We consider the second of Mullin’s sequences of prime numbers related to Euclid’s proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.

"!# $% & !# ' %() !#* + ' ' , .-" # /'0 1 2 ' % 3 * 4! 1" 4 5 *6 7 !8 9 ' # : # % .-" #!# 5 "19 1" # " ; &9< ' #*6 ; 4 = # " > ' " 7 4!? # " ' # ! @ + A0 1" !2( 2 CB+D=E" 4 = # " F 7 4 " ' G 9 ' # . H ' IGJ &"1 F K 4 L M ' '19 N /2 ' ( 19 ' , O . 4* '!L 4 " "!# # J 5 J 'G9 J 4! !# O '* !#1" 4 # " P " LQ # + ' 4 7 4 # " R 9 #S? / ' + A # F #&"1 ' T FB9 F ' U ' 7-" !# 'GV 1 /4 V H (W () H ' 8 4 L...

Let X be a d-dimensional projective arithmetic variety and L an invertible sheaf on X . We fix a C∞-hermitian metric | · | of L. In Arakelov geometry, we frequently ask whether H(X,L) has a free basis consisting of strictly small sections. However, we know few about this problem in general. For example, Zhang [7] proved that H(X,nL) possesses a free basis as above for n ≫ 1 if the following con...

Several models based on the diffusion-limited aggregation (DLA) model were proposed and their scaling properties explored by computational and theoretical approaches. In this paper, we consider a new extension of the on-lattice DLA model in which the unitary random steps are replaced by random flights of fixed length. This procedure reduces the screening for particle penetration present in the ...

Dear Weekes I suppose you begin to think I have almost forgot there is such a place as Hurst, at least that it is now inhabited by any person I know; however, I assure you that is not the case. I am sorry to hear of the Death of poor Tom Wadg's Wife; it must be a great loss to him; the Puerperal Fever is certainly a terrible disease, and since I have attended Dr. Clarke's78 Lectures on Midwifer...

15th July 1836 My dear Sharpey, The day before yesterday I got a summons from Carswell' to see him in some way or other as soon as possible and when I called on him I found you were the subject of conversation. From what he said I think there is every likelihood ofyour being chosen by the Council of the London University, but of course we can have but a very imperfect notion at present. The Fac...

This short paper deals with the classical finite-horizon linear-quadratic regulator problem with the terminal state constrained to be zero, for both continuous and discrete-time systems. Closed-form expressions for the optimal state and costate trajectories of the Hamiltonian system, as well as the corresponding control law, are derived through the solutions of two infinite-horizon LQ problems,...

We study the variations of the quadratic performance associated to a linear differential system of retarded type for small values of the delays. From an interpretation of delays as singular perturbations of abstract evolution operators, we revisit the usual theory of representation and optimal control of retarded systems. This leads to a new parameterization of associated Riccati operators for ...