On the adjunction process over a surface in char.p

Let K_s be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P²,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then

1. (K_s?L)² is spanned at each point by global sections. Let \(\phi :S \to P^N _F \) be the map given by the sections Γ(K_s?L)², and let φ=s o r its Stein factorization.
2. r:S→S′=r(S) is the contraction of a finite number of lines, E_i for i=1,...r, such that E_i·E_i=K_S·E_i=?L·E_i=?1.
3. If h°(L)≥6 and L·L≥9, then s is an embedding.

     

New families of graceful banana trees

Consider a family of stars. Take a new vertex. Join one end-vertex of each star to this new vertex. The tree so obtained is known as abanana tree. It is proved that the banana trees corresponding to the family of stars

1. (K_1,1, K_1,2,…, K_{1,t ?1}, (α + l) K_1,t, K_{1,t + 1}, …, K_1,n), α ? 0
2. (2K_1,1, 2K_1,2,…, 2K_{1,t? 1}, (α + 2)K_1,t, 2K_{1,t + 1, …}, 2K_1,n), 0 ? α <t and
3. (3K_1,t, 3K_1,2, …, 3K_1,n) are graceful.

     

On a certain type of commutators of operators

LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.

1. C +G is a commutatorAB-BA with self-adjointA.
2. There exists an infinite orthonormal sequencee _j inH such that |Σ _j ⁿ =1 (Ce _j, e_j)| is bounded.
3. C is not of the formC ₁ ⊕C ₂ whereC ₁ has finite dimensional domain andC ₂ satisfies inf {|(C ₂ x, x)|: ‖x‖=1}>0.
4. 0 is in the convex hull of the set of limit points of spC.

     

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp ⁿ.

1. E can be generated overF by some elementy whose minimal polynomial has the specific formT ^pn?T?z.
2. A formula for the genus ofE is given.
3. IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).
4. We present a new example of a function fieldE/K whose gap numbers are nonclassical.

     

On the cyclicity and cocyclicity ofG-modules

LetG be a linear algebraic group over an algebraically closed fieldK. We call a (rational)G-module cyclic if it is generated by one element, and call it cocyclic if its dual is cyclic. We callG a c.c. group if the cyclicity is equivalent to the cocyclicity for anyG-module. IfG is not a c.c. group, the critical number ofG is the greatest integerc(G) such that the cyclicity is equivalent to the cocyclicity for anyG-module of dimension ≤c(G). In this paper we deduce some equivalent conditions for cyclicity and cocyclicity, and use them to prove the following main results:

1. A completely reducibleG is a c.c. group. The inverse holds for a connectedG in case charK>0, and also in case charK=0 with an exception thatG has a non-trivial unipotent quotient group.
2. \(\mathbb{G}_a \) is a c.c. group if charK=0; ( \(\mathbb{G}_a \) )=2 if charK>0.
3. IfG is reductive of typeA ₁ with charK=p>0, then $$c(G) = \left\{ \begin{gathered} \min \left\{ {2p - 1,p + 4} \right\}in case G is simply connected, \hfill \\ min\left\{ {2p - 1,p + 17} \right\}otherwise \hfill \\ \end{gathered} \right.$$

     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

Pancyclism in Chvátal-Erdös's graphs

LetG = (X, E) be a simple graph of ordern, of stability numberα and of connectivityk withα ≤ k. The Chvátal-Erdös's theorem [3] proves thatG is hamiltonian. We have investigated under these conditions what can be said about the existence of cycles of lengthl. We have obtained several results:

1. IfG ≠ K _k,k andG ≠ C ₅,G has aC _n?1 .
2. IfG ≠ C ₅, the girth ofG is at most four.
3. Ifα = 2 and ifG ≠ C ₄ orC ₅,G is pancyclic.
4. Ifα = 3 and ifG ≠ K _3,3,G has cycles of any length between four andn.
5. IfG has noC ₃,G has aC _n?2 .

     

On projective planes admitting elations and homologies

We consider projective planes Π of ordern with abelian collineation group Γ of ordern(n?1) which is generated by (A, m)-elations and (B, l)-homologies wherem =AB andA εl. We prove

1. Ifn is even thenn=2^e and the Sylow 2-subgroup of Γ is elementary abelian.
2. Ifn is odd then the Sylow 2-subgroup of Γ is cyclic.
3. Ifn is a prime then Π is Desarguesian.
4. Ifn is not a square thenn is a prime power.

     

Group inverse for a class of 2×2 anti-triangular block matrices over skew fields

Suppose K is a skew field. Let K ^m×n denote the set of all m×n matrices over K. In this paper, we give necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following three cases, respectively:

1. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)$ ;
2. $\mathrm{rank}(S)=\mathrm{rank}(AB^{\pi})$ ;
3. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)=\mathrm{rank}(AB^{\pi})$ ,

where A,B,C??K ^n×n , B ^# exists, R(B)=R(C), N(B)=N(C) and S=B ^?? AB ^?? . The paper??s conclusions generalized some related results of Zhao and Bu (Electron. J. Linear Algebra 21:63?C75, 2010).     

