Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen

In [8] the author extended the concept of neighbouring functions (cp. [9]) to the case of several variables. Using these results it is shown that under some weak conditions a multiplicative functionf in two variables has a mean-value different from zero if and only if the two multiplicative functionsf ₁(n)=f(n, 1) andf ₂(n)=f(1,n) have mean-values different from zero. Applications to theorems ofDelange [3],Elliott [6] andDaboussi [1] are given.     

Generalized controllability and inertia theory. II

In a paper of the same title, Carlson and Hill [2] established results in inertia theory and controllability for a large class of linear transformations on the space of n × n hermitian matrices. Extensions of some of those results and some generalizations of theorems of Wimmer [12] and of Carlson and Loewy [3] are given here.     

Homology and cohomology with compact supports forq-convex spaces

Summary In this paper we give an elementary proof of basic vanishing properties for homology and cohomology with compact supports of q-complete spaces which follow from the results of H.Hamm [16], [17] and K.-H.Fieseler-L.Kaup [13]. At the same time we obtain new finiteness results for the homology and the cohomology with compact supports in the q-convex case, which is not treated in [16], [17] and [13]. Our work extends to general q-complete spaces recent papers of M.Coltoiu-N.Mihalache [8] and M.Coltoiu [7] which treated the case of Stein spaces (q=0). A typical result is the following: if X is a q-complete space of dimension n, then H_i (X, Z)=0 for i>n+q and H_n+q (X, Z) is free, if X is also purely dimensional and locally a set-theoretic complete intersection, then H _c ⁱ (X, Z)=0 for ic ⁿ –q(X, Z) is free. The vanishing of the cohomology with compact supports for q-complete spaces has as consequence Lefschetz-type theorems for singular spaces (the homology statements) proved by C.Okonek [24] using Goresky-MacPherson stratified Morse theory.     

On some theorems on circulant matrices

In [2] some theorems on n × n circulant matrices were introduced under the hypothesis n a prime number. We extend these theorems to the case n =2?r where r is a prime greater then 2.     

On the strong convergence of sequences of Halpern type in Hilbert spaces

ABSTRACT

In this paper, we introduce a concept of A-sequences of Halpern type where A is an averaging infinite matrix. If A is the identity matrix, this notion become the well-know sequence generated by Halpern's iteration. A necessary and sufficient condition for the strong convergence of A-sequences of Halpern type is given whenever the matrix A satisfies some certain concentrating conditions. This class of matrices includes two interesting classes of matrices considered by Combettes and Pennanen [J. Math. Anal. Appl. 2002;275:521–536]. We deduce all the convergence theorems studied by Cianciaruso et al. [Optimization. 2016;65:1259–1275] and Muglia et al. [J. Nonlinear Convex Anal. 2016;17:2071–2082] from our result. Moreover, these results are established under the weaker assumptions. We also show that the same conclusion remains true under a new condition.     

On the generalised jump of a function by its Fourier coefficients

Summary Author studies the summability (C,1+α+ρ) of the sequence nB_n(x) under weaker conditions than those ofMinakishisundaram [3] and thus generalises his theorems. on the ? Jump of a function ? and by applying a tauberian theorem obtaing a criteria for the (C, α+ρ) summability of the conjugate series.     

Orthogonal hyperspheres

Umbilical projection ([12], [14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres [17] andn+1 intersecting hyperspheres in ann-space [18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space [23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2 ⁿ pairs ofsemi-inverse [22] simplexes, perspective from their radical centreH, such that the 2 ⁿ primes of perspectivity coincide with their 2 ⁿ hyperplanes of similitude and form anS-configuration (S-C) [15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic [25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2 ⁿ Brocard diameters of the 2 ⁿ pairs of complementary simplexes of intersection concur atH. The hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net [15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2 ⁿ pairs ofcomplementary hyperspheres of intersection form the 2 ⁿ vertices of the dual [15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.Due inspiration is derived from the works ofCourt ([2]–[9]) on mutually orthogonal circles and spheres. Presented by G. Hajós     

Solutions in the large of Liénard systems with forcing terms

Summary By the use of general methods of functional analysis existence theorems in the large are proved for periodic solutions of systems of ordinary differential equations(1) of the Liénard type with forcing terms. These theorems extend previous results of the same authors in[7]. Dedicated to Prof.Dario Graffi on his 70th birthday Entrata in Redazione il 5 settembre 1975.     

Zero-sum problems with subgroup weights

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider ℤ _p ^d for a general d and subgroups of Z* _p as weights.     

