On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

Some new invariant pairs (t,3) for projective hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair (t,r): t is the number of neighbours of a given point on a given line passing through it and r is the order of the underlying projective plane. The Drake-Lenz method [2],[3] of using auxiliary matrices for the constructions of projective Hjelmslev planes has become standard by now. This paper is intended to give some new constructions of projective Hjelmslev planes with invariant pairs (t,3) by making use of the generalization and improvement of the Drake-Lenz theorem [3] obtained by the author in [6] and [7]. The results of this paper add 8 new values to the list ([5], example 3.7(ii)) of invariant pairs (t,3) with t 1,000 for projective Hjelmslev planes.     

Posets with large dimension and relatively few critical pairs

The dimension of a poset (partially ordered set)P=(X, P) is the minimum number of linear extensions ofP whose intersection isP. It is also the minimum number of extensions ofP needed to reverse all critical pairs. Since any critical pair is reversed by some extension, the dimensiont never exceeds the number of critical pairsm. This paper analyzes the relationship betweent andm, when 3tmt+2, in terms of induced subposet containment. Ifmt+1 then the poset must containS _t , the standard example of at-dimensional poset. The analysis form=t+2 leads to dimension products and David Kelly's concept of a split. Whent=3 andm=5, the poset must contain eitherS ₃, or the 6-point poset called a chevron, or the chevron's dual. Whent4 andm=t+2, the poset must containS _t , or the dimension product of the Kelly split of a chevron andS _t–3, or the dual of this product.     

The Second Lacuna in the Orders of Movement Groups of the Spaces U n

We prove the nonexistence of symmetrically linearly connected spaces of hyperplane elements admitting a movement group G _r with n ² - n + 2 r n² - 1, n 5.     

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.     

Functional inequality characterizing nonnegative concave functions in (0, ∞) k

Summary In the present note we prove that every functionf: (0, ) [0, ) satisfying the inequalityaf(s) + bf(t) f(as + bt), s, t > 0, for somea andb such that 0 <a < 1 <a + b must be of the formf(t) = f(1)t, (t > 0). This improves our recent result in [2], where the inequality is assumed to hold for alls, t 0, and gives a positive answer to the question raised there.An analogue for functions of several real variables of the above result characterizes concave functions. Conjugate functions and some relations to Hölder's and Minkowski's inequalities are mentioned.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Extreme Geodesic Graphs

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u–v geodesic of G, while for S V(G), the set I[S] is the union of all sets I[u, v] for u, v S. A set S of vertices of G for which I[S] = V(G) is a geodetic set for G, and the minimum cardinality of a geodetic set is the geodetic number g(G). A vertex v in G is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in G is its extreme order ex(G). A graph G is an extreme geodesic graph if g(G) = ex(G), that is, if every vertex lies on a u–v geodesic for some pair u, v of extreme vertices. It is shown that every pair a, b of integers with 0 a b is realizable as the extreme order and geodetic number, respectively, of some graph. For positive integers r, d, and k 2, it is shown that there exists an extreme geodesic graph G of radius r, diameter d, and geodetic number k. Also, for integers n, d, and k with 2 d > n, 2 k > n, and n – d – k + 1 0, there exists a connected extreme geodesic graph G of order n, diameter d, and geodetic number k. We show that every graph of order n with geodetic number n – 1 is an extreme geodesic graph. On the other hand, for every pair k, n of integers with 2 k n – 2, there exists a connected graph of order n with geodetic number k that is not an extreme geodesic graph.     

Time-optimal controls of the equation of evolution in a separable and reflexive Banach space

Let a variable, closed, bounded, and convex subset ofX, a separable and reflexive Banach space, be denoted byG(t). Suppose thatG(t) varies upper-semicontinuously with respect to inclusion ast varies in [0,T]. We say that the strongly measurable mapu from [0,T] toX is an admissible control if, for almost everyt in [0,T],u(t) is an element ofU, a closed, bounded, and convex subset ofX, and u _p M ^1p, where p>1 andM>0.Ifx ^u is the weak solution todx/dt+A(t)x=u(t), 0tT, whereA(t) is as defined by Tanabe in Ref. 1, we say that the responsex ^u to the controlu hits the target in timeT ^u ifx ^u (0)=0 andx ^u (T ^u ) is an element ofG(T ^u ). If there is a control with this property, then there is a time-optimal control.     

