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

Black Carbon Concentrations and Diesel Vehicle Emission Factors Derived from Coefficient of Haze Measurements in California: 1967-2003

We have derived ambient black carbon (BC) concentrations and estimated emission factors for on-road diesel vehicles from archived Coefficient of Haze (COH) data that was routinely collected beginning in 1967 at 11 locations in the San Francisco Bay Area. COH values are a measure of the attenuation of light by particles collected on a white filter, and available data indicate they are proportional to BC concentrations measured using the conventional aethalometer. Monthly averaged BC concentrations are up to five times greater in winter than summer, and, consequently, so is the population?s exposure to BC. The seasonal cycle in BC concentrations is similar for all Bay Area sites, most likely due to area-wide decreased pollutant dispersion during wintertime. A strong weekly cycle is also evident, with weekend concentrations significantly lower than weekday concentrations, consistent with decreased diesel traffic volume on weekends. The weekly cycle suggests that, in the Bay Area, diesel vehicle emissions are the dominant source of BC aerosol. Despite the continuous increase in diesel fuel consumption in California, annual Bay Area average BC concentrations decreased by a factor of ~;;3 from the late 1960s to the early 2000s. Based on estimated annual BC concentrations, on-road diesel fuel consumption, and recent measurements of on-road diesel vehicle BC emissions, diesel BC emission factors decreased by an order of magnitude over the study period. Reductions in the BC emission factor reflect improved engine technology, emission controls and changes in diesel fuel composition. A new BC monitoring network is needed to continue tracking ambient BC trends because the network of COH monitors has recently been retired

Sensitivity analysis of ozone formation and transport for a Central California air pollution episode

CMAQ-HDDM is used to determine spatial and temporal variations in ozone limiting reagents and local vs upwind source contributions for an air pollution episode in Central California. We developed a first- and second- order sensitivity analysis approach with the Decoupled Direct Method to examine spatial and temporal variations of ozone-limiting reagents and the importance of local vs upwind emission sources in the San Joaquin Valley of central California for a five-day ozone episode (29th July-3rd Aug, 2000). Despite considerable spatial variations, nitrogen oxides (NO{sub x}) emission reductions are overall more effective than volatile organic compound (VOC) control for attaining the 8-hr ozone standard in this region for this episode, in contrast to the VOC control that works better for attaining the prior 1-hr ozone standard. Inter-basin source contributions of NO{sub x} emissions are limited to the northern part of the SJV, while anthropogenic VOC (AVOC) emissions, especially those emitted at night, influence ozone formation in the SJV further downwind. Among model input parameters studied here, uncertainties in emissions of NO{sub x} and AVOC, and the rate coefficient of the OH + NO{sub 2} termination reaction, have the greatest effect on first-order ozone responses to changes in NO{sub x} emissions. Uncertainties in biogenic VOC emissions only have a modest effect because they are generally not collocated with anthropogenic sources in this region

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