A singularly perturbed linear two-point boundary-value problem

<p>We consider the following singularly perturbed linear two-point boundary-value problem:p> <p>Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x���1 (1a)p> <p>By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)p> <p>Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.p> <p>A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.p> <p>Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).p>

Arithmetic of cyclotomic fields and Fermat-type equations

<p>Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.p> <p>The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.p>

Influence of composition on the structure and properties of Fe-Pd-P and Ni-Pd-P amorphous alloys

<p>Ternary alloys of nickel-palladium-phosphorus and iron-palladium- phosphorus containing 20 atomic % phosphorus were rapidly quenched from the liquid state. The structure of the quenched alloys was investigated by X-ray diffraction. Broad maxima in the diffraction patterns, indicative of a glass-like structure, were obtained for 13 to 73 atomic % nickel and 13 to 44 atomic % iron, with palladium adding up to 80%.p> <p>Radial distribution functions were computed from the diffraction data and yielded average interatomic distances and coordination numbers. The structure of the amorphous alloys could be explained in terms of structural units analogous to those existing in the crystalline Pd<sub>3sub>P, Ni<sub>3sub>P and Fe<sub>3sub>P phases, with iron or nickel substituting for palladium. A linear relationship between interatomic distances and composition, similar to Vegard's law, was shown for these metallic glasses.p> <p>Electrical resistivity measurements showed that the quenched alloys were metallic. Measurements were performed from liquid helium temperatures (4.2°K) up to the vicinity of the melting points (900°K- 1000°K). The temperature coefficient in the glassy state was very low, of the order of 10<sup>-4sup>/°K. A resistivity minimum was found at low temperature, varying between 9°K and 14°K for Ni_x-Pd_80-x -P<sub>20sub> and between 17°K and 96°K for Fe_x-Pd_80-x -P<sub>20sub>, indicating the presence of a Kondo effect. Resistivity measurements, with a constant heating rate of about 1.5°C/min,showed progressive crystallization above approximately 600°K.p> <p>The magnetic moments of the amorphous Fe-Pd-P alloys were measured as a function of magnetic field and temperature. True ferromagnetism was found for the alloys Fe<sub>32sub>-Pd<sub>48sub>-P<sub>20sub> and Fe<sub>44sub>-Pd<sub>36sub>-P<sub>20sub> with Curie points at 165° K and 380° K respectively. Extrapolated values of the saturation magnetic moments to 0° K were 1.70 µ<sub>Bsub> and 2.10 µ<sub>Bsub> respectively. The amorphous alloy Fe<sub>23sub>-Pd<sub>57sub>-P<sub>20sub> was assumed to be superparamagnetic. The experimental data indicate that phosphorus contributes to the decrease of moments by electron transfer, whereas palladium atoms probably have a small magnetic moment. A preliminary investigation of the Ni-Pd-P amorphous alloys showed that these alloys are weakly paramagnetic.p>

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

<p>Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A<sub>i+1sub> = A<sub>isub>B - BA<sub>isub>, i = 0, 1, 2,…, with A = A<sub>osub>. Let f<sub>ksub> (A, B; σ) = A<sub>2K+1sub> - <sup>σsup>1<sup>Asup>2K-1 <sup>+sup> <sup>σsup>2<sup>Asup>2K-3 -… +(-1)<sup>Ksup>σ<sub>Ksub>A<sub>1sub> where σ = (σ<sub>1sub>, σ<sub>2sub>,…, σ<sub>Ksub>), σ<sub>isub> belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f<sub>nsub>(A, B; σ) = 0 if σ<sub>isub> is the i<sup>thsup> elementary symmetric function of (β<sub>4sub>- β_s)<sup>2sup>, 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β<sub>4sub> are the characteristic roots of B. In this thesis we discuss relations involving f<sub>ksub>(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f<sub>ksub>(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X<sub>1sub>, X<sub>2sub>…X<sub>rsub> belong to the radical of the algebra generated by A and B over F, where X<sub>isub> has the form f<sub>2sub>(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f<sub>ksub>(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g<sub>ksub>(w; σ) = w<sup>2K+1sup> - σ<sub>1sub>w<sup>2K-1sup> + σ<sub>2sub>w<sup>2K-3sup> - …. +(-1)<sup>Ksup> σ<sub>Ksub>w over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231]. p>

A generalized Hausdorff dimension for functions and sets

<p>Let E be a compact subset of the n-dimensional unit cube, 1<sub>nsub>, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the numberp> <p>d<sub>Csub>(E) = sup(β:H<sub>β, Csub>(E) > 0),p> <p>where H<sub>β, Csub> is the outer measure p> <p>inf(Ʃm(C<sub>isub>)<sup>βsup>:UC<sub>isub> Ↄ E, C<sub>isub> ϵ C) . p> <p>Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’<sub>nsub>, of 1<sub>nsub>, whose closure intersects 1<sub>nsub> - 1’<sub>nsub>. For n = 2, the outer measure H<sub>β, Csub> is adopted in place of the usual: p> <p>Inf(Ʃ(diam. (C<sub>isub>))<sup>βsup>: UC<sub>isub> Ↄ E, C<sub>isub> ϵ C), p> <p>for the purpose of studying the influence of the shape of the covering sets on the dimension d<sub>Csub>(E).p> <p>If E is a closed set in 1<sub>1sub>, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),p> <p>d<sub>Csub>(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)p> <p>where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown thatp> <p>d<sub>Csub>(E) = sup (d<sub>Csub>(μ):μ ϵ M(E)).p> <p>This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d<sub>Csub>(E) as a function of the covering class C is reduced to the study of d<sub>Csub>(f) where f ϵ Ӻ. Specifically, the set of points in 1<sub>1sub>,p> <p>(*) {d<sub>Bsub>(F), d<sub>Csub>(f)): f ϵ Ӻ}p> <p>is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1<sub>2sub>, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.p> <p>In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1<sub>2sub> with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formulap> <p>d<sub>Csub>(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ Cp> <p>wherep> <p>∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).p> <p>A characterization of the equivalence d<sub>Csub><sub>1sub>(f) = d<sub>Csub><sub>2sub>(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C<sub>isub> (I = 1, 2). p>

On the Cesari fixed point method in a Banach space

<p>In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.p> <p>The following is my formulation of the Cesari fixed point method:p> <p>Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that p�� is compact and such that for every x in P��, P<sup>-1sup>x���Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г. p> <p>Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:p> <p>(i) Py = PWy.p> <p>(ii) y = (P + (I - P)W)y.p> <p>Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:p> <p>(1) For each x in P��, P + (I - P)W is a contraction from P<sup>-1sup>x���Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P<sup>-1sup>x���Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).p> <p>(2) The function y just defined is continuous from P�� into B.p> <p>(3) There are no fixed points of PWy on the boundary of P��, so that the (finite- dimensional) fixed point index i(PWy, int P��) is defined.p> <p>Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int P��).p> <p>The three theorems of this thesis can now be easily stated.p> <p>Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.p> <p>Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P<sub>1sub>) and (Г, W, P<sub>2sub>). Assume that P<sub>2sub>P<sub>1sub>=P<sub>1sub>P<sub>2sub>=P<sub>1sub> and assume that either of the following two conditions holds:p> <p>(1) For every b in B and every z in the range of P<sub>2sub>, we have that ‖b=P<sub>2sub>b‖ ≤ ‖b-z‖p> <p>(2)P<sub>2sub>Г is convex.p> <p>Then i(Г, W, P<sub>1sub>) = i(Г, W, P<sub>2sub>).p> <p>Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).p> <p>Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.p>