examples and counterexamples for existence of categorical quotients abstract we give examples for existence and non existence of categorical quotients for algebraic group actions in the categories of algebraic varieties and prevarieties all our examples are subtorus actions on toric varieties

a link invariant from the symplectic geometry of nilpotent slices we define an invariant of oriented links in s 3 using the symplectic geometry of certain spaces which arise naturally in lie theory more specifically we present a knot as the closure of a braid which in turn we view as a loop in configuration space fix an affine subspacesm of the lie algebra sl2m c which is a transverse slice to the adjoint action at a nilpotent matrix with two equal jordan blocks the adjoint quotient map restricted to sm gives rise to a symplectic fibre bundle over configuration space an inductive argument constructs a distinguished lagrangian submani

positivity of character subbundles and minimum principle for noncompact group actions based on the results of berndtsson on the positivity of holomorphic vector bundles associated to holomorphic fibration we prove that under the setting of the compact lie group actions the character subbundles of such bundles are holomorphic vector bundles and positive in the sense of nakano the second main purpose of the present paper is to establish berndtsson s integral form of kiselman s minimum principle in the setting of the noncompact lie group actions

linear invariants of complex manifolds and their plurisubharmonic variations abstract for a bounded domain d and a real number p 0 denote by a p d the linear space of l p integrable holomorphic functions on d equipped with the l p pseudonorm we prove that two bounded hyperconvex domains d 1 c n and d 2 c m are biholomorphic in particular n m if there is a linear isometry between a p d 1 and a p d 2 for some 0 p 2 the same result holds for p 2 p 2 4 provided that the p bergman kernels on d 1 and d 2 are exhaustive with this as a motivation we show that for all p 0 the p bergman kernel on a strongly pseudoconvex domain with c 2 boundary or a simply connected homogeneous regular domain is exhaustive these results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear invariants of complex manifolds the second part of the present work devotes to studying variations of these invariants we show that the direct image sheaf of the twisted relative pluricanonical bundle associated to a holomorphic family of stein manifolds or compact kahler manifolds is positively curved with respect to the canonical singular finsler metric

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

an oka principle for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on normal stein spaces x and y which are locally g biholomorphic over a common categorical quotient q when is there a global g biholomorphism x y if the actions of g on x and y are what we with justification call generic we prove that the obstruction to solving this local to global problem is topological and provide sufficient conditions for it to vanish our main tool is the equivariant version of grauert s oka principle due to heinzner and kutzschebauch we prove that x and y are g biholomorphic if x is k contractible where k is a maximal compact subgroup of g or if x and y are smooth and there is a g diffeomorphism x y over q which is holomorphic when restricted to each fibre of the quotient map x q we prove a similar theorem when is only a g homeomorphism but with an assumption about its action on g finite functions when g is abelian we obtain stronger theorems our results can be interpreted as instances of the oka principle for sections of the sheaf of g biholomorphisms from x to y over q this sheaf can be badly singular even for a low dimensional representation of sl2 our work is in part motivated by the linearisation problem for actions on n it follows from one of our main results that a holomorphic g action on n which is locally g biholomorphic over a common quotient to a generic linear action is linearisable

infinite dimensionality of the automorphism groups of homogeneous stein manifolds we show that the group rm aut mathcal o x of holomorphic automorphisms of a stein manifold x with dim x 2 is infinite dimensional provided x is a homogeneous space of a holomorphic action of a complex lie group

on the holomorphic separability of discrete quotients of complex lie groups let g be a complex lie group and h g be a closed complex subgroup of g we refer to the quotient x g h as a complex homogeneous manifold it is natural to ask under which conditions a complex homogeneous manifold x is stein first we give some known results matsushima m proved that if g is reductive then x g h is a stein manifold if and only if the group h is reductive in particular if dime h 0 then g h is stein if and only if h is a finite group barth and otte bo showed that for g reductive a complex homogeneous manifold g h being holomorphically separable implies that h is algebraic in g as the following example shows the condition of holomorphical separability is not sufficient for steinness let g sl2 c

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

simple and robust methods for support vector expansions most support vector sv methods proposed in the recent literature can be viewed in a unified framework with great flexibility in terms of the choice of the kernel functions and their constraints we show that all these problems can be solved within a unique approach if we are equipped with a robust method for finding a sparse solution of a linear system moreover for such a purpose we propose an iterative algorithm that can be simply implemented finally we compare the classical sv approach with other recently proposed cross correlation based alternative methods the simplicity of their implementation and the possibility of exactly calculating their computational complexity constitute important advantages in a real time signal processing scenario

