smooth and proper noncommutative schemes and gluing of dg categories abstract in this paper we discuss different properties of noncommutative schemes over a field we define a noncommutative scheme as a differential graded category of a special type we study regularity smoothness and properness for noncommutative schemes admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes in this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules as a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme moreover if the algebra has finite global dimension then the full subcategory is admissible we also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance as another application of the main theorem we obtain in characteristic 0 an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme

a modified log rank test for competing risks with missing failure type summary we propose a modified log rank test for the analysis of competing risks survival data when failure type is missing for some individuals the proposed test reduces to a standard log rank test when all failure types are known the test arises from a partial likelihood constructed under semiparametric assumptions on the relationship between cause specific hazards

a numerical verification method of bifurcating solutions for 3 dimensional rayleigh benard problems this paper is the three dimensional extension of the two dimensional work in nakao et al reliable comput 9 5 359 372 2003 and watanabe et al j math fluid mech 6 1 20 2004 on a computer assisted proof of the existence of nontrivial steady state solutions for rayleigh benard convection based on the fixed point theorem using a newton like operator the differences are emerging of complicated types of bifurcation direct attack on the problem without stream functions and increased complexity of numerical computation the last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case actually this work should be the first result for the three dimensional navier stokes problems which seems to be very difficult to solve by theoretical approaches

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

g twisted braces and orbifold landau ginzburg models given an algebra with a group g action we construct brace structures for its g twisted hochschild cochains an an application we construct g frobenius algebras for orbifold landau ginzburg b models and present explicit orbifold cup product formula for all invertible polynomials

rigorous computation of smooth branches of equilibria for the three dimensional cahn hilliard equation in this paper we propose a new general method to compute rigorously global smooth branches of equilibria of higher dimensional partial differential equations the theoretical framework is based on a combination of the theory introduced in global smooth solution curves using rigorous branch following van den berg et al math comput 79 271 1565 1584 2010 and in analytic estimates and rigorous continuation for equilibria of higher dimensional pdes gameiro and lessard j diff equ 249 9 2237 2268 2010 using this method one can obtain proofs of existence of global smooth solution curves of equilibria for large continuous parameter ranges and about local uniqueness of the solutions on the curve as an application we compute several smooth branches of equilibria for the three dimensional cahn hilliard equation

categorical smooth compactifications and generalized hodge to de rham degeneration we disprove two unpublished conjectures of kontsevich which state generalized versions of categorical hodge to de rham degeneration for smooth and for proper dg categories but not smooth and proper in which case degeneration is proved by kaledin in algebra geometry and physics in the 21st century birkhauser springer cham pp 99 129 2017 in particular we show that there exists a minimal 10 dimensional a infty algebra over a field of characteristic zero for which the supertrace of mu 3 on the second argument is non zero as a byproduct we obtain an example of a homotopically finitely presented dg category over a field of characteristic zero that does not have a smooth categorical compactification giving a negative answer to a question of toen this can be interpreted as a lack of resolution of singularities in the noncommutative setup we also obtain an example of a proper dg category which does not admit a categorical resolution of singularities in the terminology of kuznetsov and lunts int math res not 2015 13 4536 4625 2015 that is it cannot be embedded into a smooth and proper dg category

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

on the use of cause specific failure and conditional failure probabilities examples from clinical oncology data abstract nonparametric maximum likelihood estimation of the probability of failing from a particular cause by time t in the presence of other acting causes i e the cause specific failure probability is discussed a commonly used incorrect approach is to take 1 minus the kaplan meier km estimator 1 km whereby patients who fail of extraneous causes are treated as censored observations examples showing the extent of bias in using the 1 km approach are presented using clinical oncology data this bias can be quite large if the data are uncensored or if a large percentage of patients fail from extraneous causes prior to the occurrence of failures from the cause of interest each cause specific failure probability is mathematically defined as a function of all of the cause specific hazards therefore nonparametric estimates of the cause specific failure probabilities may not be able to identify categorized covariate effects on the cause specific hazards these effects would be correctly identified

deformation theory of objects in homotopy and derived categories i general theory abstract this is the first paper in a series we develop a general deformation theory of objects in homotopy and derived categories of dg categories namely for a dg module e over a dg category we define four deformation functors def h e codef h e def e codef e the first two functors describe the deformations and co deformations of e in the homotopy category and the last two in the derived category we study their properties and relations these functors are defined on the category of artinian not necessarily commutative dg algebras

