A curiosity: “supersmooth” varieties
Theory
A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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Search results for “sinh”
132 results
An example of a non-reduced Picard scheme
Theory
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
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Finite flat commutative group schemes embed locally into abelian schemes
Theory
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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Some examples of algebraic groups
Theory
Two pathological phenomena for algebraic groups over general bases — a group degenerating between the multiplicative and additive group across a DVR, and a non-affine identity component — built as centralizers in SL_n.
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The Steinberg Representation
Theory
An introduction to the Steinberg representation of a finite group of Lie type — its alternating-sum construction from parabolic inductions, worked out explicitly for SL_2.
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Change of Basis
Theory
The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
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Quadratic Forms
Theory
Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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CMI 2000, The Millennium Prize Problems
Problem sheet
The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.
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Векторное произведение
Other
Ориентация тройки векторов, определение векторного произведения через длину, перпендикулярность и правую тройку, геометрический смысл (площадь параллелограмма), свойства с доказательствами и полный вывод формулы через определитель.
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ISE TCAD 7.0 on Apple Silicon: Running a 2001 Semiconductor Package on a MacBook Air M1
Notes
How to bring up a twenty-year-old Unix TCAD package under emulation on an ARM Mac, drive it from Jupyter, and parse its data formats in Python
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Топология-2. Лекция 3
Lecture
Сингулярные гомологии — стандартные симплексы и вложения гиперграней, сингулярные симплексы и цепи, дифференциал; нулевые гомологии и компоненты линейной связности, гомологии точки, приведённые гомологии; цепные отображения и отображения гомологий, функториальность, цепное отображение, индуцированное непрерывным отображением; цепные гомотопии и призма; теорема о гомотопической инвариантности с полным доказательством через призменный оператор; подкомплексы, факторкомплексы и гомологии пары.
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