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For $n\ge 2$, the rational homotopy group of a based topological space $X$ is the vector spaceHere ${\pi }_n(X)$ is an abelian group, so this tensor product is defined. For a finitely generated abelian group ${\pi }_n(X)\cong {\mathbb{Z}}^r\oplus T$ with $T$ finite, the result is ${\mathbb{Q}}^r$: finite-order classes disappear and the free generators acquire rational coefficients. More generally every torsion element maps to zero, since $ma=0$ implies $a\otimes q=(ma)\otimes (q/m)=0$ for $m\ne 0$.

For example, ${\pi }_3(S^2)\cong \mathbb{Z}$ gives ${\pi }_3(S^2)\otimes \mathbb{Q}\cong \mathbb{Q}$, whereas ${\pi }_4(S^3)\cong \mathbb{Z}/2$ becomes zero. The general tensor product definition also handles groups that are not finitely generated: for an Eilenberg–MacLane space $K(\mathbb{Q},n)$, its degree-$n$ rational homotopy group is still $\mathbb{Q}$.

For a simply connected CW complex, these vector spaces are the homotopy groups of its rationalization of a topological space. An arbitrary nonabelian fundamental group cannot be tensored with $\mathbb{Q}$ by this abelian group formula; the nilpotent-group localization used more generally in rational homotopy theory is a separate construction.

For a based space $X$, adjunction gives

If $A\subseteq X$ is contractible and $(X,A)$ has the homotopy extension property, the quotient mapis a homotopy equivalence. Extend a contraction of $A$ to $X$; its endpoint is constant on $A$ and therefore factors through $q$ to supply a homotopy inverse.

The Sidorenko conjecture is the assertion that every finite bipartite graph $H$ satisfiesfor every nonempty finite simple graph $G$. Here $t(H,G)$ is the homomorphism density: the probability that a uniformly random map from $V(H)$ to $V(G)$ sends every edge to an edge. The host graph $G$ need not be bipartite. In the equivalent graphon formulation, the edge constraints are integrated against one symmetric measurable kernel.^[ref]

For the three-vertex graph path, writing $n=|V(G)|$ givesby the Cauchy-Schwarz inequality. The example explains the comparison: correlations between adjacent edge constraints can increase the homomorphism count above the independent-edge heuristic. Bipartiteness is necessary for an assertion covering all hosts, since a nonbipartite $H$ has no homomorphism to a bipartite graph of positive edge density.

Let $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ be the homogeneous polynomial representation of SU2. An element $g$ of SU(2) has eigenvalues $z,z^{-1}$ with $|z|=1$. Its character isTo see this, choose an eigenbasis in the defining two-dimensional representation. The degree-$n$ monomials form a basis of the symmetric power; their eigenvalues are the products of $n-k$ copies of $z$ and $k$ copies of $z^{-1}$. Adding them computes the trace.

For $z\ne \pm 1$, the geometric series givesWith $z=e^{i\theta }$ this becomes $\sin ((n+1)\theta )/\sin \theta$. The original finite sum also handles the apparent singularities: ${\chi }_{V_n}(I)=n+1$ and ${\chi }_{V_n}(-I)=(-1{)}^n(n+1)$.

Holographic D.C. conductivity describes the zero-frequency electric response of a boundary system using a bulk gauge field. In a clean normal state at nonzero charge density, the electric current generally overlaps with conserved momentum. A constant electric field then continuously accelerates that momentum, producing a zero-frequency delta function rather than an ordinary finite D.C. response.

Introducing relaxation of momentum can broaden this contribution and yield finite conductivity when no other protected dissipationless channel remains. That qualification matters: a superfluid can retain a delta function even when translations are broken. One must specify the phase, conserved quantities and frequency limit before interpreting a bulk calculation as a measured D.C. conductivity.

For a bulk gauge field, the leading boundary value of its time component is the dual chemical potential, while the normalizable radial coefficient determines the charge density.

Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form $d{s}^2=L^2[d{Z}^2/Z^2+g_{ij}(Z,x)d{x}^{i}d{x}^j/Z^2]$.

