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Author: Grafiati

Published: 10 December 2022

Last updated: 29 January 2023

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1

Gill, John. "Complex dynamics of the limit periodic system Fn(z) = Fn−1(⨍n(z)), ⨍n → ⨍." Journal of Computational and Applied Mathematics 32, no. 1-2 (November 1990): 89–96. http://dx.doi.org/10.1016/0377-0427(90)90420-5.

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2

Čermák, Jan, Stanislav Šabata, Vratislav Blechta, and Bernard L. Shaw. "Nickel(0) and Palladium(0) Complexes with (Z,Z)-Ph2PCH2C(t-Bu)=N-N=C(t-Bu)CH2PPh2. Alkyne-to-Alkenyl Conversion in [Pd(MeOCOCCCOOMe){(Z,Z)-Ph2PCH2C(t-Bu)=N-N=C(t-Bu)CH2PPh2}]." Collection of Czechoslovak Chemical Communications 65, no. 1 (2000): 17–22. http://dx.doi.org/10.1135/cccc20000017.

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Novel zero-valent metal complexes [ML2{(Z,Z)-Ph2PCH2C(t-Bu)=N-N=C(t-Bu)CH2PPh2}] (M = Ni, L = CO, t-BuNC) with the azine diphosphine chelating in a nine-membered ring were prepared. The nickel dicarbonyl complex was found to be in an equilibrium with its isomer containing (E,Z)-chelated azine diphosphine, the equilibrium being on the side of the (Z,Z)-isomer below 60 °C. A similar palladium complex (M = Pd, L2 = CH3OCOC≡CCOOCH3) prepared earlier was found to undergo a rearrangement with simultaneous alkyne-to-alkenyl conversion of the acetylenedicarboxylate dimethyl ester ligand.

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3

AOUF, M. K. "COEFFICIENT ESTIMATES FOR BOUNDED STARLIKE FUNCTIONS OF COMPLEX ORDER." Tamkang Journal of Mathematics 25, no. 2 (June 1, 1994): 113–23. http://dx.doi.org/10.5556/j.tkjm.25.1994.4433.

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Let $F(b,M,n)$($b\neq 0$, complex, $M >1/2$, and $n$ is a positive integer) denote the classof functions $f(z)=z+\sum_{k=n+1}^\infty a_kz^k$ analytic in $U=\{z: |z|< 1\}$ which satisfy for fixed $M$, $f (z)/z \neq 0$ in $U$ and \[ \left|\frac{b-1+\frac{zf'(z)}{f(z)}}{b}-M\right|<M, \quad z\in U.\] Also let $F^*(b,M,n)$ ($b\neq 0$, complex, $M >1/2$, and $n$ is a positive integer) denote the class of functions $f(z)=1/z+\sum_{k=n}^\infty a_kz^k$ analytic in the annulus $U^* = \{z : 0 < |z| < 1\}$ which satisfy \[ \left|\frac{b-1+\frac{zf'(z)}{f(z)}}{b}-M\right|<M, \quad z\in U^*.\] In this paper we obtain bounds for the coefficients of functions of the above classes.

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4

Bozkurt, Durmuş, and Burcu Bozkurt-Altındağ. "Integer powers of certain complex tridiagonal matrices and some complex factorizations." Filomat 31, no. 15 (2017): 4715–24. http://dx.doi.org/10.2298/fil1715715b.

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In this paper, we obtain a general expression for the entries of the rth power of a certain n x n complex tridiagonal matrix where if n is even, r ? Z or if n is odd, r ? N. In addition, we get the complex factorizations of Fibonacci polynomials, Fibonacci and Pell numbers.

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5

Li, Jing, Jianjun Zhang, and Liangwen Liao. "On Growth of Meromorphic Solutions of Complex Functional Difference Equations." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/828746.

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The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form(∑λ∈I‍αλ(z)(∏ν=1n‍f(z+cν)lλ,ν))/(∑μ∈J‍βμ(z)(∏ν=1n‍f(z+cν)mμ,ν))=Q(z,f(p(z))), whereI={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ⋃‍{0}, ν=1,2,…,n}andJ={μ=(mμ,1,mμ,2,…,mμ,n)∣mμ,ν∈ℕ⋃‍{0}, ν=1,2,…,n}are two finite index sets,cν (ν=1,2,…,n)are distinct complex numbers,αλ(z)(λ∈I)andβμ(z)(μ∈J)are small functions relative tof(z),andQ(z,u)is a rational function inuwith coefficients which are small functions off(z),p(z)=pkzk+pk-1zk-1+⋯+p0∈ℂ[z]of degreek≥1. We also give some examples to show that our results are sharp.

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6

MALİK, Shabir, and Ashish KUMAR. "On the maximum modulus of a complex polynomial." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 3 (September 30, 2022): 666–81. http://dx.doi.org/10.31801/cfsuasmas.989344.

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In this paper we impose distinct restrictions on the moduli of the zeros of p(z)=n∑v=0avzvp(z)=∑v=0navzv and investigate the dependence of ∥p(Rz)−p(σz)∥‖p(Rz)−p(σz)‖, R>σ≥1R>σ≥1 on MαMα and Mα+πMα+π, where Mα=max1≤k≤n|p(ei(α+2kπ)/n)|Mα=max1≤k≤n|p(ei(α+2kπ)/n)| and on certain coefficients of p(z)p(z). This paper comprises several results, which in particular yields some classical polynomial inequalities as special cases. Moreover, the problem of estimating p(1−wn)p(1−wn), $0<w\leq$ given $p(1)=0$ is considered.

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7

Zhang, Jianjun. "On Complex Differential-Difference Equations of Malmquist Type." Journal of Mathematics 2022 (November 8, 2022): 1–10. http://dx.doi.org/10.1155/2022/9554312.

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In this paper, we shall give some properties of complex differential-difference equation of the form ∑ μ = 1 n α μ z ∏ i = 1 t f z + c i a i 0 μ f ′ z + c i a i 1 μ ⋯ f k i z + c i a i k i μ / ∑ ν = 1 m β ν z ∏ i = 1 t f z + c i b i 0 ν f ′ z + c i b i 1 ν ⋯ f k i z + c i b i k i ν = R z , f z , where a i j μ μ = 1,2 , … , n and b i j ν ν = 1,2 , … , m i = 1,2 , … , t ; j = 0,1 , … , k i are non-negative integers, c i i = 1,2 , … , t are distinct, nonzero complex numbers, α μ z μ = 1,2 , … , n and β ν z ν = 1,2 , … , m are small functions relative to f z , and R z , f z is a rational function in f z with coefficients, which are small functions of f z . We also consider related complex functional equations in the paper. These results are improvements of some several previous results.

