The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

On impulsive semidynamical systems: Relation between topological entropy and various types of expansivity

In this paper, we generalize the notion of expansivity, C-W expansivity and N-expansivity for impulsive dynamical systems and we prove C-W expansivity implies positive [Formula: see text]-topological entropy. Also, we define new version of topological entropy for impulsive semiflows and we can bound this topological entropy of expansive impulsive semiflows from lower by the exponential growth rate of periodic orbits.

On Two-Stage Guessing

Stationary memoryless sources produce two correlated random sequences Xn and Yn. A guesser seeks to recover Xn in two stages, by first guessing Yn and then Xn. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in n) of any positive ρ-th moment of the total number of guesses when Yn is obtained by applying a deterministic function f component-wise to Xn. We prove that, depending on f, the least exponential growth rate in the two-stage setup is lower than when guessing Xn directly. We further propose a simple Huffman code-based construction of a function f that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the ρ-th moment of the total number of guesses required to recover Xn when Stage 1 need not end with a correct guess of Yn and without assumptions on the stationary memoryless sources producing Xn and Yn.

Speed and strength of an epidemic intervention

An epidemic can be characterized by its strength (i.e., the reproductive number R ) and speed (i.e., the exponential growth rate r ). Disease modellers have historically placed much more emphasis on strength, in part because the effectiveness of an intervention strategy is typically evaluated on this scale. Here, we develop a mathematical framework for the classic, strength-based paradigm and show that there is a dual speed-based paradigm which can provide complementary insights. In particular, we note that r = 0 is a threshold for disease spread, just like R = 1 [ 1 ], and show that we can measure the strength and speed of an intervention on the same scale as the strength and speed of an epidemic, respectively. We argue that, while the strength-based paradigm provides the clearest insight into certain questions, the speed-based paradigm provides the clearest view in other cases. As an example, we show that evaluating the prospects of ‘test-and-treat’ interventions against the human immunodeficiency virus (HIV) can be done more clearly on the speed than strength scale, given uncertainty in the proportion of HIV spread that happens early in the course of infection. We also discuss evaluating the effects of the importance of pre-symptomatic transmission of the SARS-CoV-2 virus. We suggest that disease modellers should avoid over-emphasizing the reproductive number at the expense of the exponential growth rate, but instead look at these as complementary measures.

Estimating the basic reproduction number for COVID-19 in Western Europe

Objective To estimate the basic reproduction number (R0) for COVID-19 in Western Europe. Methods Data (official statistics) on the cumulative incidence of COVID-19 at the start of the outbreak (before any confinement rules were declared) were retrieved in the 15 largest countries in Western Europe, allowing us to estimate the exponential growth rate of the disease. The rate was then combined with estimates of the distribution of the generation interval as reconstructed from the literature. Results Despite the possible unreliability of some official statistics about COVID-19, the spread of the disease appears to be remarkably similar in most European countries, allowing us to estimate an average R0 in Western Europe of 2.2 (95% CI: 1.9–2.6). Conclusions The value of R0 for COVID-19 in Western Europe appears to be significantly lower than that in China. The proportion of immune persons in the European population required to stop the outbreak could thus be closer to 50% than to 70%.

Impact of environmental temperature on Covid-19 spread: Model and analysis of measurements recorded during the second pandemic in Cyprus

The paper investigates the effect of the environmental temperature on the spread of COVID-19. We study the daily numbers of the cases infected and deaths caused by Covid-19 during the second wave of the pandemic within 2020, and how they were affected by the daily average-high temperature for the districts of the Republic of Cyprus. Among the findings of the paper, we show that (i) the average ratio of the PCR to rapid positive tests is ∼2.57±0.25, as expected from the tests’ responses, indicating that PCR overestimates positivity by ∼2.5 times; (ii) the average age of deaths caused by Covid-19 increases with rate about a year of age per week; (iii) the probability of a person infected by Covid-19 to develop severe symptoms leading to death is strongly depended on the person’s age, while the probability of having a death on the age of ∼67 or younger is less than 1/1000; (iv) the number of infected cases and deaths declined dramatically when the environmental temperature reaches and/or climbs above the critical temperature ofTC=30.1±2.4 C0; (v) the observed negative correlation between the exponential growth rate of the infected cases and the environmental temperature can be described within the framework of chemical kinetics, with at least two competing reactions, the connection of the coronavirus towards the receptor and the dissolution of the coronavirus; the estimated activation energy difference corresponding to the competing chemical reactions, 0.212±0.25 eV, matches the known experimental value; and (vi) the infected cases will decline to zero, when the environmental temperature climbs above the critical temperature within the summery days of 2021, which is expected for the Republic of Cyprus by the 16thof May, 2021.

Cross-Cultural Mentoring in Counselor Education: A Call to Action

Given the lack of formalized cross-cultural mentorship guidelines within professional counseling associations and accreditation programs, the recruitment and retention of marginalized graduate students, may be in jeopardy. The authors explored the value of mentoring for graduate students, the exponential growth rate of diversity within graduate programs, and how the disparity of marginalized faculty members creates a need and opportunity for cross-cultural mentorship. Recommendations for the creation of holistic cross-culturalmentorship guidelines for faculty-student dyads are provided.

Chaos exponents of SYK traversable wormholes

In this paper we study the chaos exponent, the exponential growth rate of the out-of-time-ordered four point functions, in a two coupled SYK models which exhibits a first order phase transition between the high temperature black hole phase and the low temperature gapped phase interpreted as a traversable wormhole. We see that as the temperature decreases the chaos exponent exhibits a discontinuous fall-off from the value of order the universal bound 2π/β at the critical temperature of the phase transition, which is consistent with the expected relation between black holes and strong chaos. Interestingly, the chaos exponent is small but non-zero even in the wormhole phase. This is surprising but consistent with the observation on the decay rate of the two point function [1], and we found the chaos exponent and the decay rate indeed obey the same temperature dependence in this regime. We also studied the chaos exponent of a closely related model with single SYK term, and found that the chaos exponent of this model is always greater than that of the two coupled model in the entire parameter space.

Numerically statistical investigation of the partly super-exponential growth rate in the COVID-19 pandemic (throughout the world)

A number of particles in a multiplying medium under rather general conditions is asymptotically exponential with respect to time t with the parameter λ, i.e., with the index of power {\lambda t}. If the medium is random, then the parameter λ is the random variable. To estimate the temporal asymptotics of the mean particles number (via the medium realizations), it is possible to average the exponential function via the corresponding distribution. Assuming that this distribution is Gaussian, the super-exponential estimate of the mean particle number could be obtained and expressed by the exponent with the index of power {t{\rm E}\lambda+t^{2}{\rm D}\frac{\lambda}{2}}. The application of this new formula to investigation of the COVID-19 pandemic is performed.