the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

universality of 2 qudit ternary reversible gates in this paper we investigate the synthesis of ternary reversible circuits in the absence of ancilla bits we demonstrated that 2 qudit ternary swap not and 1 controlled not gates are universal for realization of arbitrary ternary n qudit reversible circuits without ancilla qudits and all even ternary n qudit reversible circuits can be constructed by ternary not and ternary 1 controlled not gates without ancilla qudits the realization approach is constructive this result is significantly different from the binary case

an algebraic approach to strong stabilizability of linear nd mimo systems although some necessary conditions for the strong stabilizability of linear multidimensional nd multiple input multiple output mimo systems have been available recently very little is known about sufficient conditions for the same problem this note presents two sufficient conditions for strong stabilizability of some classes of linear nd mimo systems obtained using an algebraic approach a simple necessary and sufficient condition is also given for the strong stabilizability of a special class of linear nd mimo systems an advantage of the proposed algebraic approach is that a stable stabilizing compensator can be constructed for an nd plant satisfying the sufficient conditions for the strong stabilizability presented in this note

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

obstructions to the existence of sasaki einstein metrics we describe two simple obstructions to the existence of ricci flat kahler cone metrics on isolated gorenstein singularities or equivalently to the existence of sasaki einstein metrics on the links of these singularities in particular this also leads to new obstructions for kahler einstein metrics on fano orbifolds we present several families of hypersurface singularities that are obstructed including 3 fold and 4 fold singularities of ade type that have been studied previously in the physics literature we show that the ads cft dual of one obstruction is that the r charge of a gauge invariant chiral primary operator violates the unitarity bound

on a class of fully nonlinear elliptic equations on closed hermitian manifolds ii l estimate we study a class of fully nonlinear elliptic equations on closed hermitian manifolds under the assumption of the cone condition we derive the l estimate directly as an application we solve the complex quotient equations on closed kahler manifolds 2016 wiley periodicals inc

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

on the strong stabilizability of mimo n dimensional linear systems a plant is strongly stabilizable if there exists a stable compensator to stabilize it based on some theorems in complex analysis of several variables proved in this paper we present necessary conditions for the strong stabilizability of complex and real n d multi input multi output mimo shift invariant linear plants for the real case the condition is a generalization of the parity interlacing property of youla bongiorno and lu automatica j ifac 10 1974 pp 159 173 for the strong stabilizability of a real one dimensional mimo plant these conditions are also sufficient for the cases of n d plants with a single output miso or with a single input simo for general n d mimo plants we do not know if the conditions are sufficient or not a useful sufficient but not necessary condition for the strong stabilizability of a class of n d n geq 2 mimo plants is given

model matching with strong stability in switched linear systems abstract this work deals with model matching by output dynamic feedback in switched linear systems the plant and the model are assumed to be subject to different switching signals a necessary and sufficient condition for achieving model matching with asymptotic stability of both the closed loop compensated system and the compensator for a sufficiently large dwell time is proven such condition is obtained by specializing the structural condition with a suitable redefinition of the robust controlled invariant subspace involved capable of capturing not only the structural aspect of the problem but also the stability aspects the effect of the combined action of nonzero initial states and nonzero inputs is dealt with the solution to a more demanding problem formulation where the dwell time is assumed to be fixed and given is also discussed

the j flow and stability we study the j flow from the point of view of an algebro geometric stability condition in terms of this we give a lower bound for the natural associated energy functional and we show that the blowup behavior found by fang and lai 11 is reflected by the optimal destabilizer finally we prove a general existence result on complex tori

i inverness for polynomial matrices of non constant rank abstract let p1 pt be polynomials in c n with a variety v of common zeros contained in a suitable open set u explicit formulas are provided to construct rational functions 1 s such that i 1spi i 1 and such that the singularities of the i are contained in u this result is applied to compute rational functions valued 1 inverses of matrices with polynomial coefficients which do not have constant rank while retaining control over the location of the singularities of the rational functions themselves

