Journal articles: 'Smooth numbers'

1

Baker, Roger. "Smooth numbers in Beatty sequences." Acta Arithmetica 200, no. 4 (2021): 429–38. http://dx.doi.org/10.4064/aa210322-22-6.

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2

Laishram, Shanta, and M. Ram Murty. "Grimm's conjecture and smooth numbers." Michigan Mathematical Journal 61, no. 1 (March 2012): 151–60. http://dx.doi.org/10.1307/mmj/1331222852.

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3

Oehring, Charles. "Singular numbers of smooth kernels." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (May 1988): 511–14. http://dx.doi.org/10.1017/s0305004100065129.

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In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

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4

Qin, Zhenzhen, and Tianping Zhang. "Kloosterman sums over smooth numbers." Journal of Number Theory 182 (January 2018): 221–35. http://dx.doi.org/10.1016/j.jnt.2017.06.011.

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5

CROOT, ERNIE. "SMOOTH NUMBERS IN SHORT INTERVALS." International Journal of Number Theory 03, no. 01 (March 2007): 159–69. http://dx.doi.org/10.1142/s1793042107000833.

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We show that for any ∊ > 0, there exists c > 0, such that for all x sufficiently large, there are x1/2 ( log x)- log 4 - o(1) integers [Formula: see text], all of whose prime factors are [Formula: see text].

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6

Shparlinski, Igor E. "Character sums with smooth numbers." Archiv der Mathematik 110, no. 5 (March 5, 2018): 467–76. http://dx.doi.org/10.1007/s00013-018-1168-y.

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7

kinsley, Anto A., and J. Joan princiya. "Center Smooth Sets and Center Smooth Numbers of Graphs." Journal of Physics: Conference Series 1770, no. 1 (March 1, 2021): 012071. http://dx.doi.org/10.1088/1742-6596/1770/1/012071.

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8

Heath-Brown, D. R. "The differences between consecutive smooth numbers." Acta Arithmetica 184, no. 3 (2018): 267–85. http://dx.doi.org/10.4064/aa170913-11-7.

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9

Shparlinski, Igor E. "Character sums over shifted smooth numbers." Proceedings of the American Mathematical Society 135, no. 09 (May 2, 2007): 2699–706. http://dx.doi.org/10.1090/s0002-9939-07-08785-0.

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10

Oehring, Charles. "Singular numbers of smooth kernels. II." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (January 1989): 165–67. http://dx.doi.org/10.1017/s0305004100001493.

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Reade[10] has recently improved Weyl's classical estimate λn = o(n−3/2) for the eigenvalues of a symmetric kernel K∈C1 by relaxing the Cl hypothesis to the assumptions that K∈L2[0, 2π]2, that K is absolutely continuous in each variable separately, and that both ∂K/∂s and ∂K/t belong to L2[0, 2π]2. The conclusion of his theorem, that is, of course, stronger than λn = o(n−3/2).

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11

Banks, William D., and David J. Covert. "Sums and products with smooth numbers." Journal of Number Theory 131, no. 6 (June 2011): 985–93. http://dx.doi.org/10.1016/j.jnt.2010.11.001.

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12

Armentano, Diego. "Stochastic perturbations and smooth condition numbers." Journal of Complexity 26, no. 2 (April 2010): 161–71. http://dx.doi.org/10.1016/j.jco.2010.01.003.

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13

Greene, Robert E., and Helga Schirmer. "Smooth realizations of relative Nielsen numbers." Topology and its Applications 66, no. 1 (September 1995): 93–100. http://dx.doi.org/10.1016/0166-8641(95)00002-x.

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14

Reading, Christopher, and Gabriela Jofré. "Smooth snake population decline and its link with prey availability." Amphibia-Reptilia 41, no. 1 (June 12, 2020): 43–48. http://dx.doi.org/10.1163/15685381-20191237.

