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.

For a homogeneous incompressible flow, strict geostrophic balance means that the Coriolis acceleration is exactly balanced by the pressure gradient, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects material acceleration rather than merely assuming that its size is small. With constant vertical rotation, ${\pi }_z=0$ and taking the curl gives ${\mathbf{u}}_z=0$, the Taylor–Proudman theorem. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.

If a map on a normed vector space satisfies $\Vert T(x)\Vert \le q\Vert x\Vert$ for every $x$, with $0\le q<1$, induction gives $\Vert T^n(x)\Vert \le q^n\Vert x\Vert$. The geometric series then proves absolute convergence of its orbit series whenever the ambient normed vector space is complete. The scalar map $T(x)=\sin (qx)$ satisfies the bound because $|\sin u|\le |u|$.

For $|r|<1$, let $S_N$ contain the first $N$ terms, with $S_0=0$. The remainder after those terms isThis is the exact absolute error. A convenient bound replaces the denominator by $1-|r|$, giving $|R_N|\le |a||r{|}^N/(1-|r|)$ through the triangle inequality. The exact denominator is preferable when the sign or complex phase of $r$ is known.

For $a>0$ and $0<r<1$, the omitted fraction of the entire sum is $R_N/(a/(1-r))=r^N$. Capturing at least $99$ percent therefore means $r^N\le 0.01$. Taking logarithms and applying the ceiling function gives the smallest integer $N$:This gives $7,21,90,$ and $459$ terms for $r=0.5,0.8,0.95,$ and $0.99$, respectively. The difference is a concrete reason to care about the common ratio, even when all four series converge.

An absolute target has a different scale. For $a=1$, $r=0.8$, and required absolute error at most $0.01$, the total sum is $5$ andThe first acceptable term count is $N=28$: the remainder is approximately $0.0096714$, while at $N=27$ it is approximately $0.0120893$. Counting terms and checking the tail avoids an ambiguous instruction to stop when a term merely looks small.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

A polarization is an involutive complex rank-half distribution on which the symplectic form vanishes. Polarized prequantum sections satisfy ${\nabla }_{X}s=0$ for $X\in \mathcal{F}$. A real polarization is tangent to a Lagrangian foliation; a complex Kähler polarization uses the antiholomorphic tangent spaces. Only observables whose Hamiltonian flows preserve the chosen distribution act directly on the polarized space. Singular leaves, holonomy and changes of polarization are genuine quantization difficulties.

Let a uniformly expanding interval map have cylinders ${\Delta }_w$ whose maximum length at level $n$ is at most $C{\lambda }^{-n}$, with $\lambda >1$. Suppose an invariant probability density $h$ is Lipschitz continuous with constant $L$ and satisfies $h\ge h_{\ast }>0$. Its cylinder probabilities are $\nu ([w])={\int }_{{\Delta }_w}hdx$. Define the piecewise constant logarithmic averageThe average of $h$ over ${\Delta }_w$ differs from $h(x)$ by at most $L\mathrm{diam}({\Delta }_w)$. Since the logarithm has derivative at most $1/h_{\ast }$ on this range,Thus the cylinder data recover the logarithm of the density uniformly at a geometric rate.

Conversely, take consistent positive cylinder probabilities and suppose these $g_n$ converge uniformly to a finite function $g$. Put $h=e^g$. If $\Vert g_n-g{\Vert }_{\infty }\le {\delta }_n\to 0$, integrating over every level-$n$ cylinder compares its $h$-mass with its assigned probability within factors $e^{\pm {\delta }_n}$. Subdivide any fixed cylinder into these finer cylinders and sum: consistency gives the same multiplicative bounds for the fixed cylinder. Letting $n\to \infty$ proves ${\int }_{{\Delta }_w}hdx=\nu ([w])$.

Shrinking cylinders determine the Borel probability measure, so the geometric image of the symbolic law is exactly $h(x)dx$. Use a consistent convention at branch endpoints; bounded limiting cylinder densities imply zero mass at individual endpoints. If the symbolic law is shift invariant, the coding relation transfers that invariance to the interval map. This converse is much stronger than inspecting a density ratio along just one itinerary.

A geometric morphism $p:\mathcal{D}\to \mathcal{F}$ is localic when $\mathcal{D}$ is, over $\mathcal{F}$, the topos of sheaves on an internal locale of $\mathcal{F}$. Equivalently, subobjects of the objects $p^{\ast }X$, with $X\in \mathcal{F}$, form a separating family in $\mathcal{D}$.

A geometric morphism $h$ is hyperconnected when its inverse image functor of a geometric morphism is full and faithful and its essential image is closed under subobjects. It then identifies all subobjects of $h^{\ast }X$ with subobjects of $X$, and its image is closed under quotients as well, by descending the kernel-pair relation.

For a geometric morphism $f:\mathcal{E}\to \mathcal{F}$, the displayed internal frame represents $X\mapsto {\mathrm{Sub}}_{\mathcal{E}}(f^{\ast }X)$. The localic part is ${\mathrm{Sh}}_{\mathcal{F}}(L)\to \mathcal{F}$. Its basic open families pull back to the subobjects of $f^{\ast }X$ they represent; gluing these gives a full faithful, subobject-closed inverse image, hence the hyperconnected geometric morphism part. The represented subobject functor recovers the internal frame, proving uniqueness up to equivalence over $\mathcal{F}$.