analytic formulas for the regulator of a number field letr rk andw wk be the regulator and the number of roots of unity in the number fieldk we determine allk for whichr w 1 8 there are nine such fields sincew 2 we have allk withr 1 4 in particular we find the first three minima ofr over allk there are three main ingredients to the proof i a new analytic formula theorem a below ii a refinement of remak s geometric approach iii a refinement of zimmert s analytic method

oeljeklaus toma manifolds and arithmetic invariants we consider oeljeklaus toma manifolds coming from number fields with precisely one complex place our general theme is to relate the geometry to the arithmetic we show that just knowing the fundamental group allows us to recover the number field we also show that this fails if there are more complex places the first homology turns out to be related to an interesting ideal we compute the volume in terms of the discriminant and regulator of the number field is there a conceptual reason for this we explore this and see what happens if we entirely experimentally regard them as baby siblings of hyperbolic manifolds coming from number fields

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

the computation of visible surface representations a computational theory of visible surface representations is developed the visible surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of 1 integrating multiscale constraints on surface depth and orientation from multiple visual sources 2 interpolating dense piecewise smooth surfaces from these constraints 3 detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation and 4 structuring large scale distributed surface representations to achieve computational efficiency visible surface reconstruction is an inverse problem a well posed variational formulation results from the use of a controlled continuity surface model discontinuity detection amounts to the identification of this generic model s distributed parameters from the data finite element shape primitives yield a local discretization of the variational principle the result is an efficient algorithm for visible surface reconstruction

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

pkcs a polynomial kernel family with compact support for scale space image processing in a scale space framework the gaussian kernel has some properties that make it unique however because of its infinite support exact implementation of this kernel is not possible to avoid this drawback there exist two different approaches approximating the gaussian kernel by a finite support kernel or defining new kernels with properties closed to the gaussian in this paper we propose a polynomial kernel family with compact support which overcomes the gaussian practical drawbacks while preserving a large number of the useful gaussian properties the new kernels are not obtained by approximating the gaussian though they are derived from it we show that for a suitable choice of kernel parameters this family provides an approximated solution of the diffusion equation and satisfies some other basic constraints of the linear scale space theory the construction and properties of the proposed kernel are described and an application in which handwritten data are extracted from noisy document images is presented

global bifurcation of periodic solutions to ordinary differential equations abstract general homotopy continuation and bifurcation results are discussed and proved for a class of semiflows these results and further conley homotopy index methods are applied to obtain a type of global bifurcation for periodic solutions of second order ordinary differential equations

on a free boundary problem for the curvature flow with driving force we study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles our first main result concerns the classification of solutions every solution falls into one of the three categories namely area expanding area bounded and area shrinking types we then study in detail the asymptotic behavior of solutions in each category among other things we show that solutions are asymptotically self similar both in the area expanding and the area shrinking cases while solutions converge to either a stationary solution or a traveling wave in the area bounded case we also prove results on the concavity properties of solutions one of the main tools of this paper is the intersection number principle however in order to deal with solutions with free boundaries we introduce what we call the extended intersection number principle which turns out to be exceedingly useful in handling curves with moving endpoints

an entire solution of a bistable parabolic equation on r with two colliding pulses we consider semilinear parabolic equations of the form equation a ut uxx f u x r t i where i 0 i 0 or i i solutions defined for all x t r2 x t r2 are referred to as entire solutions assuming that f c1 r f c1 r is of a bistable type with stable constant steady states 0 and 0 0 we show the existence of an entire solution u x t u x t of the following form for t t u t u t has two humps or pulses one near the other near as t increases the humps move toward the origin x 0x 0 eventually colliding and forming a one hump final shape of the solution with respect to the locally uniform convergence the solution u t u t is a heteroclinic orbit connecting the stable steady state 0 to the unstable ground state of the equation uxx f u 0uxx f u 0 we find the solution u as the limit of a sequence of threshold solutions of the cauchy problem for equation a

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

noise resistant invariants of curves projective invariants are shape descriptors that are independent of the point of view from which the shape is seen and therefore are of major importance in object recognition they make it possible to match an image of an object to one stored in a database without the need for searching for the correct viewpoint an invariant representation of a general curve is obtained the calculation is local and does not suffer from the occlusion problem of global descriptors to make the method robust differentiation techniques that give much more reliable results than previous ones are developed these differentiation methods are useful in many other applications as well

blow up of solutions with sign changes for a semilinear diffusion equation abstract this paper is concerned with the initial boundary value problem formula with the dirichlet neumann or periodic boundary condition here 0 is a parameter and f is an odd function of u satisfying f 0 0 and some convexity condition let z u be the number of times of sign changes for u c 0 1 it is shown that there exists an increasing sequence of positive numbers k k 0 1 2 such that any solution with z u 0 k blows up in finite time if k and there exists a global solution with z u 0 k if k

