve postoperative optimal functional recovery, but there is a probability of tumor recurrence months or years after surgical resection. this study aims to evaluate the prevalence of recurrent meningioma and the correlation between tumor recurrence and certain factors. materials and met...

Deene transportation polytope Tn;m to be a polytope of nonnegative n m matrices with row sums equal to m and column sums equal to n. We present an eecient algorithm for computing the numbers f k of the k-dimensional faces for the transportation polytope T n;n+1. The construction relies on the new recurrence relation for the numbers f i , which is of independent interest.

Let LN+1 be a linear differential operator of order N + 1 with constant coefficients and real eigenvalues 1, . . . , N+1, let E( N+1) be the space of all C∞-solutions of LN+1 on the real line. We show that for N 2 and n = 2, . . . , N , there is a recurrence relation from suitable subspaces En to En+1 involving real-analytic functions, andwithEN+1=E( N+1) if and only if contiguous eigenvalues a...

In this paper explicit expressions and some recurrence relations are derived for marginal and joint moment generating functions of generalized order statistics from Erlang-truncated exponential distribution. The results for k-th record values and order statistics are deduced from the relations derived. Further, a characterizing result of this distribution on using the conditional expectation of...

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations r t xΔ t γ Δ p t f x τ t 0 on a time scale T; here γ > 0 is a quotient of odd positive integers with r and p real-valued positive rdcontinuous functions defined on T. Our results not only extend some results established by Hassan in 2008 but also unify ...