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Abstract The relationship between the numbers of smooth snakes, Coronella austriaca, and common lizards, Zootoca vivipara, was investigated in a 6.5 ha area of lowland heath within Wareham Forest in southern England. With the exception of 2002 the numbers of lizards, small mammals and individual smooth snakes captured, or observed, were recorded during each of 21 annual surveys between May and October 1997-2018. Smooth snake diet was investigated annually between 2004 and 2015 by analysing faecal samples and showed that lizards, particularly the common lizard, and pigmy shrews, Sorex minutus, were important prey species. There was no significant correlation between the occurrence of any small mammal species and either snake numbers or their presence in smooth snake diet. Over the study period there was an overall decline in the number of smooth snakes captured whilst there was an overall increase in the number of common lizard sightings. The frequency of common lizards found in the diet of smooth snakes was positively correlated with their abundance within the study area. There was a significant correlation between the decline of smooth snake numbers and the subsequent increase in the number of common lizard sightings suggesting that lizard abundance may be controlled by snake numbers. Conversely, we found no evidence indicating that smooth snake numbers were dependent on lizard numbers suggesting that factors other than prey availability e.g. habitat change due to cattle grazing, blocking ground water drainage ditches, or climatic variables, were impacting on snake numbers, particularly between 2012 and 2018.

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15

Bugeaud, Yann. "On the Zeckendorf Representation of Smooth Numbers." Moscow Mathematical Journal 21, no. 1 (February 2021): 31–42. http://dx.doi.org/10.17323/1609-4514-2021-21-1-31-42.

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16

Balog, A., J. Brüdern, and T. D. Wooley. "On smooth gaps between consecutive prime numbers." Mathematika 46, no. 1 (June 1999): 57–75. http://dx.doi.org/10.1112/s0025579300007567.

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17

Gorodetsky, Ofir. "Smooth numbers and the Dickman ρ function." Journal d'Analyse Mathématique 151, no. 1 (December 2023): 139–69. http://dx.doi.org/10.1007/s11854-023-0328-6.

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AbstractWe establish an asymptotic formula for ψ(x, y) whose shape is xρ(log x/ log y) times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for ψ(x, y). With this formula at hand we prove oscillation results for ψ(x, y), which resolve a question of Hildebrand on the range of validity of ψ(x, y) ≍ xρ(log x/ log y). We also address a question of Pomerance on the range of validity of ψ(x, y) ≥ xρ(log x/ log y).Along the way we improve classical estimates for ψ(x, y) and, on the Riemann Hypothesis, uncover an unexpected phase transition of ψ(x, y)at y = (log x)3/2+o(1).

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18

BUGEAUD, YANN, and HAJIME KANEKO. "On the digital representation of smooth numbers." Mathematical Proceedings of the Cambridge Philosophical Society 165, no. 3 (August 29, 2017): 533–40. http://dx.doi.org/10.1017/s0305004117000640.

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AbstractLet b ⩾ 2 be an integer. Among other results we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of b cannot simultaneously be divisible only by very small primes and have very few nonzero digits in its representation in base b.

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19

Gong, Ke. "On certain character sums over smooth numbers." Glasnik Matematicki 44, no. 2 (December 9, 2009): 333–42. http://dx.doi.org/10.3336/gm.44.2.06.

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20

Booker, Andrew R., and Carl Pomerance. "Squarefree smooth numbers and Euclidean prime generators." Proceedings of the American Mathematical Society 145, no. 12 (August 31, 2017): 5035–42. http://dx.doi.org/10.1090/proc/13576.

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21

Hernández-Gutiérrez, Rodrigo, and Logan C. Hoehn. "Smooth fans that are endpoint rigid." Applied General Topology 24, no. 2 (October 2, 2023): 407–22. http://dx.doi.org/10.4995/agt.2023.17922.

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Let X be a smooth fan and denote its set of endpoints by E(X). Let E be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan X such that E(X) is homeomorphic to E and for every homeomorphism h : X → X , the restriction of h to E(X) is the identity. On the other hand, we also prove that if X is any smooth fan such that E(X) is homeomorphic to complete Erdős space, then X is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik.

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22

Vakil, Ravi. "The Characteristic Numbers of Quartic Plane Curves." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 1089–120. http://dx.doi.org/10.4153/cjm-1999-048-1.

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AbstractThe characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

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23

Sultanov, A. Ya, G. A. Sultanova, and O. A. Monakhova. "On the group of automorphisms of the algebra of plural numbers." Differential Geometry of Manifolds of Figures, no. 54(2) (2023): 63–70. http://dx.doi.org/10.5922/0321-4796-2023-54-2-6.

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The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element. In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order. In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.

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24

Mustaţă, Mircea, and Wenliang Zhang. "Estimates forF-jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers." Nagoya Mathematical Journal 210 (June 2013): 133–60. http://dx.doi.org/10.1215/00277630-2077035.

