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Suppose a topological space admits an unbounded continuous function $f$ to $\mathbb{R}$. Set $u=|f|$ and choose $x_n$ so that $t_n=u(x_n)$ increases to infinity. There is a continuous function $h:[0,\infty )\to \mathbb{R}$ with $h(t_n)=(-1{)}^{n}n$: interpolate linearly between successive prescribed values and extend constantly below $t_1$. Thenis continuous and takes values strictly between $-1$ and $1$, while its values on the even and odd chosen sequences tend to $1$ and $-1$. Thus neither bound is attained. In particular, unbounded continuous witnesses to noncompactness of a real subspace give this stronger failure of the extreme value theorem for every noncompact real subspace.

In a finite perfect-information game without chance moves, solve terminal decision points first. At each earlier decision point, its player chooses an action maximizing its own payoff among the already solved continuations. Record a choice in every subgame, including those not reached by the root choice. This constructs a subgame perfect equilibrium. Tied choices may yield several equilibrium strategies and several outcomes. The name refers to Ernst Zermelo; the computational procedure is an application of backward induction.

Two players alternate proposals for dividing a pie, with a fixed final round. Agreement ends the game; rejection shrinks the next available pie by a factor $0<\alpha <1$. Final rejection gives both players zero. Offers range continuously over the entire available pie. Backward induction in games determines immediate agreement and the proposer's share through the continuous-offer bargaining acceptance lemma and the finite-horizon bargaining parity effect. Discrete offer grids can change both the boundary behavior and the exact shares.

A leading fixed-discharge, level-surface approximation to a shallow erodible-bed flow gives perturbation equations $C_x=4U-C$ and $U_t+3\beta U_x=-4U+C$. The displayed Fourier growth rate has negative real part for every nonzero wave number, independently of bedload parameter $\beta$. Bedload changes the propagation speed but not this damping. The zero-wave-number mode is neutral, so absence of growing modes should not be confused with strict asymptotic decay of every spatially uniform perturbation.

At an endpoint of a second-derivative variational problem, a free value $y$ gives the first displayed condition, while a free slope $y^{\prime }$ gives the second. Each condition is required only when its corresponding endpoint variation is free; a prescribed value or slope instead forces that variation to vanish. For $F=K(y^{\prime \prime }{)}^2/2+qy$, the two conditions become $y^{\prime \prime \prime }=0$ and $y^{\prime \prime }=0$, respectively. A clamped beam has both value and slope fixed, a pinned beam fixes value but leaves slope free, and a free end leaves both free.

For a smooth functional $\int F(x,y,y^{\prime },y^{\prime \prime })dx$, let $P=F_{y^{\prime }}$ and $Q=F_{y^{\prime \prime }}$. Along an extremal, the Euler-Lagrange equation for a second-derivative functional is $F_y-P^{\prime }+Q^{\prime \prime }=0$. Differentiating $H=y^{\prime \prime }Q+y^{\prime }(P-Q^{\prime })-F$ by the chain rule givesConsequently $H$ is constant whenever $F$ has no explicit dependence on $x$. This extends the Beltrami identity to a second-derivative variational problem. If $F$ is independent of $y^{\prime \prime }$, the expression reduces to $y^{\prime }{F}_{y^{\prime }}-F$.

With values $y(0)=y(\pi /2)=y^{\prime }(\pi /2)=0$ and $y^{\prime }(0)=1$, the higher-order Euler-Lagrange equation is $y^{(4)}-y=0$. Write $c=\cosh (\pi /2)$ and $s=\sinh (\pi /2)$. Solving the four linear boundary equations gives $y=[c(\cos x-\cosh x)+(1-s)\sin x+(1+s)\sinh x]/2$. The added function $x^2$ does not affect the first variation.

A sublattice is a subgroup of a Euclidean lattice. It is a Euclidean lattice in its real linear span, which may have smaller dimension than the original ambient Euclidean space. The equal-parity integer points in ${\mathbb{Z}}^d$ form a full-rank sublattice of index $2^{d-1}$: reduction of all coordinate differences modulo two gives the quotient.

There are infinitely many primes in each of the classes one and three modulo four. For the first, a hypothetical finite list $p_1,\dots ,p_r$ is contradicted by any prime divisor $q$ of $(2p_1\cdots p_r{)}^2+1$: it is odd, not on the list, and admits a square root of minus one, so $q\equiv 1(\mathrm{mod}4)$. For the second, $4p_1\cdots p_r-1$ is three modulo four; its prime factorization must include a prime that is three modulo four, and none of the listed primes divides it. These special congruence classes therefore admit elementary product-plus-or-minus-one arguments without the general theorem on primes in arithmetic progressions.

