Mathematics

A law or statistical object is scale invariant if its form is unchanged by rescaling lengths, with any specified scaling weight for the fields. A dimensionless field's three-dimensional nonzero-mode power spectrum obeys $P(\lambda k)={\lambda }^{-3}P(k)$ when its statistics have this scaling. The meaning of invariance must specify whether zero modes and regulators are being transformed.

Infinity denotes unbounded growth or an ideal quantity larger than every finite value; it is not a real number.

A cyclic vector gives a companion matrix, whose characteristic relation annihilates its generator and hence every iterate. If there is no cyclic vector spanning the whole space, a nonzero proper cyclic subspace is invariant. Induction annihilates the restriction and the quotient; the quotient polynomial first sends the whole space into the invariant subspace, and the restriction polynomial then kills it. The block-triangular characteristic polynomial is the product of these two polynomials.

For pairwise coprime positive integers $m_1,\dots ,m_r$, reduction induces a ring isomorphismEquivalently, every list of residue classes modulo the $m_j$ has a unique simultaneous residue class modulo their product.

For a positive integer $b={\prod }_i{p}_i^{n_i}$, every rational number $a/b$ has a unique representation $a/b=n+{\sum }_i{q}_i/p_i^{n_i}$ with $n\in \mathbb{Z}$ and $0\le q_i<p_i^{n_i}$. With $B_i=b/p_i^{n_i}$, the coefficients satisfy $q_i\equiv a{B}_i^{-1}(\mathrm{mod}{p}_i^{n_i})$. Their existence and uniqueness use modular multiplicative inverses and the Chinese remainder theorem.

A nonzero polynomial of degree $d$ over a field has at most $d$ roots. In particular, if a polynomial congruence $f(x)\equiv 0(\mathrm{mod}p)$ has degree $d$ and not all coefficients are divisible by the prime $p$, it has at most $d$ incongruent solutions modulo $p$.

For coprime $m,n$, reduction gives $(\mathbb{Z}/mn\mathbb{Z}{)}^{\times }\cong (\mathbb{Z}/m\mathbb{Z}{)}^{\times }\times (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.

The order of a unit modulo a product of coprime moduli is the least common multiple of the orders of its components.

These permutations have no cycle of length two, while fixed points and longer cycles are allowed. Inclusion-exclusion over specified two-cycles gives the count shown: only disjoint specified pairs can coexist. Their fraction among all permutations tends to $e^{-1/2}$. This condition differs from being a derangement and from refusing to express a permutation as a product of transpositions.

A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted.

A dimensionless variable is unchanged by a consistent change of physical units. Scaling a variable by a characteristic quantity with the same dimensions produces a dimensionless ratio.

A transcendental equation contains a nonalgebraic function of its unknown, such as an exponential or logarithm, and generally cannot be reduced to a polynomial equation.

Reflecting an interval repeatedly across its endpoints maps a reflected path to a free path in an image interval. A return trajectory from $q\in (0,L)$ ends at $q+2rL$ after an even number of wall reflections, or $-q+2rL$ after an odd number. An infinite square well has reflection amplitude $-1$, so the corresponding Dirichlet heat kernel on an interval is the difference of direct and mirrored free kernels. Individual straight paths illustrate the geometry; the path integral sums all time-sliced paths with the same endpoints.

For a compact continuous module over the Iwasawa algebra of a Zp-extension, lifts of a basis of $M/(p,T)M$ generate $M$. Their generated image is compact. The quotient $Q$ satisfies $Q=(p,T)Q$; in every finite continuous quotient, $(p,T)$ acts nilpotently, since the group action is pro-$p$. Thus all finite quotients of $Q$ vanish, and compactness gives $Q=0$.

For a positively graded ring with degree-zero part a field, put $R_{+}={⨁}_{d>0}{R}_d$. A bounded-below graded module $M$ with $M=R_{+}M$ is zero: a nonzero homogeneous element of least degree would have to be a sum of positive-degree coefficients times elements of lower degree. Hence homogeneous lifts of a basis of $M/R_{+}M$ generate $M$.

A nonzero number $T$ is a period of a function $f$ when $f(x+T)=f(x)$ throughout its domain.

A quasi-steady approximation neglects a field's time derivative because that field relaxes much faster than the variables that drive it. The rapidly adjusting field then satisfies a steady equation with the slow variables treated as instantaneous parameters.

