Homothetic floating bodies

We show that a convex bodyK in ⁿ is homothetic to an ellipsoid if there is a sequence { _k }k converging to 0 so thatK is homothetic to its floating bodiesK _k.Supported by NSF grant DMS-9108003.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Some applications of the nash-williams lemma to the edge-reconstruction conjecture

It is shown that a graph whose vertices have one of four possible degrees is edgereconstructible if either 8 ord 2log ₂ 68, where andd are the minimum and the average degree of a graph respectively.     

Rarefied gas flow through long slots

The transition flow of a gas between two reservoirs through a two-dimensional slot is studied. The width of the slot is much smaller than the length but finite. The ratio of the equilibrium number densities in the reservoirs is near 1 and the equilibrium temperatures are the same. Galerkin's method of solution is applied to the linearized BGK equation and the total mass flow is evaluated as a function of the inverse Knudsen number and the length-to-half width ratioL. Limits 1, 1 andL 1 are investigated and good agreement with other authors is found. Simple analytical expressions are obtained for these cases.

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Riassunto Si studia la corrente gassosa tra due serbatoi uniti da un canale bi-dimensionale nel regime di transizione. La larghezza del canale è molto più piccola della sua lunghezza ma è finita. II rapporto tra le densità di molecole nei serbatoi è prossimo a 1 e le temperature d'equilibrio sono le stesse. II metodo di soluzione di Galerkin viene applicato all'equazione di Bhatnagar, Gross e Krook linearizzata e si valuta la protata in funzione del numero inverso di Knudsen e del rapportoL tra lunghezza e semi-larghezza. Si studiano i limiti 1, 1 eL1 e si trova un boun accordo con precedenti lavori. Per questi casi si trovano semplici espressioni analitiche.

     

Generalization of Gδ*-Diagonals and ωΔ-Spaces

In this paper we introduce the concepts of a quasi-G*-diagonal and quasi--space as generalizations of the concepts of G*-diagonal and -space respectively. It is shown that a quasi-Moore space may be characterized in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact -space with quasi-G-diagonal is metrizable and every collectionwise normal quasi--space is metrizable.     

Bounds for bounded motion around a perturbed fixed point

We consider a dissipative map of the plane with a bounded perturbation term. This perturbation represents e.g. an extra time dependent term, a coupling to another system or noise. The unperturbed map has a spiral attracting fixed point. We derive an analytical/numerical method to determine the effect of the additional term on the phase portrait of the original map, as a function of the bound on the perturbation. This method yields a value _c such that for< _c the orbits about the attractor are certainly bounded. In that case we obtain a largest region in which all orbits remain bounded and a smallest region in which these bounded orbits are captured after some time (the analogue of basin and attractor respectively).The analysis is based on the Lyapunov function which exists for the unperturbed map.     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

Point homogeneous flat affine planes

All flat projective planes (P,) whose automorphism group contains a 2-dimensional, connected closed subgroup fixing at most one line 1, are classified, except the following 2 classes: (1) – L₂, and not transitive on P1, and (2) S₁xR.Research supported in part by National Research Council of Canada (NRC)Herrn Professor Dr. Werner Burau zum 70. Geburtstag     

G8 estimator of solutions of systems of linear algebraic equations

For linear forms of regularized solutions (x, c)=Re c' · Re[I + i)+A'A_n ^–1]^–1 A'_nb of systems of equations Ax=b, where A is an n×m matrix, x, c, b are vectors, and _n is a sequence of constants, we propose the estimator , where is any measurable solution of the equation ()Re[1+₁a(())]²+ (₁–₂)(1+₁(gq()))=, a(y)=n^–1 Sp[Iy+^–1Z_s'Z^s+ iI]^–1, , _i=n_n ²_n ^–1s_n ^–1, ⁿ=mI_n ²_n ^–1s_n ^–1, X_i are independent observations on the matrix A. Under certain conditions, it is proved that G₈ is a consistent estimator for n and 0.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 111–119, 1988.     