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AbstractGiven an ideal a on a smooth variety in characteristic zero, we estimate theF-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

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25

Román, Gábor. "On sums of monotone functions over smooth numbers." Acta Universitatis Sapientiae, Mathematica 13, no. 1 (August 1, 2021): 273–80. http://dx.doi.org/10.2478/ausm-2021-0016.

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Abstract In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers ( A i ) i = 1 ∞ ⊆ d m n ( f ) \left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right) , which requirements are sufficient for the asymptotic ∑ n ∈ A N P ( n ) ≤ N θ f ( n ) ∼ ρ ( 1 / θ ) ∑ n ∈ A N f ( n ) \sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} } to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

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26

Munsch, Marc, and Igor E. Shparlinski. "On smooth square‐free numbers in arithmetic progressions." Journal of the London Mathematical Society 101, no. 3 (January 23, 2020): 1041–67. http://dx.doi.org/10.1112/jlms.12297.

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27

Balog, Antal, and Carl Pomerance. "The distribution of smooth numbers in arithmetic progressions." Proceedings of the American Mathematical Society 115, no. 1 (January 1, 1992): 33. http://dx.doi.org/10.1090/s0002-9939-1992-1089401-4.

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28

Banks, William D., John B. Friedlander, Moubariz Z. Garaev, and Igor E. Shparlinski. "Character sums with exponential functions over smooth numbers." Indagationes Mathematicae 17, no. 2 (June 2006): 157–68. http://dx.doi.org/10.1016/s0019-3577(06)80013-3.

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29

Cascini, Paolo, and Luca Tasin. "On the Chern numbers of a smooth threefold." Transactions of the American Mathematical Society 370, no. 11 (May 17, 2018): 7923–58. http://dx.doi.org/10.1090/tran/7216.

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30

MATOMÄKI, KAISA. "A NOTE ON SMOOTH NUMBERS IN SHORT INTERVALS." International Journal of Number Theory 06, no. 05 (August 2010): 1113–16. http://dx.doi.org/10.1142/s1793042110003381.

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31

Matomäki, Kaisa. "Another note on smooth numbers in short intervals." International Journal of Number Theory 12, no. 02 (February 18, 2016): 323–40. http://dx.doi.org/10.1142/s1793042116500196.

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In this paper, we prove that, for any positive constants [Formula: see text] and [Formula: see text] and every large enough [Formula: see text], the interval [Formula: see text] contains numbers whose all prime factors are smaller than [Formula: see text].

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32

Emala, C. W., A. Aryana, M. A. Levine, R. P. Yasuda, S. A. Satkus, B. B. Wolfe, and C. A. Hirshman. "Basenji-greyhound dog: increased m2 muscarinic receptor expression in trachealis muscle." American Journal of Physiology-Lung Cellular and Molecular Physiology 268, no. 6 (June 1, 1995): L935—L940. http://dx.doi.org/10.1152/ajplung.1995.268.6.l935.

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Airway smooth muscle from asthmatic humans and from the Basenji-greyhound dog (BG) dog is hyporesponsive to beta-adrenergic agonist stimulation. Because adenylyl cyclase is under dual regulation in airway smooth muscle, we compared muscarinic receptor-coupled inhibition of adenylyl cyclase in airway smooth muscle from BG and mongrel dogs. Inhibition of forskolin-stimulated adenylyl cyclase activity by the muscarinic M2 agonist oxotremorine was greater in airway smooth muscle membranes from BG compared with mongrel controls. Quantitative immunoprecipitation studies showed increased numbers of m2 but not m3 muscarinic receptors in the BG airway smooth muscle. The enhanced ability of muscarinic agonists to inhibit adenylyl cyclase in BG airway smooth muscle may be due to the greater numbers of muscarinic m2 receptors, which may account in part for impaired airway smooth muscle relaxation in the BG model of airway hyperresponsiveness.

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33

Moon, H. K., T. O’Connell, and R. Sharma. "Heat Transfer Enhancement Using a Convex-Patterned Surface." Journal of Turbomachinery 125, no. 2 (April 1, 2003): 274–80. http://dx.doi.org/10.1115/1.1556404.

