On the Exponential Integrability of Conjugate Functions

We relate the exponential integrability of the conjugate function $${\tilde{f}}$$ f ~ to the size of the gap in the essential range of f. Our main result complements a related theorem of Zygmund.

AGREEMENT THEOREMS FOR SELF-LOCATING BELIEF

In this paper, I first outline Aumann’s famous “no agreeing to disagree” theorem, and a second related theorem. These results show that if two or more agents, who have epistemic and credal states that are defined over algebras that do not include any self-locating propositions, have certain information about one another’s epistemic and credal states, then such agents must assign the same credence to certain propositions. I show, however, that both of these theorems fail when we consider agents who have epistemic and credal states that are defined over algebras that do include self-locating propositions. Importantly, these theorems fail for such agents even when we restrict our attention to the credences that such agents have in non-self-locating propositions. Having established this negative result, I then outline and prove three agreement theorems that hold for such agents.

AN EXAMPLE CONCERNING BOUNDED LINEAR REGULARITY OF SUBSPACES IN HILBERT SPACE

We study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.

Normal Family of Meromorphic Functions concerning Shared Values

We obtain a normal criterion of meromorphic functions concerning, shared values. Let ℱ be a family of meromorphic functions in a domain D and let k,n≥k+2 be positive integers. Let a≠0,b be two finite complex constants. If, for each f∈ℱ, all zeros of f have multiplicity at least k+1 and f+a(f(k))n and g+a(g(k))n share b in D for every pair of functions f,g∈ℱ, then ℱ is normal in D. This result generalizes the related theorem according to Xu et al. and Qi et al., respectively. There is a gap in the proofs of Lemma 3 by Wang (2012) and Theorem 1 by Zhang (2008), respectively. They did not consider the case of f(z) being zerofree. We will fill the gap in this paper.

Sparse fusion systems

We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman–Thompson p-Nilpotency Theorem for fusion systems and a related theorem of Stellmacher. We then define a more restrictive class of saturated fusion systems, called extremely sparse systems, that are constrained for all primes.

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

Commutative 2-local ring spectra

A theorem is proved characterising representable, multiplicative commutative cohomology theories that split as sums of singular cohomologies after localisation at 2. This theorem is shown to be equivalent to one proved by Würgler and Pazhitnov and Rudyak, for which we provide a simplified proof. We also provide a simple proof of a related theorem of Boardman.

Quasi-bounded and singular thermal potentials

In his monograph [2], Doob presented the theories of quasi-bounded and singular functions associated with the Laplace and heat equations. In particular, on p. 712, he proved in the classical case that a potential is quasi-bounded if and only if its associated measure vanishes on polar sets. However, on p. 714, he proved only weaker results for the thermic case, namely that a potential is quasi-bounded (i) if its associated measure vanishes on semi-polar sets, and (ii) only if its measure vanishes on polar sets. The main purpose of this note is to establish that the exact analogue of the result for the classical case holds for the thermic case. We also use a related theorem on thermal potentials, to prove a second characterization of those which are quasi-bounded, analogous to one due to Arsove and Leutwiler for the classical case. We also give the corresponding characterizations of singular thermal potentials.