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In the alternating group $A_6$, of order $360$, a Sylow subgroup of order five is generated by a five-cycle. There are $6\cdot 4!=144$ five-cycles, four in each such subgroup, giving $36$ subgroups. A Sylow subgroup of order nine is generated by disjoint three-cycles, hence is $C_3\times C_3$. The unordered partitions of six letters into two triples give all such subgroups by Sylow conjugacy, and their single three-cycles recover the partition uniquely. There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)/2=10$ partitions.

For a bottom-forced gravity wave with signed surface coefficient $b$, evaluate the first-order velocity field at a parcel's mean position $(\bar{x},\bar{y})$ and integrate, with $\Theta =k(\bar{x}-ct)$. The horizontal and vertical displacement coefficients areThus the pathline is an ellipse with horizontal and vertical semiaxes $|A|,|B|$ whenever both are nonzero, and a line segment if one vanishes. The identity $A^2-B^2=b^2[(g/(k{c}^2){)}^2-1]$ determines which axis is longer. Also $B(0)=b$ and $B(-d)=a$, verifying the surface and bottom displacements. Closed first-order orbits do not rule out a higher-order mean drift.

Set $p=(0,0)$, $q=(1,0)$, ${\epsilon }_m=1/(m+2)$, and let $A_m$ be the polygonal arc joining $({\epsilon }_m,0)$, $({\epsilon }_m,m)$, $(1-{\epsilon }_m,m)$, and $(1-{\epsilon }_m,0)$. In the planar subspace $Z=\{p,q\}\cup {\bigcup }_m{A}_m$, each arc is clopen. The component of $p$ is therefore just $\{p\}$. Any clopen neighborhood of $p$ contains the left endpoints of all sufficiently high arcs, hence those entire connected arcs, and by closedness also the limit $q$ of their right endpoints. Thus its quasi-component is $\{p,q\}$. Higher arcs escape to infinity while their endpoints approach the two distinguished points.

For $u\in C^2(D)\cap C(\overline{D})$ on a bounded domain in ${\mathbb{R}}^n$ with $\Delta u\le 0$, the minimum lies on the boundary. Define $u_{\epsilon }=u-\epsilon |x{|}^2$, so $\Delta u_{\epsilon }\le -2n\epsilon <0$. An interior minimum would have positive semidefinite Hessian matrix and nonnegative trace, a contradiction. Thus $u_{\epsilon }$ attains its minimum on the boundary. Boundedness controls $|u-u_{\epsilon }|$ uniformly by a constant times $\epsilon$; letting $\epsilon \downarrow 0$ proves the weak minimum principle for the original superharmonic function. The perturbation handles the non-strict inequality, including constant functions.

On three players, let $v(S)=1$ exactly when $S$ contains player 1 and at least one of players 2 and 3, and zero otherwise. This simple cooperative game is superadditive: disjoint coalitions cannot both contain player 1, so at most one can have value one. Its core of a cooperative game is exactly $\{(1,0,0)\}$, since singleton nonnegativity and the two winning-pair constraints force the other two coordinates to zero. In a random player ordering, player 1 is pivotal unless it comes first, whereas each other player is pivotal in just one of the six orders. Thus its Shapley value is $(2/3,1/6,1/6)$, outside the core: coalition $\{1,2\}$ receives only $5/6<1$. Convex cooperative games have the stronger property that their Shapley value belongs to the core of a convex game.

For vector subspaces $U\subseteq W\subseteq V$ of a finite-dimensional vector space, such a $W^{\prime }$ always exists. Extend a basis of $U$ to one of $W$, then extend again to one of $V$. Let $W^{\prime }$ be spanned by the original basis of $U$ together with the vectors added in the second extension. Uniqueness of coordinates gives the displayed intersection and sum. Equivalently, choose a direct sum complement $C$ with $V=W\oplus C$ and put $W^{\prime }=U\oplus C$. Necessarily $\dim W^{\prime }=\dim V-\dim W+\dim U$. In $M_2(\mathbb{R})$, the skew-symmetric subspace $U$ inside the traceless subspace $W$ admits $W^{\prime }=\mathrm{span}\{I,\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\}$.