If a primitive polynomialhas a rational root $r/s$ in lowest terms, then $r\mid a_0$ and $s\mid a_n$. In particular, a monic integer polynomial can have only integer rational roots dividing its constant term.

A finite family of pairwise commuting diagonalizable operators on a finite-dimensional complex vector space is simultaneously diagonalizable. In particular, commuting Hermitian operators admit an orthonormal common eigenbasis.

Suppose $D$ has $n$ distinct eigenvalues over a field and $C$ commutes with $D$. Each eigenspace of $D$ is one-dimensional. Since commuting maps preserve eigenspaces, $C$ acts as a scalar on each of them. The eigenvectors of $D$ give a common eigenbasis and hence simultaneous diagonalization. For real matrices, the result gives a real eigenbasis only if the eigenvalues of $D$ are real; otherwise it applies over $\mathbb{C}$.

For commuting diagonalizable operators $A_1,\dots ,A_r$ on a finite-dimensional vector space, the joint spectrum is the set of tuples $(a_1,\dots ,a_r)$ for which there is a common nonzero eigenvector $v$ with $A_{i}v=a_{i}v$. Commuting self-adjoint operators on a finite-dimensional inner-product space admit a simultaneous orthonormal eigenbasis.

For a holomorphic function near the spectrum, the spectrum of its functional-calculus value is exactly the image of the original spectrum. If a scalar is absent from that image, the reciprocal of the shifted function is holomorphic on a smaller spectral neighborhood and supplies an inverse. If it is attained at $\lambda$, a divided difference factors $f(x)-f(\lambda )$ through $x-\lambda$. These factors commute, so invertibility of their product would contradict $\lambda \in \sigma (x)$.

For a closed linear operator and $z$ in its resolvent, the displayed identity follows by factoring $(A-z{)}^{-1}-wI$ for $w\ne 0$. Its inverse exists exactly when $A-(z+1/w)I$ has a bounded inverse. Products involving $A$ are bounded after composing with a resolvent, since $A(A-\lambda {)}^{-1}=I+\lambda (A-\lambda {)}^{-1}$. Zero may be in the bounded resolvent's spectrum when $A$ is unbounded.

An element $a$ is a non-zero-divisor when multiplication by $a$ is injective, equivalently when $ab=0$ implies $b=0$.

A variable is a symbol that represents an element whose value may vary or remain unspecified.

A constant is a fixed value within the problem or model under discussion.

If two variables satisfy $y=kx$ for a fixed constant $k$, then $k$ is their constant of proportionality.

A normed algebra is an algebra over a field with a norm satisfying $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. Its completion is a Banach algebra, since multiplication extends continuously. Completeness is additional structure, rather than part of the definition of a normed algebra.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

Suppose a ring has an underlying real normed vector space and its norm is multiplicative, without initially assuming real bilinearity of multiplication. For fixed $x$, distributivity makes $y\mapsto xy$ additive, while $\Vert xy-xz\Vert =\Vert x\Vert \Vert y-z\Vert$ makes it continuous. Additivity gives rational scalar compatibility, and continuity extends it to every real scalar. Apply the same argument in the other variable to obtain a bilinear map. The norm identity also excludes nonzero zero divisors.

A normed division algebra is a nonzero unital normed algebra in which every nonzero element has a two-sided multiplicative inverse. The complex Gelfand-Mazur theorem forces such an algebra to be $\mathbb{C}$, even when completeness is not initially assumed: embed it in its Banach algebra completion and use nonemptiness of the Banach-algebra spectrum.

For an additive semigroup $D$ with identity zero, the semigroup algebra $k[D]$ has a $k$-basis of formal symbols $t^q$, $q\in D$, and multiplication $t^q{t}^{q^{\prime }}=t^{q+q^{\prime }}$. For $D=\mathbb{Z}[1/2{]}_{\ge 0}$ it is the integral domain ${\bigcup }_{h\ge 0}k[t^{1/2^h}]$. Its positive-exponent ideal is a nonzero idempotent ideal, and the strictly ascending chain $(t)⊊(t^{1/2})⊊\cdots$ shows that the algebra is not a Noetherian ring.

A semiring has a commutative monoid operation $(S,+,0)$ and a monoid operation $(S,\cdot ,1)$, with multiplication distributing over addition on both sides and $0$ absorbing multiplication. Additive inverses are not required. The natural numbers with their usual addition and multiplication are a basic example. Regarding multiplication as composition yields a one-object category with commutative-monoid enrichment.