The Baumslag-Solitar group is

Suppose an urn starts with $m$ green balls and $m+2$ red balls. Drawing green removes one green ball, while drawing red adds one green and one red ball. The difference remains two, and $h(g)=1/(g+1)$ is harmonic for the green-ball chain. Stopping $h$ on hitting $0$ or $N$ and then letting $N$ tend to infinity gives termination Probability

A bounded operator $T$ on a separable Hilbert space is trace class when $|T{|}^{1/2}$ is Hilbert--Schmidt. Its trace norm is

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

In the single-doublet Standard Model this physical excitation is a spin zero excitation of a scalar field. Its mass depends on the Higgs potential, so the theory's gauge symmetry alone does not specify where to search. The Higgs boson discovery in 2012 tested the existence of a new particle resonance; establishing its detailed identity requires testing its interactions and spin as well.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

The phrase “absorbing” describes a rearrangement of degrees of freedom, not a collision in which a gauge boson swallows another particle. The gauge symmetry remains a redundancy of the description. A chosen vacuum expectation value leaves a smaller subgroup manifest, while gauge-invariant observables describe the physical masses.

A one-field calculation isolates the mechanism. In an Abelian Higgs model, write a complex scalar field locally as $\Phi =(v+H)e^{i\theta }/\sqrt{2}$, use $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$, and define $B_{\mu }=A_{\mu }-{\partial }_{\mu }\theta /e$. ThenThe second term contains a Proca field mass $m_B=ev$ and its interactions with the radial Higgs boson. A local gauge transformation removes $\theta$ from the scalar field but leaves its information in the longitudinal vector polarization. The electroweak symmetry breaking calculation applies this idea to three broken directions.

If a well-order is the union of suborders of types $\xi$ and $\eta$, its type is at most $\xi \#\eta$.

For a basis $(v_i)$, the matrix $A$ of a Hermitian form has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^{T}A\overline{y}$.

A function $d\in {\omega }^{\omega }$ dominates $h\in {\omega }^{\omega }$ when $d(n)>h(n)$ for all but finitely many $n$. A dominating real over a ground model dominates every such function belonging to that model.

With transverse and axial moments $I,J$, the Hamiltonian is

For $e_k=x^{\ast }-x_k$,If $A$ is diagonal with entries ${\lambda }_i$, a coordinate permutation turns this matrix into blocks

For the quadratic objective $F(x)=x^{T}Ax/2-b^{T}x$ with a real symmetric matrix $A$, the Heavy-ball method takesMultiplying by $-A$ and adding $b$ givesUnder the usual zero-initial-momentum choice $x_{-1}=x_0$, one has $r_{-1}=r_0$, so induction expresses every residual as a polynomial in $A$ applied to $r_0$. It belongs to the associated Krylov subspace. With two arbitrary starting iterates, the general expression instead combines polynomials applied to both starting residuals; a single-residual polynomial claim then needs an additional relation between them.

For $D>0$ and $t>0$, the Euclidean heat kernel for $u_t=D\Delta u$ on ${\mathbb{R}}^d$ isIt is the fundamental solution from a unit Dirac delta distribution. Its integral is one, its coordinate variances are $2Dt$, and $K_t\ast K_s=K_{t+s}$. For suitable initial data, $u(\cdot ,t)=K_t\ast u_0$ solves the heat equation. The approximate identity property recovers the initial data in Lp norm for $1\le p<\infty$, and locally uniformly for bounded continuous data.^[ref]

The formula quantifies spreading: the width grows like $\sqrt{Dt}$ while the peak falls like $t^{-d/2}$. Its strict positivity yields infinite propagation speed for nonzero nonnegative initial data. On an interval, boundary conditions require a different kernel, such as the Dirichlet heat kernel on an interval or Neumann heat kernel on an interval; their mass behavior records whether the boundary absorbs or reflects. The Gaussian heat kernel for standard Brownian motion uses $D=1/2$, which explains its different factor of two.