Number Of Nontrivial Solutions Research Articles

Counting certain quadratic partitions of zero modulo a prime number

Abstract Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

Generalized linking theorem with applications to nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.

Nonlinear planar secondary resonance analyses of suspended cables with thermal effects

This paper deals with a geometrically nonlinear model of suspended cables in thermal environments. In particular, the secondary resonant behaviors with thermal effects are investigated and discussed analytically and numerically. First, considering the influence of temperature changes, a dimensionless coefficient is generated. The nonlinear dynamic equations of motion in thermal environments are established, and the governing PDEs are reduced to a set of ODEs by using the Galerkin procedure. Four super and sub-harmonic resonant cases are studied, and frequency responses equations and steady-state solutions for each case are obtained. Extensive numerical results show that the hardening/softening behaviors, the response amplitudes, the number of nontrivial solutions, the range of resonant responses, the jump points, and the peak values are all closely connected to temperature. In some specific warming/cooling conditions, the nonlinear system just shows linear vibration behaviors or even just opposite spring characteristics. The time–displacement curves and phase portraits are extremely sensitive to temperature changes, but the number of cluster points in Poincaré sections seem to be independent of temperature changes.

Threshold functions for systems of equations on random sets

Abstract We present a unified framework to deal with threshold functions for the existence of certain combinatorial structures in random sets. More precisely, let M ⋅ x = 0 be a linear system defining a fixed structure (k-arithmetic progressions, k-sums, B h [ g ] sets or Hilbert cubes, for example), and A be a random set on 1 , … , n where each element is chosen independently with the same probability. We show that, under certain natural conditions, there exists a threshold function for the property “ A m contains a non-trivial solution of M ⋅ x = 0 ” which only depends on the dimensions of M. We study the behavior of the limiting distribution of the number of non-trivial solutions in the threshold scale, and show that it follows a Poisson distribution in terms of volumes of certain convex polytopes arising from the linear system under study.

The Equation in ℤ/pℤ

We present a simple and character-free approach to the determination of the number of nontrivial solutions of the equation in ℤ/pℤ, where m and k are integers not divisible by p.

Phase transitions in a slender cylinder composed of an incompressible elastic material. II. Analytical solutions for two boundary-value problems

In this sequel paper, we solve two boundary-value problems for the asymptotic model equation derived in the first paper. One of the purposes is to compare the analytical solutions with the experimental results obtained by a number of other authors. With the help of a phase-plane analysis on this model, first we manage to solve a force-controlled problem. We find that it is necessary to divide the external stress into seven intervals. Analytical solutions in all intervals are obtained. An interesting finding is that the value of the radius–length ratio has a great influence on the solutions. In particular, it influences the number of non-trivial solutions. In this regard, the potential application of this result to the cold drawing industry is pointed out. Our results also yield the possible nucleation stress which turns out to be dependent on the radius–length ratio a . A simple approximate formula for the nucleation stress is given, which shows that it is a decreasing function of a . This seems to capture the size effect observed in experiments: the smaller the diameter is, the larger the nucleation stress is. We also obtain the analytic solutions for a displacement-controlled problem with the help of those of a force-controlled problem. The engineering stress–strain curve plotted from the solution of the preferred configurations seems to capture the key features of the curve measured in a few experiments.

Equal sums of three powers

in a region maxxi ≤ B. Here d will be a fixed positive integer. There are 6B+O(B) trivial solutions in which x4, x5, x6 are a permutation of x1, x2, x3. We write N (0) d (B) for the number of non-trivial solutions, and our task is to estimate this quantity. When d = 2 it is a simple exercise with the circle method to show that Nd(B) ∼ cB for an appropriate positive constant c, so that the non-trivial solutions dominate the trivial ones. For larger values of d, it follows from Hua’s inequality [11] that

On a reaction-diffusion system involving the critical exponent

In this paper we study the existence and multiplycity of the nontrivial solutions for the following elliptic system with Dirichlet boundary conditions and critical nonlinearity { -D u = l u + W(x) u |u|2* - 2 -kv -D v = d u - g v u = v = 0 in W in W in ¶ W where WIRN (N ³ 3) is a bounded regular domain, W(×) I Lµ (W) with the property that there exists h >0 such that W(×) ³h a.e. in W and l,d,g are real parameters. We show that the number of nontrivial solutions, in a left neighbourhood of each lU j, j=1,2,..., is at least twice the multiplicity of lU j, where the set {lU j}j IN* represents the spectrum of a certain integro-differential operator.

Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

In this paper we study the existence of nontrivial solutions for the boundary value problem{−Δu−λu−u|u|2⁎−2=0inΩu=0on∂Ωwhen Ω⊂Rn is a bounded domain, n ⩾ 3, 2⁎=2n(n−2) is the critical exponent for the Sobolev embedding H01(Ω)⊂Lp(Ω), λ is a real parameter.We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λj⁎, λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ∈]λj⁎, λj[, the number of nontrivial solutions is at least twice the multiplicity of λj.The same kind of results holds also when Ω is a compact Riemannian manifold of dimension n ⩾ 3, without boundary and Δ is the relative Laplace-Beltrami operator.

Positive solutions of a non-linear eigenvalue problem with discontinuous non-linearity

SynopsisWe seek non-trivial solutions (u,λ)∈C1([0,1])×[0,∞ with u(x)≧0 for all x ∈[0,1], of the nonlinear eigenvalue problem –u″(x)=λf(u(x)) for x ∈ (0,1) and u(0)=u(1)=0,where f:[0,∞)→[0,∞) is such that f(p) = 0, for p ∈ [0,1), and f(p) = K(p), for p ∈ (1,∞), and K: [1, ∞)→(0, ∞) is assumed to be twice continuously differentiable. (The value ƒ(1) is only required to be positive.)Existence and multiplicity theorems are given in the cases where ƒ is asymptotically sub-linear and ƒ is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term ƒ we obtain the precise number of non-trivial solutions for given values of λ ∈ [0, ∞).