Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ? and α_j, β_j: ? → ? are described.     

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.     

Boundary blow-up solutions of <Emphasis Type="Italic">p</Emphasis>-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ _p u = a(x)u ^m ?b(x)f(u) with p >  1 and 0 <  m <  p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Oscillatory hyper Hilbert transforms along general curves

We consider the oscillatory hyper Hilbert transform H _γ,α,β f(x) = ∫ ₀ ^∞ f(x - Γ(t))e^it-β t^-(1+α)dt; where Γ(t) = (t, γ(t)) in ?² is a general curve. When γ is convex, we give a simple condition on γ such that H _γ,α,β is bounded on L ² when β > 3α, β > 0: As a corollary, under this condition, we obtain the L ^p -boundedness of H _γ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H _γ,α,β is bounded on L ²: As an application, we construct a class of strictly convex curves along which H _γ,α,β is bounded on L ² only if β > 2α > 0.     

On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λ_j,φ_j} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u ³, u ², (u - p), u ²(u ² - p), are completed and some conjectures are presented.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Classification and nondegeneracy of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>+1) Toda system with singular sources

We consider the following Toda system

Open image in new window

where γ _i >?1, δ ₀ is Dirac measure at 0, and the coefficients a _ij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:

$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$

This generalizes the classification result by Jost and Wang for γ _i =0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γ _i +γ _i+1+?+γ _j ?? for all 1≤i≤j≤n, then any solution u _i is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.

     

Minimum Difference Representations of Graphs

Define a k-minimum-difference-representation (k-MDR) of a graph G to be a family of sets \({\{S(v): v\in V(G)\}}\) such that u and v are adjacent in G if and only if min{|S(u)?S(v)|, |S(v)?S(u)|} ≥ k. Define ρ _min(G) to be the smallest k for which G has a k-MDR. In this note, we show that {ρ _min(G)} is unbounded. In particular, we prove that for every k there is an n ₀ such that for n > n ₀ ‘almost all’ graphs of order n satisfy ρ _min(G) > k. As our main tool, we prove a Ramsey-type result on traces of hypergraphs.     

Many parameter Hölder perturbation of unbounded operators

If \({u \mapsto A(u)}\) is a C ^0,α -mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parameterized by u in an (even infinite dimensional) space, then any continuous (in u) arrangement of the eigenvalues of A(u) is indeed C ^0,α in u.     

Search-extension method for finding multiple solutions of nonlinear problem

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) = 0 in Ω, u = 0 on (?)Ωare approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u3 in a square and L-shape domain are presented. The results show that there exist at least 3k - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -Δ(Conjecture 1).     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

Principal Functions for Bi-free Central Limit Distributions

We find the principal function of the completely non-normal operator l(v₁) + l(v₁)* + i(r(v₂) + r(v₂)*) on a subspace of the full Fock space \({\mathcal{F}}({\mathcal{H}})\) which arises from a bi-free central limit distribution. As an application, we find the essential spectrum of this operator.