Hypercubic 4 and 5-Designs from Double-Error-Correcting BCH Codes

An ordered orthogonal array OOA(, k, n) is a binary 2 ^k × n matrix with the property that for each complete -set of columns, each possible -tuple occurs in exactly 2 ^k– rows of those columns (for definition of a complete -set, see below). Constructions of OOA(, k, n) for = 4 and = 5 are given.     

Randwertprobleme und Methode der inneren Parallelen: eine gemeinsame Erweiterung zweier Ungleichungen von Pólya

Zusammenfassung Die Methode der inneren Parallelen von Makai und Pólya wird hier auf das Randwertproblem (3) angewendet. Die sich ergebenden Schranken (11) und (11) für das FunktionalQ() enthalten als Grenzfälle die von Pólya [9] erhaltenen Schranken (2) und (1) für die Torsionssteifigkeit (Fall=0) bzw. für den Grundton einer Membran (Fall ₁.

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Summary We apply here the method of interior parallels of Makai and Pólya to the boundary value problem (3). The resulting bounds (11) and (11) for the functionalQ() contain as limit cases the bounds (2) and (1) obtained by Pólya [9] for torsional rigidity (case=0) and for the lowest eigenfrequency of a membrane (case ₁).

     

Noise reduction by gradual memorization

Zusammenfassung Der vorliegende Beitrag beschreibt ein einfaches, nichtlineares, abgetastetes Filter, das «Gradueller Speicher» genannt wird. Das Filter besteht aus einem Speicher, welches neue Eingangswerte nur graduell, d. h. schrittweise, speichert: Bei jedem neuen Eingangswert wird er um einen konstanten Schritt ±s in Richtung des letzteren nachgestellt. Es wird gezeigt, dass eine solche «Graduelle Speicherung» eine Verminderung von Gaussschem Rauschen um einen Faktor gestattet. (=/s, d. h. ist gleich der Streuung des Eingangsrauschens geteilt durch den Schritt s.) Die Wahrscheinlichkeitsfunktion des Speicherausganges wird berechnet und durch eine Gauss-Funktion angenähert. eine Formel wird hergeleitet, welche die richtige Wahl von s in praktischen Fällen gestattet. Im letzten Abschnitt wird das dynamische Verhalten eines graduellen Speichers betrachtet, die Reaktionszeitt _r , welche der Speicher braucht, um einer plötzlichen Änderung U_o im wahren Eingangssignal nachzufolgen, wird berechnet.t _r wird umso grösser, je grösser die Streuung des Eingangsrauschens ist.

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List of Symbols Real valueX=U ₀+U Input signal (true value + noise) - Real valueU ₀ Normalized valueu ₀=U ₀/s True input signal - Real valueU Normalized valueu=U/s Input noise - Real value s Step of memory - Real valueY=U ₀+R Output signal (true value + noise) - Real valueR Normalized valuer=R/s Output noise - Real value Normalized value Average momentary value of output signal - Real value Normalized value =/s Standard deviation of Gaussian noise at input - Normalized value Standard deviation of Gaussian noise at output - Normalized valueb Parameter of logistical distribution - Real value U ₀ Normalized valuea=U ₀/s Step of true input value - Real valuet Normalized value =t/t Time - Real value t=T Normalized value 1 Sampling interval - Real value Normalized value Average momentary velocity of output signal - Real valueU _tol Prescribed tolerance interval for output signal - Real value Prescribed probability (that the output signal remains within the tolerance interval) - Real valuew(U) Normalized valuew(u) Probability density function (input) - Real value (R) Normalized value (r) Probability (density) function (output) - Normalized valueq _r =1-P _r Cumulative distribution function - Normalized valueK=r/ Momentary value of output noise, expressed in multiples of the standard deviation - Real valueQ Normalized valueQ=/ Noise reduction factor - Real valuet _r Normalized value _r =t _r /t Time of reaction - Normalized valuez Increase in reaction time due to input noise     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Combinatorics of Maximal Minors

We continue the study of the Newton polytope _m,n of the product of all maximal minors of an m × n-matrix of indeterminates. The vertices of _m,n are encoded by coherent matching fields = (), where runs over all m-element subsets of columns, and each is a bijection [m]. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Gröbner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of _m,n for m n by means of generalized permutations, which are bijections between (n – m + 1)–element subsets of columns and (n – m + 1)–element submultisets of rows.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

