Commutativity of quaternion‐matrix–valued functions and quaternion matrix dynamic equations on time scales

In this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also a corresponding existence and uniqueness theorem is proved. In addition, the commutativity of quaternion‐matrix–valued functions is investigated and some sufficient and necessary conditions of commutativity and noncommutativity are established on time scales. Also the fundamental solution matrices of some basic quaternion matrix dynamic equations are obtained. Examples are provided to illustrate the results, which are completely new on hybrid domains particularly when the time scales are the quantum case and the discrete case ; , both of which are significant for the study of quaternion q‐dynamic equations and quaternion difference dynamic equations. Finally, we present several applications including multidimensional rotations and transformations of the submarine, the gyroscope, and the planet whose dynamical behaviors are depicted by quaternion dynamics on time scales and the corresponding iteration numerical solution for homogeneous quaternion dynamic equations are provided on various time scales.     

A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

General soliton and (semi-)rational solutions to the nonlocal Mel'nikov equation on the periodic background

In this paper, the Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi-)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram-type determinants. When N is even, soliton, line breather and (semi-)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi-rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.     

随机环境中受控分枝过程的极限定理

讨论了随机环境中受控分枝过程{Z_n:n∈N}的极限问题.给出了过程在{S_n:n∈N}下的规范化过程{W_n:n∈N}几乎处处收敛、L~1收敛和L~2收敛的充分条件,以及过程{W:n∈N}的极限非退化于0的充分条件和必要条件,得到了过程在{I_n:n∈N}下的规范化过程{W_n:n∈N}几乎处处收敛和L~1收敛的充分条件.     

Exact N-Wave Solutions of Generalized Burgers Equations

Exact N-Wave solutions for the generalized Burgers equation where j, α, β, and γ are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.     

Selbstadjungierte Relations-Erweiterungen für eine Klasse von symmetrischen Differential-Funktional-Operatoren

Symmetry- and selfadjointness-conditions are derived for ordinary differential-integral-interface operators under integral-interface conditions. Criteria for the existence of selfadjoint extensions in L×L are given. These extensions are characterized in a constructive way. The main tools are some extension-theorems for linear relations (subspaces), wich are developed in section 2.     

A Nonlinear Elliptic System Arising in Electron Density Theory

Of concern is the nonlinear system

in R³ Here λ_i,C_j are given positive constants. Such a system arises in spin polarized ground state electron density theory. The parametres λ_:1:, λ_:2 should be such that

where N_1>,N₂ are given positive numbers satisfying where Z is some give positive number. (Here N$sub:1$esub:, N$sub:2$esub:, Z are, respectively, the number of spin up electrons, spin down electrons, and protons in the system.) We establish existence for this problem. Uniqueness remains open.     

Indivisible sets and well‐founded orientations of the Rado graph

Every set can been thought of as a directed graph whose edge relation is ∈ . We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ?₁ that are distinct up to siblinghood.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered.     

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

In this paper, the partially party‐time () symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called x‐nonlocal DS equations, while a fully symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x‐nonlocal DS equations, the usual (2 + 1)‐dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)‐dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.     

Intersection Numbers For Subspace Designs

Intersection numbers for subspace designs are introduced and q‐analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q‐analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2‐ subspace design. Furthermore, several simplified or alternative proofs concerning intersection numbers of ordinary block designs are discussed.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

Covering arrays for some equivalence classes of words

Covering arrays for words of length over a ‐letter alphabet are arrays with entries from the alphabet so that for each choice of columns, each of the ‐letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size of a covering array. Definitive results for , as well as general results, are provided.     

Microstructure of a-SiOx:H

A set of a-SiO_x:H (0.52 <x< 1.58) films are fabricated by plasma-enhanced-chemical-vapor-deposition (PECVD) method at the substrate temperature of 250°C. The microstructure and local bonding configurations of the films are investigated in detail using micro-Raman scattering, X-ray photoelectron spectroscopy (XPS) and Fourier transform infrared spectroscopy (FTIR). It is found that the films are structural inhomogeneous, with five phases of Si, Si₂O:H, SiO:H, Si₂O₃:H and SiO₂ that coexist. The phase of Si is composed of nonhydrogenated amorphous silicon (a-Si) clusters that are spatially isolated. The average size of the clusters decreases with the increasing oxygen concentration x in the films. The results indicate that the structure of the present films can be described by a multi-shell model, which suggests that a-Si cluster is surrounded in turn by the subshells of Si₂O:H, SiO:H, Si₂O₃:H, and SiO₂.     

Definable types in algebraically closed valued fields

In 15 , Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any that there is a pair of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an ‐type over M realized in N which is not definable.     

Conservation Laws for the Fully Nonlinear Long Wave Equations

The fully nonlinear long wave equations describe the motion over a flat bottom of a two-dimensional inviscid fluid with a free surface in a gravitational field in the long wave approximation. These equations are shown to possess an infinite number of conservation laws (in two space dimensions) in the form The conserved densities T and the fluxes ?X and ?Y are polynomials in the height h and the horizontal and vertical components of velocity, u and v, and also in integrals of powers of u. The method of proof is a modification of the method recently devised by D. J. Benney to prove that these same equations possess an infinite number of conservation laws (in one space dimension) in the form where T and X are polynomials in the height h and integrals of powers of u. Conservation laws which explicitly contain x and t are also given.     

Nonlocal boundary value problem for fractional differential equations with p‐Laplacian

In this paper, we are concerned with the existence of positive solutions for the following nonlocal BVP of fractional DEs with p‐Laplacian operator By using the fixed point theorem in a cone, multiplicity solutions of the BVP are obtained. An example is also given to show the effectiveness of the obtained result. Copyright © 2013 John Wiley & Sons, Ltd.