Hurwitz Type Research Articles

Tricuspidization of Quadricuspid Aortic Valve

A 65-year-old woman presented with New York Heart Association class II-III symptoms, no overt signs of heart failure, and echocardiographic findings of a quadricuspid aortic valve, Hurwitz type C, with severe aortic regurgitation, dilated left ventricle (7 cm), and moderate left ventricular dysfunction (45%). She subsequently underwent tricuspidization of the valve at the level of the abnormal commissure with subcommissural annuloplasty. At her 6-week follow-up visit, the patient was in New York Heart Association class I, with reduction of left ventricular diastolic dimensions, trace aortic regurgitation, and good mobility of the leaflets.

Integral extensions and the [formula omitted]-invariant

In this note we compare the a-invariant of a homogeneous algebra B to the a-invariant of a subalgebra A. In particular we show that if A⊂B is a finite homogeneous inclusion of standard graded domains over an algebraically closed field with A normal and B of minimal multiplicity then A has minimal multiplicity. In some sense these results are algebraic generalizations of Hurwitz type theorems.

Applications of hybrid universality to multivariable zeta-functions

In the present paper, we obtain new results on universality as applications of hybrid universality and almost-periodicity in its half-plane of absolute convergence. By using these, we show the universality for Euler products of Igusa type, Euler–Zagier multiple zeta-functions and Tornheim–Hurwitz type of double zeta-functions.

GENERALIZED GAMMA FUNCTIONS WITH CHARACTERS

The main objective of this paper is to define Γ-functions Γ[χ](v) with characters χ and study their properties. To this end, we ought to introduce L-functions [Formula: see text] of Hurwitz type. We prove that [Formula: see text] holds, which combines the theory at "s = 0" and the theory at "s = 1".

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.

On Multiple Interpolation Functions of the -Genocchi Polynomials

Recently, many mathematicians have studied various kinds of the -analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.", Kim defined new generating functions of -Genocchi, -Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates -Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

New notion of index for hedgehogs of [formula omitted] and applications

The aim of this paper is to introduce new tools for studying the following two important and difficult problems in R 3 : (1) The Minkowski problem (to prescribe the Gauss curvature) for hedgehogs (i.e., for Minkowski differences of convex bodies); (2) The search for Sturm–Hurwitz type theorems (relating number of zeros to expansions in spherical harmonics). First, (1) we give a brief survey of hedgehog theory and a short introduction to these problems; (2) we recall briefly the main results already obtained (one of which is a counter-example to a conjecture of A.D. Alexandrov) and we explain why new tools are necessary for going further. Finally, we introduce a new notion of index for studying hedgehogs and we give first geometrical applications.

The universality for linear combinations of Lerch zeta functions and the Tornheim–Hurwitz type of double zeta functions

In this paper, we consider the universality for linear combinations of Lerch zeta functions. J. Kaczorowski, A. Laurincikas and J. Steuding treated universal Dirichlet series with the case that the compact sets \({\mathcal{K}_l}\) are disjoint. But we consider the both cases that the compact subset \({\mathcal{K}_l}\) is disjoint and not disjoint. Next, we will show the non-trivial zeros of the Tornheim–Hurwitz type of double zeta functions in the region of absolute convergence. Moreover we show the universality for the Tornheim–Hurwitz type of double zeta function.

Zeros and the universality for the Euler-Zagier-Hurwitz type of multiple zeta-functions

In this paper, we show relations between the zero-free region and the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions. Roughly speaking these relations imply that we can obtain the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions by their zero-free property, and vice versa. Moreover, we obtain the non-trivial zeros, joint denseness and functional independence for the Euler–Zagier–Hurwitz type of multiple zeta-functions.

Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials

The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the - -Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these - -Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these - -Euler polynomials.

Q-Hardy–Berndt type sums associated with q-Genocchi type zeta and q- l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q -analogue of the Genocchi zeta function, q -analogue Hurwitz type Genocchi zeta function, and q -Genocchi type l -function. We define partial zeta function. By using this function, we construct p -adic interpolation functions which interpolate generalized q -Genocchi numbers at negative integers. We also define p -adic meromorphic functions on C p . Furthermore, we construct new generating functions of q -Hardy–Berndt type sums and q -Hardy–Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.

On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type

In this paper, we consider certain analogues of Eisenstein series involving hyperbolic sine and cosine functions. Using a result of Hurwitz, we prove some evaluation formulas for these series at positive integers. We can regard these formulas as analogues of the famous formulas for certain series given by Cauchy, Ramanujan, Berndt, and so on, as well as those for the Eisenstein series given by Hurwitz.

