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Let (A,S) be an Artin group and X a subset of S; denote by AX the subgroup of A generated by X. When A is of spherical type, we prove that the normalizer and the commensurator of AX in A are equal and are the product of AX by the quasi-centralizer of AX in A. Looking the associated monoids A+ and AX+, we described the quasi-centralizer of AX+ in A+ thanks to results in Coxeter groups. These two results generalize earlier results of Paris [J. Algebra 196 (1997) 369–399]. Finally, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin group and in the Coxeter group.

It is known that if an equation describes non-trivial one-parameter families of pseudo-spherical surfaces, its conservation laws, (generalized, nonlocal) symmetries and Backlund transformations can be studied by geometrical means [4, 10]. In this letter it is pointed out that there exist correspondences, or 'generalized Backlund transformations', between arbitrary solutions (satisfying some genericity conditions) of any two single equations describing pseudo-spherical surfaces. Then, the notion of a hierarchy of equations of pseudo-spherical type is introduced, and a theorem stating that there also exist correspondences between arbitrary solutions of any two such hierarchies is presented. A full account of these results appears elsewhere [12, 13].

We consider extremum problems for entire functions of exponential spherical type related to important extremum problems on the optimal point (the Chernykh point) in the sharp jackson inequality in the spaceL2(ℝn) and the connection between codes and designs on the torusTn.

We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As a consequence we obtain a classification of twin buildings whose rank 2 residues correspond to split algebraic groups over a field of cardinality at least 4. A similar result follows for twin buildings whose rank 2 residues are finite.

We consider the point-line geometries that arise as a shadow space in a spherical building with a diagram of type An, Bn, Cn, Dn or En, and determine in which cases the geometry is spanned by the set of points on an apartment. It turns out that this happens precisely in the cases corresponding to a minimal weight.

Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial., Comment: 24 pages, 8 figures, 2 tables. Major revision

For an incidence geometry Δ, its geometric cycle code H (Δ) is the top-dimensional homology group over F 2 recognized as a subspace of the chain group [5]. Minimal supports, the subchamber systems corresponding to minimal-weight vectors, of H (Δ) are worthy of study, because those for buildings Δ of spherical type coincide with apartments of Δ (Theorem 1). The minimal supports of H (Δ) for the sporadic A 7 -geometry are also determined (Theorem 2).

We prove that the Hecke algebra H ( W , W J ) \mathcal {H}(W,{W_J}) , where W is a Weyl group of spherical type and W J {W_J} is a standard parabolic subgroup of W of corank ≥ 2 \geq 2 , is noncommutative.

We define the notion of d i -regularity and regularity in generalized polygons, thus generalizing the notion of ‘regular point’ in a generalized quadrangle or hexagon. We show that a thick generalized polygon cannot contain too many regular points unless it is a projective plane, quadrangle or hexagon. For certain polygons (thick ‘odd’gons and 8-gons), we show that even a certain number of d2-regular points cannot exist. As an application, we present a geometric and rather elementary proof of the non-existence of thick buildings of spherical type H3.

Let \(Z\) be a homogeneous space \(Z=G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H\). The investigation concerns the quantitative decay of matrix coefficients on \(Z\) under the assumption that \(Z\) is of spherical type, that is, minimal parabolic subgroups have open orbits on \(Z\).

We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG 1(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y).

We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to establish the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin–Tits groups of spherical type.

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR n to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.

The results of experimental investigations of the strength of spherical metal shells under internal hydrostatic loading are analyzed. The characteristic features of fracture of a spherical pressure vessel at a test temperature of 210–215 K are revealed.

We provide a new presentation for simply connected Kac-Moody groups of 2- spherical type and for their universal central extensions. Under mild local restrictions, these results extend to the more general class of groups of Kac-Moody type (i.e. groups endowed with a root datum).

Morphological characters of haemocytes and seasonal changes in blood indices were studied in Bellamya bengalensis. Two types of haemocytes were identified: I (Ia and Ib) and II. Spherical type I haemocytes (ca. 93%) are cells capable of spread-ing on the substratum and forming numerous pseu-dopodia. Subtypes Ia and Ib are most probably two different development stages of the same cell type. Type II haemocytes are oval cells of stable shape, rarely forming pseudopodia. A wide individual var-iation and significant seasonal changes were found, involving the number of haemocytes and haemato-logical indices.

