Heuristic Arguments Research Articles

UNLIKELY INTERSECTIONS IN FINITE CHARACTERISTIC

We present a heuristic argument based on Honda–Tate theory against many conjectures in ‘unlikely intersections’ over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. Using methods of additive combinatorics, we answer a related question of Chai and Oort where the ambient Shimura variety is a power of the modular curve.

A Reaction-Diffusion Algorithm for Segmentation of Three-Dimensional Sinusoidal Networks in Rats Fed a High-Fat and High-Cholesterol Diet: New Insights and Evaluation

Microstructures in the liver are primarily composed of hepatocytes, hepatic blood, and biliary vessels. Because each hepatocyte comes in contact with both vessels, these vessels form three-dimensional (3D) periodic network patterns. Confocal microscope images are useful for observing 3D structures; however, it is necessary to explicitly describe the vessel structures using 3D images of sinusoidal endothelial cells. For this purpose, we propose a new approach for image segmentation based on the Turing reaction-diffusion model, in which temporal and spatial patterns are self-organized. Turing conditions provided reliable tools for describing the 3D structures. Moreover, using the proposed method, the sinusoidal patterns of rats fed a high-fat/high-cholesterol diet were examined; these rats exhibited pathological features similar to those of human patients with nonalcoholic steatohepatitis related to metabolic syndrome. The findings showed that the parameter in diffusion terms differed significantly among the experimental groups. This observation provided a heuristic argument for parameter selection leading to pattern recognition problems in diseased rats.

PSC Bridge Subjected to Combined Post Tension and Post Compression—A Case Study

Starting from the ideas of Conventional Post Tensioning we present a heuristic argument of advantages of combined actions of post compression along with post tensioned technique for a PSC member through a Design Example. Our aim was to assess the characterization of a pre stressed member if it was to be under the Load effects of post compressing a bar with post tensioned method through hydraulic jacks as the reinforcements in the tensioned zone of conventional PSC bridge were to be compressed in order to induce internal tensile stress similar to internal compressive stresses developed due to conventional post tensioned design. The results ultimately concluded that post compressing a Slender bar by a pre stressing force in the compression zone by a value equal to 0.1 - 0.7 times the pre stressing force in the tension zone would eventually lead to cancelling out of tensile and compressive stresses, thereby forming the desired section which is comparatively smaller in size but can account for sustainability. The anchorage at the top end was provided by special slender steel rods to eliminate the compressive stresses. All the dead loads were counteracted by the action of prestress and the bridge section was able to carry only live load which is deduced through examples in the article.

A stratification approach using logit-based models for confounder adjustment in the study of continuous outcomes.

The control of confounding is an area of extensive epidemiological research, especially in the field of causal inference for observational studies. Matched cohort and case-control study designs are commonly implemented to control for confounding effects without specifying the functional form of the relationship between the outcome and confounders. This paper extends the commonly used regression models in matched designs for binary and survival outcomes (i.e. conditional logistic and stratified Cox proportional hazards) to studies of continuous outcomes through a novel interpretation and application of logit-based regression models from the econometrics and marketing research literature. We compare the performance of the maximum likelihood estimators using simulated data and propose a heuristic argument for obtaining the residuals for model diagnostics. We illustrate our proposed approach with two real data applications. Our simulation studies demonstrate that our stratification approach is robust to model misspecification and that the distribution of the estimated residuals provides a useful diagnostic when the strata are of moderate size. In our applications to real data, we demonstrate that parity and menopausal status are associated with percent mammographic density, and that the mean level and variability of inpatient blood glucose readings vary between medical and surgical wards within a national tertiary hospital. Our work highlights how the same class of regression models, available in most statistical software, can be used to adjust for confounding in the study of binary, time-to-event and continuous outcomes.

Accurate and efficient calculation of response times for groundwater flow

We study measures of the amount of time required for transient flow in heterogeneous porous media to effectively reach steady state, also known as the response time. Here, we develop a new approach that extends the concept of mean action time. Previous applications of the theory of mean action time to estimate the response time use the first two central moments of the probability density function associated with the transition from the initial condition, at t = 0, to the steady state condition that arises in the long time limit, as t→∞. This previous approach leads to a computationally convenient estimation of the response time, but the accuracy can be poor. Here, we outline a powerful extension using the first k raw moments, showing how to produce an extremely accurate estimate by making use of asymptotic properties of the cumulative distribution function. Results are validated using an existing laboratory-scale data set describing flow in a homogeneous porous medium. In addition, we demonstrate how the results also apply to flow in heterogeneous porous media. Overall, the new method is: (i) extremely accurate; and (ii) computationally inexpensive. In fact, the computational cost of the new method is orders of magnitude less than the computational effort required to study the response time by solving the transient flow equation. Furthermore, the approach provides a rigorous mathematical connection with the heuristic argument that the response time for flow in a homogeneous porous medium is proportional to L2/D, where L is a relevant length scale, and D is the aquifer diffusivity. Here, we extend such heuristic arguments by providing a clear mathematical definition of the proportionality constant.

