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The following quote is from Joseph Dauben (in his article Georg Cantor: The Personal Matrix of His Mathematics, Isis, Vol. 69, No. 4, Dec., 1978, pp. 534-550). When he suddenly suffered his first breakdown in May 1884, Cantor had just returned from an apparently successful, quite enjoyable trip to Paris.
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Browse other questions tagged ag.algebraic-geometry algebraic-curves complete-intersection or ask your own question. Featured on Meta Opt-in alpha test for a new Stacks editor
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Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list in my teaching, to motivate general education undergraduates, and early year majors, suggesting to them an idea of what research mathematicians do. Meaning of "not too famous" Examples of problems ...
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1 Answer1. This question is answered mostly by simple algebra and calculus, so here is a sketch of how the second expression is derived. Set y 1 = u and y 2 = 1 − u in the conjugate function expression for the log-exp-sum. Simplify what is obtained after using the conjugate function expression to replace the log-exp-sum in the original ...
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This is a generalisation from $\mathbb{R}^n$ to $\mathbb{C}^n$ of the inequality which was proved in Minimising the squared sum minus the sum of squares inequalities oc.optimization-and-control convex-optimization
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I studied mathematical logic using a book not written in English. I would now like to study it again using a textbook in English. But I hope I can read a text that is similar to the one I used before, so I ask here for recommendations. Any recommendation will be appreciated. The characters of the mathematical logic book I used before is as follows:
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A common definition of the logarithm for (finite dimensional) matrices is via the Dunford-Taylor integral: ln. . ( T) := 1 2 π i ∮ Γ ln. . ( z) ( z − T) − 1 d z, Where Γ is a simple closed smooth curve that contains all the eigenvalues of T in anticlockwise direction. Note that the above is a seemingly natural generalization of ...
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Derivative of log determinant [closed] Ask Question Asked 2 months ago. Active 2 months ago. Viewed 158 times -2 $\begingroup$ Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it's on-topic for MathOverflow ...
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Derivative of log determinant and inverse. Ask Question Asked 8 years, 2 months ago. Active 5 years, 4 months ago. Viewed 15k times 1 1 $\begingroup$ I have a matrix $\Sigma$ with element $(i,j)$ $$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$ The matrix is positive definite and symmetric (it is a covariance matrix). ...
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$\begingroup$ There is an implicit convention to use trigraphs rather than digraphs to denote standard functions ($\exp$, $\cos$, $\tan$, $\log$, $\operatorname{det}$, $\lim$, $\sup$, $\operatorname{adj}$, $\operatorname{vol}$, etc.), except in those rare cases in which there is no obvious pronounceable trigraph available (e.g. $\operatorname{tr}$ for the trace, or $\operatorname{st}$ for the ...
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5 Answers5. In my classes I use it to indicate anxiety. So an equals with ! over it means "we want to show this equality is true". An equals without ! means "I am asserting this is true". I don't know how universal this convention is, though. I do know I'm not the only person to use this convention.
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$\begingroup$ There is an implicit convention to use trigraphs rather than digraphs to denote standard functions ($\exp$, $\cos$, $\tan$, $\log$, $\operatorname{det}$, $\lim$, $\sup$, $\operatorname{adj}$, $\operatorname{vol}$, etc.), except in those rare cases in which there is no obvious pronounceable trigraph available (e.g. $\operatorname{tr}$ for the trace, or $\operatorname{st}$ for the ...
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I am looking for a reference to study logarithm of an invertible triangular matrix. What is a good algorithm? Are there any good reference which studies this topic both theoretically and from an algorithm view point? I am also looking for structure of the logarithm of an upper/lower triangular invertible matrix.
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− log det is a smooth convex function on the PSD cone (this is a standard fact, and follows from the Chandler Davis theorem -- see, e.g., my arXiv preprint on "another proof of the Davis theorem", or see Boyd and Vanderberghe's convex optimization for deep significance of this fact in convex programming), so the set log
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Active 4 years, 5 months ago. Viewed 546 times. 2. Please prove or disprove that, for symmetric matrix A = A T, we have. max x ∈ { ± 1 } n x T A x ≥ Tr ( A) linear-algebra convex-optimization. Share. Improve this question. edited Dec 29 '16 at 10:15.
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Derivative of log determinant [closed] Ask Question Asked 2 months ago. Active 2 months ago. Viewed 158 times -2 $\begingroup$ Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it's on-topic for MathOverflow ...
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3 Answers3. this identity is known as Jacobi's formula. Another proof, using the characteristic polynomial where is the -th characteristic coefficient. Namely, assume that is invertible. Then Thus Where is the adjugate of which satifies Cramer's rule . Since invertible matrices are dense, the formula follows.
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A common definition of the logarithm for (finite dimensional) matrices is via the Dunford-Taylor integral: ln. . ( T) := 1 2 π i ∮ Γ ln. . ( z) ( z − T) − 1 d z, Where Γ is a simple closed smooth curve that contains all the eigenvalues of T in anticlockwise direction. Note that the above is a seemingly natural generalization of ...
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The theory of fields is an undecidable theory, and so one cannot give a computable procedure for deciding whether a given statement in the formal language of fields is true or not in all fields.This is a sense in which this theory is not fully understood. Indeed, the situation is that we have proved that we can have no computably complete understanding of the theory of fields.
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