Hierarchies of evolution equations of pseudo-spherical type are introduced, generalizing the notion of a single equation describing pseudo-spherical surfaces due to S.S. Chern and K. Tenenblat, and providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of $sl(2,{\bf R})$--valued linear problems. As an application, it is shown that there exists a local correspondence between any two (suitably generic) solutions of arbitrary hierarchies of equations of pseudo-spherical type., Comment: 30 pages

When a solar probe penetrates into the corona for the in-situ observation of the solar wind, the temperature of the carbon-carbon heat shield is expected to rise up to over 2000 K and the sublimation will occur at its surface. In the present study, rarefied flows around a solar probe with sublimation gas injection in the solar wind are numerically studied by the DSMC method. Two types of the heat shield configuration, that is, the inclined disk type and the balloon-like spherical type, are considered. Results show that the velocity distribution function of the solar wind captured by the sensor of the probe becomes different from that of the undisturbed freestream due to the gas-surface interaction and the molecular collision between the solar wind and the sublimation gas cloud around the probe. Such discrepancy depends on the total sublimation rate, the shape of the heat shield and the location of the sensor. Consequently, the balloon type heat shield and the sensor located on the top of the boom extended from the base of the body seem promising because of smaller disturbance at observation of the solar wind.

In this very short note we show that the residual nilpotence of pure Artin groups of spherical type is easily deduced from the faithfulness of the Krammer-Digne representations., Comment: 2 pages ; corrected references

In this note we prove the uniqueness of U in a group G with a spherical split-BN-pair of rank $ \geq 3 $ ,i.e., if G has such a BN-pair with $ B = U \cdot H, U $ a nilpotent normal subgroup of B, and $ B_{G} = 1 $ , then $ U = U^{+} $ and $ G_0 $ is a normal subgroup of G. Here $ G_0 $ is the corresponding group of Lie-type and $ U^+ $ the subgroup of $ G_0 $ generated by all root-subgroups corresponding to positive roots.

The geometries studied in this paper are obtained from buildings of spherical type by removing all chambers at non-maximal distance from a given element or flag. I consider a number of special cases of the above construction chosen among those which most frequently appear in the literature, proving that the resulting geometry is always simply connected but for three cases of small rank defined over GF(2) and GF(4). I also compute the universal cover in those exceptional cases.

Two different types of colonies from human thyroid papillary carcinomas were observed in collagen gel culture. One was a branching type, with many three-dimensional outgrowths. The other was a spherical type, with no outgrowths. As viewed by electron microscopy, the tumor cells of spherical-type colonies had more microvilli on their surfaces than did cells in the branching-type colonies. Histochemically, the tumor cells of spherical-type colonies had stronger positive reactions than did tumor cells of branching-type colonies. This suggests that tumor cells of spherical-type colonies have a higher potential for dissemination and invasion than do tumor cells of branching-type colonies. In fact, cases with higher percentages of spherical-type colonies could show extracapsular invasion histopathologically. Our results indicate that collagen gel culture is a useful method for studying the heterogeneity of tumor cells as well as their invasive ability in vitro.

We show that in general the subgeometry of a building of spherical type induced by all objects in general position with respect to a given residue is a residually connected geometry with a Buekenhout-Tits diagram resembling the original diagram, but with certain strokes replaced by the corresponding ‘affine’ strokes. The exceptions (where connectedness fails) are discussed in some detail.

We consider locally finite geometries of rank 3 belonging to certain diagrams where all strokes represent classes of non-trivial designs, at least one of them consisting of symmetric designs other than projective planes. The gonality diagram of such geometry is of spherical type, whereas the diameter diagram is of affine type. In cases like this, the criteria available from 24 cannot tell us anything about the finiteness or infiniteness of a geometry. In the main theorem of this paper we prove that, certain imply finiteness additional conditions on the parameters of the designs associated with the strokes of the diagram, a geometry is necessarily finite.