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8

Dragomir, S. S. "New Inequalities of CBS-Type for Power Series of Complex Numbers." Fasciculi Mathematici 57, no. 1 (December 1, 2016): 37–51. http://dx.doi.org/10.1515/fascmath-2016-0015.

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Abstract Let $f(\lambda ) = \sum\nolimits_{n = 0}^\infty {a_n \lambda ^n } $ be a function defined by power series with complex coefficients and convergent on the open disk D(0, R) ⊂ ℂ, R > 0. In this paper we show amongst other that, if α, z ∈ ℂ are such that |α|, |α| |z|2 < R, then $$\left| {f(\alpha )f(\alpha z^2 ) - f^2 (\alpha z)} \right| \le f_A \left( {\left| \alpha \right|} \right) f_A \left( {\left| \alpha \right|\left| z \right|^2 } \right) - \left| {f_A (\left| \alpha \right|z)} \right|^2 .$$ where $f_A (z) = \sum\nolimits_{n = 0}^\infty {\left| {\alpha _n } \right|z^n } $ . Applications for some fundamental functions defined by power series are also provided.

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9

Jiao, Xiaoxiang, Chiakuei Peng, and Xiaowei Xu. "Lagrangian submanifolds in complex projective space CP n." Frontiers of Mathematics in China 7, no. 6 (October 22, 2012): 1129–40. http://dx.doi.org/10.1007/s11464-012-0244-z.

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10

Raston, CL, CR Whitaker, and AH White. "Lewis-Base Adducts of Main Group Metal(I) Compounds. XI. Di-μ-iodo-bis(N,N,N′,N″,N″-pentamethyldiethylenetriamine-N,N′,N″-sodium)." Australian Journal of Chemistry 42, no. 8 (1989): 1393. http://dx.doi.org/10.1071/ch9891393.

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The synthesis and structural characterization of the 1:1 adduct of sodium(I) iodide and N,N,N′,N″,N″- pentamethyldiethylenetriamine (pmdeta), a novel coordination complex of sodium(I), is recorded. Single-crystal X-ray structure determination shows the compound to be a �,�′- diiodo-bridged dimer, with the tridentate base making up the five-coordinate environment of sodium: [( pmdeta )NaI2Na( pmdeta )]. Crystals are triclinic, P1, a 10.113(2),b 9.470(2), c 8.793(4) � , α 114.48(2), β 92.09(2), γ 96-65(1)°, Z= 1 dimer ; R was 0.037 for 1837 'observed' reflections.

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11

Li, Hong, and Hong Yan Xu. "Notes on Solutions for Some Systems of Complex Functional Equations in ℂ 2." Journal of Function Spaces 2021 (July 2, 2021): 1–9. http://dx.doi.org/10.1155/2021/5424284.

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The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 1 + P 1 z f 2 z 1 + c 1 , z 2 + c 2 m 1 = Q 1 z , ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 2 + P 2 z f 1 z 1 + c 1 , z 2 + c 2 m 2 = Q 2 z , where P 1 z , P 2 z , Q 1 z , and Q 2 z are polynomials in ℂ 2 ; n 1 , n 2 , m 1 , and m 2 are positive integers, and c = c 1 , c 2 ∈ ℂ 2 . We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.

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12

Ge, Shulin, Tengfei Kang, Lili Lin, Xiying Zhang, Peng Zhao, Xiaohua Liu, and Xiaoming Feng. "Chiral N,N′-dioxide/Sc(OTf)3 complex-catalyzed asymmetric dearomatization of β-naphthols." Chem. Commun. 53, no. 86 (2017): 11759–62. http://dx.doi.org/10.1039/c7cc06388a.

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13

Huang, Zhi-Bo, and Ran-Ran Zhang. "Some Properties on Complex Functional Difference Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/283895.

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We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form∑λ∈Iαλ(z)(∏j=0nf(z+cj)λj)=R(z,f∘p)=((a0(z)+a1(z)(f∘p)+ ⋯ +as(z) (f∘p)s)/(b0(z)+b1(z)(f∘p)+ ⋯ +bt(z)(f∘p)t)), whereIis a finite set of multi-indexesλ=(λ0,λ1,…,λn),c0=0,cj∈ℂ∖{0} (j=1,2,…,n)are distinct complex constants,p(z)is a polynomial, andαλ(z) (λ∈I),ai(z) (i=0,1,…,s), andbj(z) (j=0,1,…,t)are small meromorphic functions relative tof(z). We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.

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14

Darus, Maslina, and Imran Faisal. "Some subclasses of analytic functions of complex order defined by new differential operator." Tamkang Journal of Mathematics 43, no. 2 (June 30, 2012): 223–42. http://dx.doi.org/10.5556/j.tkjm.43.2012.740.

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Let \hskip 2pt $\mathcal{A}(n)$ \hskip 2pt denote \hskip 2pt the \hskip 2pt class \hskip 2pt of \hskip 2pt analytic \hskip 2pt functions \hskip 2pt $f$ \hskip 2pt in \hskip 2pt the \hskip 2pt open \hskip 2pt unit \hskip 2pt disk \hskip 2pt $U=\{z:|z|<1\}$ \hskip 2pt normalized \hskip 2pt by \hskip 2pt $f(0)=f'(0)-1=0.$ \hskip 2pt In \hskip 2pt this \hskip 2pt paper, \hskip 2pt we \hskip 2pt introduce \hskip 2pt and \hskip 2pt study \hskip 2pt the \hskip 2pt classes \hskip 2pt $S_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho)$ \hskip 2pt and \hskip 2pt $R_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho)$ \hskip 2pt of \hskip 2pt functions \hskip 2pt $f\in\mathcal{A}(n)$ with $(\mu)z(D^{\mho+2}_{\lambda}(\alpha, \omega)f(z))'+(1-\mu)z(D^{\mho+1}_{\lambda}(\alpha, \omega)f(z))'\neq0$ and satisfy some conditions available in literature, where $f\in\mathcal{A}(n), \alpha, \omega, \lambda, \mu \geq0, \mho\in \mathbb{N}\cup\{0\},\,\,z\in U,$ and $D^{m}_{\lambda}(\alpha, \omega)f(z): \mathcal{A}\rightarrow \mathcal{A},$ is the linear fractional differential operator, newly defined as follows $$D^{m}_{\lambda}(\alpha, \omega)f(z) = z+ \sum\limits_{k=2}^{\infty}a_{k}(1+(k-1)\lambda \omega^{\alpha})^{m}z^{k}\cdot$$ Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes $S_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho, \omega)$ and $R_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho, \omega)$ are given.

