85,261 research outputs found
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.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Noboru Endou and Artur Korniłowicz. 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.Jarosław 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 Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama and Artur Korniłowicz. 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
An integral equation method for solving neumann problems on simply and multiply connected regions with smooth boundaries
This research presents several new boundary integral equations for the solution of Laplace’s equation with the Neumann boundary condition on both bounded and unbounded multiply connected regions. The integral equations are uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. The complete discussion of the solvability of the integral equations is also presented. Numerical results obtained show the efficiency of the proposed method when the boundaries of the regions are sufficiently smooth
Inequalities and majorisations for the Riemann-Stieltjes integral on time scales
We prove dynamic inequalities of majorisation type for functions on time
scales. The results are obtained using the notion of Riemann-Stieltjes delta
integral and give a generalization of [App. Math. Let. 22 (2009), no. 3,
416--421] to time scales.Comment: Submitted 30-Apr-2009; revised 15-Feb-2010; accepted 24-Mar-2010; for
publication in Math. Inequal. App
Notes on the Riemann Hypothesis
These notes were written from a series of lectures given in March 2010 at the
Universidad Complutense of Madrid and then in Barcelona for the centennial
anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an
introduction to the Riemann Hypothesis and a panoramic view of the world of
zeta and L-functions. We first review Riemann's foundational article and
discuss the mathematical background of the time and his possible motivations
for making his famous conjecture. We discuss some of the most relevant
developments after Riemann that have contributed to a better understanding of
the conjecture.Comment: 2 sections added, 55 pages, 6 figure
Laughlin states on higher genus Riemann surfaces
Considering quantum Hall states on geometric backgrounds has proved over the
past few years to be a useful tool for uncovering their less evident
properties, such as gravitational and electromagnetic responses, topological
phases and novel geometric adiabatic transport coefficients. One of the
transport coefficients, the central charge associated with the gravitational
anomaly, appears as a Chern number for the adiabatic transport on the moduli
spaces of higher genus Riemann surfaces. This calls for a better understanding
of the QH states on these backgrounds. Here we present a rigorous definition
and give a detailed account of the construction of Laughlin states on Riemann
surfaces of genus . By the first principles construction we prove
that the dimension of the vector space of Laughlin states is at least
for the filling fraction . Then using the path
integral for the 2d bosonic field compactified on a circle, we reproduce the
conjectured -degeneracy as the number of independent holomorphic
blocks. We also discuss the lowest Landau level, integer QH state and its
relation to the bosonization formulas on higher genus Riemann surfaces.Comment: 33 pages, 2 figures, v2: Fay's conventions for the
-differential and the Arakelov metric are adopted, resulting in
slight modifications of the affected formulas. Several other cosmetic changes
and fixed typos, v3: further corrections, version to appear in Commun. Math.
Phy
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