On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

Decompositions of complete graphs into isomorphic bipartite subgraphs

Let |E(G)|= andf, a 1-1 mapping ofV(G) into {0,1,...,}. Thenf is called a -valuation ofG if the induced function given by , for alluvE(G) is 1-1. A -valuationf is called an -valuation ofG if there exists a nonnegative number such that for everyuvE(G) withf(u)<f(v),f(u)<f(v). Let denote the graph of then-dimensionalG-cube. ForG=K _{3, 3},K _{4, 4}, andP _k ,it is shown that for any positive integern, then-dimensionalG-cube has an -valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2

Let {B _t ,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p,q for all , provided the integrand u belongs to the space . Moreover, if u is bounded and belongs to for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p, ^H .     

Irrationalité de Valeurs Associées a l'Exponentielle de Carlitz

Lete and be the Carlitz-module analogues of their usual counterparts. We have proved in [4]-that these elements of are algebraically independent over whenq3. We study here the remaining caseq=2 and prove among other things that 1,e, are linearly independent over .

     

Lattices of dominions in quasivarieties of Abelian groups

Let M be any quasivariety of Abelian groups, (H) be the dominion of a subgroup H of a group G in M, and L_q(M) be the lattice of subquasivarieties of M. It is proved that (H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = { (H) | N L_q(M)} forms a lattice under set-theoretic inclusion and the map : L_q(M) L(G,H,M) such that (N) = (H) for any quasivariety N L_q(M)is an antihomomorphism of the lattice L _q (M) onto the lattice L(G, H, M).__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 238–251, March–April, 2005.     

Attracting properties for one dimensional flows of a general barotropic viscous fluid. Periodic flows

Summary We consider the motion of a barotropic compressible fluid in a one dimensional bounded region with impermeable boundary, see equation (1.1). Here, u(t, q) denotes the velocity and v(t, q) the specific volume. The quantity log v(t, q) measures the displacement of v(t, q) with respect to the equilibrium v 1. For the sake of brevity we denote here different norms by the simbol . We show that there is a positive constant r₀=r₀(), a small ball B₁ (r) (with radius R₁ (r), ), and a large ball B(r) (with radius R(r), ) such that the following holds, for each r [0, r₀ [(i) If f(t) < r for all t 0, and if (u(0), log v(0))R(r) (i.e. (u(0), log v(0)) B(r)) then, for sufficiently large values of t, (u(t), log v(t))R₁ (r); (ii) The solutions starting at time t=0 from the large ball B(r) have all the same asymptotic behaviour (see (1.11)); (iii) If f is T-periodic then there is a (unique) T-periodic solution (u(t), log v(t)) inside the small ball B₁ (r). This periodic solution atracts all solutions which intersect the large ball B(r). Periodic solutions had been previously studied only for very specific pressure laws, namely p(v)-log v and p(v)-v^–1.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small and < N, a {v _{+ 1}, v ; N, q}-minihyper consists of a sum of -spaces, we show that the excess points of an s-cover with excess of PG(N, q), (s + 1)|(N + 1), form a sum of s-spaces, and that no maximal partial s-spreads with deficiency of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

Games of Chains and Cutsets in the Boolean Lattice

B. Bajnok and S. Shahriari proved that in 2^[n], the Boolean lattice of order n, the width (the maximum size of an antichain) of a non-trivial cutset (a collection of subsets that meets every maximal chain and does not contain or [n]) is at least n-1. We prove that, for n5, in the Boolean lattice of order n, given -1 disjoint long chains, we can enlarge the collection to a cutset of width n-1. However, there exists a collection of long chains that cannot be so extended.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

The Rank of a Direct Power of a Small-Cancellation Group

We construct an example of a finitely generated group G such that rank((G )ⁿ)=2 for all n1. For each n, we construct a finitely presented group G _n such that rank((G _n )ⁿ)=2. We conjecture that if G is a word-hyperbolic group then rank(G ⁿ ) as $ n. For each m we give an example of a residually finite group K _m such that K _m has exactly two relators, but K _m has no proper subgroups of index $ m. We construct a finitely generated group D such that there is an epimorphism DD×D.     

On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

We describe dual spaces of classes of the Hardy–Sobolev type of functions holomorphic in a polydisk for 0 < p 1 and q (0, ) and for p (1, ) and q = 1.