Abstract The aim of the current study was to analyze players’ interactions in serve and return situations examine way points were decided due positions surfaces. Fourteen ATP 1000 tournament matches analyzed (7 on hard surface; 7 clay surface). type surface did not affect ended (χ²=9.26; p > 0.05) but influence vertical positioning (χ²=270.86; < 0.001). positioning, both lateral (χ²=92.68...

A class G of graphs is said to be χ-bounded if there is a function f : N → R such that for all G ∈ G and all induced subgraphs H of G, χ(H) ≤ f(ω(H)). In this paper, we show that if G is a χ-bounded class, then so is the closure of G under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if G is χ-bounded...

Given data from a spherical Gaussian distribution with unknown mean vector θ, estimates of quadratic functionals are constructed by thresholding. Mean squared error bounds are derived via a comparison with those already available for a suitable noncentral χ variate. By way of illustration, the resulting inequalities are used to yield an optimal rate adaptivity result for estimation of integrate...

For a finite commutative ring R, let a,b,c∈R be fixed elements. Consider the equation ax+by=cz where x, y, and z are idempotents, units, any element in respectively. We say that R satisfies r-monochromatic clean condition if, for r-colouring χ of elements there exist x,y,z∈R with χ(x)=χ(y)=χ(z) such holds. define m(a,b,c)(R) to least positive integer r does not satisfy condition. This means exi...

The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

A set of problems is offered to accompany the set of lectures with the above title. Topic 1: Coherent States 1.1 Let {|n〉}, n = 0, 1, 2, 3, . . ., denote a complete set of orthonormal vectors in an abstract Hilbert space , with 〈m|n〉 = δmn. Next, let |χ〉 = √ 1 − |χ|2 |0〉 + χ|1〉 , for all complex χ, 0 ≤ |χ| ≤ 1, denote a family of two-dimensional vectors. a) If χ→ χ in the sense of complex numbe...

(1) (c, χ) = (7n, n), (2) (c, χ) = (5n, n), (3) (c, χ) = (4n, n), (4) (c, χ) = (2n, n), (5) (c, χ) = ((6 + 8g)n, (1 + g)n (for g ≥ 0), (6) (c, χ) = (7n+ (6 + 8g)m,n+ (1 + g)m), (7) (c, χ) = (7n+ 5m,n+m), (8) (c, χ) = (7n+ 4m,n+m), (9) (c, χ) = (7n+ 2m,n+m), (10) (c, χ) = ((6 + 8g)n+ 5m, (1 + g)n+m) (for g ≥ 0), (11) (c, χ) = ((6 + 8g)n+ 4m, (1 + g)n+m) (for g ≥ 0), (12) (c, χ) = ((6 + 8g)n+ 2m,...