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Deformation quantization and Borel's theorem in locally convex spaces

Miroslav Engliš, Jari Taskinen (2007)

Studia Mathematica

It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions...

Derivatives are Borel functions.

A. Járai (1985)

Aequationes mathematicae

Deux exemples de fonctions non mesurables

Zbigniew Grande (1979)

Colloquium Mathematicae

Diagonals of separately continuous functions of $n$ variables with values in strongly $σ$ -metrizable spaces

Olena Karlova, Volodymyr Mykhaylyuk, Oleksandr Sobchuk (2016)

Commentationes Mathematicae Universitatis Carolinae

We prove the result on Baire classification of mappings $f : X \times Y \to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $P P$ -space, $Y$ is a topological space and $Z$ is a strongly $σ$ -metrizable space with additional properties. We show that for any topological space $X$ , special equiconnected space $Z$ and a mapping $g : X \to Z$ of the $(n - 1)$ -th Baire class there exists a strongly separately continuous mapping $f : X^{n} \to Z$ with the diagonal $g$ . For wide classes of spaces...

Diferenciální počet II [Book]

Jarník, Vojtěch (1984)

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^{1}$ -functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{'} (x) = {lim}_{t \to x +} f_{+}^{'} (t)$ and $f_{-}^{'} (x) = {lim}_{t \to x -} f_{-}^{'} (t)$ for each $x$ ). Further, for each modulus $ω$ , we characterize the class $D S C_{ω}$ of functions on $ℝ$ which can be written as $f = g - h$ , where $g$ and $h$ are semiconvex with modulus $C ω$ (for some $C > 0$ ) using a new notion of...

Differentiability of Minima of Non-Differentiable Functionals.

Mariano Giaquinta, Enrico Giusti (1983)

Inventiones mathematicae

Differentiable Functions into Real Normed Spaces

Hiroyuki Okazaki, Noboru Endou, Keiko Narita, Yasunari Shidama (2011)

Formalized Mathematics

In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space [14].

Differentiable Functions on Normed Linear Spaces

Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then...

Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces

Takao Inoué, Noboru Endou, Yasunari Shidama (2010)

Formalized Mathematics

In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

Differentiation through starlike sets in $R^{m}$

Calixto Calderón (1973)

Studia Mathematica

Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

Dariusz Idczak (1998)

Czechoslovak Mathematical Journal

We give characterizations of the distributional derivatives $D^{1, 1}$ , $D^{1, 0}$ , $D^{0, 1}$ of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.

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