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For distinct interpolation nodes, write $D_i$ for the divided difference with $x_i$ omitted. Symmetry and the divided-difference recurrence give the displayed identity. Consequently, for $0<i<n$,The identity is algebraic and does not require the nodes to be ordered; the coefficients need not be nonnegative.

For a block code, separable distortion is the average of the individual symbol costs. With Hamming distortion it is the fraction of incorrectly reconstructed symbols, rather than the unnormalized number of mismatches. This normalization makes expected distortion comparable across different block lengths.^[ref]

Expected distortion is the expectation of the reconstruction cost under the joint probability distribution of $X$ and $\hat{X}$. It weights common source outcomes more strongly than rare ones. A bound on expected distortion is therefore different from a bound on the cost for every input, or on the probability of exceeding a specified cost.

The reconstruction alphabet is the set of values that the decoder may output. It need not equal the source alphabet. A restricted reconstruction alphabet can prevent zero expected distortion, even if the representation is allowed arbitrarily many bits.

Dissociation of an ionic compound separates its constituent ions. It differs from creating charged species by ionization of neutral molecules. Dissolved ions can also associate into ion pairs, so a dissociated description does not exclude all subsequent association.

With weak kinematic viscosity $\nu$, zero buoyancy diffusion and locally unchanged plane internal gravity wave polarization, the displacement pseudomomentum of an internal gravity wave obeys $P_t+{\partial }_z(c_{gz}P)=-\nu K^{2}P$. For constant wavenumbers and a stationary wave envelope, $P\propto |\hat{\sigma }{|}^2$ proves the displayed amplitude decay. Decay is along the group velocity, so $c_{gz}<0$ describes a source above the receiving level. In a slowly varying shear flow, use wave action and the changing intrinsic frequency rather than treating $P/|\hat{\sigma }{|}^2$ as constant.

Plumbing identifies a base disk times a fiber disk in one oriented rank-two disk bundle with such a neighborhood in another, exchanging base and fiber coordinates. Their zero sections then intersect transversely once. For disk bundles over $S^2$ arranged along a chain with Euler numbers $-a_i$, the intersection form has diagonal entries $-a_i$, adjacent entries one and other entries zero. The same manifold has a Kirby diagram consisting of adjacent Hopf-linked unknots with framings $-a_i$. Its boundary is described by the corresponding negative continued fraction, with the orientation fixed by the surgery convention.

Let $R$ be a one-dimensional normal Noetherian local domain. Choose $0\ne x$ in its maximal ideal $\mathfrak{m}$ and a minimal $n$ with ${\mathfrak{m}}^n\subseteq (x)$. If $n>1$, choose $y\in {\mathfrak{m}}^{n-1}\setminus (x)$ and put $z=y/x$. Then $z\notin R$ but $z\mathfrak{m}\subseteq R$. If $z\mathfrak{m}\subseteq \mathfrak{m}$, the determinant trick makes $z$ integral, contrary to normality. Thus $zu$ is a unit for some $u\in \mathfrak{m}$, and $v=u(zv)/(zu)$ for every $v\in \mathfrak{m}$ proves $\mathfrak{m}=(u)$. The case $n=1$ is already principal. Such a local domain is a discrete valuation ring.

For a normalized discrete valuation, a uniformizer $\pi$, and a chosen set $A$ of representatives of the residue field, every existing nonzero element has a unique displayed expansion, with its first digit not in the maximal ideal. Starting from $y_0={\pi }^{-v(x)}x$, choose $a_j$ representing $y_j$ modulo the maximal ideal and put $y_{j+1}=(y_j-a_j)/\pi$. The error after $r$ digits has valuation at least $v(x)+r$, so the sums converge to $x$ even without completeness. Conversely arbitrary digit sequences need not converge in an incomplete field: in $\mathbb{Q}$ with its three-adic valuation, the digits of $\sqrt{10}\in {\mathbb{Q}}_3$ give a counterexample. Hensel lemma gives this square root, but it is not rational.

A field with a surjective discrete valuation, complete for the associated non-Archimedean metric. Its valuation ring is a discrete valuation ring, and its maximal ideal is generated by a uniformizer. Completeness allows successive congruence solutions to converge to actual roots, which underlies the Hensel lemma. Non-Archimedean local fields are examples; completeness alone does not require the residue field to be finite.

Let the numerical support of a probability distribution be $x_1<\cdots <x_n$. Its cumulative distribution function counts how many values have been passed:It is zero below $x_1$, equals $j/n$ on $x_j\le t<x_{j+1}$ for $1\le j<n$, and is one at and above $x_n$. Every jump is $1/n$, even when the horizontal gaps differ.

For inverse transform sampling, take an ideal continuous uniform distribution for $U$ on $[0,1)$. Partition that interval into $n$ intervals of width $1/n$ and outputEach interval has probability $1/n$, proving the desired discrete uniform distribution. Using the floor function, the index is $j=1+\lfloor nU\rfloor$; for consecutive integer values the output is $X=a+\lfloor nU\rfloor$. With the usual quantile function convention $Q(u)=\inf \{t:F_X(t)\ge u\}$, one instead writes $Q(u)=x_{\lceil nu\rceil }$ for $0<u\le 1$, using the ceiling function. The endpoint conventions differ only at a finite set of points with probability zero under the ideal continuous uniform distribution.

