Hankel, Toeplitz, and Hermitian-Toeplitz Determinants for Certain Close-to-convex Functions

Let f be analytic in $$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$ D = { z ∈ C : | z | < 1 } , and be given by $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ f ( z ) = z + ∑ n = 2 ∞ a n z n . We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional $$J_{2,3}(f).$$ J 2 , 3 ( f ) .

The Perceptual Organisation of Visual Elements: Lines

The aim of this study is to verify the conditions under which a series of visual stimuli (line segments) will be subjectively perceived as visual lines or surfaces employing four experiments. Two experiments were conducted with the method of subjective evaluation of the line segments, and the other two with the Osgood semantic differential. We analysed five variables (thickness, type, orientation, and colour) potentially responsible for the lines’ categorisation. The four experiments gave similar results: higher importance of the variables thickness and type; general lower significance of the variable colour; and general insignificance of the variable orientation. Interestingly, for the variable type, straight lines are evaluated as surfaces more frequently than curved lines and perceived as geometrical, flat, hard, static, rough, sharp, bound, sour, frigid, masculine, cold and passive. Curved lines are prevalently evaluated as lines, and categorised as organic, rounded, soft, dynamic, fluffy, blunt, free, sweet, sensual, feminine, warm and active. These results highlight the specificity of perceptual characteristics for the considered variables and confirm the relevance of the characteristics of variables such as thickness and type.

The sharp bound of the third Hankel determinant for functions of bounded turning

We find the sharp bound for the third Hankel determinant $$\begin{aligned} H_{3,1}(f):= \left| {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } \\ {a_{2} } & {a_{3} } & {a_{4} } \\ {a_{3} } & {a_{4} } & {a_{5} } \\ \end{array} } \right| \end{aligned}$$ H 3 , 1 ( f ) : = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 for analytic functions f with $$a_n:=f^{(n)}(0)/n!,\ n\in \mathbb N,\ a_1:=1,$$ a n : = f ( n ) ( 0 ) / n ! , n ∈ N , a 1 : = 1 , such that $$\begin{aligned} {{\,\mathrm{Re}\,}}f'(z)>0,\quad z\in \mathbb D:=\{z \in \mathbb C: |z|<1\}. \end{aligned}$$ Re f ′ ( z ) > 0 , z ∈ D : = { z ∈ C : | z | < 1 } .

On the Hosoya Indices of Bicyclic Graphs with Small Diameter

Let G be a graph. The Hosoya index of G , denoted by z G , is defined as the total number of its matchings. The computation of z G is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.

Wiener index in graphs with given minimum degree and maximum degree

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.

The Szemerédi–Petruska conjecture for a few small values

Let H be a 3-uniform hypergraph of order n with clique number $$\omega (H)=k$$ ω ( H ) = k . Assume that the union of the k-cliques of H equals its vertex set, the intersection of all maximum cliques of H is empty, but the intersection of all but one k-clique is non-empty. For fixed $$m=n-k$$ m = n - k , Szemerédi and Petruska conjectured the sharp bound $$n\hbox {\,\,\char 054\,\,}{m+2\atopwithdelims ()2}$$ n 6 m + 2 2 . In this note the conjecture is verified for $$m=2,3$$ m = 2 , 3 and 4.

Can Coherent Predictions be Contradictory?

We prove the sharp bound for the probability that two experts who have access to different information, represented by different $\sigma$-fields, will give radically different estimates of the probability of an event. This is relevant when one combines predictions from various experts in a common probability space to obtain an aggregated forecast. The optimizer for the bound is explicitly described. This paper was originally titled ‘Contradictory predictions’.

Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

GEODESICS IN THE SIERPINSKI CARPET AND MENGER SPONGE

In this paper, we study geodesics in the Sierpinski carpet and Menger sponge, as well as in a family of fractals that naturally generalize the carpet and sponge to higher dimensions. In all dimensions, between any two points we construct a geodesic taxicab path, namely a path comprised of segments parallel to the coordinate axes and possibly limiting to its endpoints by necessity. These paths are related to the skeletal graph approximations of the Sierpinski carpet that have been studied by many authors. We then provide a sharp bound on the ratio of the taxicab metric to the Euclidean metric, extending Cristea’s result for the Sierpinski carpet. As an application, we determine the diameter of the Sierpinski carpet taken over all rectifiable curves. For other members of the family, we provide a lower bound on the diameter taken over all piecewise smooth curves.