kahler einstein metrics and volume minimization abstract we prove that if a polarized q fano variety admits a kahler einstein metric then the normalized volume functional over the associated affine cone is globally minimized at the canonical divisorial valuation obtained by blowing up the vertex this is also generalized to the logarithmic and the orbifold settings a refinement of this result together with the result in 41 gives an equivalent characterization of k semistability for any smooth fano manifold we also prove that the valuation associated to the reeb vector field of a smooth sasaki einstein metric minimizes the normalized volume over the corresponding kahler cone these results strengthen and generalize the minimization result of martelli sparks yau

on explicit solutions to the bezout equation explicit constructions are given for the solution of the bezout equation for matrices whose coefficients are multivariate polynomials or entire functions of paley wiener type this procedure also provides the 1 inverse when the bezout equation has no solutions

on a class of fully nonlinear flows in kahler geometry in this paper we study a class of fully nonlinear metric flow on kahler manifolds which includes the j flow as a special case we provide a sufficient and necessary condition for the long time convergence of the flow generalizing the result of songweinkove as a consequence under the given condition we solved the corresponding euler equation which is fully nonlinear of monge ampere type as an application we also discuss a complex monge ampere type equation including terms of mixed degrees which was first posed by chen

robustness analysis of uncertain linear singular systems with output feedback control the robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered the problem is transformed into a robust nonsingularity problem based on the linear fractional transformation lft approach the robustness bounds to preserve regularity impulse immunity and stability are found in terms of the structured singular value spl mu with respect to parametric uncertainties the lft approach provides a unified framework for robustness analysis of both uncertain linear continuous discrete time singular systems

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

fully non linear elliptic equations on compact hermitian manifolds we derive a priori estimates for solutions of a general class of fully non linear equations on compact hermitian manifolds our method is based on ideas that have been used for different specific equations such as the complex monge ampere hessian and inverse hessian equations as an application we solve a class of hessian quotient equations on kahler manifolds assuming the existence of a suitable subsolution the method also applies to analogous equations on compact riemannian manifolds

k semistability for irregular sasakian manifolds we introduce a notion of k semistability for sasakian manifolds this extends to the irregular case of the orbifold k semistability of ross thomas our main result is that a sasakian manifold with constant scalar curvature is necessarily k semistable as an application we show how one can recover the volume minimization results of martelli sparks yau and the lichnerowicz obstruction of gauntlett martelli sparks yau from this point of view

convergence of the j flow on toric manifolds we show that on a kahler manifold whether the j flow converges or not is independent of the chosen background metric in its kahler class on toric manifolds we give a numerical characterization of when the j flow converges verifying a conjecture in 19 m lejmi and g szekelyhidi the j flow and stability in this case we also strengthen existing results on more general inverse sigma k equations on kahler manifolds

immersed hypersurfaces with constant weingarten curvature in a recent paper korevaar 1 5 used the alexandrov reflection principle to show that closed embedded hypersurfaces in n 1 lhn 1 or the upper hemisphere of s 1 are umbilic spheres provided a certain function f of the principal curvatures t t 22 is constant he only had to assume that f is positive on the positive cone 212i 0 vi and that f is elliptic on the component f of 21 f 0 which contains here f is said to be elliptic if 0f d21 0 for all i 1 inn this generalization of earlier sphere theorems see 9 and 10 for references cannot be extended to hypersurface immersions in view of recent counterexamples 1 11 however assuming additional curvature conditions walter derived in 10 global results for hypersurface immersions in a space n l c of constant curvature c which have a constant higher mean curvature function h here h is the r th symmetric function of the principal curvatures it was shown that such hypersurfaces are of constant mean curvature h1 provided they have nonnegative sectional curvature and non negative principal curvatures as a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified here we show that it is not necessary to assume all principal curvatures to be non negative moreover we extend waiter s result to general symmetric functions f f 2 let m be a smooth connected and compact manifold without boundary then we have the following result

