In this paper we give an explicit method for constructing every rational elliptic surface that has a four-torsion section. We use this construction to study the moduli space of birational equivalence classes of Galois covers that arise from rational elliptic surfaces with a relative dihedral group action. As a conclusion we find that the moduli space for such Galois covers whose Galois groups are the dihedral group of order eight is a nodal rational curve.

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The problem of determining the least degree of plane curves vanishing at given points with certain multiplicities is fascinating and difficult. In 1959,Nagata gave a conjecture on the lower bound of the degree. Since then, many mathematicians have focused on solving the conjecture for several special cases. In this paper, we show several results on determining the least degree for certain cases of equal multiplicities.

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Multi-poly-Bernoulli numbers, defined by using multiple polylogarithms, are generalizations of Kaneko's poly-Bernoulli numbers. We consider multi-poly-Bernoulli numbers of negative index in particular, and give the generating function of them. As a main result, we give a formula which expresses multi-poly-Bernoullinumbers of negative index as a sum of powers.

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In this paper, we prove that the Sasano systems of types B₃⁽¹⁾ and D₄⁽²⁾ have no meromorphic solution at t =∞, which implies the non-existence of rational solutions of the systems. Sasano systems of types B₃⁽¹⁾ and D₄⁽²⁾ are the systems of ordinary differential equations whose Bäcklund transformation groups are isomorphic to the affine Weyl groups of types B₃⁽¹⁾ and D₄⁽²⁾.

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It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a result of Paulsen and Smith who proved the same for Banach algebras acting on Hilbert spaces.

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In this paper we obtain a necessary and sufficient condition for the four isotropy irreducible representations in SU (3)/T (k, l), (k, l) ∈ Z² (|k|+|l| ≠ 0), to be mutually inequivalent, and then completely estimate the Ricci curvature and the scalar curvature on SU (3)/T (k, l) with an arbitrarily given SU (3)-invariant Riemannian metric under the above inequivalent condition.

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Using the probability inequalities and the weak invariance principles, the limit behavior of the complete moment convergence for negatively associated (NA) random variables is investigated. Namely, it is shown how fast the series generated by the moments of the sums for NA sequences tends to infinity as ε goes to zero.

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We study Kubo-Martin-Schwinger(KMS) states on finite-graph C^∗-algebras with sinks and sources. We compare finite-graph C^∗-algebras with C^∗-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C^∗-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

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The Catalan constant, G=Σ^∞_n=1[(−1)^k/(2n−1)²], has attracted much attention and has appeared in various contexts. It is the special value at s=2 of the Dirichlet L-function associated with the Gaussian field with its character odd. In this paper, we appeal to the Barnes G-function and a form of the functional equation for the Riemann zeta-function,to deduce an expression for the integral in t/sin t and one for G. These give rise to a corollary which reveals the intrinsicity of the Catalan constant G to the Barnes G-function. We also use one of Ramanujan's formulas to derive a theorem.

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Acyclic categories were introduced by Kozlov and can be viewed as generalized posets. Similar to posets, one can define their incidence algebras and a related topological complex. We consider the incidence algebra of either a poset or acyclic category as the quotient of a path algebra by the parallel ideal. We show that this ideal has a quadratic Gröbner basis with a lexicographic monomial order if and only if the poset or acyclic category is lexshellable.

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In this paper, we study blow ups of solutions to the second sound equation ∂_t²u=u∂_x(u∂_xu), which is more natural than the second sound equation in Landau-Lifshitz's text in large time. We assume that the initial data satisfies u(0,x)≥ δ>0 for some δ. We give sufficient conditions that two types of blow up occur: one of the two types is that L^∞-norm of ∂_tu or ∂_xu goes up to the infinity; the other type is that u vanishes, that is, the equation degenerates.

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We define the poly-Cauchy numbers by generalizing the Cauchy numbers of the first kind (the Bernoulli numbers of the second kind when divided by a factorial). We study their characteristic properties which are analogous to those of the poly-Bernoulli numbers introduced by Kaneko, and generalize those of the classical Cauchy numbers of the first kind as well as of the second kind.

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Let p be an odd prime number, and a prime number with ≠ p. Let h_n⁻ be the relative class number of the pⁿ⁺¹th cyclotomic field Q(ζ_pⁿ⁺¹). When p = 3 or 5, it is known that h_n⁻ for any n ≥ 0 if ² 1 mod 9 (respectively ⁸ 1 mod 100). We show some corresponding results for the case p = 7.

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Starting with a representation φ of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in V_E:=E⊕V. By the adjunction of a unit element to V_E, we obtain a clan V_E⁰ with unit element. By computing the determinant of the right multiplication operators of V_E⁰, we get an explicit expression of the basic relative invariants of V_E⁰ in terms of the Jordan algebra principal minors of V and the quadratic map associated with φ. For the dual clan of V_E⁰, we also obtain an explicit expression of the basic relative invariants in a parallel way.

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The aim of this paper is to prove a well-known conjecture on the regularity index of n+3 almost equimultiple fat points in Pⁿ.

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Based on the analogy between knots and primes, Hillman, Matei and Morishita defined the Iwasawa invariants for sequences of cyclic covers of links with an analog of Iwasawa's class number formula of number fields. In this paper, we consider the existence of covers of links with prescribed Iwasawa invariants, discussing analogies in number theory. We also propose and consider a problem analogous to Greenberg's conjecture.

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In this paper we show the existence of a non-trivial solution to a fractional Schrödinger equation, which turns out to be a global minimizer of a variational problem. We give a sharp condition for the existence of global minimizers. To do this we develop a new compactification scheme of angularly regular solutions.

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In this paper we investigate irregular Gabor frames. It is shown that for any Schwartz function the family of its time-frequency shifts constitutes a weighted Gabor frame if only the lattice of sampling time-frequency values is dense enough. The proof uses classical analysis tools only and is based on the localization of the kernel of the Gabor transform. Further, the method allows us to find the density bound explicitly. A numerical example is presented.

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