ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

A checking test for a monotone read-once function f depending essentially on all its n variables is a set of vectors M distinguishing f from all other monotone read-once functions of the same variables. We describe an inductive procedure for obtaining individual lower and upper bounds on the minimal number of vectors T(f) in a checking test for any function f. The task of deriving the exact value of T(f) is reduced to a combinatorial optimization problem related to graph connectivity. We show that for almost all functions f expressible by read-once conjunctive or disjunctive normal forms, T(f) ~n / ln n. For several classes of functions our results give the exact value of T(f).

In [9], the notion of c-differentials was introduced as a potential expansion of differential cryptanalysis against block ciphers utilizing substitution boxes. Drawing inspiration from the technique of higher order differential cryptanalysis, in this paper we propose the notion of higher order c-derivatives and differentials and investigate their properties. Additionally, we consider how several classes of functions, namely the multiplicative inverse function and the Gold function, perform under higher order c-differential uniformity.

Have decidual natural killer (dNK) cells a different microRNA (miRNA or miR) expression pattern compared to NK cells circulating in the peripheral blood (pb) of healthy pregnant women in the first trimester of gestation? dNK cells have a unique miRNA profile, showing exclusive expression of a set of miRNAs and significant up- or down-regulation of most of the miRNAs shared with pbNK cells. dNK cells differ from pbNK cells both phenotypically and functionally, and their origin is still debated. Many studies have indicated that miRNAs regulate several important aspects of NK cell biology, such as development, activation and effector functions. Decidua basalis and peripheral blood specimens were collected from women (n = 7) undergoing voluntary termination of gestation in the first trimester of pregnancy. dNK and pbNK cells were then highly purified by cell sorting. miRNAs expression was analysed by quantitative RT-PCR (qRT-PCR)-based arrays using RNA purified from freshly isolated and highly purified pbNK and dNK cells. Results from arrays were validated by qRT-PCR assays. The bioinformatics tool ingenuity pathway analysis (IPA) was applied to determine the cellular network targeted by validated miRNAs and the correlated biological functions. Herein, we identified the most differentially expressed miRNAs in NK cells isolated from peripheral blood and uterine decidua of pregnant women. We found that 36 miRNAs were expressed only in dNK cells and two miRNAs only in pbNK cells. Moreover, 48 miRNAs were commonly expressed by both NK cell preparations although at different levels: 28 were upregulated in dNK cells, while 15 were downregulated compared to pbNK cells. Validation of a selected set (n = 11) of these miRNAs confirmed the differential expression of nine miRNAs: miR-10b and miR-214 expressed only in dNK cells and miR-200a-3p expressed only in pbNK cells; miR-130b-3p, miR-125a-5p, miR-212-3p and miR-454 were upregulated while miR-210-3p and miR-132 were downregulated in dNK cells compared to pbNK cells. IPA network analysis identified a single network connecting all the miRNAs as well as their significant involvement in several classes of functions: 'Organismal injury, Reproductive system disease, Inflammatory disease' and 'Cellular development'. These miRNAs target molecules such as argonaute 2, tumour protein p53, insulin and other genes that belong to the same network and significantly influence cell differentiation and pregnancy. In the present study, the cellular network and biological functions modulated by miRNAs differentially expressed in dNK and pbNK cells were identified by IPA considering only molecules and relationships that were with confidence 'experimentally observed' in leucocytes. The decidual and pbNK cells that were analysed here are a heterogeneous population and further study will help to disentangle whether there are differences in miRNA production by the different subsets of NK cells. This is the first study describing a different miRNA expression profile in dNK cells compared to matched pbNK cells during the first trimester of pregnancy. Our findings improved the body of knowledge on dNK cell biology and strongly suggest further investigation into the roles of miRNAs that are differentially expressed in human dNK compared to pbNK cells. Our results suggest that specific miRNAs can modulate dNK cell origin and functions, highlighting a potential role of this miRNA signature in human development and diseases. This work was supported by grants from the Istituto Pasteur, Fondazione Cenci Bolognetti, the European NoE EMBIC within FP6 (Contract number LSHN-CT-2004-512040), Istituto Italiano di Tecnologia, and Ministero dell'Istruzione, dell'Università e della Ricerca (Ricerche Universitarie), and from Università Politecnica delle Marche. There are no conflicts of interest to declare.

