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We prove that for any compact set E subset of R-2, dim(H)(E) > 1, there exists x is an element of E such that the Hausdorff dimension of the pinned distance set
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We study the pinned distance set Δ_x(E)={|x - y| : y ∈ E} in even dimensions. We utilize the orthogonal projection method as in [X. Du et al., Math Ann 380 (2021)] to show that for any compact subset E ? ?~d, where d is an even ...
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We study the pinned distance set Δ_x(E)={|x - y| : y ∈ E} in even dimensions. We utilize the orthogonal projection method as in [X. Du et al., Math Ann 380 (2021)] to show that for any compact subset E ? ?~d, where d is an even integer and d/2 < dim_H(E) 收起
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In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In p...
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In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical distance problems due to Iosevich and Rudnev (2007) [13] and the cubic distance problems due to Iosevich and Koh (2008) [12]. Moreover, our results are higher-dimensional version of Vu's work (Vu, 2008 [24]) on two dimensions. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a generalization of the work by Chapman et al. (2012) [2] who studied the pinned distance problems related to spherical distances. Discrete Fourier analysis and exponential sum estimates play an important role in our proof.
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It is well known that SiC doping in superconducting MgB_2 improves the upper critical magnetic field (B_(c2)) and the critical current density (J_c) under high magnetic field. However, the relationship between SiC doping and the f...
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It is well known that SiC doping in superconducting MgB_2 improves the upper critical magnetic field (B_(c2)) and the critical current density (J_c) under high magnetic field. However, the relationship between SiC doping and the flux pinning mechanism has not been clarified. In this study, several MgB_2 samples with and without SiC doping were prepared by the conventional in situ powder-in-tube method. The critical current densities and the force-displacement characteristics of fluxoids in samples were investigated by an ac inductive measurement (Campbell's method). The Labusch parameter (a_L) and the interaction distance (d_i) were estimated from the obtained force-displacement profile. It was found that SiC doping enhances the values of a_L, but does not change the characteristics of the magnetic field dependence of a_L apparently. Namely, a_L vs. B~(3/2) characteristics in the pure samples and SiC doped samples are almost the same. Such a result of a_L properties implies that the pinning mechanism in the SiC doped samples could be consistent with the conventional pinning theory. On the other hand, d_i, which is considered to be proportional to the size of pinning potential, decreases rapidly with increasing magnetic field, especially in the pure samples. For high magnetic field region, the variations of d_i were deduced to be caused by flux creep. The depth of pinning potential, U_0, was estimated by using the values of a_L and d_i. The values of U_0 give evidence of that SiC doping can prevent the flux bundles moving to another pinning center under high magnetic field.
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We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime...
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We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if dim_H(A) > 1, then the set of distances spanned by points of A has Hausdorff dimension at least 40/57 > 0.7 and there are many y ∈ A such that the pinned distance set {|x - y|:x ∈ A} has Hausdorff dimension at least 29/42 and lower box-counting dimension at least 40/57. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.
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We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A subset of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage...
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We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A subset of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\subset {\mathbb {R}}<^>2$$\end{document} is a Borel set of Hausdorff dimension s>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>1$$\end{document}, then its distance set has Hausdorff dimension at least 37/54 approximate to 0.685\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$37/54\approx 0.685$$\end{document}. Moreover, if s is an element of(1,3/2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (1,3/2]$$\end{document}, then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x-y|:x is an element of A}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ |x-y|:x\in A\}$$\end{document} has Hausdorff dimension >= 23s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge \tfrac{2}{3}s$$\end{document} and packing dimension at least 14(1+s+3s(2-s))>= 0.933\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{1}{4}(1+s+\sqrt{3s(2-s)}) \ge 0.933$$\end{document}. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension >1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$>1$$\end{document}. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.
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The capillary force of the axisymmetric liquid bridge between two continuously fully wet disks is investigated. The energy minimization method shows excellent consistency with the method based on the Young-Laplace equation. The ca...
