The papers in this volume reflect much of the current unease of a world that perceives itself once more at the edge of chaos. The authors present different vistas of that experience and their inherent dialectic, expressed in numerous and ceaseless conflicts between ordering and disordering processes. They can be read as comments on the ongoing processes that lead toward greater complexity.

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

Two of the most difficult problems in the foundations of physics are (1) what gives rise to the arrow of time and (2) what the ontology of quantum mechanics is. They are difficult because the fundamental dynamical laws of physics do not privilege an arrow of time, and the quantum-mechanical wave function describes a high-dimensional reality that is radically different from our ordinary experiences. -/- In this paper, I characterize and elaborate on the ``Wentaculus” theory, a new approach to time’s (...) arrow in a quantum universe that offers a unified solution to both problems. Central to the Wentaculus are (i) Density Matrix Realism, the idea that the quantum state of the universe is objective but can be impure, and (ii) the Initial Projection Hypothesis, a new law of nature that selects a unique initial quantum state. On the Wentaculus, the quantum state of the universe is sufficiently simple to be a law, and the arrow of time can be traced back to an exact boundary condition. It removes the intrinsic vagueness of the Past Hypothesis, eliminates the Statistical Postulate, provides a higher degree of theoretical unity, and contains a natural realization of “strong determinism." I end by responding to four recent objections. In a companion paper, I elaborate on Density Matrix Realism. (shrink)

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function but by a density matrix. I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories (...) with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research. (Bibliographic note: the thesis was submitted for the Master of Science in mathematics at Rutgers University.). (shrink)

Discusses the conflict between subjective time and historical time, looks at how fiction and historical writings create a model of temporal experience, and considers the question of sense and reference.

The concept of time-coarsened density matrix for open systems has frequently featured in equilibrium and non-equilibrium statistical mechanics, without being probed as to the detailed consequences of the time averaging procedure. In this work we introduce and prove the need for a selective and non-uniform time-sampling, whose form depends on the properties of the bath. It is also applicable when an open microscopic sub-system is coupled to another finite system. By use of a time-periodic minimal coupling model between these (...) two systems, we present detailed quantitative consequences of time coarsening, which include initial state independence of equilibration, deviations from long term averages, their environment size dependence and the approach to classicality, as measured by a Leggett–Garg type inequality. An interacting multiple qubit model affords comparison between the time integrating procedure and the more conventional environment tracing method. (shrink)

In this paper, I characterize and elaborate on the “Wentaculus” theory, a new approach to time’s arrow in a quantum universe that offers a unified solution to the problems of what gives rise to the arrow of time and what the ontology of quantum mechanics is.

Discusses the conflict between subjective time and historical time, looks at how fiction and historical writings create a model of temporal experience, and considers the question of sense and reference.

In the first two volumes of this work, Paul Ricoeur examined the relations between time and narrative in historical writing, fiction, and theories of literature. This final volume, a comprehensive reexamination and synthesis of the ideas developed in volumes 1 and 2, stands as Ricoeur's most complete and satisfying presentation of his own philosophy. Ricoeur's aim here is to explicate as fully as possible the hypothesis that has governed his inquiry, namely, that the effort of thinking at work in (...) every narrative configuration is completed in a refiguration of temporal experience. To this end, he sets himself the central task of determing how far a poetics of narrative can be said to resolve the "aporias"—the doubtful or problematic elements—of time. Chief among these aporias are the conflicts between the phenomenological sense of time and the cosmological sense on the one hand and the oneness or unitary nature of time on the other. In conclusion, Ricoeur reflects upon the inscrutability of time itself and attempts to discern the limits of his own examination of narrative discourse. "As in his previous works, Ricoeur labors as an imcomparable mediator of often estranged philosophical approaches, always in a manner that compromises neither rigor nor creativity."—Mark Kline Taylor, _Christian Century _ "In the midst of two opposing contemporary options—either to flee into ever more precious readings... or to retreat into ever more safe readings... —Ricoeur's work offers an alternative option that is critical, wide-ranging, and conducive to new applications."—Mary Gerhart, _Journal of Religion_. (shrink)

In the first two volumes of this work, Paul Ricoeur examined the relations between time and narrative in historical writing, fiction, and theories of literature. This final volume, a comprehensive reexamination and synthesis of the ideas developed in volumes 1 and 2, stands as Ricoeur's most complete and satisfying presentation of his own philosophy. Ricoeur's aim here is to explicate as fully as possible the hypothesis that has governed his inquiry, namely, that the effort of thinking at work in (...) every narrative configuration is completed in a refiguration of temporal experience. To this end, he sets himself the central task of determing how far a poetics of narrative can be said to resolve the "aporias"—the doubtful or problematic elements—of time. Chief among these aporias are the conflicts between the phenomenological sense of time and the cosmological sense on the one hand and the oneness or unitary nature of time on the other. In conclusion, Ricoeur reflects upon the inscrutability of time itself and attempts to discern the limits of his own examination of narrative discourse. "As in his previous works, Ricoeur labors as an imcomparable mediator of often estranged philosophical approaches, always in a manner that compromises neither rigor nor creativity."—Mark Kline Taylor, _Christian Century _ "In the midst of two opposing contemporary options—either to flee into ever more precious readings... or to retreat into ever more safe readings... —Ricoeur's work offers an alternative option that is critical, wide-ranging, and conducive to new applications."—Mary Gerhart, _Journal of Religion_. (shrink)

In volume 1 of this three-volume work, Paul Ricoeur examined the relations between time and narrative in historical writing. Now, in volume 2, he examines these relations in fiction and theories of literature. Ricoeur treats the question of just how far the Aristotelian concept of "plot" in narrative fiction can be expanded and whether there is a point at which narrative fiction as a literary form not only blurs at the edges but ceases to exist at all. (...) Though some semiotic theorists have proposed all fiction can be reduced to an atemporal structure, Ricoeur argues that fiction depends on the reader's understanding of narrative traditions, which do evolve but necessarily include a temporal dimension. He looks at how time is actually expressed in narrative fiction, particularly through use of tenses, point of view, and voice. He applies this approach to three books that are, in a sense, tales about time: Virgina Woolf's _Mrs. Dalloway;_ Thomas Mann's _Magic Mountain;_ and Marcel Proust's _Remembrance of Things Past_. "Ricoeur writes the best kind of philosophy—critical, economical, and clear."—Eugen Weber, _New York Times Book Review _ "A major work of literary theory and criticism under the aegis of philosophical hermenutics. I believe that... it will come to have an impact greater than that of Gadamer's Truth and Method—a work it both supplements and transcends in its contribution to our understanding of the meaning of texts and their relationship to the world."—Robert Detweiler, _Religion and Literature_ "One cannot fail to be impressed by Ricoeur's encyclopedic knowledge of the subject under consideration.... To students of rhetoric, the importance of Time and Narrative... is all too evident to require extensive elaboration."—Dilip Parameshwar Gaonkar, _Quarterly Journal of Speech_. (shrink)

_Time and Narrative_ builds on Paul Ricoeur's earlier analysis, in _The Rule of Metaphor_, of semantic innovation at the level of the sentence. Ricoeur here examines the creation of meaning at the textual level, with narrative rather than metaphor as the ruling concern. Ricoeur finds a "healthy circle" between time and narrative: time is humanized to the extent that it portrays temporal experience. Ricoeur proposes a theoretical model of this circle using Augustine's theory of time and Aristotle's theory of plot (...) and, further, develops an original thesis of the mimetic function of narrative. He concludes with a comprehensive survey and critique of modern discussions of historical knowledge, understanding, and writing from Aron and Mandelbaum in the late 1930s to the work of the Annales school and that of Anglophone philosophers of history of the 1960s and 1970s. "This work, in my view, puts the whole problem of narrative, not to mention philosophy of history, on a new and higher plane of discussion."—Hayden White, _History and Theory _ "Superb.... A fine point of entrance into the work of one of the eminent thinkers of the present intellectual age."—Joseph R. Gusfield, _Contemporary Sociology _. (shrink)

The two books make a notable contribution in drawing together many of the philosophical problems about time, and the associated literature. The expositions are also valuable for their interdisciplinary strengths, especially in the history and philosophy of science and (to a lesser extent) in theology, and for the clarity and thoroughness of Craig's approach. However, the two books do not present, as might at first appear, a side by side exposition of the respective strengths and weaknesses of the A-series and (...) the B-series views of time. They are, rather, one interconnected defence of the A-series view. Some of the strengths and weaknesses of Craig's exposition and defence of the A-series view are noted. (shrink)

The present study examined whether cinematographic editing density affects viewers’ perception of time. As a second aim, based on embodied models that conceive time perception as strictly connected to the movement, we tested the hypothesis that the editing density of moving images also affects viewers’ eye movements and that these later mediate the effect of editing density on viewers’ temporal judgments. Seventy participants watched nine video clips edited by manipulating the number of cuts (slow‐ and fast‐paced editing (...) against a master shot, unedited condition). For each editing density, multiple video clips were created, representing three different kinds of routine actions. The participants’ eye movements were recorded while watching the video, and the participants were asked to report duration judgments and subjective passage of time judgments after watching each clip. The results showed that participants subjectively perceived that time flew more while watching fast‐paced edited videos than slow‐paced or unedited videos; by contrast, concerning duration judgments, participants overestimated the duration of fast‐paced videos compared to the master‐shot videos. Both the slow‐ and the fast‐paced editing generated shorter fixations than the master shot, and the fast‐paced editing led to shorter fixations than the slow‐paced editing. Finally, compared to the unedited condition, editing led to an overestimation of durations through increased eye mobility. These findings suggest that the editing density of moving images by increasing the number of cuts effectively altered viewers’ experience of time and add further evidence to prior research showing that performed eye movement is associated with temporal judgments. (shrink)

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S=∫ L 1 Φ d 4 x+∫ L 2 $\sqrt{-g}$ d 4 x where the volume element Φ d 4 x is independent of the metric. For global scale invariance, a “dilaton” φ has to be introduced, with non-trivial potentials V(φ)=f 1 eαφ in L 1 and U(φ)=f 2 e 2αφ in L 2 . This leads to (...) non-trivial mass generation and a potential for φ which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for “Quintessential models,” which are scale invariant but formulated with the use of volume element Φ d 4 x alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants. (shrink)

No one theory of time is pursued in the essays of this volume, but a major theme that threads them together is Wolfson’s signature idea of the timeswerve as a linear circularity or a circular linearity, expressions that are meant to avoid the conventional split between the two temporal modalities of the line and the circle.

This anthology is the most ambitious, comprehensive, and authoritative English-language sourcebook of Korean civilization ever assembled. Encompassing social intellectual, religious, and literary traditions from ancient times through World War II, this collection reveals the grand corpus of thought, beliefs, and customs unique to the Korean people. Volume I features three major periods of Korean history: the Three Kingdoms and Unified Silla (57 B.C.-935), Koryo (918-1392), and Early Choson (1392-1600). Each section begins with a broad historical introduction to provide (...) context and perspective, and contains representative writings from the era, with commentary, background, and analysis. (shrink)

In 1962, the philosopher Richard Taylor used six commonly accepted presuppositions to imply that human beings have no control over the future. David Foster Wallace not only took issue with Taylor's method, which, according to him, scrambled the relations of logic, language, and the physical world, but also noted a semantic trick at the heart of Taylor's argument. _Fate, Time, and Language_ presents Wallace's brilliant critique of Taylor's work. Written long before the publication of his fiction and essays, Wallace's thesis (...) reveals his great skepticism of abstract thinking made to function as a negation of something more genuine and real. He was especially suspicious of certain paradigms of thought-the cerebral aestheticism of modernism, the clever gimmickry of postmodernism-that abandoned "the very old traditional human verities that have to do with spirituality and emotion and community." As Wallace rises to meet the challenge to free will presented by Taylor, we witness the developing perspective of this major novelist, along with his struggle to establish solid logical ground for his convictions. This volume, edited by Steven M. Cahn and Maureen Eckert, reproduces Taylor's original article and other works on fatalism cited by Wallace. James Ryerson's introduction connects Wallace's early philosophical work to the themes and explorations of his later fiction, and Jay Garfield supplies a critical biographical epilogue. (shrink)

This volume explores how functional brain imaging techniques like positron emission tomography have influenced cognitive studies. The first chapter outlines efforts to relate human thought and cognition in terms of great books from the late 1800s through the present. Chapter 2 describes mental operations as they are measured in cognitive science studies. It develops a framework for relating mental operations to activity in nerve cells. In Chapter 3, the PET method is reviewed and studies are presented that use PET (...) to map the striate cortex and to activate extrastriate motion, color, and form areas. Chapter 4 shows how top down processes involving attention can lead to activation of these same areas in the detection of targets, visual search, and visual imagery. This chapter reveals complex networks of activations. Chapters 5 and 6 deal with the presentation of words. Chapter 5 illustrates PET studies of the anatomy of visual word processing and shows how the circuitry used for generating novel uses of words changes as the task becomes automated. Chapter 6 applies high density electrical recording to explore these activations in real time and to show how a constant anatomy can be reprogrammed by task instructions to produce and perform different cognitive tasks. Chapter 7 shows how studies of brain lesions and PET converge on common networks underlying attentional functions such as visual orienting, target detection, and maintenance of the alert state. Chapters 8 and 9 apply the network approach to examine normal development of attention in infants and pathological conditions resulting from brain damage, and psychiatric pathologies of depression, schizophrenia, and attention deficit disorder. In Chapter 10, new developments such as functional MRI are discussed in terms of future developments and integration of cognitive neuroscience. (shrink)

The two books make a notable contribution in drawing together many of the philosophical problems about time, and the associated literature. The expositions are also valuable for their interdisciplinary strengths, especially in the history and philosophy of science and (to a lesser extent) in theology, and for the clarity and thoroughness of Craig's approach. However, the two books do not present, as might at first appear, a side by side exposition of the respective strengths and weaknesses of the A-series and (...) the B-series views of time. They are, rather, one interconnected defence of the A-series view. Some of the strengths and weaknesses of Craig's exposition and defence of the A-series view are noted. (shrink)

The susceptible-transmissible-removed (STR) model is a deterministic compartment model, based on the susceptible-infected-removed (SIR) prototype. The STR replaces 2 SIR assumptions. SIR assumes that the emigration rate (due to death or recovery) is directly proportional to the infected compartment’s size. The STR replaces this assumption with the biologically appropriate assumption that the emigration rate is the same as the immigration rate one infected period ago. This results in a unique delay differential equation epidemic model with the delay equal to the (...) infected period. Hamer’s mass action law for epidemiology is modified to resemble its chemistry precursor—the law of mass action. Constructing the model for an isolated population that exists on a surface bounded by the extent of the population’s movements permits compartment density to replace compartment size. The STR reduces to a SIR model in a timescale that negates the delay—the transmissible timescale. This establishes that the SIR model applies to an isolated population in the disease’s transmissible timescale. Cyclical social interactions will define a rhythmic timescale. It is demonstrated that the geometric mean maps transmissible timescale properties to their rhythmic timescale equivalents. This mapping defines the hybrid incidence (HI). The model validation demonstrates that the HI-STR can be constructed directly from the disease’s transmission dynamics. The basic reproduction number (R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0$$\end{document}) is an epidemic impact property. The HI-STR model predicts that R0∝ρnB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0 \propto \root \mathfrak{B} \of {\rho_n}$$\end{document} where ρn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_n$$\end{document} is the population density, and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{B}}$$\end{document} is the ratio of time increments in the transmissible- and rhythmic timescales. The model is validated by experimentally verifying the relationship. R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{R}}_0$$\end{document}’s dependence on ρn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_n$$\end{document} is demonstrated for droplet-spread SARS in Asian cities, aerosol-spread measles in Europe and non-airborne Ebola in Africa. (shrink)