### Probability of the moderate deviations for the sum-functions of spacing

JPRM-Vol. 1 (2006), Issue 1, pp. 217 – 230 Open Access Full-Text PDF
Sherzod Mira, Zam Mirakhmedov, Syed Ikram Abbas Tirmizi
Abstract: Let $$0=U_{0,n}\leq U_{1,n}\leq … \leq U_{n,n}=1$$ be an ordered sample from uniform [0,1] distribution $$D_{in}=U_{i,n}-U_{i-1,n}$$, $$i=1,2,..,n$$, $$n=1,2,…$$ , be their spacings,and let $$f_{1n},…,f_{nn}$$ be a set of measurable functions. In this paper theorems on the probabilities of deviations in the moderate zones for $$R_{n}D=f_{1n}(nD_{1n}, … , f_{nn}(nD_{nn}$$ are presented. Application of these results to study an intermediate efficiencies of the tests based on statistic $$R_{n}(D)$$ are also considered.

### Open-loop control of quantum particle motion: effective splitting in momentum space

JPRM-Vol. 1 (2006), Issue 1, pp. 208 – 216 Open Access Full-Text PDF
Babar Ahmad, Sergei Borisenok, Saifullah, Yuri Rozhdestvensky
Abstract: In this paper an effective quantum particle beam-splitter in the momentum space is realized in the frame of open-loop control scheme. We demonstrate for small interaction time that the splitting effect $$±40\hbar k$$ with summarized relative intensity in both main components is about 50 per cent from initial intensity of the atomic beam.

### On some numerical invariants associated to a compact sets (in metric spaces)

JPRM-Vol. 1 (2006), Issue 1, pp. 203 – 207 Open Access Full-Text PDF
Abstract: In this note we associate to a compact subset of a metric space a subset of natural numbers. We give some interesting properties of this last subset. We also introduce a configuration matrix for a given compact set and propose a conjecture related to this.

### On the arithmetic of the rational function field $$K(X)$$

JPRM-Vol. 1 (2006), Issue 1, pp. 198 – 202 Open Access Full-Text PDF
Abstract: Let $$K$$ be a commutative field. In this paper we study the action of the automorphism group of the rational function field $$K(X)$$ on the set of all valuations of $$K(X)$$ which are trivial on $$K$$. We apply this study in finding a classification of some simple algebraic extension of $$K$$.

### On some subfields of $$K((X))$$

JPRM-Vol. 1 (2006), Issue 1, pp. 194 – 197 Open Access Full-Text PDF
Shaheen Nazir,  Angel Popescu
Abstract:  Let K be a commutative field and let $$K((X))$$ be the field of Laurent series in one variable $$X$$, consider with its natural $$X-$$adic topology. In this paper we prove that any closed subfield $$K ⊂ L ⊂ K((X))$$ is of the form $$L = K((f))$$ and $$K((X))$$ is algebraic over L of degree $$ord_{X}(f)$$. Some other properties of $$L$$ are studied.

### A cycle or Jahangir ramsey unsaturated graphs

JPRM-Vol. 1 (2065), Issue 1, pp. 187 – 193 Open Access Full-Text PDF
Kashif Ali, Surahmat
Abstract: A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some result concerning both Ramsey saturated and unsaturated graph. In particular, we show that a cycle $$C_{n}$$ and a Jahangir $$J_{m}$$ Ramsey unsaturated or saturated graphs of $$R(C_n, W_m)$$ and $$R(P_n, J_m)$$, respectively. We also suggest an open problems.

### Monotonic surfaces for computer graphics

JPRM-Vol. 1 (2006), Issue 1, pp. 170 – 186 Open Access Full-Text PDF
Malik Zawwar Hussain, Maria Hussain
Abstract: Computer graphics environment requires realistic visual models of data generated. These data can be either 2D or 3D and corresponding visual models are called curves and surfaces. In this paper, a piecewise rational cubic function [8] has been extended to rational bicubic function. Simple constraints are derived on the free parameters in the description of rational bicubic function to preserve the shape of monotonic data.

### New subclass of starlike functions of complex order

JPRM-Vol. 1 (2006), Issue 1, pp. 157 – 169 Open Access Full-Text PDF
Yasar Polatoglu, H. Esra Ozkan
Abstract: The aim of the present paper is to investigate a new subclass of starlike functions of complex order $$b\neq 0$$. Let $$f(z)=z+a_{2}z^{2}+…$$ be an analytic function in the unit disc $$D=\{z| |z|<1\}$$ which satisfies $$1+\frac{1}{b}(z\frac{f'(z)}{f(z)}-1)=\frac{1+A\omega z}{1+B\omega z}$$, for some $$\omega \in \Omega$$ and for all $$z \in D$$. Then f is called a Janowski starlike function of complex order b, where A and B are complex numbers such that $$Re(1-A\overline{B})\geq |A-B|, im(1-A\overline{B}<|A-B|, |B|<1$$ and $$\omega(z)$$ ) is a Schwarz function in the unit disc D [1], [10], [12]. The class of these functions is denoted by $$S^{∗}(A, B, b)$$. In this paper we will give the representation theorem, distortion theorem, two point distortion theorem, Koebe domain under the montel normalization, and coefficient inequality for this class.

### Vertex-magic total labelings of disconnected graph

JPRM-Vol. 1 (2006), Issue 1, pp. 147 – 156 Open Access Full-Text PDF
Slamin, A.C. Prihandoko, T.B. Setiawan, F. Rosita, B. Shaleh
Abstract: Let G be a graph with vertex set $$V = V (G)$$ and edge set $$E = E(G)$$ and let $$e = |E(G)|$$ and $$v = |V (G)|$$. A one-to-one map $$λ$$ from $$V ∪ E$$ onto the integers $$\{1, 2, …, v + e\}$$ is called vertex magic total labeling if there is a constant $$k$$ so that for every vertex $$x$$, $$λ+\sum λ(xy)=k$$. where the sum is over all vertices $$y$$ adjacent to $$x$$. Let us call the sum of labels at vertex x the weight $$w_{λ}(x)$$ of the vertex under labeling $$λ$$; we require $$w_{λ}(x) = k$$ for all $$x$$. The constant $$k$$ is called the magic constant for $$λ$$. In this paper, we present the vertex magic total labelings of disconnected graph, in particular, two copies of isomorphic generalized Petersen graphs $$2P(n, m)$$, disjoint union of two non-isomorphic suns $$S_{m} ∪ S_{n}$$ and t copies of isomorphic suns $$tS_{n}$$.
### Upper bounds for the size Ramsey numbers for $$P_3$$ versus $$C^{t}_{3}$$ or $$P_{n}$$
Abstract: In this paper, we derive an upper bound for the size Ramsey number for a path $$P_3$$ versus a friendship graph $$C_{3}^{t}$$. Furthermore, some minimal Ramsey graph for a combination $$(P_{3}, C_{3}^{t})$$is presented. We also give an upper bound of the size Ramsey number for $$P_3$$ versus $$P_n$$.