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By using the relations between supersingular elliptic curves defined over a finite field of characteristic p > 3 and the p-th division polynomial, we present a property about the reduction modulo a prime p > 3 of the p-th division polynomial based on a heuristic argument and numerical evidence. We prove this property determining the p-th division polynomial of supersingular elliptic curves. As a consequence of this result, we present a criterion to discard supersingular elliptic curves.

Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics; however, the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on $$\mathbb {Z}^2$$ with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integrable probability literature to study coalescence structure of finite geodesics, thereby making rigorous a heuristic argument due to Newman (Auffinger et al., 50 Years of First-passage Percolation, American Mathematical Soc., 2017).

In this paper, we provide new results and algorithms (including backtracking versions of Nesterov accelerated gradient and Momentum) which are more applicable to large scale optimisation as in Deep Neural Networks. We also demonstrate that Backtracking Gradient Descent (Backtracking GD) can obtain good upper bound estimates for local Lipschitz constants for the gradient, and that the convergence rate of Backtracking GD is similar to that in classical work of Armijo. Experiments with datasets CIFAR10 and CIFAR100 on various popular architectures verify a heuristic argument that Backtracking GD stabilises to a finite union of sequences constructed from Standard GD for the mini-batch practice, and show that our new algorithms (while automatically fine tuning learning rates) perform better than current state-of-the-art methods such as Adam, Adagrad, Adadelta, RMSProp, Momentum and Nesterov accelerated gradient. To help readers avoiding the confusion between heuristics and more rigorously justified algorithms, we also provide a review of the current state of convergence results for gradient descent methods. Accompanying source codes are available on GitHub.

Recently, Martinez-Penas and Kschischang (IEEE Trans. Inf. Theory, 2019) showed that lifted linearized Reed-Solomon codes are suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed-Solomon codes. Compared to the construction by Martinez-Penas-Kschischang, interleaving allows to increase the decoding region significantly (especially w.r.t. the number of insertions) and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. The proposed decoder is a list decoder that can also be interpreted as a probabilistic unique decoder. Although our best upper bound on the list size is exponential, we present a heuristic argument and simulation results that indicate that the list size is in fact one for most channel realizations up to the maximal decoding radius., Comment: 6 pages, 2 figures, accepted at ISIT 2021

Hierarchical structures include spatial systems (e.g., a continent), clusters of a spatial system (e.g., countries of a continent), zones of a cluster (e.g., states of a country), and so on. Variable-specific classification of the zones $({{X_i}})$(Xi) of a cluster of zones $(X)$(X) within a spatial system is the main focus of this paper. Variable-specific (e.g., GDP, population, trade, resources, economic activity etc) classification of zones is done by computing the levels of interaction between the ith and jth zones. Based on a heuristic argument, we proposed a modified gravity model for the computation of levels of interaction between the zones. This argument is based on the following two facts: (i) the level of interaction between the zones ${X_i}$Xi and ${X_j}$Xj, with masses $m{X_i}$mXi and $m{X_j}$mXj is direction-dependent, and (ii) the level of interactions between the zones ${X_i}$Xi and ${X_j}$Xj, with masses $m{X_i}$mXi and $m{X_j}$mXj, situated at strategically insignificant locations would be much different (lesser) from that of the zones ${X_i}$Xi and ${X_j}$Xj with similar masses $m{X_i}$mXi and $m{X_j}$mXj but situated at strategically highly significant locations. Based on this argument, we provide a modified gravity model by incorporating the $d{X_{ij}} \ne d{X_{ji}}$dXij≠dXji, and the product of location significance indexes $({\varphi {X_i}\varphi {X_j}})$(ϕXiϕXj) of the corresponding zones. This modified gravity model yields the level of interaction between the two zones that satisfies $F{X_{ij}} \ne F{X_{ji}}$FXij≠FXji. We demonstrate this modified gravity model on the 28 states of India, whereby the areal extents (land resources) of each state is considered as a parameter mass. The levels of interactions are presented for all pairs of states.