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15

Fischbacher, T., H. Nicolai, and H. Samtleben. "Non-Semisimple and Complex Gaugings of N=16 Supergravity." Communications in Mathematical Physics 249, no. 3 (April 7, 2004): 475–96. http://dx.doi.org/10.1007/s00220-004-1081-z.

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16

Dubi, Chen. "Triangular realization of rational functions of N complex variables." Multidimensional Systems and Signal Processing 19, no. 1 (April 10, 2007): 123–29. http://dx.doi.org/10.1007/s11045-007-0026-z.

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17

NOOR, KHALIDA INAYAT. "ON A GENERALIZATION OF CLOSE-TO-CONVEXITY OF COMPLEX ORDER." Tamkang Journal of Mathematics 29, no. 2 (June 1, 1998): 73–87. http://dx.doi.org/10.5556/j.tkjm.29.1998.4241.

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The class $V_k$ of bounded boundary rotation is used to generalize the concept of close-to-convexity of complex order. A function $f:f(z)=z+\sum_{n=2}^\infty a_nz^n$, analytic in the unit disc $E$, belongs to $T_k(b)$, $b \neq 0$ (complex) if and only if there exists a function $g \in V_k$ such that \[ Re\left\{1+\frac{1}{b}\left(\frac{f'(z)}{g'(z)}-1\right)\right\}>0, \quad z\in E.\] Some basic properties, rate growth of Hankel determinant and radii problems for the functions in $T_k(b)$ are studied.

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18

Canty, AJ, BW Skelton, PR Traill, and AH White. "Structural Chemistry of the Platinum Group Metals: the Palladacyclopentane Complex (Butane-1,4-diyl)(N,N,N',N'-tetramethylethylenediamine)palladium(II)." Australian Journal of Chemistry 47, no. 11 (1994): 2119. http://dx.doi.org/10.1071/ch9942119.

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A single-crystal X-ray structure determination of the reagent (butane-1,4-diyl)(N,N,N′,N′- tetramethylethylenediamine)palladium(II) has shown that crystals are orthorhombic, Pbca, a 15.832(3), b 14.128(9), c 11.451(3) Ǻ, Z 8. The square planar complex has puckered PdCH2CH2CH2CH2 and PdNMe2CH2CH2NMe2 rings in which the outer carbon atoms deviate by c. �0.35 Ǻ from the 'PdC2N2' mean plane.

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19

Cui, Y. U., Guo-Xin Sun, Zhen-Wei Zhang, and Si-Xiu Sun. "Preparation, characterization and crystal structure of the complex formed between N,N,N',N'-tetrabutylmalonamide and lanthanum(III)nitrate." Journal of the Serbian Chemical Society 71, no. 5 (2006): 513–19. http://dx.doi.org/10.2298/jsc0605513c.

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The complex [La(NO 3)3(TBMA)2] (TBMA = N,N,N?,N?-tetrabutylmalonamide) was prepared by the extraction method and characterized by single-crystal X-ray diffraction. The complex consists of two bidentate TBMA extractant and three bidentate nitrate ions coordinated to a La3+ ion. The lanthanum(III) is 10-coordinate, being bonded to four oxygen atoms of the malonamides and six oxygen atoms of the three nitrates. The crystal data are as follows C38H76N7O13La, Mr = 977.97, triclinic system, space group P 1, a = 1.3025(12), b = 1.4353(14), c = 0.4871(13) nm, ? = 70.025(17), ? = 71.411(19), ? = 84.017(18)?, V = 2.476(4) nm3, Z = 2, F(000) = 1028, D = 1.312 g.cm-3, R 1 = 0.0833, wR 2 = 0.1471.

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20

Ibrahim, Rabha W. "Ulam Stability for Fractional Differential Equation in Complex Domain." Abstract and Applied Analysis 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/649517.

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The present paper deles with a fractional differential equationzαDzαu(z)+zu'(z)+(z2-a2)u(z)=∑n=0∞anzn+α,1<α≤2, wherez∈U:={z:|z|<1}in sense of Srivastava-Owa fractional operators. The existence and uniqueness of holomorphic solutions are established. Ulam stability for the approximation and holomorphic solutions are suggested.

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21

Labisbal, Elena, Antonio Sousa-Pedrares, Werner Kaminsky, and Douglas X. West. "Structure of N-Methylisatin N(4)-Dimethylthiosemicarbazone and its Electrochemically Synthesized 6-Coordinate Cadmium(II) Complex." Zeitschrift für Naturforschung B 57, no. 8 (August 1, 2002): 908–13. http://dx.doi.org/10.1515/znb-2002-0811.

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N-Methylisatin N(4)-dimethylthiosemicarbazone, HMIs4DM, prepared by condensation of N-methylisatin and N(4)-dimethylthiosemicarbazide crystallizes in the orthorhombic space group Pbc21, with a = 7.539(1), b = 11.566(4), c = 29.320(1) Å, V = 2556.6(9) Å3 and Z = 8. Cadmium metal was oxidized in the presence of HMIs4DM in an acetonitrile solution, which produced a complex of the formula [Cd(MIs4DM)2]. [Cd(MIs4DM)2] crystallizes in the orthorhombic space group Pbcn with a = 15.529(5), b = 10.672(4), c = 16.247(3) Å , V = 2692.7(14) Å3 and Z = 8. [Cd(MIs4DM)2] is symmetrical with two identical MIs4DM ligands that are at an angle of about 80°.

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22

Roesky, Herbert W., Jörg Sundermeyer,, Jürgen Schimkowiak, Peter G. Jones, Mathias Noltemeyer, Tina Schroeder, and George M. Sheldrick. "Facile Synthesis and Crystal Structure of [(PhSO2 N)2 WVI Cl2 (CH3CN)2] - the Oxidative Imination of W(CO)6 by N,N-Dichlorophenylsulphonamide." Zeitschrift für Naturforschung B 40, no. 6 (June 1, 1985): 736–39. http://dx.doi.org/10.1515/znb-1985-0608.