For $S=\{1,2,4,8\}$, the four intervals $[0,1/4)$, $[1/4,1/2)$, $[1/2,3/4)$, and $[3/4,1)$ select $1,2,4,8$, respectively. Thus $U=3/10$ selects $2$, while $U=9/10$ selects $8$. The upper half of the interval selects values at least four, recovering probability $1/2$. Equal widths refer to the sampling probability coordinate, not to distances between output values. This is an exact mathematical construction assuming an ideal continuous uniform distribution; a finite computer random-number source needs its own justification of equal output chances.

Suppose the support of a probability distribution consists of all consecutive integers from $a$ to $b$, including both endpoints, with $a\le b$. Its cardinality is $n=b-a+1$, and a discrete uniform distribution on it satisfiesFor a short derivation, set $K=X-a$, whose possible values are $0,\dots ,n-1$. The finite sums giveand henceAdding $a$ shifts the expected value by $a$ and leaves the variance unchanged. For completeness, the first sum follows by pairing $k$ with $n-1-k$. For the second, sum $(k+1{)}^3-k^3=3k^2+3k+1$ from $k=0$ to $n-1$: the left side is $n^3$, so inserting the first sum gives the stated result.

The edge case $a=b$ has $n=1$: $X=a$ with probability one, expected value $a$, and variance zero. On an arithmetic progression $\{a,a+h,\dots ,a+(n-1)h\}$ with $h>0$, the same argument instead gives expected value $a+h(n-1)/2$ and variance $h^2(n^2-1)/12$. Equal spacing is the extra hypothesis behind these formulas; the discrete uniform distribution on $\{1,2,4,8\}$ requires the general sums above.

For $U_t=-cDU$ with $c>0$, a positive discrete norm matrix $H$ satisfying the displayed identity gives $d(U^{T}HU)/dt=-c(U_N^2-U_0^2)$. Homogeneous inflow data $U_0=0$ therefore leave only the nonpositive outflow flux. Inhomogeneous inflow contributes a controlled boundary forcing term. This estimate includes the numerical boundary closure, unlike an interior-only Fourier stability analysis. For any dissipative semidiscrete operator in the $H$ norm, Backward Euler method steps preserve the contraction by the identity $\Vert U^{n+1}{\Vert }_H^2-\Vert U^n{\Vert }_H^2=2k⟨U^{n+1},A{U}^{n+1}{⟩}_H-\Vert U^{n+1}-U^n{\Vert }_H^2$.

For discrete opfibrations $A\to D\leftarrow B$ with weakly filtered categories, the map taking a pair to its connected components of a category is a bijection. For surjectivity, send two base objects in the same component to a common target and lift both arrows. For injectivity, choose common targets upstairs for two representatives; a cocone over the resulting finite connected diagram downstairs makes their lifted composites agree. Hence the representatives have a common target in the pullback in a category. Over a filtered category, the component functor on the slice of discrete opfibrations preserves all finite categorical limits: it preserves pullbacks and takes the terminal object to a singleton.

An integrable predictable strategy whose discounted gains ${\varphi }_n\Delta {\tilde{S}}_n$ are integrable preserves the martingale property of discounted prices in finite discrete time. Bounded holdings suffice. Without integrability the claim is false: let ${\tilde{S}}_0=3/2$, ${\tilde{S}}_1=1+U$, ${\tilde{S}}_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ is an independent fair sign. Holding zero stock first and $1/U$ shares in the second period, funded by the bank, gives zero-initial-cost terminal wealth $\epsilon /(2U)$, which is not integrable. Replication pricing requires the corresponding true-martingale condition, not merely formal self-financing.

A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.

In the semistrip Laplace spectral global relation, reality makes $W(\pm \kappa )$ real for $\kappa >0$. Its imaginary parts at those two arguments remove the unknown vertical-side normal derivative. With $G_j^s(\kappa )={\int }_0^{\infty }\sin (\kappa x)g_j(x)dx$ and $S_j(\kappa )={\int }_0^{\infty }\sin (\kappa x)q_y(x,j)dx$, the result isThe lower outward normal derivative is $-q_y$, so its Dirichlet-to-Neumann map coefficient is $-S_0$, whereas the upper coefficient is $S_l$.

// Target: analysis.bigb

For $q_t=q_{xx}+\alpha q_x$, the exponential change of unknown gives $u_t=u_{xx}-{\alpha }^{2}u/4$. Its Fourier sine transform satisfies a closed scalar evolution with a boundary forcing term. The initial transform uses $e^{\alpha x/2}{q}_0(x)$; ordinary integrable-transform arguments require sufficient decay of this weighted datum. For $q_0=e^{-ax}$ this requires $a>\alpha /2$. In contrast, the unweighted sine transform of $q$ couples to its cosine transform through the drift term, so it does not directly diagonalize this half-line problem.

For a primitive Dirichlet character modulo $N>1$ of parity $a\in \{0,1\}$, put ${\Theta }_{\chi }(\tau )={\sum }_n{n}^a\chi (n)e^{\pi i{n}^2\tau }$, extending the character by zero on nonunits. The square root is analytic with positive value when $-i\tau$ is positive real. Apply Poisson summation separately in each residue class modulo $N$, and use the finite Fourier transform of a primitive Dirichlet character. The Fourier transforms of $e^{\pi i\tau x^2}$ and $x{e}^{\pi i\tau x^2}$ are $(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$ and $(t/\tau )(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$. These give the formula, including its odd-character phase.