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Journal articles on the topic 'Convex domain'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Nikolov, Nikolai, Peter Pflug, and W{\l}odzimierz Zwonek. "An example of a bounded $\mathsf C$-convex domain which is not biholomorphic to a convex domain." MATHEMATICA SCANDINAVICA 102, no. 1 (March 1, 2008): 149. http://dx.doi.org/10.7146/math.scand.a-15056.

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We show that the symmetrized bidisc is a $\mathsf C$-convex domain. This provides an example of a bounded $\mathsf C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.

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2

Bourchtein, Ludmila, and Andrei Bourchtein. "Logarithmically Convex Reinhardt Domains." Ciência e Natura 25, no. 25 (December 9, 2003): 07. http://dx.doi.org/10.5902/2179460x27233.

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The domains of certain types, such as Reinhardt ones, are important in different problems of theory of functions of several complex variables. For instance, any power series of several complex variables converges in the complete logarithmically convex Reinhardt domain. In this article we prove the logarithmic convexity of complete convex Reinhardt domain.

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3

Pflug, Peter, and Włodzimierz Zwonek. "Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains." Forum Mathematicum 30, no. 1 (January 1, 2018): 159–70. http://dx.doi.org/10.1515/forum-2016-0217.

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Abstract We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains. The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain Ω in {\mathbb{R}^{n}} with the Kobayashi distance of the tube domain over the domain Ω is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tube domains with a smooth convex bounded base are also studied in detail.

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4

Jimbo, Shuichi, and Jian Zhai. "Instability in a geometric parabolic equation on convex domain instability on convex domain." Journal of Differential Equations 188, no. 2 (March 2003): 447–60. http://dx.doi.org/10.1016/s0022-0396(02)00103-1.

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5

Nikolov, Nikolai, and Pascal J. Thomas. "Convex characterization of linearly convex domains." MATHEMATICA SCANDINAVICA 111, no. 2 (December 1, 2012): 179. http://dx.doi.org/10.7146/math.scand.a-15223.

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6

Tereshchenko, Vasyl, Sergiy Pilipenko, and Andriy Fisunenko. "Domain Triangulation between Convex Polytopes." Procedia Computer Science 18 (2013): 2500–2503. http://dx.doi.org/10.1016/j.procs.2013.05.428.

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7

Filipsson, Lars. "ℂ-convexity in infinite-dimensional Banach spaces and applications to Kergin interpolation." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–9. http://dx.doi.org/10.1155/ijmms/2006/80846.

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We investigate the concepts of linear convexity andℂ-convexity in complex Banach spaces. The main result is that anyℂ-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given aℂ-convex domainΩin the Banach spaceXand a pointp∉Ω, there is a complex hyperplane throughpthat does not intersectΩ. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined inℂ-convex domains.

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8

Groemer, H. "Stability Theorems for Convex Domains of Constant Width." Canadian Mathematical Bulletin 31, no. 3 (September 1, 1988): 328–37. http://dx.doi.org/10.4153/cmb-1988-048-3.

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AbstractIt is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

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9

YANG, YUNLONG, and DEYAN ZHANG. "DEFORMING A CONVEX DOMAIN INTO A DISK BY KLAIN’S CYCLIC REARRANGEMENT." Bulletin of the Australian Mathematical Society 97, no. 2 (February 20, 2018): 313–19. http://dx.doi.org/10.1017/s0004972717001113.

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For a convex domain, we use Klain’s cyclic rearrangement to obtain a sequence of convex domains with increasing area and the same perimeter which converges to a disk. As a byproduct, we give a proof of the classical isoperimetric inequality in the plane.

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10

SINGH, I. V., B. K. MISHRA, and MOHIT PANT. "AN EFFICIENT PARTIAL DOMAIN ENRICHED ELEMENT-FREE GALERKIN METHOD CRITERION FOR CRACKS IN NONCONVEX DOMAINS." International Journal of Modeling, Simulation, and Scientific Computing 02, no. 03 (September 2011): 317–36. http://dx.doi.org/10.1142/s1793962311000475.

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In the present work, an efficient partial domain, intrinsic, enriched element-free Galerkin criterion has been extended to simulate the cracks lying in nonconvex domains. According to this criterion, only a part of the domain near the crack tip has been enriched. A linear ramp function has been used to avoid the sudden truncation of the enrichment effect. Some cases of cracks lying in convex as well as in nonconvex domains have been solved by both full and partial domain enrichment criteria under plane stress conditions. For the cracks lying in convex domain, the results obtained by full domain enrichment criterion are found in good agreement with those obtained by partial domain enrichment criterion, whereas for cracks lying in nonconvex domain, the results obtained by full domain enrichment criterion are found to be misleading. The partial domain enrichment not only accurately simulates the cracks in nonconvex domains but also reduces the computational cost of the method.

