A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

D-strong and almostD-strong near-rings

In this paper,D-strong and almostD-strong near-rings have been defined. It has been proved that ifR is aD-strongS-near ring, then prime ideals, strictly prime ideals and completely prime ideals coincide. Also ifR is aD-strong near-ring with identity, then every maximal right ideal becomes a maximal ideal and moreover every 2-primitive near-ring becomes a near-field. Several properties, chain conditions and structure theorems have also been discussed.Most of the parts of this paper are included in author's doctoral dissertation at Sukhadia University Udaipur (1983). The author expresses his gratitude to Dr.S. C. Choudhary for his kind guidance.     

Equivalence of two series of spherical representations of a free group

Summary The spherical principal series of a non-commutative free group may be analytically continued to yield a series of uniformly bounded representations, much as the spherical representations (in1/2) + it of SL (2,R) may be analytically continued in the strip 0 < Rez < 1. This series of uniformly bounded representations was constructed and studied by A. M.Mantero and A.Zappa. Independently T.Pytlik and R.Szwarc introduced and studied representations of the free group which contain a series of subrepresentations indexed by spherical functions. Both series consist of irreducible representations and include the spherical complementary series. The aim of this paper is to prove that the non-unitary uniformly bounded representations of the two series are also equivalent.     

A robust and efficient algorithm for planar competitive location problems

In this paper we empirically analyze several algorithms for solving a Huff-like competitive location and design model for profit maximization in the plane. In particular, an exact interval branch-and-bound method and a multistart heuristic already proposed in the literature are compared with uego (Universal Evolutionary Global Optimizer), a recent evolutionary algorithm. Both the multistart heuristic and uego use a Weiszfeld-like algorithm as local search procedure. The computational study shows that uego is superior to the multistart heuristic, and that by properly fine-tuning its parameters it usually (in the computational study, always) find the global optimal solution, and this in much less time than the interval branch-and-bound method. Furthermore, uego can solve much larger problems than the interval method.     

On the coefficients of functions analytic in the unit disc having fast rates of growth

Summary The present paper is concerned with functions analytic in the unit disc having rapidly increasing maximum moduli. To study the precise rates of growth of such functions the concept of index is introduced. Several growth parameters in terms of the index are defined for a function analytic in the unit disc and their characterizations in terms of the Taylorseries development of the function are obtained. The results in the present paper improve and refine the earlier results ofSons (J. Math. Anal. Appl,24 (1968), pp. 296–306),MacLane (Asymptotic values of [holomorphic functions,Rice University Studies, Houston, 1963), andKapoor andJuneja (Indian J. Pura Appl. Kath.,7 (3) (1976), pp. 241–248). Entrata in Redazione il 7 giugno 1978.     

Union congruence in a subspace of a Finsler space

Summary The union curves of a Riemannian space were studied bySpringer [8],Misra [2] andUpadhyay [9]. In a Finsler space, these curves have been studied byPrakash-Behari [4],Sinha [7],Mishra-Sinha [3] andSingh [6]. In the present paper, we wish to extend the concept of union curves in the Finsler space and as such the concept of union congruence has been discussed. The two types of union curves of the Finsler subspace are the particular cases of these curves. These are also generalization of union curves and union curvatures of a vector-field of Finsler space. It has also been shown that the λ-geodesics [5] are special case of these curves. Entrata in Redazione il 14 settembre 1970.     

Positive functionals on BP*-algebras

A new class of locally convex algebras, called BP^*-algebras, is introduced. It is shown that this class properly includes MQ^*-algebras which were introduced and studied by the first author andR. Rigelhof [10]. Among other results, it is proved that each positive functional on a BP^*-algebraA is admissible but not necessarily continuous as shown by an example. However, ifA, in addition, is either (i) a Q-algebra, or (ii) has an identity and is barrelled, or (iii)A is endowed with the inductive limit topology, then each positive functional onA is continuous.This work was supported by an N.R.C. Grant.     

