By Jost J., Xin Y. L.

We receive a Bernstein theorem for detailed Lagrangian graphs in for arbitrary merely assuming bounded slope yet no quantitative limit.

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B,3 + J. L{i) a,b,c > O. 1) the same general type as for R(i). This is the case of the potential induced in a solenoid or electromagnet by a current III. i(t). Capacitor. Same situation with i,v interchanged, the relation now being dv C(v) dt 3. i, C(v) > O. •• ,bR G the follow- be the branches 3. A Structural Property 55 of G -~. R Let f designate any chain of 5(m. 1) We propose to describe an algebraic method for arriving at the same result. "~R be a base for cycles and let Since the yls are independent the matrix [€hk) the branches be so ordered that the matrix is already nonsingular.

2. Prove that the union of any number and the intersection of a finite number of open sets are open sets. Show also that the union of a finite number and the intersection of any number of closed sets are closed sets. 3. point x Let R of R such that every Show that closed sets property: 4. A C R. be a space and F A limit point of Y(x,r) of R closed sets A of a set A F characterized by the property least upper bound of all numbers as inf d(x,y) is sup d(x,y) Show that 6. d(x,A) A). for all A x,y N Show that A is the Hence a closed set is in let and > N yof sup N inf N x denote the =--sup (-N) to a set A; the diameter A • is defined deAl of A A.

IV. GRAPHS. 2) The number 39 of components of a graph is a topological p invariant. Characteristic. It is by definition the number (number of nodes - number of branches). 3) X(G) is a topological invariant. 4) ~O - and ~O by ~l' The characteristic of a loop is zero, that of an arc or of a tree is unity. For a loop or an arc the proof is immediate. 2). bi + ni removing X(~'). and Let J( be a be a node of order be the unique branch ending at what is left is still a tree J(' and ni. Upon X ( ~) P By repeating the process one arrives at a single point X(~) = X(P) = 1.

### A Bernstein theorem for special Lagrangian graphs by Jost J., Xin Y. L.

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