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.     

Strip generation algorithms for constrained two-dimensional two-staged cutting problems

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate of length L and width W, as to maximize the sum of the profits of the pieces to be cut. Each piece type i, i = 1, …, n, is characterized by a length l_i, a width w_i, a profit (or weight) c_i and an upper demand value b_i. The upper demand value is the maximum number of pieces of type i which can be cut on rectangle (L, W). In this paper, we study the two-staged fixed orientation C_TDC, noted FC_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two guillotine cuts, and each piece has a fixed orientation. We solve the FC_2TDC problem using several approximate algorithms, that are mainly based upon a strip generation procedure. We evaluate the performance of these algorithms on instances extracted from the literature.     

Space–time symmetry violation,configuration mixing model and E-infinity theory

We have studied the time reversal symmetry violation on the bases of the configuration mixing model and E-infinity theory. With the use of the Cabibbo angle approximation, we have presented the transformation matrix in terms of the golden ratio (?), and shown that the time reversal symmetry violation is described by the configuration mixing of the unstable and stable manifolds (W^u, W^s). The magnitude of the mixing for the weak interaction field is given by the expression sin² θ_T(theor) ≈ sin⁴ θ_C(theor) ≈ (?)¹² = 3.105 × 10^?3, which is compared to the Kaon decay experiment ～2.3 × 10^?3. We have also discussed the space–time symmetry violation by using the CPT theorem.     

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m²) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D₁, … ,D_k, F), where D_js are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D₁ + FG₁, … ,D_k + FG_k) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km²) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.     

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.     

The Exponent of Discrepancy Is at Least 1.0669

LetP⊂[0, 1]^dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑_z∈P w(z)=1. TheL₂-discrepancy of the weighted set (P, w) is defined as theL₂-average ofD(x)=vol(B_x)−w(P∩B_x) overx∈[0, 1]^d, where vol(B_x) is the volume of thed-dimensional intervalB_x=∏^d_k=1 [0, x_k). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε^−ppoints in [0, 1]^dwithL₂-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL₂-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

An extended rational interpolation method

To interpolate function, f(x), a ? x ? b, when we have some information about the values of f(x) and their derivatives in separate points on {x₀, x₁, … , x_n} ? [a, b], the Hermit interpolation method is usually used. Here, to solve this kind of problems, extended rational interpolation method is presented and it is shown that the suggested method is more efficient and suitable than the Hermit interpolation method, especially when the function f(x) has singular points in interval [a, b]. Also for implementing the extended rational interpolation method, the direct method and the inverse differences method are presented, and with some examples these arguments are examined numerically.     

The maximal and minimal ranks of a quaternion matrix expression with applications

In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X₁, X₂) = C₇ ? A₄X₁B₄ ? A₅X₂B₅ where X₁, X₂ are variant quaternion matrices subject to quaternion matrix equations A₁X₁ = C₁, A₂X₁ = C₂, A₃X₁ = C₃, X₂B₁ = C₄, X₂B₂ = C₅, X₂B₃ = C₆. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper.     

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).     

Integrals,Quantum Galois Extensions,and the Affineness Criterion for Quantum Yetter–Drinfel'd Modules

In this paper we shall generalize the notion of an integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of an integral of a threetuple (H, A, C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ: C → Hom(C, A) of (H, A, C) if and only if any representation of (H, A, C) is injective in a functorial way, as a corepresentation of C. In particular, the quantum integrals associated to Yetter–Drinfel'd modules are defined. Let now A be an H-bicomodule algebra, ^H$\text{Y}$$\text{D}$_A the category of quantum Yetter–Drinfel'd modules, and B = {a ∈ A|∑S^− 1(a_〈1〉)a_{〈 − 1〉} ⊗ a_〈0〉 = 1_H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ∈ ^H$\text{Y}$$\text{D}$_A. We shall prove the following affineness criterion: if there exists γ: H → Hom(H, A) a total quantum integral and the canonical map β: A ⊗ _B A → H ⊗ A, β(a ⊗ _B b) = ∑S^− 1(b_〈1〉)b_{〈 − 1〉} ⊗ ab_〈0〉 is surjective (i.e., A/B is a quantum homogeneous space), then the induction functor – ⊗ _B A: $\text{M}$_B → ^H$\text{Y}$$\text{D}$_A is an equivalence of categories. The affineness criteria proven by Cline, Parshall, and Scott, and independently by Oberst (for affine algebraic groups schemes) and Schneider (in the noncommutative case), are recovered as special cases.     

