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Uncertainty: Motivations and Known Results
"... Computing statistics is important. Traditional data processing in science and engineering starts with computing the basic statistical characteristics such as the population mean and population variance: E = 1 n · n� i=1 xi V = 1 n · n� (xi − E) 2. Additional problem. Traditional engineering statisti ..."
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Computing statistics is important. Traditional data processing in science and engineering starts with computing the basic statistical characteristics such as the population mean and population variance: E = 1 n · n� i=1 xi V = 1 n · n� (xi − E) 2. Additional problem. Traditional engineering statistical formulas assume that we know the exact values xi of the corresponding quantity. In practice, these values come either from measurements or from expert estimates. In both case, we get only approximations �xi to the actual (unknown) values xi.
Known results for Permutation Pattern
, 2013
"... A permutation pi is said to contain another permutation σ, in symbols σ pi, if there exists a subsequence of entries of pi that has the same relative order (orderisomorphic) as σ, and in this case σ is said to be a pattern of pi. Otherwise, pi is said to avoid the permutation σ. Example A permutat ..."
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A permutation pi is said to contain another permutation σ, in symbols σ pi, if there exists a subsequence of entries of pi that has the same relative order (orderisomorphic) as σ, and in this case σ is said to be a pattern of pi. Otherwise, pi is said to avoid the permutation σ. Example A permutation contains the pattern 1 2 3 (resp. 3 2 1) if it has an increasing (resp. decreasing) subsequence of length 3. Permutation pi = 3 9 1 8 6 7 4 5 2 (written in oneline notation) contains the pattern σ = 5 1 3 4 2, as can be seen by considering the subsequence 9 1 6 7 2.
1. VARIANTS OF KNOWN RESULTS................ 5
, 2011
"... ABSTRACT: We consider the problem of predicting as well as the best linear combination of d given functions in least squares regression under L ∞ constraints on the linear combination. When the input distribution is known, there already exists an algorithm having an expected excess risk of order d/n ..."
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ABSTRACT: We consider the problem of predicting as well as the best linear combination of d given functions in least squares regression under L ∞ constraints on the linear combination. When the input distribution is known, there already exists an algorithm having an expected excess risk of order d
0.3 Known Results.......................................... 3
, 2003
"... We generalize certain parts of the theory of group rings to the twisted case. Let G be a finite group acting (possibly trivially) on a field L of characteristic coprime to the order of the kernel of this operation. Let K ⊆ L be the fixed field of this operation, let S be a discrete valuation ring wi ..."
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We generalize certain parts of the theory of group rings to the twisted case. Let G be a finite group acting (possibly trivially) on a field L of characteristic coprime to the order of the kernel of this operation. Let K ⊆ L be the fixed field of this operation, let S be a discrete valuation ring with field of fractions K, maximal ideal generated by pi and integral closure T in L. We compute the colength of T oG in a maximal order in LoG. Moreover, if S/piS is finite, we compute the S/piSdimension of the center of T o G/Jac(T o G). If this quotient is split semisimple, this yields a
Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
, 2000
"... In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in conver ..."
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Cited by 628 (41 self)
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In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly
◮ Mathematical problem ◮ Known results ◮ New results
, 2014
"... Model statement: Consider a population of sequences of fixed length N composed of twoletter alphabet, say, {0, 1}, therefore 2N different sequences. The population is subject to two evolutionary forces. First evolutionary force is selection, which is included in the system through the Malthusian fi ..."
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Model statement: Consider a population of sequences of fixed length N composed of twoletter alphabet, say, {0, 1}, therefore 2N different sequences. The population is subject to two evolutionary forces. First evolutionary force is selection, which is included in the system through the Malthusian fitness, defined here for simplicity as m(particular sequence σ) = m(Hσ), where Hσ is the Hamming norm of this sequence, i.e., number of 1s in sequence σ. In this way we do not distinguish between sequences with the same number of 1s and hence reduce the dimensionality of the problem from 2N × 2N to (N + 1) × (N + 1). Hence, we consider at this point only permutation invariant fitness landscapes M = diag(m0,...,mN) or m = (m0,...,mN)
Learning with local and global consistency.
 In NIPS,
, 2003
"... Abstract We consider the general problem of learning from labeled and unlabeled data, which is often called semisupervised learning or transductive inference. A principled approach to semisupervised learning is to design a classifying function which is sufficiently smooth with respect to the intr ..."
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Cited by 673 (21 self)
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to the intrinsic structure collectively revealed by known labeled and unlabeled points. We present a simple algorithm to obtain such a smooth solution. Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data.
The rendering equation
 Computer Graphics
, 1986
"... ABSTRACT. We present an integral equation which generallzes a variety of known rendering algorithms. In the course of discussing a monte carlo solution we also present a new form of variance reduction, called Hierarchical sampling and give a number of elaborations shows that it may be an efficient n ..."
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Cited by 912 (0 self)
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ABSTRACT. We present an integral equation which generallzes a variety of known rendering algorithms. In the course of discussing a monte carlo solution we also present a new form of variance reduction, called Hierarchical sampling and give a number of elaborations shows that it may be an efficient
A fast iterative shrinkagethresholding algorithm with application to . . .
, 2009
"... We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterat ..."
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Cited by 1058 (9 self)
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We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast
A Critical Point For Random Graphs With A Given Degree Sequence
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 the ..."
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Cited by 507 (8 self)
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then almost surely all components in such graphs are small. We can apply these results to G n;p ; G n;M , and other wellknown models of random graphs. There are also applications related to the chromatic number of sparse random graphs.
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