stability and stabilization of multidimensional input output systems in this paper we discuss stability and stabilization of continuous and discrete multidimensional input output io behaviors of dimension r which are described by linear systems of complex partial differential resp difference equations with constant coefficients where the signals are taken from various function spaces in particular from those of polynomial exponential functions stability is defined with respect to a disjoint decomposition of the r dimensional complex space into a stable and an unstable region with the standard stable region in the one dimensional continuous case being the set of complex numbers with negative real part a rational function is called stable if it has no poles in the unstable region an io behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region this is equivalent to the stability of its transfer matrix and an additional condition the system is called stabilizable if there is a compensator io system such that the output feedback system is well posed and stable we characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable io system parametrization the theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional io maps as developed for instance by bose guiver shankar sule xu lin ying zerz and quadrat and of course the seminal papers of vidyasagar youla and others in the one dimensional case in contrast to the existing literature the theorems and proofs of this paper do not need or employ the so called fractional representation approach i e various matrix fraction descriptions of the transfer matrix thus avoiding the often lengthy matrix computations and seeming to be of interest even for one dimensional systems at least to the author an important mathematical tool new in systems theory is gabriel s localization theory which only in the case of ideal convex shankar sule unstable regions coincides with the usual one algorithmic tests for stability stabilizability and ideal convexity and the algorithmic construction of stabilizing compensators are addressed but still encounter many difficulties see in particular the open problems listed by xu e

a direct approach to compensator design for distributed parameter systems we present a direct approach to finite order compensator design for distributed parameter systems i e one that is not based on reduced order modelling instead we use a parametrization around an initial compensator which displays both controller order and closed loop stability in a convenient way the main result is an existence theorem which holds for a wide class of linear time invariant systems parabolic delay damped hyperbolic the most important assumptions are bounded inputs and outputs finitely many unstable modes completeness of eigenvectors an example is included to illustrate the feasibility of our method for purposes of design

a behavioral approach to delay differential systems we will study linear time invariant delay differential systems from the behavioral point of view as it was introduced for dynamical systems by willems dynam report 2 1989 pp 171 269 a ring cal h which lies between bbb r s z z 1 and bbb r s z z 1 will be presented whose elements can be interpreted as a generalized version of delay differential operators on cal c infty bbb r bbb r in this framework a behavior is the kernel of such an operator using the ring cal h an algebraic characterization of inclusion respectively equality of the behaviors under consideration is given finally controllability of the behaviors is characterized in terms of the rank of the associated matrices in the case of time delay state space systems this criterion becomes the known hautus criterion for spectral controllability

feedback stabilization of mimo 3 d linear systems the authors solve the open problem of the existence of double coprime factorizations for a large class of multi input multi output mimo three dimensional 3 d linear systems it is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal mimo 3 d plant are simple then the plant has a double coprime factorization the authors then give a parameterization of all stabilizing compensators for a mimo 3 d system in this class the key result developed in the paper is a novel and constructive technique of replacing an unstable polynomial with a stable polynomial step by step an illustrative example is also provided

stabilization of time delay systems using finite dimensional compensators for linear time invariant systems with one or more noncommensurate time delays necessary and sufficient conditions are given for the existence of a finite dimensional stabilizing feedback compensator in particular it is shown that a stabilizable time delay system can always be stabilized using a finite dimensional compensator the problem of explicitly constructing finite dimensional stabilizing compensators is also considered

fredkin gates for finite valued reversible and conservative logics the basic principles and results of conservative logic introduced by fredkin and toffoli in 1982 on the basis of a seminal paper of landauer are extended to d valued logics with a special attention to three valued logics different approaches to d valued logics are examined in order to determine some possible universal sets of logic primitives in particular we consider the typical connectives of ukasiewicz and godel logics as well as chang s mv algebras as a result some possible three valued and d valued universal gates are described which realize a functionally complete set of fundamental connectives two no go theorems are also proved

realizing ternary quantum switching networks without ancilla bits this paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits the results we established are twofold the first shows that ternary swap ternary not and ternary toffoli gates are universal for the realization of arbitrary n n ternary quantum switching networks without ancilla bits the second result proves that all n n quantum ternary networks can be generated by not controlled not multiply two and toffoli gates our approach is constructive