Supplement to the paper “the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)”

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let tL₁(0,1), let N_t be the smallest rearrangement-invariant order ideal vector lattice in L₁(0,1), containing t. In the paper one investigates the properties of the image E(N_t¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, gL₁(0,1), such that E(N_t¦A)N_g. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(tA)QE(t*A*), where is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1Q2⁵. One formulates the problem of the smallest value of Q.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 137–141, 1986.     

Maximum principle and second-order conditions for minimax problems of optimal control

In this paper, we study the optimal control problem of minimizing the functionalJ(x, u)=max_t1tt2(x(t),t). We formulate and prove necessary optimality conditions for this problem. We establish the equivalence between the initial minimax problem and a problem involving a terminal functional and phase constraints.     

On trees and tree dimension of ordered sets

We call an ordered set (X, ) a tree if no pair of incomparable elements ofX has an upper bound. It is shown that there is a natural way to associate a tree (T, ) with any ordered set (X, ), and (T, ) can be characterized by a universal property. We define the tree dimensiontd(X, ) of an ordered set as the minimal number of extensions of (X, ) which are trees such that the given order is the intersection of those tree orders. We give characterizations of the tree dimension, relations between dimension and tree dimension, and removal theorems.     

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

It is shown that for a compact Desarguesian projective Klingenberg plane P with incidence structure P=#x2119;, , I and neighbour relation , where two distinct points always lie on some line, exactly one of the following holds: P is a non-discrete connected or totally disconnected ordinary projective plane with =id, P is a finite projective plane with =id, P is a finite projective Hjelmslev plane with id, or P is a non-discrete totally disconnected ordinary projective plane with id.Dedicated to H. Salzmann on his 60th birthdayThe author wishes to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial assistance in the writing of this paper.     

Estimates in Minkowski's nonhomogeneous conjecture for small dimensions

One gives estimates from above for the nonhomogeneous arithmetic minimum M _n which are better for n, I.I · I0 than the known ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 106, pp. 104–133, 1981.     

Weighted inequalities for integral operators with some homogeneous kernels

In this paper we study integral operators of the form

Automorphism groups of comparability and covering graphs

Let G be a group and H a subgroup of G. It is shown that there exists a partially ordered set (X, ) such that G is isomorphic to the group of all automorphisms of the comparability graph of (X, ) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (X, ). There also exists a partially ordered set (Y, ) such that G is isomorphic to the group of all automorphisms of the covering graph of (Y, ) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (Y, ). In this representation X and Y can be taken to be finite if G is finite and of the same cardinality as G if G is infinite.     

On well-ordered mono-unary algebras

Given a linearly ordered set (A, _R ) and an R-monotone function f: AA, we give a necessary and sufficient condition on A, f, _R , involving generating sets and forbidden subalgebras, for _R to be a well-ordering.Partially supported by Hungarian National Foundation for Scientific Research Grant nr. 1813.     

One-third-integrality in the max-cut problem

Given a graphG = (V, E), the metric polytopeS (G) is defined by the inequalitiesx(F) – x(CF) |F| – 1 for , |F| odd,C cycle ofG, and 0 x _e 1 fore E. Optimization overS (G) provides an approximation for the max-cut problem. The graphG is called 1/d-integral if all the vertices ofS(G) have their coordinates in{i/d 0 i d}. We prove that the class of 1/d-integral graphs is closed under minors, and we present several minimal forbidden minors for 1/3-integrality. In particular, we characterize the 1/3-integral graphs on seven nodes. We study several operations preserving 1/d-integrality, in particular, thek-sum operation for 0 k 3. We prove that series parallel graphs are characterized by the following stronger property. All vertices of the polytopeS (G) {x x u} are 1/3-integral for every choice of 1/3-integral bounds, u on the edges ofG. Research by this author was partially done at CWI in Amsterdam.Research by this author was done at the Institut für Diskrete Mathematik of Bonn, supported by the A. von Humboldt Foundation.Deceased on April 2nd, 1995.     

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ (S denotes a monoid, and is a right congruence onS), and determined conditions onS under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which is generated by a single pair of elements ofS), and further results on such collapsing of flatness properties are obtained. An observation which is used extensively in this study is the fact that forw andt inS withwtt,S/(wt,t) is flat if and only ift is a regular element ofS.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A4494.Research supported by Estonian Research Foundation Grant No. 930.