A nonzero polynomial is factorwise separable if every irreducible factor is a separable polynomial. Unlike the convention that all roots must be distinct, this condition allows repeated irreducible factors. Its splitting field is a separable field extension precisely when the polynomial is factorwise separable: in one direction every root has a separable minimal polynomial; in the other, adjoining finitely many roots with separable minimal polynomials gives a separable extension. Thus the implication from a separable splitting extension to a polynomial with distinct roots needs the extra squarefree hypothesis. The example $(X-1{)}^2$ already distinguishes the conventions.

Consider an implicit centered heat stencil of width $\Delta x$ weighted by $\theta$, combined with an explicit centered stencil of width $2\Delta x$ weighted by $1-\theta$. With $\mu =\kappa \Delta t/(\Delta x{)}^2$, its amplification factor is $G(\xi )=[1-\mu (1-\theta ){\sin }^2\xi ]/[1+4\mu \theta {\sin }^2(\xi /2)]$. For $0\le \theta <1/2$, contraction in the discrete ${\ell }^2$ norm holds exactly under the displayed condition. It is unconditional for $1/2\le \theta \le 1$. Put $v={\sin }^2(\xi /2)$: the condition $G\ge -1$ reduces to $4\mu v[(1-2\theta )-(1-\theta )v]\le 2$. Maximizing this concave quadratic proves the threshold; the Parseval identity converts the multiplier bound into contraction.

Choosing uniformly among currently unused neighbours gives a self-avoiding walk with a path-dependent probability: $a_j$ is the number of choices available at step $j$. In the square lattice, a four-step straight path has probability $1/(4\cdot 3^3)$, whereas the path $(0,0),(1,0),(1,1),(0,1),(-1,1)$ has probability $1/(4\cdot 3^2\cdot 2)$. There is no trapping at that length, and these unequal probabilities disprove uniformity.

When a trapped attempt is restarted, the final law is the one-attempt law conditional on successful completion. This divides successful-path probabilities by a common normalizing constant and does not remove their unequal products. The local uniformity of each move therefore does not imply uniformity over complete paths.

A self-avoiding polygon on a lattice is a closed nearest-neighbour walk whose vertices are distinct except for the final repetition of the starting point. On the square lattice a nondegenerate polygon has even perimeter at least four, since each step changes the parity of the sum of the coordinates. One can generate such a polygon by generating a self-avoiding walk with $M-1$ edges, retaining it only when its endpoint is a neighbour of its origin, and appending the closing edge. The resulting distribution inherits the bias of the walk sampler unless an additional correction is applied.

A nonnegative second variation requires its highest-derivative coefficient $P=f_{vv}$ to be nonnegative. If it were negative on an interval, a smooth compactly supported factor times $\sin (nx)$ would have negative derivative energy of order $n^2$, while its zero-order energy stays bounded. Large $n$ would contradict nonnegativity. This is a local necessary condition; it does not rule out negative directions caused by the lower-order terms.

For fixed-endpoint variations, a second variation is coercively positive if the displayed estimate holds for one constant $c>0$ and every admissible variation. The endpoint Poincaré inequality makes this equivalent to a positive lower bound in the $W^{1,2}$ norm. It is stronger than strict positivity on each nonzero variation: $q(\phi )=\int {\phi }^2$ is strictly positive, but high-frequency variations make its ratio to $\int {\phi }^{\prime 2}$ tend to zero.

For $q=\int (p{\phi }^{\prime 2}+2r\phi {\phi }^{\prime }+s{\phi }^2)$ with continuous coefficients, coercive positivity implies $p\ge c$. Indeed, if $p-c<0$ on an interval, multiply a smooth function supported there by $\sin (nx)$. Its negative highest-derivative contribution has order $n^2$, while the mixed term has order at most $n$ and the remaining term is bounded. This contradicts the displayed inequality for large $n$. In particular coercive positivity implies the Strengthened Legendre condition, whereas mere strict positivity does not.

No cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work. If the reservoir loses heat $Q>0$ at temperature $T$, its entropy change is $-Q/T$ while the cyclic device returns to its starting state. A sole-effect conversion would therefore reduce total entropy.

With the square root positive for $Z$ on the positive imaginary axis, the displayed conformal map takes the upper half-plane bijectively to $\{\mathrm{Re}z>0\}\setminus [1,\infty )$. Squaring, subtracting from one, taking the positive-real-part square root, and reciprocating identifies each intermediate slit domain. The boundary prevertices $-1,0,1$ correspond to the lower infinite end, the slit tip at one, and the upper infinite end. The inverse is $\sqrt{1-z^{-2}}$ with positive imaginary part.

