In this paper we study non-Galois, totally and tamely ramified fields with prime power discriminants and present a method for the constructions of such fields. We also study the Galois groups of the Galois closure extensions of such fields.

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For any McKay-Thompson series which appear in Moonshine, the Hecke-type Faber polynomial P_n(X) of degree n is defined. The Hecke-type Faber polynomials are of course special cases of the Faber polynomials introduced by Faber a century ago. We first study the locations of the zeros of the Hecke-type Faber polynomials of the 171 monstrous types, as well as those of the 157 non-monstrous types. We have calculated, using a computer, the zeros for all n ≤ 50. These results suggest that in many (about 13%) of the cases, we can expect that all of the zeros of P_n(x) are real numbers. In particular, we prove rigorously that the zeros of the Hecke-type Faber polynomials (of any degree) for the McKay-Thompson series of type 2A are real numbers. We also discuss the effect of the existence of harmonics, and the effect of a so-called dash operator. We remark that by the dash operators, we obtain many replicable functions (with rational integer cofficients) which are not necessarily completely replicable functions. Finally, we study more closely the curves on which the zeros of the Hecke-type Faber polynomials for type 5B lie in particular in connection with the fundamental domain (on the upper half plane) of the group Γ₀(5), which was studied by Shigezumi and Tsutsumi. At the end, we conclude this paper by stating several observations and speculations.

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A probabilistic construction of the heat semigroup and kernel associated with certain non-commutative harmonic oscillators is given. As an application, the unitarily equivalence of the non-commutative harmonic oscillators with the ordinary harmonic oscillators is shown.

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We consider a CR mapping between real generic manifolds in the complex space and the induced mapping between the conormal bundles. With a pair of projections we can make the fibers be totally real and the induced mapping be CR. In particular, we obtain the explanation of a result by Chirka and Rea about constancy of normal rank.

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In this paper, we give a classification of real hypersurfaces in a non-flat complex space form such that the (pseudo-)holomorphic sectional curvatures with respect to the generalized Tanaka-Webster connection are constant.

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We discuss the spectral gaps of the generalized Kronig-Penney Hamiltonians which possess two point interactions in the basic cell [0, 2π). We determine whether or not the j th spectral gap of the Hamiltonian is absent for a given j ∈ N.

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We say that a C^*-algebra A is approximately square root closed if any normal element in A can be approximated by a square of a normal element in A. We study when A is approximately square root closed, and have an affirmative answer for AI-algebras, Goodearltype algebras over the torus, purely infinite simple unital C^*-algebras, etc.

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We investigate values of the multiple sine function of period (1,...,1) at rational numbers. We show explicit formulas using special values of Dirichlet L-functions. We also study the case of the multiple gamma function.

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Adams gave the notion of a Hopf algebroid generalizing the notion of a Hopf algebra and showed that certain generalized homology theories take values in the category of comodules over the Hopf algebroid associated with each homology theory. A Hopf algebra represents an affine group scheme which is a group in the category of a scheme and the notion of comodules over a Hopf algebra is equivalent to the notion of representations of the affine group scheme represented by a Hopf algebra. On the other hand, a Hopf algebroid represents a groupoid in the category of schemes. Therefore, it is natural to consider the notion of comodules over a Hopf algebroid as representations of the groupoid represented by a Hopf algebroid. This motivates the study of representations of groupoids, and more generally categories, for topologists. The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category, using the notion of a fibered category.

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Deformations of the multiple gamma and sine functions with respect to their periods are studied. To describe such deformations explicitly, a new class of generalized gamma and sine functions are introduced. In particular, we study the deformations from the viewpoint of multiplication formulas and Raabe's integral formulas for these gamma and sine functions. This new class of gamma functions contains Milnor's type multiple gamma functions as a special case.

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An array code/linear array code is a subset/subspace, respectively, of the linear space Mat_m×s(Fq), the space of all m × s matrices with entries froma finite field Fq endowed with a non-Hamming metric known as the RT-metric or $\\ ho$-metric or m-metric. In this paper, we obtain a sufficient lower bound on the number of parity check digits required to achieve minimum $\\ ho$-distance at least d in linear array codes using an algorithmic approach. The bound has been justified by an example. Using this bound, we also obtain a lower bound on the numberBq(m × s, d) where Bq(m × s, d) is the largest number of code matrices possiblein a linear array code V ⊆ Mat _{m × s} (Fq) having minimum $\\ ho$-distance at least d.

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We consider an effective lower bound of the Siegel-Tatuzawa type for general L-functions with three standard assumptions. We further assume three hypotheses in this paper that are essential in developing our argument.Under these assumptions and hypotheses, we prove a theorem of Siegel-Tatuzawa type for general L-functions. In particular, we prove such a theorem for symmetric power L-functions under certain assumptions.

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In general, the properties of a given connection and the conjugate connection in a vector bundle E over a closed Riemannian manifold (M, g) are different. It is interesting to find whether or not the conjugate connection of a Yang-Mills connection in a vector bundle E over (M, g) is a Yang-Mills connection. In this paper, we get the fact that a connection D in a vector bundle E over (M, g) is a Yang-Mills connection if and only if the conjugate connection is a Yang-Mills connection.

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We explain the construction of a Hilbert space on quadrics arising by the method of pairing polarizations. Then we introduce a family of measures and operators from function spaces on these quadrics to L₂(Sⁿ) which are defined by fiber integration. We compare the quantization operators and characterize them in the framework of pseudo-differential operator theory. An asymptotic property of the reproducing kernel of the Hilbert spaces consisting of holomorphic functions defined on quadrics is proved. This is a generalization of the Segal-Bargmann space and its reproducing kernel. Next we treat the case of the complex projective space and we explain that the space corresponding to the quadric is a matrix space consisting of rank-one complex matrices whose square is zero. Most of the theorems can be stated in the same way parallel to the sphere case.

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Let M² be an oriented 2-manifold and f : M²→R³ a C^∞-map. A point p ∈M² is called a singular point if f is not an immersion at p. The map f is called a front (or wave front), if there exists a unit C^∞-vector field ν such that the image of each tangent vector df (X) (X ∈ TM²) is perpendicular to ν, and the pair (f, ν) gives an immersion into R³ × S². In a previous paper, we gave an intrinsic formulation of wave fronts in R³. In this paper, we investigate the behavior of cuspidal edges near corank-one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation.

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We consider a Markov chain on a finite state space and obtain an expression of the joint distribution of the cover time and the last point visited by the Markov chain. As a corollary, we obtain the spectral representation of the distribution of the cover time.

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