Bernoulli's inequality states that $(1+x{)}^r\ge 1+rx$ for every integer $r\ge 0$ and real $x\ge -1$. In particular, $(1-x{)}^r\ge 1-rx$ for $0\le x\le 1$.

An algebraic operation combines one or more elements of a set to produce another element.

A binary operation on a set $S$ is a function $S\times S\to S$.

A binary operation is commutative if $a\ast b=b\ast a$ for every pair in its domain. A particular pair may commute even when the operation is not commutative. For example, ordinal addition and ordinal multiplication are not generally commutative.

A binary operation $\ast$ is associative when $(x\ast y)\ast z=x\ast (y\ast z)$ for every $x,y,z$.

A semigroup is a set equipped with an associative operation.

An element of a semigroup is idempotent when multiplying it by itself leaves it unchanged. Every element of a left-zero or right-zero semigroup is idempotent, whereas a group has only its identity as an idempotent. The Ellis–Numakura lemma guarantees such an element in a nonempty compact one-sided topological semigroup. Additive idempotence on ultrafilters gives the idempotent ultrafilter relation $p+p=p$.

A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

A semigroup presentation is finite when both its generator set and its list of defining equalities are finite. A finite transition table for a Turing machine can be encoded this way: state-and-tape words simulate transitions, boundary rules supply blank cells, and halting-state rules erase the configuration to one distinguished symbol.

A subsemigroup is a nonempty subset of a semigroup closed under its associative operation. In a compact Hausdorff space carrying a left-topological semigroup, descending chains of nonempty compact subsemigroups have a nonempty compact subsemigroup as their intersection. This supplies the minimal object used in the Ellis–Numakura lemma.

A monoid is a semigroup with a two-sided identity element.

A left monoid action assigns each $m\in M$ a map $a\mapsto m\cdot a$ with $1\cdot a=a$ and $(mn)\cdot a=m\cdot (n\cdot a)$. It need not be invertible or injective. A left M-set is equivalently a covariant functor from the one-object category determined by $M$.

A right monoid action has $x1=x$ and $(xm)n=x(mn)$. It is a contravariant functor from the one-object category of the monoid to sets. The regular right action is multiplication on $M$ itself. Products of right actions use the diagonal action.

Give $M\times X$ the diagonal right monoid action. On its equivariant maps into $Y$, define $(ek)(g,x)=e(kg,x)$. The evaluation map of an exponential object is $e(1,x)$ and currying of $h:Z\times X\to Y$ is $\hat{h}(z)(g,x)=h(zg,x)$. Both maps are equivariant, and they are inverse under evaluation. The left multiplication $kg$ in this right action is essential in a noncommutative monoid.

For a group, evaluation at the identity identifies the monoid exponential with all set functions $X\to Y$. Its inverse is $e_f(g,x)=f(x{g}^{-1})g$, and the resulting function action is the displayed formula. Fixed functions are precisely equivariant maps. The inverse on the input is necessary for equivariance of evaluation and for the right-action law.

An equivariant map $h:A\to B$ between left M-sets obeys $h(m\cdot a)=m\cdot h(a)$. These are the natural transformations of the corresponding functors from the one-object monoid category.

An M-set is a set carrying a monoid action. This terminology here uses left actions. Morphisms are equivariant maps of monoid sets, and the resulting functor category $[M,\mathbf{Set}]$ is an elementary topos.

A monoid homomorphism $h:M\to N$ preserves multiplication and identity: $h(xy)=h(x)h(y)$ and $h(1_M)=1_N$. It therefore preserves every finite ordered product, including the empty product. The monoids with these maps form the category $\mathbf{Mon}$.

A monoid automorphism is a bijective operation- and identity-preserving self-map of a monoid. Its inverse is also a monoid homomorphism. Permuting prime numbers and extending by prime factorization gives all automorphisms of the multiplicative monoid of positive natural numbers; extending by $0\mapsto 0$ gives automorphisms of the multiplicative monoid including zero.

The free monoid on an alphabet consists of all finite words in its letters, including the empty word, under concatenation. Distinct words represent distinct elements.

A commutative monoid is a monoid whose operation is commutative. The natural numbers with addition form a commutative monoid that is not an abelian group, because positive elements have no additive inverses.

