The Growth and Distortion Theorems for Slice Monogenic Functions

The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\mathbb D\subset \mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.Comment: 24 pages; Accepted by Pacific Journal of Mathematics for publicatio

Extremal functions of boundary Schwarz lemma

In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Carath\'eodory theorem for univalent holomorphic self-mappings of the open unit disk $\mathbb D\subset \mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma

Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains

In this paper, we generalize a recent work of Liu et al. from the open unit ball $\mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Carath\'eodory metric near the boundary of $C^2$ domains and the well-known Graham's estimate on the boundary behavior of the Carath\'eodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.Comment: Accepted by CAOT for publicatio

Julia theory for slice regular functions

Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quaternions. Together with some explicit examples, it shows that the slice derivative of a slice regular self-mapping of $\mathbb B$ at a boundary fixed point is not necessarily a positive real number, in contrast to that in the complex case, meaning that its commonly believed version turns out to be totally wrong.Comment: To appear in Transactions of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1412.420

DG Poisson algebra and its universal enveloping algebra

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.Comment: Accepted by Science China Mathematic

Universal enveloping algebras of Poisson Ore extensions

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra. As consequences, we observe certain ring-theoretic invariants of the universal enveloping algebras that are preserved under iterated Poisson-Ore extensions. We apply our results to iterated quadratic Poisson algebras arising from semiclassical limits of quantized coordinate rings and a family of graded Poisson algebras of Poisson structures of rank at most two.Comment: 13 page

Universal enveloping algebras of differential graded Poisson algebras

In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra $A$, we construct two isomorphic differential graded algebras: $A^e$ and $A^E$. It is proved that the category of differential graded Poisson modules over $A$ is isomorphic to the category of differential graded modules over $A^e$, and $A^e$ is the unique universal enveloping algebra of $A$ up to isomorphisms. As applications of the universal property of $A^e$, we prove that $(A^e)^{op}\cong (A^{op})^e$ and $(A\otimes_{\Bbbk}B)^e\cong A^e\otimes_{\Bbbk}B^e$ as differential graded algebras. As consequences, we obtain that ``$e$'' is a monoidal functor and establish links among the universal enveloping algebras of differential graded Poisson algebras, differential graded Lie algebras and associative algebras.Comment: 37 pages, the abstract is rewritten, another construction of the universal enveloping algebra is given and several typos are fixe

Robust time-varying formation design for multi-agent systems with disturbances: Extended-state-observer method

Robust time-varying formation design problems for second-order multi-agent systems subjected to external disturbances are investigated. Firstly, by constructing an extended state observer, the disturbance compensation is estimated, which is a critical term in the proposed robust time-varying formation control protocol. Then, an explicit expression of the formation center function is determined and impacts of disturbance compensations on the formation center function are presented. With the formation feasibility conditions, robust time-varying formation design criteria are derived to determine the gain matrix of the formation control protocol by utilizing the algebraic Riccati equation technique. Furthermore, the tracking performance and the robustness property of multi-agent systems are analyzed. Finally, the numerical simulation is provided to illustrate the effectiveness of theoretical results.Comment: 14 pages, 5 figure

Boundary Julia theory for slice regular functions

The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B\subset \mathbb H$ and of the right half-space $\mathbb H_+$. Especially, we find a new phenomenon that the classical Hopf lemma about $f'(\xi)>1$ at the boundary point may fail in general in quaternions, and its quaternionic variant should involve the Lie bracket reflecting the non-commutative feature of quaternions.Comment: This paper has been rewritten and retitled as "Julia theory for slice regular functions"(see arXiv:1502.02368

Pulling back error to the hidden-node parameter technology: Single-hidden-layer feedforward network without output weight

According to conventional neural network theories, the feature of single-hidden-layer feedforward neural networks(SLFNs) resorts to parameters of the weighted connections and hidden nodes. SLFNs are universal approximators when at least the parameters of the networks including hidden-node parameter and output weight are exist. Unlike above neural network theories, this paper indicates that in order to let SLFNs work as universal approximators, one may simply calculate the hidden node parameter only and the output weight is not needed at all. In other words, this proposed neural network architecture can be considered as a standard SLFNs with fixing output weight equal to an unit vector. Further more, this paper presents experiments which show that the proposed learning method tends to extremely reduce network output error to a very small number with only 1 hidden node. Simulation results demonstrate that the proposed method can provide several to thousands of times faster than other learning algorithm including BP, SVM/SVR and other ELM methods.Comment: 7 page