We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon o... more We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincare theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabil...
We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each ... more We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end, and an internal damper at an arbitrary point along the bar's length. The system is described by four independent parameters and exhibits a variety of behaviors including rigid motion, super stability/instability and zero damping. The solution is obtained by applying the Laplace transform to the equation of motion and computing the Green's function of the transformed problem. This leads to an unconventional eigenvalue-like problem with the spectral variable in the boundary conditions. The eigenmodes of the problem are necessarily complex-valued and are not orthogonal in the usual inner product. Nonetheless, in generic cases we obtain an explicit eigenmode expansion for the response of the bar to initial conditions and external force. For some special values of parameters the system of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at all. We thorough...
We consider abstract equations of the form Ax=-z on a locally convex space, where A generates a p... more We consider abstract equations of the form Ax=-z on a locally convex space, where A generates a positive semigroup and z is a positive element. This is an abstract version of the operator Lyapunov equation A*P+PA=-Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that -A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that z is an order unit, or that the space contains any order units. As an application, we generalize Wonham's theorem on the operator Lyapunov equations with detectable right hand sides to reflexive Banach spaces.
Synchronization of Huygens ’ clocks and the Poincare ́ method
We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon o... more We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincare ́ theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destab...
We give a short new proof of large N duality between the Chern-Simons invariants of the 3-sphere ... more We give a short new proof of large N duality between the Chern-Simons invariants of the 3-sphere and the Gromov-Witten/Donaldson-Thomas invariants of the resolved conifold. Our strategy applies to more general situations, and it is to interpret the Gromov-Witten, the Donaldson-Thomas, and the Chern-Simons invariants as different characterizations of the same holomorphic function. For the resolved conifold, this function turns out to be the quantum Barnes function, a natural q-deformation of the classical one that in its turn generalizes the Euler gamma function. Our reasoning is based on a new formula for this function that expresses it as a graded product of q-shifted multifactorials. Copyright q 2008 Sergiy Koshkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Conormal bundles to knots and the Gopakumar-Vafa conjecture
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating... more We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in it have bounded geometry, and that the moduli spaces of holomorphic curves ending on the Lagrangians are compact in the Gromov topology.
We study geometric variational problems for a class of nonlinear σ-models in quantum field theory... more We study geometric variational problems for a class of nonlinear σ-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers in each 2-homotopy class. Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G→G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.
We develop a general form of the Ritz method for trial functions that do not satisfy the essentia... more We develop a general form of the Ritz method for trial functions that do not satisfy the essential boundary conditions. The idea is to treat the latter as variational constraints and remove them using the Lagrange multipliers. In multidimensional problems in addition to the trial functions boundary weight functions also have to be selected to approximate the boundary conditions. We prove convergence of the method and discuss its limitations and implementation issues. In particular, we discuss the required regularity of the variational functional, the completeness of systems of the trial functions, and conditions for consistency of the equations for the trial solutions. The discussion is accompanied by a detailed examination of examples, both analytic and numerical, to illustrate the method.
We describe a new method for finding analytic solutions to some initial-boundary problems for par... more We describe a new method for finding analytic solutions to some initial-boundary problems for partial differential equations with constant coefficients. The method is based on expanding the denominator of the Laplace transformed Green's function of the problem into a convergent geometric series. If the denominator is a linear combination of exponents with real powers one obtains a closed form solution as a sum with finite but time dependent number of terms. We call it a d'Alembert sum. This representation is computationally most effective for small evolution times, but it remains valid even when the system of eigenmodes is incomplete and the eigenmode expansion is unavailable. Moreover, it simplifies in such cases. In vibratory problems d'Alembert sums represent superpositions of original and partially reflected traveling waves. They generalize the d'Alembert type formulas for the wave equation, and reduce to them when original waves can undergo only finitely many reflections in the entire course of evolution. The method is applied to vibrations of a bar with dampers at each end and at some internal point. The results are illustrated by computer simulations and comparisons to modal and FEM solutions.
We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CWcomple... more We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CWcomplexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variational problems of mathematical physics.
We introduce a natural generalization of the golden cryptography, which uses general unimodular m... more We introduce a natural generalization of the golden cryptography, which uses general unimodular matrices in place of the traditional Q matrices, and prove that it preserves the original error correction properties of the encryption. Moreover, the additional parameters involved in generating the coding matrices make this unimodular cryptography resilient to the chosen plaintext attacks that worked against the golden cryptography. Finally, we show that even the golden cryptography is generally unable to correct double errors in the same row of the ciphertext matrix, and offer an additional check number which, if transmitted, allows for the correction.
This paper describes a new method for generating stationary integervalued time series from renewa... more This paper describes a new method for generating stationary integervalued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p, p−1) series from lifetime distributions with constant hazard rates after lag p are given as an illustration.
Euler used intrinsic equations expressing the radius of curvature as a function of the angle of i... more Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light rays normal to it, and study the Euler's problem for general caustics. The resulting curves are characterized when the rays are at a constant angle to the curve, generalizing the case of evolutes. Aside from analogs of classical solutions we encounter some new types of curves. We also consider caustics of parallel rays reflected by a curved mirror, where Euler's problem leads to a novel pantograph equation, and describe its analytic solutions.
We extend the notion of triangle to "imaginary triangles" with complex valued sides and... more We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles, and apply the theory of plane Cremona transformations to find "Pythagorean theorems" for them, which are interpreted as the implicit equations of their parametrizing curves.
Optimal allocation in annual plants with density-dependent fitness
Theory in Biosciences
We study optimal two-sector (vegetative and reproductive) allocation models of annual plants in t... more We study optimal two-sector (vegetative and reproductive) allocation models of annual plants in temporally variable environments that incorporate effects of density-dependent lifetime variability and juvenile mortality in a fitness function whose expected value is maximized. Only special cases of arithmetic and geometric mean maximizers have previously been considered in the literature, and we also allow a wider range of production functions with diminishing returns. The model predicts that the time of maturity is pushed to an earlier date as the correlation between individual lifetimes increases, and while optimal schedules are bang-bang at the extremes, the transition is mediated by schedules where vegetative growth is mixed with reproduction for a wide intermediate range. The mixed growth lasts longer when the production function is less concave allowing for better leveraging of plant size when generating seeds. Analytic estimates are obtained for the power means that interpolate between arithmetic and geometric mean and correspond to partially correlated lifetime distributions.
In Book I of the Elements, Euclid defines angle as “the inclination to one another of two lines w... more In Book I of the Elements, Euclid defines angle as “the inclination to one another of two lines which meet one another and do not lie in a straight line” [8]. “Lines” in Euclid can be curves. So he allows curvilinear angles, but they are considered only once in the Elements [4]. Namely, in Proposition 16 of Book III, Euclid characterizes a tangent to a circle as the line perpendicular to its diameter, and notes that “the angle of the semicircle is greater, and the remaining angle less than any acute rectilineal angle.” The “remaining angle,” the one between the circle and its tangent, became controversial already in antiquity. This is because Euclid’s remark put it in tension with the definition of ratio, foundational to many parts of the Elements that anticipate what we now call Real Analysis. The definition, commonly attributed to Euclid’s predecessor Eudoxus of Cnidus, reads: “Magnitudes are said to have a ratio to one another which can when multiplied exceed one another.” A horn angle, however, no matter how many times multiplied, does not exceed any acute rectilinear angle—it is infinitesimal. Although this is presented as a definition, Euclid later wields it as an axiom, assuming that segments, areas, etc., do always have a ratio. This assumption also goes back to Eudoxus. Archimedes articulates it explicitly as Axiom V in On the Sphere and Cylinder, and that all non-zero real numbers have ratios is now called the axiom of Archimedes. Euclid does not give a name to his “remaining angle,” but a fifth century philosopher Proclus mentions one when commenting on the bisection of rectilinear angles. “Bisection of any kind of angle,” he writes, “is not a matter for elementary treatise... Thus, it is difficult to say if it is possible to bisect so-called horn-like angle” [8, I.9], [13]. A medieval mathematician Nemorarius called it “the angle of contingence” (c. 1220), and Cardano “the angle of contact” (1550). Its infinitesimal nature attracted the attention of many prominent later mathematicians, including Vieta, Galileo, Newton, Leibniz, and more recently, Hilbert and Klein [4]. At the end of nineteenth century, horn angles were placed into the general context of non-Archimedean analysis and geometry. In the twentieth century, the American mathematician Edward Kasner, and his students, studied the horn angles in the context of conformal geometry [9–12]. To readers interested in the history of horn angles, we recommend a series of 1996 posts by historian Jose Cabillon available online [4], which gives a comprehensive review, and includes an extensive bibliography of original sources. For a more recent work on horn angles, see [2, 6, 15]. Our present goal will be to do what Proclus found difficult to say could be done, to bisect the line-circle and circle-circle horn angles. First, we will do it in the spirit of the ancient Greeks, which will give us a new perspective on conic sections. Then we will briefly discuss Kasner’s conformal approach.
Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new con... more Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning.
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