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Статті в журналах з теми "Holomorphic images"

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Автор: Grafiati

Опубліковано: 6 вересня 2023

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1

Bernal-González, L., M. C. Calderón-Moreno, and J. A. Prado-Bassas. "Holomorphic Operators Generating Dense Images." Integral Equations and Operator Theory 60, no. 1 (November 14, 2007): 1–11. http://dx.doi.org/10.1007/s00020-007-1547-4.

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2

Fernández, José L., and Domingo Pestana. "Radial images by holomorphic mappings,." Proceedings of the American Mathematical Society 124, no. 2 (1996): 429–35. http://dx.doi.org/10.1090/s0002-9939-96-02971-1.

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3

Forstneric, Franc, and Jörg Winkelmann. "Holomorphic discs with dense images." Mathematical Research Letters 12, no. 2 (2005): 265–68. http://dx.doi.org/10.4310/mrl.2005.v12.n2.a11.

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4

Chen, Bo-Yong, and Xu Wang. "Holomorphic Maps with Large Images." Journal of Geometric Analysis 25, no. 3 (April 8, 2014): 1520–46. http://dx.doi.org/10.1007/s12220-014-9482-5.

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5

Gillet, H., and C. Soulé. "Direct images of Hermitian holomorphic bundles." Bulletin of the American Mathematical Society 15, no. 2 (October 1, 1986): 209–13. http://dx.doi.org/10.1090/s0273-0979-1986-15476-5.

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6

Giannotti, Cristina, and Andrea Spiro. "On the Singular Loci and the Images of Proper Holomorphic Maps From Pseudoellipsoids." MATHEMATICA SCANDINAVICA 117, no. 2 (December 14, 2015): 170. http://dx.doi.org/10.7146/math.scand.a-22865.

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7

Baouendi, M. S., and Linda Preiss Rothschild. "Images of real hypersurfaces under holomorphic mappings." Journal of Differential Geometry 36, no. 1 (1992): 75–88. http://dx.doi.org/10.4310/jdg/1214448443.

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8

D'Angelo, John. "A monotonicity result for volumes of holomorphic images." Michigan Mathematical Journal 54, no. 3 (November 2006): 623–46. http://dx.doi.org/10.1307/mmj/1163789918.

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9

Ebenfelt, Peter, and Linda P. Rothschild. "Images of real submanifolds under finite holomorphic mappings." Communications in Analysis and Geometry 15, no. 3 (2007): 491–507. http://dx.doi.org/10.4310/cag.2007.v15.n3.a2.

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10

Xiao, Feng, Shui Qing Miao, and Li Guo. "Color Image Enhancement on YIQ Color Space." Applied Mechanics and Materials 631-632 (September 2014): 478–81. http://dx.doi.org/10.4028/www.scientific.net/amm.631-632.478.

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Анотація:

The three components of the color images enhance the images directly in the RGB color space, will cause distortion of the image. So our paper will convert true color image from RGB space to YIQ space, the holomorphic filtering and histogram equalization are executed on Y component to make the image enhancement, the Y component contains a lot of image information; finally, the image is converted from YIQ color space to RGB color space once again. Experimental results show that the approach which were proposed in the paper, Combined with the method of holomorphic filtering and histogram equalization to overcome the uneven illumination, the image is dark and other shortcomings to achieve satisfactory enhancement.

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11

Isaev, A. V. "The images of levi-nondegenerate manifolds under holomorphic mappings." Complex Variables, Theory and Application: An International Journal 27, no. 3 (May 1995): 217–33. http://dx.doi.org/10.1080/17476939508814819.

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12

Carmen Calderón-Moreno, María Del. "Holomorphic Differential Operators and Plane Sets with Dense Images." Complex Variables, Theory and Application: An International Journal 47, no. 2 (February 2002): 167–76. http://dx.doi.org/10.1080/02781070290010913.

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13

Lebl, Jiří, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang. "CR singular images of generic submanifolds under holomorphic maps." Arkiv för Matematik 52, no. 2 (October 2014): 301–27. http://dx.doi.org/10.1007/s11512-013-0193-0.

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14

GAUSSIER, HERVÉ, and KANG-TAE KIM. "COMPACTNESS OF CERTAIN FAMILIES OF PSEUDO-HOLOMORPHIC MAPPINGS INTO ${\mathbb C}^n$." International Journal of Mathematics 15, no. 01 (February 2004): 1–12. http://dx.doi.org/10.1142/s0129167x04002168.

