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Uniformly defining complexity classes of functions [chapter]

Sven Kosub, Heinz Schmitz, Heribert Vollmer
1998 Lecture Notes in Computer Science  
Each such class is defined in our model by a certain family of functions.  ...  We introduce a general framework for the definition of function classes.  ...  With a number of examples we made clear that virtually all (polynomial time) function classes which are of current topical interest in complexity theory can be defined using our machinery.  ... 
doi:10.1007/bfb0028595 fatcat:uboyazksqbfnho7hevvn64yvoe

Properties of Certain Class of Uniformly Starlike and Convex Functions Defined by Convolution

M. K. Aouf, A. O. Mostafa, A. A. Hussain
2015 International Journal of Open Problems in Complex Analysis  
The aim of this paper is to obtain the modified Hadamard products and properties associated with generalized fractional calculus operators for functions belonging to the class T S γ (f, g; α, β) of β-uniformly  ...  univalent functions defined by convolution.  ...  Open problem The authors suggest to study the properties of the class S λ p (f, g; α, β) when the functions f (z) and g(z) are p-valent functions. Acknowledgment.  ... 
doi:10.12816/0016283 fatcat:7oxc3sgzardxnemoytx4rgiypm

On Learnability, Complexity and Stability [chapter]

Silvia Villa, Lorenzo Rosasco, Tomaso Poggio
2013 Empirical Inference  
We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik.  ...  We survey classic results characterizing learnability in term of suitable notions of complexity, as well as more recent results that establish the connection between learnability and stability of a learning  ...  In this case, a new notion of complexity needs to be defined since the VC-dimension of real valued function classes is not defined.  ... 
doi:10.1007/978-3-642-41136-6_7 dblp:conf/birthday/VillaRP13 fatcat:672cm6k3rnfnbd2azofttt54au

On Learnability, Complexity and Stability [article]

Silvia Villa, Lorenzo Rosasco, Tomaso Poggio
2013 arXiv   pre-print
We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik.  ...  We survey classic results characterizing learnability in term of suitable notions of complexity, as well as more recent results that establish the connection between learnability and stability of a learning  ...  In this case, a new notion of complexity needs to be defined since the VC-dimension of real valued function classes is not defined.  ... 
arXiv:1303.5976v1 fatcat:irmbuvhr4zaafmgz7w2xjpi47m

Convexity and Spirallikeness Conditions for Two New General Integral Operators

H. Özlem Güney, Serap Bulut
2013 Journal of Mathematics  
We define two new general integral operators for certain analytic functions in the unit disc and give some sufficient conditions for these integral operators on some subclasses of analytic functions.  ...  In particular, we get the classes US 0,1,1 , , (1, 0, 1) ≡ UST, UK 0,1,1 , , ( , , 1) ≡ UCV (23) of uniformly starlike and uniformly convex functions, respectively, first defined by Goodman [5] .  ...  (v) For = 1, = 0, = = 1, and = 1, we have the classes US 0,1,1 , , (1, , 1) ≡ UST ( ) , UK 0,1,1 , , ( , , 1) ≡ UCV ( ) (22) of uniformly starlike and uniformly convex functions of order (0 ≤ < 1), respectively  ... 
doi:10.1155/2013/841837 fatcat:l6rwcnzlvfcstebpbmblhfxcwi

Page 8550 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Relations between the class of functions with the above property and classes of uniformly starlike, and uniformly convex functions are found.” 30 FUNCTIONS OF A COMPLEX VARIABLE 2000m:30019 30C45 33C20  ...  starlike and convex functions of complex order.  ... 

