A lumped thermal model assigns one representative temperature and heat capacity to each region of an apparatus. For a single body of heat capacity $C$ connected to a bath by thermal conductance $G$, with supplied power $P$, an elementary balance isThe model stores internal energy in the body and treats exchange through a thermal resistance, leaving internal spatial temperature gradients unresolved.

A small solid specimen with rapid internal equilibration can approximate this behavior. A constant heating step gives a final rise $P/G$ and a thermal time constant $C/G$ when $C$ and $G$ are constant. Comparing thermometers at different positions tests the assumption of nearly uniform specimen temperature. If appreciable internal gradients persist, the apparatus needs several coupled thermal regions or a spatial heat equation model. Fitting a single exponential is useful only after this physical approximation has been checked.

The classical electrostatic self-energy functional of an electron number density is the expression in the title. The factor one-half prevents counting each pair twice. Its functional derivative is the Hartree potential; integrating density times that potential gives $2E_H$. The functional includes an unphysical one-electron self-interaction unless other terms cancel it. It differs from the Hartree energy atomic unit.

For Hartree direct integrals $J_{ij}$ and orbital eigenvalues ${ϵ}_i$, the product-state energy is $E={\sum }_i{ϵ}_i-{\sum }_{i<j}{J}_{ij}$. Each pair enters two orbital mean fields but only once in the total interaction energy. Orbital energies are therefore not independent additive particle energies.

For integers $N>m\ge 1$, compare the decreasing function $1/x$ with unit rectangles: on $n-1<x<n$, $1/(x+1)<1/n<1/x$. Integrating and summing over $n=m+1,\dots ,N$ proves the displayed strict bounds. Adding a known harmonic number $H_m$ gives bounds on $H_N$ without computing every omitted term. A requested rounding is certified when both bounds round to the same value; a midpoint estimate alone does not establish the rounding.

Suppose the inner and outer boundary values on spheres of radii $0<a<b$ have convergent Legendre polynomial expansions with coefficients $u_n,v_n$. For each degree set $F_n(r)=r^n-a^{2n+1}{r}^{-n-1}$ and $H_n(r)=r^n-b^{2n+1}{r}^{-n-1}$. ThenBoth radial terms solve $(r^2{R}_n^{\prime }{)}^{\prime }=n(n+1)R_n$, so the displayed angular expansion is harmonic wherever the series and required derivatives converge. The two radial factors take endpoint values $(1,0)$ and $(0,1)$, respectively, proving the boundary matching. For continuous data with the usual harmonic-extension convergence, the maximum principle for harmonic functions gives uniqueness. The construction separates the angular orthogonality from the two-boundary radial interpolation.

On $0<x<a$, $0<y<b$, the bounded harmonic function with value $F$ on the open upper edge and zero on the other three open edges has the sine-hyperbolic series in the title. Separation of variables gives the modes, and the Fourier sine series of a constant selects only odd $n$. The series converges exponentially on compact subsets of the interior, justifying its derivatives in Laplace's equation.

For $F\ne 0$, the upper corners have incompatible one-sided boundary limits, so the solution cannot extend continuously to them. Assigning isolated corner values does not alter the bounded interior Dirichlet problem. For numerical evaluation, the stable mode ratio isThis avoids overflow from separately evaluating large hyperbolic sines.

At low density, sequential insertion gives a configurational integral approximated by ${\prod }_{j=0}^{N-1}(a-4j\pi r^2)$. Expanding its logarithm to first order in occupied area gives $N\ln a-2\pi r^{2}N(N-1)/a$. Replacing $N-1$ by $N$ and matching a free-volume expression gives $(a-Nb{)}^N$, with $b=2\pi r^2$. This matching is controlled only at low density; its pole at packing fraction $1/2$ is not a physical close-packing limit.

A circular free-area model for a disc surrounded by fixed neighbours a centre-to-centre distance $R$ away. The inscribed circle available to the moving centre has radius $R-2r$. It is an approximation to the actual curved cage boundary, not the area of a disc whose radius is $R-r$.

A complete integral depends on as many independent parameters as configuration coordinates and supplies a locally invertible canonical transformation to constants of motion. For a type-one generating function $G(q,Q,t)$, require a nondegenerate mixed-derivative matrix in the relevant local chart. Solving $G_t+H(q,G_q,t)=0$ makes the transformed Hamiltonian zero, so both new canonical coordinates and new canonical momenta are constant. An arbitrary parameter-deficient solution does not ensure such a transformation.

A halohydrin with bromine on the carbon adjacent to OH. Its relative stereochemistry can select between properly aligned epoxide closure and semipinacol rearrangement.

A particle entering a uniform axial magnetic field on the tube axis follows a transverse circle through that entry point. Its guiding centre is one Larmor radius from the axis, so the maximum radial displacement from the axis is twice the Larmor radius. For tube radius $a$, clearance requires $v_{\perp }\le |q|Ba/(2m)$ when the transit samples a complete gyration. If the transit is shorter, the actual sampled arc determines clearance instead. The axis is revisited after each full cyclotron period, not after each half-period.

Neuregulin 1 is a signaling protein ligand acting through ErbB receptors. The human NRG1 gene produces several protein isoforms through alternative promoter usage and alternative splicing; these products differ in their structure and tissue distribution.^[ref]

Its developmental roles include ventricular trabeculation. A targeted mutation in mice revealed that loss of neuregulin 1 disrupts normal development of several organ systems, providing experimental evidence of requirement rather than expression alone.^[ref]

Let $J$ be the plane rotation matrix through $\pi /2$, generating a cyclic group of order four. A quadratic form $a{x}^2+b{y}^2$ has its two coefficients exchanged by a quarter-turn and unchanged by a half-turn. Thus the pullback action on this two-dimensional space gives the displayed group representation: $D(J^k)=D(J{)}^k$, with $D(J{)}^2=I$. It is not a faithful representation, because its kernel of a group homomorphism is $\{I,J^2\}$. Its two invariant coefficient lines correspond to $x^2+y^2$ and $x^2-y^2$, on which a quarter-turn acts by $+1$ and $-1$, respectively. Restricting to this diagonal space is valid for these signed coordinate permutations; a general linear transformation can introduce an $xy$ term.