On the Liouville function at polynomial arguments

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

The use of effective distance functions has been explored for many data mining problems including clustering, nearest neighbor search, and indexing. Recent research results show that if the Pearson variation of the distance distribution converges to zero with increasing dimensionality, the distance function will become unstable (or meaningless) in high dimensional space even with the commonly used Lp metric on the Euclidean space. This result has spawned many subsequent studies. We first comment that although the prior work provided the sufficient condition for the unstability of a distance function, the corresponding proof has some defects. Also, the necessary condition for unstability (i.e., the negation of the sufficient condition for the stability) of a distance function, which is required for function design, remains unknown. Consequently, we first provide in this paper a general proof for the sufficient condition of unstability. More importantly, we go further to prove that the rapid degradation of Pearson variation for a distance distribution is in fact a necessary condition of the resulting unstability. With the result, we will then have the necessary and the sufficient conditions for unstability, which in turn imply the sufficient and necessary conditions for stability. This theoretical result derived leads to a powerful means to design a meaningful distance function. Explicitly, in light of our results, we design in this paper a meaningful distance function, called Shrinkage-Divergence Proximity (abbreviated as SDP), based on a given distance function. It is empirically shown that the SDP significantly outperforms prior measures for its being stable in high dimensional data space and robust to noise, and is thus deemed more suitable for distance-based clustering applications than the priorly

holds, and we shall deal mainly with such series in this paper. Part II is devoted to a study of the convergence properties of these and related series. In particular, it is shown that the of points for which a regular hyperdirichlet series is convergent and absolutely convergent are interiors of half-planes (which may be the whole plane or the empty set) plus certain portions of their boundaries and possibly additional depending primarily upon the roots of the Pn(z) and-of their partial sums. These exceptional sets are shown to have outer logarithmic capacity zero, to be of first category with respect to the plane and certain subsets of the plane, and to have Hausdorff measure zero with respect to any measure function h(t) for which the integral

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.

We show that on any translation surface, if a regular point is contained in some simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in $${\mathbb {RP}}^1$$. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist on some translation surfaces. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa’s classifications of periodic points and of $$\text {GL}(2,{\mathbb {R}})$$ orbit closures in hyperelliptic components, as well as a recent result of Eskin–Filip–Wright.

On the equivalence of the Choquet, pan- and concave integrals on finite spaces

A long term goal in the theory of systems and control is to develop a systematic methodology for the design of feedback control schemes capable of shaping the response of complex dynamical systems, in both an equilibrium and a nonequilibrium setting. In this talk, we will focus primarily on periodic steady-state behavior, a phenomenon that is pervasive in nature and in man-made systems. We will begin with an analysis of the asymptotically stable oscillation in the classical voltage controlled oscillator (VCO), followed by an analysis of Brockett's recent design of a feedback law which creates an asymptotically stable oscillation in a three dimensional, nonholonomic model of an AC controlled rotor with a constant steady-state angular velocity. We will show how to design feedback laws for stabilizable n-dimensional systems so that the existence, periods and stability of periodic responses can be analyzed and shaped when the nonlinear feedback system is driven with an arbitrary periodic input. This design is one application of a general theory developed, jointly with R. Brockett, for the existence of oscillations in a nonlinear dynamical system. The sufficient conditions use a multi-valued analogue of Liapunov functions, in much the same way as the angular variable in polar coordinates is multi-valued. For the VCO the angular variable is the output of an integral controller, while for the AC motor it measures the rotation of the magnetic field. In the general case, the sufficient conditions can be checked point-wise, just as in Liapunov theory, and therefore do not require the knowledge of the trajectories of the system or a cross-section for the dynamics. Moreover, these results can be readily used in the theory of output regulation to shape the nonequilibrium steady-state response of dissipative nonlinear feedback systems. Finally, using the recent solution of the Poincaré Conjecture and more, we show these sufficient conditions are necessary for the existence of an asymptotically stable oscillation - a satisfying result in the spirit of the converse theorems of Liapunov theory.

In the present paper, we have introduced a new subclass of analytic functions involving integral operator defined by polylogarithm function. Necessary and sufficient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated.

This paper considers the existence of solutions and the equivalence of four kinds of vector variational inequalities (WI). More precisely, a sufficient condition is provided under which the solution sets of these VVIs are nonempty and equal. An example is given, showing that such a sufficient condition is essential to ensure the results. Actually, the main theorems in this paper can be regarded as a suitable correction and a refinement of recent results due to Chang et al. (Ref.1).

We define weighted geometric mean method of convergence for sequences in by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of convergence, convergence, convergence, convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence.

The integration of functional modules at the molecular level into RNA nanostructures holds great potential for expanding their applications. However, the quantitative integration of nucleoside analogue molecules into RNA nanostructures and their impact on the structure and function of RNA nanostructures remain largely unexplored. Here, we report a transcription-based approach to controllably integrate multiple nucleoside analogues into a 2000 nucleotide (nt) single-stranded RNA (ssRNA) origami nanostructure. The resulting integrated ssRNA origami preserves the morphology and biostability of the original ssRNA origami. Moreover, the integration of nucleoside analogues introduced new biomedical functions to ssRNA origamis, including innate immune recognition and regulation after the precise integration of epigenetic nucleoside analogues and synergistic effects on tumor cell killing after integration of therapeutic nucleoside analogues. This study provides a promising approach for the quantitative integration of functional nucleoside analogues into RNA nanostructures at the molecular level, thereby offering valuable insights for the development of multifunctional ssRNA origamis.

A recent result by Kardoš et al. (2023) related to the famous Berge–Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say O , in a bridgeless cubic graph, there exists a 1-factor intersecting all cycles in O in at least one edge. This remarkable result opens up natural generalizations in the case of an r -regular graph G and a t -factor F , with r and t being positive integers. In this paper, we start the study of this problem by proving necessary and sufficient conditions on G , t and r to assure the existence of a suitable F for any possible choice of the set O . First of all, we show that G needs to be 2-connected. Under this additional assumption, we highlight how the ratio t r seems to play a crucial role in assuring the existence of a t -factor F with the required properties by proving that t r ≥ 1 3 is a further necessary condition. We suspect that this condition is also sufficient, and we confirm it in the case t r = 1 3 , generalizing the case t = 1 and r = 3 proved by Kardoš, Máčajová, Zerafa, and in the case t r = 1 2 with t even. Finally, we provide further results for the case where even cycles are included.

Nonlocal variational theory and optimal control problems

will be considered. Conditions will be established in order that the solutions of (1.1) essentially behave asymptotically as those of y (n) = h(t). The part of the forcing function h(t) which is dominant for large values of t will be called the primary part of the forcing term. The forcing functions considered here will, generally, have primary part tmebt, m and b real. Sufficient conditions on f and h in order to guarantee the existence of solutions of (1.1) for large values of t will be tacitly assumed. In [1, ??3.7 and 3.11], Cesari discusses known asymptotic results for equation (1.1) when the equation is linear and the primary part of h(t) is a constant. Recent results on the asymptotic behavior of the solutions of a homogeneous equation are given in [2]-[4], [6], [7]. ?2 of this paper is concerned with integrable forcing terms; that is, f00h(t)dt is finite. The results of the remainder of the paper are developed for h(t) of the form h(t) = H(t)tmebt + R(t),

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1, 1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M G Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical self-adjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical self-adjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces.