solution bounds of the continuous riccati matrix equation a new approach is proposed for estimating the solution of the continuous algebraic riccati equation care upper and lower solution bounds of the care are presented comparisons show that the present bounds are more general and or tighter than existing results

new upper solution bounds of the continuous algebraic riccati matrix equation by utilizing some linear algebraic techniques new upper bounds of the solution of the continuous algebraic riccati equation care are derived according to the present bounds iterative procedures are also developed for obtaining more precise estimations comparing with existing results the obtained bounds are less restrictive

slope inequality for families of curves over surfaces in this paper we investigate the general notion of the slope for families of curves f x rightarrow y the main result is an answer to the above question when dim y 2 and we prove a lower bound for this new slope in this case over fields of any characteristic both the notion and the slope inequality are compatible with the theory for dim y 0 1 in a very natural way and this gives a strong evidence that the slope for an n fold fibration of curves f x rightarrow y may be k x y n mathrm ch n 1 f omega x y rather than the usual stability methods the whole proof of the slope inequality here is based on a completely new method using characteristic p 0 geometry a simpler version of this method yields a new proof of the slope inequality when dim y 1

counterexamples to the kawamata viehweg vanishing on ruled surfaces in positive characteristic we give counterexamples to the kawamata viehweg vanishing theorem on ruled surfaces in positive characteristic and prove that if there is a counterexample to the kawamata viehweg vanishing theorem on a geometrically ruled surface f x c then either c is a tango curve or all of sections of f are ample

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

some bounds of the solution of the algebraic riccati equation bounds on extremal eigenvalues and lower and upper bounds of the trace for the solution of the algebraic riccati equation are presented through two examples it is shown that in some special cases the presented bounds can be better than the results recently published

uniruled surfaces of general type we give a systematic construction of uniruled surfaces in positive characteristic using this construction we find surfaces of general type with non trivial global vector fields surfaces with arbitrarily non reduced picard schemes as well as surfaces with inseparable canonical maps in particular we show that some previously known pathologies are not sporadic but exist in abundance

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

severi inequality for varieties of maximal albanese dimension in this paper we prove the general severi inequality for varieties of maximal albanese dimension suppose that x is an n dimensional projective normal minimal and mathbb q gorenstein variety of general type in characteristic zero if x is of maximal albanese dimension then k n x ge 2 n chi omega x

improved upper bounds for the solution of the continuous algebraic riccati matrix equation in this paper we present upper bounds for the solution matrix of the continuous algebraic riccati matrix equation care by utilizing some matrix inequalities and linear algebraic techniques also for the derived each bounds iterative algorithms are developed to obtain tighter solution estimates having compared some appearing results in the literature the obtained bounds are less restrictive and more efficient finally numerical examples are given to illustrate the effectiveness of the proposed results

chern slopes of simply connected complex surfaces of general type are dense in 2 3 we prove that for any number r 2 2 3 there are spin resp nonspin and minimal simply connected complex surfaces of general type x with c 2 x c2 x arbitrarily close to r in particular this shows the existence of simply connected surfaces of general type arbitrarily close to the bogomolov miyaoka yau line in addition we prove that for any r2 1 3 and any integer q 0 there are minimal complex surfaces of general type x with c 2 x c2 x arbitrarily close to r and 1 x isomorphic to the fundamental group of a compact riemann surface of genus q

new results for the bounds of the solution for the continuous riccati and lyapunov equations this paper presents upper and lower matrix bounds for the solution of the continuous algebraic matrix riccati equation furthermore a new lower matrix bound for the solution of the continuous algebraic lyapunov equation is also developed these are new results

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

new lower solution bounds of the continuous algebraic riccati matrix equation in optimal and robust control problems the so called continuous algebraic riccati equation care plays an important part by utilizing some matrix inequalities and linear algebraic techniques new lower matrix bounds for the solution of the care are derived following the derived bounds iterative algorithms are then developed to obtain sharper solution estimates in comparison to existing results the obtained bounds are less restrictive finally we give numerical examples to demonstrate the effectiveness of our results

brief lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations in this paper we propose lower matrix bounds for the continuous algebraic riccati and lyapunov matrix equations we give comparisons between the parallel estimates finally we give examples showing that our bounds can be better than the previous results for some cases

a note on eigenvalue bounds in algebraic riccati equation in this note an upper bound on the maximum eigenvalue of the solution matrix k of the algebraic riccati equation is established the approach outlined also results in several lower bounds which are more general than those derived in 1 for some of the largest eigenvalues of k