Uniform Polylogarithmic Space Completeness

Flavio Ferrarotti, Senén González, Klaus-Dieter Schewe, José María Turull-Torres
2022 Frontiers in Computer Science  
Moreover, we provide evidence that uniformly complete problems can help us to understand the still unclear relationship between complexity classes such as PolyL and polynomial time.  ...  Thus, we propose an alternative notion of completeness inspired by the concept of uniformity studied in circuit complexity theory.  ...  Of course, we also need to reference some related complexity classes. They are formally defined as follows. Definition 2.6.  ... 
doi:10.3389/fcomp.2022.845990 fatcat:b4xvuchu4rgttdvjubzzgnkuxu

Entire functions with prescribed singular values [article]

Luka Boc Thaler
2020 arXiv   pre-print
with both the Speiser class and the Eremenko-Lyubich class of entire functions.  ...  This new class is closed under the composition and its is dense in the space of all non-vanishing entire functions.  ...  This project was supported by the research program P1-0291 from ARRS, Republic of Slovenia.  ... 
arXiv:1908.06026v3 fatcat:bzwaoxtrvrb7nnyjkhxxkof6qy

Page 330 of Automation and Remote Control Vol. 35, Issue 2 [page]

1974 Automation and Remote Control  
We define on the set R an arbitrary function y(t) with values on the set £. Then the superposition of the functions f(£,t) and € = y(t) form a complex function of the form f[¢(t), t].  ...  the complex Fourier series (4) of the superposition f[¢(t), t], formed by the functions f(£, t) and & = y(t), converges to it uniformly, Using the results of Kolmogorov [1] (see also [2-4]), we can indicate  ... 

Page 3560 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
Note that the class UCV(1/2,1/2) corresponds to the class of uniformly convex functions, as introduced by Good- man.  ...  classes of analytic functions are introduced and investigated.” 97f:30019 30045 Parvatham, R. (6-MADR-R; Madras); Premabai, Millicent (6-MADRAA; Madras) On uniformly starlike functions of order (a, f).  ... 

A CHARACTERIZATION OF UNIFORMLY DISTRIBUTED SEQUENCES OF INTEGERS

Miyoko Uchiyama, Saburo Uchiyama
1962 Journal of Faculty of Science Hokkaido University Series I Mathematics  
The main purpose of this note is to present another characterization of uniformly distributed sequences of integers, making use of a kind of integrals defined over the space of integers.  ...  This, as well as the criterion quoted above, will have some analogy with the well-known characterization of uniform distribution $(mod 1)$ of sequences of real numbers (cf. [3; Chap. IV]).  ...  .$ for every measurable set $E$ of positive measure. But this is in substance contained in the proof of Proposition 4. (Of course, a similar result holds for sin $2\pi\alpha x.$ )  ... 
doi:10.14492/hokmj/1530691262 fatcat:56i32mahtjaatd2r7c3q6imiba

Page 5069 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Let Ky(a,f) be the class of normalized close-to-convex functions defined in the open unit disc D by | (22) ' | arg $ g(z) where g € S*(), the class of analytic normalized starlike functions of order f,  ...  The authors give sufficient coefficient conditions for a certain class of hypergeometric functions to be uniformly convex of a positive order.  ... 

Page 5323 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
of uniformly convex functions of order a and uniformly starlike functions of order a, respec- tively, which are the generalisations of the uniformly convex and the uniformly starlike functions, respectively  ...  In this note a general class of functions is introduced which includes as special cases a number of classes of univalent functions studied over the years and defined via the Ruscheweyh derivative.  ... 

Elements of harmonic analysis, 4 [article]

Stephen Semmes
2004 arXiv   pre-print
These informal notes briefly discuss Fourier inversion in terms of Gauss--Weierstrass kernels and summability.  ...  R n which are uniformly bounded and converge to another bounded continuous function h on R n uniformly on compact subsets of R n , then lim j→∞ λ(h j ) = λ(h). (3.6) As a basic class of examples, if f  ...  As another class of examples, if a ∈ R n , then we get a bounded measure on R n , the Dirac mass at a, defined by evaluating a given function at a.  ... 
arXiv:math/0403109v1 fatcat:mxzrveoydbfn7ary7wl3w2wo6a

Page 80 of Annals of Mathematics Vol. 37, Issue 1 [page]

1936 Annals of Mathematics  
The metric of G2 defines an invariant metric on G/H. Under that metric f(a) is uniformly continuous.  ...  We now identify the group G/H with the class of functions f(zay) and define f(a) on G/H by its values on the co-sets of H in G. b.  ... 
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