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AbstractThe reaction of N,N-dichlorophenylsulphonamide with tungsten hexacarbonyl in refluxing CCl4 leads in good yield to the yellow polymeric complex [(PhSO2N)2WCl2]x, which may be converted to the octahedral nitrene complex [(PhSO2N)2WCl2(CH3CN)2] by recrystallisation from acetonitrile. Crystals of the acetonitrile complex are triclinic, space group P 1̅, a = 820.7(2), b = 1128.8(3), c = 1286.9(3) pm, α = 89.25(3), β = 89.54(3). γ = 72.67(2)°, Z = 2, R = 0.027 for 2527 unique observed reflections.

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23

Roesky, Herbert W., Jörg Sundermeyer, Mathias Noltemeyer, George M. Sheldrick, Karen Meyer-Bäse, and Peter G. Jones. "Darstellung und Struktur des N-Thiobis-N′ -(phenylsulfonyl)schwefeldiimids / Synthesis and Structure of N-Thiobis-N′-(phenylsulfonyl)sulfurdiimide." Zeitschrift für Naturforschung B 41, no. 1 (January 1, 1986): 53–58. http://dx.doi.org/10.1515/znb-1986-0111.

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The reaction of S4N4 with PhSO2NCl2 proceeds with formation of (PhSO2NSN)2S 2. The main product 2 was characterized by an X-ray structure determination. [P21/c, a = 996.8(2), b = 1733.0(2), c = 1131.2(1) pm, β = 114.69(1)°, Z = 4, R = 0.044 for 1911 reflections.] 2 contains a sulfur-nitrogen chain with a planar central N4S3-Unit. The substituents at the NSN-groups are in cis-trans positions.2 reacts with AgAsF6 to yield the 1:1 complex 3; attempts to recrystallise this from CH2Cl2 led to 2 · CH2Cl2, which was also subjected to a crystal structure determination [P1̄, a = 554.4(2), b = 1313.5(4), c = 1541.3(4) pm, α = 111.05(2), β = 94.61(2), γ = 92.24(2)°. Z = 2. R = 0.088 for 1728 reflections]. Minor differences between the two forms of 2 are observed.

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24

Li, Song-Lin, and Thomas C. W. Mak. "Synthesis and Crystal Structural Characterization of Mixed-Ligand Cadmium(II) Complexes of 3-Triphenylphosphoniopropanoate and N ,N ,N′ ,N′-Tetramethylethylenediamine (tmen)." Australian Journal of Chemistry 50, no. 1 (1997): 79. http://dx.doi.org/10.1071/c96050.

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Two mixed-ligand cadmium(II) complexes have been synthesized from the reaction of 3-triphenyl- phosphoniopropanoate, Ph3P+ (CH2)2CO2¯, and Me2N(CH2)2NMe2 (abbreviated as tmen) with Cd(ClO4)2.x H2O and CdI2, and structurally characterized by single-crystal X-ray analysis: [Cd { Ph3P(CH2)2CO2 } 2(tmen)(H2O)] (ClO4)2.2H2O (1), space group P -1 with a 13·461(5), b 13·694(5), c 18·229(4) Å, α 92·34(2), β 110·51(2), γ 117·44(2)° and Z 2; [CdI2 { Ph3P(CH2)2CO2 } (tmen)] (2), space group P 21/c with a 16·768(6), b 11·741(3), c 16·915(5) Å, β 112·28(2)° and Z 4. In complex (1), the cadmium(II) atom is in a distorted pentagonal bipyramidal environment with two chelated betaine ligands and one nitrogen atom of the chelated tmen ligand defining the equatorial plane, and an aqua ligand and the other tmen nitrogen atom occupying the axial positions. One of the betaine ligands acts in the symmetric and the other in the asymmetric chelate mode. In complex (2), the distorted octahedral coordination environment about the cadmium(II) atom involves a chelated tmen ligand, an asymmetrically chelated betaine ligand, and two cis-related iodo ligands.

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25

Jonsson, Jakob. "3-torsion in the Homology of Complexes of Graphs of Bounded Degree." Canadian Journal of Mathematics 65, no. 4 (August 1, 2013): 843–62. http://dx.doi.org/10.4153/cjm-2013-008-4.

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AbstractFor δ ≥ 1 and n ≥ 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When δ = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n − 4)/3 ≤ d ≤ (n − 6)/2. This paper establishes similar bounds for δ ≥ 2. Specifically, there is 3-torsion in degree d wheneverThe procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3zis a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

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26

Gui, Xian Min, Hong Yan Xu, Wen Ju Tang, and Hua Wang. "Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ²." Open Mathematics 19, no. 1 (January 1, 2021): 1416–34. http://dx.doi.org/10.1515/math-2021-0113.

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Abstract By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we mainly investigate the existence and the forms of entire solutions for the partial differential-difference equation of Fermat type α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 m + f ( z 1 + c 1 , z 2 + c 2 ) n = 1 {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{m}+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{n}=1 and α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 2 + [ γ 1 f ( z 1 + c 1 , z 2 + c 2 ) − γ 2 f ( z 1 , z 2 ) ] 2 = 1 , {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{2}+{\left[{\gamma }_{1}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-{\gamma }_{2}f\left({z}_{1},{z}_{2})]}^{2}=1, where m , n m,n are positive integers and α , β , γ 1 , γ 2 \alpha ,\beta ,{\gamma }_{1},{\gamma }_{2} are constants in C {\mathbb{C}} . We give some results about the forms of solutions for these equations, which are great improvements of the previous theorems given by Xu and Cao et al. Moreover, it is very satisfactory that we give the corresponding examples to explain the conclusions of our theorems in each case.

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LEBRUN, CLAUDE. "FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY." International Journal of Mathematics 06, no. 03 (June 1995): 419–37. http://dx.doi.org/10.1142/s0129167x95000146.

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Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M4n, g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

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28

Bulygina, L. A., N. S. Khrushcheva, V. I. Sokolov, and A. A. Khodak. "Synthesis and catalytic activity of a complex of 1,1´-bis-isoquinoline N,N´-dioxide with PdCl2." Russian Chemical Bulletin 64, no. 2 (February 2015): 429–31. http://dx.doi.org/10.1007/s11172-015-0880-z.

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29

Weiss, Armin, and Stefan Dick. "Der zweikernige Fe(III)-Komplex von N,N,N',N'-Tetrakis(2-pyridylmethyl)-2-hydroxy-1,3-diaminopropan - Molekül- und Kristallstruktur und seine Wechselwirkung mit Montmorillonit / The Binuclear Fe(III) Complex of N,N,N',N'-Tetrakis(2-pyridylmethyl)-2-hydroxy-1,3-diaminopropane - Molecular and Crystal Structure and its Interaction with Montmorillonite." Zeitschrift für Naturforschung B 49, no. 8 (August 1, 1994): 1051–58. http://dx.doi.org/10.1515/znb-1994-0807.

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Yellow crystals of [Fe2(HPTP)(OH)(NO3)2](Cl04)2 (HPTP = N,N,N',N'-tetrakis(2-pyridyl- methyl)-2-hydroxy-1,3-diaminopropane) were grown by slow diffusion of diethylether into a solution of the complex salt in acetonitrile. Crystal data: space group R3̅, a = 4280,1(15) pm, c = 1147,6(7) pm, Z = 18. The complex exhibits a binuclear Fe(III) core with (μ-oxo)(μ-alkoxo) ligands as bridging units. The nitrate anions coordinate each with a short and a long Fe-O bond. The coordination spheres of the iron atoms can be looked at as distorted edge-shearing pentagonal bipyramides. The complex cation was introduced into the interlayer region of montmorillonite by cation exchange in acetonitrile. The results of X-ray, IR- and UV/VIS experiments suggest that the cation exchange is reversible. In the interlayer region the dinuclear cations are arranged in a rather loosely packed way, thus creating channels similar to those in pillared smectites or zeolites, respectively.

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30

Alarcón, Antonio, Josip Globevnik, and Francisco J. López. "A construction of complete complex hypersurfaces in the ball with control on the topology." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 751 (June 1, 2019): 289–308. http://dx.doi.org/10.1515/crelle-2016-0061.

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AbstractGiven a closed complex hypersurface {Z\subset\mathbb{C}^{N+1}} ({N\in\mathbb{N}}) and a compact subset {K\subset Z}, we prove the existence of a pseudoconvex Runge domain D in Z such that {K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of {\mathbb{C}^{N+1}}. For {N=1}, we derive the existence of complete properly embedded complex curves in the unit ball of {\mathbb{C}^{2}}, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc {\mathbb{D}\subset\mathbb{C}} into the unit ball of {\mathbb{C}^{2}}. These are the first known examples of complete bounded embedded complex hypersurfaces in {\mathbb{C}^{N+1}} with any control on the topology.

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31

Bairagi, S. D., Vinay Jain, T. K. Mishra, and L. Saha. "On the location of the zeros of certain polynomials." Publications de l'Institut Math?matique (Belgrade) 99, no. 113 (2016): 287–94. http://dx.doi.org/10.2298/pim1613287b.

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We extend Aziz and Mohammad?s result that the zeros, of a polynomial P(z) =?n j=0 ajzj, taj ? aj?1 > 0, j=2,3,...,n for certain t(>0), with moduli greater than t(n?1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P(z), of degree n, with complex coefficients, does not vanish in the disc |z?aei?| < a/(2n); a > 0, max |z|=a |P(z)| = |P(aei?)|, for r < a < 2,r being the greatest positive root of the equation xn?2xn?1+1=0, and finally obtained an upper bound, for moduli of all zeros of a polynomial,(better, in many cases, than those obtainable from many other known results).

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32

Schnorr, René, Marcel Handke, and Berthold Kersting. "Synthesis, characterization and molecular structure of a dinuclear uranyl complex supported by N,N′,N″,N′″-tetra-(3,5-di-tert-butylsalicylidene)-1,2,4,5-phenylenetetraamine." Zeitschrift für Naturforschung B 70, no. 10 (October 1, 2015): 757–63. http://dx.doi.org/10.1515/znb-2015-0107.

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AbstractThe preparation, characterization and the molecular structure of a dinuclear uranyl complex [(UO2)2L(OCMe2)2] supported by the bis-salophen ligand N,N′,N″,N′″-tetra-(3,5-di-tert-butylsalicylidene)-1,2,4,5-phenylenetetraamine (L4–) is described. [(UO2)2L(OCMe2)2] was prepared by reaction of uranyl nitrate with the neutral, protonated form of the ligand (H4L) in acetone. From a saturated acetone solution [(UO2)2L(OCMe2)2]·1.5(OCMe2) crystallizes triclinically, space group P1̅ with a = 1522.7(2), b = 1751.4(2), c = 1815.4(2) Å, α = 109.16(1), β = 99.29(1), γ = 105.29(1)° and Z = 2. Each uranium atom is surrounded in a distorted pentagonal bipyramidal fashion by two O and two N atoms of the salicylidene units, one O atom of an acetone ligand, and the two oxo groups. The cyclic voltammogram of [(UO2)2L(OCMe2)2] shows two quasi-reversible redox processes centered at +0.57 V and +0.82 V vs. Fc+/Fc attributed to the sequential oxidation of the coordinating phenolates to phenoxyl radicals. The crystal structure of an ethanol solvate of H4L was also determined by X-ray crystallography. H4L·5EtOH: triclinic, space group P1̅, a = 1003.4(3), b = 1187.7(3), c = 1905.1(5) Å, α = 75.75(2), β = 78.74(2), γ = 66.66(2)°, Z = 1.

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33

Zhang, Xiaomei, and Xiang Chen. "Uniqueness of difference polynomials." AIMS Mathematics 6, no. 10 (2021): 10485–94. http://dx.doi.org/10.3934/math.2021608.

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<abstract><p>Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.</p></abstract>

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34

Paul, Navendu, Rudra Sarkar, and Sabyasachi Sarkar. "Zinc protoporphyrin–trimethylamine-N-oxide complex involves cholesterol oxidation causing atherosclerosis." JBIC Journal of Biological Inorganic Chemistry 26, no. 2-3 (March 13, 2021): 367–74. http://dx.doi.org/10.1007/s00775-021-01861-z.

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35

Bagno, Eli, and Riccardo Biagioli. "Colored-descent representations of complex reflection groups G(r, p, n)." Israel Journal of Mathematics 160, no. 1 (August 2007): 317–47. http://dx.doi.org/10.1007/s11856-007-0065-z.

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36

Mudrov, A. "Star product on complex sphere $$\mathbb {S}^{2n}$$ S 2 n." Letters in Mathematical Physics 108, no. 6 (March 22, 2018): 1443–54. http://dx.doi.org/10.1007/s11005-018-1074-z.

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37

Barcelo, Hélène, and Shelly Smith. "The discrete fundamental group of the order complex of B n." Journal of Algebraic Combinatorics 27, no. 4 (September 13, 2007): 399–421. http://dx.doi.org/10.1007/s10801-007-0094-z.

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38

Szczepański, Jerzy. "Chebyshev Polynomials and Continued Fractions Related." Science, Technology and Innovation 7, no. 4 (December 31, 2019): 1–8. http://dx.doi.org/10.5604/01.3001.0013.6863.

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Let $p$, $q$ be complex polynomials, $\deg p>\deg q\geq 0$. We consider the family of polynomials defined by the recurrence $P_{n+1}=2pP_n-qP_{n-1}$ for $n=1, 2, 3, ...$ with arbitrary $P_1$ and $P_0$ as well as the domain of the convergence of the infinite continued fraction $$f(z)=2p(z)-\cfrac{q(z)}{2p(z)-\cfrac{q(z)}{2p(z)-...}}$$ null

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39

Boeré, René T., Vicki Klassen, and Gotthelf Wolmershäuser. "Superamidines 2. Synthesis of the bulky ligand N,N'-bis-(2,6-diisopropylphenyl)-trifluoroacetamidine and its molybdenum carbonyl complex." Canadian Journal of Chemistry 78, no. 5 (May 1, 2000): 583–89. http://dx.doi.org/10.1139/v00-063.

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N,N'-bis-(2,6-diisopropylphenyl)trifluoroacetamidine has been prepared for the first time from 2,6-diisopropylaniline and the trifluoroacylation reagent TFAP via the imidoylchloride. The crystal structure of the amidine was determined, indicating that it crystallizes in the Z-anti tautomer, in contrast to the nonfluorinated analogue, which is E-anti in the solid state. In solution, as indicated by NMR spectroscopy, it exists in two isomeric forms. The amidine reacts with Mo(CO)6 to produce a coordination complex with Mo(CO)3 in which the ligand is also in the Z-anti geometry, the metal is η6-coordinated to the imino-2,6-diisopropylphenyl ring, and the amino N-H unit is directed towards the metal, as determined by a single-crystal X-ray structure. Unlike the analogous nonfluorinated acetamidine, there is no indication of an intermediate in which the neutral amidine is coordinated in a monodentate fashion to an Mo(CO)5 unit, which we now attribute to the predominant geometry of the ligand, both in the solid state and in solution, being Z-anti. The high steric bulk of this superamidine ligand apparently prevents the formation of a metal-metal bonded Mo2(amidinate)4 as observed previously in a redox reaction between N,N'-diphenylbenzamidine and Mo(CO)6 under similar thermal reaction conditions.Key words: trifluoromethyl, superamidine, amidine, molybdenum, carbonyl, coordination.

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40

Labisbal, Elena, Antonio Sousa, Alfonso Castiñeiras, Jose A. García-Vázquez, Jaime Romero, Gordon A. Bain, and Douglas X. West. "Electrochemical Synthesis of a 6-Coordinate Cadmium(II) Complex with N-Methylisatin N(4)-cyclohexylthiosemicarbazone." Zeitschrift für Naturforschung B 55, no. 2 (February 1, 2000): 162–66. http://dx.doi.org/10.1515/znb-2000-0205.

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Abstract Cadmium metal was oxidized in the presence of N-methylisatin N(4)-cyclohexyl-thiosemicarbazone (HMeIs4Chex) in an acetonitrile solution, which produced a complex of the formula [Cd(MeIs4Chex)2]. The two MeIs4Chex ligands are at an angle close to 90° from each other, and there is hydrogen bonding to two adjacent molecules by the anionic ligands remaining NH groups. The complex crystallizes in the monoclinic space group P21/c with a = 11.820(4), b = 11.491(4), c = 27.834(4) Å, β = 106.82(2)°, V = 3618.7(17) Å3 and Z = 4.

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41

Schwarz, Simon, Montserrat Galceran Mestres, Elke Niquet, Carlos F. Barboza da Silva, and Joachim Strähle. "Synthese und Struktur der Nitridokomplexe (Ph3Sb)2Cl3Os≡N-RhCl(COD), (Me2PhP)2(PhCN)Cl2Re≡N-RhCl2(C5Me5), [(Me2PhP)3(PhCN)ClRe≡N-OsCl2(CO)3][OsCl3(CO)3] und [(Me2PhP)3(PhCN)ClRe≡N-ReCl4-N≡ReCl(NCPh)(PMe2Ph)3][OsCl4(CO)2] / Synthesis and Structure of the Nitrido Complexes (Ph3Sb)2Cl3Os≡N-RhCl(COD), (Me2PhP)2(PhCN)Cl2Re≡N-RhCl2(C5Me5), [(Me2PhP)3(PhCN)ClRe≡N-OsCl2(CO)3][OsCl3(CO)3], and [(Me2PhP)3(PhCN)ClRe≡N-ReCl4-N≡ReCl(NCPh)(PMe2Ph)3][OsCl4(CO)2]." Zeitschrift für Naturforschung B 59, no. 2 (February 1, 2004): 167–73. http://dx.doi.org/10.1515/znb-2004-0208.

