A graph-theoretical representation of PL-manifolds — A survey on crystallizations

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati.The original concept is that of contracted triangulation, which was introduced with the main goal of finding a minimal atlas for topological manifolds ([P₁ 1968], [P₂ 1974], [P₃ 1974], [FG₂ 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P₄ 1975], [F₁ 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject.For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

The Århus integral of rational homology 3-spheres III: Relation with the Le–Murakami–Ohtsuki invariant

Continuing the work started in [Å-I] and [Å-II], we prove the relationship between the Århus integral and the invariant (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in [LMO]. The basic reason for the relationship is that both constructions afford an interpretation as integrated holonomies. In the case of the Århus integral, this interpretation was the basis for everything we did in [Å-I] and [Å-II]. The main tool we used was formal Gaussian integration. For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map j_m, as formal negative dimensional integration. The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

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.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

Surgery on 3-manifolds with \mathbb{S} 1-actions

We study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable -manifold. As a consequence, we give alternative proofs of some classical results due to W. Heil and L. Moser. Moreover, we completely specify the Seifert invariants for the considered manifolds. Finally we classify the manifolds obtained by surgery along certain Seifert links and determine geometric presentations of their fundamental groups.Work performed under the auspices of C.N.R. (National Research Council) of Italy and partially supported by Ministero della Ricerca Scientifica e Tecnologica within the projects Geometria Reale e Complessa and Topologia.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p q and it is also independent of the classification of the arc-transitive graphs of order pq for p q.     

Shape-preserving Kolmogorov widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ₀ and let ₊ ^s be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t ₀,..., t _s I, the corresponding divided differences [x; t ₀,...,t _s ] of order s are nonnegative. Let ₊ ^s B _{p +} ^s B _p, 1 p where B _p is a unit ball in the space L _p, and let ₊ ^s L _{q +} ^s L _q, 1 q . For every s 3 and 1 q p , we determine the exact orders of the shape-preserving Kolmogorov widths {x - y} \right\ L_q , $$]]>, where M ⁿ is the collection of all affine linear manifolds M ⁿ in L _q such that dim M ⁿ n and M ⁿ ₊ ^s L _q .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.     

Anwendungen der Nielsenschen Kürzungsmethode in Gruppen mit einer definierenden Relation

LetG=a ₁,b ₁, ...,a _q ,b _q | (W ₁ (a ₁,b ₁) ...W _q (a _q b _q ))=1,q1, 2,W _j (a _j ,b _j )1. We solve the isomorphism problem forG inasmuch as we can decide in finitely many steps, if any arbitrary one-relator group is isomorphic toG or not. FurthermoreG turns out to have a finitely generated automorphism group. Forq=1 this was proved byS. J. Pride. Forq2 the proof is based onNielsen's reduction method. There are some other interesting results on subgroup problems for one-relator groups obtained by this method.     

The Volterra Operator is not Supercyclic

It is shown that the classical Volterra operator, which is cyclic, is not supercyclic on any of the spaces L^p[0, 1], 1 p < . This solves a question posed by Héctor Salas. This contrasts with the fact that the derivative operator, the left inverse of the Volterra operator, although unbounded, is hypercyclic on L^p[0, 1].     

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.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Non removable edges in 3-connected cubic graphs

LetG be a cyclicallyk-edge-connected cubic graph withk 3. Lete be an edge ofG. LetG be the cubic graph obtained fromG by deletinge and its end vertices. The edgee is said to bek-removable ifG is also cyclicallyk-edge-connected. Let us denote by S _k (G) the graph induced by thek-removable edges and by N _k (G) the graph induced by the non 3-removable edges ofG. In a previous paper [7], we have proved that N ₃(G) is empty if and only ifG is cyclically 4-edge connected and that if N ₃(G) is not empty then it is a forest containing at least three trees. Andersen, Fleischner and Jackson [1] and, independently, McCuaig [11] studied N ₄(G). Here, we study the structure of N _k (G) fork 5 and we give some constructions of graphs such thatN _k (G) = E(G). We note that the main result of this paper (Theorem 5) has been announced independently by McCuaig [11].

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Résumé SoitG un graphe cubique cyliquementk-arête-connexe, aveck 3. Soite une arête deG et soitG le graphe cubique obtenu à partir deG en supprimante et ses extrémités. L'arêtee est ditek-suppressible siG est aussi cycliquementk-arête-connexe. Désignons par S _k (G) le graphe induit par les arêtesk-suppressibles et par N _k (G) celui induit par les arêtes nonk-suppressibles. Dans un précédent article [7], nous avons montré que N ₃(G) est vide si et seulement siG est cycliquement 4-arête-connexe et que si N ₃(G) n'est pas vide alors c'est une forêt possédant au moins trois arbres. Andersen, Fleischner and Jackson [1] et, indépendemment, McCuaig [11] ont étudié N ₄(G). Ici, nous étudions la structure de N _k (G) pourk 5 et nous donnons des constructions de graphes pour lesquelsN _k (G) = E(G). Nous signalons que le résultat principal de cet article (Théorème 5) a été annoncé indépendamment par McCuaig [11].

     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

Set systems with no union of cardinality 0 modulom

Letq be a prime power. It is shown that for any hypergraph = {F ₁,...,F _d(q–1)+1} whose maximal degree isd, there exists Ø ₀ , such that 0 (modq).     

Note on the existence of large minimal blocking sets in galois planes

A subsetS of a finite projective plane of orderq is called a blocking set ifS meets every line but contains no line. For the size of an inclusion-minimal blocking setq+ +Sq +1 holds ([6]). Ifq is a square, then inPG(2,q) there are minimal blocking sets with cardinalityq +1. Ifq is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3q points. In the present note we show that inPG(2,q),q1 (mod 4) there are minimal blocking sets having more thanqlog₂ q/2 points. The blocking sets constructed in this note contain the union ofk conics, whereklog₂ q/2. A slight modification of the construction works forq3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog₂ q for some constantc.As a by-product we construct minimal blocking sets of cardinalityq +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of parabolas, they are not classical.     

Characterization of {(q + 1) + 2, 1;t,q}-min · hypers and {2(q + 1) + 2, 2; 2,q}-min · hypers in a Finite projective geometry

In this paper, we shall characterize all {(q + 1) + 2, 1;t, q}-min · hypers and all {2(q + 1) + 2, 2; 2,q}-min · hypers for any integert 2 and any prime powerq 3. In the next paper [8], we shall characterize all {2(q + 1) + 2, 2;t, q}-min · hypers for any integert 3 and any prime powerq 5 using the results in this paper.