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The heat transfer rate from a smooth wall in an internal cooling passage can be significantly enhanced by using a convex patterned surface on the opposite wall of the passage. This design is particularly effective for a design that requires the heat transfer surface to be free of any augmenting features (smooth). Heat transfer coefficients on the smooth wall in a rectangular channel, which had convexities on the opposite wall were experimentally investigated. Friction factors were also measured to assess the thermal performance. Relative clearances δ/d between the convexities and the smooth wall of 0, 0.024, and 0.055 were investigated in a Reynolds number ReHD range from 15,000 to 35,000. The heat transfer coefficients were measured in the thermally developed region using a transient thermochromic liquid crystal technique. The clearance gap between the convexities and the smooth wall adversely affected the heat transfer enhancement NuHD. The friction factors (f ), measured in the aerodynamically developed region, were largest for the cases of no clearance δ/d=0). The average heat transfer enhancement Nu¯HD was also largest for the cases of no clearance δ/d=0, as high as 3.08 times at a Reynolds number of 11,456 in relative to that Nuo of an entirely smooth channel. The normalized Nusselt numbers Nu¯HD/Nuo, as well as the normalized friction factors f/fo, for all three cases, decreased with Reynolds numbers. However, the decay rate of the friction factor ratios f/fo with Reynolds numbers was lower than that of the normalized Nusselt numbers. For all three cases investigated, the thermal performance Nu¯HD/Nuo/f/fo1/3 values were within 5% to each other. The heat transfer enhancement using a convex patterned surface was thermally more effective at a relative low Reynolds numbers (less than 20,000 for δ/d=0) than that of a smooth channel.

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34

Mustaţă, Mircea, and Wenliang Zhang. "Estimates for F-jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers." Nagoya Mathematical Journal 210 (June 2013): 133–60. http://dx.doi.org/10.1017/s0027763000010758.

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AbstractGiven an ideal a on a smooth variety in characteristic zero, we estimate theF-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

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35

Jensen, B. L., B. M. Sumer, and J. Fredsøe. "Turbulent oscillatory boundary layers at high Reynolds numbers." Journal of Fluid Mechanics 206 (September 1989): 265–97. http://dx.doi.org/10.1017/s0022112089002302.

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This study deals with turbulent oscillatory boundary-layer flows over both smooth and rough beds. The free-stream flow is a purely oscillating flow with sinusoidal velocity variation. Mean and turbulence properties were measured mainly in two directions, namely in the streamwise direction and in the direction perpendicular to the bed. Some measurements were made also in the transverse direction. The measurements were carried out up to Re = 6 × 106 over a mirror-shine smooth bed and over rough beds with various values of the parameter a/ks covering the range from approximately 400 to 3700, a being the amplitude of the oscillatory free-stream flow and ks the Nikuradse's equivalent sand roughness. For smooth-bed boundary-layer flows, the effect of Re is discussed in greater detail. It is demonstrated that the boundary-layer properties change markedly with Re. For rough-bed boundary-layer flows, the effect of the parameter a/ks is examined, at large values (O(103)) in combination with large Re.

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36

Fu, Lie, and Grégoire Menet. "On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four." Mathematische Zeitschrift 299, no. 1-2 (January 12, 2021): 203–31. http://dx.doi.org/10.1007/s00209-020-02682-7.

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AbstractWe extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

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37

Núñez-Betancourt, Luis, and Emily E. Witt. "Generalized Lyubeznik numbers." Nagoya Mathematical Journal 215 (September 2014): 169–201. http://dx.doi.org/10.1017/s0027763000010941.

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AbstractGiven a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. Thesegeneralized Lyubeznik numbersare defined in terms ofD-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generatedK-algebras. These new invariants are indicators ofF-singularities in characteristicp >0 and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.

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38

Chen, Jiaming, Alex Küronya, Yusuf Mustopa, and Jakob Stix. "Convex Fujita numbers are not determined by the fundamental group." Advances in Geometry 24, no. 4 (October 1, 2024): 577–90. http://dx.doi.org/10.1515/advgeom-2024-0029.

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Abstract We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension d ≥ 2 and an arbitrary projective group as fundamental group.

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39

Núñez-Betancourt, Luis, and Emily E. Witt. "Generalized Lyubeznik numbers." Nagoya Mathematical Journal 215 (September 2014): 169–201. http://dx.doi.org/10.1215/00277630-2741026.

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AbstractGiven a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of D-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated K-algebras. These new invariants are indicators of F-singularities in characteristic p > 0 and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.

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40

Han, J. C., P. R. Chandra, and S. C. Lau. "Local Heat/Mass Transfer Distributions Around Sharp 180 deg Turns in Two-Pass Smooth and Rib-Roughened Channels." Journal of Heat Transfer 110, no. 1 (February 1, 1988): 91–98. http://dx.doi.org/10.1115/1.3250478.