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

some algebraic aspects of the strong stabilizability of time delay linear systems this note considers some algebraic aspects of the output feedback strong stabilizability of time delay linear systems a sufficient condition for the existence of a stable finite dimensional stabilizing compensator for a single input single output siso time delay system is given an illustrative example is also given and two open problems are posed

strategies for designing geometric transformations on quantum images three design strategies for constructing new geometric transformations on quantum images from other transformations are proposed the strategies focus on the affected areas in the images the separability and smoothness of the transformations by exploiting a representation of images on quantum computers extensively the complexity in terms of the number of basic gates and the depth of the corresponding circuits are analyzed to understand the advantages and disadvantages of each strategy examples to demonstrate the applicability of the proposed strategies are presented the strategies provide high level tools to explore and analyze transformations which are necessary to build practical image processing applications on quantum computers

implementing an arbitrary reversible logic gate the 2w reversible logic gates of width w i e reversible logic gates with w inputs and w outputs together with the action of cascading form a group r we define a subgroup k consisting of the switched gates there are 2w 1 2 such gates they partition the group r into 2w 1 1 double cosets this allows us to decompose an arbitrary reversible gate into the cascade of three gates a switched gate an upside down simple control gate and a second switched gate we present an algorithm to perform this factorization and thus provide a method of implementing an arbitrary reversible gate into hardware the algorithm can be used to automate the implementation of a reversible function in future c mos technologies realizing low cost computing

frankel conjecture and sasaki geometry abstract we classify simply connected compact sasaki manifolds of dimension 2 n 1 with positive transverse bisectional curvature in particular the kahler cone corresponding to such manifolds must be bi holomorphic to c n 1 0 as an application we recover the theorem of mori and siu yau on the frankel conjecture and extend it to certain orbifold version the main idea is to deform such sasaki manifolds to the standard round sphere in two steps both fixing the complex structure on the kahler cone first we deform the metric along the sasaki ricci flow and obtain a limit sasaki ricci soliton with positive transverse bisectional curvature then by varying the reeb vector field which essentially decreases the volume functional we deform the sasaki ricci soliton to a sasaki einstein metric with positive transverse bisectional curvature i e a round sphere the second deformation is only possible when one treats simultaneously regular and irregular sasaki manifolds even if the manifold one starts with is regular quasi regular i e kahler manifolds orbifolds

boolean lie algebras and the leibniz rule using internal negations acting on boolean functions the notion of boolean lie algebra is introduced the underlying lie product is the boolean analogue of the poisson bracket the structure of a boolean lie algebra is determined it turns out to be solvable but not nilpotent we prove that the adjoint representation of an element of the boolean lie algebra acts as a derivative operator on the space of boolean functions the adjoint representation is related to the previously known concept of the sensitivity function using the notion of adjoint representation we give the definition of a temporal derivative applicable to iterative dynamics of boolean mappings

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

minimizing normalized volumes of valuations for any mathbb q gorenstein klt singularity x o we introduce a normalized volume function widehat mathrm vol that is defined on the space of real valuations centered at o and consider the problem of minimizing widehat mathrm vol we prove that the normalized volume has a uniform positive lower bound by proving an izumi type estimate for any mathbb q gorenstein klt singularity furthermore by proving a properness estimate we show that the set of real valuations with uniformly bounded normalized volumes is compact and hence reduce the existence of minimizers for widehat mathrm vol to a conjectural lower semicontinuity property for a smooth point we use an inequality of de fernex ein musta to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of widehat mathrm vol finally the relation to fujita s work on divisorial stability is pointed out

generating the group of reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits such insight is gained from constructing chains of maximal subgroups the subgroup of control gates plays a prominent role as it is a sylow 2 subgroup

the decomposition of an arbitrary reversible logic circuit the 2w reversible logic circuits of width w i e reversible logic circuits with w inputs and w outputs together with the action of cascading form a group g isomorphic to the symmetric group we define two conjugate subgroups g1 and g2 together they partition the group g into 2w 1 1 double cosets these allow us to decompose an arbitrary member of g into a cascade of three simpler members this decomposition is a far relative of the well known lu decomposition of a square matrix

