Stability of Taylor–Couette and Dean flow: A semi-analytical study

An alternative method is presented for solving the eigenvalue problem that governs the stability of Taylor–Couette and Dean flow. The eigenvalue problems defined by the two-point boundary value problems are converted into initial value problems by applying unit disturbance method developed by Harris and Reid [27] in 1964. Thereafter, the initial value problems are solved by differential transform method in series and the eigenvalues are computed by shooting technique. Critical wave number and Taylor number for Taylor–Couette flow are computed for a wide range of rotation ratio (μ), −4 ? μ ? 1 (first mode) and −2 ? μ ? 1 (second mode). The radial eigenfunction and cell patterns are presented for μ = −1, 0, 1. Also, we have computed critical wave number and Dean number successfully.     

Long range variations on the Fibonacci universal code

Fibonacci coding is based on Fibonacci numbers and was defined by Apostolico and Fraenkel (1987) [1]. Fibonacci numbers are generated by the recurrence relation Fi=Fi−1+Fi−2∀i?2 with initial terms F₀=1, F₁=1. Variations on the Fibonacci coding are used in source coding as well as in cryptography. In this paper, we have extended the table given by Thomas [8]. We have found that there is no Gopala-Hemachandra code for a particular positive integer n and for a particular value of a∈Z. We conclude that for n=1,2,3,4, Gopala-Hemachandra code exists for a=−2,−3,…,−20. Also, for 1?n?100, there is at most m consecutive not available (N/A) Gopala-Hemachandra code in GH−(4+m) column where 1?m?16. And, for 1?n?100, as m increases the availability of Gopala-Hemachandra code decreases in GH−(4+m) column where 1?m?16.     

Existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval

In this paper we consider the following m-point fractional boundary value problem with p-Laplacian operator on infinite interval where 0<????1, 2<????3, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, ?? _p (s)=|s| ^p?2 s,p>1, (?? _p )^?1=?? _q , $\frac{1}{p}+\frac{1}{q}=1$ . 0<?? ₁<?? ₂<?<?? _m?2<+??, ?? _i ??0, i=1,2,??,m?2 satisfies $0 <\sum_{i=1}^{m-2}\beta_{i}\xi_{i}^{\alpha-1} < \Gamma(\alpha)$ . We establish solvability of the above fractional boundary value problems by means of the properties of the Green function and some fixed-point theorems.     

Existence of positive solutions for the singular fractional differential equations

In this paper, we consider the following two-point fractional boundary value problem. We provide sufficient conditions for the existence of multiple positive solutions for the following boundary value problems that the nonlinear terms contain i-order derivative where n?1<α≤n is a real number, n is natural number and n≥2, α?i>1, i∈N and 0≤i≤n?1. ${}^{c}D_{0^{+}}^{\alpha}$ is the standard Caputo derivative. f(t,x ₀,x ₁,…,x _i ) may be singular at t=0.     

New ideas for decomposing nonlinearities in differential equations

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence {c_j} satisfying 1 = c₁ ? c₂ ? ? and j ? c_j for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910-955] are derived as particular cases.     

The period function of classical Liénard equations

In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree 2?−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ?, can be reduced to the study of slow-fast Liénard equations close to their limiting layer equations. We show that near the central system of degree 2?−1 the number of critical periods is at most 2?−2. We show the occurrence of slow-fast Liénard systems exhibiting 2?−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2?−2 is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter ? ? 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h_i = x_i+1 − x_i, with 0 = x₀ < x₁ < ? < x_N = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small ?, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L², for large as well as small values of N, and over a wide range of values of ?. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness.     

