effect of colored noise on networks of nonlinear oscillators we discuss noise induced pattern formation in different two dimensional networks of nonlinear oscillators namely a sequence of biochemical reactions and the lorenz system the main focus of the work is on the dependence of these patterns on the correlation time i e the color of exponentially correlated gaussian noise it is seen that in the nonchaotic case the homogeneity or average cluster size goes through a minimum with higher correlation time while in its chaotic regime the lorenz system shows a higher degree of synchronization when the correlation time of the noise is increased in order to elucidate the origin of this phenomenon the effect of colored noise on the individual oscillator is investigated it is shown that the specific dependence of the network s homogeneity on the noise correlation time arises from an interplay of the collective behavior and the properties of the single oscillators doi 10 1103 physreve 64 021105

multiple scales scheme for bifurcation in a delayed extended van der pol oscillator abstract the van der pol vdp oscillator is an essential nonlinear oscillator which has been extensively studied however the results of delayed generalized vdp oscillator are relatively less the issue of bifurcation control for a delayed extended vdp oscillator is fully considered in this paper firstly a nonlinear time delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator and the results of stability of such oscillator are obtained by means of stability switches theory secondly a general explicit formula is derived to establish the properties of hopf bifurcation by utilizing the method of multiple scales moreover the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically finally the effectiveness of theoretical results is verified via a numerical example

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

twisted fourier mukai functors due to a theorem by orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of fourier mukai type we extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor as an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

derived categories of torsors for abelian schemes abstract in the first part of our paper we show that there exist non isomorphic derived equivalent genus 1 curves and correspondingly there exist non isomorphic moduli spaces of stable vector bundles on genus 1 curves in general neither occurs over an algebraically closed field we give necessary and sufficient conditions for two genus 1 curves to be derived equivalent and we go on to study when two principal homogeneous spaces for an abelian variety have equivalent derived categories we apply our results to study twisted derived equivalences of the form d b j d b j when j is an elliptic fibration giving a partial answer to a question of c ld raru

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

on the recursive sequence xn 1 α xn 1xn in this paper we study the global stability the permanence and the oscillation character of the recursive sequence xn 1 xn 1xn n 0 1 where is a negative number and the initial conditions x 1 and x0 are negative numbers

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

equivalences of derived catgories of sheaves on smooth stacks we extend orlov s representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities

on the difference equation xn 1 α xn 1xn in this paper we study the behavior of the solutions of the difference equation x n 1 a x n 1x n n 0 1 where a is a negative number included are results which considerably improve and correct those in the recently published paper a e hamza on the recursive sequence x n 1 a x n 1x n j math anal appl 322 2006 668 674 we also refute conjecture 2 in g ladas open problems and conjectures j difference equ appl 7 2 2001 477 482

on a nonlinear second order system of difference equations we prove that all positive solutions of the next system of difference equationsx n 1 a y n px n 1 q y n 1 a x n py n 1 q email protected n 0 where parameters a p and q are positive numbers are bounded if one of the following conditions is satisfied p 2 4 or p 1 q q 1 completely describing the boundedness character of the system in this case

characterization of escape times of josephson junctions for signal detection the measurement of the escape time of a josephson junction might be used to detect the presence of a sinusoidal signal embedded in noise when use of standard signal processing tools can be prohibitive due to the extreme weakness of the source or to the huge amount of data in this paper we show that the prescriptions for the experimental setup and some physical behaviors depend on the detection strategy more specifically by exploitation of the sample mean of escape times to perform detection two resonant regions are identified at low frequencies there is a stochastic resonance or activation phenomenon while near the plasma frequency a geometric resonance appears furthermore detection performance in the geometric resonance region is maximized at the prescribed value of the bias current the naive sample mean detector is outperformed in terms of error probability by the optimal likelihood ratio test the latter exhibits only geometric resonance showing monotonically increasing performance as the bias current approaches the junction critical current in this regime the escape times are vanishingly small and therefore performance is essentially limited by measurement electronics the behavior of the likelihood ratio and sample mean detector for different values of incoming signal to noise ratio is discussed and a relationship with the error probability is found detectors based on the likelihood ratio test could be employed also to estimate unknown parameters in the applied input signal as a prototypical example we study the phase estimation problem of a sinusoidal current which is accomplished by using the filter bank approach finally we show that for a physically feasible detector the performances are found to be very close to the cramer rao theoretical bound applications might be found for example in some astronomical detection problems where the all sky gravitational and or radio wave search for pulsars requires the analysis of nearly sinusoidal long lived waveforms at very low signal to noise ratio or to analyze weak signals in the subterahertz range where the traditional electronics counterpart is difficult to implement

