29,227 research outputs found
Riemann Integral of Functions from R into n-dimensional Real Normed Space
In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21]Miyajima Keiichi - Faculty of Engineering, Ibaraki University, Hitachi, JapanKorniĆowicz Artur - Institute of Informatics, University of BiaĆystok, Sosnowa 64, 15-887 BiaĆystok PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Artur KorniĆowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from R into real normed space. Formalized Mathematics, 19(1):17-22, 2011, doi: 10.2478/v10037-011-0003-8.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan PopioĆek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992
Riemann Integral of Functions R into C
In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].Miyajima Keiichi - Faculty of Engineering, Ibaraki University, Hitachi, JapanKato Takahiro - Faculty of Engineering, Graduate School of Ibaraki University, Hitachi, JapanShidama Yasunari - Shinshu University, Nagano, JapanAgnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121-124, 1993.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Noboru Endou and Artur KorniĆowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.JarosĆaw Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama and Artur KorniĆowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Riemann Integral of Functions from R into Râż
In this article, we define the Riemann Integral of functions from R into Râż, and prove the linearity of this operator. The presented method is based on [21].Miyajima Keiichi - Ibaraki University, Hitachi, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Artur KorniĆowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.JarosĆaw Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan PopioĆek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
More on the Continuity of Real Functions
In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.Narita Keiko - Hirosaki-city, Aomori, JapanKornilowicz Artur - Institute of Informatics, University of BiaĆystok, Sosnowa 64, 15-887 BiaĆystok, PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Artur KorniĆowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan PopioĆek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and PaweĆ Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992
A rigorous real time Feynman Path Integral and Propagator
We will derive a rigorous real time propagator for the Non-relativistic
Quantum Mechanic transition probability amplitude and for the
Non-relativistic wave function. The propagator will be explicitly given in
terms of the time evolution operator. The derivation will be for all
self-adjoint nonvector potential Hamiltonians. For systems with potential that
carries at most a finite number of singularity and discontinuities, we will
show that our propagator can be written in the form of a rigorous real time,
time sliced Feynman path integral via improper Riemann integrals. We will also
derive the Feynman path integral in Nonstandard Analysis Formulation. Finally,
we will compute the propagator for the harmonic oscillator using the
Nonstandard Analysis Feynman path integral formuluation; we will compute the
propagator without using any knowledge of classical properties of the harmonic
oscillator
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
Asymptotic Euler-Maclaurin formula over lattice polytopes
An asymptotic expansion formula of Riemann sums over lattice polytopes is
given. The formula is an asymptotic form of the local Euler-Maclaurin formula
due to Berline-Vergne. The proof given here for Delzant lattice polytopes is
independent of the local Euler-Maclaurin formula. But we use it for general
lattice polytopes. As corollaries, an explicit formula for each term in the
expansion over Delzant polytopes in two dimension and an explicit formula for
the third term of the expansion for Delzant polytopes in arbitrary dimension
are given. Moreover, some uniqueness results are given.Comment: 35 pages. Results in the previous version are generalized to lattice
polytopes. Some further results are added. The title is changed. The
organization is changed to clarify the discussion
Fractional Noether's Theorem with Classical and Riemann-Liouville Derivatives
We prove a Noether type symmetry theorem to fractional problems of the
calculus of variations with classical and Riemann-Liouville derivatives. As
result, we obtain constants of motion (in the classical sense) that are valid
along the mixed classical/fractional Euler-Lagrange extremals. Both Lagrangian
and Hamiltonian versions of the Noether theorem are obtained. Finally, we
extend our Noether's theorem to more general problems of optimal control with
classical and Riemann-Liouville derivatives.Comment: This is a preprint of a paper whose final and definite form will be
published in: 51st IEEE Conference on Decision and Control, December 10-13,
2012, Maui, Hawaii, USA. Article Source/Identifier: PLZ-CDC12.1832.45c07804.
Submitted 08-March-2012; accepted 17-July-2012. arXiv admin note: text
overlap with arXiv:1001.450
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