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11

Liang, Jihua, and Hui Kou. "Convex power domain and vietoris space." Computers & Mathematics with Applications 47, no. 4-5 (February 2004): 541–48. http://dx.doi.org/10.1016/s0898-1221(04)90044-2.

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12

Agrawal, Om P., and Yufeng Xu. "Generalized vector calculus on convex domain." Communications in Nonlinear Science and Numerical Simulation 23, no. 1-3 (June 2015): 129–40. http://dx.doi.org/10.1016/j.cnsns.2014.10.032.

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13

Zając, Sylwester, and Paweł Zapałowski. "Complex geodesics in convex domains and ℂ-convexity of semitube domains." Advances in Geometry 21, no. 2 (April 1, 2021): 149–62. http://dx.doi.org/10.1515/advgeom-2020-0009.

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Abstract In this paper the complex geodesics of a convex domain in ℂ n are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂ n . The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed.

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14

Shimizu, Satoru. "A Remark on Homogeneous Convex Domains." Nagoya Mathematical Journal 105 (March 1987): 1–7. http://dx.doi.org/10.1017/s0027763000000696.

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In this note, by a homogeneous convex domain in Rn we mean a convex domain Ω in Rn containing no complete straight lines on which the group G(Ω) of all affine transformations of Rn leaving Ω invariant acts transitively. Let Ω be a homogeneous convex domain. Then Ω admits a G(©)-invariant Riemannian metric which is called the canonical metric (see [11]). The domain Ω endowed with the canonical metric is a homogeneous Riemannian manifold and we denote by I(Ω) the group of all isometries of it. A homogeneous convex domain Ω is called reducible if there is a direct sum decomposition of thé ambient space Rn = Rn1 × Rn2, ni > 0, such that Ω = Ω1 × 02 with Ωi a homogeneous convex domain in Rni; and if there is no such decomposition, then Ω is called irreducible.

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15

Youness, Ebrahim A. "Stability inE-convex programming." International Journal of Mathematics and Mathematical Sciences 26, no. 10 (2001): 643–48. http://dx.doi.org/10.1155/s0161171201006317.

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We define and analyze two kinds of stability inE-convex programming problem in which the feasible domain is affected by an operatorE. The first kind of this stability is that the set of all operatorsEthat make an optimal set stable while the other kind is that the set of all operatorsEthat make certain side of the feasible domain still active.

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16

Cavoretto, R., A. De Rossi, and E. Perracchione. "Partition of unity interpolation on multivariate convex domains." International Journal of Modeling, Simulation, and Scientific Computing 06, no. 04 (December 2015): 1550034. http://dx.doi.org/10.1142/s1793962315500348.

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In this paper, we present an algorithm for multivariate interpolation of scattered data sets lying in convex domains [Formula: see text], for any [Formula: see text]. To organize the points in a multidimensional space, we build a [Formula: see text]-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function (RBF) approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in [Formula: see text], where [Formula: see text] can be any convex domain, like a 2D polygon or a 3D polyhedron. Finally, an application to topographical data contained in a pentagonal domain is presented.

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17

Khan, Safeer Hussain. "Fixed Point Approximation of Nonexpansive Mappings on a Nonlinear Domain." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/401650.

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We use a three-step iterative process to prove some strong andΔ-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.

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18

Abu-Muhanna, Yusuf, and Glenn Schober. "Harmonic Mappings onto Convex Domains." Canadian Journal of Mathematics 39, no. 6 (December 1, 1987): 1489–530. http://dx.doi.org/10.4153/cjm-1987-071-4.

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Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and

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19

Peres, Asher, and Daniel R. Terno. "Convex probability domain of generalized quantum measurements." Journal of Physics A: Mathematical and General 31, no. 38 (September 25, 1998): L671—L675. http://dx.doi.org/10.1088/0305-4470/31/38/003.

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20

Zhu, Baocheng, and Wenxue Xu. "Reverse Bonnesen-style inequalities on surfaces of constant curvature." International Journal of Mathematics 29, no. 06 (June 2018): 1850040. http://dx.doi.org/10.1142/s0129167x18500404.

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This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface [Formula: see text] of constant curvature [Formula: see text] by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in [Formula: see text] and Bottema’s result in the Euclidean plane [Formula: see text].

