Abstract This paper is concerned with stochastic H ₂/H _∞ control problem for Poisson jump-diffusion systems with (x, u, v)-dependent noise, which are driven by Brownian motion and Poisson random jumps. A stochastic bounded real lemma (SBRL for short) for Poisson jump-diffusion systems is firstly established, which stands out on its own as a very interesting theoretical problem. Further, sufficient and necessary conditions for the existence of a state feedback H ₂/H _∞ control are given based on four coupled matrix Riccati equations. Finally, a discrete approximation algorithm and an example are presented.

The authors study, by applying and extending the methods developed by Cazenave (2003), Dias and Figueira (2014), Dias et al. (2014), Glassey (1994–1997), Kato (1987), Ohta and Todorova (2009) and Tsutsumi (1984), the Cauchy problem for a damped coupled system of nonlinear Schrödinger equations and they obtain new results on the local and global existence of H ¹-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.

This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using Čech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The Roper-Suffridge operator on the unit ball B ⁿ in C ⁿ is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ _G,γ (f)](z) preserves the properties of S_Ω ^*(A,B), as well as strong and almost spirallikeness of type β and order α on B ⁿ. Thus, the mappings in S_Ω ^*(A,B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.

This paper deals with some parabolic Monge-Ampère equation raised from mathematical finance: V _s V _yy+ry V _y V _yy−θV _y ²= 0 (V _yy < 0). The existence and uniqueness of smooth solution to its initial-boundary value problem with some requirement is obtained.

Let F be a finitely generated free group. Martino and Ventura gave an explicit description for the fixed subgroups of automorphisms of F. The author generalizes their results to injective endomorphisms.

Applying Nevanlinna theory of the value distribution of meromorphic functions, the author studies some properties of Nevanlinna counting function and proximity function of meromorphic solutions to a type of systems of complex differential-difference equations. Specifically speaking, the estimates about counting function and proximity function of meromorphic solutions to systems of complex differential-difference equations can be given.

Metric n-Lie algebras have wide applications in mathematics and mathematical physics. In this paper, the authors introduce two methods to construct metric (n+1)-Lie algebras from metric n-Lie algebras for n ≥ 2. For a given m-dimensional metric n-Lie algebra (g, [, · · ·, ], B _g), via one and two dimensional extensions L = g +Fc and g₀ = g + Fx ⁻¹ +Fx ⁰ of the vector space g and a certain linear function f on g, we construct (m+1)- and (m+2)-dimensional (n+1)-Lie algebras (L, [, · · ·, ] _cf) and (g₀, [, · · ·, ]1), respectively. Furthermore, if the center Z(g) is non-isotropic, then we obtain metric (n+1)-Lie algebras (L, [, · · ·, ] _cf, B) and (g₀, [, · · ·, ]₁, B) which satisfy B|_g×g = B _g. Following this approach the extensions of all (n + 2)-dimensional metric n-Lie algebras are discussed.

This paper is concerned with a kind of mean value problem of Kloosterman sums, which will lead to a sum of Kloosterman sums over short intervals.

Let {X, X _k: k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums ${S_n} = \sum\limits_{i = 1}^n {{X_i}} $, in a unified form in which x may be a random variable of an arbitrary type, which state that under some suitable conditions, for some constants T > 0, a and τ > 1/2 and for every fixed γ > 0, the relation $P\left( {{S_n} - na \in \left( {x,\;x + T]} \right)} \right)\~nF\left( {\left( {x + a,\;x + a + T} \right]} \right)$ holds uniformly for all x ≥ γn ^τ as n→∞, that is, $\mathop {\lim }\limits_{n \to + \infty } \mathop {\sup }\limits_{x \geqslant \gamma {n^\tau }} \left| {\frac{{P\left( {{S_n} - na \in \left( {x,\;x + T} \right]} \right)}}{{nF\left( {\left( {x + a,\;x + a + T} \right]} \right)}} - 1} \right| = 0$. The authors also discuss the case where X has an infinite mean.

The author discusses 2-adjacency of two-component links and study the relations between the signs of the crossings to realize 2-adjacency and the coefficients of the Conway polynomial of two related links. By discussing the coefficient of the lowest m power in the Homfly polynomial, the author obtains some results and conditions on whether the trivial link is 2-adjacent to a nontrivial link, whether there are two links 2-adjacent to each other, etc. Finally, this paper shows that the Whitehead link is not 2-adjacent to the trivial link, and gives some examples to explain that for any given two-component link, there are infinitely many links 2-adjacent to it. In particular, there are infinitely many links 2-adjacent to it with the same Conway polynomial.

In the dual risk model, the surplus process of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the authors assume that the surplus process is the sum of a compound Poisson process and an independent Wiener process. The dual of the jump-diffusion risk model under a threshold dividend strategy is discussed. The authors derive a set of two integro-differential equations satisfied by the expected total discounted dividend until ruin. The cases where profits follow an exponential or mixtures of exponential distributions are solved. Applying the key method of the Laplace transform, the authors show how the integro-differential equations are solved. The authors also discuss the conditions for optimality and show how an optimal dividend threshold can be calculated as well.

Let f be a holomorphic Hecke eigenform of weight k for the modular group Γ = SL ₂(Z) and let λ _f (n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate $\sum\limits_{n \leqslant x} {{\lambda _f}\left( {{n^2}} \right)} \ll {x^{\frac{1}{2}}}{k^{\frac{1}{2}}}{\left( {\log \left( {x + k} \right)} \right)^6}$ is established, which improves the previous result.