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The Euclidean metric is the constant Riemannian metric on ${\mathbb{R}}^n$ given in Cartesian coordinates by $g={\sum }_i(d{x}^i{)}^2$. Its Levi-Civita connection has zero Christoffel symbols in those coordinates, so it is flat and its geodesics are straight lines. The Euclidean norm is the pointwise length determined by this metric.

For a smooth function $f$ on a Riemannian manifold, its Riemannian gradient is the unique vector field satisfying $g({\nabla }_{g}f,V)=df(V)$ for every vector field $V$. It depends on the Riemannian metric, and its negative generates the downward gradient flow.

Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

The Riemann zeta function satisfies ${\int }_0^T|\zeta (1/2+it){|}^{4}dt\ll T(\log T{)}^4$ for $T\ge 2$. Squaring a short Dirichlet polynomial produces the divisor function; the divisor-square summatory bound controls the off-diagonal and diagonal terms.

The trivial zeros of the Riemann zeta function are $s=-2,-4,-6,\dots$. The remaining zeros are the Nontrivial zeros of the Riemann zeta function.

The completed zeta function $\Lambda (s)={\pi }^{-s/2}\Gamma (s/2)\zeta (s)$ satisfies $\Lambda (s)=\Lambda (1-s)$ and has poles at zero and one. It is also sometimes denoted $\xi (s)$; this differs from the entire xi function obtained by multiplying by $s(s-1)/2$. The distinction matters in the constant term of a nonholomorphic Eisenstein series.

Abel summation applied to the counting function $\lfloor w\rfloor$ gives the displayed identity for $\Re s>1$ and every $x>0$. Since $0\le \{w\}<1$, the integral is locally uniformly convergent and holomorphic for $\Re s>0$. This supplies a meromorphic continuation of the Riemann zeta function to that half-plane, with its sole pole at one and residue one. Keeping the fractional endpoint term makes the formula valid at noninteger cutoffs.

For $s=\sigma +it$, $\sigma >0$, and $x\ge |t|/\pi$, the displayed approximation holds away from the pole. Apply the Van der Corput sum-integral lemma to $-t\log w/(2\pi )$ on $w\ge x$, where its derivative has modulus at most one half. Abel summation with $w^{-\sigma }$ converts the bounded unweighted discrepancy to $O(x^{-\sigma })$. Locally uniform convergence of this weighted discrepancy extends the identity from $\sigma >1$ to $\sigma >0$. It turns estimates for finite exponential sums into estimates for the Riemann zeta function.

For large positive $t$ and $0<\sigma \le 1$, a fixed sufficiently large $C$ gives the displayed upper bound; to the right of one use $|\zeta (\sigma +it)|\ll {\log }^{2/3}t$. Its nonlinear dependence on $1-\sigma$ permits a wider zero-free region of the Riemann zeta function through the Landau zero-free-region theorem. Its proof uses estimates for exponential sums; its use as an upper-bound input is separate from a proof of a zero-free region.

For $T/2\le \Im s\le T$ and $0<\Re s=\sigma \le 1$, the Hardy-Littlewood approximation to the Riemann zeta function truncates zeta at $T$ with error $O(T^{-\sigma })$. Multiply the finite sums and use $a_n={\sum }_{m\mid n,m\le M,n/m\le T}\mu (m)$. All coefficients with $1<n\le \min (M,T)$ vanish by the Möbius divisor-sum identity. The total error is bounded using ${\sum }_{m\le M}{m}^{-\sigma }\ll M^{1-\sigma }\log M$.

For a continuous real function $f$ on $[a,b]$, uniform step approximation on a compact interval gives a finite step function $g={\sum }_j{c}_j{1}_{I_j}$ with small uniform error. Each cosine integral over an interval has absolute value at most $2/|\omega |$. ThereforeFirst choose $g$ to make the first term small, then let $|\omega |$ grow with $g$ fixed. This proves the cosine version of the Riemann-Lebesgue lemma without differentiating $f$; the sine and complex-exponential versions follow in the same way.

The difference between a weighted integral and a discrete approximation is a quadrature error. For a Hölder continuous function on a partition of an interval with mesh $\Delta$, a left endpoint approximation has error of order ${\Delta }^{\alpha }$. For a normalized window indicator function, only the two endpoint cells contribute and the error is bounded by a constant times $\Delta /h$.

The local resistive-force theory coefficients ${\xi }_{\parallel }$ and ${\xi }_{\perp }$ describe drag along and across a filament tangent. For very slender bodies their leading logarithmic values have ratio ${\xi }_{\perp }/{\xi }_{\parallel }\simeq 2$. Geometry, ends and nonlocal fluid interactions affect the corrections. Anisotropy enables propulsion by rotating helices.

A material point at body-frame position $\mathbf{r}$ has velocity relative to stationary ambient fluid $\mathbf{U}+\mathbf{\Omega }\times \mathbf{r}+({\partial }_t\mathbf{r}{)}_{\mathrm{body}}$. The translational, rotational and deformation terms must all be retained before assigning perturbation orders.

For $y=ϵg(x,t)$ on $0\le x\le L$, uniform local drag and force/torque balance give $U_1=0$, $V_1=-4⟨g_t{⟩}_x+6⟨x{g}_t{⟩}_x/L$, and ${\Omega }_1=6⟨g_t{⟩}_x/L-12⟨x{g}_t{⟩}_x/L^2$. The tangent can be replaced by the undeformed direction at this order because the velocity is already first order.

For a small-amplitude planar filament without mean axial drift, resistive-force theory gives the leading time-averaged force on the filament as $F_x=({\xi }_{\perp }-{\xi }_{\parallel })T^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_x{y}_{t}dxdt$. The parallel and perpendicular drag coefficients of a slender filament must differ to obtain propulsion at this order. A right-going wave gives negative axial force; a left-going wave gives positive axial force.

The leading viscous dissipation due to a small planar filament stroke is $\dot{W}={\xi }_{\perp }{T}^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_t^{2}dxdt$. It follows from the work against the local resistive-force theory force. The transverse velocity controls the leading power; the quadratic longitudinal velocity contributes only at higher order.

If $K$ is a finitely generated group, then the semidirect product $K⋊H$ is a residually finite group exactly when both $K$ and $H$ are residually finite groups. To separate an element whose $H$ coordinate is nontrivial, use a finite quotient of $H$. For $1\ne k\in K$, choose a finite-index normal subgroup $U$ omitting $k$, and intersect all subgroups of index at most $[K:U]$ to obtain a characteristic finite-index subgroup $C\le U$. The induced map into $(K/C)⋊\mathrm{im}(H\to \mathrm{Aut}(K/C))$ has finite target and separates $k$. More generally, finite generation can be replaced by a separating family of finite-index normal subgroups invariant under the given $H$ action.

For quiver representations $X=(X_i,f_{\rho })$ and $Y=(Y_i,g_{\rho })$, a morphism consists of maps $h_i:X_i\to Y_i$ satisfying $h_{t(\rho )}{f}_{\rho }=g_{\rho }{h}_{s(\rho )}$ for every arrow. Composition is vertexwise. These families are the kernel of the extension complex of quiver representations.

The dimension vector of a finite-dimensional quiver representation is $({\dim }_k{X}_i{)}_{i\in Q_0}$. Direct sums add dimension vectors. Fixing the vector gives an affine space of possible arrow matrices, the quiver representation space.