asymptotic and oscillatory properties of the fourth order nonlinear difference equations in this article a class of fourth order difference equations with quasi differences and a deviating argument is considered we state new oscillation theorems for the super linear and for the half linear case and we complete the existing results in the literature

fano varieties of cubic fourfolds containing a plane we prove that the fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a k3 surface on the other hand we show that the fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a k3 surface the moduli spaces of complexes and of sheaves are related by wall crossing in the derived category of twisted sheaves on the corresponding k3 surface

derived categories and kummer varieties we prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding kummer varieties are equivalent as well the second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two kummer surfaces in terms of hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces

effects of a periodic drive and correlated noise on birhythmic van der pol systems abstract this paper considers the dynamics of a van der pol birhythmic oscillator submitted both to colored noise and harmonic excitation applying the quasi harmonic assumption to the corresponding langevin equation we derive an approximated fokker planck equation that is compared with the results of computer simulations we thus derive both the effects of the correlation time and the harmonic excitation on the parameter space where birhythmicity appears in this region we find that the multi limit cycle van der pol oscillator reduces to an asymmetric bistable system where the sinusoidal drive intensity plays the role of asymmetric parameter and noise can lead to stochastic bifurcations consisting in a qualitative change of the stationary amplitude distribution under both influence of noise and harmonic excitation the dynamics can be well characterized through the concepts of pseudo potential that regulates the low noise arrhenius like behavior

on the global periodicity of discrete dynamical systems and application to rational difference equations a new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here this condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations

on the nonlinear difference equation system xn 1 a yn m xn yn 1 a xn m yn in this paper we study the boundedness persistence and global asymptotic stability of positive solutions of the system of two difference equations x n 1 a y n mx n y n 1 a x n my n n 0 1 where a 0

application of difference equations in divisibility abstract we present an easy applicable method which for a large class of integer sequences a n shows when all their terms are divisible by a natural number m 1

global stability of some symmetric difference equations suppose r 0 1 m n and 1 k1 k2 k2m 1 and let s2m 1 1 2 2m 1 we show that every positive solution to the difference equation yn p2m 12m 1 yn k1r yn k2r yn k2m 1r p2m2m 1 yn k1r yn k2r yn k2m 1r n n0 where p2m 12m 1 x1 x2 x2m 1 r 1rodd2m 1 t1 t2 tr s2m 1t1 t2 trxt1xt2 xtr and p2m2m 1 x1 x2 x2m 1 1 r 2reven2m t1 t2 tr s2m 1t1 t2 trxt1xt2 xtr converges to one this result confirms a quite recent conjecture posed by liu and yang 2010 in 10 we also prove another result regarding a related equation

on the eventual periodicity of xn 1 max 1xn anxn 1 with a period five parameter in this paper we will study the max difference equation x n 1 max 1x n a nx n 1 when a n is a period five sequence of real numbers greater than one we will prove that every solution is eventually periodic with period twenty

on the rational difference equation y n 1 a y n y n k we find conditions for the global asymptotic stability of the unique negative equilibrium y 1 a of the equation yn 1 a yn yn k where y k y k 1 y0 a r and k 1 2 3 4

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

on the recursive sequence xn axn 1 bxn 2c dxn 1xn 2 in this paper we study the invariant intervals the global attractivity of the equilibrium points and the asymptotic behavior of the solutions of the difference equationx n ax n 1 bx n 2c dx n 1x n 2 n 1 2 where a 0 b c d 0

