All square chiliagonal numbers

A square chiliagonal number is a number which is simultaneously a chiliagonal number and a perfect square (just as the well-known square triangular number is both triangular and square). In this work, we determine which of the chiliagonal numbers are perfect squares and provide the indices of the corresponding chiliagonal numbers and square numbers. The study revealed that the determination of square chiliagonal numbers naturally leads to a generalized Pell equation x² ? Dy² = N with D = 1996 and N = 996², and has six fundamental solutions out of which only three yielded integer values for use as indices of chiliagonal numbers. The crossing/independent recurrence relations satisfied by each class of indices of the corresponding chiliagonal numbers and square numbers are obtained. Finally, the generating functions serve as a clothesline to hang up the indices of the corresponding chiliagonal numbers and square numbers for easy display and this was used to obtain the first few sequence of square chiliagonal numbers.     

Analogues to Fermat primes related to Pell��s equation

The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² ? 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell??s equation x ² ? dy ² = 1, given by ${x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}The coefficients of some weight 3 modular forms give reason to study primes of the form p = 2x ² − 1 = 2dy ² + 1. If x _a , y _a are the positive solutions of Pell’s equation x ² − dy ² = 1, given by x_a + y_a ?d = (x₁ + y₁ ?d)^a{x_a + y_a \sqrt{d} = (x_1 + y_1 \sqrt{d})^a}, and if p_a = 2 x_a² - 1{p_a = 2 x_a^2 - 1} is prime, then a = 2 ^m is a power of 2. So there are analogues to the Fermat numbers 2 ^a + 1.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

Asymptotic analysis of differential-delay equations and nonuniqueness of periodic solutions

We study the differential-delay equation x′(t) = ?αf(x(t–1)), where α is a positive parameter and f is an odd function which decays like x^?r at infinity. In particular, we consider the case r ? 2, and prove the existence of periodic solutions with special symmetries which are different from previously known periodic solutions of minimal period 4. For r = 2 we prove sharp asymptotic estimates for the minimal periods of these solutions. Our results disprove a conjecture of R. D. Nussbaum.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation \frac?u?t = \mathop?ⁿ_i=1\frac?²?x²_iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u, but much less on those of the Hermite heat equation \frac?U?t = \mathop?ⁿ_i=1(\frac?²?x²_i - x²_i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

On the Solutions of a Second Order Differential Equation Arising in the Theory of Resonant Oscillations in a Tank

The properties of the solutions of the differential equation y″ = y² ? x² ? c subject to the condition that y is bounded for all finite x discussed. The arguments of Holmes and Spence have been used by Ockendon, Ockendon, and Johnson to show that there are no solutions if c is large and negative. Numerically we find that solutions exist provided c is greater than a critical value c* and estimate this value to be c* = ?…. As x tends to + ∞ the solutions are asymptotic to . The relation between A₊ and ?₊ are found analytically as A₊ → ∞. This problem arises as a connection problem in the theory of resonant oscillations of water waves.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Size of regulators and consecutive square–free numbers

We prove that, for almost all D, the fundamental solution ${x_{0}+y_{0}\sqrt D}$ of the associated Pell equation x ²?Dy ² = 1 is greater than D ^1.749···. We also show a strong link between this question and the error term in the asymptotic formula of the number of pairs of consecutive square-free numbers.     

Incompressible navier-stokes and euler limits of the boltzmann equation

We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?^N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?^N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t₀, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t₀ which are close, to O(?²) in a suitable norm, to the local Maxwellian [p/(2πT)^d/2]exp{?[v ? ?u(x,t)]²/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.     

Boundary-value problems for the wave equation with Lévy Laplacian in the Gâteaux class

We present the solutions of the initial-value problem in the entire space and the solutions of the boundary-value and initial-boundary-value problems for the wave equation

Quadratic invariant curves for a planar mapping

A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a planar mapping admitting quadratic invariant curves y=x ²+C, except for the case -3/4≥C≤0. This remaining case is now resolved, and we describe the solutions of the functional equation K(x ²+C)+k(x)=x by iterations of y.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Asymptotic expansions of the pressure and the free boundary for flows through porous media

In this paper we discuss the free boundary value problem for the pressure head u of a compressible fluid flowing through a homogeneous porous medium. This process is governed by the partial differential equation ∈?_tu-? _x ² u=0, where ∈ is proportional to the compressibility of the fluid. We shall show that the pressure as well as the free boundary converge to the corresponding stationary solutions when ∈ tends to zero and shall furthermore estimate the error in terms of powers of ∈. Roughly speaking in the case of water, for example, this means that if we neglect its compressibility, which indeed is very small, we can estimate the error.     

On –δu = K(x)u5 in ℝ3

In this paper, we study the equation –Δu = K(x)u⁵ in ?³ and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|^?1. © 1993 John Wiley & Sons, Inc.     

On Diophantine equation 3a 2 x 4 − By 2 = 1

Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ ? 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field ${{\rm Q}(\sqrt{1-3a^{2}})}Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ − 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field Q(?{1-3a²}){{\rm Q}(\sqrt{1-3a^{2}})} is not divisible by 2, the equation 3a ² x ⁴ − By ² = 1 has at most two solutions. However, both solutions occur in only one case, a = 1, b = 2, as solved by Bumby. The proof utilizes the law of quadratic reciprocity that seems very rare in solving Diophantine equations, and the solution will be also obtained effectively through the proof when it exists.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

Time-dependent gradient perturbations of fractional Laplacian

We construct a fundamental solution of the equation ${\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0}We construct a fundamental solution of the equation ?_t - D^a/2 - b(·, ·) ·?_x = 0{\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0} for a ? (1, 2){\alpha \in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.     

Pseudospherical Surfaces and Evolution Equations

We consider evolution equations, mainly of type u_t = F(u, u_x,..., ?^ku/?x^k), which describe pseudo-spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.