Let H0 (resp. H∞) denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k ≥ 1 let Hk denote the class of k-hyponormal pairs in H0. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T1, T2) ∈ H1, the pair (T 2 1 , T2) may fail to be in H1. Conversely, we find a pair (T1, T2) ∈ H0 such tha...

Let Tt be the semigroup of linear operators generated by a Schrödinger operator −A = ∆− V , where V is a nonnegative polynomial. We say that f is an element of H1 A if the maximal function Mf(x) = supt>0 |Ttf(x)| belongs to L1. A criterion on functions F which implies boundedness of the operators F (A) on H1 A is given.

and Applied Analysis 3 Lemma 8 (Schmitt [27]). Let V be a real reflexive Banach space. Let G : R × V to V be completely continuous, and let a, b ∈ R (a < b) be such that the solution of (10) is, a priori, bounded inV for λ = a and λ = b; that is, there exists anR > 0 such that G (a, u) ̸ = u ̸ = G (b, u) (13) for all u with ‖u‖ ≥ R. Furthermore, assume that deg (I − G (a, ⋅) , BR (0) , 0) ̸ = deg ...

Let Y a random variable with probability density function (pdf) p(y|θ), where θ is an unknown parameter vector. Given an outcome y of Y , we aim at maximizing the likelihood function L(θ) ≡ p(y|θ) wrt θ over a given search space Θ. This is the very principle of maximum likelihood (ML) estimation. Unfortunately, except in not very exciting situations such as, e.g. estimating the mean and varianc...

Following [3] we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if A is a residually finite, finitely generated modal algebra such that HSP(A) has equationally definable principal congruences, then the profinite completion of A is the MacNeille completion of A, and ♦ is smooth. Specific examples of such modal algebras are the free...

Yale University and University of California, Berkeley Let Sn n ≥ 0 be a random walk having normally distributed increments with mean θ and variance 1, and let τ be the time at which the random walk first takes a positive value, so that Sτ is the first ladder height. Then the expected value EθSτ, originally defined for positive θ, may be extended to be an analytic function of the complex variab...

There is a simple example for Max Cut that shows our analysis is nearly tight. Consider G = C5, the cycle on 5 nodes {0, 1, 2, 3, 4}, with all edge weights being 1. The optimum solution is 4 with an optimum cut being, say, S = {1, 3} with δ(S) = {(0, 1), (1, 2), (2, 3), (3, 4)} (we can do no better because C5 is not bipartite). Next we define a feasible SDP solution with value greater than 4. L...

Let D be a negative discriminant, and let Θ(D) be the complex vector space generated by the binary theta series θf attached to the positive definite binary quadratic forms f(x, y) = ax + bxy + cy whose discriminant D(f) = b − 4ac equals D/t, for some integer t. It is a well-known classical fact that Θ(D) is a subspace of the space M1(|D|, ψD) of modular forms of weight 1, level |D| and Nebentyp...

The bootstrap is a powerful tool for inference in statistical models. This note describes briefly what the bootstrap is and how to use it. Suppose X1, . . . , Xn is an i.i.d. sample drawn from an unknown distribution P . Consider testing the null hypothesis H0 : θ = θ0 versus the alternative H1 : θ 6= θ0 where θ is a parameter related to the distribution P . The first step is to construct a tes...

Definition. An algebraic integer is the root of a monic polynomial in Z[x]. An algebraic number is the root of any non-zero polynomial in Z[x]. We are interested in studying the structure of the ring of algebraic integers in an algebraic number field. A number field is a finite extension of Q. We’ll assume that the number fields we consider are all subfields of C. Definition. Suppose that K and...