The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model. Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4. Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be q(1-q)/4. The Lagrangian of the RG fixed-point theory for r retarded and r advanced replicas is proposed to be the GL(r|r)_4 Wess-Zumino-Witten model deformed by a truly marginal perturbation. The latter emerges from the non-linear sigma model by a natural scenario of spontaneous symmetry breaking., v1: 40 pages, 4 figures; v2: 50 pages (four new sections: 4.3, 4.4, 4.12, 4.13); v3: 60 pages (added section on spontaneously broken symmetry), published version

We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. The typical mass (number of sites $s$) in the largest cluster is proportional to $L^2$, and the mean mass of the remaining (smaller) clusters is also proportional to $L^2$. The normalized (per site) density $n_s$ of clusters of size (mass) $s$ is proportional to $s^{-\tau}$, while the volume fraction $P_k$ occupied by the $k$th largest cluster scales as $k^{-q}$. We put forward a heuristic argument that $\tau=2$ and $q=1$. However, the numerically measured values are $\tau\approx1.83$ and $q\approx1.20$. We suggest that these are effective exponents that drift towards their asymptotic values with increasing $L$ as slowly as $1/\ln L$ approaches zero., Comment: 14 pages, 13 figures

Many-body localization is a fascinating theoretical concept describing the intricate interplay of quantum interference, i.e. localization, with many-body interaction induced dephasing. Numerous computational tests and also several experiments have been put forward to support the basic concept. Typically, averages of time-dependent global observables have been considered, such as the charge imbalance. We here investigate within the disordered spin-less Hubbard ($t-V$) model how dephasing manifests in time dependent variances of observables. We find that after quenching a N\'eel state the local charge density exhibits strong temporal fluctuations with a damping that is sensitive to disorder $W$: variances decay in a power law manner, $t^{-\zeta}$, with an exponent $\zeta(W)$ strongly varying with $W$. A heuristic argument suggests the form, $\zeta\approx\alpha(W)\xi_\text{sp}$, where $\xi_\text{sp}(W)$ denotes the noninteracting localization length and $\alpha(W)$ characterizes the multifractal structure of the dynamically active volume fraction of the many-body Hilbert space. In order to elucidate correlations underlying the damping mechanism, exact computations are compared with results from the time-dependent Hartree-Fock approximation. Implications for experimentally relevant observables, such as the imbalance, will be discussed., Comment: 7+3 pages, 6+4 figures

This paper describes a non-interactive process allowing a prover to convince a verifier that a modulus n is the product of two primes (p, q) of about the same size. A further heuristic argument conjectures that \(p-1\) and \(q-1\) have sufficiently large prime factors for cryptographic applications.

The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.

Combinatorics, Probability & Computing, 29 (6), ISSN:0963-5483, ISSN:1469-2163

International audience; Regularization of the classical Laplacian matrices was empirically shown to improve spectral clustering in sparse networks. It was observed that small regularizations are preferable, but this point was left as a heuristic argument. In this paper we formally determine a proper regularization which is intimately related to alternative state-of-the-art spectral techniques for sparse graphs.

This article describes a heuristic argument for understanding certain physical systems in terms of properties that resemble the beliefs and goals of folk psychology. The argument rests on very simple assumptions. The core of the argument is that predictions about certain events can legitimately be based on assumptions about later events, resembling Aristotelian ‘final causation’; however, more nuanced causal entities (resembling fallible beliefs) must be introduced into these types of explanation in order for them to remain consistent with a causally local Universe.