A Proof of Friedman’s Ergosphere Instability for Scalar Waves

Let $(\mathcal{M}^{3+1},g)$ be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $\mathscr{E}$ and no future event horizon $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic argument that the energy of certain solutions $\phi$ of $\square_{g}\phi=0$ grows to $+\infty$ as time increases. In this paper, we provide a rigorous proof of Friedman's instability. Our setting is, in fact, more general. We consider smooth spacetimes $(\mathcal{M}^{d+1},g)$, for any $d\ge2$, not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $\partial\mathscr{E}$ of $\mathscr{E}$ on a small neighborhood of a point $p\in\partial\mathscr{E}$. This condition always holds if $(\mathcal{M},g)$ is analytic in that neighborhood of $p$, but it can also be inferred in the case when $(\mathcal{M},g)$ possesses a second Killing field $\Phi$ such that the span of $\Phi$ and the stationary Killing field $T$ is timelike on $\partial\mathscr{E}$. We also allow the spacetimes $(\mathcal{M},g)$ under consideration to possess a (possibly empty) future event horizon $\mathcal{H}^{+}$, such that, however, $\mathcal{H}^{+}\cap\mathscr{E}=\emptyset$ (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira, Cardoso and Crispino. Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes.

Nonlinear compliance of elastic layers to indentation.

Thin samples adherent to a rigid substrate are considerably less compliant to indentation when compared to specimens that are not geometrically confined. Analytical corrections to this so-called substrate effect exist for various types of indenters but are not applicable when large deformations are possible, as is the case in biological materials. To overcome this limitation, we construct a nonlinear scaling model characterized by one single exponent, which we explore employing a parametric finite element analysis. The model is based on asymptotes of two length scales in relation to the sample thickness, i.e., indentation depth and radius of the contact area. For small indentation depth, we require agreement with analytical, linear models, whereas for large indentation depth and extensive contact area, we recognize similarity to uniaxial deformation, indicating a divergent force required to indent nonlinear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that nonlinear effects originating from either the material or geometric confinement are clearly separated only in thin samples. Furthermore, in this regime the scaling model can be derived by following a heuristic argument extending a linear model to large indentation depths. Lastly, in a large indentation setting where the contact is small in comparison with sample thickness, we observe nonlinear effects independent of material type that we attribute to a higher-order influence of geometrical confinement. In this regime, we define a scalar as the ratio of strains along principal axes as obtained by comparison with the case of a point force on a half-space. We find this scalar to be in quantitative agreement with the scaling exponent, indicating an approach to distinguish between nonlinear effects in the scaling model. While we conjecture our findings to be applicable to other flat-ended indenters, we focus on the case of a flat-ended cylinder in normal contact with a thin layer. The analytical solution for small indentation associated with this problem has been given by Hayes et al. (J Biomech 5:541-551, 1972), for which we provide a convenient numerical implementation.

Digital quantum simulator in the presence of a bath

For a digital quantum simulator (DQS) imitating a target system, we ask the following question: Under what conditions is the simulator dynamics similar to that of the target in the presence of coupling to a bath? In this paper, we derive conditions for close simulation for three different physical regimes, replacing previous heuristic arguments on the subject with rigorous statements. In fact, we find that the conventional wisdom that the simulation cycle time should always be short for good simulation need not always hold up. Numerical simulations of two specific examples strengthen the evidence for our analysis, and go beyond to explore broader regimes.

Bogolyubov inequality for the ground state and its application to interacting rotor systems

We have formulated and proved the Bogolyubov inequality for operators at zero temperature. So far this inequality has been known for matrices, and we were able to extend it to certain class of operators. We have also applied this inequality to the system of interacting rotors. We have shown that if: {\em i)} the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases sufficiently fast with a distance, and {\em iii)} there is an energy gap over the ground state, then the spontaneous magnetization in the ground state is zero, i.e. there is no LRO in the system. We present also heuristic arguments (of perturbation-theoretic nature) suggesting that one- and two-dimensional system of interacting rotors has the energy gap independent of the system size if the interaction is sufficiently small.