Randomized ray tracing in room acoustics can be used to compute echograms, but the results computed at the receptor are affected by statistical errors [1]. To decrease these statistical errors, the usual solution is to increase the number of rays, but this also increases the computation time. Another solution is to increase the receptor size, but this lowers the spatial resolution of the computed sound field, which is particularly important for the early part of the echogram. In order to decrease the computation time and keep a sufficient resolution, the method proposed in this paper is based on a progressive modification of the receptor size (spherical type) as long as the ray's travel grows. At the same time, the number of rays into the room can be decreased as the sound ray lengths increase while keeping the statistical errors more or less constant. The consequence is that the sound field spatial resolution evolves from “precise” at the beginning of the echogram to “rough” at its end. The first applica...

Giant vortex structures can occur in type-II nanometer-scale spherical superconducting samples. These structures have been studied within the framework of the nonlinear Ginzburg–Landau theory. Different types of boundary condition were analyzed.

This paper is concerned with the local recognition of certain graphs and geometries associated with exceptional groups of Lie type. The local approach to geometries is inspired by group theory. Finite simple groups are often characterized by local information, for example, the fusion pattern of involutions centralizing a given involution. The main results here, although of a geometric nature, are a contribution to obtaining a characterization of a group of exceptional Lie type by the fusion pattern of root subgroups centralizing a given root subgroup. Let d be the shadow space of a (thin or thick) building of spherical type M,, (where n indicates the rank) with respect to a given node r of A4 (cf. Tits [ 13, 143). We shall view d as a space, i.e., as a set of points together with a collection of subsets of size at least two of the point set, called lines. Thus, the points of A are the vertices of type r of the building in question and the lines are the residues of flags of cotype {r}. The local recognitions we intend to discuss are based on the fact that up to (nonspecial) isomorphisms, the building is uniquely determined by d. If p is a point of A, the se1 A, r(p) of points collinear with p (including p) constitutes a subspace in the sense that each line bearing two distinct points of A,,(p) is entirely contained in A Gl(p). A space with this property is often called a gamma space. Since A affords a group of automorphisms which is transitive on the point set if n > 2 (cf. [ 13]), we can associate with A a space A ,c, such that for each point p of A the subspace A .,(p) of A is isomorphic to A G r. (Here and elsewhere, a subspace X is regarded as a space by taking 348 0021-8693/89 $3.00

Let\(E_\sigma (f;l_q )_{L_2 } (R_m )\) be the best mean-square approximation of a functionf(x)e L2(Rm) (m=1, 2, ...) by integral functions of the exponential spherical type (in the sense of thelq metric, 0 0, whereω(f,Π/σ; lp)L2(Rm) is the spherical (in the sense of the metriclp, 0

After a short description of clinical facts, electron microscopic findings on four malignant melanomas of the iris are described. Though all tumors had been classified as being of the spindle cell type, the ultrastructure of the tumor cells shows remarkable differences between the four cases. This can be seen especially by comparison of the nucleoli, the chromatin, and the cytoplasmatic structures. All four tumors have the same spherical type of melanosomes. No morphologic features pointing to relative clinical harmlessness of iris melanomas were found.

The Ornstein-Zernike (OZ) equation is considered for the wall-particle distribution functiong0(x) in the case of a flat, impenetrable wall atx = 0 and a fluid of hard-core particles whose centers are constrained by the wall to occupy the semiinfinite spacex >σ/2, whereσ is the particle diameter. A solution is given in terms of the wall-particle direct correlation function c0(x) forx >σ/2, the bulk-fluid direct correlation function cB(t), and pB, the average bulk density. Explicit formulas for the contact surface density, total excess surface density, and the Laplace transform of the fluid density near the wall are given. For mean spherical type approximations, c0(x) forx >σ/2 and cB(t) are both prescribed functions; for this case, a closed-form solution is obtained. An example is discussed and additional equations that enable one to go beyond the approximations considered above are introduced.

