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The Dold manifold P (m,n) is obtained from the product Sm × CPn of the m-dimensional sphere and n-dimensional complex projective space by identifying (x, [z1, . . . , zn+1]) with (−x, [z1, . . . , zn+1]), where z denotes the complex conjugate of z. We answer the parallelizability question for the Dold manifolds P (m,n) and, by completing an earlier (2008) result due to Peter Novotný, we solve the vector field problem for the manifolds P (m, 1).

We find full invariants for detecting non--zero elements in ${L}_0(F(t))\otimes \Q$, this group plays an important role in topology in the work done by Casson and Gordon., Comment: 8 pages

If p ( s ) p(s) is a positive polynomial of degree 2 d 2d , then its outer factor q ( s ) q(s) has the property that the magnitude of each of its coefficients is larger than the magnitude of the corresponding coefficient of any other factor. In fact, this extremal property holds over vector-valued factorizations r ( s ) ∗ r ( s ) = p ( s ) r(s)^{*}r(s)=p(s) . Corollaries include a result for symmetric functions and complex conjugate pairs.

We propose a method of constructing explicit Banach spaces not isomorphic to their complex conjugates as subspaces of a natural class of Banach spaces. In particular, it is shown that Lp, for 1 < p < 2, contains real subspaces with at least two non-isomorphic complex structures.

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

We show that for compactly supported perturbations of the wave equation, solutions whose total energy decays to zero must be smooth, provided that there are no trapped rays. This paper discusses solutions to perturbed wave equations where the perturbation is restricted to a compact subset of space. The basic intuitive idea which lies behind the results and proofs is that a solution represents a wave which moves freely except when it lies in the compact set. 'The expected global behavior is that the wave will stay in this region for a while and then travel off to infinity as a free wave. Since energy is not lost in this last phase, we do not expect that the total energy of a wave will decay to zero as time goes to infinity. In fact, for solutions which are not smooth we can make this argument precise by keeping track of the singularities to show that a finite amount of energy does radiate to infinity. 'In this way we prove that waves whose total energy decays to zero must be smooth. Let L 83-82 8 2 L d 2 E a..(t, x) + E a.(t, x) dx + a(t, x) d+ b(t, x). We suppose that a.. is a positive definite symmetric matrix and that 1. L = a2/8t2_X82/8x2 for IxI > R. 2. The coefficients of L are real valued and smooth. 3. n, the number of space dimensions, is odd and greater than 2. The reality hypothesis allows us to restrict attention to real solutions and to avoid annoying complex conjugates and real parts. 'It is not essential. We are interested in solutions u(t, x) of the equation Lu = 0 which have the property that Received by the editors December 19, 1973. AMS (MOS) subject classifications (1970). Primary 35L15; Secondary 35L45, 35P25, 35B40, 35D10. IResearch partially supported by the National Science Foundation under grant NSF GP 34260. Copyright

Two n-dimensional unitary representations which differ by complex conjugation or tensoring with a character induce topologically equivalent actions on the Grassmann manifold of complex m-planes in n-space. This paper shows under modest dimension hypotheses that only such projectively equivalent linear representations of compact connected Lie groups can give topologically conjugate actions.

The Sturm-Liouville eigenvalue problem − y + q ( t ) y = λ r ( t ) y - y + q(t)y = \lambda r(t)y , t ∈ [ a , b ] t \in [a, b] , where y y is required to satisfy a pair of homogeneous separated boundary conditions at t = a t = a , t = b t = b is considered when no sign restrictions are imposed upon the coefficients q q , r r . It will be shown that the general eigenvalue problem above can admit at most finitely many nonsimple real eigenvalues (in some cases none at all). Moreover, we will show by means of an example that nonsimple real eigenvalues may occur in the case when each of q q and r r changes sign in ( a , b ) (a, b) and under Dirichlet boundary conditions.

Let μ \mu be a positive Borel measure on a compact subset K K of the complex plane. Denote the weak-star closure in L ∞ ( μ ) {L^\infty }\left ( \mu \right ) of R ( K ) R\left ( K \right ) by R ∞ ( K , μ ) {R^\infty }\left ( {K,\mu } \right ) . Given f ∈ R ∞ ( K , μ ) f \in {R^\infty }\left ( {K,\mu } \right ) , denote the weak-star closure in L ∞ ( μ ) {L^\infty }\left ( \mu \right ) of the algebra generated by R ∞ ( K , μ ) {R^\infty }\left ( {K,\mu } \right ) and the complex conjugate of f f by A ∞ ( f , μ ) {A^\infty }\left ( {f,\mu } \right ) . This paper determines the structure of A ∞ ( f , μ ) {A^\infty }\left ( {f,\mu } \right ) . As a consequence, a solution is obtained to a problem concerned with minimal normal extensions of functions of a subnormal operator.

