By A. M. Vinogradov

This publication is devoted to basics of a brand new conception, that is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This concept grew up from the classical geometry of PDE's originated through S. Lie and his fans via incorporating a few nonclassical principles from the speculation of integrable platforms, the formal thought of PDE's in its glossy cohomological shape given by means of D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). the most results of this synthesis is Secondary Calculus on diffieties, new geometrical items that are analogs of algebraic forms within the context of (nonlinear) PDE's. Secondary Calculus strangely finds a deep cohomological nature of the overall idea of PDE's and exhibits new instructions of its extra growth. contemporary advancements in quantum box concept confirmed Secondary Calculus to be its typical language, promising a nonperturbative formula of the speculation. as well as PDE's themselves, the writer describes present and strength functions of Secondary Calculus starting from algebraic geometry to box thought, classical and quantum, together with components comparable to attribute sessions, differential invariants, concept of geometric constructions, variational calculus, keep an eye on thought, and so forth. This booklet, centred mostly on theoretical facets, kinds a traditional dipole with Symmetries and Conservation legislation for Differential Equations of Mathematical Physics, quantity 182 during this related sequence, Translations of Mathematical Monographs, and indicates the speculation ""in action"".

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**Example text**

Finally, concerning the second one, take a ≥ 0 and b ≥ b ≥ 0: if a ≤ l ≤ b then A(l) + B(b − l) ≥ A(l) + B(b − l) ≥ D(a, b) ; on the other hand, if b ≤ l ≤ b then A(l) + B(b − l) ≥ A(b) + B(0) ≥ D(a, b) . It follows that D(a, b ) ≥ D(a, b), so D is nondecreasing also in its second variable and the proof is completed. Chapter 3 Optimal Connected Networks In this chapter we consider the problem of ﬁnding an optimal network under the additional constraint that the admissible networks are assumed connected.

In order to ﬁnd a competitor to µ, that will lead to a contradiction, we ﬁx an ε > 0 (that we will eventually lead to 0) and we let Cε to stand for the open ε−tubular neighborhood of ∆0 , that is the set of those points of Ω the minimal distance of which from θi is strictly less than ε and is reached at some point of ∆0 . Notice that, up to an arbitrary small movement of P and Q, ∆0 can be chosen in such a way that, for all ε > 0 except for countably many, 1 µ ∂Cε = 0 , and ∀ n ∈ N , µn ∂Cε = 0 .

23) where the function J: R+ × R+ → R is given by J(a, b) := inf A a + l + B b − l : 0 ≤ l ≤ b . 24) Before giving the proof, we shortly discuss the above formula. 19. 2), is that, roughly speaking, one can “walk on the railway”: in other words, the cost δ¯Σ of some path θ is not necessarily given by the cost of moving by own means out of the network and by train along it, but moving by own means out of the network and possibly in some part of it, and by train along the remaining part. The basic idea of the proof is then easily imagined: instead of walking on the network, one can just walk very close to it, which is possible since the dimension N is larger than 1.