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

a categorical invariant for cubic threefolds abstract we prove a categorical version of the torelli theorem for cubic threefolds more precisely we show that the non trivial part of a semi orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class

dynamics of a higher order rational difference equation abstract in this paper we will investigate a nonlinear rational difference equation of higher order our concentration is on invariant intervals periodic character the character of semicycles and global asymptotic stability of all positive solutions of x n 1 x n x n k bx n cx n k n 0 1 it is worth to mention that our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

global attractivity in a recursive sequence in this paper we study the invariant intervals the globally attractivity of the two equilibrium points and the oscillatory behavior of the solutions of the difference equationx n ax n 1 bx n 2c x n 2 n 1 2 where a b c 0

a gabriel theorem for coherent twisted sheaves the aim of this work is to give a generalization of gabriel s theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes we start by showing that we can recover a noetherian scheme x from the category coh x of coherent twisted sheaves over x where lies in the cohomological brauer group of x this follows from the bijective correspondence between closed subsets of x and serre subcategories of finite type of coh x moreover any equivalence between coh x and coh y where x and y are noetherian schemes and alpha in br x br y induces an isomorphism between x and y

non uniqueness of fourier mukai kernels we prove that the kernels of fourier mukai functors are not unique in general on the other hand we show that the cohomology sheaves of those kernels are unique we also discuss several properties of the functor sending an object in the derived category of the product of two smooth projective schemes to the corresponding fourier mukai functor

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

global attractivity of the difference equation xn 1 α xn k xn in this paper we investigate the global stability of all negative solutions of the difference equationx n email protected x n kx n n 0 1 where a 1 is an integer and the initial conditions x k x 0 are arbitrary real numbers we show that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

on a system of difference equations with maximum in this paper we study the periodic character of the solutions of the system of difference equations with maximum x n 1 max a y n x n 1 y n 1 max b x n y n 1 where a b are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers we prove that every solution is eventually periodic

on the existence of fourier mukai functors a theorem by orlov states that any equivalence f d b mathrm coh x rightarrow d b mathrm coh y between the bounded derived categories of coherent sheaves of two smooth projective varieties x and y is isomorphic to a fourier mukai transform phi e r pi 2 e mathop otimes limits l l pi 1 where the kernel e is in d b mathrm coh x times y in the case of an exact functor which is not necessarily fully faithful we compute some sheaves that play the role of the cohomology sheaves of the kernel and that are isomorphic to the latter whenever an isomorphism f cong phi e exists we then exhibit a class of functors that are not full or faithful and still satisfy the above result

local existence of lipschitz continuous solutions of systems of nonlinear functional equations with iterated deviations abstract some conditions which guarantee the local existence of lipschitz continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

stochastic bifurcations and coherencelike resonance in a self sustained bistable noisy oscillator we investigate the influence of additive gaussian white noise on two different bistable self sustained oscillators duffing van der pol oscillator with hard excitation and a model of a synthetic genetic oscillator in the deterministic case both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state we find that under the influence of noise their dynamics can be well characterized through the concept of stochastic bifurcation consisting in a qualitative change of the stationary amplitude distribution for the duffing van der pol oscillator analytical results obtained for a quasiharmonic approach are compared with the result of direct computer simulations in particular we show that the dynamics is different for isochronous and anisochronous systems moreover we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line this effect is similar to coherence resonance however in the case of anisochronous systems this effect breaks down and a new phenomenon anisochronous based stochastic bifurcation occurs

on the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

reconstruction and finiteness results for fourier mukai partners abstract we develop some methods for studying the fourier mukai partners of an algebraic variety as applications we prove that abelian varieties have finitely many fourier mukai partners and that they are uniquely determined by their derived category of coherent d modules we also generalize a famous theorem due to bondal and orlov

equivalences of twisted k3 surfaces we prove that two derived equivalent twisted k3 surfaces have isomorphic periods the converse is shown for k3 surfaces with large picard number it is also shown that all possible twisted derived equivalences between arbitrary twisted k3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a k3 surface

a refined derived torelli theorem for enriques surfaces we prove that two general enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their kuznetsov components are equivalent we apply the same techniques to give a new simple proof of a conjecture by ingalls and kuznetsov relating the derived categories of the blow up of general artin mumford quartic double solids and of the associated enriques surfaces

on the difference equation yn 1 a ynyn k with a 0 abstract we find conditions for the global asymptotic stability of the unique negative equilibrium y 1 a of the equation 0 1 y n 1 a y n y n k where y k y k 1 y0 0 a

normal forms for rational difference equations with applications to the global periodicity problem we propose a classification and derive the associated normal forms for rational difference equations with complex coefficients as an application we study the global periodicity problem for second order rational difference equations with complex coefficients we find new necessary conditions as well as some new examples of globally periodic equations

on the asymptotic behavior and periodic nature of a difference equation with maximum in this paper we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation x n max ax n 1 1xn 2 a n 0 where a 0 and 0 a 1 we prove that every positive solution of this difference equation approaches x 1 or is eventually periodic with a period 2 3 or 4

