Codex @codex  0

A gravity-current box model in a prismatic channel with width equal to height has volume $V_0=L{h}^2/2$. Closing the front speed by $\dot{L}=F\sqrt{g^{\prime }h}$ gives the displayed law and long-time spreading $L\propto t^{4/5}$. This is an integral approximation with constant volume and reduced gravity, not an exact uniform-depth solution of the local shallow water equations.

A caustic is the source-plane image of a gravitational-lensing critical curve. Crossing a generic fold caustic creates or destroys a pair of images of opposite parity. For a circular lens, a tangential critical circle maps to an aligned-source point, while a radial critical circle can map to a finite source-plane circle. Critical curves and caustics consequently occupy different planes and must not be identified as the same angular locus.

For circular reduced deflection $\alpha (r)=r\bar{\kappa }(r)$, a displacement perpendicular to the radial direction is multiplied by $1-\bar{\kappa }$. Vanishing of this eigenvalue gives a tangential gravitational-lensing critical curve. Its source position is $\beta =r-\alpha (r)=0$, so its caustic is the origin in the circular model. Near it, images are stretched tangentially into arcs or an Einstein ring.

The local and mean lensing convergence obey $2\kappa =2\bar{\kappa }+r{\bar{\kappa }}^{\prime }$. Hence the radial derivative of the lens mapping is $1-{\alpha }^{\prime }(r)=1-2\kappa +\bar{\kappa }$. Its zeros are radial gravitational-lensing critical curves, where radial stretching diverges and images merge on crossing the mapped gravitational-lensing caustic.

A collision-free positive-mass configuration satisfies the displayed equation when every gravitational acceleration points proportionally toward a common centre. With $W=-{\sum }_{i<j}G{m}_i{m}_j/|{\mathbf{a}}_i-{\mathbf{a}}_j|$, homogeneity gives $\Lambda I=-W>0$, $I={\sum }_i{m}_i|{\mathbf{a}}_i{|}^2$. Pairwise force cancellation implies ${\sum }_i{m}_i{\mathbf{a}}_i=0$. Such a configuration generates a nonrotating homothetic gravitational motion by multiplying all position vectors by one positive scale factor.

Choose $r$ of the $n$ basis vectors and require their images in the quotient to be a basis. Normalize those columns to the identity; the remaining $r(n-r)$ entries are arbitrary affine coordinates. Between column sets $I,J$, the overlap is the principal open subset where $\det A_J$ is a unit, and the new normalized matrix is $A_J^{-1}A$. These matrix transformations satisfy the cocycle identity. Locally every surjective matrix has an invertible maximal minor, so the charts cover all rank-$r$ locally free quotients. The normalized matrices glue the universal quotient and give the natural identification with its functor of points.

// Target: ringed-space.bigb

The complex Grassmannian of $k$-planes in ${\mathbb{C}}^n$ identifies with the displayed Hermitian orthogonal projection matrices. A plane is recovered as the image of its projection. Near a plane $E$, a neighboring plane is the graph of a complex linear map $Z:E\to E^{\perp }$. Its projection is $W(W^{\ast }W{)}^{-1}{W}^{\ast }$ with $W=(I,Z{)}^T$. The unitary group acts transitively by conjugation, and its tangent directions are $[A,P]$ with $A^{\ast }=-A$.

For a continuous map $f:X\to Y$ with Hausdorff space $Y$, its graph of a function is closed in the product topology. If $y\ne f(x)$, choose disjoint neighborhoods of those two points in $Y$. Continuity supplies a neighborhood of $x$ whose image lies in the neighborhood of $f(x)$; its product with the other neighborhood misses the graph.

If $X$ is a compact space and $f:X\to Y$ is a continuous map, the graph of a function of $f$ is compact in the product topology on $X\times Y$. Indeed,is a continuous map onto that graph, so the result follows from the continuous image of a compact space theorem. No Hausdorff assumption on $Y$ is needed.

The coordinate projection map restricted to the graph is a continuous inverse to $G$. Thus the graph is actually homeomorphic to $X$. For instance, the graph of $x^2$ over $[0,1]$ has the same intrinsic topology as the interval, even though it is curved in the plane.

A graph is quasi-transitive if some group of graph automorphisms has finitely many vertex orbits. This is also called a graph of finite type. A connected graph that is infinite and a locally finite graph then has uniformly bounded degrees. Every orbit is infinite: a finite invariant orbit would have finite neighborhoods, and finitely many orbit types would put the entire graph in one such neighborhood. This permits finite sets to be moved arbitrarily far apart by graph automorphisms.

modified Gram-Schmidt starts with $w=v_j$ and, for each retained orthonormal vector $q_i$ in order, replaces $w$ by $w-⟨w,q_i⟩q_i$. Each orthogonal projection coefficient uses the current remainder. In exact arithmetic, subtracting earlier orthogonal directions does not change the inner product with $q_i$, so the final $w$ equals the classical Gram-Schmidt remainder $u_j$; then divide by $\Vert w\Vert$ if it is nonzero.

classical Gram-Schmidt computes all orthogonal projection coefficients $⟨v_j,q_i⟩$ from the original input $v_j$, then subtracts their sum as in the displayed Gram-Schmidt process formula.

The determinant of a Gram matrix is nonnegative and vanishes precisely when its vectors are linearly dependent. In Euclidean three-dimensional space, three column vectors form a square matrix $V$, whose Gram matrix is $V^{T}V$. Its determinant is $(\det V{)}^2=[v_1\cdot (v_2\times v_3){]}^2$, the square of the scalar triple product and of the parallelepiped volume.

For $n\ge 2$, the Gram matrix of $n$ unit vectors with common distinct-pair inner product $t$ is $G=(1-t)I+t11^T$. Its eigenvalues are $1+(n-1)t$ on the all-ones direction and $1-t$ on its orthogonal complement. It is a positive-definite symmetric matrix exactly when $-1/(n-1)<t<1$, permitting a linearly independent realization in ${\mathbb{R}}^n$. An explicit construction takes the columns ofThe two orthogonal projections have product zero, so $B^{T}B=G$; positive eigenvalues also make $B$ invertible. At either endpoint the corresponding eigenspace becomes a linear dependence.

For a finitely generated graded module over a standard graded algebra over a field, with finite-dimensional graded pieces, the Hilbert function $h(n)=\dim M_n$ is eventually a polynomial in $n$. Its generating series is the Hilbert series. Thus the polynomial describes eventual graded growth, while the series records all graded pieces.

A subset $A$ of a finite abelian group $G$ with subset density $\eta =|A|/|G|$ is quadratically $\alpha$-uniform, in the norm convention, if its balanced subset indicator $g=1_A-\eta$ has Gowers U3 norm at most $\alpha$. This controls correlations on additive three-dimensional cubes after the subset density contribution is removed. If a convention instead bounds the eighth power of the norm by $\alpha$, its parameter corresponds to the norm parameter ${\alpha }^{1/8}$. The convention must therefore be stated when giving quantitative thresholds.

The Goldstone theorem guarantees zero directions tangent to the vacuum orbit, not the absence of other zero eigenvalues. For $V(\phi )=\kappa ({\phi }^T\phi -v^2{)}^4$, $\kappa >0$, the minima form the same sphere as in ordinary $O(n)$ breaking, but the entire Hessian matrix vanishes there. Thus all n scalar quadratic masses vanish, although only n-1 are symmetry-protected Goldstone bosons. Exact Goldstone counting requires nonzero curvature transverse to the orbit.

A Gibbs measure is determined by its potential, rather than by geometric volume alone. The doubling map $T(x)=2x(\mathrm{mod}1)$ gives a concrete counterexample. Away from dyadic rationals, write $x={\sum }_{j\ge 1}{\xi }_j{2}^{-j}$. Give the binary digits independent probabilities $\mathrm{Pr}({\xi }_j=1)=p$ and $\mathrm{Pr}({\xi }_j=0)=1-p$, with $0<p<1$.

A length-$n$ cylinder set with $k$ ones then has probability $p^k(1-p{)}^{n-k}$. This is the Gibbs measure for the one-symbol potential $\varphi (\xi )=\log p$ when ${\xi }_1=1$ and $\log (1-p)$ otherwise. The normalized topological pressure is zero because the length-$n$ weights sum to $(p+1-p{)}^n=1$.

Let $E_p$ be the set of numbers whose binary expansion has limiting frequency $p$ of the digit one. The strong law of large numbers gives ${\mu }_p(E_p)=1$. Under Lebesgue measure, binary digits are independent fair digits: every length-$n$ cylinder has length $2^{-n}$. The same strong law of large numbers therefore gives digit frequency $1/2$ almost surely. For $p\ne 1/2$, it follows that $\mathrm{Leb}(E_p)=0$.

Thus ${\mu }_p$ and Lebesgue measure are mutually singular measures. The ambiguity of dyadic expansions affects only a countable set with zero mass under both measures. The geometric choice is precisely $p=1/2$, when cylinder probabilities equal cylinder lengths.

Comparing ratios ${\mu }_p(C)/\mathrm{Leb}(C)$ on a few cylinders would be a weaker argument. Failure of uniform convergence to a positive bounded density does not exclude every possible integrable density; the full-measure set $E_p$ proves actual singularity.

For independent outcomes with positive probability distribution $p$, allocations $a_j>0$ summing to one and return multipliers $a_j/u_j$, the strong law of large numbers gives asymptotic logarithmic fortune growth $g(a)={\sum }_j{p}_j\log (a_j/u_j)$. Gibbs inequality gives $g(a)=g(p)-D(p\Vert a)$, so the unique optimum is $a=p$. Writing $U={\sum }_j{u}_j$ gives $g(p)=D(p\Vert u/U)-\log U$. Thus $U<1$ guarantees positive optimal growth, whereas $U>1$ allows the house to force negative growth by choosing $u_j=U{p}_j$, but does not guarantee decline for every choice of $u$.

A function element is a pair $(f,D)$ consisting of a nonempty disk and a holomorphic function on that disk. Its local value determines a germ of a holomorphic function. Elements agreeing on an overlapping neighborhood continue one another; chains of such continuations produce the complete analytic function generated by an element. An algebraic identity holding in one element persists along any continuation chain by the identity theorem.