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

Immunohistochemical characterization of the 'intimal proliferation' phenomenon in Sneddon's syndrome and essential thrombocythaemia

Cellular changes were immunocytochemically characterized in skin vessels of five patients with idiopathic generalized racemose livedo (Sneddon's syndrome), and one patient with localized racemose livedo associated with essential thrombocythaemia. Antibodies against alpha-smooth muscle-actin, tropomyosin, desmin, vimentin, factor VIII-related antigen, human endothelial cells (CD31), human macrophages (CD68), and HLA-DR positive cells (CR3/43) were used. Conventional light microscopy showed, in all cases, intimal thickening of ascending arteries and arterioles as a result of an accumulation of cells and extracellular hyalinized material. None of the specimens showed infiltration with polymorphonuclear leucocytes or macrophages. The cells in the region of the intimal hyperplasia showed intense positive immunostaining for alpha-smooth muscle actin and tropomyosin. Staining for the intermediate filament desmin was localized to the resident smooth muscle cells of the media, whereas staining for vimentin was found in all types of cells in both the intima and media. Positive immunostaining for factor VIII-related antigen and CD31 was strictly confined to the endothelial cells lining the narrowed lumina of the vessels. No positive staining with either antibody was observed in totally occluded vessels. Cells in the subintimal space did not show reactivity for CD68 in any of the specimens, but two cases showed solitary cells with positive staining for HLA-DR in this region. There were no differences in staining pattern between Sneddon's syndrome and essential thrombocythaemia with any of the antibodies. Our results support the assumption that the 'intimal proliferation' in both diseases is caused by colonization of the subendothelial space with contractile cells of possible smooth muscle origin.(ABSTRACT TRUNCATED AT 250 WORDS

Acoustic transducer apparatus with reduced thermal conduction

A horn is described for transmitting sound from a transducer to a heated chamber containing an object which is levitated by acoustic energy while it is heated to a molten state, which minimizes heat transfer to thereby minimize heating of the transducer, minimize temperature variation in the chamber, and minimize loss of heat from the chamber. The forward portion of the horn, which is the portion closest to the chamber, has holes that reduce its cross-sectional area to minimize the conduction of heat along the length of the horn, with the entire front portion of the horn being rigid and having an even front face to efficiently transfer high frequency acoustic energy to fluid in the chamber. In one arrangement, the horn has numerous rows of holes extending perpendicular to the length of horn, with alternate rows extending perpendicular to one another to form a sinuous path for the conduction of heat along the length of the horn

Guadalupe pluton–Mariposa Formation age relationships in the southern Sierran Foothills: Onset of Mesozoic subduction in northern California?

We report a new 153 ± 2 Ma SIMS U-Pb date for zircons from the hypabyssal Guadalupe pluton which crosscuts and contact metamorphoses upper crustal Mariposa slates in the southern Sierra. A ~950 m thick section of dark metashales lies below sandstones from which clastic zircons were analyzed at 152 ± 2 Ma. Assuming a compacted depositional rate of ~120 m/Myr, accumulation of Mariposa volcanogenic sediments, which overlie previously stranded Middle Jurassic and older ophiolite + chert-argillite belts in the Sierran Foothills, began no later than ~160 Ma. Correlative Oxfordian-Kimmeridgian strata of the Galice Formation occupy a similar position in the Klamath Mountains. We speculate that the Late Jurassic was a time of transition from (1) a mid-Paleozoic–Middle Jurassic interval of mainly but not exclusively strike-slip and episodic docking of oceanic terranes; (2) to transpressive plate underflow, producing calcalkaline igneous arc rocks ± outboard blueschists at ~170–150 Ma, whose erosion promoted accumulation of the Mariposa-Galice overlap strata; (3) continued transpressive underflow attending ~200 km left-lateral displacement of the Klamath salient relative to the Sierran arc at ~150–140 Ma and development of the apparent polar wander path cusps for North and South America; and (4) then nearly orthogonal mid and Late Cretaceous convergence commencing at ~125–120 Ma, during reversal in tangential motion of the Pacific plate. After ~120 Ma, nearly head-on subduction involving minor dextral transpression gave rise to voluminous continent-building juvenile and recycled magmas of the Sierran arc, providing the erosional debris to the Great Valley fore arc and Franciscan trench

Integration of genetic and physical maps of the Primula vulgaris S locus and localization by chromosome in situ hybridization

•Heteromorphic flower development in Primula is controlled by the S locus. The S locus genes, which control anther position, pistil length and pollen size in pin and thrum flowers, have not yet been characterized. We have integrated S-linked genes, marker sequences and mutant phenotypes to create a map of the P. vulgaris S locus region that will facilitate the identification of key S locus genes. We have generated, sequenced and annotated BAC sequences spanning the S locus, and identified its chromosomal location. •We have employed a combination of classical genetics and three-point crosses with molecular genetic analysis of recombinants to generate the map. We have characterized this region by Illumina sequencing and bioinformatic analysis, together with chromosome in situ hybridization. •We present an integrated genetic and physical map across the P. vulgaris S locus flanked by phenotypic and DNA sequence markers. BAC contigs encompass a 1.5-Mb genomic region with 1 Mb of sequence containing 82 S-linked genes anchored to overlapping BACs. The S locus is located close to the centromere of the largest metacentric chromosome pair. •These data will facilitate the identification of the genes that orchestrate heterostyly in Primula and enable evolutionary analyses of the S locus