A commutative monoid is cancellative when $a+c=b+c$ implies $a=b$. Cancellation makes the natural map into its Grothendieck group injective.

A cancellative commutative monoid is torsion-free in this sense when $na=nb$ implies $a=b$ for every positive integer $n$. Its Grothendieck group is then a torsion-free abelian group, because $n(a-b)=0$ forces $na=nb$. This is the condition inherited by a submonoid of a torsion-free abelian group.

This category has commutative monoids and their homomorphisms. Pointwise addition makes each hom-set a commutative monoid, and composition is additive in both variables. Finite cartesian products are also coproducts, using the coordinate injections with zero in the other coordinates and summing images. Therefore it is a semi-additive category, but it is not an additive category. The reflexive pair from the submonoid $\{(m,n):m\le n\le 2m\}$ of ${\mathbb{N}}^2$ to $\mathbb{N}$, using the two projections and section $m\mapsto (m,m)$, has an arrow from $1$ to $2$ and none from $2$ to $1$ in its represented graph. Thus a reflexive pair here need not yield a groupoid.

An additive semigroup is a semigroup whose operation is written as addition.

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.

A left-topological semigroup requires only that every right translation $x\mapsto xa$ be continuous.

Every nonempty compact Hausdorff left-topological semigroup contains an idempotent. Choose a minimal nonempty compact subsemigroup $K$. For $a\in K$, minimality gives $Ka=K$; the nonempty compact subsemigroup $\{x\in K:xa=a\}$ is therefore all of $K$, so $a^2=a$.

A matrix is strictly diagonally dominant when $|a_{ii}|>{\sum }_{j\ne i}|a_{ij}|$ in every row.

Lie theory studies continuous symmetries through Lie groups, Lie algebras, and their representations.

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

For a finite-dimensional vector space $V$, the affine group consists of invertible affine maps $x\mapsto Ax+b$, where $A$ belongs to the general linear group and $b\in V$. Its product is $(A,b)(A^{\prime },b^{\prime })=(A{A}^{\prime },b+A{b}^{\prime })$, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.

A rational representation is a group homomorphism whose matrix coefficients are regular functions on the affine algebraic group. For the general linear group, those functions are polynomials in matrix entries and the inverse determinant. The word rational does not permit poles at points of the group.

For finite-dimensional rational representations of complex $G{L}_m$ or $S{L}_m$, average a Hermitian inner product over the compact group $U(m)$ or $SU(m)$ using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.

The matrix coefficients of a rational $G{L}_m$ action belong to its coordinate ring $\mathbb{C}[g_{ab},\det (g{)}^{-1}]$. Multiplying the action by a sufficiently large determinant twist clears all determinant denominators and produces a polynomial representation. Irreducible rational actions are indexed by decreasing integer highest weights, with negative coordinates allowed.

For a dominant integer tuple $\lambda$, choose $k=\max (0,-{\lambda }_m)$ and define $D_{\lambda }={\det }^{-k}\otimes D_{\lambda +k(1,\dots ,1)}$, using the polynomial Schur module on the right. The alternant character formula for the general linear group shows that a larger admissible shift yields the same irreducible representation.

The dual of $D_{\lambda }$ has dominant label $-w_0\lambda =(-{\lambda }_m,\dots ,-{\lambda }_1)$. Substitution $x_i\mapsto x_i^{-1}$ in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.

The irreducible rational $G{L}_m$ modules are $D_{\lambda }(V)=(\det V{)}^{{\lambda }_m}\otimes D_{\lambda -{\lambda }_m(1,\dots ,1)}(V)$. The second factor is a Schur module. Clearing determinant denominators reduces completeness to the Schur algebra classification of polynomial modules. Distinct dominant integer tuples have distinct highest torus weights. Their characters are symmetric Laurent polynomials; only polynomial modules have characters defined at every singular endomorphism.

Split a rational $S{L}_m$ representation by its finite scalar center: on $W_k$, $\rho (\zeta I)={\zeta }^{k}I$ for ${\zeta }^m=1$. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of $\rho {|}_{W_k}$ can be represented on $S{L}_m$ by a sum of homogeneous polynomials $P_d$ with $d\equiv k(\mathrm{mod}m)$, by averaging over the scalar center. Its extension is ${\sum }_d(\det g{)}^{(k-d)/m}{P}_d(g)$, a regular function on $G{L}_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

Tensoring a rational representation of the general linear group with a determinant power shifts every highest-weight coordinate by the same integer $r$. Positive twists can clear determinant denominators; negative twists supply rational representations that do not extend to singular matrices. Twisting preserves irreducibility and dimension.

