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Register a new userA Fresh Look at the Hot Hand Paradox
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S. Redner
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We discuss the hot hand paradox within the framework of the backward Kolmogorov equation. We use this approach to understand the apparently paradoxical features of the statistics of fixed-length sequences of heads and tails upon repeated fair coin flips. In particular, we compute the average waiting time for the appearance of specific sequences. For sequences of length 2, the average time until the appearance of the sequence HH (heads-heads) equals 6, while the waiting time for the sequence HT (heads-tails) equals 4. These results require a few simple calculational steps by the Kolmogorov approach. We also give complete results for sequences of lengths 3, 4, and 5; the extension to longer sequences is straightforward (albeit more tedious). Finally, we compute the waiting times $T_{nrm H}$ for an arbitrary length sequences of all heads and $T_{nrm(HT)}$ for the sequence of alternating heads and tails. For large $n$, $T_{2nrm H}sim 3 T_{nrm(HT)}$.
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The apparent unification of gauge couplings around 10^16 GeV is one of the strong arguments in favor of Supersymmetric extensions of the Standard Model (SM). In this contribution a new analysis, using the latest experimental data, is performed. The strong coupling alpha_s emerges as the key factor for evaluating the results of the fits, as the experimental and theoretical uncertainties in its measurements are substantially higher than for the electromagnetic and weak couplings. The present analysis pays special attention to numerical and statistical details. The results, combined with the current limits on the supersymmetric particle masses, favor a value for the SUSY scale <~ 150 GeV and for alpha_s = 0.118-0.119.
Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of ${0,1}^G$. When $G$ = $(mathbb N,+)$ and $F_n = {1,2,...,n}$, the $(F_n)$-normality coincides with the classical notion. We prove that: $bullet$ If $(F_n)$ is a F{o}lner sequence in $G$, such that for every $alphain(0,1)$ we have $sum_n alpha^{|F_n|}<infty$, then almost every $xin{0,1}^G$ is $(F_n)$-normal. $bullet$ For any F{o}lner sequence $(F_n)$ in $G$, there exists an Cham-per-nowne-like $(F_n)$-normal set. $bullet$ There is a natural class of nice F{o}lner sequences in $(mathbb N,times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice F{o}lner sq. $bullet$ Let $Asubsetmathbb N$ be a classical normal set. Then, for any F{o}lner sequence $(K_n)$ in $(mathbb N,times)$ there exists a set $E$ of $(K_n)$-density $1$, such that for any finite subset ${n_1,n_2,dots,n_k}subset E$, the intersection $A/{n_1}cap A/{n_2}capldotscap A/{n_k}$ has positive upper density in $(mathbb N,+)$. As a consequence, $A$ contains arbitrarily long geometric progressions, and, more generally, arbitrarily long geo-arithmetic configurations of the form ${a(b+ic)^j,0le i,jle k}$. $bullet$ For any F{o}lner sq $(F_n)$ in $(mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $bullet$ For any nice F{o}lner sequence $(F_n)$ in $(mathbb N,times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schrodinger equation (as analogue of the time-dependent Schrodinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.
An alternative approach to the calculation of tunneling actions, that control the exponential suppression of the decay of metastable phases, is presented. The new method circumvents the use of bounces in Euclidean space by introducing an auxiliary function, a tunneling potential $V_t$ that connects smoothly the metastable and stable phases of the field potential $V$. The tunneling action is obtained as the integral in field space of an action density that is a simple function of $V_t$ and $V$. This compact expression can be considered as a generalization of the thin-wall action to arbitrary potentials and allows a fast numerical evaluation with a precision below the percent level for typical potentials. The method can also be used to generate potentials with analytic tunneling solutions.
A significant number of high power proton beams are available or will go online in the near future. This provides exciting opportunities for new fixed target experiments and the search for new physics in particular. In this note we will survey these beams and consider their potential to discover new physics in the form of axion-like particles, identifying promising locations and set ups. To achieve this, we present a significantly improved calculation of the production of axion-like particles in the coherent scattering of protons on nuclei, valid for lower ALP masses and/or beam energies. We also provide a new publicly available tool for this process: the Alpaca Monte Carlo generator. This will impact ongoing and planned searches based on this process.
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