Vector-Valued Entire Functions of Several Variables: Some Local Properties

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:Cn→R+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable.

Monitoring experts

We study the design of contracts that incentivize experts to collect information and truthfully report it to a decision maker. We depart from most of the previous literature by assuming that the transfers cannot depend on the realized state or on the ex post payoff of the decision maker. The contract thus has to induce the experts to “monitor each other” by making the transfers contingent on the entire vector of reports. We characterize the least costly contract that implements any given vector of efforts and derive the cost function for the decision maker. We then study properties of optimal contracts by comparing the value of information and its cost.

Entire functions of bounded index in frame

We introduce a concept of entire functions having bounded index in a variable direction, i.e. in a frame. An entire function $F\colon\ \mathbb{C}^n\to \mathbb{C}$ is called a function of bounded frame index in a frame $\mathbf{b}(z)$,if~there exists $m_{0} \in\mathbb{Z}_{+}$ such that for every $m \in\mathbb{Z}_{+}$ and for all $z\in \mathbb{C}^{n}$one has $\displaystyle\frac{|{\partial^{m}_{\mathbf{b}(z)}F(z)}|}{m!}\leq\max_{0\leq k \leq m_{0}} \frac{|{\partial^{k}_{\mathbf{b}(z)}F(z)}|}{k!},$where $\partial^{0}_{\mathbf{b}(z)}F(z)=F(z),$ $\partial^{1}_{\mathbf{b}(z)}F(z)=\sum_{j=1}^n \frac{\partial F}{\partial z_j}(z)\cdot b_j(z),$ \ $\partial^{k}_{\mathbf{b}(z)}F(z)=\partial_{\mathbf{b}(z)}(\partial^{k-1}_{\mathbf{b}(z)}F(z))$ for $k\ge 2$ and $\mathbf{b}\colon\ \mathbb{C}^n\to\mathbb{C}^n$ is a entire vector-valued function.There are investigated properties of these functions. We established analogs of propositions known for entire functions of bounded index in direction. The main idea of proof is usage the slice $\{z+t\mathbf{b}(z)\colon\ t\in\mathbb{C}\}$ for given $z\in\mathbb{C}^n.$We proved the following criterion (Theorem 1) describing local behavior of modulus $\partial_{\mathbf{b}(z)}^kF(z+t\mathbf{b}(z))$ on the circle $|t|=\eta$: {\it An entire~function$F\colon\ \mathbb{C}^n\to\mathbb{C}$ is of bounded frame index in the frame $\mathbf{b}(z)$ if and only iffor each $\eta>0$ there exist$n_{0}=n_{0}(\eta)\in \mathbb{Z}_{+}$ and $P_{1}=P_{1}(\eta)\geq 1$such that for every $z\in \mathbb{C}^{n}$ there exists $k_{0}=k_{0}(z)\in \mathbb{Z}_{+},$\$0\leq k_{0}\leq n_{0},$ for which inequality$$\max\left\{\left|{\partial_{\mathbf{b}(z)}^{k_{0}} F(z+t\mathbf{b}(z))}\right|\colon\ |t|\leq\eta \right\}\leqP_{1}\left|\partial_{\mathbf{b}(z)}^{k_{0}}{F(z)}\right|$$holds.

Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

We consider a class of vector-valued entire functions $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables. Let $|\cdot|_p$ be a norm in $\mathbb{C}^p$. Let $\mathbf{L}(z)=(l_{1}(z),\ldots,l_{n}(z))$, where $l_{j}(z)\colon \mathbb{C}^{n}\to \mathbb{R}_+$ is a positive continuous function.An entire vector-valued function $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$ is said to be ofbounded $\mathbf{L}$-index (in joint variables), if there exists $n_{0}\in \mathbb{Z}_{+}$ such that $\displaystyle \forall z\in G \ \ \forall J \in \mathbb{Z}^n_{+}\colon \quad\frac{|F^{(J)}(z)|_p}{J!\mathbf{L}^J(z)}\leq \max \left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^K(z)} \colon K\in \mathbb{Z}^n_{+}, \|K\|\leq n_{0} \right \}.$ We assume the function $\mathbf{L}\colon \mathbb{C}^n\to\mathbb{R}^p_+$ such that $0< \lambda _{1,j}(R)\leq\lambda _{2,j}(R)<\infty$ for any $j\in \{1,2,\ldots, p\}$ and $\forall R\in \mathbb{R}_{+}^{p},$where $\lambda _{1,j}(R)=\inf\limits_{z_{0}\in \mathbb{C}^{p}} \inf \left \{{l_{j}(z)}/{l_{j}(z_{0})}\colon z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \},$ $\lambda _{2,j}(R)$ is defined analogously with replacement $\inf$ by $\sup$.It is proved the following theorem:Let $|A|_p=\max\{|a_j|\colon 1\leq j\leq p\}$ for $A=(a_1,\ldots,a_p)\in\mathbb{C}^p$. An entire vector-valued function $F$ has bounded $\mathbf{L}$-index in joint variables if and only if for every $R\in \mathbb{R}^{n}_+$ there exist $n_{0}\in \mathbb{Z}_{+}$, $p_0>0$ such that for all $z_{0}\in \mathbb{C}^{n}$ there exists $K_{0}\in \mathbb{Z}_{+}^{n}$, $\|K_0\|\leq n_{0}$, satisfying inequality $\displaystyle\!\max\!\left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^{K}(z)} \colon \|K\|\leq n_{0},z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \}%\leq \nonumber\\\label{eq:5}\leq p_{0}\frac{|F^{(K_0)}(z_0)|_p}{K_0!\mathbf{L}^{K_0}(z_0)},$ where $\mathbb{D}^{n}[z_{0},R]=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^{n}\colon |z_1-z_{0,1}|<r_{1},\ldots, |z_n-z_{0,n}|<r_{n}\}$ is the polydisc with $z_0=(z_{0,1},\ldots,z_{0,n}),$\ $R=(r_{1},\ldots,r_{n})$. This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable.

Evidentials and their pivot in Tibetic and neighboring Himalayan languages

This paper focuses on a specific type of perspective-indexing constructions in Tibetic and neighboring languages, namely a type of verbal marker that is consistently construed from the perspective of the speaker in statements, the addressee in questions, and the source (= the original/reported speaker) in reported speech clauses. As these markers indicate how one obtained the information profiled in a sentence and may thus be viewed as a type of evidential, they cannot at the same time establish reference to any participant of the current speech act and thus by default reflect the perspective of the ‘informant’ of the respective sentence type. If we define the encountered distinctions in relation to a cause-effect vector in the sense of DeLancey (1986), these languages all contain what we may call an ‘insider’ marker indicating access to the entire vector including its causal origin and an ‘outsider’ marker indicating access only to its effect end. Whereas the insider markers typically occur when the informant is the subject and the outsider markers when s/he is not, the present paper discusses the different ways in which Tibetic and neighboring languages deviate from this basic pattern, and argues that these differences reflect the fact that the markers in the latter languages were only secondarily evidentialized in reported speech clauses, likely due to contact with Tibetic.

Locating Separatrices and Basins of Stability in Biodynamic Systems

In biomechanics, a separatrix or recovery envelope exists between standing and falling. Standing with postural sway is a distinctly different type of motion than falling. A comparable problem to standing postural sway is the challenge of maintaining torso stability. In this case, a separatrix exists delineating stable torso sway from unstable and potentially injurious motion. An approach is presented for identifying separatrices in state space generated from noisy time series data sets representative of those generated from experiments. We demonstrate how Lagrangian coherent structures (LCS), ridges in the state space distribution of finite-time Lyapunov exponents (FTLE), can be used to locate these separatrices. As opposed to previous approaches which required an entire vector field, this method can be performed using a single trajectory that evolves over time.