Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

On the duality of generalized Lie and Hopf algebras

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H  , there is a natural isomorphism of Lie algebras Q(H)^?≅P(H^°)$Q{(H)}^{?}\cong P(H^{°})$, where Q(H)^?$Q{(H)}^{?}$ is the dual Lie algebra of the Lie coalgebra of indecomposables of H  , and P(H^°)$P(H^{°})$ is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras.     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

The Drazin inverse of a singular,unreduced tridiagonal matrix

For a tridiagonal, singular matrix A   we present a method for the computation of the polynomial p(λ)$p(\lambda )$ such that A^D=p(A)$A^D=p(A)$ holds, where A^D$A^D$ is the Drazin inverse of A. The approach is based on the recursion of characteristic polynomials of leading principal submatrices of A.     

Decay rate estimations for linear quadratic optimal regulators

Let u(t)=−Fx(t)$u(t)=-Fx(t)$ be the optimal control of the open-loop system x^′(t)=Ax(t)+Bu(t)$x^{\prime }(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system x^′(t)=(A−BF)x(t)$x^{\prime }(t)=(A-BF)x(t)$. Main attention is given to the case of a skew-Hermitian matrix A. Given an operator A, for a class of cases, we find a matrix B that provides an almost optimal decay rate.     

Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}$\mathrm{Int}(A)=\{f\in B[x]|f(A)\subseteq A\}$. Restricting the coefficients to elements of k  , we obtain the commutative ring _Intk(A)={f∈k[x]|f(A)⊆A}${\mathrm{Int}}_k(A)=\{f\in k[x]|f(A)\subseteq A\}$; this makes Int(A)$\mathrm{Int}(A)$ a left _Intk(A)${\mathrm{Int}}_k(A)$-module. Previous researchers have noted instances when a D-module basis for A   is also an _Intk(A)${\mathrm{Int}}_k(A)$-basis for Int(A)$\mathrm{Int}(A)$. We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)$\mathrm{Int}(A)$, its localizations at primes of D, and finite residue rings of A.     

Orthogonally additive polynomials on Fourier algebras

We show that an n-homogeneous polynomial P   on the Fourier algebra A(G)$A(G)$ of a locally compact group G   can be represented in the form P(f)=〈T,fⁿ〉$P(f)=〈T,f^n〉$(f∈A(G))$(f\in A(G))$ for some T   in the group von Neumann algebra VN(G)$\mathit{VN}(G)$ of G if and only if it is orthogonally additive and completely bounded.     

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are _A∞$A_{\infty }$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg?(_∂φ(f(z))/z)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.     

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.     

The classification of Leonard triples of Racah type

Let K$\mathbb{K}$ denote an algebraically closed field of characteristic zero. Let V   denote a vector space over K$\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A^?$A^{?}$, A^ε$A^{\epsilon }$ in End(V)$\mathrm{End}(V)$ such that for each B∈{A,A^?,A^ε}$B\in \{A,A^{?},A^{\epsilon }\}$ there exists a basis for V with respect to which the matrix representing B   is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the _Z3${\mathbb{Z}}_3$-symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra U(_sl2)$U({\mathit{sl}}_2)$.