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21

FERRADA-SALAS, ÁLVARO, RODRIGO HERNÁNDEZ, and MARÍA J. MARTÍN. "ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS." Bulletin of the Australian Mathematical Society 96, no. 2 (August 7, 2017): 256–62. http://dx.doi.org/10.1017/s0004972717000685.

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The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying $$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$ for some $\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$, along with their rotations, play an important role among those functions that are harmonic and orientation-preserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ are convex.

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22

SABER, SAYED. "The ∂ Cauchy-Problem on Weakly q-Convex Domains in CPn." Kragujevac Journal of Mathematics 44, no. 4 (December 2020): 581–91. http://dx.doi.org/10.46793/kgjmat2004.581s.

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Let D be a weakly q-convex domain in the complex projective space ℂPn. In this paper, the (weighted) ∂-Cauchy problem with support conditions in D is studied. Speciﬁcally, the modiﬁed weight function method is used to study the L2 existence theorem for the ∂-Neumann problem on D. The solutions are used to study function theory on weakly q-convex domains via the ∂-Cauchy problem.

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23

Picchi Scardaoni, Marco, and Marco Montemurro. "Convex or non-convex? On the nature of the feasible domain of laminates." European Journal of Mechanics - A/Solids 85 (January 2021): 104112. http://dx.doi.org/10.1016/j.euromechsol.2020.104112.

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24

Ramachandran, C., R. Ambrose Prabhu, and Srikandan Sivasubramanian. "Starlike and convex functions with respect to symmetric conjugate points involving conical domain." Mathematica Slovaca 68, no. 1 (February 23, 2018): 89–102. http://dx.doi.org/10.1515/ms-2017-0083.

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AbstractEnough attentions to domains related to conical sections has not been done so far although it deserves more. Making use of the conical domain the authors have defined a new class of starlike and Convex Functions with respect to symmetric points involving the conical domain. Growth and distortion estimates are studied with convolution using domains bounded by conic regions. Certain coefficient estimates are obtained for domains bounded by conical region. Finally interesting application of the results are also highlighted for the function Ωk,βdefined by Noor.

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25

Marshall, Nicholas F. "Stretching Convex Domains to Capture Many Lattice Points." International Mathematics Research Notices 2020, no. 10 (May 23, 2018): 2918–51. http://dx.doi.org/10.1093/imrn/rny102.

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Abstract We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant 1. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the (d − 1)-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes and Freitas, van den Berg, Bucur and Gittins, Ariturk and Laugesen, van den Berg and Gittins, and Gittins and Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

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26

Coodey, M., and S. Simons. "The convex function determined by a multifunction." Bulletin of the Australian Mathematical Society 54, no. 1 (August 1996): 87–97. http://dx.doi.org/10.1017/s0004972700015100.

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We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

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27

IBRAHIMOU, BOUBAKARI, and OMER YAYENIE. "CONVEX STANDARD FUNDAMENTAL DOMAIN FOR SUBGROUPS OF HECKE GROUPS." Bulletin of the Australian Mathematical Society 83, no. 1 (September 14, 2010): 96–107. http://dx.doi.org/10.1017/s0004972710001681.

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AbstractIt is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

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28

Yayenie, Omer. "NONEXISTENCE OF H-CONVEX CUSPIDAL STANDARD FUNDAMENTAL DOMAIN." Bulletin of the Korean Mathematical Society 46, no. 5 (September 30, 2009): 823–33. http://dx.doi.org/10.4134/bkms.2009.46.5.823.

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29

Achchab, B., A. Agouzal, N. Debit, M. Kbiri Alaoui, and A. Souissi. "Nonlinear parabolic inequalities on a general convex domain." Journal of Mathematical Inequalities, no. 2 (2010): 271–84. http://dx.doi.org/10.7153/jmi-04-24.

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30

TANIGUCHI, Takeo, and Chikashi OHTA. "Automatic mesh generation of 3-dimensional convex domain." Doboku Gakkai Ronbunshu, no. 432 (1991): 137–44. http://dx.doi.org/10.2208/jscej.1991.432_137.

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31

Makhota, Alla Aleksandrovna. "On completeness of exponential systems in convex domain." Ufimskii Matematicheskii Zhurnal 10, no. 1 (2018): 76–79. http://dx.doi.org/10.13108/2018-10-1-76.

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32

Shen, Chong, and Fu-Gui Shi. "Characterizations of L-convex spaces via domain theory." Fuzzy Sets and Systems 380 (February 2020): 44–63. http://dx.doi.org/10.1016/j.fss.2019.02.009.

