Modular Anti-Inverses of Prime Numbers and Two Prime-Generating Algorithms Based Upon Them

It is well known that for any prime number p the integers {2,3,4,...p-2} group into pairs (called "inverse pairs" or "modular inverses") for which the product of each pair is ≡ +1 (mod p). In a similarly way they also form pairs (we call "anti-inverses") such that the product is ≡ −1 (mod p). Further, we find that for all primes that are ≡ +1 (mod 4) there are two and only two integers a and b ∈ {2, 3, 4, ...p − 2} which are self-anti-inverse, i.e. a 2 ≡ b 2 ≡ −1 (mod p). These serve as self-anti-inverses uniquely to a single p. Deeper investigation of these primes and their self-anti-inverses reveals a triplet of integers (K ab , Ka, K b) from which p, a and b can be generated. Two prime-generating algorithms, one based on the self-anti-inverses, and one based on the triplet of K's, are described

Modular Anti-Inverses of Prime Numbers and Two Prime-Generating Algorithms Based Upon Them

It is well known that for any prime number p the integers {2,3,4,...p-2} group into pairs (called "inverse pairs" or "modular inverses") for which the product of each pair is ≡ +1 (mod p). In a similarly way they also form pairs (we call "anti-inverses") such that the product is ≡ −1 (mod p). Further, we find that for all primes that are ≡ +1 (mod 4) there are two and only two integers a and b ∈ {2, 3, 4, ...p − 2} which are self-anti-inverse, i.e. a 2 ≡ b 2 ≡ −1 (mod p). These serve as self-anti-inverses uniquely to a single p. Deeper investigation of these primes and their self-anti-inverses reveals a triplet of integers (K ab , Ka, K b) from which p, a and b can be generated. Two prime-generating algorithms, one based on the self-anti-inverses, and one based on the triplet of K's, are described

Biogenic 2‐methyl‐3‐buten‐2‐ol increases regional ozone and HO x sources

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/95042/1/grl23505.pd

Scaling and efficiency of PRISM in adaptive simulations of turbulent premixed flames

The dominant computational cost in modeling turbulent combustion phenomena numerically with high fidelity chemical mechanisms is the time required to solve the ordinary differential equations associated with chemical kinetics. One approach to reducing that computational cost is to develop an inexpensive surrogate model that accurately represents evolution of chemical kinetics. One such approach, PRISM, develops a polynomial representation of the chemistry evolution in a local region of chemical composition space. This representation is then stored for later use. As the computation proceeds, the chemistry evolution for other points within the same region are computed by evaluating these polynomials instead of calling an ordinary differential equation solver. If initial data for advancing the chemistry is encountered that is not in any region for which a polynomial is defined, the methodology dynamically samples that region and constructs a new representation for that region. The utility of this approach is determined by the size of the regions over which the representation provides a good approximation to the kinetics and the number of these regions that are necessary to model the subset of composition space that is active during a simulation. In this paper, we assess the PRISM methodology in the context of a turbulent premixed flame in two dimensions. We consider a range of turbulent intensities ranging from weak turbulence that has little effect on the flame to strong turbulence that tears pockets of burning fluid from the main flame. For each case, we explore a range of sizes for the local regions and determine the scaling behavior as a function of region size and turbulent intensity

Measurement of spin parameters in inclusive Lambda and K(S) production using a polarized proton beam

A polarized proton beam incident on a Beryllium target was used for inclusive $\Lambda$(1116 meV) production at beam momenta of 13.3 GeV and 18.5 GeV. The beam polarization was transverse to the beam direction with magnitude 0.63(0.40) at 13.3(18.5)GeV. The trigger condition favoured forward produced $\Lambda$'s with moderately high p\sb{\rm T}(p\sb{\rm T}\sim 1GeV). The $\Lambda$ polarization was measured and found to be in agreement with results from earlier experiments which used unpolarized proton beams. Analyzing power (A) and depolarization (D\sb{\rm NN}) of the $\Lambda$'s were both measured and compared with a hyperon polarization model in which the polarization arises from a Thomas precession effect. There is good agreement with its predictions: A = 0 and D\sb{\rm NN} = 0. In particular, our measurement of D\sb{\rm NN} = $-$0.009 $\pm$ 0.015 supports the idea that the valence quarks carry all of the hadron spin, since this assumption is implicit in the model's use of SU(6) wave functions to form final state hadrons from beam fragments and sea quarks. The analyzing power of K\sb{\rm s} was also measured at 13.3(18.5)GeV and found to be $-$0.094 $\pm$ 0.012($-$0.076 $\pm$ 0.015). We use the same model to predict A of K\sb{\rm s}, taking into account K\sb{\rm s}productgion from various sources (K\sp\circ, K\sp\circ and K*) and find good agreement with the data. Finally a small sample of $\bar\Lambda$ was isolated from the 18.5GeV sample and was found to have A = 0.03 $\pm$ 0.1, consistent with the model's prediction of zero