Small Denominators Research Articles

Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

Dirichlet-Neumann problem for the typeless high order partial differential equation with deviating over the space argument is studied in the domain, which is the Cartesian product of the segment $(0,T)$ and the unit circle $\Omega=\mathbb R/(2\pi \mathbb Z)$. Dirichlet-Neumann problem for hyperbolic equations and their systems in case with absent argument deviation $h$ has been studied by the authors before. Correct solvability conditions have been established for these problems for almost all (with respect to Lebesgue measure) numbers $T>0$ and for almost all (with respect to Lebesgue measure) vectors, constructed by coefficients of the equation.
 In this paper, the solvability conditions of the problem for $h\ne0$ are described and the influence of the deviation $h$ on the solvability of the problem is studied. The solution of the problem is constructed in the form of the series with respect to the systems of orthogonal functions. Metric estimations (of exponential type) are proved for small denominators appearing during construction of the problem solution. These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. The obtained results are based on the fact that the corresponding characteristic determinant permits factorization in the form of the product of hyperbolic functions with integer parameters.

The Effect of Money Denomination on Prosocial Behavior: Through the Perspective of Metaphorical Cognitive Theory.

Based on metaphorical cognitive theory, this research did four experiments to examine whether and how one important feature of money, denomination, could influence prosocial behavior. Study 1 was an experiment with a sample size of 209 undergraduates (Mage = 18.97) showed that a larger denomination enhanced the probability of participants engaging in prosocial behavior rather than with a smaller denomination. Study 2 collecting data from 269 undergraduates (Mage = 18.50) further showed that larger denominations condition inspired more prosocial behavior than the control condition; and the small denominations condition produced similar levels of prosocial behavior to the control condition. Study 3 used single factor design with a sample size of 192 undergraduates (Mage = 20.49) repeated the results of Study 2. Furthermore, Study 3 excluded an important alternative explanation that the value rather than the denomination influenced prosocial behavior. Last, Study 4 applied a factorial design experiment with a sample size of 132 undergraduates (Mage = 20.92) which demonstrated that generosity mediated the effect of denomination on prosocial behavior; the effect of denomination on prosocial behavior did not depend on money priming methods. Finally, theoretical and practical implications were discussed.

A tale of two rhythms: Locked clocks and chaos in biology.

The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.

Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet--Fourier space scale. The proofs are based on the differentiable Nash--Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.

Exponential stability of fast driven systems, with an application to celestial mechanics

We construct a normal form suited to fast driven systems. We call so systems including actions I, angles ψ, and one fast coordinate y, moving under the action of a vector-field N depending only on I and y and with vanishing I-components. In the absence of the coordinate y, such systems have been extensively investigated and it is known that, after a small perturbing term is switched on, the normalised actions I turn to have exponentially small variations compared to the size of the perturbation. We obtain the same result of the classical situation, with the additional benefit that no trapping argument is needed, as no small denominator arises. We use the result to prove that, in the three-body problem, the level sets of a certain function called Euler integral have exponentially small variations in a short time, closely to collisions.

Updating an Empirically Based Tool for Analyzing Congenital Heart Surgery Mortality.

STAT Mortality Categories (developed 2009) stratify congenital heart surgery procedures into groups of increasing mortality risk to characterize case mix of congenital heart surgery providers. This update of the STAT Mortality Score and Categories is empirically based for all procedures and reflects contemporary outcomes. Cardiovascular surgical operations in the Society of Thoracic Surgeons Congenital Heart Surgery Database (January 1, 2010 - June 30, 2017) were analyzed. In this STAT 2020 Update of the STAT Mortality Score and Categories, the risk associated with a specific combination of procedures was estimated under the assumption that risk is determined by the highest risk individual component procedure. Operations composed of multiple component procedures were eligible for unique STAT Scores when the statistically estimated mortality risk differed from that of the highest risk component procedure. Bayesian modeling accounted for small denominators. Risk estimates were rescaled to STAT 2020 Scores between 0.1 and 5.0. STAT 2020 Category assignment was designed to minimize within-category variation and maximize between-category variation. Among 161,351 operations at 110 centers (19,090 distinct procedure combinations), 235 types of single or multiple component operations received unique STAT 2020 Scores. Assignment to Categories resulted in the following distribution: STAT 2020 Category 1 includes 59 procedure codes with model-based estimated mortality 0.2% to 1.3%; Category 2 includes 73 procedure codes with mortality estimates 1.4% to 2.9%; Category 3 includes 46 procedure codes with mortality estimates 3.0% to 6.8%; Category 4 includes 37 procedure codes with mortality estimates 6.9% to 13.0%; and Category 5 includes 17 procedure codes with mortality estimates 13.5% to 38.7%. The number of procedure codes with empirically derived Scores has grown by 58% (235 in STAT 2020 vs 148 in STAT 2009). Of the 148 procedure codes with empirically derived Scores in 2009, approximately one-half have changed STAT Category relative to 2009 metrics. The New STAT 2020 Scores and Categories demonstrated good discrimination for predicting mortality in an independent validation sample (July 1, 2017-June 30, 2019; sample size 46,933 operations at 108 centers) with C-statistic = 0.791 for STAT 2020 Score and 0.779 for STAT 2020 Category. The updated STAT metrics reflect contemporary practice and outcomes. New empirically based STAT 2020 Scores and Category designations are assigned to a larger set of procedure codes, while accounting for risk associated with multiple component operations. Updating STAT metrics based on contemporary outcomes facilitates accurate assessment of case mix.