Sur les Suites Presque Echangeables DansL q , 1≦q<2

Let 1≦q<p<2. We construct a bounded sequence (X _n ) _n∈N inL _q which defines a typeσ onL _q , such that:

1. (X _n ) _n∈N is equivalent to the unit vector basis ofl _p .
2. The l-conic classK ₁(σ) generated byσ is not relatively compact for the topology of uniform convergence on bounded sets ofL _q .
3. (X _n ) _n∈N has no almost exchangeable subsequence after any change of density.

This sequence does not verify the two natural conditions inL _q -spaces that ensure the existence of an almost symmetric subsequence.     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

О рядах по системе Фра нклина

In this paper some basis properties are proved for the series with respect to the Franklin system, which are analogous to those of the series with respect to the Haar system. In particular, the following statements hold:

1. The Franklin series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) converges a.e. onE if and only if \(\mathop \Sigma \limits_{n = 0}^\infty a_n^2 f_n^2 (x)< + \infty \) a.e. onE;
2. If the series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) , with coefficients ¦a _n ¦↓0, converges on a set of positive measure, then it is the Fourier-Franklin series of some function from \(\bigcap\limits_{p< \infty } {L_p } \) ;
3. The absolute convergence at a point for Fourier—Franklin series is a local property;
4. If an integrable function (fx) has a discontinuity of the first kind atx=x ₀, then its Fourier-Franklin series diverges atx=x ₀.

     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

Algorithms for the vector maximization problem

We consider a convex setB inR ⁿ described as the intersection of halfspacesa _i ^T x ≦b _i (i ∈ I) and a set of linear objective functionsf _j =c _j ^T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

1. To decide if a given point inB is efficient.
2. To find an efficient point inB.
3. To decide if a given efficient point is the only one that exists, and if not, find other ones.
4. The solutions of the above problems do not depend on the absolute magnitudes of thec _j. They only describe the relative importance of the different activitiesx _i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.

     

Degenerate complementary cones induced by aK 0-matrix

In this paper we prove two results concerning the unionC of all the degenerate complementary cones associated with the linear complementarity problem (M, q) whereM is aK ₀-matrix.

1. C is the same as the set of allq ∈R ⁿ for which (M, q) has infinitely many solutions.
2. C is the same as the boundary of the set of allq ∈ R ⁿ for which (M, q) has a solution, an easily observable geometric result for a 2 × 2K ₀-matrix.

     

Diameters of the pieces in Borsuk's covering

Letd _n be the smallest number such thatS ^n?1 can be covered byn+1 sets each of diameter ?d _n . (Thusd ₂=2π/3.) THEOREM 1.Let C be convex body in E ⁿ ,n?2,of diameter D.Suppose each point in ?C has a spherical image of a diameter <1/2(π?d _n ).Then:

1. There exists a greatest positive number Δ=Δ(C)such that, for any a, b∈[0, Δ)and any interior normals n _a and n _b of ?C at a and b respectively, one has: the number Δis
2. C can be covered by n+1sets whose diameters are

A convex bodyA inE ⁿ will be calledr-touched if for anyx∈?A there exists a ballB of radiusr\s>0 satisfyingB?A,x∈B. THEOREM 2.Every r-touched convex body in E ⁿ ,n?2,of diameter D can be covered by n+1sets of diameters i=1, 2, ...,n+1. A similar Hadwiger estimate is better for any fixed dimension and sufficiently smallr. Our estimate is better for any fixedr and sufficiently high dimension.     

Noise correlation bounds for uniform low degree functions

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by δ are called δ-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics. In our main result we show that $\operatorname{\mathbb {E}}[f_{1}(X_{1}^{1},\ldots,X_{1}^{n}) \ldots f_{k}(X_{k}^{1},\ldots,X_{k}^{n})]$ is close to 0 under the following assumptions:

the vectors $\{ (X_{1}^{j},\ldots,X_{k}^{j}) : 1 \leq j \leq n\}$ are independent identically distributed, and for each j the vector $(X_{1}^{j},\ldots,X_{k}^{j})$ has a pairwise independent distribution;

the functions f _i are uniform;

the functions f _i are of low degree.

We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.     

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Torsion points of elliptic curves over large algebraic extensions of finitely generated fields

The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ ₁, …,σ _e)of elements of the abstract Galois group G(K)of K we have:

1. If e=1,then E _tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l.
2. If e≧2,then E _tor(K(σ))is finite.
3. If e≧1,then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order.

HereK(σ) is the fixed field in the algebraic closureK ofK, ofσ ₁, …,σ _e, and “almost all” is meant in the sense of the Haar measure ofG(K).     

Minimal nil-transformations of class two

On a metric minimal flow (X, a) which is a torus (K) extension of its largest almost periodic factorZ=X/K, the following conditions are equivalent.

1. (X, a) is a nil-transformation of the form (N/Γ,a) whereK is central inN and [N, N]?K.
2. E(X), the enveloping group of (X, a) is a nilpotent group of class 2.
3. Any minimal subset Ω ofX×X is invariant under the diagonal action ofK and the quotient Ω/K=Z ₁, is the largest almost periodic factor of Ω.

The enveloping groups of such flows are described and a corollary on cocycles of the circle into itself is deduced. Finally general minimal niltransformations of class two are shown to be of the form considered in condition (i) above (possibly with a different nilpotent group) and consequently we deduce that the class of minimal flows which are group factors of nil-transformations of class 2 is closed under factors.