On ${\mathbb{R}}^2$, the coordinate seminorms $p_x$ and $p_y$ measure horizontal and vertical components separately. Their kernels of seminorms are the vertical and horizontal axes. Neither is a norm, but together they form a separating family of seminorms. A basic zero neighbourhood isEach condition alone is an unbounded strip; their intersection is an open rectangle. For $a=b=1$, this is the unit open ball of the maximum norm $\Vert (x,y){\Vert }_{\infty }=\max \{|x|,|y|\}$. For unequal tolerances it is the unit open ball of the weighted norm $\max \{|x|/a,|y|/b\}$. Thus the two coordinate measurements generate the usual topology of the plane.

For a numerical check, take $a=1$, $b=1/2$: $(0.8,0.4)$ passes both tests, $(0.8,0.7)$ passes only the horizontal test, and $(1.2,0.4)$ passes only the vertical test. A direction missed by one seminorm is detected by the other.

The kernel of a seminorm is its zero set, a linear subspace by absolute homogeneity and the triangle inequality. The estimate $|p(x)-p(y)|\le p(x-y)$ shows that adding an element of $\ker p$ does not change $p$. Hence $\Vert x+\ker p\Vert =p(x)$ defines a norm on the quotient vector space $E/\ker p$.

On $E$ itself, $d_p(x,y)=p(x-y)$ is a pseudometric; it is a metric exactly when $p$ is a norm. In particular, zero distance in one pseudometric need not mean equality of vectors. A separating family of seminorms removes the common zero-distance ambiguity.

A rectangular band is a semigroup satisfying the displayed identities; these also imply associativity. Every nonempty such algebra is a product $B\times C$ with $(b,c)\ast (b^{\prime },c^{\prime })=(b,c^{\prime })$. Fixing $e$, the factor sets are $B=\{x\ast e\}$ and $C=\{e\ast x\}$, and $x\mapsto (x\ast e,e\ast x)$ has inverse $(b,c)\mapsto b\ast c$. Homomorphisms between nonempty products are exactly pairs of maps on the two factors.

An atmospheric thermal inversion is a region where temperature increases outward. In a strongly irradiated atmosphere it can arise when shortwave opacity exceeds thermal infrared opacity and stellar energy is absorbed high in the atmosphere.

If a self-orthogonal binary subspace has odd length $n$ and dimension $(n-1)/2$, its dual code is obtained by adjoining the all-ones word. Self-orthogonality makes every codeword even-weight, so $1\in C^{\perp }$ but $1\notin C$. Comparing dimensions proves the formula. The dimension hypothesis alone is insufficient: $C=⟨100⟩$ at length three is a counterexample.

For a smooth projective algebraic curve $C$, the normal bundle of its diagonal in $C\times C$ is $TC$: on the diagonal the quotient of $TC\oplus TC$ by the diagonal subbundle is identified with $TC$ by taking the difference. The self-intersection formula therefore gives ${\Delta }_C^2=\mathrm{deg}TC=2-2g(C)$.

In oriented Euclidean four-space, gauge curvature is self-dual when $F=\ast F$. The Hodge star squares to one on two-forms, so this picks its positive eigenspace; the negative eigenspace is anti-self-duality of gauge curvature. The Bianchi identity then implies the Yang-Mills equations, and the action saturates the positive-charge Yang-Mills instanton Bogomolny bound.

For a binary self-dual code, every word $c$ satisfies $c\cdot c=0$. In characteristic two this is ${\sum }_i{c}_i=0$, so the all-ones word is orthogonal to every codeword and lies in the dual code, hence the code itself. The dual code dimension identity gives even length and half-length dimension.

For $n$ studies conditionally independent given the truth of one relationship, the event that at least one is positive has probability $1-{\beta }^n$ under the alternative and $1-(1-\alpha {)}^n$ under the null. Its Bayes factor therefore gives the displayed posterior odds. For interior error probabilities these tend back to the prior odds $R$. If power exceeds size, the Bayes factor decreases to one: the quotient of geometric sums is a weighted average of decreasing powers $[\beta /(1-\alpha ){]}^k$.

This result concerns the coarsened selection event. Full observed results instead multiply their individual Bayes factors, and concordant evidence can accumulate without this bound. It is a useful example of information discarded by publication bias.

A small planar crack of area $S$ opening normally by $d(t)$ has volume change $V=Sd$ and the displayed seismic moment tensor. Its principal dipoles are $\lambda V,\lambda V,(\lambda +2\mu )V$: two act in the crack plane and the largest acts normally for positive opening and ordinary positive Lamé parameters. The definition follows by contracting the isotropic elastic stiffness tensor with two copies of the crack normal.

