Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Equilateral Sets in <Emphasis Type="Italic">l</Emphasis><Stack><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript></Stack>

We show that for every odd integer p 1 there is an absolute positive constantc_p, so that the maximum cardinality of a set of vectors in Rⁿ such that the l_p distance between any pair is precisely 1, is at most c_p n log n. We prove some upper bounds for other l_p norms as well.     

Maximal subgroups of <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">D</Emphasis>) with finite conjugacy classes

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D ^* is an FC-group, then D is a field. Let M be a maximal subgroup of GL _n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M _n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)] < ∞.     

Relative <Emphasis Type="Italic">FI</Emphasis>-injective and <Emphasis Type="Italic">FI</Emphasis>-flat modules

Let R be a ring, n a fixed nonnegative integer and FP _n (F _n ) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext ¹(N,M) = 0 (resp., Tor ₁(F,N) = 0) for any N ∈ FP _n . It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FP _n -precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F _n -preenvelope K → F of a right R-module K with F projective if and only if M ∈^⊥ F _n . These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

On <Emphasis Type="Italic">CEP</Emphasis>-subgroups of <Emphasis Type="Italic">n</Emphasis>-periodic products

There is a well-known fact, that any group G ₁ is a CEP-subgroup both for the direct product G ₁ × G ₂ and the free productG ₁ * G ₂ of G ₁ with any group G ₂. The paper gives a necessary and sufficient condition providing that a multiplier G _i of a n-periodic product Π _i∈I ⁿ G _i of any family of groups {G _i } _i∈I is a CEP-subgroup. Particularly, the found criterionmeans that any group G ₁ of odd period n ≥ 665 is a CEP-subgroup of the n-periodic product Π _i∈I ⁿ G _i for any group G ₂.     

Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L _p -spaces of vector functions on \mathbbRⁿ {\mathbb{R}^n} (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> integrability of trigonometric series with special varying coefficients

Very recently, Yu, Le and Zhou introduced the so called △B1^＊ and △B2^＊ conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.     

A new characterization of the simple group <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

The simple group A ₁(p ⁿ ) is proved to be uniquely determined by the set of the orders of the maximal abelian subgroups of A ₁(p ⁿ ).     

On approximation of functions by trigonometric polynomials with incomplete spectrum in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, 0 < <Emphasis Type="Italic">p</Emphasis> < 1

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L _p , 0 < p <1, by trigonometric polynomials that are constructed by the system {e^ikx}_{k ? \mathbbZ\B} \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} . Bibliography: 13 titles.     

Intersections of Primary Subgroups in Nonsoluble Finite Groups Isomorphic to <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

Given a finite group G with socle isomorphic to L _n (2 ^m ), we describe (up to conjugacy) all ordered pairs of primary subgroups A and B in G such that A ∩ B ^g ≠ 1 for all g ∈ g.     

On Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Sets,Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Functions and <Emphasis Type="Italic">E</Emphasis>-Epigraphs

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R ⁿ is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.     

Factorization of the r-matrix for the quantum algebra <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

We propose a method for construction of the general solution of the Yang–Baxter equation with the U _q (sℓ _n ) symmetry algebra. This method is based on the factorization property of the corresponding L-operator. We present a closed-form expression for the universal R-matrix in the form of a difference operator acting on the space of functions of n(n − 1) variables. Bibliography: 16 titles.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.     

Representations of algebraic groups of type <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> with small weight multiplicities

Lower estimates for the maximal weight multiplicities in irreducible representations of the algebraic groups of type C _n in characteristic p ≤ 7 are found. If n ≥ 8 and p ≠ 2 , then for an irreducible representation either such a multiplicity is at least n− 4 − [n]₄,where [n]₄ is the residue of n modulo 4, or all the weight multiplicities are equal to 1.For p = 2, the situation is more complicated, and for every n and l there exists a class of representations with the maximal weight multiplicity equal to 2 ^l . For symplectic groups in characteristic p > 7 and spinor groups similar results were obtained earlier. Bibliography: 15 titles.     

On Stability of Cones in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript> with Zero Scalar Curvature

In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rⁿ⁺¹. If Mⁿ⁻¹ is a compact hypersurface of the sphere Sⁿ(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an ε for which the truncate cone C(M)_ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mⁿ⁻¹ of the sphere with zero scalar curvature and S₃ different from zero, for which all truncated cones based on M are stable. Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70.     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.