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Fermat Numbers F n = 2 2n + 1

R. D. Carmichael
1919 The American mathematical monthly  
By means of this result it is easy to establish Pepin's theorem concerning prime numbers F.: For n > 1, F, is prime if and only if it divides kl(Fn-l) + 1, where k is any quadratic non-residue of F,, as  ...  Consider the possibility that a = n-1. Then from (1) it follows that k2 + 1 is divisible by p and hence that k > 22"; whence we have a contradiction. Therefore, a _ n -2.  ... 
doi:10.2307/2971748 fatcat:azv2qzojrrbnpknwrnjcgs4dtm

n-Transitivity and the complementation property

Leo Livshits, Gordon MacDonald
2001 Linear Algebra and its Applications  
We investigate the structure of (minimal) n-transitive semigroups of matrices of rank at most n in M k (F), by considering an equivalent problem regarding certain families of (n − k)-dimensional subspaces  ...  of F k .  ...  It follows that L has the complementation property in F 4 if and only if for every choice of non-zero x, z in F 2 there exists B ∈ B such that Bx is not a multiple of z.  ... 
doi:10.1016/s0024-3795(01)00246-4 fatcat:gsmonicvrjabfk6ewslvc4b3wi

On AGT description of $ \mathcal{N} = 2 $ SCFT with N f = 4

Gaston Giribet
2010 Journal of High Energy Physics  
We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of N=2 SCFT.  ...  We also make some comments on elementary surface operators and WZW theory.  ...  The N f = 4 theory has SO(8) global symmetry, and it turns out that the SO(8) triality mixes with the SL(2, Z) duality symmetry of the theory in a funny way.  ... 
doi:10.1007/jhep01(2010)097 fatcat:hshq2d2emna4znk4zbratyn5xe

Fibonacci groups, F(2,n), are hyperbolic for n odd and n >= 11 [article]

Christopher Chalk
2020 arXiv   pre-print
We do this by applying a curvature argument to an arbitrary van Kampen diagram of F(2,n) and show that it satisfies a linear isoperimetric inequality. It then follows that F(2, n) is hyperbolic.  ...  We prove that the Fibonacci group, F(2,n), for n odd and n >= 11 is hyperbolic.  ...  I would also like to thank Caitlin for her generous help and advice.  ... 
arXiv:2005.10653v1 fatcat:etnqyg4lbzcxxkdc5a4dzkih74

The quasiconformal subinvariance property of John domains in ^n and its application [article]

M. Huang, Y. Li, S. Ponnusamy, X. Wang
2013 arXiv   pre-print
The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in ^n.  ...  But we get from (3.8), (3.19) and Theorem H that Mod f (G 1 ), f (B 2 ); f B(z 1,0 , d D (z 1,0 )) ≤ Mod (f (G 1 ), f (B 2 ); R n ) ≤ ψ n 1 4 ρ 18 , which implies φ n 4 7 cρ 11 ≤ µ 3 (n, ρ)Kψ n 1 4 ρ  ...  Then Mod (E, F ; R n ) ≤ µ 3 (n, c)Mod (E, F ; G) for every pair of continua E and F in G, where µ 3 (n, c) is a constant depending on n and c. Definition 2.2.  ... 
arXiv:1108.5550v2 fatcat:grjuoszycbejjgtmiaynartwzu

Some properties of S(n)-θ-closed spaces

Shouli Jiang, Ivan Reilly, Shuquan Wang
1999 Topology and its Applications  
Recently Dikranjan and Giuli characterized S(n)-θ-closed spaces in terms of special covers and filters. They also gave some interesting results and examples, as well as listing several open problems.  ...  We give conditions that make an S(n)-θ-closed space Katetov S(n). Furthermore, we provide examples which answer three of their questions in the general case.  ...  To prove that (X, τ ) is an S(n)-space, it is enough to show that ad θ n { F α } = ad θ n {V ∩ F : V is a τ neighbourhood of x and FF α } = {p}. It is clear that ad θ n { F α } ⊃ {p}.  ... 
doi:10.1016/s0166-8641(98)00015-7 fatcat:7y53byyilzcfbkf5w2kcvsowla

Exact partition functions for deformed N = 2 $$ \mathcal{N}=2 $$ theories with N f = 4 $$ {\mathcal{N}}_f=4 $$ flavours

Matteo Beccaria, Alberto Fachechi, Guido Macorini, Luigi Martina
2016 Journal of High Energy Physics  
combinations of Eisenstein series and Jacobi theta functions with well defined modular properties.  ...  We consider the Ω-deformed N=2 SU(2) gauge theory in four dimensions with N_f=4 massive fundamental hypermultiplets.  ...  The total prepotential is therefore remarkably simple and reads F (α,β) (ν) = −β ∆ m log ν Λ − β log 1 + N n=1 M 2n (q) ν 2n . (2.6) The properties of this result have been discussed in the introduction  ... 
doi:10.1007/jhep12(2016)029 fatcat:wr2uvczzgjctnftiu5qy4fy3ia

Drazin-monotonicity characterizations for property-n matrices

Hans Joachim Werner
1985 Linear Algebra and its Applications  
A square matrix A is said to have prmerty n if there exists a nonnegative power of A.  ...  In this paper, necessary and sufficient conditions for such matrices to have a nonnegative Drazin inverse are presented.  ...  Then A has property n if and only if A has property n(w) for each integer w >, ind(A). Proof. Suppose that A is Drazin-monotone, and let A have property n.  ... 
doi:10.1016/0024-3795(85)90259-9 fatcat:w65uahxqazfslidenmmj7q7p2a

On the relevance of the differential expressions f^2+f'^2, f+f" and f f"- f'^2 for the geometrical and mechanical properties of curves [article]

James Bell Cooper
2011 arXiv   pre-print
We present a unified approach to known and new properties of curves by showing the ubiquity of the expressions in the title in the analytic treatment of their mechanical and geometric properties  ...  cubic), n = 1 3 Cayley's sextic, n = 1 2 is the cardioid and n = 2 the leminiscate.  ...  Then if we compute the coefficients of the two fundamental forms, we find that E = f 2 , F = 0, G = f 2 + f2 and L = f 2 /(f 2 + f2 ) 1/2 , M = 0, N = (f2f f ′′ )/(f 2 + f2 ) 1/2 .  ... 
arXiv:1102.1579v1 fatcat:m2zxzstnxvhffcqgr27qvecp44

Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2

C.J Everett
1977 Advances in Mathematics  
If so, the list of parity sequences for m = 1, 2,... in (3) would be rather remarkable, in view of the following property, which it does indeed have; namely, the 2* parity sequences for the integers m  ...  If we set A, = A(29 and define TZN = 2N + {(2N+1 -2N) -(AN+1 -AN)} = AN + (2N+1 -AN+,) 3 2N, (10) then for M = 2N + l,..., nN, it is For the function f defined in Eq.  ... 
doi:10.1016/0001-8708(77)90087-1 fatcat:pwzefdmmubb55pll4wmuhcfidu

Topological properties of some algebraically defined subsets ofβN

Neil Hindman, Dona Strauss
2017 Topology and its Applications  
ideal; and N * +N * .  ...  Let S be a discrete semigroup and let the Stone-Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center  ...  F contains the open subsets of βN, because βN has a basis of c clopen sets, and F obviously contains the closed subsets of βN. It is also obvious that F is closed under countable unions.  ... 
doi:10.1016/j.topol.2017.02.001 fatcat:kk6wc6yozvbu7nmabmltbp5u6m

Weaknesses in the SL2($$ \mathbb{F}_{2^n } $$) Hashing Scheme [chapter]

Rainer Steinwandt, Markus Grassl, Willi Geiselmann, Thomas Beth
2000 Lecture Notes in Computer Science  
We show that for various choices of the parameters in the SL2(IF2n ) hashing scheme, suggested by Tillich and Zémor, messages can be modified without changing the hash value.  ...  Moreover, we would like to thank the referees for several useful remarks and references.  ...  Further properties of the group SL 2 (IF p n ) are summarized as follows: Remark 1. For any subfield IF p m ≤ IF p n of IF p n , SL 2 (IF p m ) and all its conjugates are subgroups of SL 2 (IF p n ).  ... 
doi:10.1007/3-540-44598-6_18 fatcat:hdfdsqtfv5ferjze5i2aq3rnvq

On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

Juan Matías Sepulcre, Tomás Vidal
2014 Mediterranean Journal of Mathematics  
R(z) = cw 1 a z 1 e γz + cw 2 a z 2 e γz + . . . + cw n a z n e γz and it is associated to the functional equation v 1 f (b 1 z) + v 2 f (b 2 z) + . . . + v n f (b n z) = 0, z ∈ C, (4.4) with v j = cw  ...  to it.  ... 
doi:10.1007/s00009-014-0444-8 fatcat:vrouftgdyjclrf54zqez5pjg6i

The Synthesis of Sodium(and Potassium) N, N'-bis-(o-hydro xybenzoyl)-ethylenediaminato-Cuprate(II) and Its Properties

Heijiro OJIMA
1967 Nippon kagaku zassi  
3) PH(電 解 液) ○:[CuHBen】2㍉ △:[NiHBen]2ww 電 解 波:HNO、 一 一N4NO・-NaOH系(0・05F) 図3[MHBcn]2一 の ロ 紙 電 気 泳 動 ③ ㌻ き N l C O / 捕 \ 一 \ / 瑞 O N - C 牟 ' ◎ ね)くc、 ll cH2・ 一閲一CH2 ゆ 2一 図5〔NiHBen]2 お よ び 類 似 錯 体 のd→4吸 収  ...  な お,Na2[CuHBen]の0・005F溶 液(O」N 硝 酸 ナ ト リウ ム 中)のPH値 は9・0± 二 〇・01を 示 し ・平 衡(3)が OO.5停 >0 .4 詮 費 ごo・3 醤0.2 蕊 α1 図4 OlO÷ 《4筆8101214 PH Na2[CuHBen]のd→d吸 収 帯(540mμ)のpH依 存 性 0 bD o 一 4 ウ 一 15 2◎2530 ヲ(cm-1×10-  ... 
doi:10.1246/nikkashi1948.88.3_329 fatcat:lfvgrwp525gb3jle4ptoznvq2u

Duadic Codes over F 2 + u F 2

San Ling, Patrick Solé
2001 Applicable Algebra in Engineering, Communication and Computing  
Duadic codes over F 2 +uF 2 are introduced as abelian codes by their zeros. This is the function field analogue of duadic codes over Z 4 introduced recently by Langevin and Solé.  ...  We classify them in modest lengths and exhibit interesting non-cyclic examples.  ...  Notations and Definitions Let R be the commutative ring F 2 + uF 2 := F 2 [X]/(X 2 ). This ring is endowed with the obvious addition and multiplication, with the property that u 2 = 0.  ... 
doi:10.1007/s002000100079 fatcat:btccoloy4vhgtdoy76vkfcarvi
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