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Анотація:

We present a normal family theorem for injective almost holomorphic maps from a manifold with almost complex structures into [Formula: see text]. Our theorem implies a new consequence even for the holomorphic mappings of a complex manifold into [Formula: see text], which can be seen as a generalization of the convergence theorem for Frankel's scaling sequence whose images are not necessarily convex. Moreover, our method is closer in spirit to the circle of ideas centered around the classical Montel theorem.

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15

Giang, Ha Huong. "Uniqueness theorem for holomorphic mappings on annuli sharing few hyperplanes." Ukrains’kyi Matematychnyi Zhurnal 73, no. 2 (February 22, 2021): 249–60. http://dx.doi.org/10.37863/umzh.v73i2.107.

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Анотація:

UDC 517.5 We prove a uniqueness theorem of linearly nondegenerate holomorphic mappings from annulus to complex projective space with different multiple values and a general condition on the intersections of the inverse images of these hyperplanes.

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16

Rudin, Walter. "Tangential H∞-images of boundary curves." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 1 (July 1988): 115–18. http://dx.doi.org/10.1017/s0305004100065282.

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Анотація:

1.Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixedn≥ 1. We define (Γ, μ) to be aFatou pair inΩ if(a) Γ is a continuous family of boundary curves γwin Ω, one ending at each w ∈ ∂Ω,(b) μ is a positive finite Borel measure on ∂Ω, and(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:(a) The map(w, t) → γw(t)is continuous, from ∂Ω × [0, 1) into Ω, andfor every w in the boundary ∂Ω of Ω.(c) For everyf ∈ H∞(Ω)(the class of all bounded holomorphic functions in Ω), the limitexists a.e. [μ].

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17

Alexander, H. "On the Volumes of Images of Holomorphic Mappings in C n." Proceedings of the American Mathematical Society 98, no. 3 (November 1986): 461. http://dx.doi.org/10.2307/2046202.

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18

Bernal-González, Luis. "Holomorphic Functions Having Large Images under the Action of Differential Operators." Journal of Mathematical Analysis and Applications 230, no. 2 (February 1999): 390–99. http://dx.doi.org/10.1006/jmaa.1998.6201.

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19

Alexander, H. "On the volumes of images of holomorphic mappings in ${\bf C}\sp n$." Proceedings of the American Mathematical Society 98, no. 3 (March 1, 1986): 461. http://dx.doi.org/10.1090/s0002-9939-1986-0857941-3.

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20

Betsakos, Dimitrios. "On the images of horodisks under holomorphic self-maps of the unit disk." Archiv der Mathematik 102, no. 1 (January 2014): 91–99. http://dx.doi.org/10.1007/s00013-013-0600-6.

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21

Przytycki, Feliks, and Jan Skrzypczak. "Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps." Mathematische Annalen 290, no. 1 (March 1991): 425–40. http://dx.doi.org/10.1007/bf01459252.

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22

Gupta, Purvi, and Rasul Shafikov. "Polynomially convex embeddings of odd-dimensional closed manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 777 (May 13, 2021): 273–99. http://dx.doi.org/10.1515/crelle-2021-0021.

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Анотація:

Abstract It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

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23

Pashaev, Oktay K. "Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators." International Journal of Geometric Methods in Modern Physics 18, no. 05 (March 5, 2021): 2150075. http://dx.doi.org/10.1142/s0219887821500754.

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Анотація:

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

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24

Radha, R., та D. Venku Naidu. "Image ofLp(ℝn)under the Hermite Semigroup". International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–13. http://dx.doi.org/10.1155/2008/287218.

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It is shown that the Hermite (polynomial) semigroup{e−tℍ:t>0}mapsLp(ℝn,ρ)into the space of holomorphic functions inLr(ℂn,Vt,p/2(r+ϵ)/2)for eachϵ>0, whereρis the Gaussian measure,Vt,p/2(r+ϵ)/2is a scaled version of Gaussian measure withr=pif1<p<2andr=p′if2<p<∞with1/p+1/p′=1. Conversely ifFis a holomorphic function which is in a “slightly” smaller space, namelyLr(ℂn,Vt,p/2r/2), then it is shown that there is a functionf∈Lp(ℝn,ρ)such thate−tℍf=F. However, a single necessary and sufficient condition is obtained for the image ofL2(ℝn,ρp/2)undere−tℍ,1<p<∞. Further it is shown that ifFis a holomorphic function such thatF∈L1(ℂn,Vt,p/21/2)orF∈Lm1,p(ℝ2n), then there exists a functionf∈Lp(ℝn,ρ)such thate−tℍf=F, wherem(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1and1<p<∞.

