symplectic cut-and-gluing formulae of the relative Gromov–Witten invariants, we get a recursive formula for the Hurwitz number of triple ramified coverings of a Riemann surface by a Riemann surface.

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

This paper revisits the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so we employ several construction methods ― some conventional and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated ‘λ-fold triple systems’. Finally, we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for ‘perfect grouping systems’ which we introduce.

Using both the exact enumeration method (microcanonical transfer matrix) for small systems (up to 9×9 lattices) and the Wang-Landau Monte Carlo algorithm for large systems (up to 30×30 lattices), we obtain the exact and approximate densities of states g(M,E), as a function of magnetization M and exchange energy E, for the triangular-lattice Ising model in the presence of an external uniform magnetic field. The method for evaluating the exact density of states g(M,E) of the triangular-lattice Ising model is introduced for the first time. Based on the density of states g(M,E), we investigate the properties of the various thermodynamic quantities as a function of temperature T and magnetic field h and find the phase diagram of the Ising antiferromagnet in the magnetic field. In addition, the zero-temperature thermodynamic properties are studied by reference to the density of states at the corner or along the edge line on the magnetization-energy (ME) diagram.

Loading noise generated by steady aerodynamic force exerted on the rotating body surface is theoretically analyzed and its radiation characteristics is examined as a fundamental research of helicopter rotor noise. For simplicity, the force exerted on each blade is not distributed but concentrated at one point and the noise is evaluated by using Lowson' exact formula with a discussion of the physical meaning of each term in the formula. For a single point force rotating with various angular frequencies, we investigated the radiation characteristics and theoretically explained the physical behavior at near and far-field. By investigating the amplitude of acoustic pressure with various distances, we observed the different decreasing ratio at near- and far-field with the discussion of the effect of acceleration of angular frequency. Finally, the phenomenon that the noise level is reduced everywhere as the number of blade increases is explained with the suggestion of a noise reduction idea, the limitations of this study, and the future research topics.

The main purpose of this paper is to study on generating functions of the -Genocchi numbers and polynomials. We prove new relation for the generalized -Genocchi numbers which is related to the -Genocchi numbers and -Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define -Genocchi zeta and -functions, which are interpolated -Genocchi numbers and polynomials at negative integers. We also give some applications of generalized -Genocchi numbers.

In this paper, we consider the Euler-Bernoulli beam equation with memory and boundary output feedback control term. We prove the existence of solutions using the Galerkin method and then investigate the exponential stability of solutions by using multiplier technique.