There exists no established theory for high temperature superconductivity, since Müller and Bednorz's discovery in 1986, although some partial theories have been proposed. We will give a consistent mathematical theory of high temperature superconductivity based on a theory of stochastic processes of Schrödinger and Nagasawa.

In this paper, we study the existence and global exponential stability of positive almost periodic solutions for the generalized model of hematopoiesis with multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to ensure that all solutions of this model converge exponentially to the positive almost periodic solution.

We study the non-topological radial solutions of the Abelian Chern-Simons equation with two Higgs particles. We establish the non-degeneracy property of linearized equation and the uniqueness property for the corresponding non-topological radial solutions.

In this paper, we deduce some results which are generalizations of Petrenko, Fuchs, Niino, Bergweiler, Eremenko and Marchenko's work. We will estimate the deviation b(a, G) of a holomorphic curve G(z) with respect to a small holomorphic curve a(z) and give an estimation of L(r, a, G).

In this paper we consider the 'natural' random walk on a planar graph and scale it by a small positive number δ. Given a simply connected domain D and its two boundary points a and b, we start the scaled walk at a vertex of the graph nearby a and condition it on its exiting D through a vertex nearby b, and prove that the loop erasure of the conditioned walk converges, as δ ↓ 0, to the chordal SLE₂ that connects a and b in D, provided that an invariance principle is valid for both the random walk and the dual walk of it. Our result is an extension of one due to Dapeng Zhan [12] where the problem is considered on the square lattice. A convergence to the radial SLE₂ has been obtained by Lawler, Schramm and Werner [3] for the square and triangular lattices and by Yadin and Yehudayoff [10] for a wide class of planar graphs. Our proof, though an adaptation of that of [3] and [10], involves some new ingredients that arise from two sources: one for dealing with a martingale observable that is different from that used in [3] and [10] and the other for estimating the harmonic measures of the random walk started at a boundary point of a domain.

Let S and T be reduced divisors on P² which have no common components, and Δ = S + 2T. We assume deg Δ = 6. Let π : X → P² be a normal triple cover with branch divisor Δ, i.e. π is ramified along S (resp. T) with the index 2 (resp. 3). In this note, we show that X is either a P¹-bundle over an elliptic curve or a normal cubic surface in P³. Consequently, we give a necessary and sufficient condition for Δ to be the branch divisor of a normal triple cover over P².

We state and prove several characterizations of Thom's regularity condition for stratified maps. In particular we extend to stratified maps some characterizations of Whitney (a) regularity, due to the second author.

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.

We introduce a discrete Laplacian A on the complete graph with N vertices, that is, K_N. We obtain the best constants of three kinds of discrete Sobolev inequalities on K_N. The background of the first inequality is the discrete heat operator (d/dt + A + a₀I) ··· (d/dt + A + a_M−1I) with positive distinct characteristic roots a₀, ..., a_M−1. The second one is the difference operator (A + a₀I) ··· (A + a_M−1I) and the third one is the discrete polyharmonic operator A^M. Here A is an N × N real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is 1 = ^t(1, 1, ..., 1). A discrete heat kernel, a Green's matrix and a pseudo Green's matrix are obtained by means of A.

Let K be a field and I a monomial ideal of the polynomial ring S = K[x₁, ..., x_n]. We show that if either: 1) I is almost complete intersection, 2) I can be generated by less than four monomials; or 3) I is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on [n], then Stanley's conjecture holds for S/I.

We construct a bilinear form associated to a sub-conformal structure on a manifold M. In the case the sub-conformal structure corresponds to a partially-integrable CR structure we obtain a conformal Lorentz structure which coincides with Fefferman's construction on a circle bundle over M. The main contribution is the use of invariant forms with values in a vector space instead of the full information contained in the Cartan connection in order to simplify the construction.

We give a shorter proof of the following theorem of Kathryn Mann [M]: the identity component of the group of the compactly supported C^r diffeomorphisms of Rⁿ cannot admit a nontrivial C^p-action on S¹, provided n ≥ 2, r ≠ n + 1 and p ≥ 2. We also give a new proof of another theorem of Mann [M]: any nontrivial homomorphism from the group of the orientation preserving C^r diffeomorphisms of the circle to the group of C^p diffeomorphisms of the circle is the conjugation of the standard inclusion by a C^p diffeomorphism, if r ≥ p, r ≠ 2 and p ≠ 1.

In this article, we study the local dynamical structure of a rational mapping F of P² at a fixed indeterminate point p. In the previous paper, using a sequence of points which is defined by blow-ups, we have constructed an invariant family of holomorphic curves at p. In this paper, using the same sequence of points, we approximate a set of points whose forward orbits stay in a neighborhood of p. Moreover, for a specific rational mapping we construct a family {W_j}_{j$\in$ {1,2}^N} of center manifolds of p. The main result of this paper is to give the asymptotic expansion of the defining function of W_j.

Planar elastica problem is a classical but has broad connections with various fields, such as elliptic functions, differential geometry, soliton theory, material mechanics, etc. This paper regards classical elastica as a theory corresponding to Lebesgue L² case, and extends it to L^p cases. For the sake of the effect of p-Laplacian, novel curious solutions appear especially for cases p > 2. These solutions never appear in 1 < p ≤ 2 cases and we call them flat-core solutions according to Takeuchi [6, 7].

We correct a result in "Extremal disks and extremal surfaces of genus three", Kodai Math. J. 28, no. 1 (2005), 111-130. In the paper we have shown that there exist 16 compact Riemann surfaces of genus three up to conformal equivalence in which two extremal disks are isometrically embedded. However we have three more of them up to conformal equivalence. In the present paper we give these three surfaces and show that they are hyperelliptic. We also determine the groups of automorphisms of them.

The aim of this article is to establish some conditions under which the attractors of iterated function systems become dendrites. We associate to an attractor of an iterated function system (IFS) some graphs and we prove that for a large class of IFSs their attractors are dendrites if the associated graphs are trees. We also give some examples of such sets.

This paper is concerned with a 2nth-order nonlinear difference equation. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Dirichlet boundary value problem and give some new results. The existing results are generalized and significantly complemented.

We settle a question posed by Umehara and Yamada, which generalizes a completeness lemma useful in differential geometry.