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The reaction of (Ph3Sb)2Cl3OsN with [Rh(μ-Cl)(COD)]2 in benzonitrile yields the heteronuclear nitrido complex (Ph3Sb)2Cl3Os≡N-RhCl(COD) (1). It crystallizes as dark brown blocks in the triclinic space group P1 with a = 1080.09(5), b = 1173.15(5), c = 1702.28(9) pm, α = 101.214(4), β = 92.923(8), γ = 92.836(5)֯, and Z = 2. The complex fragment RhCl(COD) is coordinated to the osmium nitrido complex via a linear nitrido bridge Os≡N-Rh, with a bond angle Os-N-Rh = 175.3(5)° and distances Os-N = 168.6(6) and Rh-N = 184.6(6) pm. The heteronuclear complexes (Me2PhP)2(PhCN)Cl2Re≡N-RhCl2(C5Me5) (2) and [(Me2PhP)3(PhCN)ClRe≡NOsCl2( CO)3][OsCl3(CO)3] (3) are obtained by the reaction of (Me2PhP)3Cl2ReN with [RhCl(μ- Cl)(C5Me5)]2 and [OsCl(μ-Cl)(CO)3]2, respectively, in benzonitrile. 2 forms orange red crystals with the composition 2・C6H5CN in the monoclinic space group P21/c and a =910.23(7), b =1255.4(1), c = 3761.5(3) pm, β = 95.617(9)° and Z = 4. The nitrido bridge Re≡N-Rh to the complex fragment [RhCl2(C5Me5)] has a bond angle Re-N-Rh of 171.9(3)° and distances Re-N = 169.4(5) and Rh-N = 201.9(5) pm. 3 forms yellow orange plates with the monoclinic space group P21/c and a = 1113.74(5), b = 2038.5(2), c = 2130.1(1) pm, β = 90.298(6)° and Z = 4. The nitrido bridge within the complex cation [(Me2PhP)3(PhCN)ClRe≡N-OsCl2(CO)3]+ is characterized by a bond angle Re-N-Os of 167.7(4)° and distances Re-N = 169.5(8) and Os-N = 206.1(8) pm. The ligands of the octahedral anion [OsCl3(CO)3]− are in a facial arrangement. The synthesis of 3 yields [{(Me2PhP)3(PhCN)ClRe≡N}2ReCl4][OsCl4(CO)2] (4) as a byproduct. It crystallizes as red brown blocks with the composition 4·CH2Cl2 in the space group P1 and a = 1444.2(1), b = 1536.3(1), c = 2150.5(2) pm, α =87.41(1), β =88.54(1), γ =62.453(8)° and Z =2. In the two symmetry independent, centrosymmetric cations [(Me2PhP)3(PhCN)ClRe≡N-ReCl4-N≡ReCl(NCPh)(PMe2Ph)3]2+ of 4 two nitrido complexes [(Me2PhP)3(PhCN)ClReN]+ coordinate with their terminal nitrido ligands a central, square planar ReCl4 unit. The resulting nitrido bridges with bond angles Re(1)-N(1)- Re(2) = 167.0(4)° and Re(3)-N(2)-Re(4) = 168.9(4)°, respectively, exhibit distances Re(1)-N(1) = 172.4(7) and Re(2)-N(1) = 197.1(7) or Re(3)-N(2) = 171.4(6) and Re(4)-N(2) = 197.1(6) pm. The two carbonyl ligands of the anion [OsCl4(CO)2]2− are in cis position.

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42

Mir, Abdullah, Adil Hussain, and Imtiaz Hussain. "Upper bound estimates for the polar derivative of a polynomial with restricted zeros." Publications de l'Institut Math?matique (Belgrade) 112, no. 126 (2022): 103–10. http://dx.doi.org/10.2298/pim2226103m.

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The polar derivative of a polynomial P(z) of degree n with respect to a complex number ? is a polynomial nP(z) + (??z)P?(z) of degree at most n?1 and is denoted by D?P(z). We consider the class of polynomials P(z) = a0 + ?n v=? avzv, ? ? 1, of degree n such that P(z) ?0 in |z| < k, k ? 1 and establish some upper bound estimates for the maximum modulus of D?P(z) on the unit disk by involving some of the coefficients of P(z). The obtained results refine and generalize some well known polynomial inequalities.

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43

Chō, Muneo, and Biljana Nacevska. "Spectral properties of n-normal operators." Filomat 32, no. 14 (2018): 5063–69. http://dx.doi.org/10.2298/fil1814063c.

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For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

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44

Ren, Ying Hui, Hai Xia Ma, Kang Zhen Xu, Jie Huang, Du Bei Zhang, and Ji Rong Song. "Synthesis and Crystal Structure of Copper (II) Complex Derived from N-(4,6-dimethylpyrimidin)-N’-(ethoxycarbonyl)thiourea." Advanced Materials Research 396-398 (November 2011): 2103–6. http://dx.doi.org/10.4028/www.scientific.net/amr.396-398.2103.

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A new copper(Ⅱ) complex derived from N-(4,6-dimethylpyrimidin)-N’-(ethoxycarbonyl) thiourea was synthesized. The crystal is of monoclinic, P21/n, a=10.228(2), b= 11.429(2), c= 13.390(3) Ǻ, β =104.05(3)º, V=1518.3(5) Ǻ3 and Z = 2. Its crystal and molecular structure shows two N-(4,6-dimethylpyrimidin)-N’-(ethoxycarbonyl)thiourea molecules coordinate to copper(Ⅱ) through single dentate of the nitrogen atom.

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45

Sznajder, Roman, and Kanchan Basnyat. "Dynamics of a certain sequence of powers." International Journal of Mathematics and Mathematical Sciences 24, no. 4 (2000): 283–88. http://dx.doi.org/10.1155/s0161171200003136.

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For any nonzero complex numberzwe define a sequencea1(z)=z,a2(z)=za1(z),…,an+1(z)=zan(z),n∈ℕ. We attempt to describe the set of thesezfor which the sequence{an(z)}is convergent. While it is almost impossible to characterize this convergence set in the complex plane𝒞, we achieved it for positive reals. We also discussed some connection to the Euler's functional equation.

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46

Trześniewska-Ofiara, Zuzanna, Katarzyna Sikora, Anna Żyła, Brygida Beck, Mariola Mendrycka, and Agnieszka Woźniak-Kosek. "Brain abscess caused by Nocardia abscessus." Lekarz Wojskowy 100, no. 3 (October 12, 2022): 189–94. http://dx.doi.org/10.53301/lw/149958.