Isodynamic & isogonic simplexes

Summary A simplex in an n-space and its tangential simplex formed of the n+1 tangent hyperplanes of its circumhypersphere (0) at its vertices are polar reciprocal of each other w. r. t. (0). The n+1 joins of their corresponding vertices, in general, do not concur [3, p.41, Ex.7;4;18;24;25]. But when n=2, the3 joins of the corresponding vertices of a triangle and its tangential triangle always concur at itsLemoine point L as its3 symmedians which are the isogonal conjugates of its medians w. r. it [7] such that its circumcircle coincides with the polar conic of L w. r. t. it [9]. For n=3, the4 joins of the corresponding vertices of a tetrahedron and its tangential tetrahedron concur at itsLemoine point L, if and only if it is isodynamic [5], as its4 Lemoinians [17] (called symmedians byCourt [6] which join its vertices to theLemoine points of its opposite faces (but not as the isogonal conjugates of its medians w. r. t. it) such that its circumsphere coincides with the polar quadric [16] of L w. r. t. it. The purpose of this paper is to develope analogonsly the theory of an isodynamic simplex, in an n-space (n>3), which is in rerspective with its tangential simplex, called isogonic under the circumstances. Their relationship as cevian & anticevian simplexes, and their association with S-configurations and related cevian quadrics are also pointed out. To Eurico Bompiani on his scientific Jubilee     

Modelling NASDAQ Series by Sparse Multifractional Brownian Motion

The objective of this paper is to compare the performance of different estimators of Hurst index for multifractional Brownian motion (mBm), namely, Generalized Quadratic Variation (GQV) Estimator, Wavelet Estimator and Linear Regression GQV Estimator. Both estimators are used in the real financial dataset Nasdaq time series from 1971 to the 3rd quarter of 2009. Firstly, we review definitions, properties and statistical studies of fractional Brownian motion (fBm) and mBm. Secondly, a numerical artifact is observed: when we estimate the time varying Hurst index H(t) for an mBm, sampling fluctuation gives the impression that H(t) is itself a stochastic process, even when H(t) is constant. To avoid this artifact, we introduce sparse modelling for mBm and apply it to Nasdaq time series.     

Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum

Summary The starting point of the present paper is a simple method ofPayne-Weinberger to obtain lower bounds for the first eigenvalue of a membrane through comparison with the Rayleigh principle for homogeneous strings. A generalization of this idea to auxiliary non-homogeneous strings leads to a maximum principle (7) (of which an inequality ofBarta-Pólya is a special case) or (7); in its form (8), it was essentially known (although with some unnecessary restrictive hypotheses) toM. H. Protter in 1958. — This principle is closely related to theThomson principle of boundary value problems; in particular, the allowed discontinuities of the concurrent vector fields (in the formulation (7)) are the same as inThomson's principle.—Someapplications of the principle to the calculation of rather sharp lower bounds are indicated, as well as an intuitivephysical interpretation (in terms of variation of masses and constraints). It is valid inthree dimensions as well; we indicate its extension toSchrödinger's equation.     

On a Product of Finite Monoids

m , we associate with a finite monoid S ₀ and m finite commutative monoids S ₁ ,...,S _m , a product ◊ _m (S _m , ..., S ₁ , S ₀ ). We give a representation of the free objects in the pseudovariety ◊ _m (W _m , ..., W ₁, W ₀) generated by these (m + 1) -ary products where S ₁ ∈W _i for all 0 ≤i≤m . We then give, in particular, a criterion to determine when an identity holds in ◊ _m (J ₁, ..., J ₁, J ₁) with the help of a version of the Ehrenfeucht-Fra?ssé game (J ₁ denotes the pseudovariety of all semilattice monoids.). The union U _{m > 0} ◊ _m (J ₁, ..., J ₁, J ₁) turns out to be the second level of the Straubing's dot-depth hierarchy of aperiodic monoids.     

Valiron summability of the derived fourier series and its coniugate series

Summary Valiron [12] obtained very extensive generalisations ofBorel’s method. In this paper the author has consideredValiron summability. The results proved are equivalent toBorel’s for k=1.     

