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In a complete Non-Archimedean absolute value field, a series converges if and only if its terms tend to zero: the ultrametric inequality bounds every finite tail by its largest term. The displayed sequence tends to zero in ${\mathbb{Q}}_p$. But each block $p^k\le n<p^{k+1}$ contributes $p-1$ to the real sum of its p-adic absolute values, so that real sum diverges.

For a positive even weight on a symmetric interval, any orthogonal polynomial of degree $k$ has the parity of $k$. Induct on $k$. The reflected polynomial $q_k(x)=(-1{)}^k{p}_k(-x)$ has the same leading coefficient. By substitution $x\mapsto -x$ in the weighted inner product and the inductive parity of all lower polynomials, it is orthogonal to them. Hence $q_k-p_k$ has lower degree and is orthogonal to the entire space containing it. Its squared norm is zero, forcing $q_k=p_k$. The base case is a constant. This proof needs neither monic nor unit-norm normalization; it only compares polynomials with the same leading coefficient.

An orthonormal family of orthogonal polynomials also gives each member unit norm in the chosen weighted inner product. Its pairwise inner products are the Kronecker delta. Dividing every nonzero orthogonal polynomial by its norm produces such a family; the sign or complex phase still requires a convention, such as a positive real leading coefficient.

Assume a positive even weight on a symmetric domain has finite polynomial moments. Subtracting the orthogonal projection onto a finite-dimensional subspace of $x^n$ onto the lower-degree polynomials gives the unique monic orthogonal polynomial. Reflection produces another such monic polynomial, $(-1{)}^n{\varphi }_n(-x)$, so uniqueness gives the parity identity. Every coefficient with the opposite parity to its degree therefore vanishes.

For the inner product obtained by integrating with weight $|x|$ on $[-1,1]$, odd moments vanish and the moment of $x^{2m}$ is $1/(m+1)$. The first monic orthogonal polynomials are $1$, $x$ and $x^2-1/2$. Positive definiteness follows because the weight is positive almost everywhere and a nonzero polynomial cannot vanish almost everywhere.

A system of ordinary differential equations specifies derivatives of several unknown functions of one independent variable. A vector-valued form ${\mathbf{y}}^{\prime }=\mathbf{f}(t,\mathbf{y})$ bundles its scalar equations. A scalar second-order ordinary differential equation $y^{\prime \prime }=F(t,y,y^{\prime })$ becomes the first-order system $(y_1^{\prime },y_2^{\prime })=(y_2,F(t,y_1,y_2))$ by introducing position and velocity variables. Conversely the first equation identifies $y_2=y_1^{\prime }$ and the second recovers the original equation. Its initial condition must specify both variables.

Where $y\ne 0$, its logarithmic derivative $q=y^{\prime }/y$ changes this nonlinear second-order ordinary differential equation into the first-order linear differential equation $q^{\prime }=a(x)q$. For continuous $a$, initial values $y(x_0)=y_0\ne 0$ and $y^{\prime }(x_0)=v_0$ therefore giveDifferentiation proves both the equation and the initial data. The exponential preserves the sign of $y_0$, so the solution never reaches the singular value zero on any finite interval where $a$ is continuous. For $a(x)=2x$ and $y_0=v_0=1$, the expression reduces to $y(x)=\exp ({\int }_0^x{e}^{t^2}dt)$.

An autonomous differential equation for $f(\eta )$ can be reduced by one order on a monotone solution branch by using $f$ as the new independent variable. Setting $G(f)=df/d\eta$ and applying the chain rule gives the displayed derivative identities. The absence of explicit $\eta$ dependence makes the resulting equation involve only $f,G$ and derivatives of $G$. A point with $G=0$ is singular for this coordinate change; for example $f\sim b{\eta }^2/2$ gives $G\sim \sqrt{2bf}$ on an appropriate branch, so $G_f$ need not remain finite there.

A group element has finite order if some positive integer power equals the identity element. Its order of a group element is the least such positive integer.

For equality-constrained optimization, restrict the Hessian matrix $H$ of the optimization Lagrangian to feasible tangent directions. If the columns of $Z$ form a basis of the nullspace of the constraint Jacobian matrix $A$, the resulting quadratic form has matrix $Z^{T}HZ$. Positive definiteness gives positive quadratic curvature in every nonzero tangent direction. Nonsingularity of this reduced matrix, together with full row rank of $A$, characterizes nonsingularity of the corresponding equality-constrained saddle-point matrix.

