Special Functions on the Sphere with Applications to Minimal Surfaces

A function which is homogeneous in x, y, z of degree n and satisfies V_xx + V_yy + V_zz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rⁿf_n, where f_n is a spherical surface harmonic of degree n. On a sphere, f_n satisfies ▵ f_n + n(n + 1)f_n = 0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X · n, satisfies ▵w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.     

Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (u_t + 6uu_x + u_xxx)_x ± 3u_yy ± 3u_zz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Painlevé test. As by-products, the one-soliton and two-soliton solutions and four classes of specific three-soliton solutions are explicitly presented.     

A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let X_n denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ? 0}, starting from w₀ and with drift μ < 0, to reach c ∈ [0, w₀). After the nth repair, the process takes on either the value X_n?1 + 1 or X_n?1 + 2. The probability that X_n = X_n?1 + j, for j = 1, 2, depends on whether τ ? t₀ (a fixed constant) or τ > t₀. The device is considered to be worn out when X_n ? k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field F_q is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight d_r(C), 1 ⩽ r ⩽ k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d₁(C), d₂(C), … , d_k(C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, d_r(C) = n − k + r;  r = 1, 2 , …, k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH d_r(C) = n − k + r; r = 1, 2, … , k. A linear code C with systematic check matrix [I∣P], where I is the (n − k) × (n − k) identity matrix and P is a (n − k) × k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS (N²-MDS) codes and A^μ-MDS codes. The MDS-rank of C is the smallest integer η such that d_η+1 = n − k + η + 1 and the defect vector of C with MDS-rank η is defined as the ordered set {μ₁(C), μ₂(C), μ₃(C), … , μ_η(C), μ_η+1(C)}, where μ_i(C) = n − k + i − d_i(C). We call C a dually defective code if the defect vector of the code and its dual are the same. We also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported.     

Pattern Frequency Sequences and Internal Zeros

Let q be a pattern and let S_n, q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S_n, q(c))_c ≥ 0 has internal zeros. If q is a monotone pattern it turns out that, except for q = 12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q = 1(l + 1)l…2 there are infinitely many sequences which contain internal zeros and when l = 2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.     

Computing best-possible bounds for the distribution of a sum of several variables is NP-hard

In many real-life situations, we know the probability distribution of two random variables x₁ and x₂, but we have no information about the correlation between x₁ and x₂; what are the possible probability distributions for the sum x₁ + x₂? This question was originally raised by A.N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x₁ + x₂; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n > 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x₁ + ⋯ + x_n? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not best-possible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem.     

Exact special solitary solutions with compact support for the nonlinear dispersive K(m, n) equations

The nonlinear dispersive K(m, n) equations, u_t−(u^m)_x−(uⁿ)_xxx = 0 which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. General formulas for the solutions of K(m, n) equations are established.     

On Certain C-Test Words for Free Groups

Let F_m be a free group of a finite rank m ≥ 2 and let X_i, Y_j be elements in F_m. A non-empty word w(x₁,…,x_n) is called a C-test word in n letters for F_m if, whenever (X₁,…,X_n) = w(Y₁,…,Y_n) ≠ 1, the two n-typles (X₁,…,X_n) and (Y₁,…,Y_n) are conjugate in F_m. In this paper we construct, for each n ≥ 2, a C-test word v_n(x₁,…,x_n) with the additional property that v_n(X₁,…,X_n) = 1 if and only if the subgroup of F_m generated by X₁,…,X_n is cyclic. Making use of such words v_m(x₁,…,x_m) and v_m + 1(x₁,…,x_m + 1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u₁, u₂ ∈ F_m such that every endomorphism ψ of F_m with non-cyclic image is completely determined by ψ(u₁), ψ(u₂).     