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

geometric invariants and object recognition we discuss the role of the general invariance concept in object recognition and review the classical and recent literature on projective invariance invariants help solve major problems of object recognition for instance different images of the same object often differ from each other because of the different viewpoint from which they were obtained to match the two images common methods thus need to find the correct viewpoint a difficult problem that can involve search in a high dimensional space of all possible points of view and or finding point correspondences geometric invariants are shape descriptors computed from the geometry of the shape that remain unchanged under geometric transformations such as changing the viewpoint thus they can be matched without search deformations of objects are another important class of geometric changes for which invariance is useful

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

blow up of sign changing solutions for a one dimensional nonlinear diffusion equation abstract this paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the neumann or dirichlet boundary conditions assuming that the nonlinearity satisfies a rather general growth condition we consider the blow up and global existence of sign changing solutions it is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution the solution blows up in finite time if an initial value changes its sign at most k times whereas there exist stationary solutions with more than k zeros the proof is based on the intersection number principle combined with a topological method

a remark on the energy blow up behavior for nonlinear heat equations lim t t u t h 1 0 a point a is called a blow up point of u if there exists an tn a t such that u an tn the set of all blow up points of u t is called the blow up set and denoted by s from giga and kohn 8 theorem 5 3 there are no blow up points in therefore we see from 3 and the boundedness of that s is not empty many papers are concerned with the cauchy problem for 1 see for instance 21 or the problem of finding sufficient blow up conditions on the initial data see

on mean curvature flow with driving force for symmetric motion with singular initial hypersurface abstract we consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface we study this problem by level set method we give some criteria to judge whether the interface evolution is fattening or non fattening

high order differentiation filters that work reliable derivatives of digital images have always been hard to obtain especially but not only at high orders we analyze the sources of errors in traditional filters such as derivatives of the gaussian that are used for differentiation we then study a class of filters which is much more suitable for our purpose namely filters that preserve polynomials up to a given order we show that the errors in differentiation can be corrected using these filters we derive a condition for the validity domain of these filters involving some characteristics of the filter and of the shape our experiments show a very good performance for smooth functions

decomposition of one dimensional waveform using iterative gaussian diffusive filtering methods the gaussian smoothing method is shown to have a wide transition zone around the cutoff frequency selected to filter a given dataset we proposed two iterative gaussian smoothing methods to tighten the transition zone one being approximately diffusive and the other being strictly diffusive the first version smoothes repeatedly the remaining high frequency parts and the second version requires an additional step to further smooth the resulting smoothed response in each of the smoothing operation based on the choice of the criterion for accuracy the smoothing factor and the number of iterations are derived for an infinite data length in both methods by contrast for a finite length data string results of the interior points sufficiently away from the two endpoints obtained by both methods can be shown to exhibit an approximate diffusive property the upper bound of the distance affected by the error propagation inward due to the lack of data beyond the two ends is numerically estimated numerical experiments also show that results of employing the iterative gaussian smoothing method are almost the same as those obtained by the strict diffusive version except that the error propagation distance induced by the latter is slightly deeper than that of the former the proposed method has been successfully applied to decompose the wave formation of a number of test cases including two sets of real experimental data

examples of non trivial rank in locally conformal kahler geometry we consider locally conformal kahler geometry as an equivariant homothetic kahler geometry k we show that the de rham class of the lee form can be naturally identified with the homomorphism projecting to its dilation factors thus completing the description of locally conformal kahler geometry in this equivariant setting the rank r m of a locally conformal kahler manifold is the rank of the image of this homomorphism using algebraic number theory we show that r m is non trivial providing explicit examples of locally conformal kahler manifolds with 1 nless text upshape rmfamily r m nless b 1 as far as we know these are the first examples of this kind moreover we prove that locally conformal kahler oeljeklaus toma manifolds have either r m b 1 or r m b 1 2

