We study the extended Euler sums and the alternating extended Euler sums and establish their explicit expressions in terms of Riemann zeta functions and Hurwitz zeta functions. Comparing with the existing results, ours are simpler and thus yield significantly better accuracy when Matlab is used for numerical calculation.

and extended analytically to the whole complex plane by a functional equation (see [8, p. 14]). The original Riemann hypothesis asserts that the nonreal zeros of the Riemann zeta function ζ(s) all lie on the line Re(s) = 1/2. In his monumental paper [11] of 1859, Riemann made this assertion in order to derive an expression for the deviation of the exact number of primes ≤ x, which is denoted by...

We give an unconditional version of a conditional, on the Extended Riemann Hypothesis, result of L. Babai, A. Banerjee, R. Kulkarni and V. Naik (2010) on the evasiveness of sparse graphs.

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function,...

We prove the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩ BVloc for the 3 × 3 system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most raref...

This is a sequel to papers by the last two authors making the Riemann–Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlevé equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann–Hilbert morphism is an isomorphism. As a consequence these equations have the Painlevé property and the Okamoto–Painl...

Symbolic manipulation of sparse polynomials, given as lists of exponents and nonzero coeecients, appears to be much more complicated than dealing with polyno-mials in dense encoding (see e.g. GKS 90, KT 88, P 77a, P 77b]). The rst results in this direction are due to Plaisted P 77a, P 77b], who proved, in particular, the NP-completeness of divisibility of a polynomial x n ?1 by a product of spa...

We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion h(t, z) of a finite subset E of the Riemann sphere C parameterized by a pointed hyperbolic Riemann surface (X, t0). We prove that if this holomorphic motion has a guiding quasiconformal isotopy, then it can be extended to a new holomorphic motion of E ∪ {p} for any point p...

Extended thermodynamics derives dissipative, hyperbolic field equations for monatomic gases. One example is the system of the 13-field-case, which is a dissipative extension of the Euler equations. In this paper the system is investigated by solving a Riemann problem. Additionally some model equations are introduced so as to discuss the main properties in a transparent manner. There arises an i...