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Relations and their basic properties
 Journal of Formalized Mathematics
, 1989
"... Summary. We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are ..."
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Cited by 1060 (6 self)
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, ∪, ∩ and \ are redefined. Basic facts about the above mentioned notions are presented.
Functions and their basic properties
 JOURNAL OF FORMALIZED MATHEMATICS
, 2003
"... The definitions of the mode Function and the graph of a function are introduced. The graph of a function is defined to be identical with the function. The following concepts are also defined: the domain of a function, the range of a function, the identity function, the composition of functions, the ..."
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Cited by 1336 (32 self)
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, the 11 function, the inverse function, the restriction of a function, the image and the inverse image. Certain basic facts about functions and the notions defined in the article are proved.
Basic Properties of Real Numbers
 Journal of Formalized Mathematics
, 1989
"... this paper. A real number is an element of R ..."
Basic properties of rational numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. A definition of rational numbers and some basic properties of them. Operations of addition, subtraction, multiplication are redefined for rational numbers. Functors numerator (num p) and denominator (den p) (p is rational) are defined and some properties of them are presented. Density of ra ..."
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Cited by 43 (1 self)
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Summary. A definition of rational numbers and some basic properties of them. Operations of addition, subtraction, multiplication are redefined for rational numbers. Functors numerator (num p) and denominator (den p) (p is rational) are defined and some properties of them are presented. Density
The ContextTree Weighting Method: Basic Properties
 IEEE Trans. Inform. Theory
, 1995
"... We describe a sequential universal data compression procedure for binary tree sources that performs the "double mixture." Using a context tree, this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources, and achieves a desira ..."
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Cited by 230 (18 self)
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We describe a sequential universal data compression procedure for binary tree sources that performs the "double mixture." Using a context tree, this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources, and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters. Computational and storage complexity of the proposed procedure are both linear in the source sequence length. We derive a natural upper bound on the cumulative redundancy of our method for individual sequences. The three terms in this bound can be identified as coding, parameter, and model redundancy. The bound holds for all source sequence lengths, not only for asymptotically large lengths. The analysis that leads to this bound is based on standard techniques and turns out to be extremely simple. Our upper bound on the redundancy shows that the proposed contexttree weighting procedure is optimal in the sense that it achieves the Rissanen (1984) lower bound.
Basic Properties of Periodic Functions
"... Summary. In this article we present definitions, basic properties and some examples of periodic functions according to [4]. ..."
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Summary. In this article we present definitions, basic properties and some examples of periodic functions according to [4].
Some Basic Properties
, 2000
"... This is a continuation of the paper (quantph/0009012). In this letter we extend coherent operators and study some basic properties (the disentangling formula, resolution of unity and commutation relation). We also propose some problems. ..."
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This is a continuation of the paper (quantph/0009012). In this letter we extend coherent operators and study some basic properties (the disentangling formula, resolution of unity and commutation relation). We also propose some problems.
Basic Properties of Even and Odd Functions
"... Summary. In this article we present definitions, basic properties and some ..."
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Summary. In this article we present definitions, basic properties and some
Basic Properties of Power Series
"... Abstract These notes provide a quick introduction (with proofs) to the basic properties of power series, including the exponential function and the fact that power series can be differentiated term by term. It is assumed that the reader is familiar with the following facts and concepts from analysi ..."
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Abstract These notes provide a quick introduction (with proofs) to the basic properties of power series, including the exponential function and the fact that power series can be differentiated term by term. It is assumed that the reader is familiar with the following facts and concepts from
Results 1  10
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18,950