rational points on k3 surfaces a new canonical height a fundamental tenet of diophantine geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties this is certainly true for curves where the sign of the euler characteristic of c determines whether the set of rational points on c is finite x c 0 for higher dimensional varieties there are some precise conjectures due to bombieri lang and vojta 15 which predict when the rational points on a variety should be finite or degenerate i e not zariski dense and some conjectures of manin et al 2 5 on the distribution of rational points in those cases when they are zariski dense but except for abelian varieties their subvarieties and some fano varieties varieties for which the anticanonical bundle is ample there are very few general theorems in this paper we will study the rational points on a certain class of k 3 surfaces defined over a number field k the moduli space of marked algebraic k3 surfaces is a countable union of 19 dimensional quasi projective varieties we are going to look at the 18 dimensional family studied by wehler 1 17 wehler s family consists of k3 surfaces s whose automorphism group aut s contains a subgroup d isomorphic to the free product 7 2 2 of two cyclic groups of order 2 we will use the geometric information provided by this infinite automorphism group to study the k rat ional points on s for any point pes we can look at the orbit of p under the action of d

existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

equidistribution quantitative des points de petite hauteur sur la droite projective we introduce a new class of adelic heights on the projective line we estimate their essential minimum and prove a result of equidistribution at every place for points of small height with estimates on the speed of convergence to each rational function r in one variable and defined over a number field k is associated a normalized height on the algebraic closure of k we show that these dynamically defined heights are adelic in our sense and deduce from this equidistribution results for preimages of points under r at every place of k our approach follows that of bilu and relies on potential theory in the complex plane as well as in the berkovich space associated to the projective line over open image in new window for each prime p

optimal simultaneous detection and estimation under a false alarm constraint this paper addresses the problem of finite sample simultaneous detection and estimation which arises when estimation of signal parameters is desired but signal presence is uncertain in general a joint detection and estimation algorithm cannot simultaneously achieve optimal detection and optimal estimation performance we develop a multihypothesis testing framework for studying the tradeoffs between detection and parameter estimation classification for a finite discrete parameter set our multihypothesis testing problem is based on the worst case detection and worst case classification error probabilities of the class of joint detection and classification algorithms which are subject to a false alarm constraint this framework leads to the evaluation of greatest lower bounds on the worst case decision error probabilities and a construction of decision rules which achieve these lower bounds for illustration we apply these methods to signal detection order selection and signal classification for a multicomponent signal in noise model for two or fewer signals an snr of 3 db and signal space dimension of n 10 numerical results are obtained which establish the existence of fundamental tradeoffs between three performance criteria probability of signal detection probability of correct order selection and probability of correct classification furthermore based on numerical performance comparisons between our optimal decision rule and other suboptimal penalty function methods we observe that rissanen s 1978 order selection penalty method is nearly min max optimal in some nonasymptotic regimes

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

a local property of basic rickard equivalences abstract let p be a prime and k be an algebraically closed field of characteristic p let g and g be two finite groups b and b be blocks of g and g and p e and p e be maximal brauer b g and b g pairs if kgb and k g b are basic rickard equivalent then there is a group isomorphism p p such that it induces an equivalence between the brauer categories of kgb and k g b and that for any brauer b g pair q f contained in p e and any subgroup k of the image of n g q f in aut q the block algebras k n g k q f and k n g k q f are basic rickard equivalent where q q f is the block of k c g q such that q f p e and k denotes the image of k in aut q

p adic dynamics the quadratic map overp adic numbers is studied in some detail we prove that near almost all indifferent fixed points it is topologically conjugate to a quasiperiodic linear map we also establish the existence of chaotic behavior and describe it using symbolic dynamics

