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For the displayed integral equation, reduce its polynomial modulo a prime number $p$. Singular points satisfy $y^2-x^3-c=-3x^2=2y=0$. For $p>3$, they exist exactly when $p\mid c$, and the sole point is $(0,0)$. At $p=2$, the sole point over ${\mathbb{F}}_2$ is $(0,\overline{c})$; at $p=3$, it is $(-\overline{c},0)$ because cubing is the identity on ${\mathbb{F}}_3$. For $c=17$, this gives primes $2,3,17$ and points $(0,1),(1,0),(0,0)$ respectively. This concerns the specified plane equation; intrinsic good or bad reduction of an elliptic curve can require choosing a different integral model, especially at primes two and three.

Every invertible square matrix is a product of elementary matrices. Gaussian elimination reduces $A$ to the identity matrix by elementary row operations, giving $F_r\cdots F_{1}A=I$. Reverse their order and invert them to write $A=F_1^{-1}\cdots F_r^{-1}$. Each inverse is again elementary. Conversely every elementary matrix is invertible, so every such product is invertible. The identity may be represented by the empty product.

Elbow flexion decreases the angle between upper arm and forearm at the elbow. Contributors include the brachialis and biceps brachii. Their mechanical contributions change with joint position and, for the latter, forearm orientation.

Elbow extension increases the angle between upper arm and forearm at the elbow. The triceps brachii is its principal muscular contributor. The resulting movement depends on joint position, external load and the line of pull.

The olecranon is the proximal projection of the ulna forming the point of the elbow. Attachment of the triceps brachii connects contraction to elbow extension, illustrating how a bony projection supplies a mechanical lever.

Stop the four-ball Ehrenfest urn model upon reaching count zero or four. From count $r$, the transitions are to $r-1$ with Probability $r/4$ and to $r+1$ with Probability $(4-r)/4$. First-step analysis with absorbing boundary values gives the displayed vector $h$ of probabilities of absorption at zero and vector $t$ of expected hitting times of either endpoint. For instance, $t_1=1+3t_2/4$ and $t_2=1+(t_1+t_3)/2$, with $t_1=t_3$, give $t_1=7$. From any interior state, absorption has Probability at least $1/4$ within two steps; the resulting geometric tail bound proves both almost-sure absorption and finiteness of the expectations.

Let $A$ be a doubly stochastic matrix, let $q$ be a probability vector, and put $w=Aq$. In row $i$ with $w_i>0$, define $r_i(j)=a_{ij}{q}_j/w_i$ on the support where $a_{ij}>0$. The positive row entries ${\alpha }_i=(a_{ij})$ form a probability vector. Expanding relative entropy and using the column sums givesZero-output rows contribute zero because all their $a_{ij}{q}_j$ vanish. The row sums supply the normalized comparison laws, while the column sums preserve the input entropy contribution. Equality holds exactly when $q_j=w_i$ for every positive matrix entry $a_{ij}$, including the automatic zero values in zero-output rows.

For a Lebesgue measurable set $E\subseteq \mathbb{R}$ of finite measure, the displayed function satisfies $|H_E(t)-H_E(s)|\le |t-s|$. Indeed, for $s<t$, finite additivity and monotonicity of a measure give $H_E(t)-H_E(s)=m(E\cap (s,t])\le t-s$. Thus $H_E$ is a nondecreasing Lipschitz function and is uniformly continuous even if $E$ is unbounded.

Continuity from above of a measure gives $H_E(t)\to 0$ as $t\to -\infty$, while continuity from below of a measure gives $H_E(t)\to m(E)$ as $t\to \infty$. Every strictly intermediate value is attained by the intermediate value theorem. The corresponding half-line cut of $E$ realizes any prescribed measure between zero and $m(E)$; the endpoints are realized by the empty set and $E$ itself.

A smooth embedded invariant two-torus in three-dimensional Euclidean phase space bounds a solid torus of positive volume. The vector field is tangent to its boundary, so its normal flux is zero. If the divergence is strictly negative throughout the enclosed volume, the divergence theorem gives a strictly negative flux, a contradiction. Thus a globally negative-divergence three-dimensional vector field has no such quasiperiodic invariant torus. This argument is stronger than merely noting contraction of transported phase volume; an attractor's own zero volume is not a contradiction.

For a normalized two-dimensional discrete Laplacian satisfying the discrete maximum principle, suppose quadratic functions are differentiated exactly and $L_{h}1=0$. In a domain of diameter $D$, choose $x_0$ in the domain and $\varphi (x)=|x-x_0{|}^2$. Then $0\le \varphi \le D^2$ and $L_h\varphi =4$. Applying the discrete elliptic barrier argument to zero-boundary error $L_{h}e=\tau$ gives the displayed bound. A mere sign pattern of the stencil does not determine this constant: rescaling the whole operator changes the error-to-residual ratio without changing the domain diameter.

