Schwarz-Pick Inequalities for Hyperbolic Domains in the Extended Plane

Let and be two hyperbolic simply connected domains in the extended complex plane C¯ = C¯ {}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: at a point z ₀ . The bounds depend on the densities (z ₀) and (f(z ₀)) of the Poincaré metrics and on the hyperbolic distances of the points z ₀ and f(z ₀) to the point .     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

The odd ordér analogue of Dempwolff's B-group problem

Let be a translation plane of odd order p^2R for p a prime. Let admit at least two p-groups, B_i, i=1, 2, B₁B₂ of orders > p^R/2 which fix Baer subplanes _i, i = 1,2, ₁₂ pointwise. It is shown that under these assumptions must be a Hall plane.     

Doubly β-Derived Translation Planes

Existence and uniqueness of a doubly -derived translation plane of order 49 are proved. Furthermore, we give a complete classification of those translation planes of order 49 which can be obtained from the desarguesian plane of order 49 by a mixed double derivation, namely by applying a -derivation on and a classical derivation (also called Ostrom's derivation or -derivation) on .     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

A remark on covering graphs

Every graph G may be transformed into a covering graph either by deletion of edges or by subdivision. Let _E (G) and _V (G) denote corresponding minimal numbers. We prove _E (G) = _V (G) for every graph G.     

Some extremal results on circles containing points

We define (n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y P, where every circle containingx andy contains (n) points ofP. We establish lower and upper bounds for (n) and show that [n/27]+2(n)[n/4]+1. We define for the special case where then points are restricted to be the vertices of a convex polygon. We show that .     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

Interpolation and best polynomial approximation in the domain ofp-adic integers

, , . , . Lip

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The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.     

Pictures of monotone operators

Let E be a real Banach space with dual E ^*. We associate with any nonempty subset H of E×E ^* a certain compact convex subset of the first quadrant in ², which we call the picture of H, (H). In general, (H) may be empty, but (M) is nonempty if M is a nonempty monotone subset of E×E ^*. If E is reflexive and M is maximal monotone then (M) is a single point on the diagonal of the first quadrant of ². On the other hand, we give an example (for E the nonreflexive space L ₁[0,1]) of a maximal monotone subset M of E×E ^* such that (0,1)(M) and (1,1)(M) but (1,0)(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type (NI) — a class of multifunctions from E into E ^* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S+J is surjective. Our main tool is a classical minimax theorem.     

On regular embedding integrally in R3 of metrics of class C4 of negative curvature

On the x₀y plane let there be specified a complete metric of negative curvature K by means of the line element ds²=dx²+B²(x, y) dy², and, in the strip _a={0xa, -4-bounded function B>0,K-²<0 ( and are constants). Then, the metric in strip _a is embedded in R³ by means of a surface of class C³.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 261–266, August, 1973.     

Über lokalarchimedische Anordnungen projektiver Ebenen

The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that is Archimedean, i.e. that every ternary ring of is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

Π 2 1 -Logic and uniformization in the analytical hierarchy

We give a simple proof of ₁ ¹ and ₂ ¹ uniformization results, which is based on the use of ₂ ¹ -Logic.     

О метрических свойст вах особых множеств г оломорфных функций нескольких п еременных

LetG be a domain inC ⁿ ,EG, mes E=0 for (r)=r ^2n–1(r), where (r) is a nondecreasing non-negative function (r>0). Iff(z) is holomorphic inGE and (,f, GE)(), C=const, thenf(z) is holomorphic inG.The impossibility of the relaxation of the stipulations on () and(r) is also established.The statement above is a corollary to a more general result about the representation of a holomorphic function from a certain class in the form of an integral with respect to -measure, extended over the set of singular points of the function.     

Sufficient Conditions for the Existence of Disintegrations

Let be a probability space and a partition of . A necessary and sufficient condition is given for the existence of a -additive and measurable disintegration of P on . It is also shown that P admits a -additive (but not measurable) disintegration on whenever is a standard space and the set (₁, ₂):₁ and ₂ are in the same element of } is coanalytic in ×. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using -additive disintegrations as conditional probabilities.     

Pencils of Hyperconics in Projective Planes of Characteristic Two

Let F be any field, finite or infinite, of characteristic 2. Put =PG(2,F). Let H ₁,H ₂ be hyperconics in . In this note we study the intersectionH ₁ H ₂. In particular we obtain canonical forms for H ₁,H ₂ in the cases where |H ₁ H ₂|=4,5,6. One interesting consequence is that the case |H ₁ H ₂|=6 can only occur if F contains a subfield of order 4. Related results concerning pencils of hyperconics are presented in Theorems 6 through 9. This work also leads to an extension to general fields of characteristic 2 of the well-known even intersection property for hyperovals in PG(2,4) which is pursued elsewhere ([2]).     

On the Theorem of Maschke

It will be shown that the Theorem of Maschke can be carried over to certain classes of topological groups in the following way. Let A be a locally compact abelian strictly -divisible group, and let G be a compact totally disconnected -group of automorphisms of A, where is a set of prime numbers. If a G-admissible subgroup B of A is isolated as a direct summand, then it has a G-admissible complement in A.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 151–154, August, 1968.     

О Сходимости метода подобластей

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