In this paper, the authors construct a univalent function having a relatively compact Siegel disk whose boundary is a Jordan curve of positive area. The construction is based on a general scheme in which Chéritat added Runge’s theorem, to construct a relatively compact Siegel disk and Osgood’s method for constructing a Jordan curve of positive area.

This paper constructs a finite Lie conformal superalgebra

associated to the N = 1 Bondi-Metzner-Sachs (BMS for short) superalgebra. The authors completely determine conformal derivations, the automorphism group, and the second cohomology with coefficients in trivial module. They also classify free conformal modules of rank (1 + 1) and finite irreducible conformal modules over

\mathfrak{R}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\frak R}$$\end{document}

.

Let (M, J, g) be an anti-Kähler manifold of dimension n = 2k with an almost complex structure J and a pseudo-Riemannian metric g and let T* M be its cotangent bundle with modified Riemannian extension metric

. The modified Riemannian extension metric

{\tilde{g}}_{\mathrm{\nabla },G}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{g}_{\nabla,G}$$\end{document}

is obtained by deformation in the horizontal part of the Riemannian extension known in the literature by means of the twin Norden metric G. The paper aims first to examine the curvature properties of the cotangent bundle T* M with modified Riemannian extension metric

{\tilde{g}}_{\mathrm{\nabla },G}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{g}_{\nabla,G}$$\end{document}

and second to study some geometric solitons on the cotangent bundle T* M according to the modified Riemannian extension metric

{\tilde{g}}_{\mathrm{\nabla },G}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{g}_{\nabla,G}$$\end{document}

.

In this paper, the authors consider meromorphic solutions of nonhomogeneous differential equation

where n is a positive integer, a is a nonzero constant, P_d(z, f) is a differential polynomial in f(z) of degree d with rational functions as its coefficients and d ≤ n − 1, u(z) is a nonzero rational function, v(z) is a nonconstant polynomial with v′(z) ≠ (n + 1)a, v′(z) ≠ −na and

v^{\mathrm{\prime }}(z)\ne -\frac{(n+1{)}^2}{n}a

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v^{\prime}(z)\ne-{(n+1)^{2}\over{n}}a$$\end{document}

. They prove that if it admits a meromorphic solution f(z) with finitely many poles, then where s(z) is a rational function and sⁿ[(n + 1)s′ + sv′] + (n + 1)asⁿ⁺¹ = (n + 1)u. Using this result, they also prove that if f(z) is a transcendental entire function, then fⁿ(f′ + af) + q_m(f) assumes every complex number α infinitely many times, except for a possible value q_m(0), where n, m are positive integers with n ≥ m + 1 and q_m(f) is a polynomial in f(z) with degree m.

In the present paper, by introducing a family of coupled forward-backward stochastic differential equations (FBSDEs for short), a probabilistic interpretation for a system consisting of m second order quasilinear (and possibly degenerate) parabolic partial differential equations and (m × d) algebraic equations is given in the sense of the classical solution. For solving the problem, an L^p-estimate (p > 2) for coupled FBSDEs on large time durations in the monotonicity framework is established, and a new method to analyze the regularity of solutions to FBSDEs is introduced. The new method avoids the use of Kolmogorov’s continuity theorem and only employs L²-estimates and L⁴-estimates to obtain the desired regularity.

In this paper, the authors investigate a delay differential equation of the form

where a(z) is a nonzero rational function, P(z, w) and Q(z, w) are prime polynomials in w with rational coefficients. They remove the restriction that the order of meromorphic solutions of the above difference equation is σ₂(w) < 1, and obtain the growth of transcendental meromorphic solutions. The exact forms of all transcendental entire solutions are obtained when deg_wP = deg_wQ = 0, or deg_wP = 1 and deg_wQ = 0, respectively. If deg_wP ≥ 2 and deg_wQ = 0, or deg_wQ ≥ 1 and Q(z, 0) ≢ 0, they prove that the above equation has no transcendental entire solution. They show that the existence of transcendental entire solutions of the above equation depends on the degrees of P(z, w) and Q(z, w).

This paper studies skew constacyclic codes over a family of finite rings denoted by B_k to obtain quantum codes over the fields \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{F}_{p^r}$$\end{document} and to construct Euclidean LCD skew constacyclic codes. The author investigates the structural properties of skew constacyclic codes over B_k using a decomposition approach, and also finds necessary and sufficient conditions for skew constacyclic codes that contain their duals. Finally, the author gives some examples of quantum codes obtained via the construction and LCD codes.

In this paper, the authors show that the symplectic mean curvature flow in ℂℙ² with normal curvature pinched exists for a long time and converges to a holomorphic curve.