The Experts below are selected from a list of 231 Experts worldwide ranked by ideXlab platform
Patricia Radelet-de Grave - One of the best experts on this subject based on the ideXlab platform.
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The Use of a Particular form of the Parallelogram Law of Forces for the Building of Vaults (1650–1750)
Essays on the History of Mechanics, 2003Co-Authors: Patricia Radelet-de GraveAbstract:One of the principal, if not the only, interests in the history of mechanics of Clifford Truesdell as well as of Edoardo Benvenuto, was to understand how people discovered fundamental notions, Laws and principles. Reading those texts on vaults once more, I realised that the history of the study of stability of a vault under its own weight during the period 1650-1750 was important for the discovery of other fundamental notions, so I decided to enlarge the aim of the present paper in order to account for it. I'll try to answer the question of what the study of that problem brought to the knowledge of fundamental mechanical notions.
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the use of a particular form of the Parallelogram Law of forces for the building of vaults 1650 1750
2003Co-Authors: Patricia Radelet-de GraveAbstract:One of the principal, if not the only, interests in the history of mechanics of Clifford Truesdell as well as of Edoardo Benvenuto, was to understand how people discovered fundamental notions, Laws and principles. Reading those texts on vaults once more, I realised that the history of the study of stability of a vault under its own weight during the period 1650-1750 was important for the discovery of other fundamental notions, so I decided to enlarge the aim of the present paper in order to account for it. I'll try to answer the question of what the study of that problem brought to the knowledge of fundamental mechanical notions.
Charles B. Harris - One of the best experts on this subject based on the ideXlab platform.
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Duality of the weak Parallelogram Laws on Banach spaces
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Raymond Cheng, Charles B. HarrisAbstract:Abstract This paper explores a family of weak Parallelogram Laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak Parallelogram Law if and only if its dual satisfies an upper weak Parallelogram Law, and vice versa. Connections are established between the weak Parallelogram Laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.
Mohammad Sal Moslehian - One of the best experts on this subject based on the ideXlab platform.
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Operator equalities and Characterizations of Orthogonality]{Operator equalities and Characterizations of Orthogonality in Pre-Hilbert $C^*$-Modules
arXiv: Functional Analysis, 2020Co-Authors: Rasoul Eskandari, Mohammad Sal Moslehian, Dan PopoviciAbstract:In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the Parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the Parallelogram Law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff--James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^*$-modules.
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Schatten p-norm inequalities related to an extended operator Parallelogram Law
arXiv: Functional Analysis, 2011Co-Authors: Mohammad Sal Moslehian, Masaru Tominaga, Kichi-suke SaitoAbstract:Let $\mathcal{C}_p$ be the Schatten $p$-class for $p>0$. Generalizations of the Parallelogram Law for the Schatten 2-norms have been given in the following form: If $\mathbf{A}=\{A_1,A_2,...,A_n\}$ and $\mathbf{B}=\{B_1,B_2,...,B_n\}$ are two sets of operators in $\mathcal{C}_2$, then $$\sum_{i,j=1}^n\|A_i-A_j\|_2^2 + \sum_{i,j=1}^n\|B_i-B_j\|_2^2 = 2\sum_{i,j=1}^n\|A_i-B_j\|_2^2 - 2\Norm{\sum_{i=1}^n(A_i-B_i)}_2^2.$$ In this paper, we give generalizations of this as pairs of inequalities for Schatten $p$-norms, which hold for certain values of $p$ and reduce to the equality above for $p=2$. Moreover, we present some related inequalities for three sets of operators.
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an operator extension of the Parallelogram Law and related norm inequalities
Mathematical Inequalities & Applications, 2011Co-Authors: Mohammad Sal MoslehianAbstract:We establish a general operator Parallelogram Law concerning a characterization of inner product spaces, get an operator extension of Bohr’s inequality and present several norm inequalities. More precisely, let A be a C∗ -algebra, T be a locally compact Hausdorff space equipped with a Radon measure μ and let (At)t∈T be a continuous field of operators in A such that the function t → At is norm continuous on T and the function t → ‖At‖ is integrable. If α : T × T → C is a measurable function such that α(t,s)α(s,t) = 1 for all t,s ∈ T , then we show that ∫
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an operator extension of the Parallelogram Law and related norm inequalities
arXiv: Operator Algebras, 2010Co-Authors: Mohammad Sal MoslehianAbstract:We establish a general operator Parallelogram Law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $\mu$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. If $\alpha: T \times T \to \mathbb{C}$ is a measurable function such that $\bar{\alpha(t,s)}\alpha(s,t)=1$ for all $t, s \in T$, then we show that \begin{align*} \int_T\int_T&\left|\alpha(t,s) A_t-\alpha(s,t) A_s\right|^2d\mu(t)d\mu(s)+\int_T\int_T\left|\alpha(t,s) B_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) \nonumber &= 2\int_T\int_T\left|\alpha(t,s) A_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) - 2\left|\int_T(A_t-B_t)d\mu(t)\right|^2\,. \end{align*}
Terry R. Mcconnell - One of the best experts on this subject based on the ideXlab platform.
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Note on the Parallelogram Law
2013Co-Authors: Terry R. McconnellAbstract:We note an identity in Hilbert Spaces that generalizes the usual Parallelogram Law. In geometry, this result is called Stewart’s Theorem.
Raymond Cheng - One of the best experts on this subject based on the ideXlab platform.
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Duality of the weak Parallelogram Laws on Banach spaces
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Raymond Cheng, Charles B. HarrisAbstract:Abstract This paper explores a family of weak Parallelogram Laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak Parallelogram Law if and only if its dual satisfies an upper weak Parallelogram Law, and vice versa. Connections are established between the weak Parallelogram Laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.