A functional on a Banach space is Gâteaux differentiable at $v$ when the directional limits exist and form a bounded linear functional of $h$. Fréchet differentiability is stronger, requiring a remainder uniformly small compared with the norm of the increment. A quadratic norm bound on the Taylor remainder gives the stronger property directly. Mere existence of unrelated directional limits would not establish linear differentiability.

A monotone map preserving suprema of nonempty directed sets. Between pointed domains it need not send bottom to bottom. The pointwise directed supremum of a directed family of such maps is again Scott continuous: interchange the two directed suprema in ${\sup }_f{\sup }_{x\in A}f(x)$.

Take two increasing countable chains sharing a least element $\perp$ and a greatest element $\top$, with interiors $a_1<a_2<\cdots$ and $b_1<b_2<\cdots$ and no comparisons between branches. Each entire interior chain has $\top$ as its least upper bound. A directed set meeting both interiors must contain $\top$. Otherwise it lies within one branch, where it has either a greatest member or least upper bound $\top$. This proves that the partially ordered set is a directed-complete partial order.

The only element way-below $a_n$ is $\perp$. Indeed, an element way-below $a_n$ must be below $a_n$, but every positive $a_m\le a_n$ fails the way-below relation when tested against $\{b_1,b_2,\dots \}$: its least upper bound is $\top \ge a_n$, yet it has no member above $a_m$. Nonempty directed sets always contain an element above $\perp$, so $\perp \ll a_n$. Thus the elements way-below $a_n$ have least upper bound $\perp \ne a_n$, and this is not a continuous poset.

Two projective triangles are in perspectivity from a point when the three projective lines joining corresponding vertices meet in one projective point, called their center of perspectivity.

Two projective triangles are in perspectivity from a line when the three intersections of corresponding sides lie on one projective line, called their axis of perspectivity. In the nondegenerate setting this is dual to perspectivity from a point under point-line duality.

Every element of a pointwise definable model is uniquely specified by a formula with no added parameters. Finite tuples are then uniquely specified by conjunctions of their defining formulas. For a complete theory these formulas isolate the tuple types, so such a model is an atomic model. The converse fails: a countable dense linear order without endpoints with a named point is atomic, while its other points are moved by structure automorphisms fixing that point.

An injective self-map of the complex projective line preserving all cross-ratios is a projective transformation; neither continuity nor prior surjectivity is needed. Choose the unique projectivity agreeing with the map on three distinct projective points and compose with its inverse. The resulting map fixes those three points. Normalize them to $\infty ,0,1$. Preservation of the cross-ratio $(\infty 0:1z)=z$ now fixes every other point as well, proving that the normalized map is the identity map. Undoing the composition proves the assertion.

For the two-dimensional heat equation with zero Dirichlet boundary conditions, the Crank-Nicolson diffusion scheme preserves the interval between the minimum and maximum of the initial nodal values and zero under the displayed restriction. The old-time coefficients form a convex combination, while the new-time matrix has positive diagonal, nonpositive off-diagonal entries and row diagonal-dominance margin one. A hypothetical new value above the old maximum makes the new-time row at least that value while its right side is at most the old maximum, a contradiction; apply the reversed inequality at a new minimum. In particular initial nonnegative or nonpositive values preserve their sign. The step restriction guarantees these value bounds even though scalar A-stability holds without it.

Give the real interval $[0,1]$ its usual total order. Every nonempty directed set has a least upper bound in the real interval, so it is a directed-complete partial order. If $x<y\le \sup D$, some $d\in D$ exceeds $x$; otherwise $x$ would be an upper bound for $D$. Also $0\ll 0$, since every member of a nonempty $D$ is at least zero. For $x>0$, the increasing sequence $x(1-1/n)$, $n\ge 1$, has least upper bound $x$ but never reaches it, so $x/\ll x$. Together with $x\ll y\Rightarrow x\le y$, this proves the displayed way-below relation.

For each $y>0$, the elements way-below $y$ form $[0,y)$, a directed set with least upper bound $y$; for $y=0$ they form $\{0\}$. Thus this is a continuous poset. Its only compact element is zero, so it is not an algebraic partially ordered set.

If the ambient power set of $\omega$ is not well-orderable and $a\in L$, then the full ambient power set of $a$ belongs to $L$ exactly when $a$ is finite. Finite set operations give the forward construction for finite $a$. If $a$ is infinite, its constructible well-order gives an injection $\omega \to a$. The induced injection of power sets would make the ambient power set of $\omega$ well-orderable if $\mathcal{P}(a)$ belonged to $L$.