Efficient long division via Montgomery multiply

We present a novel right-to-left long division algorithm based on the Montgomery modular multiply, consisting of separate highly efficient loops with simply carry structure for computing first the remainder (x mod q) and then the quotient floor(x/q). These loops are ideally suited for the case where x occupies many more machine words than the divide modulus q, and are strictly linear time in the "bitsize ratio" lg(x)/lg(q). For the paradigmatic performance test of multiword dividend and single 64-bit-word divisor, exploitation of the inherent data-parallelism of the algorithm effectively mitigates the long latency of hardware integer MUL operations, as a result of which we are able to achieve respective costs for remainder-only and full-DIV (remainder and quotient) of 6 and 12.5 cycles per dividend word on the Intel Core 2 implementation of the x86_64 architecture, in single-threaded execution mode. We further describe a simple "bit-doubling modular inversion" scheme, which allows the entire iterative computation of the mod-inverse required by the Montgomery multiply at arbitrarily large precision to be performed with cost less than that of a single Newtonian iteration performed at the full precision of the final result. We also show how the Montgomery-multiply-based powering can be efficiently used in Mersenne and Fermat-number trial factorization via direct computation of a modular inverse power of 2, without any need for explicit radix-mod scalings.Comment: 23 pages; 8 tables v2: Tweak formatting, pagecount -= 2. v3: Fix incorrect powers of R in formulae [7] and [11] v4: Add Eldridge & Walter ref. v5: Clarify relation between Algos A/A',D and Hensel-div; clarify true-quotient mechanics; Add Haswell timings, refs to Agner Fog timings pdf and GMP asm-timings ref-page. v6: Remove stray +bw in MULL line of Algo D listing; add note re byte-LUT for qinv_

Long's Vortex Revisited

The conical self-similar vortex solution of Long (1961) is reconsidered, with a view toward understanding what, if any, relationship exists between Long's solution and the more-recent similarity solutions of Mayer and Powell (1992), which are a rotational-flow analogue of the Falkner-Skan boundary-layer flows, describing a self-similar axisymmetric vortex embedded in an external stream whose axial velocity varies as a power law in the axial (z) coordinate, with phi=r/z^n being the radial similarity coordinate and n the core growth rate parameter. We show that, when certain ostensible differences in the formulations and radial scalings are properly accounted for, the Long and Mayer-Powell flows in fact satisfy the same system of coupled ordinary differential equations, subject to different kinds of outer-boundary conditions, and with Long's equations a special case corresponding to conical vortex core growth, n=1 with outer axial velocity field decelerating in a 1/z fashion, which implies a severe adverse pressure gradient. For pressure gradients this adverse Mayer and Powell were unable to find any leading-edge-type vortex flow solutions which satisfy a basic physicality criterion based on monotonicity of the total-pressure profile of the flow, and it is shown that Long's solutions also violate this criterion, in an extreme fashion. Despite their apparent nonphysicality, the fact that Long's solutions fit into a more general similarity framework means that nonconical analogues of these flows should exist. The far-field asymptotics of these generalized solutions are derived and used as the basis for a hybrid spectral-numerical solution of the generalized similarity equations, which reveal the existence of solutions for more modestly adverse pressure gradients than those in Long's case, and which do satisfy the above physicality criterion.Comment: 30 pages, including 16 figure

Analysis of thin-film structures with nuclear backscattering and x-ray diffraction

Backscattering of MeV ^(4)He ions and Seemann-Bohlin x-ray diffraction techniques have been used to study silicide formation on Si and SiO_2 covered with evaporated metal films. Backscattering techniques provide information on the composition of thin-film structures as a function of depth. The glancing-angle x-ray technique provides identification of phases and structural information. Examples are given of V on Si and on SiO_2 to illustrate the major features of these analysis techniques. We also give a general review of recent studies of silicide formation

Chelate-modified polymers for atmospheric gas chromatography

Chromatographic materials were developed to serve as the stationary phase of columns used in the separation of atmospheric gases. These materials consist of a crosslinked porous polymer matrix, e.g., a divinylbenzene polymer, into which has been embedded an inorganic complexed ion such as N,N'-ethylene-bis-(acetylacetoniminato)-cobalt (2). Organic nitrogenous bases, such as pyridine, may be incorporated into the chelate polymer complexes to increase their chromatographic utility. With such materials, the process of gas chromatography is greatly simplified, especially in terms of time and quantity of material needed for a gas separation

Special coatings control temperature of structures

Special coatings in the form of paints that exhibit controlled ratios of sunlight absorptivity to grey-body emissivity control the temperature of structures in space flight. These finishes exhibit good resistance to ultraviolet radiation and do not discolor

Donor behavior in indium-alloyed silicon

The anomalous doping behavior of Si regrown from In solution was studied by (1) Schottky barrier evaluation of conductivity type, (2) electron microprobe analysis for phosphorus, and (3) channeling effect measurements for interstitial In. The latter showed In present at ~ 10^19 cm^–3, but not occupying a regular substitutional or interstitial position. A correlation was found in the first two measurements between phosphorus contamination and n-type conductivity. When the In was contacted only by quartz freshly etched in HF, the n-type behavior and phosphorus contamination disappeared. The anomalous doping behavior is most likely due to phosphorus inpurity picked up by the In

Distribution of carbonate in surface sediments of the Pacific Ocean

The distribution of carbonate on the floor of the Pacific has been remapped on the basis of 1313 points from 80 references stored in the World Ocean Sediment Data Bank of Scripps Institution of Oceanography. Percent distribution maps and carbonate versus depth diagrams generally agree with previously published information and reflect the major controlling factors of carbonate sedimentation (depth, hydrography, fertility, and sedimentary processes). While carbonate distributions are of limited use in attempting to construct dissolution profiles, major trends are identifiable. In particular, the degree of lowering of the equatorial calcite compensation depth (CCD) together with an estimate of the differences in supply rates between the equator and the subtropical gyre can be used to estimate dissolution rate increase below the lysocline. There is considerable variation in the sharpness of the ‘CCD transition’ a concept defined here. This variation is thought to reflect both geographic differences in dissolution rate gradients and redeposition processes (carbonate, deep-sea sediments, calcite, and compensation depth)

Kinetics of silicide formation by thin films of V on Si and SiO_2 substrates

The reaction rate of vacuum‐evaporated films of V of the order of 1000 Å thick is investigated by MeV He backscattering spectrometry. On substrates of single‐crystal Si and for anneal times up to several hours in the temperature range 570–650°C, VSi_2 is formed at a linear rate in time. The activation energy of the process is 1.7±0.2 eV. The presence of oxygen in amounts of 10% can significantly decelerate the reaction. On substrates of SiO_2 in the temperature range 730–820°C and anneal times of several hours or less, V_3Si is formed at a square‐root rate in time. The activation energy of this process is 2.0±0.2 eV

Exploratory study on microanalysis of thin films by backscattering techniques

Solid phase epitaxial growth of Si layers was studied by backscattering spectrometry for controllable electrical characteristics. Samples were fabricated by vacuum deposition on a thin layer of Sb before deposition of the amorphous Si layer. Analysis of the resulting SPEG layer showed that Sb was present in the SPEG layer of Si. The characteristic of the SPEG layer against the Si substrate was rectifying. A scanning microprobe picture of a cleaved sample established the presence of a depletion region more than 1 micron below the surface. Hall effect data indicated that the SPEG layer was n-type, with average free carrier concentration of about 10 to the 19th power cm/3 and average electron mobility of about 40 sq cm/Vs. SPEG with Pd silicide or Ni silicide transport layers showed fast initial transient growth regimes with slower growth in the steady state regimes