### Effect of convection on wavelet estimation for a multidimensional acoustic earth

JPRM-Vol. 1 (2005), Issue 1, pp. 220 – 226 Open Access Full-Text PDF
Abstract: We extend the wavelet estimation method due to Weglein and Secrest [4] to a marine seismic exploration model by taking into account the fluid motion and obtain the wavelet amplitude of the reflected data. This consideration is important for processing marine seismic data and for modeling seismic response. In case of known source location, the method predicts the source spectrum whereas the wave-field is predicted when the source (discrete/continuously distributed) is completely unknown. Moreover, this method is independent of the information about the properties of the earth. We find that the wavelet amplitude tends to zero and as a result, the corresponding reflected waves will be suppressed when the speed of the moving fluid gets closer and closer to the speed of sound. When the speed of the fluid motion is minuscule in comparison with the wave speed in water (Mach number $$M << 1$$), it does not effect the wavelet amplitude and the fluid motion can altogether be ignored which is in accordance with the physical observation.

### Separation of variables for nonlinear wave equation in cylinder coordinates

JPRM-Vol. 1 (2005), Issue 1, pp. 206 – 2019 Open Access Full-Text PDF
Alexander Shermenev
Abstract: Some classical types of nonlinear wave motion in the cylinder coordinates are studied within quadratic approximation. When the cylinder coordinates are used, the usual perturbation techniques inevitably leads to over determined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). However, we show that these over determined systems are compatible for the special case of the nonlinear acoustical wave equation and express explicitly the coefficients of the first two harmonics as polynomials of the Bessel functions of radius and of the trigonometric functions of angle. It gives a series of solutions to the nonlinear acoustical wave equation which are found with the same accuracy as the equation is derived.

### 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
### On the defining spectrum of $$k-$$regular graphs with $$k–1$$ colors
Abstract: In a given graph $$G = (V;E)$$, a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a $$c \geq \chi(G)$$ coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by $$(d(G; c)$$. If F is a family of graphs then $$Spec_{c}(F)=\{d| \exists G, G \epsilon F, d(G,C)=d \}$$. Here we study the cases where $$F$$ is the family of $$k-$$regular (connected and disconnected) graphs on n vertices and $$c = k-1$$. Also the $$Spec_{k-1}(F)$$ defining spectrum of all $$k-$$regular (connected and disconnected) graph on n vertices are verified for $$k = 3, 4$$ and $$5$$.