Proposition 1. M is injective if and only if its singular value decomposition M = UDV H has a V that is square and invertible. In this case, MM is invertible and M = (MHM)−1MH . Proof. Let M be an r × c matrix. Suppose that M is injective, so that rank(M) = c because the kernal is zero. Then D is a c × c matrix and so V H is also c× c. V H must already be injective (lest M not be injective), an...

In [11] L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D ≡ 1 mod 4N such that ED, the D−quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ran...

BACKGROUND The purpose of this retrospective medical record review was to compare the effects of therapist-directed (protocol RT) and physician-directed (non-protocol RT) respiratory therapy on hospital stay and 30-d post-discharge readmission in COPD subjects with acute bacterial pneumonia. METHODS We reviewed 320 medical records; 244 records were usable. Information gathered included gender...

Density functional theory (DFT) methods have been applied to study C(32) fullerenes built from four-, five-, and six-membered rings. The relative energies of pure C(32) fullerenes have been evaluated to locate three most stable structures, 32:D(4d) with two squares, 1:D(3) without square and 5:C(s) with one square. Structural analysis reveals that there is a rearrangement pathway between the lo...

We show that for each positive integer a there exist only finitely many prime numbers p such appears an odd number of times in the period continued fraction or 2p. also prove if is and D=p 2p length expansion D divisible by 4, then 1 as partial quotient D. Furthermore, we give upper bound D, where non-square, factorize some family polynomials with integral coefficients connected fractions squar...

A.K. Erlang introduced the M/D/s queue in 1917, while F. Pollaczek and C.D. Crommelin formalized the theory using complex analysis and transforms. Let D(s, λ) denote the stationary probability of experiencing no waiting time in the M/D/s queue with arrival rate λ and service requirement 1. We use D(s, λ) as a vehicle to give an overview of some of the results we obtained over the last years, in...