For the defining three-dimensional special linear Lie algebra representation $V$, contraction $(uv)\otimes f\mapsto f(u)v+f(v)u$ maps $(S^{2}V)\otimes V^{\ast }$ onto $V$. The kernel contains the highest-weight vector $e_1^2\otimes e_3^{\ast }$ of Dynkin labels $(2,1)$. The Weyl dimension formula gives its irreducible module dimension $15$, the entire kernel dimension; the Weyl complete reducibility theorem supplies the displayed splitting.

The stationary law of a two-resource linear flow network is normalizable exactly whenThese are the strict offered-load constraints on the two unit-capacity resources. Summing first over the through-route count reduces normalization to two geometric series, which diverge at equality. With positive arrival rates, the nonexplosive continuous-time Markov chain is then positive recurrent.

In the stationary law of a two-resource linear flow network, the local counts are independent geometric variables with parameters $q_i={\rho }_i/(1-{\rho }_0)$. Conditional on local counts $a,b$, the through count has a negative binomial distribution with mean ${\rho }_0(a+b+1)/(1-{\rho }_0)$. With positive loads, it is correlated with each local count, even though those two local counts are independent. Degenerate zero-load cases must be distinguished from the positive-load assertion.

For a root of a root system of a complex semisimple Lie algebra, define $t_{\alpha }$ by $B(t_{\alpha },h)=\alpha (h)$ using the Killing form on its Cartan subalgebra. Then $\alpha (t_{\alpha })\ne 0$. Otherwise vectors $x\in {\mathfrak{g}}_{\alpha }$, $y\in {\mathfrak{g}}_{-\alpha }$ with $B(x,y)=1$ would generate a solvable Lie algebra with $[x,y]=t_{\alpha }$ central. The Lie theorem would make $\mathrm{ad}{t}_{\alpha }$ nilpotent, while the root-space decomposition makes it diagonalizable, forcing $t_{\alpha }$ into the zero center of a Lie algebra.

The real traceless two-by-two matrices form a three-dimensional Lie algebra. A basis adapted to conformal mechanics obeys $\{D,E\}=-E$, $\{D,K\}=K$, $\{E,K\}=2D$.

For random walk on $\{-1,1{\}}^n$ that chooses a coordinate uniformly and flips it with probability $1/2$, the Walsh characters $f_J(x)={\prod }_{j\in J}{x}_j$ form an orthonormal eigenbasis under the uniform distribution on a finite set. Flipping a coordinate in $J$ negates $f_J$, so $P{f}_J=(1-|J|/n)f_J$. Hence $1-k/n$ has multiplicity $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, and the ordinary and absolute spectral gaps both equal $1/n$.

Pulling back the final arrow of a short exact sequence in an abelian category preserves its kernel object and yields another short exact sequence in an abelian category. The new kernel inclusion is specified by its old kernel component and zero component into the new quotient. The pullback stability of epimorphisms in an abelian category ensures the new final arrow remains epic.

Two projective presentations of the same object in an abelian category, with kernels $K,K^{\prime }$ and middle objects $P,P^{\prime }$, satisfy $K\oplus P^{\prime }\cong K^{\prime }\oplus P$. Form their pullback in a category. Its two induced short exact sequences in an abelian category split because each quotient is a projective object in a category. The two resulting biproduct descriptions of the pullback give the isomorphism.

For a variety over $k$, the module of Kähler differentials ${\Omega }_{A/k}$ on each affine chart gives a quasi-coherent sheaf. Compatibility with localization glues these into ${\Omega }_{X/k}^1$. A finite presentation of an algebra gives a finite presentation of a module for its differentials, so this is a coherent sheaf on a variety.

With $m(z)=(z+1)/(1-z)$, the compact H-hull $K_L=m^{-1}(i(0,L])$ is a unit semicircular arc attached at $-1$, with a narrow passage near $1$. As $L\to \infty$, its normalized mapping-out function of a compact H-hull tends to $2$ at each fixed point of the interior bay. Points in that bay tending to $-1$ therefore have displacement approaching three. The order of these two limits is important.

An algebraic variety that is separated over its ground field. Two affine open charts have an affine intersection: their intersection is the inverse image of the closed diagonal inside the product of the two affine charts. This ensures a finite affine cover is acyclic for quasi-coherent sheaves.

Let $k(d)={\prod }_{p\mid d}g(p)/(1-g(p))$ on permitted squarefree integers. The Selberg sieve weights haveThe numerator is the total weight of the distinct integers $ev\le L$ with $e\mid d$ and $(v,d)=1$: multiplicativity gives ${\sum }_{e\mid d}k(e)={\prod }_{p\mid d}(1-g(p){)}^{-1}$. They are a subset of the terms in $J$.

For real coefficients ${\rho }_1=1$ supported on permitted divisors up to $L$, put ${\lambda }_d={\sum }_{[u,v]=d}{\rho }_u{\rho }_v$. Then ${\sum }_{d\mid n}{\lambda }_d=({\sum }_{d\mid n}{\rho }_d{)}^2$ is an upper bound for the indicator function of numbers free of the forbidden prime factors. Its support has $d\le L^2$. If the coefficients vanish on non-squarefree integers and satisfy $|{\rho }_d|\le 1$, then $|{\lambda }_d|\le 3^{\omega (d)}$: each prime factor of $d$ occurs in $u$, in $v$, or in both.

If $H=I_n-11^T/n$ and $S=X^{T}HX/n$, then $\mathrm{rank}S=\mathrm{rank}(HX)\le \min (p,n-1)$. In particular full column rank of raw $X$ does not ensure a nonsingular centered sample covariance matrix. When $S$ is singular, its Gaussian precision profile likelihood has no finite maximizer: increase precision along a nonzero null vector.

For positive integer $n$, $R(e^{i\alpha }z)=e^{in\alpha }R(z)$ and $R(1/z)=1/R(z)$. For $n>1$ its angular Jacobian of a rational map vanishes at the poles and has axial symmetry. Its combined rotational group is the group of rotations preserving an axis; for $n=1$ it enlarges to all domain rotations paired with identical target rotations.

A finite root system in a real inner product space is crystallographic when every Cartan integer $2(\beta ,\alpha )/(\alpha ,\alpha )$ is an integer. The roots of a complex semisimple Lie algebra satisfy this because each number is a weight of its sl2 subalgebra associated with a root. Reducedness is a separate axiom; the root-space reducedness lemma establishes it for these Lie-algebra roots.

The Fitting lemma supplies a nonzero nilpotent endomorphism $f$ of a non-brick indecomposable $X$. Minimize its nonzero rank; nilpotence and minimality give $f^2=0$. Set $I=\mathrm{im}f\subset K=\ker f={⨁}_j{K}_j$. Choose a nonzero component $u:I\to K_j$. The square-zero endomorphism $X\stackrel{f}{\to }I\stackrel{u}{\to }{K}_j\hookrightarrow X$ has rank at least $\dim I$, so $u$ is injective.

If ${\mathrm{Ext}}^1(I,K_j)=0$, the projection $K\to K_j$ extends to $X\to K_j$ by the long exact sequence of Ext groups, giving a module retraction and contradicting indecomposability. Thus this extension group is nonzero. Since path algebras are hereditary, $u$ induces a surjection ${\mathrm{Ext}}^1(K_j,K_j)\twoheadrightarrow {\mathrm{Ext}}^1(I,K_j)$. Hence $K_j$ is a proper indecomposable submodule with self-extensions. Iterate until a brick module is reached; dimensions strictly decrease.

This minimal-rank argument is given in section 2 of William Crawley-Boevey's quiver lectures.

Minimize $F(p)=\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over $P=\{p:|p_i{|}_2\le \lambda \}$ usingThe pointwise Euclidean projection onto a convex set sends a block $r$ to $r/\max (1,|r{|}_2/\lambda )$. The gradient of $F$ has Lipschitz constant $\Vert D{\Vert }^2$, so finite-dimensional projected gradient descent converges for $0<\tau <2/\Vert D{\Vert }^2$. On an unscaled square grid with $N>1$, $0<\tau <1/4$ is safe. Reconstruct $u^k=g-D^{\ast }{p}^k$; unused dual directions may make the minimizing $p$ nonunique while the primal minimizer is unique.

For a binary product in a category, the diagonal ${\Delta }_A:A\to A\times A$ is the unique map whose two projection composites are $1_A$. It is a split monomorphism, since either projection is a left inverse. The diagonal converts equality of two maps into a lifting problem: $⟨u,v⟩$ factors through ${\Delta }_A$ exactly when $u=v$.

For mean-zero $f$, apply the mean ergodic theorem on an ergodic product system to $f\otimes \overline{f}$, paired with $g\otimes \overline{g}$. Its correlation is $|⟨U^{n}f,g⟩{|}^2$. The Cauchy-Schwarz inequality then gives averaged absolute correlation decay, the definition of weak mixing.