Although the adiabatic heuristic argument of the fractional quantum Hall states has been successful, continuous modification of the flux/statistics of anyons is strictly prohibited due to algebraic constrains of the braid group on a torus. We have numerically shown that the adiabatic heuristic principle for anyons is still valid even though the Hamiltonians cannot be modified continuously. The Chern number of the ground state multiplet is the adiabatic invariant, while the number of the topological degeneracy behaves wildly. A generalized Streda formula is proposed that explains the degeneracy pattern. Nambu-Goldston modes associated with the anyon superconductivity are also suggested numerically., Comment: 7 pages

Evidence is provided for a conjecture that, in the continuum limit, the mean of the causal set action of a causal set sprinkled into a globally hyperbolic Lorentzian spacetime, M, of finite volume equals the Einstein Hilbert action of M plus the volume of the co-dimension 2 intersection of the future boundary with the past boundary. We give the heuristic argument for this conjecture and analyse some examples in 2 dimensions and one example in 4 dimensions., Comment: 21 pages, 4 figures. Second version with arXiv reference added and typos corrected

The classical theory of cluster relaxation is unsatisfactory because it involves the Coulomb logarithm. The Balescu-Lenard (BL) equation provides a rigorous alternative that has no ill-defined parameter. Moreover, the BL equation, unlike classical theory, includes the cluster's self-gravity. A heuristic argument is given that indicates that relaxation does not occur predominantly through two-particle scattering and is enhanced by self-gravity. The BL equation is adapted to a spherical system and used to estimate the flux through the action space of isochrone clusters with different velocity anisotropies. A range of fairly different secular behaviours is found depending on the fraction of radial orbits. Classical theory is also used to compute the corresponding classical fluxes. The BL and classical fluxes are very different because (a) the classical theory materially under-estimates the impact of large-scale collectively amplified fluctuations and (b) only the leading terms in an infinite sum for the BL flux are computed. A complete theory of cluster relaxation likely requires that the sum in the BL equation be decomposed into a sum over a finite number of small wavenumbers complemented by an integral over large wavenumbers analogous to classical theory., Accepted for publication in MNRAS. Replaced with updated manuscript including responses to referee's comments

The gauge/gravity correspondence discovered two decades ago has had a profound influence on how the basic laws in physics should be formulated. In spite of the predictive power of holographic approaches (for example, when they are applied to strongly coupled condensed-matter physics problems), the fundamental reasons behind their success remain unclear. Recently, the role of quantum entanglement has come to the fore. Here we explore a quantity that connects gravity and quantum information in the light of the gauge/gravity correspondence. This is given by the minimal cross-section of the entanglement wedge that connects two disjoint subsystems in a gravity dual. In particular, we focus on various inequalities that are satisfied by this quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures a bipartite correlation in a given mixed state. We give a heuristic argument that supports this identification based on a tensor network interpretation of holography. This predicts that the entanglement of purification satisfies the strong superadditivity for holographic conformal field theories.

The Barnich--Troessaert bracket is a proposal for a modified Poisson bracket on the covariant phase space for general relativity. The new bracket allows us to compute charges, which are otherwise not integrable. Yet there is a catch. There is a clear prescription for how to evaluate the new bracket for any such charge, but little is known how to extend the bracket to the entire phase space. This is a problem, because not every gravitational observable is also a charge. In this paper, we propose such an extension. The basic idea is to remove the radiative data from the covariant phase space. This requires second-class constraints. Given a few basic assumptions, we show that the resulting Dirac bracket on the constraint surface is nothing but the BT bracket. A heuristic argument is given to show that the resulting constraint surface can only contain gravitational edge modes., Comment: v3: typos removed, final version for Class. Quant. Grav., section on bulk-boundary dynamics added

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., Comment: 72 pages

The multiplicative decomposition of the total deformation F = F e F i 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 ( Curl F i = 0 ) where the individual deformation tensors are gradients of deformation mappings, i.e. F = D φ = D ( φ e ∘ φ i ) = ( D φ e ) ∘ φ i ( D φ i ) = F e F i . Yet, the conditions for its validity in the general incompatible case ( Curl F i ≠ 0 ) has so far remained uncertain. We show in this paper that det F i = 1 and Curl F i bounded are necessary and sufficient conditions for the validity of F = F e F i 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 = F e F p 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 det F i = 1 and Curl F i bounded for such systems is demonstrated via suitable counterexamples.