A cyclic redundancy check represents a bit string as a polynomial over the field with two elements and uses a polynomial remainder as checking information. Agree on a monic polynomial $G(x)$ of polynomial degree $r$ with constant term $1$. Under the basic systematic convention, a message polynomial $M(x)$ producesHere $\mathrm{deg}R<r$, and addition equals subtraction over the field with two elements, so $G$ divides the transmitted polynomial $C$. The receiver tests the polynomial remainder on division by the same CRC generator polynomial.

For example, $G=x^3+x+1$ and $M=x^2+1$ give $R=x^2$ and $C=x^5+x^3+x^2=x^{2}G$. If an error polynomial $E$ changes $C$ to $C+E$, the error is undetected exactly when $G$ divides $E$. Thus the CRC generator polynomial controls which bit-error patterns the cyclic redundancy check detects. This basic convention fixes the mathematics; implementations can additionally specify initial register contents and bit-order conventions.

A checksum is a short value calculated from a data block and stored or transmitted with it to help detect corruption. The receiver recomputes the checksum from the received data and compares it with the supplied value. Different data blocks can have the same checksum, so a successful comparison does not prove equality.

For example, summing four bytes modulo $256$ gives a one-byte checksum. The bytes $10,20,30,40$ give $100$; changing the first byte to $11$ changes the checksum to $101$, whereas exchanging two bytes leaves it unchanged. The particular checksum determines which error patterns are detected.

The Internet checksum groups bytes into 16-bit words, adds them with end-around carry, and complements the result. End-around carry adds an overflow bit back into the low end of the sum. Verification includes the checksum field and expects a sum with all bits set. The covered fields depend on the protocol; IPv6 has no header checksum.^[ref][ref]

The corpus spongiosum is erectile tissue surrounding the penile urethra and continuing into the glans. Its arrangement differs from the paired corpora cavernosa. Supporting the urethral passage is mechanically compatible with erectile change without requiring identical pressure in all compartments.

Let a polynomial $q$ of degree at most $n$ have errors $f(x_j)-q(x_j)=(-1{)}^{j}h$ at $n+2$ ordered distinct points. Then $|h|$ bounds below the error $E_n(f)$ of the best uniform approximation. Otherwise an improved polynomial $p$ with error strictly below $|h|$ would make $p-q$ alternate strictly in sign at all those points, giving at least $n+1$ distinct roots of a polynomial by the intermediate value theorem. Its degree is at most $n$, a contradiction. Moreover, if $q$ is not optimal, $|h|<E_n(f)$. In the equality case, an optimal $p$ gives weak alternating signs of $p-q$; its vanishing order-$(n+1)$ divided difference has same-sign summands, forcing every nodal value to vanish and hence $p=q$.

On $[0,1]$, the chord of $\sqrt{1-x}$ is $1-x$. Its nonnegative gap has maximum $1/4$ at $x=3/4$, as differentiation shows. Shifting the chord upward by $1/8$ gives error $-1/8,1/8,-1/8$ at $0,3/4,1$. The Chebyshev alternation theorem therefore proves that $9/8-x$ is the unique best linear uniform approximation, with error $1/8$. Endpoint differentiability is unnecessary.

A homogeneous equilibrium in a spatial evolution equation is a steady state independent of position. Its stability against homogeneous perturbations need not imply stability against spatially varying modes.

For the equal superposition of the first two box states with zero initial relative quantum phase, put $\Omega =(E_2-E_1)/\hslash$. The probability in the left half is $P_{\mathrm{L}}=1/2+4\cos (\Omega t)/(3\pi )$. The continuity equation and zero wall probability current implyDirect substitution of $\psi (L/2,t)=L^{-1/2}{e}^{-i{E}_{1}t/\hslash }$ and ${\partial }_x\psi (L/2,t)=-2\pi L^{-3/2}{e}^{-i{E}_{2}t/\hslash }$ into the probability current verifies the sign. Positive current means probability leaves the left half. The quantum interference controls transport even though each component has stationary probability density.

A spermatocele is a cystic collection associated with the epididymal region that contains sperm-bearing fluid. It is an extratesticular structure, distinct from a mass arising within the testis. Its name describes the contained material and origin rather than a general term for every scrotal swelling.

Epididymitis is inflammation of the epididymis. The term identifies the affected structure and inflammatory process, not one universal organism or cause. Its anatomical location distinguishes it from orchitis, which affects the testis, although the two processes can coexist.

The sum of relative atomic masses for the simplest integer composition represented by an empirical formula.

For a uniformly elliptic second-order Dirichlet form with bounded coefficients and lower-order terms, there are $a>0$ and $b\ge 0$ giving the displayed lower bound. Cauchy-Schwarz and Young's inequality absorb the drift term into part of the principal gradient energy plus an $L^2$ correction. Adding a sufficiently large nonnegative multiple of the $L^2$ inner product therefore makes the form coercive, permitting Lax-Milgram. The shift restores coercivity even when the unshifted form is not positive.