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25

Edigarian, Armen, Said El Marzguioui, and Jan Wiegerinck. "The image of a finely holomorphic map is pluripolar." Annales Polonici Mathematici 97, no. 2 (2010): 137–49. http://dx.doi.org/10.4064/ap97-2-3.

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26

Wirths, K. J., and J. Xiao. "An Image-Area Inequality for Some Planar Holomorphic Maps." Results in Mathematics 38, no. 1-2 (August 2000): 172–79. http://dx.doi.org/10.1007/bf03322440.

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27

Leibina, O. V. "On complex submanifolds whose Grassmann image has maximal holomorphic curvature." Mathematical Notes 81, no. 3-4 (April 2007): 496–502. http://dx.doi.org/10.1134/s0001434607030273.

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28

BETSAKOS, DIMITRIOS. "HOLOMORPHIC FUNCTIONS WITH IMAGE OF GIVEN LOGARITHMIC OR ELLIPTIC CAPACITY." Journal of the Australian Mathematical Society 94, no. 2 (March 8, 2013): 145–57. http://dx.doi.org/10.1017/s1446788712000559.

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Анотація:

AbstractFor holomorphic functions $f$ in the unit disk $ \mathbb{D} $ with $f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity ${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function ${\mathrm{d} }_{\mathrm{e} } f(r \mathbb{D} )/ r$ is increasing for $0\lt r\lt 1$.

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29

Cleanthous, Galatia. "Growth Theorems for Holomorphic Functions Under Geometric Conditions for the Image." Computational Methods and Function Theory 13, no. 2 (July 2, 2013): 277–94. http://dx.doi.org/10.1007/s40315-013-0021-3.

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30

Bu, Shang Quan, and Walter Schachermayer. "Approximation of Jensen measures by image measures under holomorphic functions and applications." Transactions of the American Mathematical Society 331, no. 2 (February 1, 1992): 585–608. http://dx.doi.org/10.1090/s0002-9947-1992-1035999-6.

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31

Kwon, Daesung, and Yong-Geun Oh. "Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition." Communications in Analysis and Geometry 8, no. 1 (2000): 31–82. http://dx.doi.org/10.4310/cag.2000.v8.n1.a2.

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32

Ounaies, Myriam. "Images Réciproques D'ensembles Inévitables Par Les Applications Holomorphes De C Dans C." Complex Variables, Theory and Application: An International Journal 47, no. 4 (April 2002): 361–71. http://dx.doi.org/10.1080/02781070290013776.

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33

Wan, Xueyuan, and Genkai Zhang. "The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle." Geometriae Dedicata 214, no. 1 (May 11, 2021): 489–517. http://dx.doi.org/10.1007/s10711-021-00625-y.

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Анотація:

AbstractLet $$\pi :\mathcal {X}\rightarrow M$$ π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over $$\mathcal {X}$$ X . We obtain the asymptotic of the curvature of $$L^2$$ L 2 -metric and Qullien metric on the direct image bundle $$\pi _*(L^k\otimes K_{\mathcal {X}/M})$$ π ∗ ( L k ⊗ K X / M ) up to the lower order terms than $$k^{n-1}$$ k n - 1 , for large k. As an application we prove that the analytic torsion $$\tau _k(\bar{\partial })$$ τ k ( ∂ ¯ ) satisfies $$\partial \bar{\partial }\log (\tau _k(\bar{\partial }))^2=o(k^{n-1})$$ ∂ ∂ ¯ log ( τ k ( ∂ ¯ ) ) 2 = o ( k n - 1 ) , where n is the dimension of fibers.

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34

Matsumura, Shin-ichi. "On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature." Pure and Applied Mathematics Quarterly 16, no. 5 (2020): 1419–39. http://dx.doi.org/10.4310/pamq.2020.v16.n5.a3.

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35

Isaev, A. V. "AN ESTIMATE OF THE DIMENSION OF THE IMAGE UNDER A HOLOMORPHIC MAPPING OF REAL-ANALYTIC HYPERSURFACES." Mathematics of the USSR-Izvestiya 30, no. 1 (February 28, 1988): 89–102. http://dx.doi.org/10.1070/im1988v030n01abeh000993.

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36

Yamamoto, Hiroshi. "On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces." Kodai Mathematical Journal 26, no. 1 (March 2003): 69–84. http://dx.doi.org/10.2996/kmj/1050496649.

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37

Di Salvo, Giovanni D. "Approximation and accumulation results of holomorphic mappings with dense image." MATHEMATICA SCANDINAVICA 129, no. 2 (June 5, 2023). http://dx.doi.org/10.7146/math.scand.a-136450.

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Анотація:

We present four approximation theorems for manifold–valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\mathbb{C}^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).

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38

Lebl, Jiří, Alan Noell, and Sivaguru Ravisankar. "On CR singular CR images." International Journal of Mathematics, September 6, 2021, 2150090. http://dx.doi.org/10.1142/s0129167x21500907.

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Анотація:

We say that a CR singular submanifold [Formula: see text] has a removable CR singularity if the CR structure at the CR points of [Formula: see text] extends through the singularity as an abstract CR structure on [Formula: see text]. We study such real-analytic submanifolds, in which case removability is equivalent to [Formula: see text] being the image of a generic real-analytic submanifold [Formula: see text] under a holomorphic map that is a diffeomorphism of [Formula: see text] onto [Formula: see text], what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

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39

Choi, Yun Sung, and Mingu Jung. "Boundaries for Gelfand transform images of Banach algebras of holomorphic functions." MATHEMATICA SCANDINAVICA 129, no. 1 (February 20, 2023). http://dx.doi.org/10.7146/math.scand.a-134348.

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In this paper, we study boundaries for the Gelfand transform image of infinite dimensional analogues of the classical disk algebras. More precisely, given a certain Banach algebra $\mathcal{A}$ of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$, we show that the Shilov boundary for the Gelfand transform image of $\mathcal{A}$ can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.

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40

Naumann, Philipp. "Positivity of direct images with a Poincaré type twist." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.79.

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Abstract We consider a holomorphic family $f:\mathscr {X} \to S$ of compact complex manifolds and a line bundle $\mathscr {L}\to \mathscr {X}$ . Given that $\mathscr {L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathscr {D}$ , the direct image $f_*(K_{\mathscr {X}/S}\otimes \mathscr {D} \otimes \mathscr {L})$ carries a smooth hermitian metric. If $\mathscr {L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Păun in a special situation of a big line bundle.

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41

Duan, Zhihao, Kimyeong Lee, and Kaiwen Sun. "Hecke relations, cosets and the classification of 2d RCFTs." Journal of High Energy Physics 2022, no. 9 (September 23, 2022). http://dx.doi.org/10.1007/jhep09(2022)202.

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Abstract We systemically study the Hecke relations and the c = 8k coset relations among 2d rational conformal field theories (RCFTs) with up to seven characters. We propose that the characters of any 2d RCFT — unitary or non-unitary — satisfying a holomorphic modular linear differential equation (MLDE) can be realized as either a Hecke image or the coset of a Hecke image with respect to a c = 8k theory. Benefited from the recent results on holomorphic modular bootstrap, we check this proposal for all admissible theories with up to five characters. We also find many new interesting Hecke relations. For example, the characters of WZW models (E6)2, (E7)2, ($$ {E}_{7\frac{1}{2}} $$ E 7 1 2 )2 can be realized as the Hecke images T13, T19, T19 of Virasoro minimal models Msub(7, 6), M(5, 4), Meff(13, 2) respectively. Besides, we find the characters associated to the second largest Fisher group Fi23 and the Harada-Norton group HN can be realized as the Hecke images T23, T19 of the product theories Meff(5, 2) ⊗ Meff(7, 2) and Meff(7, 2)⊗2 respectively. Mathematically, our study provides a great many interesting examples of vector-valued modular functions up to rank seven.

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42

Wei, Huaying, and Katsuhiko Matsuzaki. "Parametrization of the p-Weil–Petersson Curves: Holomorphic Dependence." Journal of Geometric Analysis 33, no. 9 (July 1, 2023). http://dx.doi.org/10.1007/s12220-023-01338-2.

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Анотація:

AbstractSimilar to the Bers simultaneous uniformization, the product of the p-Weil–Petersson Teichmüller spaces for $$p \ge 1$$ p ≥ 1 provides the coordinates for the space of p-Weil–Petersson embeddings $$\gamma $$ γ of the real line $${\mathbb {R}}$$ R into the complex plane $${\mathbb {C}}$$ C . We prove the biholomorphic correspondence from this space to the p-Besov space of $$u=\log \gamma '$$ u = log γ ′ on $${\mathbb {R}}$$ R for $$p>1$$ p > 1 . From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of p-Weil–Petersson curves. Specifically, it implies that the correspondence of the Riemann mapping parameters to the arc-length parameters preserving the images of curves is a homeomorphism with bi-real-analytic dependence of the change of parameters. This is analogous to the classical theorem of Coifman and Meyer for chord-arc curves.