Let $A=D+L+L^T$ be a real symmetric positive-definite matrix, with $D$ diagonal and $L$ strictly lower triangular. Put $B=D/\omega +L$ and $C_{\omega }=I-B^{-1}A$ for $\omega \ne 0$. The matrix norm induced by an energy norm reduces the question to $T=A^{1/2}{C}_{\omega }{A}^{-1/2}$. Direct multiplication and $B+B^T-A=(2/\omega -1)D$ giveThe left side is positive semidefinite. The matrix following the scalar coefficient on the right is positive-definite, since $D$ has positive diagonal and $B^{-1}{A}^{1/2}$ is an invertible matrix. For $0<\omega <2$, every eigenvalue of $T^{T}T$ therefore lies in $[0,1)$. Hence its spectral radius is below one and $\Vert C_{\omega }{\Vert }_A=\Vert T{\Vert }_2<1$. Iteration errors consequently decay geometrically in the energy norm. The argument proves convergence from every initial vector, without requiring the original iteration matrix to be symmetric.

The values of all closed terms in a first-order structure form a substructure when the first-order language has at least one constant. The constant ensures nonemptiness, and applying a function symbol to closed-term values again gives a closed-term value. Relations are restricted from the ambient first-order structure. Every universal sentence true in the ambient first-order structure remains true in this substructure.

Replacing one designated subformula occurrence in a propositional formula by another propositional formula preserves well-formedness. Induct on the context surrounding the occurrence: at the root the result is the replacement formula; under a connective apply the induction hypothesis to the affected child and rebuild the parent using its formation rule.

A subformula occurrence specifies a particular position in the syntax tree of a formula. Two occurrences can contain the same logical formula while being different nodes. Replacement of one occurrence does not necessarily replace the others.

A structure isomorphism is a bijective structure embedding: it preserves constants and functions, and both preserves and reflects relations. Its inverse has the same properties. It therefore preserves truth of every first-order formula, by structural induction, so it preserves realization and omission of parameter-free types.

Take independent random variables equal to $1+n$ and $1-n$ with probability $1/(2n\log n)$ each, and otherwise equal to one. Their first absolute deviations are $1/\log n\to 0$, so $X_n\to 1$ in L1 convergence. The same triangle-inequality argument gives convergence in $L^1$ of their arithmetic averages to one. For $p>1$, however, $\mathbb{E}|X_n-1{|}^p=n^{p-1}/\log n\to \infty$.

The exceptional jumps occur infinitely often almost surely by the Borel-Cantelli second lemma, since $\sum 1/(n\log n)=\infty$. If the arithmetic averages $S_n/n$ converged to any finite value along a sample path, then $X_{n+1}/n=S_n/n-[(n-1)/n][S_{n-1}/(n-1)]$ would tend to zero. At each exceptional jump its magnitude is at least one. Thus the averages have no finite almost sure limit. Independence and convergence of first-moment averages do not alone give a strong law of large numbers.

For independent identically distributed random variables, the tail-sum formula applied to $|X_1|/\lambda$ shows that ${\sum }_n\mathbb{P}(|X_n|\ge \lambda n)=\infty$ for every $\lambda >0$ when the first absolute moment is infinite. The Borel-Cantelli second lemma, followed by a countable intersection over positive integer $\lambda$, gives the first limsup assertion.

If $|S_n|/n$ had finite limsup along such a sample path, it would eventually be bounded by some finite constant $C$. Then $X_n=S_n-S_{n-1}$ would imply $|X_n|/n\le 2C$ eventually, a contradiction. This is a necessary-integrability counterpart to the strong law of large numbers; without a finite first absolute moment, the normalized sums cannot converge to any finite value.

Let $R$ be a complete discrete valuation ring, $f\in R[X]$, and $d=|f^{\prime }(a)|$ satisfy $0<d<1$ and $|f(a)|=d^2$. Suppose the second Taylor coefficient of $f$ at every point of $R$ has norm below one. One Newton iteration over a valued field step has correction $h=-f(a)/f^{\prime }(a)$ of norm $d$. Its residual has quadratic term of norm strictly below $d^2$, and all higher terms have norm at most $d^3<d^2$. The derivative changes by a term of norm below $d$, so its norm remains $d$. The strong form of Hensel lemma therefore applies from the new point, producing a root within distance $d$ of the original point. The second Taylor coefficient is $f^{\prime \prime }/2$ when that quotient is meaningful, or the division-free binomial-coefficient Taylor expression in arbitrary characteristic. A Newton sequence that reaches a root is thereafter constant; residuals decrease strictly only before that event.

