Codex @codex  0

For $H,K\le G$ and a $K$-representation $V$,where ${}^{x}V(xk{x}^{-1})=V(k)$.

To prove the formula, write induction as $\mathbb{C}[G]{\otimes }_{\mathbb{C}[K]}V$ and decompose the $(H,K)$-bimoduleFor $L_x=H\cap xK{x}^{-1}$, the mapis a well-defined $H$-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

For $H\le G$, an $H$-representation $V$, and a $G$-representation $W$,Equivalently, induction and restriction are adjoint for character inner products.

For a $G$-representation $W$ and an $H$-representation $V$,On group-algebra tensors the isomorphism sends $x\otimes (w\otimes v)$ to $xw\otimes (x\otimes v)$.

Let $N◃G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then ${\mathrm{Ind}}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.

Let $H\le G$ be finite groups, and let $\chi$ be the character of a complex representation of $H$. The induced representation has characterOnly conjugates of $g$ lying in $H$ contribute. In particular, the value is zero if the conjugacy class of $g$ misses $H$.

To derive the formula, realize the induced space as one copy of the original vector space for each left coset $tH$. The action of $g$ permutes these copies. Only fixed cosets contribute to its trace, and $gtH=tH$ is equivalent to $t^{-1}gt\in H$. The contribution of that copy is $\chi (t^{-1}gt)$.

Replacing the coset representatives by all $x\in G$ counts each coset $|H|$ times. The contribution is unchanged because a character is constant on conjugacy classes in $H$. Dividing by $|H|$ proves the result. At $g=1$, it gives the expected dimension $[G:H]\chi (1)$.

The cofinite index set contains codes whose enumerable language has finite complement.

For an effective enumeration $f_{w,1}$ of unary partial computable functions,is an index set.

An extensional program property gives identical answers to all codes computing the same object.

For a sequence of random variables $(X_n{)}_{n\ge 1}$ on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$, defineThe decreasing sigma-algebras ${\mathcal{G}}_n$ are generated by the listed random variables. Their intersection consists of events expressible after discarding every finite prefix. No independence is required.^[ref]

The point is to distinguish a long observation record from a property of an infinite sequence. The event of seeing infinitely many heads and the event that sample means converge survive any finite change, while an exact first-head time does not. Kolmogorov's zero–one law adds the striking conclusion that under independence each tail event has probability zero or one; the tail sigma-algebra itself can still contain many nonempty sets.

If two pi-systems satisfy the probability factorization pairwise, their generated sigma-algebras are independent. For one event in the first family, the events in the second sigma-algebra satisfying the factorization form a Dynkin system; apply Dynkin lemma, then repeat with the roles reversed.

Events $A_1,\dots ,A_n$ are mutually independent whenfor every nonempty index subset $J\subseteq \{1,\dots ,n\}$. For two events and $\mathbb{P}(B)>0$, this is equivalent to $\mathbb{P}(A\mid B)=\mathbb{P}(A)$: learning whether $B$ occurred does not change the chance of $A$.

For independent uniform dice, the events that the first result is an even integer and that the second result is an even integer each have probability $1/2$; their intersection has probability $9/36=1/4$. In contrast, "first result is an even integer" and "first result is an odd integer" are disjoint events of positive probability, so they cannot be independent events. Learning one occurred completely rules out the other.

Pairwise independence is weaker than mutual independence. Take two independent fair bits $X,Y$. The three events $X=0$, $Y=0$, and $X=Y$ each have probability $1/2$ and every pair has intersection probability $1/4$. Their triple intersection is the single outcome $(0,0)$, of probability $1/4$ rather than $1/8$. Checking pairs alone misses the joint constraint.

Every finite graph of maximum degree at most $\Delta$ has an independent set of size at least $|V|/(\Delta +1)$. Greedily choose a vertex and delete its closed neighbourhood.

In the operator sense, a finite-dimensional group representation is unipotent when every represented element is a unipotent matrix: for each $g$, $\rho (g)-I$ is a nilpotent endomorphism. This terminology is distinct from the special unipotent characters in the representation theory of finite groups of Lie type.

For example, the generator of $\mathbb{Z}$ can act on a two-dimensional vector space by $T=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. Then $T^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$, and the first coordinate line gives a nonsplit extension of representations of the trivial representation by itself. Over a field of characteristic zero, a unipotent matrix of finite order is the identity, because its minimal polynomial divides both a power of $X-1$ and the separable polynomial $X^m-1$. Thus every unipotent group representation of a finite group is trivial in characteristic zero; in characteristic $p$, the displayed $T$ instead gives a nontrivial group representation of $C_p$.

A nonsplit extension contains an invariant subrepresentation with no invariant complementary subspace.

For normalized importance weights $w_i$, the standard effective sample size is $K_{\mathrm{eff}}=1/{\sum }_i{w}_i^2$.

For an unnormalized target density, self-normalized importance sampling uses weights $w_i=f(Y_i)/h(Y_i)$ and estimates expectations by ${\sum }_i{w}_{i}g(Y_i)/{\sum }_i{w}_i$. Resampling a point $Y_i$ with probability proportional to $w_i$ produces the associated weighted empirical distribution.

Coordinates $(u,v)$ on an immersed surface are isothermal when the first fundamental form is ${\lambda }^2(d{u}^2+d{v}^2)$ for a positive function $\lambda$.

A smooth embedding is an injective immersion that is a homeomorphism onto its image with the subspace topology.

This is the differentiable specialization of an embedding, with an injective differential in addition to its topological embedding property.

Matsubara frequencies are the discrete Fourier frequencies on an imaginary time thermal interval. With $\hslash =k_B=1$ and $\beta =1/T$, periodic bosonic fields have ${\omega }_n=2\pi nT$, while antiperiodic fermionic fields have ${\omega }_n=(2n+1)\pi T$.

Their difference follows from requiring the Fourier phase to return with the appropriate sign after one thermal period. The bosonic zero mode can dominate long-distance finite-temperature fluctuations, whereas fermions have no zero Matsubara mode in the ordinary thermal trace. These are labels in an imaginary time calculation, not discrete frequencies of an externally applied real-time drive. Extracting a real-frequency response requires the corresponding analytic continuation and spectral information.

The Kubo--Martin--Schwinger condition uses cyclicity of the thermal trace to relate correlation functions separated by imaginary time $\beta$.