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Uniformly defining complexity classes of functions
[chapter]
1998
Lecture Notes in Computer Science
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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
2015
International Journal of Open Problems in Complex Analysis
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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]
2013
Empirical Inference
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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]
2013
arXiv
pre-print
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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
2013
Journal of Mathematics
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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
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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
2022
Frontiers in Computer Science
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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]
2020
arXiv
pre-print
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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
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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
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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
1962
Journal of Faculty of Science Hokkaido University Series I Mathematics
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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
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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
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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]
2004
arXiv
pre-print
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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
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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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