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Let be the adjacency matrix of a graph and suppose ) = exp( ). We say that we have perfect state transfer in from the vertex to the vertex at time if there is a scalar of unit modulus such that = . It is known that perfect state transfer is rare. So C.Godsil gave a relaxation of this definition: we say that we have pretty good state transfer from to if there exists a complex number of unit modulus and, for each positive real there is a time such that ‖ ‖ < . In this paper, the quantum state transfer on 1-sum of star graphs is explored. We show that there is no perfect state transfer on , but there is pretty good state transfer on if and only if =

Motivated by a question on the ranges of the commutators of dilated floor functions in [10], together with a related problem in [3], we investigate the precise ranges of certain generalized polynomials dependent on a real parameter. Our analysis requires non-trivial tools, including Kronecker’s approximation theorem. The results highlight sharp distinctions between irrational parameters and sub-unitary and supra-unitary rational parameters. We also propose several conjectures for the irrational and supra-unitary rational cases, supported by extensive computations in Wolfram Mathematica.

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

We characterize algebraic integers which are differences of two Pisot numbers. Each such number α must be real and its conjugates over Q must all lie in the union of the disc | z | < 2 and the strip | ℑ ( z ) | < 1 . In particular, we prove that every real algebraic integer α whose conjugates over Q , except possibly for α itself, all lie in the disc | z | < 2 can always be written as a difference of two Pisot numbers. We also show that a real quadratic algebraic integer α with conjugate α ′ over Q is always expressible as a difference of two Pisot numbers except for the cases α < α ′ < - 2 or 2 < α ′ < α when α cannot be expressed in that form. A similar complete characterization of all algebraic integers α expressible as a difference of two Pisot numbers in terms of the location of their conjugates is given in the case when the degree d of α is a prime number.

We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature. In particular, we prove a precise version of the Kronecker approximation theorem and a related result on coprime approximation.

In this short note, we prove that the one-dimensional Kronecker sequence i α mod 1 , i = 0 , 1 , 2 , … , is quasi-uniform over the unit interval [0, 1] if and only if α is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).

We establish Ramanujan-style congruences modulo certain primes ℓ between an Eisenstein series of weight k , prime level p and a cuspidal newform in the ε -eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for Γ 0 ( p ) . Under a mild assumption, this refines a result of Gaba–Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler–Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by ℓ . The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

Let Λ⊂Rn\\Lambda \\subset \\mathbb R^n be an algebraic lattice coming from a projective module over the ring of integers of a number field KK. Let Z⊂Rn\\mathcal Z \\subset \\mathbb R^n be the zero locus of a finite collection of polynomials such that Λ⊈Z\\Lambda \\nsubseteq \\mathcal Z or a finite union of proper full-rank sublattices of Λ\\Lambda. Let K1K_1 be the number field generated over KK by coordinates of vectors in Λ\\Lambda, and let L1,…,LtL_1,\\dots ,L_t be linear forms in nn variables with algebraic coefficients satisfying an appropriate linear independence condition over K1K_1. For each ε>0\\varepsilon > 0 and a∈Rn\\boldsymbol a \\in \\mathbb R^n, we prove the existence of a vector x∈Λ∖Z\\boldsymbol x \\in \\Lambda \\setminus \\mathcal Z of explicitly bounded sup-norm such that ‖Li(x)−ai‖>ε\\begin{equation*} \\| L_i(\\boldsymbol x) - a_i \\| > \\varepsilon \\end{equation*} for each 1≤i≤t1 \\leq i \\leq t, where ‖ ‖\\|\\ \\| stands for the distance to the nearest integer. The bound on sup-norm of x\\boldsymbol x depends on ε\\varepsilon, as well as on Λ\\Lambda, KK, Z\\mathcal Z, and heights of linear forms. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of Λ∖Z\\Lambda \\setminus \\mathcal Z under the linear forms L1,…,LtL_1,\\dots ,L_t in the tt-torus Rt/Zt\\mathbb R^t/\\mathbb Z^t.

We present methods to compute verified square roots of a square matrix A. Given an approximation X to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of X. Our approach is based on the Krawczyk method, which we modify in two different ways in such a manner that the computational complexity for an n x n matrix is reduced to n super( 3). The methods are based on the spectral decomposition or, in the case that the eigenvector matrix is ill conditioned, on a similarity transformation to block diagonal form. Numerical experiments prove that our methods are computationally efficient and that they yield narrow enclosures provided X is a good approximation. This is particularly true for symmetric matrices, since their eigenvector matrix is perfectly conditioned.

The vortex method is a numerical technique for approximating the flow of a two-dimensional incompressible, inviscid fluid. The method amounts to approximating the vorticity of the fluid by a sum of delta functions (point vortices) and to follow the movement of the point vortices. It is shown that the velocity field computed by the vortex method converges toward the velocity of the fluid in the least squares sense. The result is established for flows without boundaries and with vorticity which have compact support and is valid for arbitrary long time intervals. The basic idea in the proof is to partition the support of vorticity into blobs at time t = 0 and to show that the path of a point vortex approximates the path of the center of gravity of the blob with which it agree's initially. The error in the vortex method is proportional to the square of the initial diameter of the blobs.