Every endomorphism of the additive finite cyclic group $C_n=\mathbb{Z}/n\mathbb{Z}$, for a positive integer $n$, has the form $f_a(k)=ak(\mathrm{mod}n)$, determined by $a=f_a(1)$. This is the self-map case of homomorphism from a finite cyclic group: repeated addition forces the value at every element. Conversely, each choice of $a$ respects addition and gives a well-defined map on congruence classes.

Let $d=\gcd (n,a)$, the greatest common divisor. The image of a group homomorphism is the cyclic subgroup generated by $a$, with size $n/d$ by order of an element of a finite cyclic group. The kernel of a group homomorphism isIndeed, $ak=0(\mathrm{mod}n)$ is equivalent to $n/d$ dividing $k$, after cancelling $d$ and using coprimality. Consequently $f_a$ is a group automorphism exactly when $a$ and $n$ are coprime. In that case a modular inverse $b$ of $a$ gives $f_a^{-1}=f_b$. This connects reversible self-maps directly to the generators of a finite cyclic group.^[ref]

Bright elliptical galaxies commonly have a shallow central surface-brightness cusp inside a break radius, whereas many less luminous elliptical galaxies have a steep cusp continuing inward. The latter can have many more luminous tracers at a fixed small radius, helping central black hole dynamical measurements. This empirical distinction is not a universal classification by mass alone. Measurements of central galaxy profiles quantify the contrast.

The Faber-Jackson relation is an empirical correlation between luminosity and stellar velocity dispersion for elliptical galaxies, approximately $L\propto {\sigma }^4$ in its classical form. Slope, scatter, sample and band matter. A heuristic combines $M\propto {\sigma }^{2}R$, common mass-to-light ratio, and common characteristic surface brightness $L/R^2$. The luminosity-dispersion correlation alone does not imply those auxiliary assumptions, and extrapolation to a dwarf spheroidal galaxy is not guaranteed.

The fundamental plane relates effective radius, stellar velocity dispersion, and mean surface brightness of elliptical galaxies. Homologous virial equilibrium with constant mass-to-light ratio predicts $R_e\propto {\sigma }^2/I_e$; empirical exponents differ. The two-variable Faber-Jackson relation is a projection, not the entire plane.

Elliptic-unit distribution gives norm-compatible units indexed by auxiliary squarefree ray moduli, with norm factor $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$ at a newly added prime. Kolyvagin derivative operators turn these units into classes with controlled valuations, and suitable auxiliary primes chosen using Chebotarev density theorem bound class-group characteristic ideals. The construction supplies the difficult divisibility in the CM Iwasawa main conjecture.

Associated with a period lattice are the Weierstrass sigma, zeta, eta and elliptic functions. They satisfy $\zeta ={\sigma }^{\prime }/\sigma$ and $\wp =-{\zeta }^{\prime }$; the eta constants record quasi-period increments of the Weierstrass zeta function.

The Jacobi elliptic functions include $\mathrm{sn}(u,k)$, $\mathrm{cn}(u,k)$ and $\mathrm{dn}(u,k)$, with ${\mathrm{sn}}^2+{\mathrm{cn}}^2=1$ and ${\mathrm{dn}}^2+k^2{\mathrm{sn}}^2=1$. Their ratios give the other standard members. The Jacobi elliptic sine is the inverse of the elliptic integral of the first kind.

For $L=a^{ij}{D}_{ij}+b^i{D}_i$ with bounded coefficients and uniform ellipticity near a boundary point $y$ satisfying the interior sphere condition, suppose $u\in C^2(\Omega )\cap C^0(\Omega \cup \{y\})$ has $Lu\ge 0$ and $u(x)<u(y)$ in a tangent open ball. With $\mathbf{n}$ the outward unit normal of that ball, ${\mathrm{liminf}}_{t\downarrow 0}[u(y)-u(y-t\mathbf{n})]/t>0$. Consequently the outward normal derivative is positive whenever it exists. The given boundary continuity alone does not assert that the derivative exists. The minimum version follows by applying this to $-u$; operators with a zeroth-order term require additional sign hypotheses.

For $n\ge 2$, the standard ellipsoid covering a centrally cut half of the unit ball has center $e_1/(n+1)$ and shape matrix $D^{\prime }=\frac{n^2}{n^2-1}(I-\frac{2}{n+1}{e}_1{e}_1^T)$. Its eigenvalues are $n^2/(n+1{)}^2$ along $e_1$ and $n^2/(n^2-1)$ in the other directions. Hence its volume ratio is $R=\frac{n}{n+1}(\frac{n^2}{n^2-1}{)}^{(n-1)/2}$. Using $\log (1-a)<-a$ and $\log (1+b)<b$ gives $\log R<-1/(2(n+1))$. Affine changes of coordinates preserve the volume ratio. The resulting geometric convergence underlies the ellipsoid method; exact rational feasibility additionally needs input-dependent size, precision and lower-dimensional feasibility bounds.

A polynomial-time algorithm for integer linear feasibility gives exact rational linear programming optimization. Rational vertex bounds give a polynomial-bit bound on every finite optimal value and its denominator. An objective threshold below that bound detects unboundedness after feasibility is established. Binary search with the additional inequality $c^{T}x\le t$ localizes the finite optimum to an interval containing a unique rational of bounded denominator; continued fractions recover it exactly. A feasible optimizer can then be found by successive coordinate minimizations on a bounded optimal face. These produce faces of one fixed rational polytope, so one uniform vertex-denominator bound works throughout, avoiding exponential growth of precision requirements.

