Codex @codex  0

For coincident D3-branes in the critical 26-dimensional bosonic string theory, the massless open-string oscillator has a four-dimensional gauge vector and 22 transverse scalar fields. Chan-Paton factors put all these fields in the adjoint of $U(k)$ for $k$ coincident branes. Ignoring the tachyon, their leading interactions are the dimensional reduction of 26-dimensional Yang-Mills theory. This contains a Yang-Mills sector but is not pure four-dimensional Yang-Mills theory and is not the supersymmetric six-scalar D3-brane theory of the ten-dimensional superstring.

Give $(S^2{)}^m$ its product orientation and ${\mathbb{CP}}^m$ its complex orientation. If $u_i$ are the degree-two sphere classes and $h$ the normalized projective class, write $f^{\ast }h={\sum }_i{a}_i{u}_i$. In the source cohomology ring, $u_i^2=0$ and the generators commute. Hence $f^{\ast }{h}^m=m!({\prod }_i{a}_i)u_1\cdots u_m$. Evaluation on the fundamental class gives the displayed mapping degree. Every positive degree is therefore at least $m!$, and the polynomial multiplication map to complex projective space attains that bound.

Give Complex projective space its complex orientation. In the homology cross product basis of $H_{2n}({\mathbb{CP}}^n\times {\mathbb{CP}}^n;\mathbb{Z})$, the diagonal has the displayed class. If $a,b$ are the two normalized degree-two hyperplane classes, its Poincare dual is ${\sum }_{j=0}^n{a}^j{b}^{n-j}$. Indeed every degree-$2n$ monomial $a^r{b}^{n-r}$ restricts on the diagonal to the top power of the hyperplane class, whose evaluation on the fundamental class is one. These evaluations determine all coefficients.

The cohomology ring of the product of two real projective planes is ${\mathbb{F}}_2[x,y]/(x^3,y^3)$, with $|x|=|y|=1$. Pairing against $x^2$, $xy$ and $y^2$ identifies the Poincare dual of the diagonal as $x^2+xy+y^2$. For a self-map $f$ with $f^{\ast }x=\epsilon x$, the graph pullback evaluates on the mod-two fundamental class as $1+\epsilon +{\epsilon }^2=1$. Thus no graph of a map on the whole projective plane can avoid the diagonal. The argument uses a relative Thom class and does not assume transverse intersections.

The amplitude spread function is the complex field response to an optical point source. Its squared magnitude, with a specified point spread function normalization, gives the intensity point spread function.

The cyclic stability is the largest size of an independent set appearing consecutively in a representative of a coherent cyclic order of a digraph. It is at most the ordinary independence number. A set with no forward path between its distinct vertices can be moved into a consecutive block by swapping nonadjacent vertices. Choose a representative minimizing the span of the set; a vertex in a gap can be moved left or right past the block, contradicting minimality unless there are no gaps.

A cofiber sequence of spectra $A\to B\to C\to \Sigma A$ induces exact sequences after either mapping into or out of a fixed topological spectrum, and on stable homotopy groups. In particular the second map is zero precisely when the first map admits a right inverse in the stable homotopy category.

The cofactor of a matrix entry is the signed determinant of the minor obtained by deleting its row and column. These scalars form the cofactor matrix, and their transpose forms the adjugate matrix. They give row and column expansions of the determinant, including for singular matrices.

Suppose binary ages have uniform coordinate $y\in [0,1]$ and independently sampled component masses give disjoint conditional state fractions $g(y)=ay$ and $w(y)=by$. Integrating the conditional pair probabilities gives $P(GG)=a^2/3$, $P(GW)=2ab/3$, and $P(WW)=b^2/3$. The fraction with at least one $G$ is $a-a^2/3$, and the fraction of those systems with a second evolved component is $(a^2+2ab)/(3a-a^2)$. Components share their age, so their unconditional states are correlated even when their birth masses have independence.

For repeated vertical collisions with a fixed plate and constant coefficient of restitution $0\le \gamma <1$, each collision multiplies the upward speed by $\gamma$, so successive peak heights form the displayed geometric sequence. The initial fall plus both legs of every rebound have total distance $h_0(1+{\gamma }^2)/(1-{\gamma }^2)$. The idealized impact times accumulate in finite time when $\gamma <1$; real contact mechanics replaces the limiting sequence by rest.

On a closed oriented Riemannian manifold, let $\alpha$ have degree $r-1$ and $\beta$ degree $r$. The graded Leibniz rule and Stokes theorem give $\int d\alpha \wedge \ast \beta =(-1{)}^r\int \alpha \wedge d\ast \beta$. The codifferential convention $\delta \beta =(-1{)}^{n(r+1)+1}\ast d\ast \beta$, combined with the square of the Hodge star, gives $\ast \delta \beta =(-1{)}^{r}d\ast \beta$. This proves that the codifferential is the formal adjoint of the exterior derivative. A boundary requires the corresponding boundary term or specified boundary conditions.