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The detailed mass transfer distributions around the sharp 180 deg turns in a two-pass, square, smooth channel and in an identical channel with two rib-roughened opposite walls were determined via the napthalene sublimation technique. The top, bottom, inner (divider), and outer walls of the test channel were napthalene-coated surfaces. For the ribbed channel tests, square, transverse, brass ribs were attached to the top and bottom walls of the channel in alignment. The rib height-to-hydraulic diameter ratios (e/D) were 0.063 and 0.094; the rib pitch-to-height ratios (P/e) were 10 and 20. Experiments were conducted for three Reynolds numbers of 15,000, 30,000, and 60,000. Results show that the Sherwood numbers on the top, outer, and inner walls around the turn in the rib-roughened channel are higher than the corresponding Sherwood numbers around the turn in the smooth channel. For both the smooth and the ribbed channels, the Sherwood numbers after the sharp turn are higher than those before the turn. The regional averages of the local Sherwood numbers are correlated and compared with published heat transfer data.

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41

Moh'D, Fida. "Coincidence Nielsen numbers for covering maps for smooth manifolds." Topology and its Applications 157, no. 2 (February 2010): 417–38. http://dx.doi.org/10.1016/j.topol.2009.10.001.

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42

Ozawa, T. "The numbers of triple tangencies of smooth space curves." Topology 24, no. 2 (1985): 1–13. http://dx.doi.org/10.1016/0040-9383(85)90021-7.

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43

Ozawa, Tetsuya. "The numbers of triple tangencies of smooth space curves." Topology 24, no. 1 (1985): 1–13. http://dx.doi.org/10.1016/0040-9383(85)90040-0.

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44

Codecà, Paolo, and Mohan Nair. "Smooth numbers and the norms of arithmetic Dirichlet convolutions." Journal of Mathematical Analysis and Applications 347, no. 2 (November 2008): 400–406. http://dx.doi.org/10.1016/j.jmaa.2008.06.006.

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45

Pichon, T., A. Pauchet, A. Astolfi, D. H. Fruman, and J.-Y. Billard. "Effect of Tripping Laminar-to-Turbulent Boundary Layer Transition on Tip Vortex Cavitation." Journal of Ship Research 41, no. 01 (March 1, 1997): 1–9. http://dx.doi.org/10.5957/jsr.1997.41.1.1.

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It is by now well established that, for Reynolds numbers larger than those corresponding to the conditions of laminar-to-turbulent boundary layer transition over a flat plate (≈0.5 × 106) and for a variety of wing shapes and cross sections, desinent cavitation numbers divided by the Reynolds number to the power 0.4 correlate with the square of the lift coefficient. In the case of foils having an NACA 16020 cross section and for Reynolds numbers below or close to those leading to transition over a flat plate, the results are very much different from those obtained for well-developed turbulent boundary layer conditions. Thus, a research program has been conducted in order to investigate the effect of boundary layer manipulation on cavitation occurrence. It consisted in determining the critical cavitation numbers, the lift coefficients, and the velocities in the tip vortex of foils having either a smooth surface or tripping roughness (promoters) near the leading edge. Tests were performed using elliptical foils of NACA 16020 cross section having the promoters extending over 60, 80 and 90 percent of the semi-span. The region near the tip was kept smooth in order to distinguish laminar-to-turbulent transition effects from tip vortex cavitation inhibition effects associated with artificial roughness at the wing tip. Results obtained at very low Reynolds numbers, ≥ 0.24 × 106, with the foil tripped on both the pressure and suction sides collapse rather well with those previously obtained at much larger Reynolds numbers with the smooth foil, and correlate with the square of the lift coefficient. The differences between the tripped and smooth foil results are due to the modification of the lift characteristics through the modification of the wing boundary layer, as shown by flow visualization studies, and as a result of the local tip vortex intensity.

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46

Mohamed, Hany A. "Effect of Rotation and Surface Roughness on Heat Transfer Rate to Flow through Vertical Cylinders in Steam Condensation Process." Journal of Heat Transfer 128, no. 3 (April 12, 2005): 318–23. http://dx.doi.org/10.1115/1.2098862.