For $dy={\sum }_i{A}^{i}yd{B}^i$ with constant linear maps, let $K={\sum }_i\Vert A^i{\Vert }_{\mathrm{op}}^2$ and let $D_m$ be consecutive Picard iterate differences starting from $y^{(0)}=a$. The vector Itô isometry gives $\mathbb{E}|D_m(t){|}^2\le K{\int }_0^t\mathbb{E}|D_{m-1}(s){|}^{2}ds$, proving the displayed estimate by induction. The square roots of these factorial bounds are summable on bounded time intervals. The Doob L2 maximal inequality then makes ${\sum }_m\mathbb{E}{\sup }_{t\le T}|D_m(t)|$ finite. Tonelli's theorem proves absolute uniform almost-sure convergence of the iterations, in addition to their $L^2$ convergence.

An irreducible Markov chain on a finite or countably infinite state space has at most one stationary distribution. On a countably infinite state space it exists exactly when the chain is positive recurrent. In that case, for every state $i$,where $T_i^{+}$ is the first return time. The expression determines every component uniquely. The formula follows by dividing the one visit to $i$ in each return cycle by the mean recurrence time; the strong law of large numbers for successive cycles makes this occupation fraction precise.

Every irreducible finite Markov chain is positive recurrent and has exactly one stationary distribution. On an infinite state space, an irreducible null recurrent Markov chain or transient Markov chain has none. For example, the simple symmetric random walk on the integers is null recurrent. Aperiodicity affects convergence of the one-time laws, but is unnecessary for existence or uniqueness of a stationary distribution.

On states $0,\dots ,m$, take an interior step of $+1$ or $-1$ with probability $p$ each and hold with probability $1-2p$. At an endpoint take the inward step with probability $2p$ and hold with probability $1-2p$. For $m\ge 1$ and $0<p\le 1/2$, the chain is irreducible and its stationary distribution has endpoint weights $1/(2m)$ and interior weights $1/m$.

Detailed balance proves this: the doubled inward endpoint transition requires its stationary mass to be half that of an adjacent interior point, and all interior points have equal mass. The total unnormalized weight is $1+2(m-1)+1=2m$. For $m=1$ there are just two equal endpoint masses.

When $p<1/2$, positive self-loops make the chain aperiodic, so its transition probabilities converge to this law. At $p=1/2$ every move changes parity and two-step returns are possible, giving period two and no such convergence from a fixed starting state. At $p=0$ every state is absorbing and uniqueness of the stationary law is lost.

Place $N\ge 2$ books in a row and distinguish one of them. At each step select the distinguished book with Probability $p\in (0,1)$ and each of the other $N-1$ books with Probability $q=(1-p)/(N-1)$. Swap the selected book with its left neighbour; selecting the leftmost book leaves the ordering unchanged.

The distinguished book's position $I\in \{1,\dots ,N\}$ is a birth-death chain. From an interior position it moves left with Probability $p$, moves right with Probability $q$ when the book immediately to its right is selected, and otherwise holds. Missing outward moves at the endpoints become holding. Detailed balance therefore gives ${\pi }_{i}p={\pi }_{i-1}q$ for $i\ge 2$. With $r=q/p$,This finite Truncated geometric distribution is valid even when $r>1$: both numerator and denominator then have the same negative sign. The uniform case is $p=q=1/N$, when every book is selected equally often. Large selection Probability for the distinguished book biases it toward the left end. The strict range of $p$ ensures positive motion in both directions and avoids division by zero in the ratio.

Represent a finite sparse vector by a sorted doubly linked list of its nonzero coordinates. A list header stores the dimension; every ordinary linked list node stores a distinct valid coordinate index, its nonzero value, and predecessor and successor references. In a circular linked list, the endpoints connect to the header, and an all-zero vector has the header linked to itself in both directions. The representation invariant requires strictly increasing ordinary indices, valid bounds and mutually consistent forward and backward links. The abstraction function assigns zero to every coordinate absent from the list. The header is metadata, never an ordinary coordinate.

For example, $(0,3,0,0,-2,0,0,5,0)$ has dimension nine and only three stored coordinates. The header's index field can store nine even though nine is not a valid coordinate index. Keeping the header distinct is therefore essential when searching for or removing entries.