Tangential fault slip $\mathbf{b}$ in an isotropic elastic material has the displayed moment density. It is symmetric and trace-free, with eigenvalues $\mu |b|,-\mu |b|,0$. The two off-diagonal force dipoles form equal couples with opposite torques, producing zero resultant force and torque. For a uniform patch of area $A$, the scalar seismic moment is $M_0=\mu A|b|$. Traction continuity removes the single-force layer in the elastodynamic surface-jump representation.

An isotropic point moment is a dilatational gradient source. Its radial projection gives direction-independent P wave radiation, whereas its transverse projection vanishes. Thus it produces no far-field S wave in a homogeneous isotropic solid; conversion at an interface can change that conclusion.

These characteristic wave amplitudes split a one-dimensional elastic wave into forward and backward components. Their instantaneous net elastic-wave energy flux is ${\varphi }_{+}^2-{\varphi }_{-}^2$. For real-part harmonic phasors the mean flux is $(|{\varphi }_{+}{|}^2-|{\varphi }_{-}{|}^2)/2$. In a static graded medium with real positive seismic impedance, put $g=(\log Z{)}^{\prime }/2$; then ${\partial }_x{\varphi }_{\pm }\pm {\beta }^{-1}{\partial }_t{\varphi }_{\pm }=g{\varphi }_{\mp }$. Seismic impedance gradients couple the two directions while preserving net flux.

At normal incidence, a uniform lossless elastic wave layer of thickness $h_j$, speed $c_j$ and seismic impedance $q_j={\rho }_j{c}_j$ has wave phase thickness ${\delta }_j=\omega h_j/c_j$. With phasors proportional to $e^{-i\omega t}$, particle velocity $V$ and compressive traction $P=-T$, the displayed matrix maps $(V,P)$ from the top to the bottom. It has determinant one. Multiplying these matrices in physical order accounts for all coherent multiple reflections. Continuity of $V,P$ connects adjacent layers without another propagation factor.

How can an invisible change in concentration produce a readable signal? The saponification of ethyl acetate by hydroxide in water gives ethanol and an acetate ion:The rate law used for this assay is $r=k_2[\mathrm{ethyl}\mathrm{acetate}][\mathrm{O}{\mathrm{H}}^{-}]$. Each reactant has coefficient one, so $r$ equals either disappearance rate. If the two initial concentrations are equal, they remain equal under this stoichiometry, giving $-dc/dt=k_2{c}^2$ and $k_{\mathrm{obs}}=k_2$, without a factor of two.

Manual for Advanced Physical Chemistry Laboratory-II by Nasima Arshad (Unit 7, Experiment 2, printed pp. 66–67) specifies a conductivity cell holding about $60\mathrm{mL}$ in constant-temperature baths, with $0.040\mathrm{mol}{\mathrm{L}}^{-1}$ ethyl acetate and sodium hydroxide stocks equilibrated separately. Mixing $30\mathrm{mL}$ of each initiates the run. The readout is electrical resistance: every minute for ten minutes, every two minutes for ten more, then every five minutes for twenty-five. The cell is rinsed with distilled water. A matched sodium hydroxide–water dilution estimates the initial resistance; a potassium chloride standard determines the cell constant. The manual asks for comparison with the extrapolated initial signal.^[ref]

The post-mixing initial concentration is $c_0=0.040(30/60)=0.020\mathrm{mol}{\mathrm{L}}^{-1}$ for each reactant, assuming additive volumes. This matches our illustrative starting concentration, but the prescribed $k_{\mathrm{obs}}=0.50$ is not a measured value from the manual. As hydroxide is replaced by the less mobile acetate ion, electrical conductance decreases. The preceding experiment describes two platinum electrodes and an alternating-voltage Wheatstone bridge to limit polarization at the electrodes. For measured electrical resistance $R(t)$, electrical conductance is $G(t)=1/R(t)$; electrical conductivity is $\kappa (t)=K_{\mathrm{cell}}G(t)$. Keep the probe geometry fixed and use one signal convention throughout.^[ref]

For the equal-starting-concentration model, assume an affine calibration between electrical conductance and remaining hydroxide concentration, with endpoints $G_0$ and $G_{\infty }$ at the same temperature. This givesThe last plot has slope $k_2{c}_0$, with units ${\mathrm{s}}^{-1}$, whereas plotting $1/c$ against time has slope $k_2$, with units $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. These are different slopes. This endpoint transformation also shows why raw resistance must not be substituted for conductance.

Useful additional controls are repeat runs, fixed immersion depth, an independently checked final signal, and comparisons of several starting concentrations. Mixing delay obscures early conversion; temperature drift changes both the rate constant and the signal; contamination adds mobile ions. The affine calibration can fail when ionic mobilities change with composition. Near completion, $G-G_{\infty }$ becomes small, so endpoint error is strongly amplified. Those are methodological limits and proposed checks, not new experimental observations. A conductivity trace supports a calibrated rate law over its tested range; it does not independently reveal the full reaction mechanism.