Man erh~ilt also die Coxeter-Gruppen in nattirlicher Weise als gewisse Restklassengruppen der Artin-Gruppen. Ftir den Fall der Z6pfegruppen ergeben sich so gerade die symmetrischen Gruppen, was nattirlich schon lange bekannt ist. Die Coxeter-Gruppen sind seit ihrer Einftihrung durch Coxeter im Jahre 1935 eingehend studiert worden, und eine sch6ne Darstellung der dabei erhaltenen Resultate findet man bei Bourbaki [1]. Ftir die Artin- Gruppen gab es, von den freien Gruppen einmal abgesehen, nur ftir die Z6pfegruppen eine Reihe yon Untersuchungen, yon denen die wichtigste in letzter Zeit die L~Ssung des Konjugationsproblems durch Garside war. Ftir die anderen Artin-Gruppen gibt es einige vereinzelte Resultate in [2, 3 und 5]. Diese beziehen sich ebenso wie der gr6gte Teil der vorliegen- den Arbeit auf den Fall, dab die zu der Artin-Gruppe G geh~Srige Coxeter- Gruppe G endlich ist. Die endlichen Coxeter-Gruppen sind schon yon Coxeter selbstWoestimmt worden: Es sind die endlichen Spiegelungs- gruppen, also - im irreduziblen Fall - die Gruppen vom Typ A,, B

1. The development of the archesporium in aroids follows no uniform rule. In one species studied, the primary archesporial cell produces two megaspores, the outer one of which functions; in two cases there are four megaspores, the outermost one of which functions; in one case the primary cell produces two cells, one of which may be a tapetal cell; one species develops four megaspores in the same horizontal plane, and the functioning one then develops a tapetal cell at its upper end; one species develops four megaspores, the innermost of which functions; and in one case the primary sporogenous cell develops directly into an embryo sac. 2. The number of antipodals varies from 3 to 11, four species having regularly more than 3. This is a relatively primitive feature, and occurs in other primitive orders. 3. Previous to the work here reported, embryos of the spherical type lacking a suspensor had been reported by Campbell in Pistia, Dieffenbachia, and Lysichiton. The writer has confirmed Campbell'S observatio...

/quad A regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids. The locally toroidal polytopes of rank 4 have been extensively discussed, and their classification is now nearly complete. In this paper, the locally toroidal polytopes of higher rank are investigated. Again, an almost complete classification of these regular polytopes is obtained; as well as sporadic examples, there are several infinite families.

A set of axioms in terms of points and lines is presented which characterize the ‘natural’ point-line geometries (Cn,1; Dn,1; E4,1; E5,1; E6,1; E7,1; E8,1; F4,1) associated with buildings of spherical type.

We measured the spin-spin cross section of polarized 59Co for polarized neutrons of 0.39–2.88 MeV in the perpendicular geometry. Comparing our results with existing data we conclude that the spin-spin potential is of the spherical type.

An axiom system is presented to characterize the Grassmannians of buildings of spherical type having hyperlines of rank 3 only; some open problems remain in the infinite case.

A result of Tits' in his paper “A local approach to buildings” is somewhat strengthened: it is shown that each geometry of type D n or E 6 is a building. A counterexample to the corresponding statement for E 7 is given. Moreover, a proof is given of the fact that any thick finite geometry of spherical type all of whose residues of type C 3 are buildings is itself a building.

Tests are described which measured the artificially induced transition location on a slightly blunted 7-deg cone. Various distributed roughness and spherical type boundary layer trips were used. The tests were conducted at Mach 9 over a free stream Reynolds number range from 500,000/ft. to 20,000,000/ft. Test results are not included in the report.

Two distinct types of cysts of Sarcocystis from the musculature of moose (Alces alces) were compared by electron microscopy. The fusiform Type A cysts differed from the spherical Type B cysts in the appearance and thickness of the primary cyst wall, organization of cyst interior, and the presence of a secondary cyst wall around Type B. The respective merozoites also differed in size as well as in the number of rhoptries and diameter and arrangement of micronemes. Comparison of the ultrastructure of the moose sarcocysts with those described from other ungulates revealed substantial differences. It appears that two hitherto undescribed species of Sarcocystis are present in moose although cross-transmission and additional life cycle studies are necessary for a complete description.

In this note we show how one may compute the parameters of a finite Lie geometry, and we give the results of such computations in the most interesting cases. We also prove a little lemma that is useful for showing that thick finite buildings do not have quotients which are (locally) Tits geometries of spherical type.

A case of a rare congenital malformation, duplication of the rectum, has been presented. The literature has been reviewed, various theories of etiology considered and the pathology, clinical features and treatment discussed.

The proportion of polysaccharide chain segments that are in a ordered state has been compared in spherical and lenticular starch granules. The spherical type of particle was found to possess a more ordered structure. It is suggested that this variation in polymer arrangement may partly occount for differences in certain properties of the two forms of granule.