holomorphic orbi discs and lagrangian floer cohomology of symplectic toric orbifolds we develop floer theory of lagrangian torus fibers in compact symplectic toric orbifolds we first classify holomorphic orbi discs with boundary on lagrangian torus fibers we show that there exists a class of basic discs such that we have one to one correspondences between a smooth basic discs and facets of the moment polytope and b between basic orbi discs and twisted sectors of the toric orbifold we show that there is a smooth lagrangian floer theory of these torus fibers which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold we show by several examples that such bulk deformation can be used to illustrate the very rigid hamiltonian geometry of orbifolds we define its potential and bulk deformed potential and develop the notion of leading order potential we study leading term equations analogous to the case of toric manifolds by fukaya oh ohta and ono

the homotopy theory of dg categories and derived morita theory the main purpose of this work is to study the homotopy theory of dg categories up to quasi equivalences our main result is a description of the mapping spaces between two dg categories c and d in terms of the nerve of a certain category of c d bimodules we also prove that the homotopy category ho dg cat possesses internal hom s relative to the derived tensor product of dg categories we use these two results in order to prove a derived version of morita theory describing the morphisms between dg categories of modules over two dg categories c and d as the dg category of c d bi modules finally we give three applications of our results the first one expresses hochschild cohomology as endomorphisms of the identity functor as well as higher homotopy groups of the classifying space of dg categories i e the nerve of the category of dg categories and quasi equivalences between them the second application is the existence of a good theory of localization for dg categories defined in terms of a natural universal property our last application states that the dg category of continuous morphisms between the dg categories of quasi coherent resp perfect complexes on two schemes resp smooth and proper schemes is quasi equivalent to the dg category of quasi coherent resp perfect complexes on their product

standard conjecture d for matrix factorizations abstract we prove the non commutative analogue of grothendieck s standard conjecture d for the dg category of matrix factorizations of an isolated hypersurface singularity in characteristic 0 along the way we show the euler pairing for such dg categories of matrix factorizations is positive semi definite

chern characters and hirzebruch riemann roch formula for matrix factorizations we study the category of matrix factorizations for an isolated hypersurface singularity we compute the canonical bilinear form on the hochschild homology of this category we find explicit expressions for the chern character and the boundary bulk maps and derive an analogue of the hirzebruch riemann roch formula for the euler characteristic of the hom space between a pair of matrix factorizations we also establish g equivariant versions of these results

on the mixed hodge structure on the cohomology of the milnor fibre in 18 the second author introduced a mixed hodge structure on the cohomology of the milnor fibre of an isolated hypersurface singularity for the definition of a mixed hodge structure el 17 sect 3 4 the weight filtration is essentially the monodromy weight filtration 18 sect 4 and thus simplifies the jordan normal form of the unipotent part of the monodromy the significance of the hodge filtration however is not so dear and its description seems difficult to use this paper gives another description of the hodge filtration which we hope is easier to understand and apply its definition does not use resolution of singularities instead it relies on the theory of holonomic modules in one variable with regular singularities this paper arose from conversations at the 1980 arbeitstagung in bonn where varchenko s conjectured hodge filtration 22 had been discussed in brieskorn s seminar we have since learned that varchenko obtained similar results to those in this paper in the summer and autumn of 1980 23 25 in sect 1 the milnor fibration is embedded in a family of smooth projective hypersurfaces the hodge theory of a smooth projective hypersurface is explained in the fashion of brylinski 2 in sect 2 in sect 3 we recall the description of the gauss manin system from 8 our new formula for the hodge filtration is explained in sects 4 5 in sect 6 we prove that it gives the same result as 18 we prove a thom sebastiani formula for the hodge filtration in sect 7 which leads to a proof of conjecture 5 4 of 18 in sect 8 we show how to prove varchenko s result about the jordan normal forms of multiplication by f in the jacobian ring of f and the logarithm of the unipotent monodormy of f with our method in the last chapter the mixed hodge structure is calculated for two examples

homological mirror symmetry for generalized greene plesser mirrors we prove kontsevich s homological mirror symmetry conjecture for certain mirror pairs arising from batyrev borisov s dual reflexive gorenstein cones construction in particular we prove hms for all greene plesser mirror pairs i e calabi yau hypersurfaces in quotients of weighted projective spaces we also prove it for certain mirror calabi yau complete intersections arising from borisov s construction via dual nef partitions and also for certain calabi yau complete intersections which do not have a calabi yau mirror but instead are mirror to a calabi yau subcategory of the derived category of a higher dimensional fano variety the latter case encompasses kuznetsov s k3 category of a cubic fourfold which is mirror to an honest k3 surface and also the analogous category for a quotient of a cubic sevenfold by an order 3 symmetry which is mirror to a rigid calabi yau threefold