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43

MATUS DE LA PARRA, V. "Equidistribution for matings of quadratic maps with the modular group." Ergodic Theory and Dynamical Systems, May 12, 2023, 1–29. http://dx.doi.org/10.1017/etds.2023.33.

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Анотація:

Abstract We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ , given by $$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$ It was proven by Bullet and Lomonaco [Mating quadratic maps with the modular group II. Invent. Math.220(1) (2020), 185–210] that $\mathcal {F}_a$ is a mating between the modular group $\operatorname {PSL}_2(\mathbb {Z})$ and a quadratic rational map. We show for every $a\in \mathcal {K}$ , the iterated images and preimages under $\mathcal {F}_a$ of non-exceptional points equidistribute, in spite of the fact that $\mathcal {F}_a$ is weakly modular in the sense of Dinh, Kaufmann, and Wu [Dynamics of holomorphic correspondences on Riemann surfaces. Int. J. Math.31(05) (2020), 2050036], but it is not modular. Furthermore, we prove that periodic points equidistribute as well.

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44

Drinovec Drnovšek, Barbara, and Franc Forstnerič. "Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets." Journal of Geometric Analysis 33, no. 6 (March 28, 2023). http://dx.doi.org/10.1007/s12220-023-01222-z.

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AbstractWe show that if E is a closed convex set in $${\mathbb {C}}^n$$ C n $$(n>1)$$ ( n > 1 ) contained in a closed halfspace H such that $$E\cap bH$$ E ∩ b H is nonempty and bounded, then the concave domain $$\Omega = {\mathbb {C}}^n{\setminus } E$$ Ω = C n \ E contains images of proper holomorphic maps $$f:X\rightarrow {\mathbb {C}}^n$$ f : X → C n from any Stein manifold X of dimension $$<n$$ < n , with approximation of a given map on closed compact subsets of X. If in addition $$2\dim X+1\le n$$ 2 dim X + 1 ≤ n then f can be chosen an embedding, and if $$2\dim X=n$$ 2 dim X = n , then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

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"Accelerating Information Security in Cloud Computing using a Novel Holomorphic Scheme." International Journal of Innovative Technology and Exploring Engineering 9, no. 1 (November 10, 2019): 521–29. http://dx.doi.org/10.35940/ijitee.j9067.119119.

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In the past decades, many security algorithms are invented and used in many applications for encryption and decryption of information or data. All the secure algorithms were uses the security key of size 8bit, 16bit, 32bit, 64bit and 128bits for encryption and decryption but literature survey says that higher the key size, higher the security. But the higher key size has number more bits and its requires more memory for storage and also to perform each and every bit through computational operations leads to more delay. To address these issues, the key and plain text are of 128bits and converted into integer. The converted integer values of key and plaintext are encrypted and decrypted using Holomorphic through Advanced Encryption Standard (AES). The research method is sophisticated and more protected through meaning fully lesser key in size and is accomplished for encryption in terms integer key and plaintext moderately than binary bits, therefore larger size of key and plaintext can be minimized and reduced the computational complexities. Finally the cipher text is uploaded into cloud through Real Time Transport Protocol (RTP) and Real Time Transport Control Protocol (RTCP) for storage. The main problem with continuous sharing of information into cloud is security attacks so in this research work, three different multimedia signals such as ECG, EEG and biomedical images are converted into integer and encrypted using AES. The stored data can access by the authorized users and can decode the information after decryption using AES.

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Bera, Sayani, Ratna Pal та Kaushal Verma. "On the Automorphism Group of Certain Short ℂ2’s". International Mathematics Research Notices, 24 серпня 2022. http://dx.doi.org/10.1093/imrn/rnac235.

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Abstract For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not = 0$, it is known that the sub-level sets of the Green’s function $G^+_H$ associated with $H$ are Short $\mathbb {C}^2$’s. For a given $c> 0$, we study the holomorphic automorphism group of such a Short $\mathbb {C}^2$, namely $\Omega _c = \{ G^+_H < c \}$. The unbounded domain $\Omega _c \subset \mathbb {C}^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $\Omega _c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, $\textrm {Aut}(\Omega _c)$, cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short $\mathbb {C}^2$’s to be biholomorphic.

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