On steiner ratio conjectures

LetM be a metric space andP a finite set of points inM. The Steiner ratio inM is defined to be(M)=inf{L _s(P)/L _m(P) |P M}, whereL _s(P) andL _m(P) are the lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. In this paper, we study various conjectures on(M). In particular, we show that forn-dimensional Euclidean space _n ,( _n )>0.615.Supported in part by the National Science Foundation of China.     

Non-Arguesian configurations and gluings of modular lattices

In two earlier papers, we investigated non-Arguesian projectivity configurations, d=(d_ij:I,j 5), in a modular latticeL, and the associated intervalsI =u /z , 5. Here we study the relationship of d to representations of L as a gluing of proper intervals. We are primarily concerned with two extreme cases:L=J F withJ an ideal andF a filter; and, whenL is of finite length,L=U {L(x):x S(L)} with S(L), the prime skeleton ofL and eachL(x), a maximal complemented subinterval ofL.Presented by Ralph Freese.Research supported by NSERC Operating Grant A-8190 and the University of Hawaii.Research supported by NSF Grant DMS-8300107.     

General Blowings-up of ℙ2 and Injectivity of Gauss Maps

We consider the blowing-up Y _k of the projective plane along k general points P ₁,...,P _k . Let _k : Y _k ² be the projection map and E _i the exceptional divisor corresponding to P _i for 1ik. For m2 and km(m+3)/2–4 let _k be the invertible sheaf _k ^*( ²(m)) _{Y k} (–E ₁–···–E _k ) on Y _k , and let _k: Y _k ^N be the morphism corresponding to _k . As _k is a local embedding, the Gauss map _k corresponding to _k is defined on Y _k by _k (x)=(d _{x k} )(T _x (Y _k )) for all xY _k . We prove that this Gauss map _k is injective.     

The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

Structures of topological type

In this paper we continue the study of structures of various types initiated by the author in the earlier paper Structures of extensions (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure on the set X×X, and a bitopological space is a pair (X,). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures ₁ and ₂-these structures define a structure =₁×₂ on the set X×X. A settopological structure on the set X is any topological structure on the set={A¦A. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 5–62, 1979.     

Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ)

A shift-invariant system is a collection of functions {g_m,n} of the form g_m,n(k)=g_m(k–an). Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system _m,n(k)=_m(k–an) such that each functionf can be written asf= f, _m,ngm,n. The mathematical theory usually addresses this problem in infinite dimensions (typically in L² () or ²()), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in ²(). For compactly supported g_{m, n} (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper.Part of this work was done while the author was a visitor at the Department of Statistics at the Stanford University.The author has been partially supported by Erwin-Schrödinger scholarship J01388-MAT of the Austrian Science foundation FWF.     

Note on weakly mixing transformations

Summary LetT be a weakly mixing transformation with respect to a probability measureP on a metric space (X, d). Suppose further that every open ball of (X, d) has positive measure. Then we show that, for anyP-measurable setA withP(A) > 0, lim supD _k (T ⁿ A) =D _k (X) fork = 2, 3,, whereD _k (B) is the geometric diameter of orderk of a subsetB ofX. It is shown further that D _k can be replaced by essD _k , in the case whenTB is measurable wheneverB is measurable. These results complement a previous one due to R. E. Rice for strongly mixing transformations and improve a result of C. Sempi on weakly mixing transformations.     

Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix

We study the lower semicontinuous envelope in L^p(), F, of a functional F of the form F(u)=A uudx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)= Buudx with B=AP A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Arithmetical identities of the Brauer-Rademacher type

Summary LetA be a regular arithmetical convolution andk a positive integer. LetA _k (r) = {d: d ^k A(r ^k )}, and letf A _k g denote the convolution of arithmetical functionsf andg with respect toA _k . A pair (f, g) of arithmetical functions is calledadmissible if(f A _k g)(m) 0 for allm and if the functions satisfy an arithmetical functional equation which generalizes the Brauer—Rademacher identity. Necessary and sufficient conditions are found for a pair (f, g) of multiplicative functions to be admissible, and it follows that, if(f A _k g)(m) 0 f(m) for allm, then (f, g) is admissible if and only if itsdual pair (f A _k g, g ^–1 ) is admissible.Iff andg ^–1 areA _k -multiplicative (a condition stronger than being multiplicative), and(f A _k g)(m) 0 for allm, then (f, g) is admissible, calledCohen admissible. Its dual pair is calledSubbarao admissible. If (f A _k g) ^–1 (m) 0 itsinverse pair (g ^–1 , f ^–1 ) is also Cohen admissible.Ifg is a multiplicative function then there exists a multiplicative functionf such that the pair (f, g) is admissible if and only if for everyA _k -primitive prime powerp ⁱ either (i)g(p ⁱ ) 0 or (ii)g(p ) = 0 for allp havingA _k -type equal tot. There is a similar kind of characterization of the multiplicative functions which are first components of admissible pairs of multiplicative functions. IfA _k is not the unitary convolution, then there exist multiplicative functionsg which satisfy (i) and are such that neitherg norg ^–1 isA _k -multiplicative: hence there exist admissible pairs of multiplicative functions which are neither Cohen admissible nor Subbarao admissible.An arithmetical functionf is said to be anA _k -totient if there areA _k -multiplicative functionsf _T andf _V such thatf = f _T A _k f _V ^-1 Iff andg areA _k -totients with(f A _k g)(m) 0 for allm, and iff _V = g _T , then the pair (f, g) is admissible. The class of such admissible pairs includes many pairs which are neither Cohen admissible nor Subbarao admissible. If (f, g) is a pair in this class, and iff(m), (f A _k g) ^–1 (m), g ^–1 (m),f ^–1 (m) andg(m) are all nonzero for allm, then its dual, its inverse, the dual of its inverse, the inverse of its dual and the inverse of the dual of its inverse are also admissible, and in many cases these six pairs are distinct.A number of related results, and many examples, are given.     