For unequal starting concentrations $a=[\mathrm{A}{]}_0$ and $b=[\mathrm{B}{]}_0$ in a one-to-one chemical reaction, let $x$ be the consumed concentration. Then $[\mathrm{A}]=a-x$, $[\mathrm{B}]=b-x$, andThe natural logarithm acts on a dimensionless quantity; differentiating the left side with respect to $x$ gives $1/((a-x)(b-x))$, verifying the relation. Mixing $20\mathrm{mL}$ of the stated ethyl acetate stock with $40\mathrm{mL}$ of the sodium hydroxide stock would give $a=0.01333\dots$ and $b=0.02666\dots \mathrm{mol}{\mathrm{L}}^{-1}$. The difference $[\mathrm{B}]-[\mathrm{A}]=b-a$ stays constant, rather than their ratio. Neither the equal-concentration reciprocal formula nor a pseudo-first-order reaction approximation is justified solely by that twofold excess.

A Cauchy-Euler equation is a linear ordinary differential equation whose coefficient powers match their derivative orders, for exampleWith $t=\log x$ and $Y(t)=y(e^t)$, the chain rule gives $x{y}^{\prime }=Y_t$ and $x^2{y}^{\prime \prime }=Y_{tt}-Y_t$. Thus the equation becomes the constant-coefficient differential equation $Y_{tt}+(a-1)Y_t+bY=f(e^t)$. In the homogeneous case, the power-law ansatz $y=x^r$ gives the indicial equation $r(r-1)+ar+b=0$.

For $x^2{y}^{\prime \prime }-x{y}^{\prime }+y=0$, the repeated root $r=1$ yields $y=x(C_1+C_2\log x)$. The logarithm is the transformed repeated-root factor $t{e}^t$. Negative real $x$ can be handled separately using $t=\log |x|$, while complex powers require a chosen logarithm branch. The singular point at zero is not crossed automatically by the logarithmic coordinate.

For two real self-adjoint equations $(p_i{y}_i\prime )\prime +q_i{y}_i=0$ with continuous coefficients, $0<p_2\le p_1$ and $q_1\le q_2$, between consecutive zeros of a nontrivial first solution a nontrivial second solution either has a zero or is proportional to the first there. This comparison controls oscillation through the coefficients.

The chain rule with $y(x)=Y(x^2)$ gives $x{y}^{\prime \prime }-y^{\prime }=4x^3{Y}^{\prime \prime }$ and transforms the equation, away from zero, into $Y^{\prime \prime }+Y=0$. Thus its solution space is spanned by $\cos (x^2)$ and $\sin (x^2)$. Both extend smoothly through the regular singular point at zero. Their even power series allow the independent data $y(0)$ and $y^{\prime \prime }(0)$ to select the two solutions; every smooth solution has $y^{\prime }(0)=0$.

Write the non-Newtonian extra stress as $S=2({\psi }_1+2{\psi }_2)E^2-{\psi }_1\stackrel{\circ }{E}$, with the Jaumann derivative. For a steady Stokes flow, $\nabla \times \nabla \cdot \stackrel{\circ }{E}=0$. If ${\psi }_1+2{\psi }_2=0$, the extra force is a gradient and only alters pressure. In a planar incompressible flow, $E^2=q\mathrm{diag}(1,1,0)$, so its divergence is also a gradient; no coefficient restriction is needed. A Newtonian velocity with prescribed boundary velocities consequently solves the second-order model with an altered pressure. On the regular weak-flow branch, Uniqueness of Stokes flow makes each velocity correction vanish. This does not establish uniqueness of arbitrary large nonlinear branches or the absence of higher-order effects in a real material.

For a sphere of radius $a$ rotating with angular velocity $\Omega$, the primary Stokes flow has $v_{\varphi }=K\sin \theta /r^2$, $K=\Omega a^3$. In the weak, inertialess second-order fluid correction, put $c={\psi }_1+2{\psi }_2$. The stream function in the title hasBoth meridional components vanish at $r=a$ and decay at infinity. Indeed, $E_{r\varphi }=-3K\sin \theta /(2r^3)$ gives $[\nabla \times \nabla \cdot (2c{E}^2){]}_{\varphi }=72c{K}^2\sin \theta \cos \theta /r^8$, and the curl of the momentum equation reduces to $(d^2/d{r}^2-6/r^2{)}^{2}H=72c{K}^2/(\mu r^7)$. For $c>0$ the secondary flow enters towards the equator and leaves near the poles; $c<0$ reverses it and $c=0$ removes this correction. The meridional circulation is even in the sign of sphere rotation. Inertial secondary flow is excluded from this calculation.