Aside. In principle, the spec-head configuration could also come into existence via external Merge. Here, we concentrate on Movement (internal Merge). Spec-head agreement Further motivation (Kayne 1989): In French/Italian, past-participle agreement with the object (with respect to gender and number) does not arise if the object remains in the position where it is merged (13-a)/(14-a). Only if the object moves (e.g., because it is a clitic) does past-participle agreement become possible (13-b)/(14-b).
Aside instance-verifiability of divulgers A divulger L1 is instance-verifiable meaning that the correctness of any valid instance of the output L1(d, b) can be verified without knowledge of any secrets, by checking that k1(L1(d, b)) = d.
Aside faulty schemes Faulty key agreement is not studied in this report, but is defined in this section, for completeness and clarification. Faulty key agreement relaxes the condition on k in a probabilistic key agreement scheme.
2.10.1. A procedural key agreement scheme is an array [a, b, c, k] where a, b, c are random variables, and k = [k1, k2, k3, k4] is array of functions such that ki such that both k3(a, k2(b, c)), k4(k1(a, b), c) are defined with probability 1. Pr[X∈R] 1A random variable x is a restriction of random variable X if a condition like the following holds: (i) there exists a set R such that Pr[X ∈ R] /= 1 and (ii) for all S ⊆ R such that Pr[X ∈ S] is defined, then Pr[x ∈ S] = Pr[X∈S] . 2 Recall that a discrete randomΣvariable v takes countably many values v1, v2, . . . with nonzero probabilities pi such that i pi = 1. By contrast, a continuous random variable assigns probabilities to events, subsets of a universe, generally using a measurable space as the universe, taking an event to be any measurable subset of the space, and letting the probability of an event be the integral of a probability density function computed over the measurable subset. A continuous random variable has probability zero of taking any particular value. Any probabilistic key agreement is a procedural key agreement, but some procedural key agreements are not probabilistic key agreement schemes.
2.10.2. A faulty key agreement scheme is a procedural key agreement scheme which is not a probabilistic key agreement scheme.
2.10.1. Generally, what makes [a, b, c, k] a faulty key agreement scheme is that The similarity in usefulness of faulty key agreement to that of probabilistic key agreement is measured by the following probability.
Aside non-associative scheme A non-associative key agreement scheme is any scheme k that is not asso- ciative. ƒ In particular, Diffie–Xxxxxxx key agreement k (Definition 2.2.1) is not as- sociative. To see this, first k1(a, b) = k3(a, b) = ba = ab = k2(b, a) = k4(b, a), for most a and b. But if k were associative, then it would be the subscheme of a multiplicative scheme K, and would have k1(a, b) = K1(a, b) = K2(a, b) = k2(a, b), a contradiction. − − ∈ { } For a smaller non-associative scheme, consider rock, scissors, paper key agreement again. Represent it as k1(a, b) = a and k2(b, c) = c and k3(a, b) = k4(a, b) = (a b) mod 3, where all a, b, c Z/3 = 0, 1, 2 , making each ki a binary operation on Z/3. Recall that this is a key agreement scheme because k3(a, k2(b, c)) = k3(a, c) = (a c) mod 3 = k4(a, c) = k4(k1(a, b), c). Suppose that k is a subscheme of a multiplicative scheme K. Then all the sessions of k must be sessions of K, so k1(a, b) = K1(a, b) and k2(b, c) = K2(b, c). But since K is multiplicative, K is symbiotic with K1 = K2. Clearly, k1 k cannot be a subscheme of K. k2, so
Aside reduction to a category Category theory is an abstraction that unifies many notions across modern al- gebra. Perhaps, it is then unsurprising that key agreement can be abstracted into category theory. Recall that category has objects and morphisms between objects. Mor- phisms compose associatively. (There is also a unique identity morphism form each object to itself, but we shall not need that.) Write Mor(O, P ), for the set of morphisms between two objects O, P . Write f ◦ g ∈ Mor(O, P ) for the composition of morphisms, f ∈ Mor(P, Q) and g ∈ Mor(O, Q). (This convention means the objects have opposite left-to-right ordering from the morphisms in a product.) First, we construct a key agreement scheme from any category, and any four objects O1, O2, O3, O4 in the category. Define a key agreement scheme k as follows: k1 : Mor(O4, O3) × Mor(O3, O2) → Mor(O4, O2) : [a, b] '→ a ◦ b k2 : Mor(O3, O2) × Mor(O2, O1) → Mor(O3, O1) : [b, c] '→ b ◦ c k3 : Mor(O4, O3) × Mor(O3, O1) → Mor(O4, O1) : [a, e] '→ a ◦ e
