computing the visual hull of solids of revolution abstract the visual hull vh s v of an object s relative to a viewing region v is a geometric entity useful for silhouette based image understanding for instance two objects can be distinguished from their silhouettes only if their visual hulls are different an object can be exactly reconstructed from its silhouettes only if equal to its visual hull the visual hull idea also allows to define a measure of the reconstruction accuracy of an object from its silhouettes there are two main cases of visual hull the internal visual hull if v is only bounded by s and the external visual hull if v is bounded by the convex hull of s this paper addresses the problem of computing both visual hulls of solids of revolution two cases are considered smooth polynomial and piecewise linear generators the algorithm has been implemented for the latter case the surface of the visual hulls when it is not coincident with the surface of s turns out to consist of segments of cones annular rings and hyperboloids of one sheet

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

a note on decomposition numbers for groups of lie type of small rank abstract using virtual projective characters coming from the l adic cohomology of deligne lusztig varieties we determine new decomposition numbers for principal blocks of some groups of lie type of small rank including g 2 q and steinberg s triality groups d 4 3 q

on the 2 decomposition numbers of steinberg s triality groups d43 q q odd abstract we compute the 2 modular decomposition numbers of steinberg s triality groups d 4 3 q where q is a power of an odd prime except for two decomposition numbers in the steinberg character and classify all absolutely irreducible representations of d 4 3 q in non defining characteristic up to a certain degree

decomposition numbers of so7 q and sp6 q abstract we complete the l modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups so 7 q and the symplectic groups sp 6 q for all prime powers q and all odd primes l different from the defining characteristic

stability of a 4th order curvature condition arising in optimal transport theory abstract a certain curvature condition introduced by ma trudinger and wang in relation with the regularity of optimal transport is shown to be stable under gromov hausdorff limits even though the condition implicitly involves fourth order derivatives of the riemannian metric two lines of reasoning are presented with slightly different assumptions one purely geometric and another one combining geometry and probability then a converse problem is studied prove some partial regularity for the optimal transport on a perturbation of a riemannian manifold satisfying a strong form of the ma trudinger wang condition

appendix non uniqueness of supercuspidal support for finite reductive groups abstract we exhibit a simple representation of the finite symplectic group of sp 8 q in characteristic l q 2 1 having two non conjugate supercuspidal supports this answers a question of m f vigneras

a note on decomposition numbers of g2 2n abstract the decomposition numbers of the finite chevalley group g 2 p n of type g 2 defined over a finite field of characteristic r which divides p n 1 are almost determined by g hiss j algebra 120 1989 339 360 i will prove that the decomposition numbers are bounded independently of p n by using the same argument as in t okuyama k waki j algebra 199 1998 544 555 in case of p 2

what s next an interactive next best view approach shape from silhouettes algorithms use either a surface or a volumetric approach a problem with the volumetric approach is that in general we know neither the accuracy of the reconstruction nor where to locate new viewpoints for improving the accuracy in this paper we present a general approach to interactive object specific volumetric algorithms based on a necessary condition for the best possible reconstruction to have been performed the outlined approach can be applied to any class of objects as an example of this approach an interactive algorithm has been implemented for convex polyhedra

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

character tables of parabolic subgroups of steinberg s triality groups we compute the conjugacy classes of elements and the character tables of the parabolic subgroups of steinberg s triality groups d43 q where q is a power of an odd prime our main tools are clifford theory applied to the levi decomposition of the parabolic subgroups and the decomposition of restrictions of the unipotent characters of d43 q for many of the calculations we use the chevie package

the visual hull of smooth curved objects the visual hull is a geometric entity that relates the shape of an object to its silhouettes or shadows this paper develops the theory of the visual hull of generic smooth objects we show that the visual hull can be constructed using surfaces which partition the viewpoint space of the aspect graph of the object the surfaces are those generated by the visual events tangent crossing and triple point an analysis based on the shape of the object at the tangency points of these surfaces allows pruning away many surfaces and patches not relevant to the construction an algorithm for computing the visual hull is outlined

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

regularity of optimal transport on compact locally nearly spherical manifolds given a couple of smooth positive measures of same total mass on a compact connected riemannian manifold m we look for a smooth optimal transportation map g pushing one measure to the other at a least total squared distance cost directly by using the continuity method to produce a classical solution of the elliptic equation of monge ampere type satisfied by the potential function u such that g exp grad u this approach boils down to proving an textit a priori upper bound on the hessian of u which was done on the flat torus by the first author the recent local c 2 estimate of ma trudinger wang enabled loeper to treat the standard sphere case by overcoming two difficulties namely in collaboration with the first author he kept the image g m of a generic point m in m uniformly away from the cut locus of m he checked a fourth order inequality satisfied by the squared distance cost function proving the uniform positivity of the so called c curvature of m in the present paper we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in c 2 norm specifying and proving a conjecture stated by trudinger

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

regularity of optimal maps on the sphere the quadratic cost and the reflector antenna building on the results of ma et al in arch rational mech anal 177 2 151 183 2005 and of the author loeper in acta math to appear we study two problems of optimal transportation on the sphere the first corresponds to the cost function d2 x y where d is the riemannian distance of the round sphere the second corresponds to the cost function log x y known as the reflector antenna problem we show that in both cases the cost sectional curvature is uniformly positive and establish the geometrical properties so that the results of loeper in acta math to appear and ma et al in arch rational mech anal 177 2 151 183 2005 can apply global smooth solutions exist for arbitrary smooth positive data and optimal maps are holder continuous under weak assumptions on the data

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

sp6 2a is good for the mckay alperin weight and related local global conjectures abstract the so called local global conjectures in the representation theory of finite groups relate the representation theory of g to that of certain proper subgroups such as the normalizers of particular p groups recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups here we show that g sp 6 2 a is good for these reductions for the mckay conjecture the alperin weight conjecture and their blockwise versions

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

eigenvector matrices of cartan matrices for finite groups abstract let c b be the cartan matrix of a p block b of a finite group g we show that there is a unimodular eigenvector matrix u b of c b over a discrete valuation ring r if all eigenvalues of c b are integers when b is a cyclic block a tame block a p block of a p solvable group or the principal 3 block with elementary abelian defect group of order 9

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

eigenvalues of unipotent elements in cross characteristic representations of finite classical groups abstract let h be a finite classical group g be a unipotent element of h of order s and be an irreducible representation of h with dim 1 over an algebraically closed field of characteristic coprime to s we show that almost always all the s roots of unity occur as eigenvalues of g and classify all the triples h g for which this does not hold in particular we list the triples for which 1 is not an eigenvalue of g we also give estimates of the asymptotic behavior of eigenvalue multiplicities when the rank of h grows and s is fixed

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group