An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

ON EXISTENCE OF SOLUTIONS OF DIFFERENCE RICCATI EQUATION

Consider the difference Riccati equation f(z+1) =(A(z)f(z)+B(z))/(C(z)f(z)+D(z)),where A,B, C,D are meromorphic functions, we give its solution family with one-parameter H(f(z))={f_0(z),f(z)=((f_1(z)-f_0(z))(f_2(z)-f_0(z)))/(Q(z)(f_2(z)-f_1(z))+(f_2(z)-f_0(z)))}, where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f_0(z),f_1(z),f_2(z) are its three distinct meromorphic solutions.     

Hlawka’s functional inequality

The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality) $$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$ for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality: $$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$ which in particular holds true for all x, y, z from a real or complex inner product space.     

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].     

A symmetry-breaking problem in the annulus

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument     

Kernel of the second order Cauchy difference on groups

Let (G, ·) be a group, (H, +) be an abelian group, and ${f:G\rightarrow H}$ . The second order Cauchy difference of f is $$C^{(2)}f(x,y,z)=f(xyz)-f(xy)-f(yz)-f(xz)+f(x)+f(y)+f(z).$$ The functional equation $$C^{(2)}f(x,y,z)=0$$ is studied. We present its general solution on free groups. Solutions on other selected groups are also given.     

Sine-Gordon equation on the semi-axis

We investigate the sine-Gordon equation u_tt?u_xx+sinu=0 on the semi-axis x>0. We show that boundary conditions of the forms u_x(0,t)=c₁ cos(u(0,t)/2)+c₂ sin(u(0,t)/2) and u(0,t)=c are compatible with the Bäcklund transformation. We construct a multisoliton solution satisfying these boundary conditions.     

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

Sharp estimates for operator\bar \partial _M on aq-concave CR manifoldon aq-concave CR manifold

We construct integral operatorsR _r andH _r on the spaces of differential forms of the type (o, r) withr <q on a regularq-concave CR manifoldM such that $$f(z) = \bar \partial _M R_r (f)(z) + R_{r + 1} (\bar \partial _M f)(z) + H_r (f)(z),$$ for a differential formf ∈ L _(0,r) ^s (M) and forz ∈ M′ ?M, whereH _r is compact andR _r admits sharp estimates.     

Computation of multiple eigenvalues of infinite tridiagonal matrices

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of ; (B) the zeros of ; (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their ``double pairs'.

     

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .     

A remark concerning Pincherle bases

In this note we find sufficient conditions for uniqueness of expansion of any two functionsf(z) and g(z) which are analytic in the circle ¦ z ¦ < R (0 < R <∞) in series $$f(z) = \sum\nolimits_{n = 0}^\infty {(a_n f_2 (z) + b_n g_n (z))}$$ and $$g_i (z) = \sum\nolimits_{n = 0}^\infty {a_n \lambda _n f_n (z)} + b_n \mu _n f_n (x)),$$ which are convergent in the compact topology, where (f _n {z} _n=0 ^∞ and {g} _n=0 ^∞ are given sequences of functions which are analytic in the same circle while {λ _n } _n=0 ^∞ and {μ _n } _n=0 ^∞ are fixed sequences of complex numbers. The assertion obtained here complements a previously known result of M. G. Khaplanov and Kh. R. Rakhmatov.     

Bounds for Zeros of Solutions of Second Order Differential Equations with Polynomial Coefficients

We consider the equation ${u''=P(z)u\;\;(z\in\mathbb{C})}$ where P(z) is a polynomial. Let z _k (u), k = 1, 2,... be the zeros of a solution u(z) to that equation. Bounds for the sums $$\sum_{k=1}^{j} \frac {1} {|z_k(u)|}\;(j=1, 2, \ldots)$$ are established. Some applications of these bounds are also considered.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Global Solutions for Nonlinear Delay Evolution Inclusions with Nonlocal Initial Conditions

We prove a sufficient condition for the existence of global C ⁰-solutions for a class of nonlinear functional differential evolution equation of the form $ \left\{{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \right. $ \left\{\begin{array}{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \end{array}\right.     

一类无穷下级整函数的Julia集的径向分布

主要研究方程f"(z)+A(z)f'(z)+B(z)f(z)=0(A(z)),B(z)为整函数)的解、解的多项式或微分多项式这些具有无穷下级的整函数的Julia集的径向分布问题.     

The Tjon-Wu equation in Banach space settings

The authors investigate the Tjon-Wu (TW) equation: (TW)

$\text{?u}\text{?t}\text{(t, x) + u(t, x) = ∫}{}_{\mathrm{x}}{}^{\mathrm{\infty }}\text{dy}\text{y}\text{∫}{}_0{}^{\mathrm{y}}\text{u(t, y ? z) u(t, z)dz, u(0, x) = u}{}_0\text{(x)}$

, which has been obtained from a classical Boltzmann equation by applying the Abel transform. (TW) is considered as an ordinary differential equation first in the space L²={u:[0,∞)→R|∫_x^∞|u(x)|²e^xdx < + ∞}The authors establish existence and uniqueness of solutions in disks of codimension 2 around 0 and around e^?x. Asymptotic stability of these latter functions is also established. The basic tool is an unusual eigenvalue property of the nonlinear right-hand side of (TW) which leads to a reformulation of (TW) as a differential equation in l². Similar results are established in L¹ working with (TW) directly.     

Bounded mild solutions to fractional integro-differential equations in Banach spaces

We study the existence and uniqueness of bounded solutions for the semilinear fractional differential equation $$D^\alpha u(t)= Au(t)+ \int_{-\infty}^t a(t-s)Au(s)ds+ f \bigl(t,u(t) \bigr), \quad t \in\mathbb{R}, $$ where A is a closed linear operator defined on a Banach space X, α>0, a∈L ¹(?₊) is a scalar-valued kernel and f:?×X→X satisfies some Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other.