We report a phenomenological power law for the reduction of network permeability in statistically homogeneous spatial networks upon removal of a single edge. We characterize this power law for plexus-like microvascular sinusoidal networks from liver tissue, as well as perturbed two- and three-dimensional regular lattices. We provide a heuristic argument for the observed power law by mapping arbitrary spatial networks that satisfy Darcy's law on a small-scale resistor network.

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., Comment: to appear in Inventiones mathematicae

For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $\lambda(n)$ for some function $\lambda$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$ has a giant component and the probability that $G(\mathcal{D})$ has no giant component are bounded away from $1$., Comment: 42 pages, to appear in Probability Theory and Related Fields

We illustrate an alternative derivation of the viscous regulariza- tion of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc., 324:331{351]. We propose and discuss a new on-smooth" variant of the viscous regulariza- tion and we offer an heuristic argument that indicates that this variant should display interesting hysteretic efiects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approx- imation scheme.

Argumentation theorists often disagree about which scheme best represents a given type of argument (e.g. argument by analogy, argument from authority, inference to the best explanation). Unfortunately, authors sometimes become involved in fruitless pseudo-agreement because they fail to perceive that their supposedly competing schemes are means for achieving different (but compatible) practical or theoretical goals. This paper explains some of the different purposes that an argument scheme may serve, and it indicates how the relevant type of pseudo-disagreement may be avoided.

We study the mean square of the error term in the Gauss circle problem. A heuristic argument based on the consideration of off-diagonal terms in the mean square of the relevant Voronoi-type summation formula leads to a precise conjecture for the mean square of this discrepancy.

We uncover unusual topological features in the long-time relaxation of the $q$-state kinetic Potts ferromagnet on the triangular lattice that is instantaneously quenched to zero temperature from a zero-magnetization initial state. For $q=3$, the final state is either: the ground state (frequency $\approx 0.75$), a frozen three-hexagon state (frequency $\approx 0.16$), a two-stripe state (frequency $\approx 0.09$), or a three-stripe state (frequency $3$., 4 pages, 6 figures. Version 2: 8 pages, 11 figures. Significantly expanded compared to version 1

In The Myth of Morality Richard Joyce presents a simple and very influential argument for the truth of moral error theory. In this paper I point out that (i) the argument does not have the form Joyce attributes to it, (ii) the argument is not valid in an extensional propositional logic and (iii) on the most natural way of explicating the meanings of the involved terms, it remains invalid. I conclude that more explanation is needed if we are to accept this particular argument for moral error theory.

In this note, we apply classical results from number theory to give an affirmative, but heuristic, answer to the question of Shparlinski (Japanese J. Math., 2012) whether there exist infinitely many primes p of the form p = k2 + lk + 1, with integers k, l, such that k > 0 and . Based on a heuristic argument, we provide a formula for the number of such primes, which is surprisingly accurate as computations show.

Let σ ( n ) \sigma (n) denote the sum of all of the positive divisors of n n , and let s ( n ) = σ ( n ) − n s(n) = \sigma (n)-n denote the sum of the proper divisors of n n . The functions σ ( ⋅ ) \sigma (\cdot ) and s ( ⋅ ) s(\cdot ) were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n n with n n deficient but s ( n ) s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n n not in the range of the function s ( ⋅ ) s(\cdot ) ; these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k k -sets, where a friendly k k -set is a collection of k k distinct integers which share the same value of σ ( n ) n \frac {\sigma (n)}{n} .

Even though tools for identifying and analyzing arguments are now in wide use in the field of argumentation studies, so far there is a paucity of resources for evaluating real arguments, aside from using deductive logic or Bayesian rules that apply to inductive arguments. In this paper it is shown that recent developments in artificial intelligence in the area of computational systems for modeling defeasible argumentation reveal a different approach that is currently making interesting progress. It is shown how these systems provide the general outlines for a system of argument evaluation that can be applied to legal arguments as well as everyday conversational arguments to assist a user to evaluate an argument.

