The chain complex with $R$ in a single degree $n$ and zero differential. A map from it into a complex chooses a cycle in degree $n$.

For $n\ge 1$, the chain complex with copies of $R$ in degrees $n,n-1$ and identity differential. It is acyclic; a map from it chooses an arbitrary degree-$n$ element and its boundary.

The tensor product of homologically graded chain complexes has differential $D(x_p\otimes y)=d{x}_p\otimes y+(-1{)}^p{x}_p\otimes dy$. The sign makes the two mixed terms in $D^2$ cancel. Total degree is the sum of the two degrees.

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

The reversed dual module of a chain complex becomes a chain complex with the unsigned precomposition differential above. A functional on $C_{-i}$ is sent to a functional on $C_{1-i}$, lowering the reversed degree by one.

Let $E_1,E_2,E_3$ be independent random variables with the standard normal distribution and set $Z_1=E_1$, $Z_2=Z_1+E_2$, $Z_3=Z_2-Z_1+E_3$. The complete acyclic graph with edges $1\to 2$, $1\to 3$, $2\to 3$ is causally minimal: deleting each edge imposes a false marginal or conditional independence. Yet $Z_1\perp Z_3$ because $Z_3=E_2+E_3$, so faithfulness of a directed acyclic graph fails. Nonzero structural coefficients alone do not ensure faithfulness.

For a prime divisor $p$ of the order of a finite group, rotate the tuples $(g_1,\dots ,g_p)$ whose product is the identity. Rotation preserves that constraint even for a nonabelian group, by conjugating the product. Every nonfixed orbit has $p$ elements, while the fixed tuples correspond to $g^p=1$. There are $|G{|}^{p-1}$ tuples, so the number of fixed tuples is divisible by $p$. Since the identity is one, another solution exists and has order exactly $p$. This supplies an elementary proof of Cauchy theorem for groups.

The set $D^{+}(\Sigma )$ of events for which every past-inextendible causal curve meets $\Sigma$. This causal characterization permits an initial sign condition on a null geodesic congruence to propagate throughout the regular region by integrating a focusing inequality from $\Sigma$.

A cataclysmic variable whose white dwarf magnetic field does not dominate the large-scale accretion flow. It normally supports an accretion disk fed through Roche-lobe overflow. The term does not rule out a magnetized donor star or magnetic braking of a binary star.

A cataclysmic variable with recurrent disk outbursts associated with the hydrogen-ionization disk instability. Enhanced accretion powers the bright phase; the event does not require thermonuclear runaway on the white dwarf.

A cataclysmic variable whose persistent luminous disk resembles a dwarf nova in outburst. A sufficiently high transfer rate can keep its accretion disk on the hot stable branch of the hydrogen-ionization disk instability, avoiding ordinary dwarf-nova cycles.

For an infinite cardinal number, $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The singular cardinals hypothesis predicts its value when $\kappa$ is a singular cardinal and $2^{\mathrm{cf}(\kappa )}<\kappa$. For a regular cardinal, its value is $2^{\kappa }$.

In an $N$-dimensional real vector space, each member of a conic hull has a representation using at most $N$ generators. To prove it, take a finite positive-coefficient representation with more than $N$ terms. Its generators are linearly dependent, say ${\sum }_j{\mu }_j{s}_j=0$, with some ${\mu }_j>0$ after reversing the relation if necessary. Subtract $t{\mu }_j$ from each coefficient, taking $t={\min }_{{\mu }_j>0}{\lambda }_j/{\mu }_j$. All coefficients stay nonnegative and at least one vanishes. Iterate. The bound differs from the $N+1$ bound for a convex hull because the coefficient sum is unrestricted.

Finite canonical terms built from zero, ordinal sums and powers of omega describe exactly the ordinals below epsilon zero. Syntactic validity and comparison are decidable by recursion through the exponent subterms. Enumerating valid term codes transfers their order to a computable well-order on all natural numbers.

The reduced canonical commutation relation for a transverse field uses a transverse delta function instead of an unconstrained componentwise delta. Its Fourier transform is the spatial transverse projector. It can be derived from the two-polarization oscillator expansion or from the Dirac bracket after gauge fixing.

For a real scalar field in one spatial dimension with metric signature $(+,-)$ and potential $V$, the stress-energy conservation law is ${\partial }_t{T}^{01}+{\partial }_x{T}^{11}=0$. The force on the field to the left of a fixed cut is $-T^{11}$ at the cut when the far-left boundary flux vanishes. Finite-energy field configurations also permit this conclusion by spatial cutoffs, without assuming pointwise decay of all derivatives.

For the real scalar field expansion $\varphi =\int d^{3}p[a_{\mathbf{p}}{e}^{-ipx}+a_{\mathbf{p}}^{†}{e}^{ipx}]/((2\pi {)}^3\sqrt{2E_{\mathbf{p}}})$, the equal-time fields extract $a_{\mathbf{p}}=e^{iEt}\int d^{3}x{e}^{-i\mathbf{p}\cdot \mathbf{x}}[\sqrt{E/2}\varphi +i\pi /\sqrt{2E}]$. The two mixed canonical commutation relation terms yield $[a_{\mathbf{p}},a_{\mathbf{q}}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$, while the other oscillator commutators vanish. This verifies the normalization of the mode expansion directly from the canonical fields.

For a massive real scalar with one internal degree of freedom with $x=m/T_{\varphi }\gg 1$, fixed separately conserved visible and dark entropy ratio $\xi$, and relativistic visible bath with entropy degrees of freedom $g_{\ast s}^{\mathrm{SM}}$,Indeed, equate $s_{\mathrm{SM}}=(2{\pi }^2/45)g_{\ast s}^{\mathrm{SM}}{T}_{\gamma }^3$ to $\xi s_{\varphi }$ and use nonrelativistic entropy at zero chemical potential. For $g$ internal states the right side gains $g^{1/3}$. The nonrelativistic species cannot be assigned a constant relativistic entropy count.

The nonrelativistic entropy at zero chemical potential and $s_{\varphi }{a}^3=\mathrm{constant}$ give $x+\frac{1}{2}\log x=3\log a+\mathrm{constant}$. Using this leading nonrelativistic approximation, differentiation yieldsThis is much smaller in magnitude than the relativistic-bath cooling rate $-1$ when $x\gg 1$ and its entropy degrees of freedom are fixed. Thus cannibal phase of a decoupled sector cooling is logarithmic. Once number-changing reactions freeze out, a number-conserving nonrelativistic gas instead cools as $a^{-2}$ during adiabatic expansion.

An additive subtraction can produce a source in the renormalization-group equation. If $G_R=G_B-a\log ({\Lambda }_0/\Lambda )$ and the bare correlator is invariant under $\mathcal{D}=\Lambda {\partial }_{\Lambda }+\beta (g){\partial }_g$, then $\mathcal{D}{G}_R=a$. A vanishing operator anomalous dimension does not remove this source.