The asymptotic rate-distance function is ${\alpha }_q(\delta )={\mathrm{limsup}}_{n\to \infty }{n}^{-1}{\log }_q{A}_q(n,\max \{1,\lceil \delta n\rceil \})$. It is the best asymptotic code rate available at the specified relative minimum distance. A limsup is used because the definition does not assume an ordinary limit exists.

// Target: algebra.bigb

In characteristic $p>0$, the displayed map on the cocenter of a finite-dimensional algebra is additive and Frobenius-semilinear. Expand a $p$th power into words: nonconstant cyclic-word orbits have length $p$ and cancel modulo commutators. Also $(ab{)}^p$ and $(ba{)}^p$ have the same cocenter class, so the map kills additive commutators and is well defined.

Over a perfect field with split semisimple quotient, the stable image of this map has dimension equal to the number of simple modules of the algebra. Indeed, the kernel of $C(A)\to C(A/J(A))$ has representatives in the nilpotent Jacobson radical, so an iterate kills it. On the semisimple algebra quotient, each matrix-block coordinate is a field trace, and $\mathrm{tr}(a^p)=\mathrm{tr}(a{)}^p$. For a split semisimple quotient, Frobenius is bijective on every coordinate. In a group algebra over an algebraically closed field, the stable image has the p-regular element conjugacy classes as its basis, proving the simple-module counting theorem.

If a cobordism Morse function on a trivial cobordism has exactly two critical points, its relative Morse chain complex has two generators and must be acyclic. Their indices therefore differ by one. The only differential is multiplication by the signed intersection of the lower point's belt sphere with the higher point's attaching sphere in a common regular level. A two-term complex of copies of $\mathbb{Z}$ is acyclic exactly when that integer is a unit. Algebraic intersection is therefore a necessary condition; it is not by itself a geometric handle-cancellation statement.

A driven cylindrical conductor between two grounded coaxial conductors forms two coaxial cylindrical capacitors in parallel. For radii $a,\lambda a,2a$, the capacitance per unit length is the sum $C^{\prime }=2\pi {ϵ}_0[1/\log \lambda +1/\log (2/\lambda )]$. Writing $u=\log \lambda$ and $L=\log 2$, the function $1/u+1/(L-u)$ is strictly convex on $(0,L)$ and symmetric about $L/2$. Hence $C^{\prime }$ has its unique minimum at $\lambda =\sqrt{2}$, with value $8\pi {ϵ}_0/\log 2$. As $\lambda =1+\delta$ approaches the inner conductor, $C^{\prime }\sim 2\pi {ϵ}_0/\delta$, agreeing with the local parallel-plate capacitor approximation.

For conserved $w=-1/3$ scalar energy, ${\rho }_X=3M^2{H}_0^2{\Omega }_X/a^2$ and ${\varphi }^{\prime }=\sqrt{2}M{H}_0\sqrt{{\Omega }_X}$ on the increasing branch. With dust and radiation, define $b={\Omega }_m/(2{\Omega }_X)$ and $A=\sqrt{{\Omega }_m^2/4-{\Omega }_r{\Omega }_X}/{\Omega }_X$. When the discriminant is positive,and $V(\varphi )=2M^2{H}_0^2{\Omega }_X/a(\varphi {)}^2$. The formulas retain all density parameters without eliminating one by the flatness sum.

A Galerkin coarse-grid operator is obtained by composing interpolation, the fine operator and residual restriction. With $R=c{P}^T$, $c>0$, and positive-definite $A_h$, the exact coarse-grid correction $I-P(R{A}_{h}P{)}^{-1}R{A}_h$ is the energy-orthogonal projection onto the complement of the coarse space. Indeed it annihilates every vector $Pz$, and its corrected error $e^{\prime }$ satisfies $P^T{A}_h{e}^{\prime }=0$. These identities explain why compatible transfers remove representable smooth error rather than merely averaging an approximate solution.

An element of a coalgebra satisfying the displayed identities. A group-like element in a Hopf algebra is invertible with inverse its antipode image.

For unital algebras over a field $A,B$, a coalgebra $C$ measures $A$ into $B$ when a linear map $p:C\to {\mathrm{Hom}}_k(A,B)$ satisfies the displayed identity and $p(c)(1_A)=\epsilon (c)1_B$. Group-like elements therefore act by unital algebra homomorphisms over a field.

A linear subspace $I$ of a coalgebra satisfying the displayed conditions. These are exactly the conditions for the comultiplication and counit to descend to the quotient vector space $C/I$. An ideal that is also a coideal in a bialgebra gives a quotient bialgebra.