On graphs that can be oriented as diagrams of ordered sets

We study some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set. For each 1, M. Aigner and G. Prins have introduced a notion of a vertex colouring, here called -good colouring, such that a 1-good colouring is the usual concept and graphs that have a 2-good colouring are precisely covering graphs. We present some inequalities for the corresponding chromatic numbers , especially for x ₂. There exist graphs that satisfy these inequalities for =2 but are not covering graphs. We show also that x ₂ cannot be bounded by a function of x=x ₁. A construction of Neetil and Rödl is used to show that x ₂ is not bounded by a function of the girth.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

Über einen Homomorphiesatz von G.Köthe

The main object of this paper is to generalize a homomorphism theorem of Köthe [5] for a wide class of not necessarily locally convex topological vector spaces. A sequence =(B_n) of balanced sets B_n in a vector space E(), such that the union of the B_n's equals E and such that B_n+B_nB_n+1 for all nN, is called an absorbent sequence. E() is called -locally topological, if it possesses an absorbent sequence =(B_n) of bounded sets and if a linear mapping A from E() into any other vector space is continuous if all restrictions are continuous at 0. It is shown, that a continuous linear mapping A from a vector space E() with an absorbant sequence of compact sets into a boundedly summing -locally topological space F() is a homomorphism if A(E) is closed in F().     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Trigonometric Cesàro bases in the spaces of functions integrable with power weight

(2), k1, >0, L _p (0,), 1p L =C. , , p, k, (C, )- L _p (0,), , , {sinnx} _n ^=k/ (C, )- L _p (0,) |x|^p . , 1p, {x sinnx} _n=k , k2 2k–2–1/p<2k–1/p, (C, )- L ^p [0,] , >–(p–1)/.     

Maximal Effective Bandwidth of Constrained Traffic

We investigate the worst possible behavior of a stationary traffic source when the traffic emanating from it is required to meet certain constraints. Specifically, the peak rate of the source is required not to exceed a level and realizations must obey a leaky bucket constraint with bucket size and leak rate . The worst case source is considered to be the one with the largest effective bandwidth, a concept which arises in the large deviation theory of queueing networks and governs the asymptotic loss rate when a large number of sources send traffic to a single server queue. We conjecture the form of the worst case traffic in general and prove the conjecture for the special case when T, the time-scale parameter of the effective bandwidth, is less than both /(–) and /, the times taken respectively to fill and empty the leaky bucket.     

Packing circuits in eulerian digraphs

LetG be an eulerian digraph; let (G) be the maximum number of pairwise edge-disjoint directed circuits ofG, and (G) the smallest size of a set of edges that meets all directed circuits ofG. Borobia, Nutov and Penn showed that (G) need not be equal to (G). We show that (G)=(G) provided thatG has a linkless embedding in 3-space, or equivalently, if no minor ofG can be converted toK ₆ by –Y andY– operations.     

On the two-gap elliptic potentials

The-parametrized family of two-gap elliptic potentials is constructed so that (i) 0<<1, (ii) for rational values of such potentials are elliptic (i.e., double-periodic), (iii) within the limit0 this family degenerates to the soliton potential, (iv) within the limit1 this family degenerates to the one-gap Lamé potential.Dedicated to the memory of J.-L. Verdier     

On exact D-optimal designs for regression models with correlated observations

Let ^* be an exact D-optimal design for a given regression model Y = X + Z . In this paper sufficient conditions are given for sesigning how the covariance matrix of Z may be changed so that not only ^* remains D-optimal but also that the best linear unbiased estimator (BLUE) of stays fixed for the design ^*, although the covariance matrix of Z * is changed. Hence under these conditions a best, according to D-optimality, BLUE of is known for the model with the changed covariance matrix. The results may also be considered as determination of exact D-optimal designs for regression models with special correlated observations where the covariance matrices are not fully known. Various examples are given, especially for regression with intercept term, polynomial regression, and straight-line regression. A real example in electrocardiography is treated shortly.     

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     

Minimal Characteristic Algebras for Some Properties of Identities

We consider algebras of a type , without nullary fundamental operation symbols. A structural property p of an identity of type is hereditary if for every set I of identities of type having the property p every consequence of I (derived identity) has the property p. An algebra of type we call characteristic for a hereditary property p if for every variety V of type we have: V if and only if every identity from Id(V) has the property p. In this paper we show minimal characteristic algebras for several hereditary properties, e.g., to be regular, to be normal, etc.1991 Mathematics Subject Classification 08B05     

Complemented Topologies on Abelian Groups

A topology on a group G is complemented if there exists an indiscrete topology ' on G such that UV={0} for suitable neighborhoods of zero U and V in the topologies and . The authors give a complementation test for an arbitrary topology. Locally compact groups with complemented topologies have been described. A group all of whose continuous homomorphic images are complete is proved to be compact. A family of 2 topologies that are pairwise complementary to one another is defined for an arbitrary group.