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The capillary force of the axisymmetric liquid bridge between two continuously fully wet disks is investigated. The energy minimization method shows excellent consistency with the method based on the Young-Laplace equation. The capillary force between two disks is typically repulsive at small distances, and becomes attractive with increasing separation distance for a fixed liquid volume. In addition, the radius ratio of the two disks has a crucial effect on the capillary forces. The mean curvature, pinned contact angles, and rupture distance of the liquid bridge between two disks of various sizes are characterized. The mean curvature of the liquid bridge first decreases; after a minimum value is achieved, it gradually increases. When the two disks are of equal size, the pinned contact angles are identical. When the two disks have unequal radii, the pinning angles of the liquid at the upper and lower disks tend to coincide with the increase in separation distance. The rupture distance decreases with the increase in the ratio of the radii of the bottom disk to the top one. Finally, the force-distance curves for fully wet circular disks have been experimentally verified, and good consistency has been found.
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Background: The objective of this study was to provide preliminary measures of the distance between the supraspinatus, infraspinatus, and subscapularis footprints and the acromion, coracoacromial ligament, and coracoid process, du...
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Background: The objective of this study was to provide preliminary measures of the distance between the supraspinatus, infraspinatus, and subscapularis footprints and the acromion, coracoacromial ligament, and coracoid process, during dynamic arm elevations through the entire range-of-motion.
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? 2022 Elsevier Inc.The present work deals with the enhancement of the tensile strength of AA 6063 plates welded by the GTAW technique using AA 5356 filler metal sprayed with TiB2 nanoparticles (0, 4, 8 and 12 wt%). Advanced chara...
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? 2022 Elsevier Inc.The present work deals with the enhancement of the tensile strength of AA 6063 plates welded by the GTAW technique using AA 5356 filler metal sprayed with TiB2 nanoparticles (0, 4, 8 and 12 wt%). Advanced characterization techniques were used to observe microstructural variations and correlate them with the properties of the tensile curves. The X-ray profiles revealed the absence of a thermochemical reaction, variations in the position and intensity of the α-Al peak in the weldment. The IPF micrographs revealed the refined grains in the α-Al phase owing to the incorporation of TiB2 nanoparticles which prevented the nucleation of grain growth according to the Zener pinning effect (0 and 12 wt% TiB2 weld: 103 ± 8 and 26 ± 8 μm, respectively). The pole figures show that the weakened texture in the planes can be attributed to the severe deformation of the grains (0 and 12 wt% TiB2 weld: 6.086 and 1.842, respectively). The engendered strain fields and accumulated dislocations enhanced the load-carrying ability of α-Al weldments. These factors improved the mechanical strength of the weld. The 12 wt%TiB2/α-Al weld demonstrated the higher work hardening rate. Increases in the strain hardening exponent provided more plastic properties and a decreased relative slip distance extended the occurrence of deformation during tension.
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This paper presents two models of carrier-dependent long-distance auxin transport in stems that represent the process at different scales. A simple compartment model using a single constant auxin transfer rate produced similar dat...
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This paper presents two models of carrier-dependent long-distance auxin transport in stems that represent the process at different scales. A simple compartment model using a single constant auxin transfer rate produced similar data to those observed in biological experiments. The effects of different underlying biological assumptions were tested in a more detailed model representing cellular and intracellular processes that enabled discussion of different patterns of carrier-dependent auxin transportand signalling. The output that best fits the biological data is produced by a model where polar auxin transport is not limited by the number of transporters/carriers and hence supports biological data showing that stems have considerable excess capacity to transport auxin. All results support the conclusion that auxin depletion following apical decapitation in pea (Pisum sativum) occurs too slowly to be the initial cause of bud outgrowth. Consequently, changes in auxin content in the main stem and changes in polar auxin transport/carrier abundance in the main stem are not correlated with axillary bud outgrowth.
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