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

prime factors of dynamical sequences let f t a q t have degree d f 2 for a given rational number x0 define xn 1 f xn for each nf 0 if this sequence is not eventually periodic and if f does not lie in one of two explicitly determined a ne conjugacy classes of rational functions then xn 1 xn has a primitive prime factor in its numerator for all su ciently large n the same result holds for the exceptional maps provided that one looks for primitive prime fac tors in the denominator of xn 1 xn hence the result for each rational function f of degree at least 2 implies a new proof that there are infinitely many primes the question of primi tive prime factors of xn dxn is also discussed for d uniformly bounded

an iterative construction of irreducible polynomials reducible modulo every prime abstract we give a method of constructing polynomials of arbitrarily large degree irreducible over a global field f but reducible modulo every prime of f the method consists of finding quadratic f f x whose iterates have the desired property and it depends on new criteria ensuring all iterates of f are irreducible in particular when f is a number field in which the ideal 2 is not a square we construct infinitely many families of quadratic f such that every iterate f n is irreducible over f but f n is reducible modulo all primes of f for n 2 we also give an example for each n 2 of a quadratic f z x whose iterates are all irreducible over q whose n 1 st iterate is irreducible modulo some primes and whose n th iterate is reducible modulo all primes from the perspective of galois theory this suggests that a well known rigidity phenomenon for linear galois representations does not exist for galois representations obtained by polynomial iteration finally we study the number of primes p for which a given quadratic f defined over a global field has f n irreducible modulo p for all n 1

sequential detection of markov targets with trajectory estimation the problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered assuming a markov evolution of the system in particular the pair signal observation forms a hidden markov model hmm a sequential procedure is proposed wherein the detection part is a sequential probability ratio test sprt and the estimation part relies upon a maximum a posteriori probability map criterion gated by the detection stage the parameter to be estimated is the trajectory of the state evolution of the system itself a thorough analysis of the asymptotic behavior of the test in this new scenario is given and sufficient conditions for its asymptotic optimality are stated i e for almost sure minimization of the stopping time and for first order minimization of any moment of its distribution an application to radar surveillance problems is also examined

semi stable reduction implies minimality of the resultant abstract for a dynamical system on p n over a number field or a function field we show that semi stable reduction implies the minimality of the resultant we use this to show that every such dynamical system over a number field admits a globally minimal presentation

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

dynamical compactifications of c 2 we find good dynamical compactifications for arbitrary polynomial mappings of c 2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula for maps of low topological degree we prove that the green function is well behaved for maps of maximum topological degree we give normal forms

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

constrained epsilon minimax test for simultaneous detection and classification a constrained epsilon minimax test is proposed to detect and classify nonorthogonal vectors in gaussian noise with a general covariance matrix and in presence of linear interferences this test is epsilon minimax in the sense that it has a small loss of optimality with respect to the purely theoretical and incalculable constrained minimax test which minimizes the maximum classification error probability subject to a constraint on the false alarm probability this loss is even more negligible as the signal to noise ratio is large furthermore it is also an epsilon equalizer test since its classification error probabilities are equalized up to a negligible difference when the signal to noise ratio is sufficiently large an asymptotically equivalent test with a very simple form is proposed this equivalent test coincides with the generalized likelihood ratio test when the vectors to classify are strongly separated in term of euclidean distance numerical experiments on active user identification in a multiuser system confirm the theoretical findings

asymptotically uniformly minimax detection and isolation in network monitoring this paper addresses the problem of multiple hypothesis testing detection and isolation of mean vectors in the case of gaussian linear model with nuisance parameters an invariant constrained asymptotically uniformly minimax test is proposed to solve this problem the invariance of the test with respect to the nuisance parameters is obtained by projecting the measurement vector onto a subspace of invariant statistics the proposed test minimizes the maximum probability of false isolation uniformly with respect to the lower bounded projections of the vectors defining the alternative hypotheses this minimization is achieved provided that the signal to noise ratio snr becomes arbitrary large the asymptotic probabilities of false alarm and false isolations and their nonasymptotic bounds are analytically established to illustrate the practical relevance of the proposed test it is applied to the problem of network monitoring it is aimed to detect and isolate volume anomalies in network origin destination od traffic demands from simple link load measurements the ambient traffic i e the od traffic matrix corresponding to the nonanomalous network state is unknown and considered as a nuisance parameter an original linear parsimonious model of the ambient traffic which is indispensable for the proposed asymptotically optimal test is designed the statistical performances of this approach to detect and isolate the anomalies are evaluated by using real data from the abilene network

