where Mag(x;y) = 1/| det[Jacη](x)| is the magnification of the lensed image x of y. Suppose that the light source at y moves along a path y(t). Then the shifted center-of-light curve relative to the source’s trajectory is defined by Xcl(t) = xcl(y(t)) − y(t). It is an important fact that locally stable k-plane lensing maps η are generic and each is differentiably equivalent about every critical...

(1) Let P, Q be subsets of E2 T, p1, p2, q1 be points of E2 T, f be a map from I into (E2 T) P, and s1 be a real number. Suppose that P is an arc from p1 to p2 and q1 ∈ P and q1 ∈ Q and f (s1) = q1 and f is a homeomorphism and f (0) = p1 and f (1) = p2 and 0 ≤ s1 and s1 ≤ 1 and for every real number t such that 0 ≤ t and t < s1 holds f (t) / ∈ Q. Let g be a map from I into (E2 T) P and s2 be a ...

We consider the exact multiplicity of nodal solutions of the boundary value problem u′′ + λf(u) = 0, t ∈ (0, 1), u′(0) = 0, u(1) = 0, where λ ∈ R is a positive parameter. f ∈ C1(R,R) satisfies f ′(u) > f(u) u , if u 6= 0. There exist θ1 < s1 < 0 < s2 < θ2 such that f(s1) = f(0) = f(s2) = 0; uf(u) > 0, if u < s1 or u > s2; uf(u) < 0, if s1 < u < s2 and u 6= 0; R 0 θ1 f(u)du = R θ2 0 f(u)du = 0. ...

and Applied Analysis 3 H4 f ∈ C R,R ; there exist two constants s2 < 0 < s1 such that f s2 f 0 f s1 0, f s > 0 for s ∈ 0, s1 ∪ s1,∞ , and f s < 0 for s ∈ −∞, s2 ∪ s2, 0 . The rest of this paper is organized as follows. In Section 2, we give some notations and the main results. Section 3 is devoted to proving the main results. 2. Statement of the Main Results Let Y {u ∈ C R,R : u t u t ω }with t...

The terminology and notation used in this paper are introduced in the following papers: [22], [3], [20], [9], [5], [12], [10], [11], [15], [2], [18], [4], [1], [21], [16], [17], [14], [13], [19], [6], [7], and [8]. For simplicity, we follow the rules: X denotes a complex unitary space, s1, s2, s3 denote sequences of X, R1 denotes a sequence of real numbers, C1, C2, C3 denote complex sequences, ...

and Applied Analysis 3 where i, j, and k are nonnegative integers. Let I be a closed interval in R. By induction, we may prove that x∗jk t Pjk ( x10 t , . . . , x1,j−1 t ; . . . ;xk0 t , . . . , xk,j−1 t ) , 1.11 βjk Pjk ⎛ ⎜⎝ j terms { }} { x′ ξ , . . . , x′ ξ ; . . . ; j terms { }} { x k ξ , . . . , x k ξ ⎞ ⎟⎠, 1.12 Hjk Pjk ⎛ ⎜⎝ j terms { }} { 1, . . . , 1; j terms { }} { M2, . . . ,M2; . . . ...

Then, let q(x) = ∑ m∈M (am ·m(x)) be the weighted sum of these monomials, for a set of weights {am}m∈M . We want to choose the weights in order to give q the properties already stated, which is equivalent to finding a non-trivial solution to a certain system of equations. Denote S = {s1, . . . , s|S|} and M = {m1, . . . ,m|M |}. Then, the system of equations we would like to solve is  m1(s1...

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent. Introduction We continue with the notation and the numbering of [4]. For the moment, we shall assume that E...

Let us consider S1, S2. We introduce S1 b S2 and S2 c S1 as synonyms of S1 is finer than S2. The following four propositions are true: (3)1 ⋃ (S1 \{ / 0}) = ⋃ S1. (4) For all partitions P1, P2 of Y such that P1 c P2 and P2 c P1 holds P2 ⊆ P1. (5) For all partitions P1, P2 of Y such that P1 c P2 and P2 c P1 holds P1 = P2. (7)2 For all partitions P1, P2 of Y such that P1 c P2 holds P1 is coarser ...

while (1) { with (s1) /* send r to 2d shape */ where (s1mask) pcoord(0)]]0]conn = r; with (s2) where (s2mask) { conn = copy_spread(&conn, 1, 0); /* spread */ conn &= e; /* combine */ reduce(&conn, conn, 0, /* reduce */ CMC_combiner_logior, 0); where (pcoord(0)==0) pcoord(1)]temp = conn; /* return result */ } with (s1) { /* stop if r hasn't changed */ if (&=(temp==r)) break; r = temp; } } CM_tim...