The logical system-wide collection of open file descriptions. Each description represents an opening, including access mode, current file offset and a reference to the underlying file. Independent opens of the same file normally create different descriptions, whereas descriptor duplication can make multiple file descriptors share one description. A table is an abstract organization here, not a requirement that all entries occupy one fixed array.

Minimizing the log-cosh energy over zero-trace functions above an obstacle yields the displayed inequality for every feasible competitor. Convexity supplies the one-sided admissible segments. On the open noncontact region, compactly supported variations of both signs are feasible because the gap from the obstacle has a positive minimum on their support. Their first variations vanish, giving $-\Delta u+\tanh u=0$ there. This stationarity equation need not hold on the contact set.

For a string of positive tension $T$ with fixed endpoints $w(\pm 1)=0$, zero transverse loading and obstacle $f=h-x^2$, $0\le h\le 1$, equilibrium minimizes $T{\int }_{-1}^1(w^{\prime }{)}^2/2dx$ among functions above $f$. On a free interval $w^{\prime \prime }=0$, while in contact $w=f$. Smooth tangency of the right line through $(1,0)$ to the parabola at $x=s$ requires $h-s^2=2s(1-s)$, yielding the displayed root in $[0,1]$.

The two pieces join with equal value and slope. On the free portions $w-f=(|x|-s{)}^2\ge 0$; in contact $-T{w}^{\prime \prime }=2T$ is the nonnegative reaction. These facts verify the complementarity conditions. The associated variational inequality and the exact energy difference prove global minimality; strict convexity with fixed endpoints proves uniqueness. At $h=0$ contact reduces to one point; at $h=1$ it fills the whole interval.

The contact set is the region where the obstacle inequality $w\ge f$ is active. On its complement the obstacle supplies no reaction; its boundary is a free boundary, determined along with the unknown solution. A point can belong to the contact set while its contact force is zero, as for a taut horizontal string just touching an obstacle at one point. Thus equality of the gap alone does not imply strictly positive pressure.

The noncontact set is where the unknown function is strictly above its obstacle. For continuous functions it is open. Every smooth variation with compact support in this set remains feasible with either sign when its amplitude is small enough. The first variation therefore vanishes there, yielding the unconstrained Euler-Lagrange equation. On the complementary contact set of an obstacle problem, only the appropriate one-sided variational inequality is available globally.

Insufficient vitamin C can produce the clinical syndrome scurvy. ^[ref]

A Krylov subspace method seeks approximations in spaces generated by repeated matrix applications to a starting vector. For a linear system with starting point $x_0$ and residual $r_0=b-A{x}_0$, the affine search space is $x_0+K_k(A,r_0)$. Choosing an optimality or orthogonality condition within it gives different algorithms. The conjugate gradient method minimizes the energy norm of the error for a symmetric positive-definite matrix; other methods use residual minimization or different projections. The growth of the Krylov subspace can stop early when an invariant subspace is reached.

Recovering a derivative from values controlled only in the uniform norm is an ill-posed problem: the differentiation map is not continuous in those norms. On $[0,1]$, take $f_k(x)=\sin (kx)/k$. Then $\Vert f_k{\Vert }_{\infty }\le 1/k\to 0$, while $f_k^{\prime }(x)=\cos (kx)$ has uniform norm one for every $k$. Arbitrarily small value perturbations can therefore conceal nonvanishing derivative changes.

Numerical differentiation can still be useful when additional smoothness bounds, a suitable sampling scale, or a regularization method control the admissible changes. The basic noise amplification by differentiation of a linear interpolant gives a finite-data version: endpoint uncertainty of size $\delta$ produces derivative uncertainty of order $\delta /h$ at spacing $h$.

Bacterial density is the number of bacteria per unit length, area or volume in a spatial model. It is a field variable, distinct from the organisms themselves.

A total family has dense linear span; it is minimal when no vector lies in the closed span of all the others. Equivalently, no proper subfamily remains total. Such a family has unique bounded coordinate functionals making a biorthogonal system. Indeed put $M_i=\overline{\mathrm{span}}\{x_j:j\ne i\}$ and $d_i=\mathrm{dist}(x_i,M_i)>0$. The functional $m+t{x}_i\mapsto t$ on $M_i+\mathbb{R}{x}_i$ is bounded by $1/d_i$, so the Hahn-Banach theorem extends it to $X$. Its norm is exactly $1/d_i$, by the distance infimum. Uniqueness follows because its values on the total family determine it on a dense subspace. For a countable family, minimality and totality alone do not ensure norm-convergent expansions; the uniform partial-sum criterion for a Schauder basis supplies the additional condition.