In [1], M. B. Balk obtains a representation for polyanalytic functions which have constant modulus on some region. Balk's result suggests the more difficult problem of obtaining necessary and sufficient conditions that two polyanalytic functions have equal modulus on some region. This problem is answered in Theorem 1 and as an immediate corollary to Theorem 1, we obtain the result of Balk cited in [1]. In Theorem 2 and subsequent corollaries, the question of establishing criteria that two polyanalytic functions have equal amplitudes or equal real parts or equal imaginary parts is considered. In what follows, G will denote a region of the finite complex plane r and G* will denote the region G*= {zEr|EeG}. DEFINITION. A function f: G-,r is said to be polyanalytic on G or n-analytic on G if and only if there exist n ?1 functions fo, fl, fr-i which are analytic on G such thatf _1 is not identically zero on G whenever n > 2 and such that f(z) = EZkfk (z), for all z E G, where k=0, 1, * * * , n-I and where z denotes the complex conjugate of z. We now need the following preparatory lemma

1. Main theorem. Let x = (xo, xi, • • • ) be a sequence of complex numbers and let | x \ he the norm | x | = (^Z | x„ |2)1/2 2:0. Let an asterisk denote complex conjugation. If \x\ <

A complex vector space X will be called an F 2 n {F_{2n}} space if and only if there is a mapping ⟨ ⋅ , ⋯ , ⋅ ⟩ \langle \cdot , \cdots , \cdot \rangle from X 2 n {X^{2n}} into the complex numbers such that: ⟨ x , ⋯ , x ⟩ > 0 \langle x, \cdots ,x\rangle > 0 if x ≠ 0 ; x k → ⟨ x 1 , ⋯ , x 2 n ⟩ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle is linear for k = 1 , ⋯ , n ; ⟨ x 1 , ⋯ , x 2 n ⟩ = ⟨ x 2 n , ⋯ , x 1 ⟩ − k = 1, \cdots ,n;\langle {x_1}, \cdots ,{x_{2n}}\rangle = {\langle {x_{2n}}, \cdots ,{x_1}\rangle ^ - } where denotes complex conjugate; ⟨ x σ ( 1 ) , ⋯ , x σ ( n ) , y τ ( 1 ) , ⋯ , y τ ( n ) ⟩ = ⟨ x 1 , ⋯ , x n , y 1 , ⋯ , y n ⟩ \langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle for all permutations σ , τ \sigma ,\tau of { 1 , ⋯ , n } \{ 1, \cdots ,n\} . In the case of a real vector space the mapping is assumed to be into the reals such that: ⟨ x , ⋯ , x ⟩ > 0 \langle x, \cdots ,x\rangle > 0 if x ≠ 0 ; x k → ⟨ x 1 , ⋯ , x 2 n ⟩ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle is linear for k = 1 , ⋯ , 2 n ; ⟨ x σ ( 1 ) , ⋯ , x σ ( 2 n ) ⟩ = ⟨ x 1 , ⋯ , x 2 n ⟩ k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle for all permutations σ \sigma of { 1 , ⋯ , 2 n } \{ 1, \cdots ,2n\} . In either case, if ‖ x ‖ = ⟨ x , ⋯ , x ⟩ 1 / 2 n \left \| x \right \| = {\langle x, \cdots ,x\rangle ^{1/2n}} defines a norm, X is called a G 2 n {G_{2n}} space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an F 2 n {F_{2n}} space is a G 2 n {G_{2n}} space if and only if | ⟨ x , y , ⋯ , y ⟩ | 2 n ≦ ⟨ x , ⋯ , x ⟩ ⟨ y , ⋯ , y ⟩ 2 n − 1 |\langle x,y, \cdots ,y\rangle {|^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}} and that G 2 n {G_{2n}} spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).

In (2), the i and bi represent the real and complex zeros, respectively, and the pi and ti represent the real and complex poles, respectively. The complex poles and zeros will occur in complex conjugate pairs and the poles will have non positive real parts. Various sets of sufficient conditions on the poles and zeros, which insure that a(t) is nondecreasing, have been published previously [1-5]. The purpose of this note is to add to this body of results an extension of the theorem given in [5]. It includes as special cases most of the results given in [1; 2; and 5]. It does not, however, include the results of [3; 4]. The following notation and numbering system will be used for the poles and zeros. The real poles are numbered according to their decreasing values; that is, Pl >P2>* * * >Pm. The real parts of the complex poles and of all the zeros are denoted by axi and numbered according to their decreasing values; that is, a,i _ is2 > . . . >a,, where n=h+g+q. When several zeros and complex poles have the same real part, they are numbered in any order. If the multiplicity of any pole or zero is r, it is counted r times. The following fact is an immediate consequence of the first lemma and the last four sentences of [5].

where [ck,,(w) ]indicates the complex conjugate of [Ec,,(w)], (cf. [1, pp. 6 and 9]). Let K be a given compact subset of D. Then we might wish to try to approximate the kernel function in K by using a finite series. Because of (2) and the fact that mD(z) is continuous, it is clear that given an e >0 and a complete orthonormal sequence {I ,, }, there exists an N= N(D, K, c, {IO,,) such that m? N and z, wEK implies

We then take the natural logarithm of both sides of (2.2) and invert the similarity transformation to obtain the desired solution(s) X. As we will show rigorously, a real solution exists provided C is nonsingular and each elementary divisor (J ordan block) of C corresponding to a negative eigenvalue occurs an even number of times. This assures that the complex part of X will have complex conjugate elementary divisors (Jordan blocks). The possible nonuniqueness of the solution can arise in two ways as we will demonstrate: (1) because the matrix C has complex eigenvalues and hence provides log J with at least a countable infinity of periodic values, and (2) because the similarity transformation which relates J to C uniquely2 via (2.1) may not relate log J to X uniquely via (2.2), in which case an uncountable infinity of solutions results.

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.