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80
Real Function Continuity
, 2002
"... this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R ..."
Abstract
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Cited by 47 (8 self)
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this paper. For simplicity, we adopt the following convention: n denotes a natural number, X , X 1 , Z, Z 1 denote sets, s, g, r, p, x 0 , x 1 , x 2 denote real numbers, s 1 denotes a sequence of real numbers, Y denotes a subset of R, and f , f 1 , f 2 denote partial functions from R to R
The limit of a real function at infinity
- Journal of Formalized Mathematics
, 1990
"... Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity. ..."
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Cited by 33 (5 self)
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Summary. We introduced the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinty. We prove basic properties of halflines, sequences divergent to infinity and the limit of function at infinity.
Functional Sequence from a Domain to a Domain
, 1992
"... this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k ..."
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Cited by 25 (0 self)
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this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k
Trigonometric Functions and Existence of Circle Ratio
- Journal of Formalized Mathematics
, 1998
"... this article, we defined sinus and cosine as real part and imaginary part of exponential function on complex, and gave thier series expression either. Then we proved the differentiablity of sin, cos and exponential function of real. At last, we showed the existence of circle ratio, and some formulas ..."
Abstract
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Cited by 13 (1 self)
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this article, we defined sinus and cosine as real part and imaginary part of exponential function on complex, and gave thier series expression either. Then we proved the differentiablity of sin, cos and exponential function of real. At last, we showed the existence of circle ratio, and some formulas of sin, cos. MML Identifier: SINCOS.
Several differentiation formulas of special functions
- Part V. Formalized Mathematics
"... Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions. ..."
Abstract
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Cited by 11 (7 self)
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Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions.
The limit of a real function at a point
- Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the left-side and the right-side limit of a real function at a point. We prove a few properties of the operations on the proper and improper one-side limits and show that Cauchy and Heine characterizations of the one-side limit are equivalent. ..."
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Cited by 7 (1 self)
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Summary. We introduce the left-side and the right-side limit of a real function at a point. We prove a few properties of the operations on the proper and improper one-side limits and show that Cauchy and Heine characterizations of the one-side limit are equivalent.
Inverse Trigonometric Functions arctan and arccot
- FORMALIZED MATHEMATICS VOL. 16, NO. 2, PAGES 147–158, 2008
, 2008
"... This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot. ..."
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Cited by 7 (3 self)
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This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.
Determinant of Some Matrices of Field Elements
, 2006
"... Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matr ..."
Abstract
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Cited by 7 (1 self)
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Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.
Probability on Finite Set and Real-Valued Random Variables
"... Summary. In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems ..."
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Cited by 6 (4 self)
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Summary. In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.