On the tractability of the Brownian Bridge algorithm

The delay of circuits whose inputs have specified arrival times

Many voting systems consist of a set of players who can form coalitions; a winning coalition is one that can pass a measure. The winning and losing coalitions can be given by several classes of functions, including switching functions, threshold functions, weighted voting systems, and simple games. We define a zonotope T n T_n related to these functions and describe its coordinates in terms of a measure of voting power. We also give a voting interpretation of the coordinates of the derived zonotope T ¯ n \bar {T}_n .

Let $\Cal P$ denote the well-known class of functions of the form $p(z)=1+q_1z+\ldots$ holomorphic in the unit disc $\bold D$ and fulfilling the conditions $Rep(z)>0$ in $\bold D$. Let $0\leq b<1, b<B, 0<\alpha <1$, be fixed real numbers and $zbold F$ a given measurable subset of the unit circle $\bold T$ of Lebesgue measure $2\pi\alpha$. For each $r \in (-\pi,\pi)$, denote by $\bold F_r=\{\xi\in \bold T; e^{-iT}\xi \in \bold F\}$ the set arising by rotation of $\bold F$ through the angle $\tau$. Denote by $\Cal P(B,b,\alpha;\bold F)$ the class of functions $p\in \Cal P$ satisfying the following conditions: there exists $\tau \in (-\pi,\pi)$ such that Rep(^{i\theta})\geq b$ a.e. on $\bold T\ \bold F_r$. In the paper the properties of the class $\Cal P(B,b,\alpha;\bold F)$ for different values of the parameters $B, b, \alpha$ and measurable sets $\bold F$ are examined. This article belongs to the series of papers ([4], [5], [6]) where different classes of functions defined by conditions on the circle $\bold T$ were studied. The results of papers [5], [6] are generalized.

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect. 1. Integral and differential operators. Let F denote the class of analytic functions which are holomorphic in the unit disk E = {| z |< 1}. Let F nad G be its subclasses consisting of f ∈ F normalized by f(0) = f ′(0) − 1 = 0 and f(0) = 1, respectively. In previous papers [3, 6] we have observed an integral operator L(a) defined on F , which is represented by L(a)f(z) = a ∫

Let S be the class of functions w — /(« ) = z + a2 z2 + a3 23 + ... analytic and schlicht in the unit disc | z | < 1. Let ry denote the least upper bound of the radii of ali öpen discs contained in the map of w = f (z) that is schlicht for | z | < 1 and h a s/' (0) = 1. Then

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

Radius of convexity of partial sums of functions in the close-to-convex family

We continue the study of the so‐called Hausdorff–Zhu operators. In this paper, we study the behavior of the singular values of such operators. We prove the general fact that the sequence of singular numbers tends to zero, as well as we prove power‐type estimates for such behavior under additional conditions on the kernel of the operator. We give application of these results to the boundedness of Hausdorff–Zhu operators in general classes of analytic functions in the unit disc and also in some special classes of analytic functions defined in terms of conditions on Taylor or Fourier coefficients.

We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set. A function f defined on a domain D has an asymptotic value b∈[−∞, ∞] at a∈∂D if there exists a path γ in D ending at a such that u(p) tends to b as p tends to a along γ. The set of all points on ∂D at which f has an asymptotic value b is denoted by A(f, b). G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +∞) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein. For α>0, let Mα be the class of subharmonic functions u in R + n + 1 ≡{(x, y):x∈ℝn, y>0} satisfying the growth condition u(x, y) ⩽ C(u)y−α for 0 < y < 1 for some constant C(u) depending on u. Denote by F(u) the Fatou set of u, which consists of points on ∂ R + n + 1 where u has finite vertical limits. For β>0, denote by Hβ the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].

Let $\mathcal{H}$ denote the class of analytic functions in the unit disc on the complex plane $\mathbf{C}$. Let $\mathcal{E}$ be a subclass of $\mathcal{H}$. If the operator $I:\mathcal{E}\rightarrow \mathcal{H}$ satisfies \begin{equation*} f(z)\prec g(z) \Rightarrow I[f](z)\prec I[g](z) \end{equation*} for all $f,g\in\mathcal{E}$, then it is called subordination-preserving operator on the class $\mathcal{E}$. In this work we consider the convexity of the Bernardi operator. We prove also that the Bernardi is the subordination-preserving operator on the class of starlike functions. The applications of main results are also presented.

Some extremal bounds for subclasses of univalent functions