Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext¹-groups. A stronger condition requiring the surjectivity of the induced maps between Ext ^k -groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras. Research partially supported by NSERC of Canada and by Hungarian National Foundation for Scientific Research grant no. 1903 Research partially supported by NSERC of Canada     

Mappings of preserving <Emphasis Type="Italic">n</Emphasis>-distance one in <Emphasis Type="Italic">n</Emphasis>-normed spaces

We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\). Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.     

Convolution operators in A ?∞ for convex domains

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ ⁿ and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.     

Linear maps preserving the minimum and reduced minimum moduli

We describe linear maps from a C^∗-algebra onto another one preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the reduced minimum modulus.     

On surjective quadratic mappings

In the paper, quadratic mappings acting from one finite-dimensional space to another are studied. Sufficient conditions for the stable surjectivity of a quadratic surjective mapping (i.e., for the condition that every quadratic mapping sufficiently close to a given one is also surjective) are obtained. The existence problem for nontrivial zeros of a surjective quadratic mapping acting from Rⁿ to Rⁿ is studied. For n = 3, the absence of these zeros is proved.     

Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator S(· + p) + T(·), where p ɛ X and S and T are maximal monotone operators on the reflexive Banach space X. Then, this is used to obtain sufficient conditions for the surjectivity of S + T and for the situation when 0 belongs to the range of S + T. Several special cases are discussed, some of them delivering interesting byproducts.     

Surjectivity of partial differential operators with good fundamental solutions

We characterize the surjectivity of a linear partial differential operator with constant coefficients on E(Ω) as well as on D^′(Ω) in terms of the existence of “good” shifted fundamental solutions. This characterization complements results of Meise, Taylor, and Vogt as well as Frerick and the present author.     

Traces of Sobolev functions with one square integrable directional derivative

We consider the Sobolev spaces of square integrable functions v, from ?ⁿ or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑?_??v. In these spaces we define closed trace operators on the boundaries ?Q and on the hyperplanes {r_?? = z}, z ∈ ?\{0}, which turn out to be possibly unbounded with respect to the usual L²‐norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L², we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ?ⁿ and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.     

Extension of smooth functions from finitely connected planar domains

Consider the Sobolev space W _∞ ^k (Ω) of functions with bounded kth derivatives defined in a planar domain. We study the problem of extendability of functions from W _∞ ^k (Ω) to the whole ℝ² with preservation of class, i.e., surjectivity of the restriction operator W _∞ ^k (ℝ²) → W _∞ ^k (Ω).     

Morita Theory for Comodules Over Corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End ^𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.     

Lattices of partially local fitting classes

This article deals only with finite groups. We prove the surjectivity of the mapping from the lattice of all normal Fitting classes into the lattice of the Lockett section generated by the Fitting classes that are not Lockett classes. Moreover, we find a sufficient surjectivity condition for the mapping of the lattice of the Lockett section generated by arbitrary Fitting classes into the lattice of the Lockett section generated by ω-local Fitting classes. This confirms Lockett’s conjecture for the ω-local Fitting classes of a given characteristic.     

Spectre de l'opérateur de transfert en dimension 1

We study spectral properties of a transfer operator ℳΦ(x)=∑_ω g _ω(x)Φ(ψ_ω x) acting on functions of bounded variation. Using a symmetrical integral, we first obtain bounds on its spectral and essential spectral radii. We then consider the dynamical determinant Det^#(Id +zℳ). Our main theorem generalizes to discontinuous weights the result of Baladi and Ruelle (for continuous weights) on the link between zeroes of the sharp determinant and eigenvalues of the transfer operator. The proof is based on regularizing the weights and uses a (new) spectral result giving the surjectivity of some applications between eigenspaces of operators. Received: 8 May 2001     

Surjectivity of power maps of real algebraic groups

In this paper we study the surjectivity of the power maps g?gn for real points of algebraic groups defined over reals. The results are also applied to study the exponentiality of such groups.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

We consider the Klein?CGordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential that is constant but different on each branch. The corresponding spatial operator is self-adjoint, and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier-type inversion formula in terms of an expansion in generalized eigenfunctions. Further, we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore, the approach via the Sturm?CLiouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the generalized eigenfunctions that incorporate the tunnel effect is justified for example by the perspective to study the influence of this effect on the L ^??-time decay of solutions.     

Decision algorithms for elementary topology I. Topological syllogistics with set and map constructs,connectedness, and cardinality comparison

In this paper we give a decision procedure for a topological syllogistic, denoted ?²,∞, involving, besides the standard Boolean set operators (∩, ∩, ′ (set complementation)) and predicates ( ? (membership), = (equality)), also the Kuratowski closure operator, continuous and closed maps with the notion of direct and inverse images, surjectivity and injectivity, as well as point evaluation, the singleton operator {·}, a predicate for comparing the cardinalities of two given sets, and a predicate expressing that a given set is topologically connected. Several examples are also discussed. © 1994 John Wiley & Sons, Inc.     

SOME RESULTS ABOUT BEST COAPPROXIMATION IN L^P（S,X）

As a counterpart to best approximation in normed linear spaces, the best coapproximation was introduced by Franchetti and Furi. In this paper, we shall consider the relation between coproximinality M in X and L ^p (S,M) in L ^p (S,X). Finally we give some results in cochebyshev subspaces and additional subspaces.     

The picture of a specific maximal monotone set

Let E be a nontrivial Banach space. The concept of picture has been used to provide a new proof of the surjectivity of S+J, for E reflexive and S: E2 ^{E *} maximal monotone. It is known that if E is reflexive, then the picture of a maximal monotone subset of E×E ^* is a singleton. We calculate an example showing that in the nonreflexive case, the picture of a maximal monotone subset can be quite substantial.