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

on the two dimensional muskat problem with monotone large initial data we consider the evolution of two incompressible immiscible fluids with different densities in porous media known as the muskat problem 21 which in two dimensions is analogous to the hele shaw cell 24 we establish for a class of large and monotone initial data the global existence of weak solutions the proof is based on a local well posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function 2016 wiley periodicals inc

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

blow up rates of solutions of initial boundary value problems for a quasi linear parabolic equation abstract we consider initial boundary value problems for a quasi linear parabolic equation k t k 2 k k with zero dirichlet boundary conditions and positive initial data it has known that each of solutions blows up at a finite time with the rate faster than t t 1 in this paper it is proved that sup k t t t 1 log log t t 1 as t t under some assumptions our strategy is based on analysis for curve shortening flows that with self crossing brought by s b angenent and j j l velazquez in addition we prove some of numerical conjectures by watterson which are keys to provide the blow up rate

local invariants for recognition geometric invariants are shape descriptors that remain unchanged under geometric transformations such as projection or changing the viewpoint a new method of obtaining local projective and affine invariants is developed and implemented for real images being local the invariants are much less sensitive to occlusion than global invariants the invariants computation is based on a canonical method this consists of defining a canonical coordinate system by the intrinsic properties of the shape independently of the given coordinate system since this canonical system is independent of the original one it is invariant and all quantities defined in it are invariant the method was applied without the use of a curve parameter this was achieved by fitting an implicit polynomial to an arbitrary curve in a vicinity of each curve point several configurations are treated a general curve without any correspondence and curves with known correspondences of one or two feature points or lines experimental results for different 2d objects in 3d space are presented

transition operators of diffusions reduce zero crossing if u t x is a solution of a one dimensional parabolic secondorder linear partial differential equation pde then it is known that under suitable conditions the number of zero crossings of the function u t decreases that is does not increase as time t increases such theorems have applications to the study of blow up of solutions of semilinear pde time dependent sturm liouville theory curve shrinking problems and control theory we generalise the pde results by showing that the transition operator of a possibly time inhomogenous one dimensional diffusion reduces the number of zero crossings of a function or even suitably interpreted a signed measure our proof is completely probabilistic and depends in a transparent manner on little more than the sample path continuity of diffusion processes

transition fronts for the fisher kpp equation this paper is concerned with transition fronts for reaction diffusion equations of the fisher kpp type basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich in the paper we describe the class of transition fronts and we study their qualitative dynamical properties in particular we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate we also classify the transition fronts in the class of measurable superpositions of standard traveling fronts

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

time frequency distributions based on compact support kernels properties and performance evaluation this paper presents two new time frequency distributions tfds based on kernels with compact support kcs namely the separable cb scb and the polynomial cb pcb tfds the implementation of this family of tfds follows the method developed for the cheriet belouchrani cb tfd the mathematical properties of these three tfds are analyzed and their performance is compared to the best classical quadratic tfds using several tests on multicomponent signals with linear and nonlinear frequency modulation fm components including the noise effects instead of relying solely on visual inspection of the time frequency domain plots comparisons include the time slices plots and the evaluation of the boashash sucic s normalized instantaneous resolution performance measure that permits to provide the optimized tfd using a specific methodology in all presented examples the kcs tfds show a significant interference rejection with the component energy concentration around their respective instantaneous frequency laws yielding high resolution measure values

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

flat affine subvarieties in oeljeklaus toma manifolds the oeljeklaus toma ot manifolds are compact complex non kahler manifolds constructed by oeljeklaus and toma and generalizing the inoue surfaces their construction uses the number theoretic data a number field k and a torsion free subgroup u in the group of units of the ring of integers of k with rank of u equal to the number of real embeddings of k ot manifolds are equipped with a torsion free flat affine connection preserving the complex structure this structure is known as flat affine structure we prove that any complex subvariety of smallest possible positive dimension in an ot manifold is also flat affine this is used to show that if all elements in u setminus 1 are primitive in k then x contains no proper analytic subvarieties

dedekind zeta motives for totally real number fields let k be a totally real number field for every odd n 3 we construct an element in the category mt k of mixed tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group by a volume calculation we prove that its period is a rational multiple of pi n k mathbb q zeta k 1 n where zeta k 1 n denotes the special value of the dedekind zeta function of k we deduce that the group mathrm ext 1 mathrm mt k mathbb q 0 mathbb q n is generated by the cohomology of a quadric relative to hyperplanes and that zeta k 1 n is a determinant of volumes of geodesic hyperbolic simplices defined over k