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.     

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.     

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.     

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.     

Where Does Smoothness Count the Most for Two-Point Boundary-Value Problems?

We study the complexity of scalar 2mth order elliptic two-point boundary-value problems Lu=f, error being measured in the energy norm. Previous work on the complexity of these problems has generally assumed that we had partial information about the right-hand side f and complete information about the coefficients of L. In this paper, we study the complexity of such problems when, in addition to partial information about f, we have only partial information about the coefficients of L. More precisely, we suppose that f has r derivatives in the L_p-sense, with r⩾−m and p∈[2, ∞], and that L has the usual divergence form Lv=∑_{0⩽i, j⩽m} (−1)ⁱ Dⁱ(a_i, j D^jv), with a_i, j being r_i, j-times continuously differentiable, where r_i, j⩾0. We first suppose that continuous linear information is available. Let r=min{r, min_{0⩽i, j⩽m} {r_i, j−i}}. If r=−m, the problem is unsolvable; for r>−m, we find that the ε-complexity is proportional to (1/ε)^1/(r+m), and we show that a finite element method (FEM) is optimal. We next suppose that only standard information (consisting of function and/or derivative evaluations) is available. Let r_min=min{r, min_{0⩽i, j⩽m} {r_i, j}}. If r_min=0, the problem is unsolvable; for r_min>0, we find that the ε-complexity is proportional to (1/ε)^1/r_min, and we show that a modified FEM (which uses only function evaluations, and not derivatives) is optimal.     

Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current,ionslip, viscous and Joule heating effects: Network numerical solutions

A theoretical study of unsteady magnetohydrodynamic viscous Hartmann–Couette laminar flow and heat transfer in a Darcian porous medium intercalated between parallel plates, under a constant pressure gradient is presented. Viscous dissipation, Joule heating, Hall current and ionslip current effects are included as is lateral mass flux at both plates. The dimensionless conservation equations for the primary (x¹-direction), secondary (z¹-direction) momentum and also energy conservation equation are derived and solved using a computational technique known as Network Simulation Methodology (NSM). Velocity distributions (u¹, w¹) and temperature distribution (T¹) at the channel centre (y¹ = 0) over time (t¹) are studied graphically for the effects of Darcy number (Da), Hartmann number (Ha), transpiration (N_t), Hall current parameter (Be), ionslip parameter (Bi), pressure gradient parameter (dP/dx¹) with Prandtl number prescribed at 7.0 (electrically conducting water), Eckert number held constant at 0.25 (heat convection from the plates to the fluid) and Reynolds number (Re) fixed at 5.0 (for Re < 10, Darcian model is generally valid). Increasing Darcy number causes an increase in temperature, T¹; values are however significantly reduced for the higher Hartmann number case (Ha = 10). For the case of low transpiration (i.e. N_t = 1 which corresponds to weak suction at the upper plate and weak injection at the lower plate), both primary velocity (u¹) and secondary velocity (w¹) are increased with a rise in Darcy number (owing to a simultaneous decrease in Darcian porous drag); temperature T¹ is also increased considerably with increasing Da. However, for stronger transpiration (N_t = 10), magnitudes of u¹, w¹ and T¹ are significantly reduced and also significant overshoots are detected prior to the establishment of steady state flow. With increasing Hall current parameter, Be, (for the purely fluid regime i.e. Da → ∞), primary velocity is considerably increased, whereas secondary velocity is reduced; temperatures are decreased in the early stages of flow but effectively increased in the steady state with increasing Be. With strong Darcian drag present (Da = 0.01 i.e. very low permeability), magnitudes of u¹, w¹ and T¹ are considerably reduced and temperatures are found to be reduced for all t¹, with increasing Hall current effect (Be). Increasing ionslip current parameter (Bi) increases primary velocity (u¹), decreases secondary velocity (w¹) and also temperature (T¹) for all time (t¹), in the infinite permeability case (Da → ∞). For weakly Darcian flow, ionslip parameter (Bi) has a much reduced effect on the velocity distributions. Temperature, T¹ is strongly increased with a rise in pressure gradient parameter, dP¹/dx¹, as is primary velocity (u¹); however, secondary velocity (w¹) is reduced. The present study has applications in hybrid magnetohydrodynamic (MHD) energy generators, materials processing, geophysical hydromagnetics, etc.