the bogomolov miyaoka yau inequality for logarithmic surfaces in positive characteristic we generalize bogomolov s inequality for higgs sheaves and the bogomolov miyaoka yau inequality in positive characteristic to the logarithmic case we also generalize shepherd barron s results on bogomolov s inequality on surfaces of special type from rank 2 to the higher rank case we use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2 these examples contradict some claims by xie

bogomolov s inequality for higgs sheaves in positive characteristic we prove bogomolov s inequality for higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 this implies the miyaoka yau inequality on surfaces of non negative kodaira dimension liftable modulo p 2 this result is a strong version of shepherd barron s conjecture our inequality also gives the first algebraic proof of bogomolov s inequality for higgs sheaves in characteristic zero solving the problem posed by narasimhan

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

non classical godeaux surfaces a non classical godeaux surface is a minimal surface of general type with k2 1 but with h01 0 we prove that such surfaces fulfill h01 1 and they can exist only over fields of positive characteristic at most 5 like non classical enriques surfaces they fall into two classes the singular and the supersingular ones we give a complete classification in characteristic 5 and compute their hodge hodge witt and crystalline cohomology including torsion finally we give an example of a supersingular godeaux surface in characteristic 5

bounds in the lyapunov matrix differential equation upper and lower bounds for the trace of the solution of the lyapunov matrix differential equation are derived it is shown that they are obtained as solutions to simple scalar differential equations as a special case the bounds for the stationary solution give ones for the solution to the lyapunov algebraic equation

relative noether inequality on fibered surfaces abstract we prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic it can be viewed as a relative version of the classical noether inequality for surfaces as a consequence we give a new proof of the slope inequality for fibered surface without using any stability method the treatment is essentially different from those of xiao cornalba harris and moriwaki we also study the geography problem of surfaces in positive characteristics and show that the severi inequality is true for surfaces of general type in positive characteristic whose albanese map is generically finite moreover the geography of surfaces with albanese fibrations is studied

a new approach for upper bound estimations of the solution of the continuous riccati equation by utilizing some linear algebraic techniques a new approach for measuring upper bounds of the solution of the continuous algebraic riccati equation care is developed two upper matrix bounds are developed comparing with those appeared in the literature the obtained bounds are less restrictive or more concise therefore they can be applied easily to deal with control problems

the severi inequality k2 4χ for surfaces of maximal albanese dimension we prove the so called severi inequality stating that the invariants of a minimal smooth complex projective surface of maximal albanese dimension satisfy k 2 s ge4 chi s

on some systems of simultaneous equations in a completely distributive lattice abstract we discuss some systems of simultaneous equations in a completely distributive lattice l when the unknowns are regarded as elements of l we obtain necessary and sufficient conditions for the systems to have a solution in addition we consider an application for systems of fuzzy relation equations

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

structure of fuzzy subrings abstract conditions are given for a fuzzy subring of a commutative ring with identity to have a fuzzy coefficient field the results obtained are then used to determine structural properties of fuzzy subrings e g to determine when a fuzzy subring is a fuzzy polynomial subring or a fuzzy power series subring

kawamata viehweg vanishing on rational surfaces in positive characteristic we prove that the kawamata viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to w2 k of certain log pairs on smooth rational surfaces as a corollary the kawamata viehweg vanishing theorem holds on log del pezzo surfaces in positive characteristic

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

automorphisms of smooth canonically polarized surfaces in positive characteristic abstract this paper investigates the geometry of a smooth canonically polarized surface x defined over an algebraically closed field of characteristic p 0 in the case when the automorphism scheme of x is not smooth in particular it is shown that a smooth canonically polarized surface x with 1 k x 2 2 and non smooth automorphism scheme tends to be uniruled and simply connected and is the purely inseparable quotient of a ruled or rational surface by a rational vector field moreover restrictions on certain numerical invariants of x are obtained in order for aut x to be smooth

brief paper bounds on performance of nonstationary continuous time filters under modelling uncertainty a method for obtaining quantitative measures of the performance of non stationary kalman type filters for time varying linear continuous time systems with modelling errors is described the performance measures consist of upper and lower bounds on the trace of the estimation error covariances the mean squared error of suboptimal filters errors in system configuration noise statistics and prior information are treated and the resulting bounds are expressed in terms of the model matrices and the range of errors in these matrices

a non vanishing theorem for q divisors on surfaces abstract we prove a non vanishing theorem of the cohomology h 0 of the adjoint divisor k x l where l is the round up of a nef and big q divisor l

