conjugacy classes of g 2 and spectral rigidity the free group 2 is generated by a 1 2 0 1 and b 10 21 and setting x a exp 2 i x b exp 2 i defines a unitary character on 2 for 0 1 a program is devised to compute tr conj class summed over all primitive conjugacy classes of 2 of trace tr combined with a luo sarnak theorem this yields lower bounds for the spectral variance for a large sampling of characters in the range 0 1 the results indicate that the berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems the program is also used to compute x ln n summed over all primitive conjugacy classes of 2 of norm n x the function 0 x is asymptotic to x and the remainder can be written as x x x the values of x are computed for all traces between 3202 and 4802 here x tr 2 2 the s cluster around 0 6 attaining a maximum of 2 3 finally it is proved that the remainder 0 x x has a negative bias by showing that the mean normalized remainder converges to a negative limit

the bang bang principle of time optimal controls for the heat equation with internal controls in this paper we study a certain time optimal control problem governed by the internal controlled heat equation and with a closed ball centered at 0 as the target set we derive by making use of pontryagin s maximum principle and a special kind of unique continuation property for solutions of the heat equation the bang bang property for time optimal controls of the problem

the second moment of gl 3 times gl 2 l functions at special points for a fixed sl 3 mathbb z maass form phi we consider the family of l functions l phi times u j s where u j runs over the family of hecke maass cusp forms on sl 2 mathbb z we obtain an estimate for the second moment of this family of l functions at the special points frac 1 2 it j consistent with the lindelof hypothesis we also obtain a similar upper bound on the sixth moment of the family of hecke maass cusp forms at these special points this is apparently the first occurrence of a lindelof consistent estimate for a sixth power moment of a family of gl 2 l functions

prime geodesic theorem for the picard manifold abstract let psl 2 z i be the picard group and h 3 be the three dimensional hyperbolic space we study the prime geodesic theorem for the quotient h 3 called the picard manifold obtaining an error term of size o x 3 2 2 where denotes a subconvexity exponent for quadratic dirichlet l functions defined over gaussian integers

on optimal location of diffusion and related optimal control for null controllable heat equation abstract we consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi dimensional heat equations both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa we formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems one is seeking the optimal domain of heat diffusion with a fixed interior actuator domain this can be considered as a domain perturbation problem in shape optimization another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain the existences of these two types of shape optimization problems over some class of open sets in general r n space have been proved separately

appendix an effective zero free region all the notation in this appendix will be as in the preceding paper let f be a maass form which is a newform for fo n with eigenvalue a and central character x normalized so that f f 1 we have seen that the size of p l the first fourier coefficient of f is intimately related to the behavior of l s f near s 1 here l s f is the l series of f the adjoint square lift of f to gl 3 and the crucial question is whether or not l s f vanishes when s is real and close to 1 it was shown in theorem 0 1 that if l s f is nonzero in a sufficiently wide neighborhood of 1 then ip 1 12 0 where the implied constant depends on e and is ineffective about a year after the preceding paper was first circulated it developed from conversations involving the above three authors that by slightly modifying the techniques introduced in that paper the possibility of a siegel zero could be completely eliminated in many cases in particular theorem 0 1 is now true unconditionally in the generic situation when f is not a lift from gl 1 that is to say when the l series of f is not equal to a hecke l series defined over a quadratic field if we include all cusp forms we can still obtain the theorem in the a aspect but must restrict ourselves to either an ineffective constant or a weaker effective constant in the n aspect we have

null controllability for a semilinear parabolic equation with gradient quadratic growth abstract this article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term the technique in this paper is a combination of cole hopf transformation and some methods from a y khapalov controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term a qualitative approach siam j control optim 41 2003 1886 1900

the second moment of gl 4 gl 2 l functions at special points abstract in this paper we obtain upper bounds for the second moment of l u j 1 2 i t j where is a hecke maass form for s l 4 z and u j is taken from an orthonormal basis of hecke maass forms on s l 2 z with eigenvalue 1 4 t j 2 the bounds are consistent with the lindelof hypothesis previously these types of upper bounds are available for only g l n g l 2 where n 3

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

ergodic endomorphisms of compact abelian groups we show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which impliesr mixing for allr 1 and we characterize such maps algebraically this is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space as a simple corollary it is found that one dimensional linear cellular automata and the accumulator automata arer mixing for allr 1