The argument from design is, generically, an attempt to argue from features of the natural world to the conclusion that it is the result of intelligent or considered creative action. The chapter examines various forms of this argument, especially those involving fine tuning, and finds them wanting. An approach based on Bayesian accumulation of evidence is also critiqued. The problem with all these approaches is that they misconstrue the situation one is in when it comes to deciding on what basis a relationship to God should be built. A relationship built on impersonal foundations can only be impersonal. But it remains perfectly appropriate to embrace the human instinct to allow a sense of wonder to increase one’s openness to God. Furthermore, the very inclination to see physical events as intelligible is itself a religious response, a recognition that the world is neither arbitrary nor meaningless.

For a Markovian open quantum system it is possible, by continuously monitoring the environment, to know the stochastically evolving pure state of the system without altering the master equation. In general, even for a system with a finite Hilbert space dimension $D$, the pure state trajectory will explore an infinite number of points in Hilbert space, meaning that the dimension $K$ of the classical memory required for the tracking is infinite. However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed that tracking of a qubit ($D=2$) is always possible with a bit ($K=2$), and gave a heuristic argument implying that a finite $K$ should be sufficient for any $D$, although beyond $D=2$ it would be necessary to have $K>D$. Our paper is concerned with rigorously investigating the relationship between $D$ and $K_{\rm min}$, the smallest feasible $K$. We confirm the long-standing conjecture of Karasik and Wiseman that, for generic systems with $D>2$, $K_{\rm min}>D$, by a computational proof (via Hilbert Nullstellensatz certificates of infeasibility). That is, beyond $D=2$, $D$-dimensional open quantum systems are provably harder to track than $D$-dimensional open classical systems. Moreover, we develop, and better justify, a new heuristic to guide our expectation of $K_{\rm min}$ as a function of $D$, taking into account the number $L$ of Lindblad operators as well as symmetries in the problem. The use of invariant subspace and Wigner symmetries makes it tractable to conduct a numerical search, using the method of polynomial homotopy continuation, to find finite physically realizable ensembles (as they are known) in $D=3$. The results of this search support our heuristic. We thus have confidence in the most interesting feature of our heuristic: in the absence of symmetries, $K_{\rm min} \sim D^2$, implying a quadratic gap between the classical and quantum tracking problems., 35 pages, 3 figures, Accepted in Quantum Journal, minor changes

We point out the existence of a second feature in the matter two-point function, besides the acoustic peak, due to the baryon-baryon correlation in the early universe and positioned at twice the distance of the peak. We discuss how the existence of this feature is implied by the well-known heuristic argument that explains the baryon bump in the correlation function. A standard $\chi^2$ analysis to estimate the detection significance of the second feature is mimicked. We conclude that for realistic values of the baryon density, an SKA-like galaxy survey will not be able to detect this feature with standard correlation function analysis.

We study the Game of Life as a statistical system on an $L\times L$ square lattice with periodic boundary conditions. Starting from a random initial configuration of density $\rho_{\rm in}=0.3$ we investigate the relaxation of the density as well as the growth with time of spatial correlations. The asymptotic density relaxation is exponential with a characteristic time $\tau_L$ whose system size dependence follows a power law $\tau_L\propto L^z$ with $z=1.66\pm 0.05$ before saturating at large system sizes to a constant $\tau_\infty$. The correlation growth is characterized by a time dependent correlation length $\xi_t$ that follows a power law $\xi_t\propto t^{1/z^\prime}$ with $z^\prime$ close to $z$ before saturating at large times to a constant $\xi_\infty$. We discuss the difficulty of determining the correlation length $\xi_\infty$ in the final "quiescent" state of the system. The decay time $t_{\rm q}$ towards the quiescent state is a random variable, we present simulational evidence as well as a heuristic argument indicating that for large $L$ its distribution peaks at a value $t_{\rm q}^*(L) \simeq 2\tau_\infty\log L$., Comment: 22 pages, 10 figures