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

curvature of vector bundles associated to holomorphic fibrations let l be a semi positive line bundle over a kahler manifold x fibered over a complex manifold y assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle e over y whose fibers over points y are the spaces of global sections over xy to l kx y endowed with the l2 metric is semi positive in the sense of nakano we also discuss various applications among them a partial result on a conjecture of griffiths on the positivity of ample bundles

finite element vibration analysis of composite beams based on higher order beam theory abstract this paper presents a new finite element formulation for the free vibration analysis of composite beams based on the third order beam theory this work also studies the influence of the mass components resulting from higher order displacements on the frequencies of flexural vibration by using hamilton s principle the variational consistent equation of motion in matrix form corresponding to the third order shear deformation theory is derived the resulting mass matrices are decomposed into three parts i e the usual part including the rotary inertia corresponding to first order theory the part resulting from higher order displacement and the part resulting from the coupling between the different components of the axial displacement the numerical examples show that the higher order and coupling mass matrices have a significant influence on the frequencies of high mode flexural vibration the present element formulation for composite beams can be easily extended to composite plates and shells

complex analytic quotients of algebraic g varieties it is shown that any compact semistable quotient of a normal variety by a complex reductive lie group g in the sense of heinzner and snow is a good quotient this reduces the investigation and classification of such complex analytic quotients to the corresponding questions in the algebraic category as a consequence of our main result we show that every compact space in nemirovski s class fancyscript q g has a realisation as a good quotient and that every complete algebraic variety in fancyscript q g is unirational with finitely generated cox ring and at worst rational singularities in particular every compact space in class fancyscript q t where t is an algebraic torus is a toric variety

an equivariant version of grauert s oka principle for example a theorem of oka asserts that every holomorphic line bundle over a domain of holomorphy in 12 is holomorphicaily trivial if and only if it is topologically trivial in g3 grauer t proves that the category of bundles over a stein space x with a complex lie group as a structure group is the same up to isomorphism if considered in the topological or complex analytic category we would like to underl ine that grauert s method of proof also has far reaching consequences see e g fr1 fr2 in this article we prove the equivariant version of grauert s oka principle for a compact lie group of holomorphic t ransformations on x the precise statements can be found in w w and w for example as a consequence of the main results we obtain

homotopy principles for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on stein manifolds x and y let px x qx and py y qy be the quotient mappings when is there an equivariant biholomorphism of x and y a necessary condition is that the categorical quotients qx and qy are biholomorphic and that the biholomorphism sends the luna strata of qx isomorphically onto the corresponding luna strata of qy fix we demonstrate two homotopy principles in this situation the first result says that if there is a g diffeomorphism x y inducing which is g biholomorphic on the reduced fibres of the quotient mappings then is homotopic through g diffeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y the second result roughly says that if we have a g homeomorphism x y which induces a continuous family of gequivariant biholomorphisms of the fibres px 1 q and py 1 q for q qx and if x satisfies an auxiliary property which holds for most x then is homotopic through g homeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y our results improve upon those of our earlier paper j reine angew math 706 2015 193 214 and use new ideas and techniques

maximal complexifications of certain homogeneous riemannian manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

exact frequency and mode shape formulae for studying vibration and stability of timoshenko beam system two sets of exact natural frequency and mode shape formulae for an axially loaded timoshenko uniform single span beam carrying elastically supported end masses are derived the end masses are restrained against rotation and translation with linear springs the two equations of motion of references 1 2 are used in the problem formulation comparisons between the derived frequency equations for a typical example of a clamped free beam are made the difference between natural frequencies is significant for beams subjected to high loads and for high modes of vibration most of the well known published exact expressions for determining the natural frequencies and the mode shapes can be obtained from the present equations as special cases the influence of beam rotary inertia and end mass es geometrical parameters on the natural frequency and critical buckling load coefficients has been studied when an axial compressive load is acting at the centre of gravity of the end mass es the same critical buckling load coefficients are found for free free pinned free and pinned pinned beams and similarly for clamped sliding and sliding sliding beams in addition if the slope due to bending is allowed at one end of the beam the same critical buckling loads are found for clamped free sliding free and sliding pinned beam configurations the present work includes graphs and numerical tables describing the variation of the natural frequencies and the critical buckling load coefficients from the design point of view