For an integer linear polyhedron $P=\{x\in {\mathbb{R}}^n:Ax\ge b\}$ with coefficients bounded by $U\ge 1$, put $D=[(n+1)U{]}^{n+1}$ and $0<\epsilon <1/D$. If $x_0\in P$, every $h$ with $\Vert h{\Vert }_2\le \epsilon /(nU)$ satisfies $A_{i}h\ge -\sqrt{n}U\Vert h{\Vert }_2\ge -\epsilon$. Thus $P_{\epsilon }$ contains a positive-radius Euclidean ball and is a full-dimensional linear polyhedron. If $P$ is empty, the normalized integer Farkas infeasibility gap gives $\lambda \ge 0$, $A^T\lambda =0$, $1^T\lambda =1$ and $b^T\lambda \ge 1/D$. A point of $P_{\epsilon }$ would give $0={\lambda }^{T}Ax\ge b^T\lambda -\epsilon >0$, a contradiction. Consequently this relaxation preserves emptiness and supplies the inner-volume bound needed by the ellipsoid method, even when $P$ has empty interior.

A nonempty linear polyhedron can have no extreme point. Intersect it with an orthant containing one feasible point, and minimize the sum of the signed coordinates there. The relevant sublevel sets are compact, so the minimum face has an extreme point, which is also extreme in the orthant intersection. The added signed-coordinate rows are integer rows with entries at most one. The integer-matrix vertex coordinate bound gives $R=n!\max (1,U{)}^n$. This supplies a bounded feasibility search even when the original polyhedron contains lines.

A spheroid is an ellipsoid of revolution obtained by rotating an ellipse about a principal axis. Rotation about the shorter axis gives an oblate spheroid; rotation about the longer axis gives a prolate spheroid. Thus the oblate form is a member of the general family.

The John ellipsoid of a convex body is its inscribed ellipsoid of largest volume. For the unit ball $K$ of a $d$-dimensional normed vector space, change coordinates so that this ellipsoid is the Euclidean unit ball. The resulting coordinates are called John position. In these coordinates the John contact decomposition implies $B_2^d\subseteq K\subseteq \sqrt{d}{B}_2^d$, or $\Vert x{\Vert }_E\le |x|\le \sqrt{d}\Vert x{\Vert }_E$.

A solid ellipsoid is the filled region bounded by an ellipsoid. The open version centered at the origin iswhere $Q$ is a real symmetric positive-definite matrix; replacing $<$ by $\le$ includes the boundary. Its semi-axis lengths are ${\lambda }_i^{-1/2}$, where ${\lambda }_i$ are the eigenvalues of $Q$. In three dimensions its volume is $4\pi /(3\sqrt{\det Q})$, by a change of variables from the unit ball.

In even-dimensional symplectic geometry, a uniform dilation $x\mapsto tx$, $t>0$, multiplies the symplectic capacity by $t^2$. This concerns one common factor in every coordinate. Independent changes in the different semi-axis lengths are not a uniform dilation and have no such common square factor.

For a positive-definite symmetric matrix $A$, the ellipsoid $x^{T}Ax=1$ has nearest points to the origin exactly at $x={\lambda }_{\max }^{-1/2}u$, where $u$ ranges over unit vectors in the largest-eigenvalue eigenspace. Indeed, the spectral theorem for real symmetric matrices gives $x^{T}Ax\le {\lambda }_{\max }|x{|}^2$, with equality precisely in that eigenspace. A simple largest eigenvalue gives two opposite nearest points; multiplicity $m>1$ gives a scaled $(m-1)$-sphere of nearest points.

For an Elitzur-Vaidman bomb tester using a rotation and its inverse, let $p$ be the probability of entering the absorbing path and $q=1-p$. Conditional on a live absorber, one trial has explosion probability $p$, successful safe identification probability $pq$, and inconclusive probability $q^2$. The inconclusive projective measurement resets the input, so summing a geometric series gives the displayed success probability after at most $N$ trials. Its limit is $q/(1+q)$, approaching one half as $p$ tends to zero through positive values. Exactly zero coupling produces no information.

Circle reduction splits the metric into a ten-dimensional graviton, one vector and one scalar, and the three-form into a three-form and two-form. Seven-torus reduction gives a four-dimensional graviton, $7+21=28$ vectors, $28+35$ direct scalars and seven two-forms. The two-form scalar duality turns the latter into seven more scalars, giving $70$ in all. A remaining four-dimensional three-form has no local polarization. These are zero-mode counts without fluxes, orbifold projections or moduli freezing.

A product compactification supported by a four-form proportional to one factor's volume form. For the maximally supersymmetric membrane throat, the other factor is a round $S^7$ and the radii obey $R_{S^7}=2R_{\mathrm{AdS}}$. The flux links the curvature scales through the supergravity field equations.

A nondegenerate elementary topos is two-valued when its terminal object has exactly the empty and whole subobjects. This is a global-truth property, distinct from being a Boolean topos, which requires complements for all subobjects of all objects.