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33

Zayed, Elsayed M. E. "Hearing the shape of a general convex domain." Journal of Mathematical Analysis and Applications 142, no. 1 (August 1989): 170–87. http://dx.doi.org/10.1016/0022-247x(89)90173-x.

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34

Diederich, Klas, and Emmanuel Mazzilli. "Zero varieties for the Nevanlinna class on all convex domains of finite type." Nagoya Mathematical Journal 163 (September 2001): 215–27. http://dx.doi.org/10.1017/s0027763000007972.

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It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

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35

Shashiashvili, M. "The Skorokhod Oblique Reflection Problem in a Convex Polyhedron." gmj 3, no. 2 (April 1996): 153–76. http://dx.doi.org/10.1515/gmj.1996.153.

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Abstract The Skorokhod oblique reflection problem is studied in the case of n-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.

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36

Aramyan, R. H., and V. A. Mnatsakanyan. "CONDITIONAL MOMENTS OF THE DISTANCE DISTRIBUTION TWO RANDOM POINTS IN A CONVEX DOMAIN IN $ {\mathbb{R}}^2 $." Proceedings of the YSU A: Physical and Mathematical Sciences 54, no. 1 (251) (April 15, 2020): 3–8. http://dx.doi.org/10.46991/pysu:a/2020.54.1.003.

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In this article we define two new integral geometric concepts: conditional moments of the chord length distribution of a convex domain and conditional moments of the distance distribution of two independent uniformly distributed points in a convex domain. We also found a relation between these two concepts.

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37

Rabiei, Nima, and Elias G. Saleeby. "On the sample-mean method for computing hyper-volumes." Monte Carlo Methods and Applications 25, no. 2 (June 1, 2019): 163–76. http://dx.doi.org/10.1515/mcma-2019-2034.

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Abstract Estimating hyper-volumes of convex and non-convex sets are of interest in a number of areas. In this article we develop further a simple geometric Monte Carlo method, known also as the sample-mean method, which transforms the domain to an equivalent hyper-sphere with the same volume. We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes. We then examine the use of this method for computing the volumes of nonstar-shaped domains. In particular, we develop two algorithms, which couple the sample-mean method with algebraic and geometric techniques, to generate and compute the volumes of low-dimensional stability domains in parameter space.

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38

Einstein-Matthews, Stanley M. "Boundary behaviour of extremal plurisubharmonic functions." Nagoya Mathematical Journal 138 (June 1995): 65–112. http://dx.doi.org/10.1017/s0027763000005195.

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In [Mo.l], S. Momm studied the boundary behaviour of extremal plurisubharmonic functions by using the pluricomplex Green function gΩ of a bounded convex domain Ω in Cn to exhaust the domain by a family of sublevel sets. Let Ω be a bounded convex domain in Cn containing the origin in its interior. The pluricomplex green function of Ω with a pole is defined by(1.1)

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39

Crasta, Graziano. "Estimates for the energy of the solutions to elliptic Dirichlet problems on convex domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 1 (February 2004): 89–107. http://dx.doi.org/10.1017/s0308210500003097.

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We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain Ω ⊂ Rn. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of Ω. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.

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40

Waksman, Peter. "Plane polygons and a conjecture of Blaschke’s." Advances in Applied Probability 17, no. 04 (December 1985): 774–93. http://dx.doi.org/10.1017/s0001867800015408.

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The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

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41

Waksman, Peter. "Plane polygons and a conjecture of Blaschke’s." Advances in Applied Probability 17, no. 4 (December 1985): 774–93. http://dx.doi.org/10.2307/1427087.

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The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

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42

Modave, Axel, Jonathan Lambrechts, and Christophe Geuzaine. "Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations." Computers & Mathematics with Applications 73, no. 4 (February 2017): 684–700. http://dx.doi.org/10.1016/j.camwa.2016.12.027.

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43

Benoist, J., and A. Daniilidis. "Dual characterizations of relative continuity of convex functions." Journal of the Australian Mathematical Society 70, no. 2 (April 2001): 211–24. http://dx.doi.org/10.1017/s1446788700002615.

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AbstractVarious properties of continuity for the class of lower semicontinuous convex functions are considered and dual characterizations are established. In particular, it is shown that the restriction of a lower semicontinuous convex function to its domain (respectively, domain of subdifferentiability) is continuous if and only if its subdifferential is strongly cyclically monotone (respectively, σ-cyclically monotone).

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44

JIA, HUILIAN, and LIHE WANG. "DIVERGENCE FORM PARABOLIC EQUATIONS ON TIME-DEPENDENT QUASICONVEX DOMAINS." International Journal of Mathematics 23, no. 12 (December 2012): 1250128. http://dx.doi.org/10.1142/s0129167x12501285.