homological mirror symmetry for calabi yau hypersurfaces in projective space we prove homological mirror symmetry for a smooth d dimensional calabi yau hypersurface in projective space for any d 3 for example d 3 is the quintic threefold the main techniques involved in the proof are the construction of an immersed lagrangian sphere in the d dimensional pair of pants the introduction of the relative fukaya category and an understanding of its grading structure a description of the behaviour of this category with respect to branched covers via an orbifold fukaya cate gory a morse bott model for the relative fukaya category that allows one to make explicit computations and the introduction of certain graded categories of matrix factorizations mirror to the relative fukaya category

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

model selection for the competing risks model with and without masking the competing risks model is useful in settings in which individuals or units may die or fail because of various causes it can also be the case that for some of the items the cause of failure is known only up to a subgroup of all causes in which case we say that the failure is group masked a widely used approach for competing risks data with and without masking involves the specification of cause specific hazard rates often because of the availability of likelihood methods for estimation and testing piecewise constant hazards are used the piecewise constant rates also offer model flexibility and computational convenience however for such piecewise constant hazard models the choice of the endpoints for each interval on which the hazards are constant is usually a subjective one in this article we discuss and propose the use of model selection methods that are data driven and automatic we compare three model selection procedures based on the minimum description length principle the bayes in

derived automorphism groups of k3 surfaces of picard rank 1 we give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a k3 surface of picard rank 1 we do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible

matrix factorizations and higher residue pairings the periodic cyclic homology of any proper dg category comes equipped with a canonical pairing we show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with kyoji saito s higher residue pairing on the twisted de rham cohomology of the singularity

on a hodge theoretic property of the kunneth map in periodic cyclic homology abstract we prove that the standard kunneth map in periodic cyclic homology of differential z 2 graded algebras is compatible with a generalization of the hodge filtration and explain how this result is related to various thom sebastiani type theorems in singularity theory

hirzebruch riemann roch type formula for dg algebras for an arbitrary differential graded algebra a with finite dimensional total cohomology we introduce a pairing on the hochschild homology of a derive an explicit formula for the chern characters of perfect a modules the chern characters take values in the hochschild homology of a and prove a hirzebruch riemann roch type formula which expresses the euler characteristic of the hom complex between two perfect a modules in terms of the pairing of their chern characters

nonparametric prevalence and mortality estimators for animal experiments with incomplete cause of death data abstract age specific prevalence and mortality estimators are important descriptors of disease development and the subsequent effects of a disease on longevity nonparametric maximum likelihood estimators for the prevalence and mortality functions are available for the case in which incidental and fatal occurrences of a disease can always be distinguished this article generalizes these methods to allow the role of a disease in causing death to be uncertain for a subset of the animals dying with the disease the proposed analysis makes no assumptions about the degree of disease lethality data from sacrificed animals can be incorporated easily although sacrifices are not necessary if a certain representativeness assumption holds no restrictions are imposed on the distribution of survival times but the prevalence function is held constant over time intervals for stabilization purposes variances are estimated by inverting the observed information matrix an em algorithm simplifies the analysis when the

nonparametric estimation of the survival function when cause of death is uncertain summary a nonparametric estimator for the survival function accommodating censored survival times and uncertainty in the assignment of cause of death is proposed for example in a carcinogenicity experiment the data on each animal may consist of an observed age at death and some indication of the probability that the tumor type under study caused death an estimator of the net survival function for time to death due to the cause of interest is developed under certain assumptions the proposed estimator is consistent and asymptotically normally distributed monte carlo simulations were used to compare this estimator with the kaplan meier estimator forcing the cause of death to be specified with certainty as required by the kaplan meier estimator may result in substantial biases

an example of a non fourier mukai functor between derived categories of coherent sheaves orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor in this paper we show that this result is false without the fully faithfulness hypothesis we also show that our functor does not lift to the homotopy category of spectral categories if the ground field is mathbb q

rigorous numerics for ill posed pdes periodic orbits in the boussinesq equation in this paper we develop computer assisted techniques for the analysis of periodic orbits of ill posed partial differential equations as a case study our proposed method is applied to the boussinesq equation which has been investigated extensively because of its role in the theory of shallow water waves the idea is to use the symmetry of the solutions and a newton kantorovich type argument the radii polynomial approach to obtain rigorous proofs of existence of the periodic orbits in a weighted l1 banach space of space time fourier coefficients with exponential decay we present several computer assisted proofs of the existence of periodic orbits at different parameter values

