On the homotopy fixed point sets of circle actions on product spaces

Archiv der Mathematik - For arbitrary $$S^{1}$$ -actions on $$S^{m}_{{\mathbb {Q}}}$$ , $$S^{n}_{{\mathbb {Q}}}$$ , and $$S^{m}_{{\mathbb {Q}}}\times S^{n}_{{\mathbb {Q}}}$$ , we show the...     

A note on a functional equation on groups with involutions in two variables

Aequationes mathematicae - In this paper, we determine the central solutions $$f:G\times G\rightarrow {{\mathbb {C}}}$$ of the functional equation $$\begin{aligned} f(x_1\sigma y_1,x_2\tau...     

Fundamental Limits of Weak Recovery with Applications to Phase Retrieval

Foundations of Computational Mathematics - In phase retrieval, we want to recover an unknown signal $${{\varvec{x}}}\in {{\mathbb {C}}}^d$$ from n quadratic measurements of the form $$y_i =...     

Semilinear subelliptic problems with critical growth on Carnot groups

We prove existence and multiplicity of solutions for the semilinear subelliptic problem with critical growth in Ω, u = 0 on ∂Ω, where is a sublaplacian on a Carnot group , 2* = 2Q/(Q − 2) is the critical Sobolev exponent for and Ω is a bounded domain of .     

Almost complex structures on connected sums of almost complex manifolds and complex projective spaces

Archiv der Mathematik - For a smooth curve C in $${{\mathbb {P}}}^{r_0}$$ lying on a rational surface scroll, we try to identify those complete and base point free linear series of small degree...     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Uniqueness of minimal surfaces whose boundary is a horizontal graph and some Bernstein problems in {{\mathbb{H}^{2}\times \mathbb{R}}}

We deduce that a connected compact immersed minimal surface in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ whose boundary has an injective horizontal projection on an admissible convex curve in ${\partial_\infty{\mathbb{H}^{2}\times \mathbb{R}}}$ , and satisfies an admissible bounded slope condition, is the Morrey’s solution of the Plateau problem and is a horizontal minimal graph. We prove that there is no entire horizontal minimal graph in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ .     

A Kato class for the Khon Laplacian

Positivity - In this paper we establish an upper estimate and a 3G-theorem for the Green function of the Khon Laplacian $$\Delta _{{\mathbb {H}}}$$ on a domain D of the Heisenberg group $${{\mathbb...     

Weighted Composition Operators from the Minimal Möbius Invariant Space into the Bloch Space

Let ${{\varphi}}$ be an analytic self-map of the open unit disk ${{\mathbb{D}}}$ in the complex plane ${{\mathbb{C}, H(\mathbb{D})}}$ the space of complex-valued analytic functions on ${{\mathbb{D}}}$ , and let u be a fixed function in ${{H(\mathbb{D})}}$ . The weighted composition operator is defined by $$(uC_{\varphi}f)(z) = u(z)f({\varphi}(z)), \quad z \in \mathbb{D}, f \in H(\mathbb{D}).$$ In this paper, we study the boundedness and the compactness of the weighted composition operators from the minimal Möbius invariant space into the Bloch space and the little Bloch space.     

Sequences of Commutator Operations

Given the congruence lattice ${{\mathbb{L}}}$ of a finite algebra A with a Mal’cev term, we look for those sequences of operations on ${{\mathbb{L}}}$ that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so far delimit the number of such sequences: the number is always at most countably infinite; if it is infinite, then ${{\mathbb{L}}}$ is the union of two proper subintervals with nonempty intersection.     

Weighted Multi-Parameter Mixed Hardy Spaces and Their Applications

Czechoslovak Mathematical Journal - Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space $$H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times...     

On Optimal 4-Dimensional Metrics

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics. Supported in part by NSF grant DMS-0604735.     

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain we prove that multipole pluricomplex Green functions are dense in the cone in of negative plurisubharmonic functions with zero boundary values.

     

On special rational curves in Grassmannians

Numerical and geometric characterizations, among all morphisms , of those which are -equivalent to the canonical morphism induced by the Morita equivalence –, are presented. The author was partially supported by KBN grants 1P03A 036 26 and 115/E-343/SPB/6.PR UE/DIE 50/2005-2008. Received: 10 September 2005     

Existence,uniqueness and regularity of the projection onto differentiable manifolds

Annals of Global Analysis and Geometry - We investigate the maximal open domain $${\mathscr {E}}(M)$$ on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ can be...     

The distribution of quadratic residues and nonresidues in arithmetic progressions

We prove some results concerning the distribution of quadratic residues and nonresidues in arithmetic progressions in the setting \( {{\mathbb{F}}_p}={{\mathbb{Z}} \left/ {{p\mathbb{Z}}} \right.} \) , where p is a large prime.     

Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Simultaneous diophantine approximation in the real and <Emphasis Type="BoldItalic">p</Emphasis>-adic fields with nonmonotonic error function<Superscript>*</Superscript>

In this paper, we show that if the volume sum \( \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} \) converges for a function ψ (not necessarily monotonic), then the set of points \( \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} \) that simultaneously satisfy the inequalities \( \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 \), for infinitely many integer polynomials P has measure zero.