Homotopy classification of some four-dimensional manifolds

In this paper there is proved a generalization of the results of Whitehead and Pontryagin on the homotopy classification of closed, simply connected four-manifolds. Let W and M be compact four-dimensional simply connected oriented four-manifolds. By q_w is denoted the intersection index on the group H₂(W).Basic Result. THEOREM (Extension). Let the groups H₁(W)and H₁(M) be finite and suppose given a homotopy equivalence fWM. In order that f can be extended to a homotopy equivalence (W,W)(M,M), it is necessary and sufficient that there should exist an isomorphism , such that the diagram is commutative and *q_m=q_m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 164–171, 1976.     

Sharp Error Bounds for Asymptotic Expansions of the Distribution Functions for Scale Mixtures

Let Z be a random variable with the distribution function G(x) and let s be a positive random variable independent of Z. The distribution function F(x) of the scale mixture X = sZ is expanded around G(x) and the difference between F(x) and its expansion is evaluated in terms of a quantity depending only on G and the moments of the powers of the variable of the form s/gr - 1, where (> 0) and (= ±1) are parameters indicating the types of expansion. For = -1, the bound is sharp under some extra conditions. Sharp bounds for errors of the approximations of the scale mixture of the standard normal and some gamma distributions are given either by analysis ( = -1) or by numerical computation ( = 1).     

How to build a barricade

Given a packing of congruent copies of a strictly convex planar setC in a parallel stripS of widthw and a straight linel, let |l| denote the sum of the lengths of the segmentslC _i over alliI It is proved that Where (C) is the density of the densest packing of congruent copies ofC in the plane. Some generalizations are also considered.     

Parabelscharen und grenzkreise in der isotropen ebene

It is known by H. Sachs [5] that the classical curve theorem of ABRAMESCU also holds in isotropic geometry. Generalising an idea due to O. Röschel [2] we regard all inscribed parabolas (s, t) of a triangle (t). This triangle is formed by the tangents of three neighbouring points of a C -curve k(t) in an isotropic plane. Let U((t)) be the circumcircle of (t) and I((t)) the incircle of the triangle (t) whose midpoints of the sides are the vertices of (t). The circle U((t)) is the locus of the isotropic focal points of (s, t) and the incircle I((T)) the envelope of the isotropic axes of (s, t). We prove that the ABRAMESU-circle — lim U((t)) — is identical with the locus of the focal points of lim (s, t) and the circle lim I((t)) with the envelope of the axes of lim (s, t). The characteristic points, different from k(t), of the circles lim U((t)) and lim I((t)) determine the direction of the affine-normal of k(t).Herrn Professor Helmut Mäurer zum 60. Geburtstag gewidmet     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.     

Die windschiefen Flächen konstanten Dralls in der Normalenkongruenz einer Fläche

In the congruence of surface normals of a given surface, the ruled surfaces for which the parameter of distribution has the constant value =0 or = are known to be developables and their base curves on the surface are the lines of curvature. In this paper a characterization is given of the ruled surfaces in the congruence of normals for which the parameter of distribution has a constant value (o,). If the given surface is developable the base curves may be characterized by a simple integral representation. If the given surface is not developable a characterization of the base curves is possible by means of the strips circumscribed along the base curves. Moreover the striction lines and torsal generators of these ruled surfaces are studied.

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Herrn Professor Dr. W. Wunderlich zum 60. Geburtstag gewidmet     

On a problem of A. Pleijel

In 1955, Arne Pleijel proposed the following problem which remains unsolved to this day: Given a closed plane convex curve C and a point x() at a fixed distance above the plane, as the point x() varies, characterize the point for which the conical surface with vertex x() and base C attains its minimum, and determine the limits as 0 and of this minimum point. The purpose of this paper is to solve the cases where approach its extremities and in the course of the solution, we obtain an interesting characterization of the limit points, which we shall call the Pleijel points of C. A consequence is that the inner Pleijel point provides an upper bound for the isoperimetric defect of C. We also generalize the problem to higher dimensional spaces, and obtain the corresponding characterizations of the limiting points for convex surfaces.     

Some remarks on the existence of densities for sums with weights

One considers the weighted sums , where X_k, are independent random variables, having no densities, and _k() are nonnegative functions, defined on the interval (0, 1). One investigates the conditions under which S(()) has a density; one proves a local limit theorem; one studies the form of the distribution in the case when ()=^k and the variables X_k take only a finite number of values.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 83–94, 1986.     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.