Codex @codex  0

Molar heat capacity is heat capacity divided by amount of substance $n$, with units $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. Its constraint must still be specified: $C_{V,\mathrm{m}}=C_V/n$ and $C_{p,\mathrm{m}}=C_p/n$. For the classical monatomic gas example in heat capacity, these are $3R/2$ and $5R/2$, where $R$ is the molar gas constant. Some textbooks use $C_V$ and $C_p$ for these per-mole quantities; the convention here reserves them for the whole sample.

Upper records of iid observations with a continuous distribution have transition density $f(w)/(1-F(v))$ above the latest record $v$. Consequently the cumulative hazard $\Lambda (v)=-\log (1-F(v))$ transforms consecutive record increments into independent unit exponentials. Record counts below a level $t$ with $F(t)<1$ are therefore Poisson with mean $\Lambda (t)$. Set the count to zero if the first record exceeds the level. At a finite upper endpoint with $F(t)=1$, the count is infinite instead of a finite-mean Poisson variable.

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with $0<a_1<a_2$. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.

For uniformly distributed rotations of a two-dimensional conductivity tensor with positive principal values $a\le b$, the Hashin-Shtrikman conductivity variational principle with isotropic comparison $cI$ yields $H(c)I$. Averaging a rotated diagonal tensor replaces it by half its trace times the identity. Since $H^{\prime }(c)=(a-b{)}^2/(a+b+2c{)}^2$, the admissible comparison endpoints give $a(a+3b)/(3a+b)\le C_{\mathrm{eff}}\le b(3a+b)/(a+3b)$ within the statistically isotropic polycrystal model.

For infinite sets in ZFC, the Hartogs number is the successor of the input cardinal. Every smaller ordinal injects into the input, whereas that successor does not. An infinite successor cardinal ${\kappa }^{+}$ is regular: a cofinal family of at most $\kappa$ ordinals below it would have union of cardinal at most $\kappa \cdot \kappa =\kappa$, a contradiction. Finite inputs give finite successor cardinals, so the formulation in terms of alephs requires an infinite input. The general Hartogs theorem itself does not require choice.

For the growing matter-era spectrum $P(k,\tau )=Ak(\tau /{\tau }_{\mathrm{eq}}{)}^4$ and Gaussian Fourier window $W(kr)=e^{-(kr{)}^2/2}$, integration gives the displayed smoothed matter density variance. A physical Hubble-radius window has $r=1/(aH)=\tau /2$, making its variance independent of crossing time. The dimensionless mode variance $k^{3}P/(2{\pi }^2)$ is likewise constant at $k\tau =2$. A pure $P\propto k$ spectrum needs ultraviolet-convergent smoothing: a real-space top-hat alone leaves a logarithmic divergence.

// Target: cosmology.bigb

The Poisson kernel for the upper half-space gives an isometric bijection from finite real or complex Borel measures, with their total variation norm of a measure, onto the harmonic Hardy space of the upper half-space. The kernel has unit mass, so Tonelli theorem gives the upper norm bound. Its approximate identity property gives weak-star convergence to the original measure and hence the reverse norm bound. For surjectivity, the mean value property for harmonic functions bounds each shifted harmonic function; bounded harmonic uniqueness gives $u(\cdot ,s+t)=P_t\ast u(\cdot ,s)$. A weak-star limit of the uniformly bounded measures $u(\cdot ,s)dx$ then gives the required boundary measure. In particular $|u(x,t)|\le c_d{t}^{-d}\Vert u{\Vert }_{h^1}$, where $c_d=\Gamma ((d+1)/2)/{\pi }^{(d+1)/2}$.

On a connected compact oriented Riemannian manifold without boundary, a top-degree form is $f{\mathrm{vol}}_g$. Its exterior derivative is automatically zero, and the Hodge codifferential vanishes exactly when $df=0$. Thus it is harmonic exactly when $f$ is constant. Equivalently the Hodge star operator identifies this space with harmonic functions. Without connectedness the dimension is the number of connected components.

On a domain carrying a branch of the complex logarithm, the holomorphic function $-i\mathrm{Log}z$ has real part equal to the chosen angle and imaginary part $-\log |z|$. These are harmonic conjugates; their level sets are rays and circles, meeting orthogonally. For the principal real arctangent $\varphi =\arctan (y/x)$, its domain has the two components $x>0$ and $x<0$. The corresponding holomorphic functions are $-i\mathrm{Log}z$ and $-i\mathrm{Log}(-z)$, using the complex logarithm on the right half-plane. This formulation does not assert a globally single-valued angle on the punctured plane.

