Homogeneity in infinite permutation groups

It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:

1. G is not (k + 3)-homogeneous.
2. IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA _k .

     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

On a permutablity problem for groups

Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX ₁, X₂, . . .,X _n of sizem ofG there exits a non-identity permutation σ such that $X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0$ (respectively X₁X₂ . . .X _n = X_σ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that

1. G ∈ P(2,3) if and only ifG isomorphic to S₃, whereS _n is the symmetric group onn letters.
2. G ∈ R(2, 2) if and only if¦G¦ ≤ 8.
3. IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.

     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Bemerkungen zu einem Satz von Fejér

Последовательность {ita_k} _(n) ^k =1/∞ вещественных ч исел называется дважды мо нотонной, еслиa _k -2a _k+1 +a _k+2 ≧0 дляk≧1. В работе доказываютс я следующие утвержде ния, являющиеся обобщени ем двух теорем Фейера:

1. Если {ita_k — дважды моно тонная последовател ьность, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^n {a_\kappa z^\kappa } > 1/2$$ дляи≧ 1.
2. Если О≦β<1 и последова тельность (k+1-2β)ak} дважд ы монотонна, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {ka_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } > \beta $$ , то есть $$\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } \varepsilon S_\beta ^\kappa $$ . При помощи 2) получены о бобщения и уточнения теорем из работы [1] о линейных комбинациях некотор ых однолистных функц ий.

     

Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:

• equivalence classes of α-invariant K-connections on X
• α-invariant gauge classes of K-connections on X, and
• α-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic K ^?-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

Die Endlichen Polardreiseitgeometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. IfS contains four elements of which no three have a product of order two, then one of the following occurs.

1. G?PGL(2,n), n≡1 (mod 2).
2. G?PSL(2,n), n≡1 (mod 2) and n≥5.
3. G?PSU(3,16).
4. G/Z(G)?PSL(2,9) with ¦Z(G)¦=3.

     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:

1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝_E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;
2. Iff(x ₁,x ₂, ...,x _m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD?R ^m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝_D¦gradf¦dx≦2π ^m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;
3. LetE=Π={z∶¦z¦=1}, and letc _m(R) be the mth Laurent coefficient ofR onΠ,C _m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C _m(n) ≦ min {n/¦m¦, 1}.

     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

On an Ulam's problem

The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? ⁿ ,n≥3, such that:

1. X has not any stationary points on ? ⁿ ;
2. all orbits ofX are bounded;
3. there exists a filtration forX are proved in the present note.

     

Bounding the vertex cover number of a hypergraph

For a hypergraphH, we denote by

1. τ(H) the minimumk such that some set ofk vertices meets all the edges,
2. ν(H) the maximumk such that somek edges are pairwise disjoint, and
3. λ(H) the maximumk≥2 such that the incidence matrix ofH has as a submatrix the transpose of the incidence matrix of the complete graphK _k .

We show that τ(H) is bounded above by a function of ν(H) and λ(H), and indeed that if λ(H) is bounded by a constant then τ(H) is at most a polynomial function of ν(H).     

Nonabelian Jacobian of smooth projective surfaces-a survey

The nonabelian Jacobian J(X;L,d) of a smooth projective surface X is inspired by the classical theory of Jacobian of curves.It is built as a natural scheme interpolating between the Hilbert scheme X [d] of subschemes of length d of X and the stack M X(2,L,d) of torsion free sheaves of rank 2 on X having the determinant OX(L) and the second Chern class(= number) d.It relates to such influential ideas as variations of Hodge structures,period maps,nonabelian Hodge theory,Homological mirror symmetry,perverse sheaves,geometric Langlands program.These relations manifest themselves by the appearance of the following structures on J(X;L,d):1) a sheaf of reductive Lie algebras;2)(singular) Fano toric varieties whose hyperplane sections are(singular) Calabi-Yau varieties;3) trivalent graphs.This is an expository paper giving an account of most of the main properties of J(X;L,d) uncovered in Reider 2006 and ArXiv:1103.4794v1.     

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 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.

     

k-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to

1. all marginal worth vectors ofv belong to the core ofv; or
2. the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
3. the extreme points of the core ofv are exactly the marginal worth vectors ofv.

Examples ofk-convexn-person games are also treated.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.