If $|{\phi }_X(u)|=1$, then $e^{iuX}$ is almost surely constant. Indeed,Hence $uX\in \theta +2\pi \mathbb{Z}$ almost surely for some real $\theta$.

For $x^{\prime }(t)=\alpha x(t-T)[1-x(t)]$, linearization about $x=1$ is $u^{\prime }=-\alpha u$ because the delayed perturbation appears only in a quadratic term. The equilibrium is therefore locally asymptotically stable for every delay $T\ge 0$.

Forlinearization about $x=1$ gives $u^{\prime }=\alpha [u(t)-2u(t-T)]$. The equilibrium is stable forand loses stability through imaginary roots $\lambda =\pm i\sqrt{3}\alpha$ at equality.

Forthere are $2n$ linear characters, given by $a\mapsto {\xi }^j$, $b\mapsto 1$ for a primitive $(2n)$th root of unity $\xi$, and $n$ two-dimensional irreducible representationswhere ${\omega }^3=1$ and $\omega \ne 1$. The two-dimensional representations parametrized by $\epsilon$ and $-\epsilon$ are isomorphic. The sum of squares of irreducible degrees is $2n+4n=6n$, so this list is complete.

For complex class functions on a finite group $G$, the character inner product isIf $\psi$ is irreducible, this inner product is its multiplicity in the representation with character $\chi$.

For an irreducible character $\chi$ and a conjugacy class $C$ represented by $c$, the central group-algebra element ${\sum }_{x\in C}x$ acts as the scalarThis scalar is an algebraic integer, since the class sum acts by an integer matrix on the regular representation.

For irreducible complex characters $\chi ,\psi$ of a finite group,Equivalently, summing over conjugacy classes $C$ with representatives $c$ gives

The centre-of-momentum energy is the total energy measured in the centre-of-momentum frame. For a scattering process with total four-momentum $P$, it is the Lorentz-invariant quantity $\sqrt{P^2}=\sqrt{s}$ in the mostly-minus convention.

The undirected infinite line is both the Cayley graph of the infinite cyclic group $\mathbb{Z}$ with generating set $\{1\}$ and the Cayley graph of the infinite dihedral group $C_2\ast C_2$ with its two involutory generators. Thus an unlabelled Cayley graph need not determine its group.

For a finite group $G$ with a symmetric generating set $S$, its diameter isEquivalently, it is the diameter of the Cayley graph ${\mathrm{Cay}}_S(G)$.

For a group $G$, a subgroup $H$, and a generating set $S$, the Schreier graph has the left cosets $G/H$ as vertices and an edge from $gH$ to $sgH$ for each $s\in S$. Its radius-$m$ ball records the cosets met by $S^m$.

Mediator-outcome confounding occurs when a common cause of the mediator and outcome is not adequately controlled. If exposure itself affects such a common cause, ordinary adjustment can fail to identify natural direct and indirect effects.

For binary exposure $A$, the total causal effect on an additive scale is $\mathbb{E}[Y(1)-Y(0)]$. With mediator potential outcome $M(a)$ and nested outcome $Y(a,m)$, consistency gives $Y(a)=Y(a,M(a))$.

A congruence on a category $\mathcal{C}$ is an equivalence relation on each hom-set that is compatible with composition on both sides. The quotient category $\mathcal{C}/\sim$ has the same objects and equivalence classes of morphisms, with composition induced from $\mathcal{C}$.

The cofinality of an ordinal $\alpha$ is the least order type of an unbounded subset of $\alpha$, equivalently the least ordinal $\beta$ admitting a cofinal function $\beta \to \alpha$.

The cardinality $|X|$ of a set $X$ expresses its size up to bijection. Two sets have the same cardinality exactly when a bijection pairs their members. For a finite set, cardinality is the ordinary number of set elements. In the presence of the axiom of choice, every set can be assigned the least ordinal in its bijection class as its cardinal number.

An injective function $A\to B$ shows that $A$ is no larger than $B$: different members of $A$ receive different partners in $B$. If there are injections in both directions, the Cantor-Schröder-Bernstein theorem supplies a bijection. Infinite cardinality records this pairing structure rather than a geometric dimension or density. That is why a line of natural numbers and a grid of natural-number pairs can have the same size, while an interval of real numbers has larger cardinality.

There is a constant $C<3$ such that every cap set in ${\mathbb{F}}_3^n$ has cardinality less than $C^n$.

For the polynomial method in combinatorics, useOver ${\mathbb{F}}_3$, this is the indicator function of $x+y+z=0$. On $A^3$ it is therefore a diagonal tensor with $|A|$ nonzero diagonal entries. Every monomial in its expansion has individual exponents at most two and total degree at most $2n$, so one of its three variable blocks has degree at most $2n/3$. Grouping terms according to such a block and using the slice rank of a diagonal tensor giveswhere $m_n$ is the number of $\alpha \in \{0,1,2{\}}^n$ with ${\sum }_i{\alpha }_i\le 2n/3$. If $X_i=1-{\alpha }_i$ are independent random variables uniform on $\{-1,0,1\}$, thenAny exponential upper bound for this tail probability gives $m_n\le (3-ϵ{)}^n$ and hence the result after absorbing the factor three and finitely many small dimensions into $C<3$.

Triangulate a triangle and label every vertex by $1$, $2$, or $3$, forbidding label $i$ on the side opposite vertex $i$. Then the number of small triangles carrying all three labels is odd, and in particular at least one such triangle exists. Counting edges whose endpoint labels are $1$ and $2$ proves the parity statement: boundary incidences are odd, while a small triangle has odd incidence exactly when it has all three labels.

The reciprocal lattice of a two-dimensional triangular Bravais lattice is triangular, so its Wigner-Seitz cell is a regular hexagon. Opposite boundary points are identified modulo reciprocal-lattice translations, and the six corners fall into two inequivalent classes of three corners each, conventionally denoted $K$ and $K^{\prime }$.

A Bragg point is a crystal momentum at which two plane waves differing by a reciprocal lattice vector are degenerate. A periodic potential mixes the waves there, producing an avoided crossing and opening a band gap.