Statistical properties of nucleotides in human chromosomes 21 and 22

In this paper the statistical properties of nucleotides in human chromosomes 21 and 22 are investigated. The n-tuple Zipf analysis with n = 3, 4, 5, 6, and 7 is used in our investigation. It is found that the most common n-tuples are those which consist only of adenine (A) and thymine (T), and the rarest n-tuples are those in which GC or CG pattern appears twice. With the n-tuples become more and more frequent, the double GC or CG pattern becomes a single GC or CG pattern. The percentage of four nucleotides in the rarest ten and the most common ten n-tuples are also considered in human chromosomes 21 and 22, and different behaviors are found in the percentage of four nucleotides. Frequency of appearance of n-tuple f(r) as a function of rank r is also examined. We find the n-tuple Zipf plot shows a power-law behavior for r < 4ⁿ⁻¹ and a rapid decrease for r > 4ⁿ⁻¹. In order to explore the interior statistical properties of human chromosomes 21 and 22 in detail, we divide the chromosome sequence into some moving windows and we discuss the percentage of ξη (ξ, η = A, C, G, T) pair in those moving windows. In some particular regions, there are some obvious changes in the percentage of ξη pair, and there maybe exist functional differences. The normalized number of repeats N₀(l) can be described by a power law: N₀(l) ∼ l^−μ. The distance distributions P₀(S) between two nucleotides in human chromosomes 21 and 22 are also discussed. A two-order polynomial fit exists in those distance distributions: log P₀(S) = a + bS + cS², and it is quite different from the random sequence.     

Inverse problems for parabolic equations 2

Let u_t − u_xx = h(t) in 0 ⩽ x ⩽ π, t ⩾ 0. Assume that u(0, t) = v(t), u(π, t) = 0, and u(x, 0) = g(t). The problem is: what extra data determine the three unknown functions {h, v, g} uniquely? This question is answered and an analytical method for recovery of the above three functions is proposed.     

Calculation of a static potential created by plane fractal cluster

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: r_n = r₀ξⁿ and e_n = e₀bⁿ, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5], [6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster (r₀ ≪ r ≪ r_N−1), where r₀, r_N−1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously (b > 0) and heterogeneously (b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.     

Second-order approximation of angle-ply composite laminated thin plate under combined excitations

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω₂ ? 2ω₁ and Ω₂ ? 2ω₂) in the presence of 1:2 internal resonance ω₂ ? 2ω₁ are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.     

Discrepancy principle for DSM II

Let Ay = f, A is a linear operator in a Hilbert space H, y ⊥ N(A) ≔ {u : Au = 0}, R(A) ≔ {h : h = Au, u ∈ D(A)} is not closed, ∥f_δ − f∥ ⩽ δ. Given f_δ, one wants to construct u_δ such that lim_δ→0∥u_δ − y∥ = 0. Two versions of discrepancy principles for the DSM (dynamical systems method) for finding the stopping time and calculating the stable solution u_δ to the original equation Ay = f are formulated and mathematically justified.     

Interpolation by Integer-Valued Polynomials

LetRbe a Krull ring with quotient fieldKanda₁,…,a_ninR. If and only if thea_iare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb₁,…,b_ninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(a_i) = b_iandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringR_vwith quotient fieldKanda₁,…,a_ninSare pairwise incongruent mod allM^k_v ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of thea_iamong residue classes of the idealsM^k_v ∩ S) such that for allb₁,…,b_n ∈ R_vthere exists a polynomialf ∈ K[x] of degree at mostdwithf(a_i) = b_iandf(S) ⊆ R_v.     

Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons

Degasperis and Procesi applied the method of asymptotic integrability and obtain Degasperis–Procesi equation. They showed that it has peakon solutions, which has a discontinuous first derivative at the wave peak, but they did not explain the reason that the peakon solution arises. In this paper, we study these non-smooth solutions of the generalized Degasperis–Procesi equation u_t − u_txx + (b + 1)uu_x = bu_xu_xx + uu_xxx, show the reason that the non-smooth travelling wave arise and investigate global dynamical behavior and obtain the parameter condition under which peakon, compacton and another travelling wave solutions engender. Under some parameter condition, this equation has infinitely many compacton solutions. Finally, we give some explicit expression of peakon and compacton solutions.