By G I Pshenichnov
This quantity provides the idea of partial differential equations (PDEs) from a latest geometric standpoint in order that PDEs could be characterised through the use of both means of differential geometry or algebraic geometry. this enables us to acknowledge the richness of the constitution of PDEs. It provides, for the 1st time, a geometrical thought of non-commutative (quantum) PDEs and provides a basic program of this concept to quantum box concept and quantum supergravity
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Extra info for A theory of latticed plates and shells
The nonuniqueness of the decomposition of the original problem is a consequence of two causes. The first consists in the arbitrariness of the choice of the operators La,, which may be one-dimensional as well as multidimensional, and, moreover, may contain terms not included in the operators L, of the original problem. 2) may be satisfied, due to the arbitrariness of the right-hand sides of Eqs. 6)). 2) be satisfied, when taking the union of the boundary conditions of all the auxiliary problems.
And in this case it is not necessary to determine function ¥>i(a). It is known (also from the physical concepts) that r. occurs at the beam's cross section points having coordinates x = 0, y = ±6 (for definiteness assume a > b). Hence, _ djP __9Ctf 9Ctf OGa2 dv dv r. 40) T (240) '~ be dp ' ~ dy dy **===coo be be OP dp °j o===0oo • »= = l/c \h _ k y=b In accordance with this formula, on the basis of Eq. 40) 8v dv dv dv\ dv22 dvx dv? dP == ~dp~ dp~ dp ~dp~ == ~dp~ ~dj) w we find a2b r. = 2 0 G — 2 - r =2.
32 Chapter 1. 8: Similarly, in the plane problem for reticulated plates the order of the system of differential equations increases from four to eight and for its solution four and not two boundary conditions must be used. We now consider the type of two additional boundary conditions needed for reticulated shells' calculations when J2i j= 0. Of practical interest is the instance when the shell's lattice consists of two families of rods as only in this case will it be necessary to consider the rods' torsional rigidity in the plane tangential to the shell's middle surface.