Variational problems in SBV and image segmentation

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = _\K [u² + (u – g)²]dx + ^{n – 1}(K), u W ^1,2(\K), K closed in .We use a weak formulation of the minimum problem in a special class SBV() of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

Remarks on certain growth conditions in problems of optimal control

In the Tonelli-Nagumo existence theorem for free problems of the calculus of variations, a uniform growth condition (T ₁) was employed to prove the equiabsolute continuity of the trajectories of a minimizing sequence. This same growth condition was shown to be equivalent to a pointwise growth condition (T ₂). Cesari utilized a generalization, condition (), in order to extend the Tonelli-Nagumo existence theorem to optimal control problems.In the present paper, we show that, for problems of optimal control, condition () is equivalent to a uniform growth condition () and that both conditions () and () are stronger than a pointwise growth condition (), a generalization of condition (T ₂). Two situations are given where conditions (), (), and () are equivalent, and the result of Tonelli and Turner concerning the equivalence of conditions (T ₁) and (T ₂) is obtained as a particular case of one of these situations.This research was partially supported by the National Science Foundation, Grant No. GK-234.     

On the divisibility of homogeneous hypergraphs

We denote byK _k ,k, 2, the set of allk-uniform hypergraphsK which have the property that every element subset of the base ofK is a subset of one of the hyperedges ofK. So, the only element inK ₂ ² are the complete graphs. If is a subset ofK _k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of . We callH -free homogeneous graphsH _n have been shown to be indivisible, that is, for any partition ofH _n into two classes, oue of the classes embeds an isomorphic copy ofH _n . [5]. Here we will investigate this question of indivisibility in the more general context of-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that all-free hypergraphs for K _k with 3 are indivisible. The-free hypergraphs with K _k ² satisfy a weaker form of indivisibility which was first shown by Henson [2] to hold forH _n . The general necessary condition for homogeneous structures to be indivisible will then be used to show that not all-free homogeneous hypergraphs are indivisible.This research has been supported by NSERC grant 69–1325.     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Direct and converse imbedding theorems for seminormed spaces

. n=2 . ., 103 498      

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

On simplexes inscribed in the minor arc of a curve

On an arc L of length h, of class Cⁿ, and in Euclidean Eⁿ, the set of n+1 points (the partition of the arc) P={0, ₁ h, , _n–1h, h}, 0<₁<<_n–1<1 determines a simplex S_h(P) inscribed in the arc. For its volume V_h(P) we evaluate lim and prove that its greatest value is obtained for a unique choice of P=P_cr. The exact values for _i from P_cr are found for n=2, 3, 4, 5, 6.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 114–120, 1990.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

Stabilization theorem for the Milnor K2-functor

Let be an associative ring. For every natural number n there is a canonical homomorphism _n: K_2,n()K₂(), where K₂ is the Milnor functor and K_2,n() the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of , _n is an epimorphism. It is proved in the article that if n – 1 is greater than the stable rank of , the homomorphism _n is an isomorphism.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 131–152, 1976.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Lax Projective Embeddings of Polar Spaces

Let be a non-degenerate polar space of rank n 3 where all of its lines have at least three points. We prove that, if admits a lax embedding e : in a projective space defined over a skewfield K, then is a classical and defined over a sub-skewfield K₀ of K. Accordingly, admits a full embedding e₀ in a K₀-projective space ₀. We also prove that, under suitable hypotheses on e and e₀, there exists an embedding such that and preserves dimensions.Received: March, 2004     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

Irregular but nonfractal drums andn-dimensional weyl conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set R ⁿ (n 1) with irregular but nonfractal boundary. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN _i ()(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ _i ()=() –N _i () for large, where() is the Weyl term. This implies that for the irregular but nonfractal drum , not only the volume || _n is spectral invariant but also the area of boundary || _n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.     

On Conjectures of Mathai and Borel

Mathai has conjectured that the Cheeger–Gromov invariant ₍₂₎ = ₍₂₎ - is a homotopy invariant of closed manifolds with torsion-free fundamental group. In this paper we prove this statement for closed manifolds M when the rational Borel conjecture is known for = ₁(M), i.e. the assembly map : H _*(B, ) L_*() is an isomorphism. Our discussion evokes the theory of intersection homology and results related to the higher signature problem.     

Surfaces and the second homology of a group

LetG be a group andK(G, 1) an Eilenberg—MacLane space, i.e. ₁(K(G,1))G, _i (K(G,1))=0,i1. We give a purely algebraic proof that the second homology groupH ₂(G)=H ₂(G,)H ₂(K(G,1)) is isomorphic to the group of stable equivalence classes of continuous mapsFK(G,1) inducing surjections on fundamental groups (resp. surjections, whereF{F _g=closed orientable surface of genusg,g}. As a corollary we obtain an algebraic proof of the well-known isomorphismH ₂(G)₂(K(G,1)) (2-dimensional bordism group).     

Lattice of subgroups

Let D be a subgroup of the group G. The lattice of intermediate subgroups is studied. The subgroup F (D F G) is said to be D-complete, if D^F=u:u F>=F. Let F be the subset of all D-complete intermediate subgroups. The system {F, N_G(F)} is a fan for D in G (RZhMat, 1980, 5A208) if and only if D^x> is a D-complete subgroup for any x G. The set {F_ga} coincides with the collection of subgroups of the form D^A (1 A C G) if and only if for any x G the subgroup D, D^x is D-complete. The last condition holds, for example, for a pronormal subgroup D.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 13–19, 1980.     

Some Classes of L 1-Convergence of Fourier Series

In this paper equivalent classes of the classes M and S _pr , p > 1, > 0. r { 0,1,2, ... ,[]} defined by Shuyun [3] are obtained. Then, it is shown that the class S _pr , 1 > p 2, 0, r {0,1,2,...,[]} is a subclass of BVC _r , where S _pr is the equivalent class of the Shuyun's class S _pr , BV is the class of null sequences of bounded variation and C _r is the extension of the Garrett--Stanojevic class. As a corollary of this result, we have obtained the theorem, proved in [7].     

Parabolische Kollineationen,die eine ebene konvexe Menge invariant lassen

Collineations ₁, ₂ of PG(2, ) leaving invariant a compact convex setK ² are called parabolic if |K Fix _i|=1. Conditions are stated under which the existence of ₁, ₂ imply that K is an ellipse.

Herrn Helmut R. Salzmann zum 65. Geburtstag gewidmet