# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing four issues in March, June, September and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### Some properties of fractional Ornstein-Uhlenbeck process

JPRM-Vol. 1 (2005), Issue 1, pp. 193 – 205 Open Access Full-Text PDF
S. C. Lim
Abstract: We study the solution of fractional Langevin equation at both zero and positive temperature. The partition function and other thermodynamic properties associated with the solution (or the fractional Ornstein-Uhlenbeck process) can be obtained by using the generalized zeta function and the heat kernel techniques.

### On the conversion of rational triangular Bézier surfaces and multi-sided patches

JPRM-Vol. 1 (2005), Issue 1, pp. 184 – 192 Open Access Full-Text PDF
Germain E. Randriambelosoa, Malik Zawwar Hussain
Abstract: By a convenient parameter change we give a necessary and sufficient condition for the
conversion of a degree $$n$$ rational triangular Bézier surfaces to a degree $$n × n$$ rectangular surface whose one of the four edges is reduced to a single multiple point. Using then the method of base point we obtain the explicit expression of multi-sided patches in Bézier form.

### Continuous automorphisms and an equivalence relation In $$K[[X]]$$

JPRM-Vol. 1 (2005), Issue 1, pp. 178 – 183 Open Access Full-Text PDF
Shaheen Nazir
Abstract: Let K be an arbitrary commutative field and let $$R = K[[X]]$$ be the ring of formal power series in one variable. Let $$G_{R}$$ be the set of all power series of the form $$u = Xv$$, where $$v$$ is a unity in $$R$$. Relative to the usual composition $$G_{R}$$ becomes a topological group with respect to the $$X-$$adic topology of $$R$$. We also study an equivalence relation on $$R$$. Let $$R = K[[X]]$$ be the ring of formal power series in one variable over a fixed commutative field $$K$$. We denote by $$ord f = min \{i: a_{i} ≠ 0\}$$ for any $$f ∈ R$$ . It is well known that $$ord f$$ is a valuation on $$R$$ and $$R$$ becomes a complete topological ring relative to the topology induced by this valuation. Let  $$G_{G}=\{u∈ R: ord u=1\}$$ and for $$u,v ∈ R_{G}$$ we denote $$(uov)(X)=v(u(X))$$, a new element of $$R_{G}$$.

### Fermi, Bose and Vicious walk configurations on the directed square lattice

JPRM-Vol. 1 (2005), Issue 1, pp. 156 – 177 Open Access Full-Text PDF
F. M. Bhatti, J. W. Essam
Abstract: Inui and Katori introduced Fermi walk configurations which are non-crossing subsets of the directed random walks between opposite corners of a rectangular $$l × w$$ grid. They related them to Bose configurations which are similarly defined except that they include multisets. Bose configurations biject to vicious walker watermelon configurations. It is found that the maximum number of walks in a Fermi configuration is $$lw + 1$$ and the number of configurations corresponding to this number of walks is a w-dimensional Catalan number $$C_{l,w}$$. Product formulae for the numbers of Fermi configurations with $$lw$$ and $$lw − 1$$ walks are derived. We also consider generating functions for the numbers of $$n−$$walk configurations as a function of $$l$$ and $$w$$. The Bose generating function is rational with denominator $$(1-z)^{lw+1}$$. The Fermi generating function is found to have a factor $$(1+z)^{lw+1}$$ and the complementary factor , $$Q_{l,w}^{frmi}(z)$$is related to the numerator of the Bose generating function which is a generalized Naryana polynomial introduced by Sulanke. Recurrence relations for the numbers of Fermi walks and for the coefficients of the polynomial $$Q_{l,w}^{frmi}(z)$$ are obtained.

### Statistical space control in non-linear systems: speed gradient method

JPRM-Vol. 1 (2005), Issue 1, pp. 144 – 155 Open Access Full-Text PDF
S. V. Borisenok
Abstract: Here we discuss the application of control theory to dynamical non-linear systems in the form of speed gradient method. This approach can be practically used for the space focusing of classical or quantum particles, for instance, in nanolithography with the beams of cooled atoms. Standard speed gradient method works here not very efficient because it allows to achieve the selected level of energy (the eigenfunction of Hamiltonian for dynamical differential equation). For the practical purposes we re-formulate the mathematical task and introduce principally new type of controlled systems. We demand achievement of the space distribution of the dynamical particles. Additionally we investigate the statistical properties of the particle dynamics but not the behavior of the single particle. Efficiency of the approach was verified by means of computer simulations.

### On the arithmetical structure of some compact subsets in the $$p$$-adic complex number field

JPRM-Vol. 1 (2005), Issue 1, pp. 136 – 144 Open Access Full-Text PDF
Angel Popescu
Abstract: We use arithmetical tools (algebraic numbers, valuations, Galois groups) in order to describe the structure of all compact subsets of the $$p-$$adic complex field, which are invariant with respect to the absolute Galois group of the $$p-$$adic number field