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A real symmetric matrix whose entries are the moments $H_{ij}={\int }_0^1{x}^{i-1}{x}^{j-1}dx$. Thus it is the Gram matrix of the monomials in the real L2 inner product on $[0,1]$. For a nonzero real coefficient vector $v$, $v^{T}Hv={\int }_0^1({\sum }_i{v}_i{x}^{i-1}{)}^{2}dx>0$, so every finite Hilbert matrix is positive-definite. The moment description connects these matrices with polynomial approximation and with the shifted Hilbert quadratic-form bound.

A rectangular array whose entries are nondecreasing in each coordinate. Every subarray obtained by restricting to a row interval and a column interval has the same property. Comparing a target with the top-right entry discards a whole row or column, giving staircase matrix search in time linear in the sum of the dimensions.

For a positive small parameter $ϵ$, an order-one mismatch between the reduced solution $y_0=f(x)$ and a prescribed right endpoint value $y(1)=b$ creates a right boundary layer. Balancing $ϵy^{\prime \prime }$ against the order-one reaction term gives width $\sqrt{ϵ}$. In the inner variable $X=(1-x)/\sqrt{ϵ}$, the leading equation is $Y_0^{\prime \prime }=Y_0-f(1)$. Its boundary and matching conditions are $Y_0(0)=b$ and $Y_0\to f(1)$ as $X\to \infty$, soThe leading additive composite expansion is $f(x)+(b-f(1))e^{-(1-x)/\sqrt{ϵ}}$, obtained by subtracting the common overlap value. For the nonlinear perturbation $ϵy^{\prime \prime }=y-x^2+ϵ(2+y^2)$ with zero endpoint values, the first two outer expansion terms are $x^2-ϵx^4$, and the leading right-layer term is $1-e^{-X}$.

For $ϵy^{\prime \prime }+x^{\alpha }{y}^{\prime }+y=0$ near zero, dominant balance gives a diffusion-drift layer of thickness ${ϵ}^{1/(\alpha +1)}$ for $-1<\alpha <1$, with the reaction term negligible. At $\alpha =1$, all three terms survive on the scale $\sqrt{ϵ}$. For $\alpha >1$, diffusion and reaction first balance on wavelength $\sqrt{ϵ}$, while the drift damps the oscillation on the larger envelope scale ${ϵ}^{1/(\alpha +1)}$. Removing the first derivative gives the envelope factor $\exp [-x^{\alpha +1}/(2ϵ(\alpha +1))]$. For $\alpha \le -1$, no shrinking algebraic diffusion layer of this regular bounded type is obtained.

Split the integral at $ϵ\ll \delta \ll 1$. Replacing the numerator by one below the cutoff gives ${ϵ}^{-1}\arctan (\delta /ϵ)$, with an error bounded by $\frac{1}{2}\log (1+{\delta }^2/{ϵ}^2)$. Above the cutoff, replacing the denominator by $x^2$ has relative error $O({ϵ}^2/{\delta }^2)$, and integration by parts gives $1/\delta +\log \delta +O(1)$. The inner $-1/\delta$ cancels the outer $1/\delta$, and all errors are $o(1/ϵ)$, proving the leading constant independently of the intermediate cutoff.

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.

A twin plane is a compatible planar interface between differently oriented crystal variants. In a piecewise homogeneous elastic deformation its reference normal is the $N$ in a rank-one connection between the two variant deformation gradients. Continuity of the deformation holds along this plane even though its derivative changes. Other material planes crossing the interface can kink after deformation.

A martensitic variant is one of the lattice deformations corresponding to the low-temperature crystal phase. The parent crystal's discrete symmetries generate different variants. In a nonlinear elasticity model each representative $U_i$ supplies a hyperelastic energy well $SO(3)U_i$. A representative need not itself be a symmetric stretch tensor; multiplication by a proper rotation leaves the same well.

A function of the current state of a Markov chain can discard information needed to predict its next value. A sufficient condition preserving the Markov property is strong lumpability: all original states with the same recorded value must give the same transition probabilities to every recorded class.

For the two-column model in growth rate of two-column sticky deposition, the pair of column heights is Markov, but its maximum alone is not. Starting empty, the maximum-height histories $(1,1,2)$ and $(1,2,2)$ both end at height two. The former leaves unequal columns, so the next height increases with probability $1/2$; the latter leaves equal columns, so it increases with probability one. Each history has positive probability. Their different predictions at the same time and current value violate the Markov property. By contrast, the absolute height difference retains exactly the information needed for its own transitions.

A total function computed by a halting logspace transducer. Its output may be much longer than its work tapes. Logspace-computable functions compose using logspace composition by recomputation.

For integers $b>1$ and $n>0$, the integer logarithm is the largest nonnegative integer $i$ such that $b^i\le n$. Equivalently,The equivalence follows from the monotonicity of exponentiation in its exponent and the definition of the floor function. For example, $3^4=81\le 100<243=3^5$, so the base-three integer logarithm of $100$ is $4$.

A multiplication loop maintains $p=b^i\le n$ and accepts another factor $b$ while $pb\le n$. Repeated squaring makes the search faster: precompute $b^{2^k}$ starting at $k=0$, and consider these factors in decreasing $k$. Include a factor when the accumulated product remains at most $n$. If $L=\lfloor {\log }_{b}n\rfloor$, the loop invariant before considering $k$ is $i\le L<i+2^{k+1}$. Accepting $2^k$ exactly when $i+2^k\le L$ halves that uncertainty, proving the greedy algorithm returns $L$. This takes $O(1+\log (1+{\log }_{b}n))$ arithmetic operations; it is not the same claim as this bound on bit complexity.

For positive operands in bounded integer arithmetic, test $p\le n\mathrm{DIV}q$ before multiplying by $q$. Integer division makes this equivalent to $pq\le n$ while preventing integer overflow in a rejected candidate.

For $F(x)=1-[1+(x/\alpha {)}^{\beta -1}{]}^{-1}$, plotting the logits of $p_i=i/(n+1)$ against sorted log observations gives an approximately straight line under the model. The probability integral transform makes $F(X_{(i)})$ a uniform order statistic with expectation $i/(n+1)$, motivating these plotting positions. With those axes the slope is $\beta -1$ and the intercept is $-(\beta -1)\log \alpha$.

A bulb-volume change shifts the indicated temperature. Contraction raises the liquid column at a fixed temperature; expansion lowers it. Permanent and temporary changes require distinction when interpreting an ice-point check.^[ref]

The norm of a linear-system residual, used to measure how closely an approximate solution satisfies its linear system. It differs from the solution-error norm $\Vert x_{\ast }-x\Vert$. Their relationship depends on the matrix inverse and the condition number, so a small residual alone is not a scale-free relative-error guarantee.

At symmetric design points, the columns $1$, $x$ and $x^2-\overline{x^2}$ are mutually orthogonal. Their squared lengths give the diagonal entries of the normal-equation matrix and hence separate least-squares coefficient variances. At the seven integer points from minus three to three, these squared lengths are $7,28,84$.