Sequentially two-leveled egalitarianism for TU games: Characterization and application

A new linear value for cooperative transferable utility games is introduced. The recursive definition of the new value for an n-person game involves a sequential process performed at n − 1 stages, applying the value to subgames with a certain size k,1 ? k < n, combining with the rule of two-leveled egalitarianism (additive normalization) in order to guarantee the efficiency property for the new value, sequentially two-leveled egalitarianism, shortly S₂EG value, applied to subgames of size k + 1. The new value will be characterized in various ways. The S₂EG value differs from the Shapley value since, besides efficiency, linearity, and symmetry, it verifies an additional property with respect to so-called scale-dummy player (replacing dummy player property). Consequently, the S₂EG value of a game may be determined as the solidarity value of the per-capita game (incorporating the proportional rule due to different levels of efficiency). Various potential representations of the new value are established. In the application to a land corn production economy, it yields allocations, in which the landlord’s interest coincides with striving for a maximum production level. For economies with the linear production function, not only the unique landlord but also all the workers have incentives to increase the scale of the economy.     

Diagonalization method for a vector boundary problem of singular perturbation type

The nonlinear boundary value problem ?y″ + f(t, ?, y, y′) = 0, y(0, ?) = α(?), y(1, ?) = β(?), where ? > 0 is a small parameter and y, f are scalar functions, has been studied extensively. However, for n-dimensional vector functions y, f the problem seems open. Here we study this vector boundary problem and obtain results which are analogous to those for the scalar case. The approach in this paper is to transform the appropriate differential equation into a canonical or diagonalized system of two first-order equations.     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

Some results on Korenblum's maximum principle

Let Ap(D) (1?p<∞) be the Bergman space over the open unit disk D in the complex plane. For p?1, let cp be the largest value of c for which Korenblum's maximum principle holds. In this paper we obtain a new lower bound on cp: cp?0.23917. We also improve the lower bound on c₂ up to 0.28185.     

Elliptic curve primality tests for Fermat and related primes

We use elliptic curves with complex multiplication to develop primality tests for Fermat primes and for primes of the form ?²³−²?−1³+1 and ?²²−²?−1²+1.     

Remark on an infinite semipositone problem with indefinite weight and falling zeros

In this work, we consider the positive solutions to the singular problem $$ \left\{\begin{array}{ll} -\Delta u = am(x)u-f(u) - \dfrac{c}{u^{\alpha}} & {\rm in}\;\Omega,\\ u=0 & {\rm on}\; \partial\Omega, \end{array} \right. $$ where 0?<?α?<?1,a?>?0 and c?>?0 are constants, Ω is a bounded domain with smooth boundary $\partial\Omega$ , Δ is a Laplacian operator, and $f:[0,\infty] \longrightarrow{\mathbb R}$ is a continuous function. The weight functions m(x) satisfies m(x)?∈?C(Ω) and m(x)?>?m ₀?>?0 for x?∈?Ω and also ||m||_?∞??=?l?<?∞. We assume that there exist A?>?0, M?>?0, p?>?1 such that alu???M?≤?f(u)?≤?Au ^p for all u?∈?[0,?∞?). We prove the existence of a positive solution via the method of sub-supersolutions when $m_{0}a>\frac{2\lambda_{1} }{1+\alpha}$ and c is small. Here λ ₁ is the first eigenvalue of operator ??Δ with Dirichlet boundary conditions.     

The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(N) → ∞ as N → + ∞. Let Nν_N(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

We consider the boundary value problems: (?p(x^′′(t)))+q(t)f(t,x(t),x(t−1),x^′(t))=0, ?p(s)=|s|^p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems.     

Optimal Lehmer Mean Bounds for the Toader Mean

We find the greatest value p and least value q such that the double inequality L _p (a, b)?<?T(a, b)?<?L _q (a, b) holds for all a, b?>?0 with a?≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here ${T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}$ and L _p (a, b)?=?(a ^p+1?+?b ^p+1)/(a ^p ?+?b ^p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively.     

New bounds on binary identifying codes

The original motivation for identifying codes comes from fault diagnosis in multiprocessor systems. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.In this paper, we concentrate on identification in binary Hamming spaces. We give a new lower bound on the cardinality of r-identifying codes when r≥2. Moreover, by a computational method, we show that M₁(6)=19. It is also shown, using a non-constructive approach, that there exist asymptotically good (r,≤?)-identifying codes for fixed ?≥2. In order to construct (r,≤?)-identifying codes, we prove that a direct sum of r codes that are (1,≤?)-identifying is an (r,≤?)-identifying code for ?≥2.