The denomination–spending matching effect

Prior research has documented a denomination effect, such that consumers are less willing to spend banknotes in large denominations (e.g., $100 bill) than in small denominations (e.g., five $20 bills). This research shows that the denomination effect persists when prices are low, but when prices are high, a reverse denomination effect applies, such that consumers prefer to pay for expensive products with large denominations. To accommodate both effects in a general framework, the authors propose a denomination–spending matching effect: consumers prefer to pay with a denomination that matches the spending amount. This matching effect is generalizable to other payment means (i.e., gift cards). A denomination fit underlies this matching effect—when denomination and spending have a similar magnitude, consumers feel right about the purchase. This experience of denomination fit prompts consumers to choose products where the price matches the denomination and increases consumers’ purchase satisfaction and anticipated consumption experience.

Steam City: Railroads, Urban Space, and Corporate Capitalism in Nineteenth-Century Baltimore by David Schley

Reviewed by: Steam City: Railroads, Urban Space, and Corporate Capitalism in Nineteenth-Century Baltimore by David Schley Albert J. Churella (bio) Steam City: Railroads, Urban Space, and Corporate Capitalism in Nineteenth-Century Baltimore By David Schley. Chicago: University of Chicago Press, 2020. Pp. 318. Steam City: Railroads, Urban Space, and Corporate Capitalism in Nineteenth-Century Baltimore By David Schley. Chicago: University of Chicago Press, 2020. Pp. 318. In Steam City, David Schley draws on the work of scholars such as William G. Roy (Socializing Capital, 1997), Pauline Maier ("The Revolutionary Origins of the American Corporation," 1993), Sean Patrick Adams (Old Dominion, Industrial Commonwealth, 2004), and John Lauritz Larson (The Market Revolution in America, 2010) who have explored the evolution of the American corporation from the early republic to the Civil War. The closest parallel to Schley's book is Andrew Schocket's 2007 Founding Corporate Power in Early National Philadelphia—a fitting comparison, given the intense rivalry between Baltimore and Philadelphia, another port city in decline, relative to New York. The broad outlines of the story are clear. Elite residents of cities (also corporations) promoted the incorporation of banks and charitable institutions that fostered the public good. Internal improvements promised similar benefits, particularly as cities contested for economic access to the western hinterland. Following the Panic of 1837, critics of public ownership in the United States increasingly adopted the rhetoric of free-market capitalism, asserting that only private corporations could provide the public good of improved mobility. Following the Civil War, that perspective underpinned the concept of corporate personhood, with businesses of all types largely exempt from governmental oversight. Schley adopts a fresh and innovative approach to these developments in the United States. Steam City reflects his assertion "that corporate power, as we understand it today, rests on a spatial order that took shape in city streets during the first half century of the railroad age" (p. 220). Technology shaped that process, but this is no story of technological determinism. Human agency is everywhere in Schley's account. To the extent that railroad development in nineteenth-century Baltimore reflected the theoretical approach provided by the social construction of technology, however, a remarkably small group was doing the constructing. Business elites methodically transformed the business corporation—and the physical and rhetorical landscape in which it operated—from a servant of the community to an exploiter of the city. More than any other eastern American metropolis, Baltimore invested heavily in public-works projects that would promote the common good. It was nonetheless evident that some sectors of the public mattered more than others. The railroad was particularly burdensome to neighborhoods occupied by the working class and People of Color. At a time when hard currency was scarce, and paper currency unreliable, the quasi-public [End Page 960] nature of the Baltimore and Ohio Railroad (B&O) facilitated an expansion of Baltimore's money supply. Schley describes the rise and rapid fall of B&O notes during the early 1840s. Printed in small denominations, they facilitated commercial exchange and economic growth. Another effect was to shift the speculative risk associated with railroad financing from wealthy Baltimoreans to the city's working class. By the 1850s, the railroad controlled Baltimore, rather than the other way around. New lines that skirted or tunneled under the city reflected the potential of railroad technology to reorder channels of commerce. B&O executives argued that the inevitable forces of competitive free-market capitalism impelled them to minimize costs, maximize revenues, and direct traffic to its most profitable destination, even if that meant away from Baltimore. Beginning in 1854, the railroad's private investors engaged in a "public, political, and deliberate" campaign to wrest control from the municipal government (p. 133). They succeeded only after they won the battle for the railroad "to define itself as a private corporation acting independent of urban public interest" (p. 162). Steam City ends with the cataclysmic strikes of 1877, events precipitated by managerial determination to put the welfare of the private corporation ahead of the public good of Baltimore and its residents. When B&O managers laid off workers, implemented wage reductions, and relocated the principal facilities used to build and repair...