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The enhancement in the rate of the heat transfer resulting from rotating smooth and rough vertical cylinders, of 1.28 and 21.75μm average roughness, respectively, are experimentally studied. Experiments were carried out for cooling fluid Reynolds numbers from 3300 to 7800 with varying the rotational speed up to 280rpm. Experimental runs at the stationary case showed an acceptable agreement with the theoretical values. The experimental Nusselt number values at various rotational speeds are correlated as functions of Reynolds, Weber, and Prandtl numbers for smooth and rough surfaces. The correlated equations were compared with the correlation obtained by another author. The results show that the enhancement of the heat transfer rate becomes more appreciable for low Reynolds numbers at high rotational speeds and for high Reynolds numbers at low rotational speeds. The rotation causes an enhancement in the overall heat transfer coefficient of ∼89% at Re=7800, We=1084, and Pr=1.48 for smooth surface and of ∼13.7% at Re=4700, We=4891, and Pr=1.696 for rough surface. Also, the enhancement in the heat transfer rates utilizing rotary surface becomes more pronounced for the smooth surface compared with the rough one, therefore the choice of the heat transfer surface is very important. The present work shows a reduction in the heat transfer rate below its peak value depending on the type of the heat transfer surface. It is shown that the enhancement in the heat transfer, i.e., enhancement in the Nusselt number, depends on the Weber number value and the surface type while the Nusselt number value mainly depends on the Reynolds and Prandtl numbers. Correlated equation have been developed to represent the Nusselt number values as functions of the Weber and Reynolds numbers within the stated ranges of the parameters.

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Irfan, Muhammad, Asim Ali, Muhammad Asif Khan, Muhammad Ehatisham-ul-Haq, Syed Nasir Mehmood Shah, Abdul Saboor, and Waqar Ahmad. "Pseudorandom Number Generator (PRNG) Design Using Hyper-Chaotic Modified Robust Logistic Map (HC-MRLM)." Electronics 9, no. 1 (January 6, 2020): 104. http://dx.doi.org/10.3390/electronics9010104.

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Robust chaotic systems, due to their inherent properties of mixing, ergodicity, and larger chaotic parameter space, constitute a perfect candidate for cryptography. This paper reports a novel method to generate random numbers using modified robust logistic map (MRLM). The non-smooth probability distribution function of robust logistic map (RLM) trajectories gives an un-even binary distribution in randomness test. To overcome this disadvantage in RLM, control of chaos (CoC) is proposed for smooth probability distribution function of RLM. For testing the proposed design, cryptographic random numbers generated by MRLM were vetted with National Institute of Standards and Technology statistical test suite (NIST 800-22). The results showed that proposed MRLM generates cryptographically secure random numbers (CSPRNG).

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ATIYAH, MICHAEL, and CLAUDE LEBRUN. "Curvature, cones and characteristic numbers." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (April 25, 2013): 13–37. http://dx.doi.org/10.1017/s0305004113000169.

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AbstractWe study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulæ to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.

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Sultanov, A. Ya, and G. A. Sultanova. "On the local representation of synectic connections on Weil bundles." Differential Geometry of Manifolds of Figures, no. 53 (2022): 118–26. http://dx.doi.org/10.5922/0321-4796-2022-53-11.

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Synectic extensions of complete lifts of linear connections in tangent bundles were introduced by A. P. Shirokov in the seventies of the last century [1; 2]. He established that these connections are linear and are real realizations of linear connections on first-order tangent bundles endowed with a smooth structure over the algebra of dual numbers. He also proved the existence of a smooth structure on tangent bundles of arbitrary order on a smooth manifold M over the algebra of plural numbers. Studying holomorphic linear connections on over an algebra , A. P. Shirokov obtained real realizations of these connections, which he called Synectic extensions of a linear connection defined on M. A natural generalization of the algebra of plural numbers is the A. Weyl algebra, and a generalization of the tangent bundle is the A. Weyl bundle. It was shown in [3] that a synectic extension of linear connections defined on M a smooth manifold can also be constructed on A. Weyl bundles , where is the A. Weyl algebra. The geometry of these bundles has been studied by many authors — A. Morimoto, V. V. Shurygin and others. A detailed analysis of these works can be found in [3]. In this paper, we study synectic lifts of linear connections defined on A. Weyl bundles.

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Brüdern, Jörg, and Trevor D. Wooley. "On Waring’s problem: Three cubes and a sixth power." Nagoya Mathematical Journal 163 (September 2001): 13–53. http://dx.doi.org/10.1017/s0027763000007893.

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We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

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Journal articles on the topic 'Smooth numbers'

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