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A suitable generalized linear model is a Gamma regression with logarithmic link:withThe gamma distribution is appropriate for a positive, continuous, right-skewed payment, and its variance function allows the conditional variance to grow like the square of the conditional mean instead of assuming constant variance. The logarithmic link function also guarantees positive fitted means and represents covariate effects multiplicatively, which accords with the Box–Cox transformation evidence for a logarithmic scale.

Because model2 uses a logarithmic transformation, the logarithm of the payment ratio is the difference of two predicted log payments. The cars have equal Brand and Bonus values and differ by one Kilometres level, so its fitted mean is the displayed mileage regression coefficient, ${\hat{\beta }}_K=0.057132$.

For two independent future payments, the prediction interval must include both the standard error $0.032654$ of ${\hat{\beta }}_K$ and two independent residual errors. The estimated variance of the predicted log ratio is thereforeUsing the residual degrees of freedom shown in the output, a 95% prediction interval for a ratio of log-normal responses isWith $t_{284,0.975}\simeq 1.97$, this is approximately $[0.120,9.33]$. The interval is wide because it predicts the ratio of two individual future payments; an interval for the ratio of their fitted mean responses would omit the $2{\hat{\sigma }}^2$ term.

The graph is a profile likelihood for the parameter $\lambda$ of a Box–Cox transformation. Its maximum is close to $\lambda =0$, and the displayed likelihood interval contains zero but excludes $\lambda =1$. Since $\lambda =0$ gives a logarithmic transformation whereas $\lambda =1$ leaves the response unchanged up to an affine transformation, the plot supports fittingas the response in model2.

Kilometres is an ordinal categorical variable: its five levels have a meaningful order. Treating it as an ordinary factor would use four indicator variables and hence four unrelated regression coefficients. Converting the levels to numerical scores instead imposes a single linear trend across successive mileage bands. This is more parsimonious and gives greater precision when that trend is a reasonable approximation, although it also assumes that adjacent levels are equally spaced in their effect.

The partial recursive functions form the smallest class containing the zero function, successor function, and projection functions and closed under composition, primitive recursion, and unbounded minimization.

Fix a register machine $P$ with $k$ inputs. By the permitted result, its complete configuration after $t$ steps,is a recursive function of the inputs and time. The predicate $H(\mathbf{x},t)$ saying that this configuration is halted is therefore recursive. Its halting time is the partial recursive functionwhere unbounded minimization is undefined when no halting time exists. Extracting the output register from a configuration is recursive, sois partial recursive. Hence every partial computable function is partial recursive.

A register machine has finitely many natural-number registers, a finite set of labelled states, and instructions that increment a register, conditionally decrement and branch according to whether a register is zero, or halt. A program is a finite sequence of such labelled instructions, and a configuration records the current instruction together with every current register value.

Given $k$ inputs, place them in designated input registers and initialize the remaining registers to zero. The machine computes a partial computable function $f:{\mathbb{N}}^k\rightharpoonup \mathbb{N}$ when it halts with output $f(\mathbf{x})$ in the output register exactly for $\mathbf{x}\in \mathrm{dom}f$ and runs forever otherwise.

To encode a machine, encode every instruction by a natural number using effective encodings of its opcode, register number, and target labels. Encode the resulting finite sequence by a Gödel numbering, for example through iterated pairing or prime powers. Effective decoding gives an enumeration $(P_n)$ of machine programs, so the unary function computed by program code $n$ may be denoted $f_{n,1}$.

The Born rule says that the probability of a projective-measurement outcome is the squared norm of the state projected onto the corresponding eigenspace. The basis vectors in the state whose second qubit is one areThey are mutually orthogonal, so

The spectral projector for outcome $+1$ isFor $|1⟩=(0,1{)}^T$, the Born rule gives

The Pauli multiplication relations and $|\hat{n}|=1$ giveTherefore the possible measurement outcomes, the eigenvalues of $A$, are

In a spherical coordinate system,Since $n_x\mp i{n}_y=\sin \theta e^{\mp i\varphi }$,

Using the Pauli matrices,

Statement (iv) is false because $\alpha \beta =0$ need not imply $\beta \alpha =0$. For example, over any field takeBoth maps are idempotent and $\alpha \beta =0$, butConsequently

Statement (iii) is false over an arbitrary field. Expanding the assumed idempotence givesMultiplying this relation on the left and right by $\alpha$ showsThus $2\alpha \beta =0$, so the conclusion would be true in characteristic different from two. In characteristic two, however, take $\alpha =\beta =I$. Then $\alpha$, $\beta$, and $\alpha +\beta =0$ are idempotent, but $\alpha \beta =I\ne 0$.