detecting a suddenly arriving dynamic profile of finite duration this paper addresses the detection of a suddenly arriving dynamic profile of a finite duration often called a transient change in contrast to the traditional abrupt change detection where the post change period is assumed to be infinitely long the detection of a suddenly arriving transient change should be done before it disappears the detection of transient changes after their disappearance is considered as missed hence the traditional quickest change detection criterion minimizing the average detection delays provided a prescribed false alarm rate is compromised the proposed optimality criterion minimizes the worst case probability of missed detection provided that the worst case probability of false alarm during a certain period is upper bounded a suboptimal cusum type transient change detection algorithm based on a subclass of truncated sequential probability ratio tests is proposed the optimization of the proposed algorithm in this subclass leads to a specially designed finite moving average test the proposed method is analyzed theoretically and by simulation a special attention is paid to the case of gaussian observations with a dynamic profile

the extended brauer quotient of n interior g algebras abstract in this note we generalize the extended brauer quotient defined by l puig and y zhou in 4 section 3 to the case of n interior g algebras where n is a normal subgroup of a finite group g we use this extended brauer quotient on permutation algebras to establish some correspondences that generalize known results that hold in the group algebra case

ergodic properties of rational mappings with large topological degree let x be a projective manifold and f x x a rational mapping with large topological degree dt ek 1 f the k 1 th dynamical degree of f we give an elementary construction of a probability measure ef such that d n t fn ef for every smooth probability measure on x we show that every quasiplurisubharmonic function is ef integrable in particular ef does not charge either points of indeterminacy or pluripolar sets hence ef is f invariant with constant jacobian f ef dt ef we then establish the main ergodic properties of ef it is mixing with positive lyapunov exponents preimages of hmost h points as well as repelling periodic points are equidistributed with respect to ef moreover when dimc x 3 or when x is complex homogeneous ef is the unique measure of maximal entropy

group graded basic morita equivalences abstract we introduce group graded basic morita equivalences between algebras determined by blocks of normal subgroups and by using the extended brauer quotient we show that they induce graded basic morita equivalences at local levels

geometry of p adic siegel discs abstract we survey recent advances in the study of regular motions over p adic fields show its varied connections with dynamics and number theory and illustrate its significance to an important class of discrete dynamical systems we also show that mappings supporting quasi periodic motions can be naturally interpreted as flows with p adic time

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

preperiodic points of polynomials over global fields given a global field k and a polynomial defined over k of degree at least two morton and silverman conjectured in 1994 that the number of k rational preperiodic points of is bounded in terms of only the degree of k and the degree of in 1997 for quadratic polynomials over k q call and goldstine proved a bound which was exponential in s the number of primes of bad reduction of by careful analysis of the filled julia sets at each prime we present an improved bound on the order of s log s our bound applies to polynomials of any degree at least two over any global field k let k be a field and let k z be a rational function let denote the n iterate of under composition that is is the identity function and for n 1 n 1 we will study the dynamics on the projective line p k in particular we say a point x is preperiodic under if there are integers n m 0 such that x x the point y x satisfies n m y y and is said to be periodic of period n m note that x p k is preperiodic if and only if its orbit x n 0 is finite for example let k q and z z 29 16 then 5 4 1 4 7 4 forms a periodic cycle of period 3 and 5 4 1 4 7 4 and 3 4 each land on this cycle after one or two iterations in addition the point is of course fixed these nine q rational points are all preperiodic under meanwhile it is not difficult to see that no other point in p q is preperiodic by showing that the denominator of a rational preperiodic point must be 4 and that the absolute value must be less than 2 in general for any global field k any dimension n 1 and any morphism p p over k of degree at least two northcott proved in 1950 that the number of k rational preperiodic points of is finite 25 more precisely he showed that the preperiodic points form a set of bounded arithmetic height years later by analogy with the theorems of mazur 19 and merel 20 on k rational torsion of elliptic curves morton and silverman proposed the following conjecture 23 uniform boundedness conjecture morton and silverman 1994 given integers d n 1 and d 2 there is a constant d n d with the following property let k be a number field with k q d and let p p be a morphism of degree d defined over k then has at most preperiodic points in p k the analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve e are precisely the preperiodic points of the multiplication bytwo map 2 e e in fact taking x coordinates the map 2 induces a rational function known to dynamicists as a lattes map p p of degree 4 whose preperiodic points date june 21 2005 revised december 21 2005 2000 mathematics subject classification primary 11g99 secondary 11d45 37f10