Every one-dimensional rational $G{L}_m$ representation is ${\det }^r$ for one integer $r$. Its restriction to the diagonal torus is a Laurent monomial. Invariance under conjugation by permutation matrices forces all its exponents to agree, and density of diagonalizable invertible matrices gives the result on the whole group. For $G{L}_1$, the character identity $p(zw)=p(z)p(w)$ directly forces a Laurent polynomial to be one monomial with coefficient one.

In a polynomial representation, every matrix coefficient is a polynomial in the matrix entries. The action extends to the monoid of all matrices. The scalar action splits it into homogeneous polynomial degrees; each homogeneous degree-$n$ category is the module category of the Schur algebra $S(m,n)$. Its irreducibles are the Schur modules $D_{\lambda }({\mathbb{C}}^m)$ with $\lambda ⊢n$ and at most $m$ rows.

The symmetric algebra on $V\oplus {\Lambda }^{2}V$ contains every irreducible polynomial representation of $GL(V)$ exactly once. Its torus character is ${\prod }_i(1-x_i{)}^{-1}{\prod }_{i<j}(1-x_i{x}_j{)}^{-1}$, which equals ${\sum }_{\lambda }{s}_{\lambda }(x)$ by the Schur identity. Complete reducibility and linear independence of Schur characters identify the summands. This is a formal graded identity: each fixed scalar degree is finite, so no analytic convergence is needed. Multiplicity one refers to irreducible modules, not arbitrary reducible representations.

For a partition of an integer $\lambda ⊢n$, applying a Young symmetrizer of that shape to the tensor power of each vector space gives a Schur functor. The construction respects linear maps because their tensor powers commute with place permutations. For single rows it gives symmetric powers, and for single columns exterior powers.

A Schur module is the value of a Schur functor on a vector space. Over the complex numbers it is either zero or an irreducible polynomial representation of the general linear group. Its character is the Schur polynomial $s_{\lambda }$ in the eigenvalues, and its highest weight is $\lambda$ padded with zeros.

A column of length greater than $m$ antisymmetrizes more than $m$ vectors, giving zero. When there are at most $m$ rows, place the $i$th basis vector in every tensor position of row $i$. Row symmetrization multiplies by a nonzero factorial product, and column antisymmetrization is nonzero because the vectors in every column are distinct. This proves the precise nonvanishing criterion for a Schur module.

The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over $S(m,n)$ correspond to homogeneous degree-$n$ polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra $\mathbb{C}{S}_n$, and take its endomorphism algebra.

The commutant of the permutation action on $V^{\otimes n}$ identifies with the symmetric tensors in $\mathrm{End}(V{)}^{\otimes n}$. Polarization spanning of symmetric tensors spans it by $a^{\otimes n}$. Interpolating $(a+tI{)}^{\otimes n}$ at $n+1$ values where $a+tI$ is invertible expresses each such power using invertible ones. Hence the Schur algebra is the linear span of the diagonal general linear action.

The commuting actions of the symmetric group and the general linear group on a tensor power are mutual commutants. The displayed sum ranges over partitions with at most $\dim V$ rows. The Specht modules and Schur modules are the simple factors for the two actions. In particular a primitive Young symmetrizer selects one copy of the matching Schur module.

If $\sigma$ has ${\alpha }_r$ cycles of length $r$, contraction of matrix entries around its cycles gives $\mathrm{tr}(\sigma {\xi }^{\otimes n})={\prod }_r\mathrm{tr}({\xi }^r{)}^{{\alpha }_r}$. The formula holds for every endomorphism, not only diagonalizable ones. The Schur–Weyl duality decomposition equates it with a sum of products of Specht and Schur characters.

The standard-tableau right-ideal decomposition of $\mathbb{C}{S}_n$, tensored over that algebra with $V^{\otimes n}$, gives this direct sum of general linear group modules. With $e_t=h_t/H_{\lambda }$, the map $e_t\mathbb{C}{S}_n\otimes T\to e_{t}T$ is an isomorphism. Individual summands generally need not be symmetric group submodules; instead each is a Schur module for the commuting action.