A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

The convergence to a Gaussian process of the number of empty cells in the classical problem of distributing particles among cells

We consider a case in which n particles are distributed independently of one another in N cells. We examine the behavior of the number of empty cells, ₀ (n), as a random function of the parameter n when n, N. We prove that for suitable variation of the time parameter, ₀ (n) will converge to a Gaussian process in the following cases: a) n/N, n/N^– l _nN; b) n/N0, n²/N.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp, 97–104, July, 1968.     

Start‐up class models in multiple‐class queues with N‐policy

In this paper, we consider multipleclass queueing systems with Npolicy in which the idle server starts service as soon as the number of customers in the startup class reaches threshold N. We consider the cases of FCFS and nonpreemptive priority. We obtain the Laplace–Stieltjes transform of the waiting times of each class of customers. We also show some results for the general behavior of such systems.     

On the Approximation of Functions on Locally Compact Abelian Groups

Questions of approximative nature are considered for a space of functions L ^p(G, ), 1 p , defined on a locally compact abelian Hausdorff group G with Haar measure . The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

Die Methode der „direkten Koeffizientenanpassung“ (μ-Form) des Separable Programming

Zusammenfassung Zur approximativen Lösung von Optimierungsproblemen mit trennbaren nichtlinearen Funktionen (Separable Programming) werden häufig modifizierte Methoden der linearen Planungsrechnung verwendet. Dabei ersetzt man die nichtlinearen Funktionen durch lineare Streckenzüge. Für diese polygonale Approximation werden in der Literatur die sog.-Form und die sog.-Form vorgeschlagen, die jedoch eine aufwendige Datenorganisation erfordern. In diesem Beitrag soll eine kompaktere Organisationsform (-Form) vorgeschlagen werden. Zu ihrer Durchführung werden die Rechenregeln der Upper-Bounding-Technique und die Regeln zur nachträglichen Änderung von Matrixelementen verwendet. Der Vorteil gegenüber der-Form und der-Form liegt in einem wesentlich geringeren Speicherplatzbedarf, im niedrigeren Rechenaufwand und in einer größeren Flexibilität.

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Summary Separable programming problems can be solved by modifications of the simplex method, when the single functions are replaced by linear polygonal approximations. Two forms of different data organization are discussed in the literature: the-form and the-form. In this paper a more compact data organization (-form) is suggested. It uses the upper bounding technique and the formulas for the subsequent alteration of matrix elements. The advantage of this method over both the-form and the-form is due to much less storage space, to a reduced computation time, and to a higher flexibility.

     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Bonds for a functional on a class of functions with positive real parts

We establish sharp upper and lower bounds for the functionalJ (p)- Re{p (z)-zp'(z)/p(z), 1. where ¦z¦=r is fixed, 0 <r <1, p(z).P_n (A, B), –1 B < a 1, a certain class of regular functions in the disk with values in the right halfplane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 686–689, May, 1990.     

A proof for Theodorus' theorem by drawing diagrams

Summary In Plato's dialogue Theaetetus, the mathematician Theodorus (ca. 470-400 BC) is said to have proved the irrationality of 3, 5, ..., 17 by drawing diagrams. In this paper such a proof is presented and compared with another, more arithmetical, reconstructed proof by W. Knorr. Moreover a hypothetical early theory of similar rectangles is extracted from Euclid's Elements, which could have served as a basis for the geometrical proof of Theodorus' theorem.Herrn N. Stephanidis in Freundschaft gewidmet     

On Asymptotic Structure,the Szlenk Index and UKK Properties in Banach Spaces

Let B be a separable Banach space and let X=B ^* be separable. We prove that if B has finite Szlenk index (for all > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if > 0 there exists () > 0 so that if x _n is a sequence in the ball of X converging ^* to x so that . In addition we show that the norm can be chosen so that () c^p for some p < and c >0.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Average Number of Real Roots of Random Harmonic Equations

Let {g_k}be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let _k (t ) (k = 0, 1, 2,...) be the normalized Jacobi polynomials orthogonal with respect to the interval [ – 1, 1 ]. Then it is proved that the average number of real roots of the random equations, _k=0 ⁿ g_kk(1)=C where Cis a constant, is asymptotically equal to n/in the same interval when nis large and even for C as long as C=O (n ²).     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

Transportation cost for Gaussian and other product measures

Consider the canonical Gaussian measure _N on , a probability measure on ^N , absolutely continuous with respect to _N . We prove that the transportation cost of to _N , when the cost of transporting a unit of mass fromx toy is measured by x–y², is at most d. As a consequence we obtain a completely elementary proof of a very sharp form of the concentration of measure phenomenon in Gauss space. We then prove a result of the same nature when _N is replaced by the measure of density 2^–N exp (– _iN |x _i |). This yields a sharp form of concentration of measure in that space.Work partially supported by an NSF Grant.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

Convolutions of random measures on a compact topological semigroup

We investigate the asymptotic behavior of a sequence of convolutionsv ⁽ⁿ ₁() ** _n (), where { _n } _n=1 is some random process taking values in a semigroupM ¹(S) of probability Borel measures on a compact topological semigroupS.     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth