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In an abelian group $G$, let $A,S$ be nonempty finite sets with $|A+S|\le K|A|$. For $q\ge 2$, $0<\epsilon <1$ and $f\in {\ell }^q(G)$, there is $T\subseteq S$ withsuch that every $t\in T-T$ is an Lp almost period of the counting convolution $1_A\ast f$:Equivalently, average over $A$ by ${\mu }_A=1_A/|A|$ and replace the right side by $\epsilon \Vert f{\Vert }_{{\ell }^q}$. This fixes the same counting convention in the convolution and norm.^[ref]

The proof samples translates of $f$. A moment bound makes most samples approximate the average over $A$; small growth of $A+S$ forces many shifts in $S$ to share a good sample. Differences between these shifts give the almost periods. The original method also works in noncommutative groups, where the order of products and side of translation must be specified.^[ref]

An almost complex structure is integrable when it comes from holomorphic coordinate charts. Equivalently, its Nijenhuis tensor vanishes, or the $(0,1)$ vector fields are closed under Lie bracket.

The Nijenhuis tensor of an almost complex manifold isIt measures the failure of the $(0,1)$ tangent distribution to be closed under the Lie bracket.

For a field $k$ and an $(n+1)$-dimensional vector space $V$, the projective space $\mathbb{P}(V)$ consists of its one-dimensional linear subspaces. Choosing a basis writes its points as homogeneous coordinates $[x_0:\cdots :x_n]$, where the coordinate vector is nonzero and simultaneous multiplication by any nonzero scalar leaves the point unchanged.

The affine chart $x_i\ne 0$ has coordinates $x_j/x_i$ for $j\ne i$, identifying it with affine space ${\mathbb{A}}^n$. These $n+1$ charts cover projective space. For the real projective line, $[x:1]$ gives every real affine coordinate, and the remaining point $[1:0]$ records the direction at infinity. On the real projective plane, parallel affine lines meet on the chosen projective line at infinity.

In algebraic geometry, the scheme ${\mathbb{P}}_k^n=\mathrm{Proj}k[x_0,\dots ,x_n]$ also includes nonclosed scheme points. Its $k$-rational points have the preceding description by one-dimensional linear subspaces. This distinction matters over a general field. ^[ref]

A morphism of algebraic varieties is a map given locally by regular functions. A projective-coordinate formula defines a morphism wherever its homogeneous coordinate functions do not vanish simultaneously.

The function field $k(X)$ of an irreducible algebraic variety consists of its rational functions.

An affine algebraic set is the common zero set $V(I)$ of an ideal in a polynomial ring over a field.

A smooth algebraic curve over a field $k$ is an algebraic curve whose structural morphism to $\mathrm{Spec}k$ is smooth. Over a perfect field, this is equivalent to all its local rings being regular local rings. At each closed point, those rings have dimension one and are discrete valuation rings, so a local uniformizer measures orders of zeros and poles.

Over an imperfect field, regular local rings alone do not suffice. Let $k={\mathbb{F}}_p(s)$, let $K=k[u]/(u^p-s)$, and take $C=\mathrm{Spec}K[t]$. The polynomial ring $K[t]$ is regular. After the base change adjoining a $p$th root $v$ of $s$, the equation becomes $(u-v{)}^p=0$, giving a nonzero nilpotent element. The resulting curve is not geometrically reduced and cannot be smooth over $k$.

Thus smoothness tests regularity after extending the ground field, rather than only before that extension. The familiar nonsingular plane-curve Jacobian criterion over an algebraic closure is a geometric version of the same condition.^[ref]

If $B$ is integral over $A$ and an element is integral over $B$, then it is integral over $A$.

A ring generated over $\mathbb{Z}$ by finitely many algebraic integers is a finitely generated $\mathbb{Z}$-module.

The minimal polynomial of an algebraic element $\alpha$ over $K$ is the unique monic polynomial of least positive degree in $K[X]$ that vanishes at $\alpha$. It is irreducible, and its degree equals $[K(\alpha ):K]$.

An affine quadrupole mode has axis perturbations whose fractional sum vanishes. It changes the ellipsoidal shape without changing volume to first order.

The affine breathing mode changes all three principal axes by the same fraction and therefore changes the stellar volume and density.

If the spectral radius of $A$ is below one, $x_{n+1}=A{x}_n+b$ converges to the unique fixed point $(I-A{)}^{-1}b$.

Over an algebraically closed field of characteristic different from two, the affine surfaceis a cylinder over a nodal plane curve. Its singular locus is the line $\{(1,0,z):z\in k\}$. Indeed, the partial derivatives are $2y$, $-(x-1)(3x-1)$ and zero; combining their vanishing with the equation forces $x=1$, $y=0$.

The normalization parametrizationis birational, with inverse $t=y/(x-1)$ away from $x=1$. Thus the surface is birational to ${\mathbb{A}}^2$ despite its whole singular line. The two parameter values $t=1$ and $t=-1$ give the two branches at each node. In characteristic two they coincide, so the same equation is no longer a cylinder over a node; the characteristic restriction concerns the claimed local singularity type.

An irreducible affine plane cubic has at most one singular point. If two existed, restriction of its cubic polynomial to their joining line would have a zero of multiplicity at least two at each point. A polynomial of degree at most three cannot have those four zeros unless it vanishes identically, making the line a component and contradicting irreducibility.

Let $X=V(F)\subset {\mathbb{A}}^N$ be a hypersurface smooth at $p$, and let $H$ be its tangent hyperplane there. If $F{|}_H$ is not identically zero near $p$, the scheme-theoretic intersection $X\cap H$ is a hypersurface in $H$ of expected dimension $N-2$. Its defining equation has zero linear term at $p$, so its tangent dimension is $N-1$ and the intersection scheme is singular at that point.

Both qualifications matter. If $X$ itself is a hyperplane, then $H=X$ and the intersection is smooth of dimension $N-1$, not a proper hyperplane section. Also, singularity of the natural intersection scheme does not force singularity of its reduced support. For $X=V(z-x^2)\subset {\mathbb{A}}^3$ at the origin, $H=V(z)$ gives the doubled line $x^2=0$ in the $(x,y)$-plane; its reduced support $x=0$ is smooth. The Jacobian criterion detects the doubled scheme, rather than discarding its nilpotents.

The holographic dictionary maps observables and parameters between a boundary quantum field theory and a bulk gravitational description. In the ordinary scalar Dirichlet prescription, one near-boundary coefficient acts as the source for a boundary local field operator, while another independent coefficient is related to its expectation value after holographic renormalization.

The AdS scalar mass--dimension relation connects the bulk mass to the boundary scaling dimension. Solving the bulk field equation with a specified boundary source then determines the response and hence boundary correlation functions. Source and response assignments depend on boundary conditions: alternative quantization, logarithmic terms and local counterterms require care. They cannot always be identified just by calling every leading coefficient a source and every subleading coefficient an expectation value.

A single-trace operator contains one trace over matrix-valued gauge group indices. For adjoint matrix fields, $\mathrm{Tr}({\Phi }^2)$ is a simple example, whereas $\mathrm{Tr}({\Phi }^2)\mathrm{Tr}({\Phi }^3)$ is a multi-trace product. The trace makes these expressions invariant under simultaneous conjugation by the gauge group.

In a suitable large-rank holographic theory, normalized single-trace operators correspond at leading order to single-particle bulk excitations, while multi-trace products create multiparticle states. This is an additional dynamical interpretation, rather than the definition of taking one trace. Mixing and finite-rank trace identities qualify the separation beyond the ideal large-rank limit. The distinction organizes correlation function counting and the bulk field content in AdS/CFT correspondence.

A Witten diagram is a position-space perturbative diagram in anti-de Sitter spacetime whose external bulk-to-boundary propagators end at boundary operator insertions.