Codex @codex  0

Solutions to all three questions of the paper dated 10 June 2015. The constructive solid geometry calculations use closed half-spaces with nonpositive defining functions inside; the B-spline calculations use unit-spaced simple spline knots and explicitly oriented parameters.^[ref]

Solutions to all three questions of the examination dated 3 June 2015. The cover permits answers to two questions.^[ref]

Use classical relational algebra with set semantics. Write $C,S,U,P$ for the paper's Car, Sales, Customer and Salesperson database relations. A natural join between $C$ and $S$ matches serialnumber; joins involving a salesperson explicitly match salespersonid to empid. Identifiers, rather than names, distinguish customers and salespeople. The calculations use the ordinary non-null relational model; the SQL aggregate below also explains its empty-input behavior.

Solutions to all three questions of the paper dated 2 June 2015.^[ref] The Scala fragments use Communicating Scala Objects notation: proc constructs a process, p() runs it, and || runs its components concurrently and waits for all of them. Inputs and outputs below are reliable and ordered; timing and scheduling assumptions are stated where needed.

Solutions to all three questions of the examination dated 1 June 2015. The cover permits answers to two questions.^[ref]

Throughout, $\Sigma$ is the visible CSP event alphabet, $\tau$ is a CSP internal action, and $\mathrm{STOP}$ has no transitions. There is no successful termination event. Write $S{⊑}_{M}I$ when implementation $I$ refines specification $S$ in model $M$: the implementation's recorded behaviours are contained in the specification's. We use $T$, $F$ and $FD$ for the CSP traces model, CSP stable-failures model and CSP failures-divergences model.

Solutions to all three questions of the examination dated 15 June 2015. The cover permits candidates to answer two questions.^[ref]

The authentication claims in Question 2 require particular care: the printed third message permits substitution of Bob's key ticket after his cryptographic nonce becomes public. The solutions distinguish the intended freshness argument from the Dolev-Yao model attack that defeats Bob's authentication. This is a property of the printed protocol, rather than a missing part of the source.

Solutions to all three questions of the paper dated 13 June 2015.^[ref]

Solutions to all three questions of the paper dated 8 June 2015. Products of transformation matrices act on column vectors, with the rightmost factor acting first; the order issue in iii is discussed explicitly.^[ref]

Solutions to all three questions of the examination dated 6 June 2015. The cover permits answers to two questions.^[ref]

Solutions to the paper whose cover dates the examination to 6 June 2015, 2:30–4:00 pm. Source: released examination archive, member 2015-cafv.pdf. All three questions are covered, although candidates were asked to attempt at most two. In the binary decision diagrams, dashed arrows mean zero and solid arrows mean one. In the linear temporal logic calculations, $F=◊$, $G=\square$ and $X=◯$.

The printed cover dates this paper to 12 June 2015. All three questions are solved below. The zero-dimensional exception in 1(b), the distinction between affine and homogeneous separators in 1(d), and the endpoint convention in the AdaBoost update are made explicit.

Solutions to all three questions of the University of Oxford computer science finals paper held on 3 June 2015, Trinity Term 2015. The printed cover gives paper code A12071W1. Questions 2 and 3 are marked corrected in the source.^[ref]

The solutions use the original printed notation; the integer parity convention needed in Question 3(e) is stated there explicitly.

Solutions to the paper dated 4 June 2015. The original cover gives a ninety-minute examination with a choice of two questions; all three questions are solved here.^[ref]

Solutions to the paper dated 5 June 2015.^[ref] All thirteen numbered subparts are treated. The original cover and both question pages have been checked, including the directions and labels of the flow network diagram. The binary counter and decimal counter analyses use the conventional all-zero initial state; the effect of allowing a different initial state is stated explicitly.

A principal formula of a type is a logical formula first-order consistent with the base first-order theory that, modulo that first-order theory, implies every logical formula of the partial type. For a complete type it isolates a single point of the type space. A realized finite-tuple partial type in a finite first-order structure has such a logical formula by conjoining one distinguishing logical formula for each competing tuple with a different partial type.

For a nonzero partial isometry $u$ in a C-star algebra, put $p=u{u}^{\ast }$ and $q=u^{\ast }u$. The closed linear space $pAq$ becomes a unital C-star algebra with the displayed operations and inherited norm, with unit $u$. The map $a\mapsto u^{\ast }a$ is an isometric -isomorphism onto the ordinary corner $qAq$, with inverse $b\mapsto ub$. The identities $u{u}^{\ast }a=a$ and $u^{\ast }ub=b$ verify the inverse maps, while contractivity of both multiplications gives isometry. Products and involutions transfer directly, giving all algebra and C-norm axioms. This construction turns a rectangular operator corner into a square algebra by using the partial isometry as its unit.

If $X$ has a Pareto distribution with positive scale $x_m$ and positive shape $\alpha$, then for $z\ge 0$,Thus the rescaled logarithm has an exponential distribution with rate $\alpha$. The tail probability calculation also identifies the support as $z\ge 0$.

The complexity class of complements of formal languages in P. A deterministic polynomial-time algorithm deciding a formal language can have its accepting and rejecting outcomes interchanged, so P is closed under complement and equals co-P.

The component of osmotic pressure due to colloidal constituents such as albumin in the circulating fluid.