In this paper, we call finite dynamical system (FDS) a discrete time and finite range dynamical system. A FDS is called a finite lattice dynamical system (FLDS) when its transition function is defined on a finite lattice. Hence, the transition function of a FLDS is a lattice operator and can be represented by an union of sup-generating operators, characterized by the operator basis. The identif...

The phase diagram of the two-dimensional extended q−states Potts model is investigated in the q → 1 limit. This is equivalent to studying the phase diagram of a two-dimensional infinite interacting lattice animal. An exact solution on the Bethe lattice and a Migdal-Kadanoff renormalization group calculation predict a line of θ transitions from an extended to a compact phase in the lattice anima...

According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than f of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation p2 + q2 + r1 = ~ipqr. By relating Markoffs algorithm for finding solutions of this equation to a problem of counting lattice points in triangles, it is shown tha...

The NTRU Signature Scheme (NSS) is a digital signature scheme based on a hard lattice problem. This lattice problem also underlies the NTRU Public Key Cryptosystem described in [1]. In this section we remind the reader of the standard description of the NTRU lattice problem in terms of products of polynomials in convolution rings. For further details, see [1]. Fix three positive integer paramet...

For every cube $$Q \subset \mathbb {R}^n$$ we let $$X_Q$$ be a quasi-Banach function space over Q such that $$\Vert \chi _Q\Vert _{X_Q} \simeq 1$$ , and for $$X= \{X_Q\}$$ define $$\begin{aligned} \Vert f\Vert _{{{\,\textrm{BMO}\,}}_X}&:=\sup _Q \,\Vert f-{\textstyle \frac{1}{|Q|}\int _Qf} _{X_Q},\\ _{{{\,\textrm{BMO}\,}}_X^*}&:=\sup \,\inf _c\, f-c\Vert _{X_Q}. \end{aligned}$$ We study necessa...

Let ωk φ( f, δ)w,Lq be the Ditzian–Totik modulus with weight w, M k be the cone of k-monotone functions on (−1, 1), i.e., those functions whose kth divided differences are nonnegative for all selections of k + 1 distinct points in (−1, 1), and denote E(X, Pn)w,q := sup f ∈X infP∈Pn ∥w( f − P)∥Lq , where Pn is the set of algebraic polynomials of degree at most n. Additionally, let wα,β (x) := (1...

We study the antiferromagnetic q-state Potts model on the square lattice for q = 3 and q = 4, using the Wang-Swendsen-Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q = 3 we obtain good control up to correlation length ξ ∼ 5000; the data are consistent with ξ(β) = Ae2ββp(1 + a1e −β + . . .) as β → ∞, with p ≈ 1. The staggered susceptibility beha...

For an affine, toric I Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T 1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y to be an isolated, at least 3-dimensional singularity, Y will be rigid unless it is even Gorenstein and dimY = 3 (dimQ = 2). For this particular case, so-ca...

Maximal angular operator sends a function defined in a sector of the complex plane to a function of modulus obtained by maximizing over all admissible values of the argument for the given modulus. The compositions of the so obtained maximal angular operator with the Poisson, Stieltjes and Laplace transform (in the sectors of their respective ranges) are shown to be bounded (nonlinear) operators...

Gene delivery vectors based on adeno-associated virus (AAV) have potential utility in the treatment of many genetic disorders. Current AAV vector manufacturing methods employ helper viruses to deliver functions needed to produce replication-defective recombinant AAV (rAAV) vectors, and clearance of infectious helper virus from the drug substance is essential for ensuring the safety of rAAV-base...