The upper half-plane joined to the downward strip $-1<x<1$ has reentrant corners at $\pm 1$. The Schwarz-Christoffel formula gives $z^{\prime }(Z)=2R/(\pi Z)$, with $R\sim Z$ at upper infinity. Since $(R+{\tan }^{-1}(1/R){)}^{\prime }=R/Z$, integration gives the displayed map. The inverse trigonometric function is continued analytically in the upper half-plane so its corner values are $\pm \pi /2$. The origin is the lower channel end and infinity the upper open end. The complex potentials $2Z/\pi$ and $2\log Z/\pi$ respectively normalize unit horizontal velocity in the upper region and unit upward velocity in the channel.

The principal square root maps the upper $Z$ half-plane bijectively onto $\Re z>0$ with the ray $[1,\infty )$ removed. The prevertex zero is the slit tip, and $\pm 1$ represent the two ends at infinity. The inverse is the unique root $Z^2=1-z^{-2}$ with positive imaginary part. Pulling back ${\pi }^{-1}\arg ((Z-1)/(Z+1))$ gives the bounded harmonic function with value one on the slit banks and zero on the imaginary axis. The harmonic function $\Im [Z/(1-Z^2)]$ has zero finite boundary values but grows at infinity, explaining the role of boundedness.

For a smooth bounded domain and a bounded continuous scalar function $f$, this problem has a weak solution. Let $S$ be the zero-Dirichlet inverse of $-\Delta$, defined by the Lax-Milgram theorem. The Poincaré inequality gives $\Vert Sh{\Vert }_2\le C_P^2\Vert h{\Vert }_2$. The Rellich-Kondrachov compactness theorem makes $S:L^2\to L^2$ compact. The Nemytskii operator $N_f$ is continuous and has bounded image, so $T=S\circ N_f$ is a nonlinear compact operator mapping a sufficiently large closed $L^2$ ball into itself. The Schauder fixed-point theorem gives a fixed point, necessarily in $H_0^1$, and its defining identity is the desired weak formulation. This proves existence; uniqueness requires additional assumptions.

Let $G:[0,1{]}^2\to \mathbb{R}$ and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be continuous. For a fixed $R>0$, put $K=\max |G|$ and $M={\max }_{y\in [0,1],|s|\le R}|f(y,s)|$. The integral equationhas a continuous solution with $\Vert u{\Vert }_{\infty }\le R$ whenever $|\mu |KM\le R$. The integral kernel is $G$; it is not the null space of a linear map.

Define the right side as $Tu$. Its supremum norm is at most $|\mu |KM$, so it maps the closed radius-$R$ ball in $C([0,1])$ into itself. Uniform continuity of $f$ on its compact box gives continuity of $T$. Uniform continuity of $G$ makes the image equicontinuous as well as bounded. The Arzelà-Ascoli theorem makes the image relatively compact, and the Schauder fixed-point theorem supplies a fixed point.

No uniqueness follows. For $G=1$, $f(y,s)=1+s^2$, and $0<\mu <1/2$, every solution is constant and its value satisfies $c=\mu (1+c^2)$, which has two real roots. A suitable ball captures the small root while the other lies outside; the existence proof does not rule it out.

For a countable minimal total family in a normed space in a Banach space, let $(x_n,f_n)$ be its unique biorthogonal system and let $P_{n}x={\sum }_{j\le n}{f}_j(x)x_j$. Uniform boundedness of these projections implies $P_{n}x\to x$ for every $x$: approximate $x$ by a finite linear combination $y$, eventually fixed by $P_n$, and bound the error by $(1+{\sup }_n\Vert P_n\Vert )\Vert x-y\Vert$. Conversely, if every vector has a norm-convergent expansion in this family, applying each continuous $f_j$ shows that its coefficients must be $f_j(x)$. Thus $P_{n}x\to x$, so the projections are pointwise bounded. The Uniform boundedness principle gives their uniform operator-norm bound. This proves the criterion and uniqueness of all expansions.

A sarcoma showing fibroblastic differentiation. The microscopic diagnosis distinguishes malignant fibrous-tissue growth from a benign fibroma or nonneoplastic scar.

Removal of excreta and wastewater through drains and sewers using water as the carrying medium. Conveyance and safe final treatment are different requirements.

A historical excreta-management approach using dry Earth in receptacles, together with organized removal.

Wastewater containing domestic or other wastes, commonly including human excreta. Treatment separates and transforms contaminants before discharge or reuse, connecting waste handling with protection of water sources.

Application of sewage to land; hydraulic suitability alone does not establish microbiological safety.

Reduction or inactivation of microorganisms on materials or in an environment to a specified purpose. It differs from removal of visible dirt and from sterilization, which addresses a more comprehensive microbial endpoint.

For a binary relation $S$, no object $b$ can satisfy $S(b,p)\leftrightarrow \neg S(p,p)$ for every $p$. Putting $p=b$ would give $s\leftrightarrow \neg s$ for $s=S(b,b)$. The forward direction derives $\neg s$ by a negation introduction subproof assuming $s$; the reverse direction then gives $s$, a contradiction. Thus this argument even avoids needing the law of excluded middle. The familiar interpretation reads $S(b,p)$ as “$b$ shaves $p$”; the logical obstruction is self-application rather than a physical peculiarity of shaving.