If $u+iv$ and $p+iq$ are holomorphic functions, with the image of the first inside the domain of the second, then $p(u,v)+iq(u,v)$ is holomorphic by the chain rule. Directly, their Cauchy-Riemann equations imply $[p(u,v){]}_x=[q(u,v){]}_y$ and $[p(u,v){]}_y=-[q(u,v){]}_x$. Differentiating these relations proves harmonicity of both parts. No nonvanishing derivative hypothesis is required.

For a harmonic function $u$, the function $g=u_x-i{u}_y$ is holomorphic. It is the derivative of a single-valued holomorphic function exactly when its integrals around all closed curves vanish. A primitive then has real part $u$ after adjusting a real constant. Simply connected domains satisfy this condition; on an annulus $u=\log |z|$ fails because the derivative has period $2\pi i$.

A zero-handle in an $n$-dimensional handle decomposition is $D^0\times D^n=D^n$. Its attaching region is empty, so it starts a connected component of the manifold. In dimension four it is a four-ball; adding one-handles and two-handles gives a four-dimensional handlebody with a two-dimensional CW spine.

A framed embedding of the left-hand region in an $m$-dimensional manifold specifies an index-$\lambda$ surgery. Remove its interior and glue the right-hand region along their common boundary $S^{\lambda -1}\times S^{m-\lambda }$. The cobordism realizing this change attaches an index-$\lambda$ handle of dimension $m+1$. On an oriented surface, index-two surgery cuts out an annular neighbourhood of a simple closed curve and caps its two boundary circles.

An index-$\lambda$ handle in dimension $d$ is this product of disks, attached along $S^{\lambda -1}\times D^{d-\lambda }$. Its core disk is $D^{\lambda }\times \{0\}$, and its belt sphere is $\{0\}\times S^{d-\lambda -1}$. The complementary boundary piece is $D^{\lambda }\times S^{d-\lambda -1}$. Crossing a critical point in the Morse handle-attachment theorem produces precisely this local change.

A four-handle is an index-four handle. In a four-dimensional handle decomposition it is a four-ball attached along its whole boundary, a three-sphere. It caps a spherical boundary component left by earlier handle attachments.

The binary linear code known as the Hamming code of length seven is the kernel of a $3\times 7$ parity-check matrix whose columns are all the nonzero vectors of ${\mathbb{F}}_2^3$. Its dimension is four. No one-column or two-column sum vanishes, but some three-column sum does, so its minimum Hamming distance is three. The syndrome of a one-bit error is its column, uniquely identifying the erroneous position.

Appending an overall parity bit to a binary Hamming code of redundancy $r\ge 2$ gives the extended Hamming code with parametersThe extension preserves dimension. Every resulting codeword has even Hamming weight, while every original nonzero codeword has weight at least three. A weight-three codeword extends to weight four, proving that the minimum Hamming distance is exactly four.

For $r=3$, extending the Hamming code of length seven gives the familiar $[8,4,4]$ linear code. A membership test detects every error of weight at most three, and nearest-word decoding corrects one error. Joint single-error correction and double-error detection uses the overall parity together with the original syndrome: a nonzero original syndrome with odd overall parity indicates a single error in an original coordinate, whereas a nonzero original syndrome with even overall parity detects a double error, under the assumption that at most two bits changed.

Here $N=2^l-1$, $m=2^{l-1}$ and the weight enumerator convention is $W_C(s,t)={\sum }_{c\in C}{s}^{\mathrm{wt}(c)}{t}^{N-\mathrm{wt}(c)}$. The dual code is the binary simplex code, with enumerator $t^N+(2^l-1)s^m{t}^{m-1}$. The MacWilliams identity gives the formula. For $l=3$, the ordinary enumerator $W_C(z,1)$ is $1+7z^3+7z^4+z^7$; for $l=1$, the degenerate one-word code has enumerator $t$.

Let $\omega$ generate ${\mathbb{F}}_{2^s}^{\times }$, $s\ge 2$, and set $n=2^s-1$. The binary cyclic code defined by $c(\omega )=0$ has a parity-check matrix consisting of every nonzero vector of ${\mathbb{F}}_2^s$ once, after choosing a basis and ordering its columns. It is therefore a Hamming code. Its generator polynomial is the minimal polynomial of an algebraic element $m_{\omega }(x)={\prod }_{j=0}^{s-1}(x-{\omega }^{2^j})$, of degree $s$.

For $s\ge 2$, use one nonzero representative from each one-dimensional subspace of ${\mathbb{F}}_3^s$ as a parity-check matrix column. The rank is $s$; no zero or proportional columns imply minimum Hamming distance at least three. Two independent columns and the representative of their sum give a weight-three relation, proving equality. Every nonzero syndrome uniquely identifies one column and one nonzero error value.