a case of the dynamical mordell lang conjecture we prove a special case of a dynamical analogue of the classical mordell lang conjecture specifically let be a rational function with no periodic critical points other than those that are totally invariant and consider the diagonal action of on mathbb p 1 g if the coefficients of are algebraic we show that the orbit of a point outside the union of the proper preperiodic subvarieties of mathbb p 1 g has only finite intersection with any curve contained in mathbb p 1 g we also show that our result holds for indecomposable polynomials with coefficients in mathbb c our proof uses results from p adic dynamics together with an integrality argument the extension to polynomials defined over mathbb c uses the method of specialization coupled with some new results of medvedev and scanlon for describing the periodic plane curves under the action of on mathbb a 2

minimax emission computed tomography using high resolution anatomical side information and b spline models in this paper a minimax methodology is presented for combining information from two imaging modalities having different intrinsic spatial resolutions the focus application is emission computed tomography ect a low resolution modality for reconstruction of radionuclide tracer density when supplemented by high resolution anatomical boundary information extracted from a magnetic resonance image mri of the same imaging volume the mri boundary within the two dimensional 2 d slice of interest is parameterized by a closed planar curve the cramer rao cr lower bound is used to analyze estimation errors for different boundary shapes under a spatially inhomogeneous gibbs field model for the tracer density a representation for the minimax mri enhanced tracer density estimator is obtained it is shown that the estimator is asymptotically equivalent to a penalized maximum likelihood pml estimator with resolution selective gibbs penalty quantitative comparisons are presented using the iterative space alternating generalized expectation maximization sage fm algorithm to implement the pml estimator with and without minimax weight averaging

algebraic stability and degree growth of monomial maps given a rational monomial map we consider the question of finding a toric variety on which it is algebraically stable we give conditions for when such variety does or does not exist we also obtain several precise estimates of the degree sequences of monomial maps

on some saturated triples and hyperfocal subgroups abstract let b denote a primitive idempotent of the subalgebra k h g in kh of g stable elements where h is a normal subgroup of a finite group g which acts by conjugation on h and k is an algebraically closed field of characteristic p using a generalization of block induction to the case of primitive idempotents in algebras of the form k h g we show that when the hyperfocal subgroup is abelian the fusion system associated to b respectively to the induced stable primitive idempotent of the generalized brauer pair corresponding to the hyperfocal subgroup are the same

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

a detection optimal min max test for transient signals page s 1954 test is optimal for detecting a permanent change in distribution in the sense that it minimizes the worst case average delay to detection given an average distance between false alarms when used to detect transient signals however it in fact becomes the generalized likelihood ratio test glrt since a glrt is in almost all cases ad hoc page s test used as such cannot be said to be optimal in any explicit sense this article discusses the development of the min max test via the new ideas of baygun and hero see ibid vol 41 p 688 703 1995 for the detection of a transient

isotriviality is equivalent to potential good reduction for endomorphisms of pn over function fields abstract let k k c be the function field of a complete non singular curve c over an arbitrary field k the main result of this paper states that a morphism p k n p k n is isotrivial if and only if it has potential good reduction at all places v of k this generalizes results of benedetto for polynomial maps on p k 1 and baker for arbitrary rational maps on p k 1 we offer two proofs the first uses algebraic geometry and geometric invariant theory and it is new even in the case n 1 the second proof uses non archimedean analysis and dynamics and it more directly generalizes the proofs of benedetto and baker we will also give two applications the first states that an endomorphism of p k n of degree at least two is isotrivial if and only if it has an isotrivial iterate the second gives a dynamical criterion for whether after base change a locally free coherent sheaf e of rank n 1 on c decomposes as a direct sum l l of n 1 copies of the same invertible sheaf l

detection and parameter estimation of a transient signal using order statistics detection and parameter estimation of a transient signal in noise is a problem of many applications it is characterized by the fact that some of the measurements consist of noise only modern statistical signal processing techniques are applied on a discrete version of the received data and are implemented by digital signal processing dsp in this correspondence we show how order statistics os based signal processing which is of a discrete nature can be used for simultaneous detection and estimation of parameters such as time of arrival and signal duration of a sampled transient signal in white noise we show that the resulting processors are more robust than the conventional processors whereas their performance is about the same at the cost of increased computational complexity

degree growth of meromorphic surface maps we study the degree growth of iterates of meromorphic self maps of compact kahler surfaces using cohomology classes on the riemann zariski space we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model 0 introduction let x be a compact kahler surface and let f x x be a dominant meromorphic mapping fix a kahler class on x normalized by x 1 and define the degree of f with respect to to be the positive real number deg f f x f x where x denotes the intersection form on h 1 1 r x when x p2 and is the class of a line this coincides with the usual algebraic degree of f one can show that deg f n m 2 deg f deg f for all m n hence the limit 1 lim n deg f exists we refer to it as the asymptotic degree of f it follows from standard arguments see proposition 3 1 that 1 does not depend on the choice of that 1 is invariant under bimeromorphic conjugacy and that 1 2 where 2 is the topological degree of f

minimax optimal hypothesis testing with estimation dependent costs this paper introduces a novel framework for hypothesis testing in the presence of unknown parameters the objective is to decide between two hypotheses where each one involves unknown parameters that are of interest to be estimated the existing approaches on detection and estimation place the primary emphasis on the detection part by solving this part optimally and treating the estimation part suboptimally the proposed framework in contrast treats both problems simultaneously and in a jointly optimal manner the resulting test exhibits the flexibility to strike any desired balance between the detection and estimation accuracies by exploiting this flexibility depending on the application in hand this new technique offers the freedom to put different emphasis on the detection and estimation subproblems the proposed optimal joint detection and estimation framework is also extended to multiple hypothesis tests we apply the proposed test to the problem of detecting and estimating periodicities in dna sequences and demonstrate the advantages of the new framework compared to the classical neyman pearson approach and the glrt

dynamical mordell lang conjecture for birational polynomial morphisms on mathbb a 2 we prove the dynamical mordell lang conjecture for birational polynomial morphisms on mathbb a 2

optimal joint detection and estimation based on decision dependent bayesian cost this paper considers the statistical inference problem where both the hypothesis testing and the parameter estimation are of primary interest with unknown parameters present in each hypothesis the goal is to detect the true hypothesis and to estimate the unknown parameters simultaneously in light of the coupling nature of these two subproblems we adopt the bayesian estimation cost function that depends on both the detection result and the estimation scheme we then obtain the optimal joint detector and estimator by minimizing the bayesian estimation cost subject to the constraint on detection performance the proposed joint solution not only yields lower estimation cost compared with the method that treats the detection and estimation separately but also allows for flexible tradeoff between the performances of these two subproblems in addition we also extend our framework to the multi hypothesis scenario where k hypotheses and their associated unknown parameters are present finally we apply the proposed joint detection and estimation framework to the spectrum sensing for cognitive radio which aims to detect the presence of the primary user and to estimate the noise interference level at the same time

intersections of polynomial orbits and a dynamical mordell lang conjecture we prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection then the polynomials have a common iterate we also prove a special case of a conjectured dynamical analogue of the mordell lang conjecture