Eigenvalues of symmetrizable matrices

New perturbation theorems for matrices similar to Hermitian matrices are proved for a class of unitarily invariant norms calledQ-norms. These theorems improve known results in certain circumstances and extend Lu's theorems for the spectral norm, see [Numerical Mathematics: a Journal of Chinese Universities, 16 (1994), pp. 177–185] toQ-norms. This material is based in part upon the third author's work supported from August 1995 to December 1997 by a Householder Fellowship in Scientific Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Office of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp.     

Problemi al contorno con condizioni omogenee per le equazioni quasi-ellittiche

Summary We are concerned with non-variational boundary value problems, with omogeneus boundary conditions, for linear partial differential equations of quasi-elliptic type in a bounded domain Θ in Rⁿ. It is well known that some of difficulties which arise in treating such problems, in comparison with ? regular ? elliptic problems, are connected with the presence of angular points in Θ: let us point out withB. Pini [32] that ? a bounded domain for which it is possible to assign a correct boundary value problem for a quasi-elliptic but not elliptic equation always has angular points ?. We suppose Θ is a cartesian product of a finite number of open sets and, in order to overcome the difficulties attached to the presence of angular points in Θ, taking as a model the two previous papers[33], [34] devoted to elliptic problems with singular data, we investigate the problem within suitable Sobolev weight spaces, connected with the angular points of Θ and included in the ones we have studied in[35]. Within such spaces we get existence and uniqueness theorems.

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Lavoro eseguito con contributo del C. N. R.

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Entrata in Redazione il 30 ottobre 1971.     

Linearly independent zeros of quadratic forms over number-fields

LetK be an algebraic number-field of degree [K:Q] =n 1 and letO denote some fixed order ofK. Let, be a quadratic form which represents zero for some. For the special caseK =Q,O =Z, theorems ofCassels and ofDavenport provide estimates for the magnitude (in terms of the coefficients off(x)) of a zero and of a pair of linearly independent zeros off, respectively. Recently,Raghavan extendedCassels' result to arbitraryK. In this article, a new proof ofDavenport's theorem for a pair of linearly independent zeros is given which not only provides explicit constants in the estimates but also extends to generalK. A refinement of this proof leads to effectively computable bounds for rational representations of a numbern0 byf.     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.     

涉及四个单形的一类不等式

本文研究了涉及四个单形的一类不等式问题.利用距离几何的理论和方法获得了两个单形的棱长与另两个单形的内点、中线、高、外接超球半径、内切超球半径、旁切超球半径以及n-1维侧面的体积、外接超球半径、内切超球半径的一类新的几何不等式.推广了文献([5])中的全部结果.     

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     

Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]).     

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.     

Teoremi merceriani relativi a trasformazioni regolari non negative e anche più generali

Sunto Ad una successione (complessa) |s_n| associamo la successione trasformata |t_n| mediante una matrice di sommazione regolare C; sono noti teoremi merceriani (per es.G. H. Hardy, E. R. Love) che dalla relazione s_n — qt_n → (1 — q)l deducono s_n → l, sotto la condizione che il moltiplicatore q sia interno a determinati campi In questa NOta si considera C regolare non negativa e, mediante l'introduzione di un' opportuna ? funzione di aderenza di un insieme in un altro ? e l'applicazione del classico teorema del ? nocciolo ? diK. Knopp, si determinano campi più ampi di quello circolare stabilito daE. R. Love; si estende il risultato anche a matrici C regolari più generali mediante teoremi più recenti diR. P. Agnew eA. Robinson; inoltre si perfeziona il classico teorema diJ. Mercer-G. H. Hardy.

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Summary Let |t_n| be trasformed, by a regular summation matrix C, of a (complex) sequence |s_n|. Mercerian theorems (e. g. G. H. Hardy, E. R. Love theorems) deduce s_n → l from s_n — qt_n → (1 — q)l if q lies in a suitable domain. This paper is concerned with regular non-negative C. A suitable ? adherence-function of a set into another ? is introduced and theKnopp's ? Core theorem ? leads to wider domains thanE. R. Love circle. The result is extended to more general regular matrices C by means ofR. P. Agnew andA. Robinson theorems. The classicalJ. Mercer-G. H. Hardy theorem is also improved.

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A Giovanni Sansone nel suo 70^mo compleanno.

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Il presente lavoro è stato oggetto di una comunicazione al VI Congresso Nazionale della Unione Matematica Italiana (Napoli, 11–16 settembre 1959).     

The DJL Conjecture for CP Matrices over Special Inclines

Drew, Johnson, and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n²/4]. While this conjecture has recently been disproved for completely positive real matrices, we show that this conjecture is true for n × n completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations.