In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form $F = \sqrt {rf\left( {s - t} \right)} $, where $r = {\left\| v \right\|^2}$, $s = \frac{{{{\left| {\left\langle {z,v} \right\rangle } \right|}^2}}}{r}$, $t = {\left\| z \right\|^2}$, f(w) is a real-valued smooth positive function of w ∈ R, and z is in a unitary invariant domain M ⊂ C ⁿ. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f′, we prove that all the real geodesics of $F = \sqrt {rf\left( {s - t} \right)} $ on every Euclidean sphere S²ⁿ⁻¹ ⊂ M are great circles.

The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D ₄ with symmetry group 〈J〉 and for D ₄ ^T with symmetry group G _max, respectively, are both tau-functions of the D ₄ Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in the article by Fan-Jarvis-Ruan (2013), of the Witten Integrable Hierarchies Conjecture for all simple (ADE) singularities.

In this paper, the authors point out that the methods used by Li (2004, 2005, 2007) can be applied to study maximal functions on weighted harmonic AN groups.

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gröbner-Shirshov basis for degenerate Ringel-Hall algebras of type F ₄.

In this work, the authors introduce the concept of (p, q)-quasi-contraction mapping in a cone metric space. We prove the existence and uniqueness of a fixed point for a (p, q)-quasi-contraction mapping in a complete cone metric space. The results of this paper generalize and unify further fixed point theorems for quasi-contraction, convex contraction mappings and two-sided convex contraction of order 2.

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L _a ²(B _d) satisfies ${P_M} - \sum\limits_i {{M_{{z^i}}}} {P_M}M_{{z^i}}^* \leqslant \frac{c}{{N + 1}}{P_M}$ for some number c > 0, then there is a sequence {f _j} of multipliers and a positive number c′ such that $c'{P_M} \leqslant \sum\limits_j {{M_{{f_j}}}} M_{{f_j}}^* \leqslant {P_M}$, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebra $\dot U$ and its canonical basis $\dot B$ given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set $\tilde M$ which depends only on the root category R and prove that there is a bijection between $\tilde M$ and $\dot B$, where R is the T ²-orbit category of the bounded derived category of the corresponding Dynkin or tame quiver. The method in this paper is based on a result of Lin, Xiao and Zhang in 2011, which gives a PBW-type basis of U⁺.

Let F _q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive element $\alpha \in {F_{{q^n}}}$ such that α + α ⁻¹ is also primitive or α + α ⁻¹ is primitive and α is a normal element of ${F_{{q^n}}}$ over F _q.

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n (n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [n+1/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S _α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.

This work is devoted to studying a quasilinear elliptic boundary value problem with superlinear nonlinearities in a weighted Sobolev space in a domain of R ^N. Based on the Galerkin method, Brouwer’s theorem and the weighted compact Sobolev-type embedding theorem, a new result about the existence of solutions is revealed to the problem.