Comments on fixed point results in classes of function with values in a b–metric space

We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.

Toward a Mechanism for the Emergence of Gravity

One of the main problems that emergent-gravity approaches face is explaining how a system that does not contain gauge symmetries ab initio might develop them effectively in some regime. We review a mechanism introduced by some of the authors for the emergence of gauge symmetries in [JHEP 10 (2016) 084] and discuss how it works for interacting Lorentz-invariant vector field theories as a warm-up exercise for the more convoluted problem of gravity. Then, we apply this mechanism to the emergence of linear diffeomorphisms for the most general Lorentz-invariant linear theory of a two-index symmetric tensor field, which constitutes a generalization of the Fierz–Pauli theory describing linearized gravity. Finally we discuss two results, the well-known Weinberg–Witten theorem and a more recent theorem by Marolf, that are often invoked as no-go theorems for emergent gravity. Our analysis illustrates that, although these results pinpoint some of the particularities of gravity with respect to other gauge theories, they do not constitute an impediment for the emergent gravity program if gauge symmetries (diffeomorphisms) are emergent in the sense discussed in this paper.

NORMAL REFLECTION SUBGROUPS OF COMPLEX REFLECTION GROUPS

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.

New G2-conifolds in M-theory and their field theory interpretation

A recent theorem of Foscolo-Haskins-Nordström [1] which constructs complete G2-holonomy orbifolds from circle bundles over Calabi-Yau cones can be utilised to construct and investigate a large class of generalisations of the M-theory flop transition. We see that in many cases a UV perturbative gauge theory appears to have an infrared dual described by a smooth G2-holonomy background in M-theory. Various physical checks of this proposal are carried out affirmatively.

Reduced norms of division algebras over complete discrete valuation fields of local-global type

Let [Formula: see text] be a complete discrete valuation field whose residue field [Formula: see text] is a global field of positive characteristic [Formula: see text]. Let [Formula: see text] be a central division [Formula: see text]-algebra of [Formula: see text]-power degree. We prove that the subgroup of [Formula: see text] consisting of reduced norms of [Formula: see text] is exactly the kernel of the cup product map [Formula: see text], if either [Formula: see text] is tamely ramified or of period [Formula: see text]. This gives a [Formula: see text]-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-[Formula: see text] degree.

An innovative approach for the assessment of masonry bridges based on two new limit analysis theorems

<p>The structural safety of a masonry arch bridge is usually assessed using the so-called kinematic approach. In this paper it is proved that the adoption of the dual static method can be more convenient, since a recent theorem (the <i>Minimum Equilibrated Compression </i>theorem) makes its application straightforward for any kind of arch. Computations are checked via the kinematic method by locating the plastic hinges (as many as needed to form a collapse mechanism) in the sections with maximum compression stress. Thanks to the <i>Consecutive Plastic Hinges </i>theorem, the kinematic multiplier may be then evaluated, using familiar moments of forces, without computing the virtual displacements due to the vertical and horizontal loads acting on the arch.</p>

Some Order Preserving Inequalities for Cross Entropy and Kullback–Leibler Divergence

Cross entropy and Kullback–Leibler (K-L) divergence are fundamental quantities of information theory, and they are widely used in many fields. Since cross entropy is the negated logarithm of likelihood, minimizing cross entropy is equivalent to maximizing likelihood, and thus, cross entropy is applied for optimization in machine learning. K-L divergence also stands independently as a commonly used metric for measuring the difference between two distributions. In this paper, we introduce new inequalities regarding cross entropy and K-L divergence by using the fact that cross entropy is the negated logarithm of the weighted geometric mean. We first apply the well-known rearrangement inequality, followed by a recent theorem on weighted Kolmogorov means, and, finally, we introduce a new theorem that directly applies to inequalities between K-L divergences. To illustrate our results, we show numerical examples of distributions.

HAPPY AND MAD FAMILIES INL(ℝ)

We prove that, in the choiceless Solovay model, every set of reals isH-Ramsey for every happy familyHthat also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under$A{D^ + }$.

Compactifications of adjoint orbits and their Hodge diamonds

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.