A theorem on the absolute Cesàro summability of factored fourier series

Summary In this paper, the author proves a theorem on the absoluteCesàro summability of factoredFourier series. His theorem extends a theorem ofMatsumoto and generalizes a theorem ofPrasad andBhatt.     

Critical sets of functions

Summary With the introduction of an alternate definition for critical point, this paper studies critical sets, as defined byW. M. Whyburn, in terms of certain related domains. Critical sets are divided into four classes. Type0 has the limit point property with respect to critical sets which are not type0; type1 and2 critical sets compare, respectively, to classical minimum and maximum points; type3 includes themin-max and flex type. This paper is a result of a study of critical sets made as a dissertation problem under the direction ofW. M. Whyburn, to whom the author is indebted for many helpful suggestions.     

Fictitious states,coupled laws and local time

Summary The work of Ray and Neveu has established that, for any transition function P on a countable set E, (i) there exists a best possible entrance boundary E ⁺ supporting a right continuous, strong Markov process X with transition function P and that (ii) the points y of E ⁺ are in one-one correspondence with the extremal entrance laws g ^y of P. Here, it is shown that, if a point y of E ⁺ is regular for itself, then the derived characteristic f ^y of the local time at y is a regular extremal entrance law coupled with g ^y in the sense of Neveu. Further, coupled laws arise only in this fashion. By using excursion theory, a simple explicit formula for f ^y in terms of g ^y may be obtained. The paper contains a conjecture about the intrinsic character of the Ray-Neveu topology and an example which shows emphatically that, in general, local time is not a derivative of occupation time.     

Online stochastic optimization under time constraints

This paper considers online stochastic combinatorial optimization problems where uncertainties, i.e., which requests come and when, are characterized by distributions that can be sampled and where time constraints severely limit the number of offline optimizations which can be performed at decision time and/or in between decisions. It proposes online stochastic algorithms that combine the frameworks of online and stochastic optimization. Online stochastic algorithms differ from traditional a priori methods such as stochastic programming and Markov Decision Processes by focusing on the instance data that is revealed over time. The paper proposes three main algorithms: expectation E, consensus C, and regret R. They all make online decisions by approximating, for each decision, the solution to a multi-stage stochastic program using an exterior sampling method and a polynomial number of samples. The algorithms were evaluated experimentally and theoretically. The experimental results were obtained on three applications of different nature: packet scheduling, multiple vehicle routing with time windows, and multiple vehicle dispatching. The theoretical results show that, under assumptions which seem to hold on these, and other, applications, algorithm E has an expected constant loss compared to the offline optimal solution. Algorithm R reduces the number of optimizations by a factor |R|, where R is the number of requests, and has an expected ρ(1+o(1)) loss when the regret gives a ρ-approximation to the offline problem.     

Notiz zu den quadratfreien Zahlen in arithmetischen Progressionen

Some lower estimates for the number of squarefree (or nearly squarefree) numbers in arthmetic progressions are proved; it follows that a sharpened version of a result ofPrachar on 3-free numbers holds true. For the least squarefree number of the serieskm ²+l the estimate <k ³⁺ is proved. A very simple proof of a theorem ofErdös andPrachar is also given.     

Generalized Minkowski formulas for closed hypersuperfaces in Riemann space

Summary This paper generalizedMinkowski formulas for a closed orientable hypersurface in aRiemann space with constant curvature which have been introduced byC. C. Hsiung, and studies on some properties of the hypersurface whose the v-th mean curvature is constant, on making use of the generalized formulas. To Enrico Bompiani on his scientific Jubilee     

Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

Summary Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

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Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet     

Some properties of components of bigraphs

For small values of n it is easily seen that the components of a bigraph on 2n points are themselves bigraphs of their point set.Simmons had asked whether this was always the case and had shown that if a bigraph on 2n points existed which had a component that was not in itself a bigraph then 2n16. In this paper we answer his question affirmatively by using a result ofLovász. We also show that any two components of a bigraph intersect. To the memory of A. Rényi