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

bounds for gl 3 gl 2 l functions and gl 3 l functions in this paper we will give the subconvexity bounds for self dual gl 3 l functions in the t aspect as well as subconvexity bounds for self dual gl 3 gl 2 l functions in the gl 2 spectral aspect

approximation of time optimal controls for heat equations with perturbations in the system potential in this paper we study a certain approximation property for a time optimal control problem of the heat equation with l infty potential we prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation where the potential has a small perturbation are close to those for the original problem we also verify that for the heat equation with a small perturbation in the potential one can construct a new time optimal control problem which has the same target as that of the original problem but has a different control constraint bound from that of the original problem such that the new and the original problems share the same optimal time and meanwhile the optimal control of the new problem is close to that of the original one the main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration this theorem was first established by wa

time optimal controls of the linear fitzhugh nagumo equation with pointwise control constraints time optimal control governed by the internally controlled linear fitzhugh nagumo equation with pointwise control constraint is considered making use of ekeland s variational principle we obtain pontryagin s maximum principle for a time optimal control problem using the maximum principle the bang bang property of the optimal controls is established under appropriate assumptions

on the equivalence of minimal time and minimal norm controls for internally controlled heat equations we prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves we consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time as a consequence of this equivalence using the well known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control

an ergodic family of maps on sequence spaces abstract we study the dynamics of a family of linear maps defined on the set e of all infinite sequences a0 a1 a2 whose components belong to a finite field k identified with the ring r k x of formal power series over k we find a basis for the set of periodic points and determine their algebraic structure including conditions that the temporal period divides the spatial period the measurable dynamics are such that certain of the maps are completely mixing with respect to haar measure on r regarded as a compact abelian group in contrast to this result the periodic points are dense in the same circumstances the theorems are applied to the example of the accumulator map a on z n 2 given by ae i i 1j 0ej i 0 1 2 involving a spatially varying neighborhood several of the results are illustrated with computer drawn graphics of the dynamics

blowup time optimal control for ordinary differential equations this paper studies a time optimal control problem governed by a classical controlled system of nonlinear ordinary differential equations corresponding to certain controls the solutions of the system have the behavior of blowup namely they blow up at a finite time the goal for controlling the system is to minimize the blowup time therefore the time optimal control problem studied in this paper holds its target set outside of its state space from this perspective it essentially differentiates from the most time optimal control problems in studies from past publications the purpose of this study is to develop a new method for proving the existence of time optimal controls for such kinds of time optimal control problems through analyzing a typical problem we also aim to create a new method for establishing the pontryagin maximum principle of optimal controls the main results derived from this paper can be extended to controlled systems where nonlinear terms have a more general form by the two methods mentioned above

time optimal control of semilinear parabolic equations via bilinear controls abstract we establish the existence of the time optimal control for semilinear parabolic equations with gradient quadratic growth via bilinear controls it is worth pointing out that there is no restriction on the growth of the nonlinearity f s with respect to the variable s in the equation which is a remarkable difference compared to the semilinear parabolic system with locally distributed controls the technique used in this paper is the combination of the hopf cole transformation the a prior estimates on solutions of parabolic equations and the strategy of the stepwise control

the optimal time control of a phase field system in this paper we prove the existence of the optimal time control of a phase field system by using the carleman inequality and we establish the maximum principle for the optimal time control problem governed by the phase field system

the existence of time optimal control of semilinear parabolic equations this work is concerned with the existence of the time optimal control for some semilinear parabolic differential equations with control distributed in a subdomain

controllability for parabolic equations with nonlinear memory this paper is concerned with the controllability of a parabolic system with nonlinear memory based on the localized estimate of the solution we prove that the system with a superlinear growth memory is not controllable furthermore two controllability results for some initial data and targets are given as well

on the existence of time optimal controls with constraints of the rectangular type for heat equations this paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations an existence result of time optimal controls for such a problem is established the rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations in the finite dimensional case there is no difference between such problems with control constraints of the rectangular type and those of the ball type from the perspective of the study on the existence of optimal controls interestingly in the infinite dimensional case the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type for infinite dimensional systems the existence for time optimal controls with constraints of the ball type has already been discussed in the literature while the study of the rectangular type has not been touched upon as far as we know

predecessors of cellular automata states i additive automata abstract let e be the space of right half infinite sequences with entries in z 2 and let q be the global transition rule for the one dimensional cellular automaton q e for which e is the state space the predecessor problem for q e is to find all solutions of the equation q where is an arbitrary element of e in this paper we consider the case in which the operator q represents an additive rule the predecessor equation is solved in closed form for all two and three site rules in particular for rules 90 and 150 and a technique is introduced which generates a closed form solution for many higher site values this is demonstrated with several examples