We give integral representations of Euler type for Appell's hypergeometric functions F2, F3, Horn's hypergeometric function H2 and Olsson's hypergeometric function FP. Their integrands are the same (up to a constant factor), and only the regions of integration vary.
Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell's F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell's F2, Horn's H2 and Olsson's FP functions, which are obtained by Olsson.
In this paper, we study the mod(p) motivic cohomology of twisted complete flag varieties X over some restricted fields k. Here we take fields k such that the Milnor K-theory KMn+2 (k) / p = 0 for some n ≥ 2. For these fields, we compute the mod(p) motivic cohomologies of the Rost motives Rn and the flag variety X containing R2.
We study Pfaffian systems of confluent hypergeometric functions of two variables with rank three, by using rational twisted cohomology groups associated with Euler-type integral representations of them. We give bases of the cohomology groups, whose intersection matrices depend only on parameters. Each connection matrix of our Pfaffian systems admits a decomposition into five parts, each of which is the product of a constant matrix and a rational 1-form on the space of variables.
In this paper, we give an example of a closed unbounded operator whose square domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.
We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.
Let G = exp(g)be an exponential solvable Lie group and Ad(G) ⊂ D an exponential solvable Lie group of automorphisms of G. Assume that for every non-∗-regular orbit D · q, q ∈ g∗, of D = exp(∂) in g∗, there exists a nilpotent ideal n of g containing ∂ · g such that D · qǀn is closed in n∗. We then show that for every D-orbit Ω in g∗ the kernel kerC∗(Ω) of Ωin the C∗-algebra of G is L1-determined, which means that kerC∗(Ω) is the closure of the kernel kerL1(Ω) of Ω in the group algebra L1(G). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group D = Ad(G). We finally give an example of a non-closed non-∗-regular orbit of an exponential solvable group G and of a coadjoint orbit O ⊂ g∗, for which the corresponding kernel kerC∗(πO) in C∗(G) is not L1-determined.
We introduce a new class of domains Dn,m(μ, p), called FBH-type domains, in ℂn × ℂm, where 0 < μ ∈ ℝ and p ∈ ℕ. In the special case of p = 1, these domains are just the Fock-Bargmann-Hartogs domains Dn,m(μ) in ℂn × ℂm introduced by Yamamori. In this paper we obtain a complete description of an arbitrarily given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group Aut(Dn,m(μ, p)) of any FBH-type domain Dn,m(μ, p) with p ≠ 1 is a Lie group isomorphic to the compact connected Lie group U(n) × U(m). This tells us that the structure of Aut(Dn,m(μ, p)) with p ≠ 1 is essentially different from that of Aut(Dn,m(μ)).
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the non-commutative polynomial algebra in two letters. These give a class of relations among multiple zeta values,which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is, the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
We show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field. Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature. To this end, we have to differentiate the thermodynamic potential with respect to the temperature twice. Since there is a solution to the BCS-Bogoliubov gap equation in the form of the thermodynamic potential, we have to differentiate the solution with respect to the temperature twice. Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature twice, as well as its existence and uniqueness. We carry out its proof on the basis of fixed point theorems.
We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be three. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be two and three.
The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram,which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara-Selberg zeta function of a directed weighted graph.