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152
A Borsuk theorem on homotopy types
 Journal of Formalized Mathematics
, 1991
"... Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retrac ..."
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Cited by 108 (6 self)
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Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.
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 ..."
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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
Bounding boxes for compact sets inE 2
 Journal of Formalized Mathematics
, 1997
"... Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous ..."
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Cited by 36 (2 self)
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Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous real map from X attains minimum. • Every continuous real map from X attains maximum. Finally, for a compact set in E 2 we define its bounding rectangle and introduce a collection of notions associated with the box.
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.
The Taylor expansions
 Formalized Mathematics
"... Summary. A concept of the Maclaurin expansions is defined here. This article contains the definition of the Maclaurin expansion and expansions of exp, sin and cos functions. ..."
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Cited by 18 (1 self)
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Summary. A concept of the Maclaurin expansions is defined here. This article contains the definition of the Maclaurin expansion and expansions of exp, sin and cos functions.
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 ..."
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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. ..."
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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.