Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

In this work we consider the following class of elliptic problems $\begin{cases} −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N), \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

Locally finite Lie algebras with root decomposition

Let K be an algebraically closed field of characteristic zero, $\frak {g}$ be a countably dimensional locally finite Lie algebra over K, and $\frak {h} \subset \frak {g}$ be a (a priori non-abelian) locally nilpotent subalgebra of $\frak {g}$ which coincides with its zero Fitting component. We classify all such pairs $(\frak {g}, \frak {h})$ under the assumptions that the locally solvable radical of $\frak {g}$ equals zero and that $\frak {g}$ admits a root decomposition with respect to $\frak {h}$. More precisely, we prove that $\frak {g}$ is the union of reductive subalgebras $\frak {g}_n$ such that the intersections $\frak {g}_n \cap \frak {h}$ are nested Cartan subalgebras of $\frak {g}_n$ with compatible root decompositions. This implies that $\frak {g}$ is root-reductive and that $\frak {h}$ is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra $\frak {h}$ of $\frak {g} = \frak {sl}(\infty)$ with respect to which $\frak {g}$ does not admit a root decomposition.Work Supported in Part by the University of Hamburg and the Max Planck Institute for Mathematics, Bonn     

与一类unit型相联系的有限维单李代数的结构

Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

The Algebra of Polynomial Integro-Differential Operators is a Holonomic Bimodule over the Subalgebra of Polynomial Differential Operators

In contrast to its subalgebra $A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle $ of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra ${\mathbb{I}}_n:=K\langle x_1, \ldots ,$ $ x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle $ of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that ${\mathbb{I}}_n$ is a left (right) coherent algebra iff n?=?1; the algebra ${\mathbb{I}}_n$ is a holonomic A _n -bimodule of length 3 ⁿ and has multiplicity 3 ⁿ with respect to the filtration of Bernstein, and all 3 ⁿ simple factors of ${\mathbb{I}}_n$ are pairwise non-isomorphic A _n -bimodules. The socle length of the A _n -bimodule ${\mathbb{I}}_n$ is n?+?1, the socle filtration is found, and the m’th term of the socle filtration has length ${n\choose m}2^{n-m}$ . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra ${\mathbb{I}}_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra ${\mathbb{I}}_n$ .     

关于全纯映照模的Schwarz引理一点注记

记DC为单位圆盘,B~k C~k为开欧氏单位球,Ω是C~k(或C)中的域.记H_n(D,Ω)为满足一定条件的全纯映照族(或函数族)的全体.作者证明了若,∈Hn(D,D),则|f′(z)|≤(n|z|~(n-1))/(1-|z|~(2n))(1-|f|(z|~2),z∈DD同时,对Hn(D,B~k)中映照的模也得到类似的结果.该结论推广了Pavlovic的相应结果.     

The coherence of complemented ideals in the space of real analytic functions

We characterize when an ideal of the algebra ${A(\mathbb{R}^d)}We characterize when an ideal of the algebra A(\mathbbR^d){A(\mathbb{R}^d)} of real analytic functions on \mathbbR^d{\mathbb{R}^d} which is determined by the germ at \mathbb R^d{\mathbb {R}^d} of a complex analytic set V is complemented under the assumption that either V is homogeneous or V?\mathbbR^d{V\cap \mathbb{R}^d} is compact. The characterization is given in terms of properties of the real singularities of V. In particular, for an arbitrary complex analytic variety V complementedness of the corresponding ideal in A(\mathbbR^d){A(\mathbb{R}^d)} implies that the real part of V is coherent. We also describe the closed ideals of A(\mathbbR^d){A(\mathbb{R}^d)} as sections of coherent sheaves.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains ${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$ generated by positive regular free holomorphic functions f and certain classes of n-tuples ${\varphi = (\varphi_1, \ldots, \varphi_n)}$ of formal power series in noncommutative indeterminates Z ₁, . . . , Z _n . Noncommutative Poisson transforms are employed to show that each abstract domain ${\mathbb{D}_f^\varphi}$ has a universal model consisting of multiplication operators (M _Z1, . . . , M _{Z n} ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M _Z1, . . . , M _{Z n} and show that all pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ are compressions of ${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$ to their coinvariant subspaces. We show that the eigenvectors of ${M_{Z_1}^*, \ldots, M_{Z_n}^*}$ are precisely the noncommutative Poisson kernels ${\Gamma_\lambda}$ associated with the elements ${\lambda}$ of the scalar domain ${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra ${H^\infty(\mathbb{D}_f^\varphi)}$ . We introduce the characteristic function of an n-tuple ${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$ , present a model for pure n-tuples of operators in the noncommutative domain ${\mathbb{D}_f^\varphi(\mathcal{H})}$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ .     

Maximal Graded Subalgebras of Loop Toroidal Lie Algebras

Let be a central simple Lie algebra over a field . We study the maximal ℤⁿ-graded subalgebra of . Mathematics Subject Classifications (2000) 17B70, 17B65, 17B67.     

Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.     

Explicit Matrix Realization of Clifford Algebras

In this paper, we construct associative subalgebras ${{L_{2}}{n}(\mathbb{R})}$ of the real ${2^{n} \times 2^{n}}$ matrix algebra ${{M_{2}}{n}(\mathbb{R})}$ , which is isomorphic to the real Clifford algebra ${C \ell_{0},n}$ for every ${n \in N}$ .     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Simplices with congruent k-faces

Let S be a non-degenerate simplex in $\mathbb{R}^{2}$. We prove that S is regular if, for some k $\in$ {1,...,n-2}, all its k-dimensional faces are congruent. On the other hand, there are non-regular simplices with the property that all their (n1)-dimensional faces are congruent.     

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S.?Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

Banach-valued holomorphic functions on the maximal ideal space of H ∞

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H ^∞ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H ^∞, prove that the maximal ideal space of the algebra $H_{\mathrm{comp}}^{\infty}(A)$ of holomorphic functions on $\mathbb{D}$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H ^∞ and A.