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86 results

IMO 2011

2026Problem sheet

52nd International Mathematical Olympiad. Amsterdam, 2011.
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360

Теория вероятностей, Листок 1

2026Problem sheetСтанислав Шапошников

Задачный листок к курсу теории вероятностей. Институт и учебное заведение в исходных публикациях не указаны, поэтому здесь не названы.
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270

Determinant via Permutations

2026TheorySergey

The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.

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170

Properties of Determinants

2026TheorySergey

The basic properties of determinants with brief proofs, and their use in simplifying computations.
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260

Matrix Rank

2026TheorySergey

Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
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190

Change of Basis

2026TheorySergey

The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.
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220

Quadratic Forms

2026TheorySergey

Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.
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160

Векторное произведение

2026OtherSergey

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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90

A curiosity: “supersmooth” varieties

2026TheorySean Cotner

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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120

An example of a non-reduced Picard scheme

2026TheorySean Cotner

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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170

Some examples of algebraic groups

2026TheorySean Cotner

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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180

The Steinberg Representation

2026TheorySean Cotner

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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270

Vector calculus. Green's theorem

2026Problem sheetSergey

A contour integral as a double integral: the hypotheses of Green's theorem, orientation of the contour, area as a contour integral, closing an open arc, an annulus, a singular point inside the contour.
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260