ON THE ANALOGS OF BERNOULLI AND EULER NUMBERS, RELATED IDENTITIES AND ZETA AND L-FUNCTIONS

In this paper, by using q-deformed bosonic p-adic integral, we give -Bernoulli numbers and polynomials, we prove Witt's type formula of -Bernoulli polynomials and Gauss multiplicative formula for -Bernoulli polynomials. By using derivative operator to the generating functions of -Bernoulli polynomials and generalized -Bernoulli numbers, we give Hurwitz type -zeta functions and Dirichlet's type -L-functions; which are interpolated -Bernoulli polynomials and generalized -Bernoulli numbers, respectively. We give generating function of -Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and -Bernoulli polynomials and ordinary Bernoulli numbers of order r and -Bernoulli numbers, respectively. We also study on -Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define -partial zeta function and interpolation function.

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well.

New Changhee [formula omitted]-Euler numbers and polynomials associated with [formula omitted]-adic [formula omitted]-integrals

Using non-archimedean q -integrals on Z p defined in [T. Kim, On a q -analogue of the p -adic log gamma functions and related integrals, J. Number Theory 76 (1999) 320–329; T. Kim, q -Volkenborn integration, Russ. J. Math. Phys. 9 (2002) 288–299], we define new Changhee q -Euler polynomials and numbers which are different from those of Kim [T. Kim, p -adic q -integrals associated with the Changhee–Barnes’ q -Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004) 415–420] and Carlitz [L. Carlitz, q -Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332–350]. We define generating functions of multiple q -Euler numbers and polynomials. Furthermore we construct a multivariate Hurwitz type zeta function which interpolates the multivariate q -Euler numbers or polynomials at negative integers.

Conventional bootstrap and normal confidence interval estimation under adaptive cluster sampling

We examine the performances of bootstrap and normal confidence interval methods for estimating the parametric total when using the Horvitz–Thompson type estimator under adaptive cluster sampling via simulation. On the basis of this and previous work, we include recommendations on using confidence interval methods under adaptive cluster sampling. Two Poisson spatially clustered populations and three sample sizes were used in our simulations. The bootstrap methods (Gross, McCarthy and Snowden, and Sitter) were based on resampling the original sample of units. All four interval estimation methods performed poorly. On average, bootstrap confidence interval estimates underestimated the parametric totals, resulting in empirical coverages lower than the set confidence level. Coverages for the normal method depended on initial sample sizes and population. In general, if confidence interval estimates are required under adaptive cluster sampling, we recommend using bootstrap interval estimation with the Hansen–Hurwitz type estimator instead of interval estimation with the Horvitz–Thompson type estimator.

The Hurwitz theorem for Bessel functions and antibound states in spectral theory

An 1889 result of Hurwitz is that the Bessel function J v ( c ) ( v > 0, c > 0) has no zeros for 2 N v N + 1, with integer N . A new application of this result is given which describes the behaviour of Jost functions and antibound states for one–dimensional Schrodinger equations with exponentially decaying potential. A new Hurwitz–type theorem with similar applications is developed for hypergeometric functions. Extensions are obtained where the power series are terminated after 2 N + 1 terms and larger zero–free intervals (2 N,V ) are found.

Results of Hurwitz type for three squares

We prove a large number of results in which the generating function of r 3( n), where r 3( n) is the number of representations of n as a sum of three squares, is, when n is restricted to an arithmetic sequence, a simple infinite product. We also conjecture a large number of further results of the same sort.

Results of hurwitz type for five or more squares

Let r k(n) denote the number of representations of n as a sum of k squares. We give many cases (for k = 5, 6, 7) in which the generating function ∑ n≥0 r k(an + b)q n is a simple infinite product. For instance, $$\sum\limits_{n \geqslant 0} {r_7 \left( {24n + 23} \right)q^n } = 49728\mathop \prod \limits_{n \geqslant 1} \frac{{(1 - q^{2n} )^{10} (1 - q^{3n} )^3 }}{{(1 - q^n )^6 }}.$$

Algebraic Criteria of Absolute Stability

New algebraic criteria of absolute stability are obtained on the basis of a polynomial form of the frequency condition for absolute stability (Popov inequality). The analysis of the inequality is reduced to an analysis of the local properties of polynomials and a rational function on the negative real semiaxis. As a result, the stability criteria do not require the plotting of any sort of curves and can be expressed on the basis of algorithms of the Routh–Hurwitz type and calculations of the negative real roots of polynomials.