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

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

the asymptotic stability of xn 1 axn bxn k 0 we obtain necessary and sufficient conditions for the asymptotic stability of the difference equation xn 1 axn bxn k 0 n 0 1 2 where the coefficients a and b are real numbers and the delay k is a nonnegative integer

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

behaviour of solutions of some linear functional equations at infinity relationship between the boundedness and the existence of a finite limit at the infinity of solutions of the linear functional equationx f t email protected t f t where email protected c f r r is a continuous increasing function satisfying the condition f t t email protected r and f is a continuous function such that there is a finite limit lim t f t is studied here by using obtained results we study behaviour of bounded solutions of the functional equationx f k t email protected email protected ix f i t f t where a i i 1 email protected are real numbers such that i 0 k email protected i 1 functions f and f satisfy above mentioned conditions and lim t f t 0

on a class of third order nonlinear difference equations this paper studies the boundedness character of the positive solutions of the difference equation x n 1 a x n px n 1 qx n 2 r n n 0 where the parameters a p q and r are positive numbers

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

difference equations and nonexistence of solutions of an integral equation abstract by using difference equations we give some easily applicable sufficient conditions for the non existence of nonnegative solutions satisfying some additional conditions of the integral equation f x x a x t b x t f t p d t where p 1 and x is a measurable space with probability measure

on asymptotic behaviour of the difference equation xn 1 α xn kxn in this paper we investigate the global stability and the periodic character of positive solutions of the difference equationx n email protected x n kx nunder specified conditions

scalar extensions of derived categories and non fourier mukai functors abstract 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 this result has been extended by lunts and orlov to include functors from perfect complexes to quasi coherent complexes in this paper we show that the latter extension is false without the full faithfulness hypothesis our results are based on the properties of scalar extensions of derived categories whose investigation was started by pawel sosna and the first author

dynamics of a family of two dimensional difference systems the boundedness and global asymptotic behavior of positive solutions of the nonlinear two dimensional difference systems x n f x n t 1 y n s 1 y n j y n s 2 x n t 2 where t 1 s 1 s 2 t 2 are all positive integers are investigated some sufficient conditions are established such that the positive solutions converge to the unique equilibrium finally some examples are given to illustrate our results

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

dynamic behavior of a recursive sequence in this paper the global asymptotic stability the global attractivity the boundedness character the periodic nature and chaotic behavior of solutions of the difference equationx n 1 a x n 1x n n 0 1 are investigated where a r is a real number and the initial conditions x 1 x 0 are arbitrary real numbers as might be expected the six cases a 3 give rise to substantially different dynamic behavior

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

twisted fourier mukai partners of enriques surfaces bridgeland and maciocia showed that a complex enriques surface x has no fourier mukai partners apart from itself that is if d b x cong d b y then x cong y we extend this to twisted fourier mukai partners if alpha is the non trivial element of mathrm br x mathbb z 2 and d b x alpha cong d b y beta then x cong y and beta is non trivial our main tools are twisted topological k theory and twisted mukai lattices

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on the recursive sequence yn 1 α yn 1β yn yn 1yn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equation y n 1 y n 1 y n y n 1 y n n 0 1 2 where 0 2 and the initial conditions y 1 y0 are arbitrary positive real numbers

on the nonautonomous difference equation xn 1 an xn 1pxnq abstract in this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation x n 1 a n x n 1 p x n q n 0 1 where an is a positive bounded sequence p q 0 and x 1 x0 are positive numbers

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied

twisted fourier mukai number of a k3 surface we give a counting formula for the twisted fourier mukai partners of a projective k3 surface as an application we describe all twisted fourier mukai partners of a projective k3 surface of picard number 1

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002