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Mohammad Sal Moslehian - One of the best experts on this subject based on the ideXlab platform.
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more on reverse triangle inequality in Inner Product spaces
arXiv: Functional Analysis, 2005Co-Authors: Arsalan Hojjat Ansari, Mohammad Sal MoslehianAbstract:Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in Inner Product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if $a$ is a unit vector in a real or Complex Inner Product space $(H; )$, $r, s>0, p\in(0,s], D=\{x\in H,\|rx-sa\|\leq p\}, x_1, x_2\in D-\{0\}$ and $ \alpha_{r,s}=\min\{\frac{r^2\|x_k\|^2-p^2+s^2}{2rs\|x_k\|}: 1\leq k\leq 2 \}$, then $$\frac{\|x_1\|\|x_2\|-Re }{(\|x_1\|+\|x_2\|)^2}\leq \alpha_{r,s}.$$
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more on reverse triangle inequality in Inner Product spaces
International Journal of Mathematics and Mathematical Sciences, 2005Co-Authors: Arsalan Hojjat Ansari, Mohammad Sal MoslehianAbstract:Refining some results of Dragomir, several new reverses of the generalized triangle inequality in Inner Product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or Complex Inner Product space (H;〈.,.〉), r,s>0, p∈(0,s], D={x∈H,‖rx−sa‖≤p}, x1,x2∈D−{0}, and αr,s=min{(r2‖xk‖2−p2
Taggart Niall - One of the best experts on this subject based on the ideXlab platform.
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Comparing the orthogonal and unitary functor calculi
2021Co-Authors: Taggart NiallAbstract:The orthogonal and unitary calculi give a method to study functors from the category of real or Complex Inner Product spaces to the category of based topological spaces. We construct functors between the calculi from the Complexification-realification adjunction between real and Complex Inner Product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.Comment: 26 pages, 1 figure. v.2 updated to accepted versio
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Comparing the orthogonal and unitary functor calculi
2020Co-Authors: Taggart NiallAbstract:The orthogonal and unitary calculi give a method to study functors from the category of real or Complex Inner Product spaces to the category of based topological spaces. We construct functors between the calculi from the Complexification-realification adjunction between real and Complex Inner Product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.Comment: 26 pages, 1 figur
Silvestru Sever Dragomir - One of the best experts on this subject based on the ideXlab platform.
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operator refinements of schwarz inequality in Inner Product spaces
Linear & Multilinear Algebra, 2019Co-Authors: Silvestru Sever DragomirAbstract:Some improvements of the celebrated Schwarz inequality in Complex Inner Product spaces in terms of selfadjoint operators 0≤A≤1H are given. Applications for orthonormal families of vectors are also ...
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improving schwarz inequality in Inner Product spaces
Linear & Multilinear Algebra, 2019Co-Authors: Silvestru Sever DragomirAbstract:AbstractSome improvements of the celebrated Schwarz inequality in Complex Inner Product spaces are given. Applications for n-tuples of Complex numbers are provided.
Arsalan Hojjat Ansari - One of the best experts on this subject based on the ideXlab platform.
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more on reverse triangle inequality in Inner Product spaces
arXiv: Functional Analysis, 2005Co-Authors: Arsalan Hojjat Ansari, Mohammad Sal MoslehianAbstract:Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in Inner Product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if $a$ is a unit vector in a real or Complex Inner Product space $(H; )$, $r, s>0, p\in(0,s], D=\{x\in H,\|rx-sa\|\leq p\}, x_1, x_2\in D-\{0\}$ and $ \alpha_{r,s}=\min\{\frac{r^2\|x_k\|^2-p^2+s^2}{2rs\|x_k\|}: 1\leq k\leq 2 \}$, then $$\frac{\|x_1\|\|x_2\|-Re }{(\|x_1\|+\|x_2\|)^2}\leq \alpha_{r,s}.$$
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more on reverse triangle inequality in Inner Product spaces
International Journal of Mathematics and Mathematical Sciences, 2005Co-Authors: Arsalan Hojjat Ansari, Mohammad Sal MoslehianAbstract:Refining some results of Dragomir, several new reverses of the generalized triangle inequality in Inner Product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or Complex Inner Product space (H;〈.,.〉), r,s>0, p∈(0,s], D={x∈H,‖rx−sa‖≤p}, x1,x2∈D−{0}, and αr,s=min{(r2‖xk‖2−p2
Sever S Dragomir - One of the best experts on this subject based on the ideXlab platform.
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some reverses of the generalised triangle inequality in Complex Inner Product spaces
Linear Algebra and its Applications, 2005Co-Authors: Sever S DragomirAbstract:Some reverses for the generalised triangle inequality in Complex Inner Product spaces are given. They improve the classical Diaz-Metcalf inequalities. They are applied to obtain inequalities for Complex numbers.