logarithmic transformations of rigid analytic elliptic surfaces we give new examples of algebraic elliptic surfaces and non algebraic rigid analytic elliptic surfaces by means of logarithmic transformations in the complex analytic case it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations using rigid analytic geometry we construct similar transformations of elliptic surfaces over complete non archimedean valuation base fields these operations yield rigid analytic elliptic fibrations with multiple fibers when the resulting surface admits an ample line bundle we may algebraize the surface in the positive characteristic case we obtain new types of algebraic elliptic surfaces we also obtain a non algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case

upper matrix bounds for the discrete algebraic riccati matrix equation we propose upper matrix bounds for the discrete algebraic riccati matrix equation we show that our results are less restrictive than the previous bounds finally we give numerical examples in order to verify the effectiveness of our results

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

on torsors under elliptic curves and serre s pro algebraic structures let mathcal o k be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions k let x k be a torsor under an elliptic curve a k over k x the proper minimal regular model of x k over s hbox spec mathcal o k and j the identity component of the neron model of mathrm pic x k k 0 we study the canonical morphism q mathrm pic 0 x s rightarrow j which extends the natural isomorphism on generic fibres we show that q is pro algebraic in nature with a construction that recalls serre s work on local class field theory furthermore we interpret our results in relation to shafarevich s duality theory for torsors under abelian varieties

effective non vanishing for algebraic surfaces in positive characteristic abstract we give a theorem on the effective non vanishing problem for algebraic surfaces in positive characteristic for the kawamata viehweg vanishing the logarithmic kollar vanishing and the logarithmic semipositivity we give their counterexamples on ruled surfaces in positive characteristic

fuzzy commutative algebra and intersection equations abstract in this paper we discuss the status of fuzzy commutative algebra as it applies to intersection equations we give an up to date account of the theory of nonlinear systems of intersection equations of l singletons the concept of an algebraic l variety is presented in order to bring fuzzy commutative algebra to bear on the solution of such intersection equations

fuzzy intersection equations abstract we give necessary and sufficient conditions for the solution of a system of fuzzy intersection equations we also give an algorithm for the solution of such a system we apply the results to fuzzy graph theory and to fuzzy commutative algebra

multiple fibers on rational elliptic surfaces our main result theorem 0 1 classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic one result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the kodaira fiber types for which no examples were previously known 0 introduction we work in this paper over an algebraically closed field k of arbitrary characteristic p an elliptic surface f x b is a fibration of a complete smooth surface x over a complete smooth curve b such that the generic fiber is a smooth curve of genus one it is natural to wonder what kind of special fibers arise we answer this in the case of rational surfaces giving as a result examples of tame multiple fibers of additive type for every p o and for all such types except ib b 4 cf 5 to our knowledge the only example heretofore recognized is of multiplicity 2 and kodaira type iox due to katsura ka l1 although since our results were obtained we have seen a preprint of katsura and ueno ku giving other examples mostly for p 7 2 3 and kodaira type not ib b 0 any special fiber is an effective divisor on x and so can be written as mf menjcj where each curve cj is reduced and irreducible and the coefficients are positive integers with g c d nj 1 the positive integer m is the rnultiplicity of the fiber mf which is said to be a multiple fiber whenever m 1 the form of the divisor f enjcj is very restricted such a divisor is an indecomposable curve of canonical type mu a topological classification of all such curves has been given by kodaira ko cf 5 we refer to the classes which arise in this classification as the kodaira fiber types more generally the kodaira type of the fiber mf is mt where t is the type of f over an algebraically closed field k it is known that any multiple fiber mf of multiplicity m on a surface x is of the form mt where t is a kodaira type and m is divisible by and in characteristic 0 equal to the order of the normal bundle ox f s of e pic f of f in x but the extent to which these conditions are sufficient for there to occur on some elliptic surface a multiple fiber of type mt is not known of particular interest to us here are the tame multiple fibers of additive type these being fibers mf of multiplicity m where pic f is the additive group ga of the ground field k and m equals the order in ga f ox f s of these are possible only if p the characteristic of k is positive since ga only then has ptorsion but their occurrence is poorly understood received by the editors march 21 1987 1980 mathematzcs subject classzficatzon 1985 re szon primary 14j25 14m20 14d99

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

on the arakelov inequality in positive characteristic in this note we generalize the arakelov inequality in positive characteristic for non isotrivial semistable families of curves of g ge 2 which are liftable to w 2 k resp w k as a consequence we give an analogue of beauville s conjecture in positive characteristic there are at least 5 singular fibers for non isotrivial semistable families of curves of g ge 2 over mathbb p 1 which are liftable to w k