compact kahler quotients of algebraic varieties and geometric invariant theory abstract given an action of a complex reductive lie group g on a normal variety x we show that every analytically zariski open subset of x admitting an analytic hilbert quotient with projective quotient space is given as the set of semistable points with respect to some g linearised weil divisor on x applying this result to hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to git semistability in the sense of mumford and hausen it follows that the number of compact momentum map quotients of a given algebraic hamiltonian g variety is finite as further corollary we derive a projectivity criterion for varieties with compact kahler quotient

blowing up extremal kahler manifolds ii this is a continuation of the work of arezzo pacard singer and the author on blowups of extremal kahler manifolds we prove the conjecture stated in szekelyhidi duke math j 161 8 1411 1453 2012 and we relate this result to the k stability of blown up manifolds as an application we prove that if a kahler manifold m of dimension 2 admits a constant scalar curvature csck metric then the blowup of m at a point admits a csck metric if and only if it is k stable as long as the exceptional divisor is sufficiently small

explicit analytical expressions for frequency equation and mode shapes of composite beams abstract a systematic procedure for the derivation of exact expressions for the frequency equation and mode shapes of composite beams undergoing free vibration is presented by using the symbolic computing package reduce the effect of material coupling between the bending and torsional modes of deformation which usually exists in composite beams due to ply orientation is taken into account while developing the theory the governing differential equations of motion of the bending torsion coupled composite beam are solved analytically for bending displacements and torsional rotations in free vibration for subsequent developments the important case of the cantilever beam is chosen because of its application to aircraft wings the boundary conditions for displacements and forces for the cantilever are imposed and the frequency equation is obtained but first in the form of a determinant and then in the form of an explicit algebraic expression the expressions for the mode shapes are also derived in explicit analytical form the method is demonstrated by an illustrative example of a composite beam for which some comparative results are available in the literature the future potential of this method particularly in the context of aeroelastic optimisation of composite wings is considerable because it is very accurate computationally efficient and importantly free from ill conditioning problems usually associated with numerical matrix manipulation

an equivariant parametric oka principle for bundles of homogeneous spaces we prove a parametric oka principle for equivariant sections of a holomorphic fibre bundle e with a structure group bundle mathscr g on a reduced stein space x such that the fibre of e is a homogeneous space of the fibre of mathscr g with the complexification k mathbb c of a compact real lie group k acting on x mathscr g and e our main result is that the inclusion of the space of k mathbb c hbox equivariant holomorphic sections of e over x into the space of k hbox equivariant continuous sections is a weak homotopy equivalence the result has a wide scope we describe several diverse special cases we use the result to strengthen heinzner and kutzschebauch s classification of equivariant principal bundles and to strengthen an oka principle for equivariant isomorphisms proved by us in a previous paper

the moduli space of fano manifolds with kahler ricci solitons abstract we construct a canonical hausdorff complex analytic moduli space of fano manifolds with kahler ricci solitons this enlarges the moduli space of fano manifolds with kahler einstein metrics we discover a moment map picture for kahler ricci solitons and give complex analytic charts on the topological space consisting of kahler ricci solitons by studying differential geometric aspects of this moment map some stacky words and arguments on gromov hausdorff convergence help to glue them together in the holomorphic manner

free vibration of functionally graded beams and frameworks using the dynamic stiffness method the free vibration analysis of functionally graded beams fgbs and frameworks containing fgbs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form the usually adopted rule that the material properties of the fgb vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration the differential equations are solved in closed analytical form when the free vibratory motion is harmonic the dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam next the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation finally the wittrick williams algorithm is applied as solution technique to solve the free vibration problems of fgbs with uniform cross section stepped fgbs and frameworks consisting of fgbs some numerical results are validated against published results but in the absence of published results for frameworks containing fgbs consistency checks on the reliability of results are performed the paper closes with discussion of results and conclusions

siu s lemma optimal l 2 extension and applications to twisted pluricanonical sheaves we prove a general version of siu s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal l 2 extension theorem and an optimal l frac 2 m extension theorem for kahler fibrations these results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles

positivity of direct images of positively curved volume forms we discuss kiselman berndtsson s minimum principle for plurisubharmonic functions in terms of the positivity of direct images of certain positively curved volume forms and generalize it to holomorphically convex manifolds with compact group actions with this generalization and other techniques we establish a minimum principle for positively curved volume forms from the point of view of geometric invariant theory on stein manifolds minimum principle with some noncompact group actions is also considered