On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$\mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big)$$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

On the Convergence of Kergin and Hakopian Interpolants at Leja Sequences for the Disk

We prove that Kergin interpolation polynomials and Hakopian interpolation polynomials at the points of a Leja sequence for the unit disk $D$ of a sufficiently smooth function $f$ in a neighbourhood of $D$ converge uniformly to $f$ on $D$. Moreover, when $f$ is $C^\infty$ on $D$, all the derivatives of the interpolation polynomials converge uniformly to the corresponding derivatives of $f$

Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.Comment: 16 pages, 2 figure

Probing the ISM Near Star Forming Regions with GRB Afterglow Spectroscopy: Gas, Metals, and Dust

We study the chemical abundances of the interstellar medium surrounding high z gamma-ray bursts (GRBs) through analysis of the damped Lya systems (DLAs) identified in afterglow spectra. These GRB-DLAs are characterized by large HI column densities N(HI) and metallicities [M/H] spanning 1/100 to nearly solar, with median [M/H]>-1. The majority of GRB-DLAs have [M/H] values exceeding the cosmic mean metallicity of atomic gas at z>2, i.e. if anything, the GRB-DLAs are biased to larger metallicity. We also observe (i) large [Zn/Fe] values (>+0.6) and sub-solar Ti/Fe ratios which imply substantial differential depletion, (ii) large a/Fe ratios suggesting nucleosynthetic enrichment by massive stars, and (iii) low C^0/C^+ ratios (<10^{-4}). Quantitatively, the observed depletion levels and C^0/C^+ ratios of the gas are not characteristic of cold, dense HI clouds in the Galactic ISM. We argue that the GRB-DLAs represent the ISM near the GRB but not gas directly local to the GRB (e.g. its molecular cloud or circumstellar material). We compare these observations with DLAs intervening background quasars (QSO-DLAs). The GRB-DLAs exhibit larger N(HI) values, higher a/Fe and Zn/Fe ratios, and have higher metallicity than the QSO-DLAs. We argue that the differences primarily result from galactocentric radius-dependent differences in the ISM: GRB-DLAs preferentially probe denser, more depleted, higher metallicity gaslocated in the inner few kpc whereas QSO-DLAs are more likely to intersect the less dense, less enriched, outer regions of the galaxy. Finally, we investigate whether dust obscuration may exclude GRB-DLA sightlines from QSO-DLA samples; we find that the majority of GRB-DLAs would be recovered which implies little observational bias against large N(HI) systems.Comment: 16 pages, 9 figures. Submitted to Ap

The Equational Theory of Fixed Points with Applications to Generalized Language Theory

We review the rudiments of the equational logic of (least) fixed points and provide some of its applications for axiomatization problems with respect to regular languages, tree languages, and synchronization trees

The importance of language for language development: Linguistic determinism in the 1980s

The semantic and syntactic functions of verbs are the major aspects of linguistic complexity that contribute to the cognitive requirements for learning language between two and three years of age. Several contrastive categories of verbs emerged from our studies with action/state as the largest and most general. Contrastive subcategories of action verbs were locative/nonlocative action, durative/nondurative action, and completive/noncompletive action. The subcategories of state verbs were volitional/epistemic/notice/communication states. The psychological and linguistic validity of these semantic categories rests on their being coextensive with major grammatical developments and/or their sequential development

Kleene Algebras and Semimodules for Energy Problems

With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce generalized energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Uncovering a close connection between energy problems and reachability and B\"uchi acceptance for semiring-weighted automata, we show that these generalized energy problems are decidable. We also provide complexity results for important special cases