on generalized inverses of matrices over integral domains it is proved that a matrix a over an integral domain admits a 1 inverse if and only if a linear combination of all the r r minors of a is equal to one where r is the rank of a some results on the existence of moore penrose inverses are also obtained

r universal reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to know which reversible logic gates can be used as building block for the reversible implementation of an arbitrary boolean function and which cannot

reducibility in sasakian geometry the purpose of this paper is to study reducibility properties in sasakian geometry first we give the sasaki version of the de rham decomposition theorem however we need a mild technical assumption on the sasaki automorphism group which includes the toric case next we introduce the concept of it cone reducible and consider s 3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2 torus of reeb type in particular we can classify all such sasakian structures up to contact isotopy on s 3 bundles over a riemann surface of genus greater than zero finally we show that in the toric case an extremal sasaki metric on a sasaki join always splits

gauduchon metrics with prescribed volume form we prove that on any compact complex manifold one can find gauduchon metrics with prescribed volume form this is equivalent to prescribing the chern ricci curvature of the metrics and thus solves a conjecture of gauduchon from 1984

realization and synthesis of reversible functions reversible circuits play an important role in quantum computing this paper studies the realization problem of reversible circuits for any n bit reversible function we present a constructive synthesis algorithm given any n bit reversible function there are n distinct input patterns different from their corresponding outputs where email protected 2 n and the other 2 n n input patterns will be the same as their outputs we show that this circuit can be synthesized by at most email protected n n 1 cnot gates and 4n email protected n not gates the time and space complexities of the algorithm are w email protected 4 n and w email protected 2 n respectively the computational complexity of our synthesis algorithm is exponentially lower than that of breadth first search based synthesis algorithms

conical kahler einstein metrics revisited in this paper we introduce the interpolation degeneration strategy to study kahler einstein metrics on a smooth fano manifold with cone singularities along a smooth divisor that is proportional to the anti canonical divisor by interpolation we show the angles in 0 2 that admit a conical kahler einstein metric form a connected interval and by degeneration we determine the boundary of the interval in some important cases as a first application we show that there exists a kahler einstein metric on mathbb p 2 with cone singularity along a smooth conic degree 2 curve if and only if the angle is in 2 2 when the angle is 2 3 this proves the existence of a sasaki einstein metric on the link of a three dimensional a2 singularity and thus answers a question posed by gauntlett martelli sparks yau as a second application we prove a version of donaldson s conjecture about conical kahler einstein metrics in the toric case using song wang s recent existence result of toric invariant conical kahler einstein metrics

on the symmetry of logic embedding logical operations in non dissipative physical processes requires the use of reversible logic following feynman s 1986 approach the sixteen distinct truth tables of classical logic are shown to be contained in the 8 reversible logic operations covered by the symmetric group s8 which permute the eight values of three logical variables small subgroups of s8 are shown to cover respectively reversible logic reversible switching and reversible arithmetic a new universal primitive is found which generates a covering group of reversible logic it is shown that the octahedral group in four dimensions covers both reversible logic and switching and hence that the orthogonal group o 4 provides a covering group for quantum gates

geometric flows and strominger systems a geometric flow on 2 2 forms is introduced which preserves the balanced condition of metrics and whose stationary points satisfy the anomaly equation in strominger systems the existence of solutions for a short time is established using hamilton s version of the nash moser implicit function theorem

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

transverse fully nonlinear equations on sasakian manifolds and applications abstract we prove a priori estimates for a class of transverse fully nonlinear equations on sasakian manifolds and give some geometric applications such as the transversion calabi yau theorem for transverse balanced and strongly gauduchon metrics we also explain that similar results hold on compact oriented taut transverse hermitian foliated manifold of complex co dimension n and give some geometric applications such as the transverse calabi yau theorems for transverse hermitian and strongly gauduchon metrics