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.

     

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.

     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

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.

     

Null curves in {\mathbb{C}^3} and Calabi–Yau conjectures

For any open orientable surface M and convex domain ${\Omega\subset \mathbb{C}^3,}$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

There exists a complete bounded CMC-1 immersion ${X:M \to \mathbb{H}^3.}$

For any convex domain ${\Omega \subset \mathbb{R}^3}$ there exists a complete proper minimal immersion (X _j ) _j=1,2,3 : M → Ω with vanishing flux. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and ${\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 \,|\, (x_1,x_2) \in D\},}$ then X can be chosen so that (X ₁, X ₂) : M → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.     

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.     

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.     

Convergence of representation averages and of convolution powers

LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:

1. if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU ⁿ (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
2. If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U ⁿ converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
3. If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .

     

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.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Embeddings into power series rings

Let T be an ordered ring without divisors of zero, and letA be the set of archimedean subgroups of T generated by a Banaschewski functionτ. Let_XΠ_Δ R be the power series ring of the real numbers ? over the totally ordered semigroup Δ of archimedean classes of T, and letχ be the usual Banaschewski function on_XΠ_Δ R. The following are equivalent:

1. τ satisfies the additional condition; for convex subgroups P,Q of T, where
2. There exists a one-to-one homomorphism Γ:T→_XΠ_Δ R of ordered rings such that for every convex subgroup Q of_XΠ_Δ R, there exists a convex subgroup P of T such that \(\Gamma (P) \subseteq Q\) and \(\Gamma (\tau (P)) \subseteq \chi (Q)\) .

     

The optimal base of a matroid/with tree-type constraints

Let M be a matroid defined on a weighted finite set E=(e_1,…,e_n).l(e) is the weight of e∈E.P (X_1,…,X_m) is a set of subsets of E.X_i,X_j∈P,if X_i∩X_j≠(the empty set),then either X_i X_j or X_jX_i.For each X_i∈P,there are two associate nonnegative integers a_i and b_i with o_i≤b_i≤|X_i|.We call a base T of M a feasible base with respect to P(or simply call it a feasible base of M),if X_i∈P,a_i≤|X_i∩T|≤b_i.A base T' is called optimal if:i) This feasible,In this paper we present a polynomial algorithm to solve the optimal base problem.     

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 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.

     

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.     

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 ).} $

     

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.     

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.

     

Bases-cobases graphs and polytopes of matroids

LetM be ablock matroid (i.e. a matroid whose ground setE is the disjoint union of two bases). We associate withM two objects:

1. Thebases-cobases graph G=G(M,M ^*) having as vertices the basesB ofM for which the complementE\B is also a base, and as edges the unordered pairs (B,B′) of such bases differing exactly by two elements.
2. Thepolytope of the bases-cobases K=K(M,M ^*) whose extreme points are the incidence vectors of the bases ofM whose complement is also a base.

We prove that, ifM is graphic (or cographic), the distance between any two vertices ofG corresponding to disjoint bases is equal to the rank ofM (generalizing a result of [10]). Concerning the polytope we prove thatK is an hypercube if and only if dim(K)=rank(M). A constructive characterization of the class of matroids realizing this equality is given.     

Elliptic and hyperbolic inversions

Letx andy be orthogonal coordinates of a pointM(u=ax+iby orax+εby) of a plane whereasx′ andy′ are orthogonal coorditanes of a pointM′(v=ax′+iby′ orax′+εby′) inverse ofM in the elliptic or hyperbolic inversion $u\bar \upsilon = k$ (k positive) $\bar \upsilon $ designating the conjugate ofv whilei andε are Clifford numbers such thati ²=?1 andε ² = 1 (a andb are real)O is the origin of axises,Ox is the axis of the inversion.

1. Two inverse points are aligned withO.
2. The inverse of a conic having a centrec is a conic but this inverse is a straight line if the origin is on the conic.

There is a metric relation between sides and diagonals of a quadrilateral inscribed in an ellipse or in an hyperbola.