Multi-PeV Signals from a New Astrophysical Neutrino Flux Beyond the Glashow Resonance

The IceCube neutrino discovery was punctuated by three showers with $E_\nu$ ~ 1-2 PeV. Interest is intense in possible fluxes at higher energies, though a marked deficit of $E_\nu$ ~ 6 PeV Glashow resonance events implies a spectrum that is soft and/or cutoff below ~few PeV. However, IceCube recently reported a through-going track event depositing 2.6 $\pm$ 0.3 PeV. A muon depositing so much energy can imply $E_{\nu_\mu} \gtrsim$ 10 PeV. We show that extending the soft $E_\nu^{-2.6}$ spectral fit from TeV-PeV data is unlikely to yield such an event. Alternatively, a tau can deposit this much energy, though requiring $E_{\nu_\tau}$ ~10x higher. We find that either scenario hints at a new flux, with the hierarchy of $\nu_\mu$ and $\nu_\tau$ energies suggesting a window into astrophysical neutrinos at $E_\nu$ ~ 100 PeV if a tau. We address implications, including for ultrahigh-energy cosmic-ray and neutrino origins.Comment: 6 pages, 4 figures + 3 pages Supplementary Material; updated to agree with version published in Physical Review Letter

Is the principle of least action a must?

The least action principle occupies a central part in contemporary physics. Yet, as far as classical field theory is concerned, it may not be as essential as generally thought. We show with three detailed examples of classical interacting field theories that it is possible, in cases of physical interest, to derive the correct field equations for all fields from the action (which we regard as defining the theory), some of its symmetries, and the conservation law of energy-momentum (this last regarded as ultimately coming from experiment)Comment: RevTeX, no figures, Eq. 3 corrected, a reference added, added sundry clarifying comments, no change in the results, 23 page

Merging KK-means with hierarchical clustering for identifying general-shaped groups

Clustering partitions a dataset such that observations placed together in a group are similar but different from those in other groups. Hierarchical and $K$-means clustering are two approaches but have different strengths and weaknesses. For instance, hierarchical clustering identifies groups in a tree-like structure but suffers from computational complexity in large datasets while $K$-means clustering is efficient but designed to identify homogeneous spherically-shaped clusters. We present a hybrid non-parametric clustering approach that amalgamates the two methods to identify general-shaped clusters and that can be applied to larger datasets. Specifically, we first partition the dataset into spherical groups using $K$-means. We next merge these groups using hierarchical methods with a data-driven distance measure as a stopping criterion. Our proposal has the potential to reveal groups with general shapes and structure in a dataset. We demonstrate good performance on several simulated and real datasets.Comment: 16 pages, 1 table, 9 figures; accepted for publication in Sta

Extra-matrix Mg\u3csup\u3e2+\u3c/sup\u3e Limits Ca\u3csup\u3e2+\u3c/sup\u3e Uptake and Modulates Ca\u3csup\u3e2+\u3c/sup\u3e Uptake-independent Respiration and Redox State in Cardiac Isolated Mitochondria

Cardiac mitochondrial matrix (m) free Ca2+ ([Ca2+]m) increases primarily by Ca2+ uptake through the Ca2+ uniporter (CU). Ca2+ uptake via the CU is attenuated by extra-matrix (e) Mg2+ ([Mg2+]e). How [Ca2+]m is dynamically modulated by interacting physiological levels of [Ca2+]e and [Mg2+]e and how this interaction alters bioenergetics are not well understood. We postulated that as [Mg2+]e modulates Ca2+ uptake via the CU, it also alters bioenergetics in a matrix Ca2+–induced and matrix Ca2+–independent manner. To test this, we measured changes in [Ca2+]e, [Ca2+]m, [Mg2+]e and [Mg2+]m spectrofluorometrically in guinea pig cardiac mitochondria in response to added CaCl2 (0–0.6 mM; 1 mM EGTA buffer) with/without added MgCl2 (0–2 mM). In parallel, we assessed effects of added CaCl2 and MgCl2 on NADH, membrane potential (ΔΨm), and respiration. We found that \u3e0.125 mM MgCl2 significantly attenuated CU-mediated Ca2+ uptake and [Ca2+]m. Incremental [Mg2+]e did not reduce initial Ca2+uptake but attenuated the subsequent slower Ca2+ uptake, so that [Ca2+]m remained unaltered over time. Adding CaCl2 without MgCl2 to attain a [Ca2+]m from 46 to 221 nM enhanced state 3 NADH oxidation and increased respiration by 15 %; up to 868 nM [Ca2+]m did not additionally enhance NADH oxidation or respiration. Adding MgCl2 did not increase [Mg2+]m but it altered bioenergetics by its direct effect to decrease Ca2+ uptake. However, at a given [Ca2+]m, state 3 respiration was incrementally attenuated, and state 4 respiration enhanced, by higher [Mg2+]e. Thus, [Mg2+]e without a change in [Mg2+]m can modulate bioenergetics independently of CU-mediated Ca2+ transport

Some correlation inequalities for probabilistic analysis of algorithms

The analysis of many randomized algorithms, for example in dynamic load balancing, probabilistic divide-and-conquer paradigm and distributed edge-coloring, requires ascertaining the precise nature of the correlation between the random variables arising in the following prototypical ``balls-and-bins'' experiment. Suppose a certain number of balls are thrown uniformly and independently at random into $n$ bins. Let $X_i$ be the random variable denoting the number of balls in the $i$th bin, $i \in [n]$. These variables are clearly not independent and are intuitively negatively related. We make this mathematically precise by proving the following type of correlation inequalities: \begin{itemize} \item For index sets $I,J \subseteq [n]$ such that $I \cap J = \emptyset$ or $I \cup J = [n]$, and any non--negative integers $t_I,t_J$, \prob[\sum_{i \in I} X_i \geq t_I \mid \sum_{j \in J} X_j \geq t_J] \-5mm] \[\leq \prob[\sum_{i \in I} X_i \geq t_I] . \item For any disjoint index sets $I,J \subseteq [n]$, any $I' \subseteq I, J' \subseteq J$ and any non--negative integers $t_i, i \in I$ and $t_j, j \in J$, \prob[\bigwedge_{i \in I}X_i \geq t_i \mid \bigwedge_{j \in J} X_j \geq t_j]\-5mm]\[ \leq \prob[\bigwedge_{i \in I'}X_i \geq t_i \mid \bigwedge_{j \in J'} X_j \geq t_j] . \end{itemize} Although these inequalities are intuitively appealing, establishing them is non--trivial; in particular, direct counting arguments become intractable very fast. We prove the inequalities of the first type by an application of the celebrated FKG Correlation Inequality. The proof for the second uses only elementary methods and hinges on some {\em monotonicity} properties. More importantly, we then introduce a general methodology that may be applicable whenever the random variables involved are negatively related. Precisely, we invoke a general notion of {\em negative assocation\/} of random variables and show that: \begin{itemize} \item The variables $X_i$ are negatively associated. This yields most of the previous results in a uniform way. \item For a set of negatively associated variables, one can apply the Chernoff-Hoeffding bounds to the sum of these variables. This provides a tool that facilitates analysis of many randomized algorithms, for example, the ones mentioned above