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in L_q(–, ) of differentiable functions whose k-th derivatives belong to L_p(–, ), by splines S_m (x) with nonfixed nodes, under the extra assumption that the norms in L_s(–, ) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 501–510, May, 1971.     

Minimality of derivative chains,corresponding to a boundary value problem on a finite segment

One investigates the minimality of derivative chains, constructed from the root vectors of polynomial pencils of operators, acting in a Hilbert space. One investigates in detail the quadratic pencil of operators. In particular, for L()=L₀+L₁+²L₂ with bounded operators L₀0, L₂0 and Re L₁0, one shows the minimality in the space_173-02 of the system {x_k, _kekx_k}, where x_k are eigenvectors of L(), corresponding to the characteristic numbers _kin the deleted neighborhoods of which one has the representation L^–1()=(–_k)^–1R_K+W_K() with one-dimensional operators R_k and operator-valued functions W_K(), k=1, 2, ..., analytic for =_k.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 195–205, February, 1990.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Extendability of Ternary Linear Codes

There are four diversities for which ternary linear codes of dimension k 3, minimum distance d with gcd(3,d) = 1 are always extendable. Moreover, three of them yield double extendability when d 1 (mod 3). All the diversities are found for ternary linear codes of dimension 3 k 6. An algorithm how to find an extension from a generator matrix is also given.This research has been partially supported by Grant-in-Aid for Scientific Research of the Ministry of Education under Contract Number 304-4508-12640137     

Durch graduierte Lie-Algebren definierte Paar-Algebren

Pair algebras which have a non degenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible are characterised by pairs (C,), where C is a n×n matrix satisfying certain conditions and is a sequence of n integers equal to 0 or 1. They occur as pair algebras of type (S(C,)_–1,S(C,)₁), xuy=[[x,u],y], where (S(C,)_r)_r is the gradation induced by . in the Kac-Moody algebraS(C). If C is an affin Cartan matrix (as in the case of Lie triple systems), there exists a finite dimensional simple Lie algebrag and a Aut (g), ord =m< such that the pair algebra is isomorphic to the pair algebra (g _–1,g ₁), xuy=[[x,u],y] (product ing), whereg _i. is the eigenspace of of eigenvalue ⁱ, a primitive m-th root of unity.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

On 2-spannedness for the adjunction mapping

LetL be a line bundle on a smooth connected projective manifold X of dimension n. We extend to any dimension the definition of k-spannedness forL; this is a notion of k-th order embedding which was recently given in the case of curves and surfaces. Then, by a reduction to the surfaces case, we prove that the adjoint bundle K_x+(n–1)L is 2-spanned ifL is (at least) 3-spanned.     

Invariant subsets of strongly continuous semigroups

Let {P(t): t0} be a strongly continuous semigroup on a Banach space X and let |\| be a continuous norm on X such that |P(t)x|exp(t)|x|, XX, t0. Let C be a |\|-closed convex subset of X and suppose that for every x in D(A) there exists a sequence (x_n : n ) in D(A) with the following properties: lim|x–x_n|=0, lim|Ax–Ax_n|=0 and every x_n has a best approximation in C (with respect to |\|) which belongs to D(A). Then P(t)CC for all t0 if and only if, for every v in CD(A), the vector Av belongs to the |\|-closure of [0, ) (C-V).     

Finite Group Actions on Spin Bundles

Let L be a complex line bundle over a closed, oriented, smooth 4-manifold X with c₁(L) w₂(TX) mod 2. Let a finite group G act on X as orientation preserving isometries and on L such that the projection L X is a G-map. We investigate the action of G on the Seiberg-Witten equations, and when G = Z₂ we study the G-invariant Seiberg-Witten invariants on X and the Seiberg-Witten invariants on its quotient setting.