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In this paper, we show the [Formula: see text] regularity of divergence form parabolic equations on time-dependent quasiconvex domains. The objective is to study the optimal parabolic boundary condition for the Lp estimates. The time-dependent quasiconvex domain is a generalization of the time-dependent Reifenberg flat domain, and assesses some properties analog to the convex domain. As to the a priori estimates near the boundary, we will apply the maximal function technique, Vitali covering lemma and the compactness method.

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Qian, Qi, Shenghuo Zhu, Jiasheng Tang, Rong Jin, Baigui Sun, and Hao Li. "Robust Optimization over Multiple Domains." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 4739–46. http://dx.doi.org/10.1609/aaai.v33i01.33014739.

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In this work, we study the problem of learning a single model for multiple domains. Unlike the conventional machine learning scenario where each domain can have the corresponding model, multiple domains (i.e., applications/users) may share the same machine learning model due to maintenance loads in cloud computing services. For example, a digit-recognition model should be applicable to hand-written digits, house numbers, car plates, etc. Therefore, an ideal model for cloud computing has to perform well at each applicable domain. To address this new challenge from cloud computing, we develop a framework of robust optimization over multiple domains. In lieu of minimizing the empirical risk, we aim to learn a model optimized to the adversarial distribution over multiple domains. Hence, we propose to learn the model and the adversarial distribution simultaneously with the stochastic algorithm for efficiency. Theoretically, we analyze the convergence rate for convex and non-convex models. To our best knowledge, we first study the convergence rate of learning a robust non-convex model with a practical algorithm. Furthermore, we demonstrate that the robustness of the framework and the convergence rate can be further enhanced by appropriate regularizers over the adversarial distribution. The empirical study on real-world fine-grained visual categorization and digits recognition tasks verifies the effectiveness and efficiency of the proposed framework.

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Kapanadze, G. "Boundary Value Problems of Bending of A Plate for an Infinite Doubly-Connected Domain Bounded by Broken Lines." Georgian Mathematical Journal 7, no. 3 (September 2000): 513–21. http://dx.doi.org/10.1515/gmj.2000.513.

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Abstract A problem of bending of a plate is considered for an infinite doubly-connected domain bounded by two convex broken lines when the plate boundary is hinge-supported and normally bending moments are applied to the points at infinity. A similar reasoning is used to study a problem of bending of a plate for an infinite domain bounded by a convex polygon and a rectilinear cut or for an infinite domain with two rectilinear cuts.

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Tatarkin, Aleksandr A. "The Density of Polynomials in a Special Space of Entire Functions of Exponential Type." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 2 (210) (June 28, 2021): 34–41. http://dx.doi.org/10.18522/1026-2237-2021-2-34-41.

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The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.

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Balogh, Zoltán M., and Christoph Leuenberger. "Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 915–35. http://dx.doi.org/10.4153/cjm-1999-040-3.

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AbstractConsider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.

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Naeem, Muhammad, Saqib Hussain, Shahid Khan, Tahir Mahmood, Maslina Darus, and Zahid Shareef. "Janowski Type q-Convex and q-Close-to-Convex Functions Associated with q-Conic Domain." Mathematics 8, no. 3 (March 18, 2020): 440. http://dx.doi.org/10.3390/math8030440.

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Certain new classes of q-convex and q-close to convex functions that involve the q-Janowski type functions have been defined by using the concepts of quantum (or q-) calculus as well as q-conic domain Ω k , q [ λ , α ] . This study explores some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. A distinction of new findings with those obtained in earlier investigations is also provided, where appropriate.

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Hypiusová, Mária, and Danica Rosinová. "Discrete-Time Pole-Region Robust Controller for Magnetic Levitation Plant." Symmetry 13, no. 1 (January 16, 2021): 142. http://dx.doi.org/10.3390/sym13010142.

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Robust pole-placement based on convex DR-regions belongs to the efficient control design techniques for real systems, providing computationally tractable pole-placement design algorithms. The problem arises in the discrete-time domain when the relative damping is prescribed since the corresponding discrete-time domain is non-convex, having a “cardioid” shape. In this paper, we further develop our recent results on the inner convex approximations of the cardioid, present systematical analysis of its design parameters and their influence on the corresponding closed loop performance (measured by standard integral of absolute error (IAE) and Total Variance criteria). The application of a robust controller designed with the proposed convex approximation of the discrete-time pole region is illustrated and evaluated on a real laboratory magnetic levitation plant.

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