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Bakterie z rodzaju Nocardia zaliczane są do tlenowych promieniowców wywołujących poważne infekcje ludzi oraz zwierząt. Występuje ponad 30 gatunków chorobotwórczych dla człowieka. Najczęściej izolowane szczepy, m.in.:<i> N. nova complex, N. abscessus, N. transvalensis complex, N. farcinica, N. asteroidestyp VI (N. cyriacigeorgica)</i> i <i>N. brasiliensis</i>, są odpowiedzialne za wiele różnych chorób począwszy od postaci skórnej, poprzez płucną oraz wielonarządową. Sporadycznie mogą powodować także ropień mózgu, zwłaszcza u osób w immunosupresji. Grupą najbardziej podatną na zachorowania są pacjenci z obniżoną odpornością. Właściwa identyfikacja bakterii z rodzaju Nocardia jest trudna. Niezbędne jest wykorzystanie metod biologii molekularnej, gdyż nowe gatunki niekiedy nie wykazują żadnych różnic fenotypowych między sobą. Prawidłowa identyfikacja jest konieczna do wdrożenia właściwej antybiotykoterapii.

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Kumar Maurya, Rajesh, and Poonam Sharma. "A Class of Analytic Starlike Functions Associated with Petal Like Region on the Positive Half of Complex Plane." Journal of the Indian Mathematical Society 87, no. 3-4 (July 1, 2020): 165. http://dx.doi.org/10.18311/jims/2020/25449.

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In the light of Riemann open mapping theorem, if we map open unit disk U conformally onto a region then depending on the geometry of boundary of we can always extract a subclass of H[a, n] by subordinating various functionals of the function f ∈ H[a, n]. Depending upon the geometry of the range set attempts have been made to find some algebraic structure in such classes, for that Hankel determinant of coefficients of functions pertaining to these classes have been studied, bounds of various coefficients have been determined and also based on the subordination principle we have determined radius |z| < r ;z ∈ U for which f belongs to such a class. In this paper our focus would be on n−PS<sup>*</sup> defined as n − PS<sup>*</sup> = {f ∈ A : Re {zf<sup>'</sup>(z)/f(z)} > 0,|(zf<sup>'</sup>(z)/f(z))<sup>n</sup> - 1|<1}.

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Liu, Si Min, and Hong Yan Xu. "Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type." Journal of Mathematics 2021 (January 20, 2021): 1–14. http://dx.doi.org/10.1155/2021/4207579.

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This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 1 z 1 , z 2 / ∂ z 2 2 n 1 + f 2 z 1 + c 1 , z 2 + c 2 m 1 = 1 α ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 + β ∂ 2 f 2 z 1 , z 2 / ∂ z 2 2 n 2 + f 1 z 1 + c 1 , z 2 + c 2 m 2 = 1 and ∂ 2 f 1 z 1 , z 2 / ∂ z 1 2 2 + f 2 z 1 + c 1 , z 2 + c 2 2 = 1 ∂ 2 f 2 z 1 , z 2 / ∂ z 1 2 2 + f 1 z 1 + c 1 , z 2 + c 2 2 = 1 . Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.

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Jantaramas, Trairat, Somphong Jitman, and Pornpan Kaewsaard. "Determinants of binomial-related circulant matrices." Special Matrices 6, no. 1 (July 1, 2018): 262–72. http://dx.doi.org/10.1515/spma-2018-0021.

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Abstract Due to their rich algebraic structures and wide applications, circulant matrices have been of interest and continuously studied. In this paper, n×n complex left and right circulant matrices whose first row consists of the coefficients in the expansion of (x + zy)n−1 are focused on, where z is a nonzero complex number and n is a positive integer. In the case where z ∈ {1, −1, i, −i}, explicit formulas for the determinants of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as the special case where z = 1. Finally, some remarks and open problems are discussed.

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Canty, AJ, NJ Minchin, LM Engelhardt, BW Skelton, and AH White. "Synthetic and Structural Studies of Binuclear Organopalladium(II) Complexes Including a Bis(pyridin-2-yl)phenylmethyl Complex With Four- and Eight-Membered Palladocycle Rings, trans(N,N)-(Pd(μ-Py2PhC-N,N',C')Cl)2.½CH2Cl2.½Me2CO." Australian Journal of Chemistry 41, no. 5 (1988): 651. http://dx.doi.org/10.1071/ch9880651.

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Abstract:

The byridin-2-yl) phenylchloromethanes PyPh2CCl and Py2PhCCl undergo oxidative addition reactions with bis ( dibenzylideneacetone )palladium(0) to form {Pd(PyPh2C) Cl }2 (1) and {Pd(Py2PhC)C}2.�2CH2Cl2.�Me2CO (2), respectively. n.m.r , studies of (1) in CDCl3 indicate presence of an equilibrium between two isomers, involving the ligand (pyridin-2-yl ) diphenylmethyl in η3-coordination, {Pd(η3-PyPh2C)(μ- Cl )}2. Complex (2) has two bis (pyridin-2-yl) phenylmethyl groups present as bridging N,C′- ligands , with the groups also N,C- bidentate to each palladium, to form a binuclear compex containing four- and eight- membered palladocycles: PdC(CN)CNPdC(CN)CNBoth palladium(II) centres in (2) have square-planar 'trans-PdCN2Cl' coordination, so that (2) may be represented as trans(N,N)-{Pd(μ- Py2PhC-N,N′, C′) Cl }2.�CH2Cl2.�Me2CO. A complex similar to (2), trans(N,N)-{Pd(μ- PyPhCH -N,C?)( γmpy ) Cl }2.CH2Cl2 (3), forms on reaction of the lithium derivative of 2-benzylpyridine, Li( PyPhCH ), with dichlorobis (4-methylpyridine)palladium(II); (3) reacts with excess 2-benzylpyridine with displacement of γmpy to form trans(N,N)-{Pd(μ- PyPhCH -N,C?)(PyPhCH2) Cl }2 (4). In developing an alternative isolation procedure for (3), involving column chromatography with 4% ethyl acetate in chloroform, the coordination complex trans- bis{1-phenyl-1-(pyridin-2-yl)prop-1-en-2-olato-O,N}palladium(II), Pd{( PyPhC )C(Me)O}2 (5), was isolated in low yield. X-Ray structural studies of (2)-(5) have been completed, with all four complexes crystallizing in the monoclinic system; (2): space group P21/c, a 9.649(6), b 21.116(9), c 18.627(7)Ǻ,β 111.42(4)°, Z 4; (3): P21/c, a 13.967(2), b 13.996(3), c 18.886(2)Ǻ, β 98.74(1)°, Z 4; (4): P21/n, a 14.274(9), b 14.584(14), c 20.97(2)Ǻ, β 109.18(5)°, Z 4; (5): P21/c, a 15.184(3), b 7.887(6), c 19.240(2)Ǻ, β 97.51(1)°, Z 4.

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