We study, by using liquid-state theories and Monte Carlo simulation, the behavior of systems of classical particles interacting through a finite pair repulsion supplemented with a longer range attraction. Any such potential can be driven Ruelle-unstable by increasing the attraction at the expense of repulsion, until the thermodynamic limit is lost. By examining several potential forms, we find that all systems exhibit a qualitatively similar behavior in the fluid phase as the threshold of thermodynamic stability is approached (and possibly surpassed). The general feature underlying the approach to Ruelle instability is a pronounced widening of the liquid-vapor binodal (and spinodal) line at low temperatures, to such an extent that at the stability threshold a vanishing-density vapor would coexist with a diverging-density liquid. We attempt to rationalize the universal pathway to Ruelle instability in soft-core fluids by appealing to a heuristic argument., Comment: 24 pages, 10 figures

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.

We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.

The increasing cost of electoral campaigns raises the need for effective campaign planning and a precise understanding of the return of such investment. Interestingly, despite the strong impact of elections on our daily lives, how this investment is translated into votes is still unknown. By performing data analysis and modeling, we show that top candidates spend more money \emph{per vote} than the less successful and poorer candidates, a sublinearity that discloses a diseconomy of scale. We demonstrate that such electoral diseconomy arises from the competition between candidates due to inefficient campaign expenditure. Our approach succeeds in two important tests. First, it reveals that the statistical pattern in the vote distribution of candidates can be explained in terms of the independently conceived, but similarly skewed distribution of money campaign. Second, using a heuristic argument, we are able to predict a turnout percentage for a given election of approximately 63\%. This result is in good agreement with the average turnout rate obtained from real data. Due to its generality, we expect that our approach can be applied to a wide range of problems concerning the adoption process in marketing campaigns., 8 pages, 4 figures

A continuous family of static spherically symmetric solutions of Einstein�s vacuum field equations with a spatial singularity at the origin (Formula presented.) is found. These solutions are parametrized by a real valued parameter (Formula presented.) (ranging from 0 to 1) and such that the radial horizon�s location is displaced continuously towards the singularity ((Formula presented.)) as (Formula presented.) increases. In the extreme limit (Formula presented.), the location of the singularity and horizon merges leading to a null singularity. In this extreme case, any infalling observer hits the null singularity at the very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. An heuristic argument is provided how one might avoid the Hawking particle emission process in this extreme case when the singularity and horizon merges. The field equations due to a delta-function point-mass source at (Formula presented.) are solved and the Euclidean gravitational action corresponding to those solutions is evaluated explicitly. It is found that the Euclidean action is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions (Formula presented.).

Purpose– A standard model of equilibrium unemployment consists of static equations for real-wage ambitions (wage curve) and real-wage scope (price curve), which jointly determine the NAIRU. The heuristics of the model states that unless the rate of unemployment approaches the NAIRU from any given initial value, inflation will be increasing or decreasing over time. The paper aims to discuss these issues.Design/methodology/approach– The authors formalize this influential heuristic argument with the aid of a dynamic model of the wage-price spiral where the static theory’s equations are re-interpreted as attractor relationships.Findings– The authors show that NAIRU unemployment dynamics are sufficient but not necessary for inflation stabilization, and that the dynamic wage-price spiral model generally has a dynamically stable solution for any predetermined rate of unemployment. The authors also discuss a restricted version of the model that conforms to the accelerationist view that inflation increases/falls if unemployment is not at its “natural rate”.Research limitations/implications– To investigate the relevance of heuristical dynamics of influential macro models, explicit modelling of such dynamics is a necessary step.Practical implications– An important argument against social orders that represent an attempt to target unemployment at relatively low levels, is refuted by the analysis.Social implications– A high degree of employment is a main premise for a social order with equal income distribution and a drive for productivity growth.Originality/value– It is important that economics give a balanced view of the possibility of attaining inflation stability at low or moderate levels of unemployment. This offering is contributions to establish such a balance.