Spectral orders and convolution

. (R) _{– –} fg(y)h(x–y) dx dy _{– –} f ^{^} (x)g ^{^} (y)h ^{^} (x–y)dx dy (f,g0) —:f×gf ^{^} ×g ^{^}(f,g 0) f^{^} g^{^} f g -, X — . , - f ₁f ₂ , f ₁ ^{^} ×gf ₂×g 0g. .     

Über Untergruppen Endlicher Algebraischer Gruppen

Let k be a commutative ring, GG finite affine algebraic k-groups, and HH the dual Hopfalgebras of the affine algebras of G resp. G. The main results of this paper are: (I) If k is semilocal (e.g. k a field) there is an H-linear, HH-colinear, unitary, augmented isomorphism HHH H, where HH is the coalgebra belonging to G/G. (II) If the k-submodule of the fixelements of (HH)^* is isomorphic to k (e.g. k principal or semilocal), then HH is a Frobeniusextension of the second kind.     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

Continuation of functionals on ultradifferentiable function spaces

, c _k b _k . . . .

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This work is supported by N.B.H.M. grant No. 48/1/94-R&D-II.     

Symmetric Spectra and Topological Hochschild Homology

A functor is defined which detects stable equivalences of symmetric spectra. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bökstedts original ad hoc definition. In particular, this shows that Bökstedts definition is correct even for nonconnective, nonconvergent symmetric ring spectra.     

Some Issues in the Theory of Optimal Algorithms and Informational Complexity

The article presents the basic concepts and reviews the main results of the theory of optimal algorithms and informational complexity. Informational complexity bounds are provided for Lipschitzian multi-criterion problems that construct the approximate Pareto-optimal strategy set under different interpretations of approximation—approximation by the functional and approximation by the argument. The informational complexity is compared for the scalar global optimization problem and the problem of finding the roots of nonlinear equations by global search methods.     

Markuševič decompositions and quasireflexivity of Banach spaces

. , , . .     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Mixed-norm estimates for a class of integral operators

(, ) — R ^m ×R ⁿ . f R ^m ×R ⁿ f_p,q, f L _p (R ^m) x y, L^q(Rⁿ). ׃ _q,r cƒ _p,r , ׃ R ^m ×R ⁿ , , , q r . , ( ¦¦) K ₀ (y); p, g r , K ₀.     

Notes on best constants in some Sobolev inequalities

, . .     

On convex multipliers of convergence of some classes of bivariate fourier series

- ()N²,L _F ( ) — , 2- , {s _m() f} -L. — . (L _F( ),L _F( ) ={(k)} (kZ²) , fL_F( ) f , , L _F( ). - ={()} ={()} , n(())m()n(()+()) . R() , .. - . , . (L _F ( ),L _F ( )) , R(,)=O(1) (x).

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The author wishes to express his gratitude to S. A.Teljakovski for setting the problem and for his attention to this paper.     

Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper

Becker has shown in [1] that for the 4-th Pythagoras number of the field (X) the inequality P₄ ((X)) 36 holds. In this paper we will show P₄ ((X)) 24 and P₄ (K) 3 for all real pythagorean fields K.     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

The skew-hyperbolic motion group of the quaternion plane

Up to conjugation, there exist three different polarities of the projective plane over Hamilton's quaternions . The skew hyperbolic motion group of P₂ is introduced as the centralizer of a polarity of the third kind. According to a result of R. Löwen, the quaternion plane is characterized among the eight-dimensional stable planes by the fact that it admits an effective action of the centralizer of a polarity of the first or second kind (i.e., the elliptic or the hyperbolic motion group). In the present paper, we prove the analogous result for skew hyperbolic case.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Distribution of the number of nonappearing lengths of cycles in a random mapping

One-to-one random mappings of the set 1, 2,..., n onto itself are considered. Limit theorems are proved for the quantities _i, 0in, max _i, min _i, where _i is the number of 0in components of the vector ( ₁, ₂,..., _n) which are equal to i, 0< i< n, and ar is the number of components of dimension r of the random mapping.Translated from Matematicheskle Zametki, Vol. 23, No. 6, pp. 895–898, June, 1978.The author is grateful to V. P. Chistyakov and V. E. Stepanov for many useful remarks.     

A Characterization of Some Minihypers and its Application to Linear Codes

Any {f,r- 2+s; r,q}-minihyper includes a hyperplane in PG(r, q) if fr-1 + s 1 + q – 1 for 1 s q – 1, q 3, r 4, where i = (qi + 1 – 1)/ (q – 1 ). A lower bound on f for which an {f, r – 2 + 1; r, q}-minihyper with q 3, r 4 exists is also given. As an application to coding theory, we show the nonexistence of [ n, k, n + 1 – qk – 2 ]q codes for k 5, q 3 for qk – 1 – 2q – 1 < n qk – 1 – q – 1 when k > q – q - \sqrt q + 2$$ " align="middle" border="0"> and for when , which is a generalization of [18, Them. 2.4].     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

НАИЛУЧШЕЕ ПРИБЛИЖЕ НИЕ УГЛОМ И ПРИБЛИЖЕНИЕ УГЛОМ ИЗ СИНГУЛЯРНЫХ ИНТЕГРАЛОВ ФУНКЦИЙf∈L p (R n ),2<P<∞

. .