Subsets of Scales in Compositions Constructed by Similarity

We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.

Ubiquitous Overlap Weight and Propensity Score Residual for Heterogeneous Treatment Effect and Its Estimation

Individual responses to a treatment D=0,1 differ, depending on covariates X. Averaging such a heterogeneous effect is usually done with the density of X, but the average with `overlap weight (OW)' is done surreptitiously with X controlled, where OW is the normalized version of PS×(1-PS) with PS denoting the propensity score. This paper brings OW out of shadow to advocate more use of OW. OW attains its maximum at PS=0.5, i.e., when subjects in one group have the best overlap with the other group, and OW accords several advantages to treatment effect estimators. First, matching with OW addresses the well-known non-overlapping support problem in a built-in way, without an arbitrary user intervention. Second, inverse probability weighting with OW overcomes the too small denominator problem, and can become efficient as well. Third, regression adjustment with OW is robust to misspecified outcome regression models. Fourth, covariate balance holds exactly if OW estimated by the method of moment is used. In these advantages, PS residual `D-PS' plays a central role. We also show how seemingly unrelated estimators are in fact closely related through OW, and an empirical analysis illustrates how similar the estimates can be.

ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.

The First Boundary Problem for a Mixed Type Equation with Characteristic Degeneracy with Discontinuous Frankl Condition

For an equation of mixed elliptic-hyperbolic type of the second kind, the boundary problem in a rectangular region is studied. The criterion of uniqueness is established. The solution is built as a sum row. When substantiating convergence, the problem of small denominators. In this connection, estimates of separation from zero small denominators with the corresponding asymptotics, which and allowed us to justify the existence of a solution in the class of regular solutions and solution stability from boundary data.

Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.

The First Boundary Problem with an Integral Condition for a Mixed-Type Equation with a Characteristic Degeneration

In this paper, we study the first boundary problem for a mixed-type elliptic-hyperbolic equation of the second kind in a rectangular domain. We construct the problem solution as the sum of a biorthogonal series and prove the uniqueness criterion for it. When proving the existence of the problem solution, we encounter the problem of small denominators. In this connection, we establish estimates for the separation of denominators from zero with the corresponding asymptotics; this allows us to justify the existence of a solution in the class of regular solutions.

The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

Complex behavior of the density in composite quantum systems

In this paper, we study how the probability of presence of a particle is distributed between the two parts of a composite fermionic system. We uncover that the difference of probability depends on the energy in a striking way and show the pattern of this distribution. We discuss the main features of the latter and explain analytically those that we understand. In particular, we prove that it is a non-perturbative property and we find out a large/small coupling constant duality. We also find and study features that may connect our problem with certain aspects of non linear classical dynamics, like the existence of resonances and sensitive dependence on the state of the system. We show that the latter has indeed a similar origin than in classical mechanics: the appearance of small denominators in the perturbative series. Inspired by the proof of KAM theorem, we are able to deal with this problem by introducing a cut-off in energies that eliminates these small denominators. We also formulate some conjectures that we are not able to prove at present but can be supported by numerical experiments.

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

Price-Denomination Effect: Choosing to Pay With Denominations That Are the Same as the Product Prices.

Building on past research on judgment anchoring, we investigate the effect of price information on consumers’ choice of denomination when making a purchase. Across seven experiments, including two in the field (N = 4,020), we find that people tend to purchase with denominations that are the same as the product prices. They use larger denominations for higher priced products that are priced at the value of the denomination held, and smaller denominations for lower priced products that are priced at the value of the smaller denomination held. The effect is not explained by storage or purchase convenience. We propose the “price-denomination effect” is driven by consumers anchoring on product price and then choosing the denomination that matches the anchor. The effect replicates across participants from different continents (United States, Europe, and Africa) and samples (online panelists, and actual consumers), as well as prices in different currencies (United States $, €, and Nigerian Naira). We further demonstrate that people’s preference for denominations also affects the choice of the form of payment used: cash versus card. Consumers are more likely to use cash (vs. card) when product price is exactly the same as a denomination held. We conclude with a discussion of theoretical and practical implications.

The smallest possible money unit! When money crashes into the laws of physics

In this paper, we demonstrate that there is an absolute physical limit on how small the smallest money unit can be, no matter how much we are able to improve our technology. The smallest money unit seems to be directly linked to the smallest possible energy unit needed to store one bit. If the smallest money unit is smaller than the cost of energy of storing one bit then there seems to be an arbitrage, which will also constrain money producers such as central banks from issuing money with a smaller denomination than this minimum money unit.

On fourth and higher moments of short exponential sums related to cusp forms

We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators. We obtain the conjectured best possible bound in the case of the fourth moment.