a note on a test for competing risks with missing failure type summary the modified log rank test for competing risks with missing failure type suggested by goetghebeur ryan 1990 is derived from a partial likelihood which leaves out some information this extra information is incorporated to give a new partial likelihood and hence a new test statistic

inference based on the em algorithm for the competing risks model with masked causes of failure summary in this paper we propose inference methods based on the em algorithm for estimating the parameters of a weakly parameterised competing risks model with masked causes of failure and second stage data with a carefully chosen definition of complete data the maximum likelihood estimation of the cause specific hazard functions and of the masking probabilities is performed via an em algorithm both the e and m steps can be solved in closed form under the full model and under some restricted models of interest we illustrate the flexibility of the method by showing how grouped data and tests of common hypotheses in the literature on missing cause of death can be handled the method is applied to a real dataset and the asymptotic and robustness properties of the estimators are investigated through simulation

categorical resolution of singularities abstract building on the concept of a smooth dg algebra we define the notion of a smooth derived category we then propose the definition of a categorical resolution of singularities our main example is the derived category d x of quasi coherent sheaves on a scheme x we prove that d x has a canonical categorical resolution if the base field is perfect and x is a separated scheme of finite type with a dualizing complex

the period map for cubic fourfolds the period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20 we determine the image of this period map thus confirming a conjecture of hassett and give at the same time a new proof of the theorem of voisin that asserts that this period map is an open embedding an algebraic version of our main result is an identification of the algebra of sl 6 invariant polynomials on the representation space sym 3 6 with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20 we also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a vinberg dynkin diagram

proof of a conjecture of batyrev and nill the second named author thanks the pacific institute for the mathematical sciences and nserc for various forms of support frequent visits to the university of alberta greatly expedited the progress of this work the first named author is also indebted to the nserc for support provided by a canada research chair and discovery grant the second named author acknowledges that this material is based upon work supported by the national science foundation under award no dms 1401446 and the engineering and physical sciences research council under grant ep n004922 1

flexible maximum likelihood methods for bivariate proportional hazards models this article presents methodology for multivariate proportional hazards ph regression models the methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional ph models along with assumptions about the association among lifetimes because the models are parametric ordinary maximum likelihood can be applied it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes unlike existing semiparametric methods a bivariate clayton model 1978 biometrika 65 141 151 is used to illustrate the approach taken because a parametric assumption about association is made efficiency and robustness comparisons are made between estimation based on the bivariate clayton model and working independence methods that specify only marginal distributions for each lifetime variable

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

topological conformal field theories and calabi yau categories abstract this is the first of two papers which construct a purely algebraic counterpart to the theory of gromov witten invariants at all genera these gromov witten type invariants depend on a calabi yau a category which plays the role of the target in ordinary gromov witten theory when we use an appropriate a version of the derived category of coherent sheaves on a calabi yau variety this constructs the b model at all genera when the fukaya category of a compact symplectic manifold x is used it is shown under certain assumptions that the usual gromov witten invariants are recovered the assumptions are that open closed gromov witten theory can be constructed for x and that the natural map from the hochschild homology of the fukaya category of x to the ordinary homology of x is an isomorphism

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

analysis of competing risks survival data when some failure types are missing we propose a method to analyse competing risks survival data when failure types are missing for some individuals our approach is based on a standard proportional hazards structure for each of the failure types and involves the solution to estimating equations we present consistent and asymptotically normal estimators of the regression coefficients and related score tests an appealing feature is that individuals with known failure types make the same contributions as they would to a standard proportional hazards analysis contributions of individuals with unknown failure types are weighted according to the probability that they failed from the cause of interest efficiency and robustness are discussed results are illustrated with data from a breast cancer trial

survival with competing risks and masked causes of failures consider a life testing situation in which systems are subject to failure from independent competing risks the hazards of various risks are proportional to each other when a failure occurs immediate i e stage 1 procedures are used in an attempt to reach a definitive diagnosis if a diagnosis is not reached this phenomenon is called masking stage 2 procedures such as failure analysis or autopsy provide definitive diagnosis for a small sample of the masked cases this paper shows how stage 1 and stage 2 information can be combined to provide statistical inference about a survival functions of the individual risks b the proportions of failures associated with individual risks and c probability for a specified masked case that each of the masked competing risks is responsible for the failure