Aside. Hedging in Rough Models This thesis mainly focuses on pricing and so we only briefly discuss the issue of hedging. In classical stochastic volatility models, the task of hedging a derivative is greatly sim- plified by the Markov property. An application of Xxx’s formula readily yields the classic PDE formulation familiar from Black-Scholes. Clearly (as previously mentioned) in rough volatility models we do not have either Markovianity or Xxx’s lemma (in the conventional sense) and so the classical approach will fail. One approach found in the literature is that of Xxxxxxxx, Xxxxxxx and Xxxxxx [FHT21] where they show that, in a certain class of models, Markovianity is recovered in the en- larged space consisting of the spot and the forward variance curve. Using this they show that, formally, an option can be hedged by the stock and a variance swap (an instance of the Xxxxx-Xxxxx formula is needed). A similar approach is adopted by Xxxx and Xxxxxxxxx [ER18] where again they show a similar Markov property involving the forward variance curve and explain how to hedge an option using the spot and the forward variance curve. These two approaches are analytically rather complex and once any form of transac- tion costs are considered the problem becomes intractable. An alternative more data driven approach inspired by the Deep Hedging framework of Xxxxxxx [Bue19] was discussed in the paper [HTZ21] where a hedging portfolio was modelled as a neural network and machine learning techniques were applied to find the optimal hedge.
Aside. In retrospect, it may seem strange that the hypothesis of spec-head agreement has been entertained at all. Fact is that it was the standard analysis for quite a few years. Counter-evidence was explained away by further assumptions. Agreement without movement Some languages, among them English, have a construction where SpecT is not occupied by an argument but by an semantically empty element, an expletive (it, there, cf. (8-b)). Nevertheless, X agrees with some vP-internal argument (indicated by in (16)), and not with the expletive, see (15-a-d). This required additional (ad hoc) assumptions under the hypothesis of spec-head agreement. (15) a. There arrive-s a train.
Aside packed associated semigroups ⊆ If key agreement k is associated with semigroup S, define a subset Sk | | S, the set of sessional elements, by including each entry of each session [a, b, c, d, e, f ] of k, as mapped to elements of an associated semigroup S, via the equivalence of k to a subscheme of the multiplicative scheme kS. The sessionality of S associated with k is the cardinality Sk of the set of sessional elements. A semigroup S is a packed associated semigroup of k, if its sessionality is minimal among all associated semigroups of k.
Aside packed associated semigroups ⊆ If key agreement k is associated with semigroup S, define a subset Sk | | S, the set of sessional elements, by including each entry of each session [a, b, c, d, e, f ] of k, as mapped to elements of an associated semigroup S, via the equivalence of k to a subscheme of the multiplicative scheme kS. The sessionality of S associated with k is the cardinality Sk of the set of sessional elements. A semigroup S is a packed associated semigroup of k, if its sessionality is minimal among all associated semigroups of k. For example, for Diffie–Xxxxxxx agreement mod p, the associated semi- group from the proof Lemma 2.20.1 has sessionality of 6p − 6, while the associated semigroup from the proof of Lemma 2.18.1 has sessionality 2p − 2. Therefore, the former semigroup is not a packed associated semigroup of k, because the latter has lower sessionality.
21.1. For key agreement schemes with an infinite number of sessions, the cardinality of the set of sessional elements is likely not very informative, but a more refined approach based on functions into the semigroup may be more informative.
Aside the email Mr xxx xxx Xxxxx and Xxxxxxxxx sent to Ren Capes more or less concluded the relation between EGDI and PanGeo. Any portals that want to be sustained should become integrated into 1G-E, which should be maintained by EGS.
