A Kronecker-type identity and the representations of a number as a sum of three squares

By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss's theorem that every positive integer is representable as a sum of three triangular numbers.Comment: EYPHKA! Theorem 1.6 and Section 4 are ne

Mechanism of the Enzymic Reduction of N_2: The Binding of Adenosine 5'-Triphosphate and Cyanide to the N_2-reducing System

The in vitro reduction of N_2 is a complex process involving at least six different reactants: two proteins [1,2] for which the names azoferredoxin (AzoFd) and molybdoferredoxin (MoFd) have been proposed[3], an electron source, the electron acceptor, ATP[4], and Mg2+[5-7]. One of the goals of research in this area is to define the orderly and quantitative participation of these reactants leading to the reduction of the electron acceptor with concomitant breakdown of ATP to ADP and inorganic phosphate[7]. The work described in this paper shows that (1) AzoFd reversibly binds both ATP, a reactant in N2 reduction, and ADP, a specific inhibitor of N2 reduction, and (2) MoFd reversibly binds cyanide, which is also reduced by the N_2-reducing system. It is suggested that the binding of ATP and of cyanide are partial reactions of the N_2-reducing system

Rubidium spacecraft atomic timing system Final report

Rubidium 87 atomic time and frequency reference system for manned space fligh

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

OPTIMAL PITCH MAP GENERATION FOR SCANNING PITCH DESIGN IN SELECTIVE SAMPLING

The reverse engineering process represents one of the best known methodologies for creating three-dimensional (3D) virtual models starting from physical ones. Even if in the last few years its usage has significantly increased, the remarkable involvement of the operator has until now represented a significant constraint for its growth. Having regard to the fact that this process, and in particular its first step (that is the acquisition phase), strongly depends on the operator's ability and expertise, this paper aims at proposing a strategy for automatically supporting an "optimal" acquisition phase. Moreover, the acquisition phase represents the only moment in which there is a direct contact between the virtual model and the physical model. For this reason, designing an "optimal" acquisition phase will provide as output an efficient set of morphological data, which will turn out to be extremely useful for the following reverse engineering passages (pre-processing, segmentation, fitting, …). This scenario drives the researcher to use a selectivesampling plan, whose grid dimensions are correlated with the complexity of the local surface region analyzed, instead of a constant one. As a consequence, this work proposes a complete operative strategy which, starting from a first raw preliminary acquisition, will provide a new selectivesampling plan during the acquisition phase, in order to allow a deeper and more efficient new scansion. The proposed solution does not require the creation of any intermediate model and relies exclusively on the analysis of the metrological performances of the 3D scanner device and of the morphological behaviour of the surface acquired

Semi-Quantitative Models for Identifying Potent and Selective Transthyretin Amyloidogenesis Inhibitors

Rate-limiting dissociation of the tetrameric protein transthyretin (TTR), followed by monomer misfolding and misassembly, appears to cause degenerative diseases in humans known as the transthyretin amyloidoses, based on human genetic, biochemical and pharmacologic evidence. Small molecules that bind to the generally unoccupied thyroxine binding pockets in the native TTR tetramer kinetically stabilize the tetramer, slowing subunit dissociation proportional to the extent that the molecules stabilize the native state over the dissociative transition state—thereby inhibiting amyloidogenesis. Herein, we use previously reported structure-activity relationship data to develop two semi-quantitative algorithms for identifying the structures of potent and selective transthyretin kinetic stabilizers/amyloidogenesis inhibitors. The viability of these prediction algorithms, in particular the more robust in silico docking model, is perhaps best validated by the clinical success of tafamidis, the first-in-class drug approved in Europe, Japan, South America, and elsewhere for treating transthyretin aggregation-associated familial amyloid polyneuropathy. Tafamidis is also being evaluated in a fully-enrolled placebo-controlled clinical trial for its efficacy against TTR cardiomyopathy. These prediction algorithms will be useful for identifying second generation TTR kinetic stabilizers, should these be needed to ameliorate the central nervous system or ophthalmologic pathology caused by TTR aggregation in organs not accessed by oral tafamidis administration