The black hole quantum atmosphere

Ever since the discovery of black hole evaporation, the region of origin of the radiated quanta has been a topic of debate. Recently it was argued by Giddings that the Hawking quanta originate from a region well outside the black hole horizon by calculating the effective radius of a radiating body via the Stefan–Boltzmann law. In this paper we try to further explore this issue and end up corroborating this claim, using both a heuristic argument and a detailed study of the stress energy tensor. We show that the Hawking quanta originate from what might be called a quantum atmosphere around the black hole with energy density and fluxes of particles peaked at about 4MG, running contrary to the popular belief that these originate from the ultra high energy excitations very close to the horizon. This long distance origin of Hawking radiation could have a profound impact on our understanding of the information and transplanckian problems.

Localization and mobility edges in the off-diagonal quasiperiodic model with slowly varying potentials

We study a one-dimensional system that includes both a commensurate off-diagonal modulation of the hopping amplitude and an incommensurate, slowly varying diagonal on-site modulation. By using asymptotic heuristic arguments, we identify four closed form expressions for the mobility edges. We further study numerically the inverse participation ratio, the density of states and the Lyapunov exponent. The numerical results are in exact agreement with our theoretical predictions. Besides a metal–insulator transition driven by the strength of the slowly varying potential, another four insulator–metal transitions are found in this model as the energy is increased in magnitude from the band center (E=0) to the mobility edges (±Ec2,±Ec1).

A Further Stroll into the Eisenstein Primes

If one imagines the Eisenstein primes to be lily pads in the pond of complex numbers, could a frog hop from the origin to infinity with jumps of bounded size? If the frog was confined to the real number line, the answer is no. Good heuristic arguments exist for it not being possible in the complex plane, but there is still no formal proof for this conjecture.If the frog's journey terminates for a given hop size, it implies that a prime-free “moat” greater than the hop size completely surrounds the origin.In the earlier MONTHLY article “A Stroll Through the Gaussian Primes,” Ellen Gethner, Stan Wagon, and Brian Wick [3] explored this problem in the Gaussian primes and by computational methods proved the existence of a -moat. Additionally they proved that prime-free neighborhoods of arbitrary radius k surrounding a Gaussian prime exist.In their concluding remarks, Gethner et al. note that “Similar questions about walks to infinity may be asked for the finitely many imaginary quadratic fields of class number 1.”This paper takes up that challenge by examining similar questions about walks to infinity in the ring of Eisenstein integers. We prove that prime-free neighborhoods of arbitrary radius k surrounding an imaginary quadratic prime exist by directly generalizing Gethner's proof regarding prime-free neighborhoods surrounding a Gaussian prime. By computational means we establish the existence of a -moat in the Eisenstein primes and provide an estimate for the bounds of any k-moat. A subtle difference between the structure of Eisenstein and Gaussian integers results in our estimate being significantly different from our initial expectation.

Proof of Renormalizability of Scalar Field Theories Using the Epstein-Glaser Scheme and Techniques of Microlocal Analysis

The renormalizability of QFT's is a vastly studied issue, and particularly the results concerning a scalar eld theory are well-known through the traditionalrenormalization approach in the literature. However, in this paper we analyze the problem through a less known approach, which justies in a more rigorousand mathematically neat manner, the heuristic arguments of standard treatments of divergencies in QFT's. This paper analyzes the renormalizability ofan arbitrary Scalar Field Theory with interaction Lagrangean L(x) =: 'm(x) : using the method of Epstein-Glaser and techniques of microlocal analysis, inparticular, the concept of scaling degree of a distribution. For a renormalizability proof of perturbative models in the Epstein-Glaser scheme one rst needs to dene an n-fold product of sub-Wick monomials of the interaction Lagrangean. This time ordering is an operator-valued distribution on R4n and the basic issue is its ill-denedness on a null set. The renormalization of a theory in this scheme amounts to the problem of extension of distributions across null sets.

Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition [formula omitted

The multiplicative decomposition of the total deformation F=FeFi between an elastic (Fe) and an inelastic component (Fi) is standard in the modeling of many irreversible processes such as plasticity, growth, thermoelasticity, viscoelasticty or phase transformations. The heuristic argument for such kinematic assumption is based on the chain rule for the compatible scenario (CurlFi=0) where the individual deformation tensors are gradients of deformation mappings, i.e. F=Dφ=D(φe∘φi)=(Dφe)∘φi(Dφi)=FeFi. Yet, the conditions for its validity in the general incompatible case (CurlFi≠0) has so far remained uncertain. We show in this paper that detFi=1 and CurlFi bounded are necessary and sufficient conditions for the validity of F=FeFi for a wide range of inelastic processes. In particular, in the context of crystal plasticity, we demonstrate via rigorous homogenization from discrete dislocations to the continuum level in two dimensions, that the volume preserving property of the mechanistics of dislocation glide, combined with a finite dislocation density, is sufficient to deliver F=FeFp at the continuum scale. We then generalize this result to general two-dimensional inelastic processes that may be described at a lower dimensional scale via a multiplicative decomposition while exhibiting a finite density of incompatibilities. The necessity of the conditions detFi=1 and CurlFi bounded for such systems is demonstrated via suitable counterexamples.

Jacob\u2019s Ladder: Logics of Magic, Metaphor and Metaphysics

In this article, we discuss some issues concerning magical thinking—forms of thought and association mechanisms characteristic of early stages of mental development. We also examine good reasons for having an ambivalent attitude concerning the later permanence in life of these archaic forms of association, and the coexistence of such intuitive but informal thinking with logical and rigorous reasoning. At the one hand, magical thinking seems to serve the creative mind, working as a natural vehicle for new ideas and innovative insights, and giving form to heuristic arguments. At the other hand, it is inherently difficult to control, lacking effective mechanisms needed for rigorous manipulation. Our discussion is illustrated with many examples from the Hebrew Bible, and some final examples from modern science.

Quantum centipedes with strong global constraint

A centipede made of N quantum walkers on a one-dimensional lattice is considered. The distance between two consecutive legs is either one or two lattice spacings, and a global constraint is imposed: the maximal distance between the first and last leg is N + 1. This is the strongest global constraint compatible with walking. For an initial value of the wave function corresponding to a localized configuration at the origin, the probability law of the first leg of the centipede can be expressed in closed form in terms of Bessel functions. The dispersion relation and the group velocities are worked out exactly. Their maximal group velocity goes to zero when N goes to infinity, which is in contrast with the behaviour of group velocities of quantum centipedes without global constraint, which were recently shown by Krapivsky, Luck and Mallick to give rise to ballistic spreading of extremal wave-front at non-zero velocity in the large-N limit. The corresponding Hamiltonians are implemented numerically, based on a block structure of the space of configurations corresponding to compositions of the integer N. The growth of the maximal group velocity when the strong constraint is gradually relaxed is explored, and observed to be linear in the density of gaps allowed in the configurations. Heuristic arguments are presented to infer that the large-N limit of the globally constrained model can yield finite group velocities provided the allowed number of gaps is a finite fraction of N.

Spectral theory of extended Harper’s model and a question by Erdős and Szekeres

The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such "spectral collapse," however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erd\H{o}s and Szekeres.

The average number of divisors of the Euler function

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by \cite{LP} is indeed close to the truth.

Partition function of free conformal fields in 3-plet representation

Simplest examples of AdS/CFT duality correspond to free CFTs in d dimensions with fields in vector or adjoint representation of an internal symmetry group dual in the large N limit to a theory of massless or massless plus massive higher spins in AdSd+1. One may also study generalizations when conformal fields belong to higher dimensional representations, i.e. carry more than two internal symmetry indices. Here we consider the case of the 3-fundamental (“3-plet”) representation. One motivation is a conjectured connection to multiple M5-brane theory: heuristic arguments suggest that it may be related to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N symmetry group that has an AdS7 dual. We compute the singlet partition function Z on S1 × Sd−1 for a free field in 3-plet representation of U(N) and analyse its novel large N behaviour. The large N limit of the low temperature expansion of Z which is convergent in the vector and adjoint cases here is only asymptotic, reflecting the much faster growth of the number of singlet operators with dimension, indicating a phase transition at very low temperature. Indeed, while the critical temperatures in the vector (Tc ∼ Nγ, γ > 0) and adjoint (Tc ∼ 1) cases are finite, we find that in the 3-plet case Tc ∼ (log N)−1, i.e. it approaches zero at large N. We discuss some details of large N solution for the eigenvalue distribution. Similar conclusions apply to higher p-plet representations of U(N) or O(N) and also to the free p-tensor theories invariant under [U(N)]p or [O(N)]p with p ≥ 3.

Criticality of inhomogeneous Nariai-like cosmological models

In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions are at the borderline between gravitational collapse and de-Sitter-like exponential expansion. Our studies focus in particular on the intriguing oscillatory dynamics which we discover. Our investigations are carried out both analytically (using heuristic mode analysis arguments) and numerically (using the numerical infrastructure recently introduced by us).