Probabilistic Arguments Research Articles

Theoretical and numerical study of a fluid flow of a slightly rarefied gas-free stream past a moving wall inside a porous medium

This study investigates the boundary layer flow problem arising in the viscous fluid flow of a slightly rarefied gas-free stream over a moving plate embedded in a porous medium. Previous works were mainly focused on numerically, and some properties of the solutions were discussed from numerical results. But, numerical solutions are only sometimes guaranteed the solution’s existence. In this work, we have adopted the famous topological shooting argument to prove the existence result of the solution. We have shown that the solution may be convex or concave depending on the parameter values. Also, for these reasons, there arise some difficulties. However, we have rectified all the challenges and proved that unique solutions exist for all the particular governing parameter values. Nevertheless, we found an exact solution for some specific parameter values. Furthermore, the Haar wavelet collocation method is used to address the numerical results of the nonlinear boundary value problem. We first validate the obtained results with the existing numerical outcomes. The impacts of permeability and slip parameters on the velocity profiles and shear stress are elucidated through tables and figures, and probable arguments are explained in detail.

Is Biological Death Final? Recomputing the Drake-S Equation for Postmortem Survival of Consciousness

This participatory team science project extended Laythe and Houran’s (2022) prior application of a famous probabilistic argument known as the "Drake equation" to the question of postmortem survival. Specifically, we evaluated effect sizes from peer-reviewed, empirical studies to determine the maximum average percentage effect that ostensibly supports (i.e., "anomalous effects") or refutes (i.e., "known confounds") the survival hypothesis. But unlike the earlier application, this research included a study-specific estimate of the hypothesized variable of "living agent psi" via a new meta-analysis of empirical studies (N = 17) with exceptional subjects vs participants from the general population. Our updated analysis found that putative psi was a meaningful variable, although it along with other known confounds still did not account for 30.3% of survival-related phenomena that appear to attest directly to human consciousness continuing after physical (biological) death. Thus, the popular conventional variables that we measured here are seemingly insufficient to account for a sizable portion of the purported empirical data that has been interpreted as evidence of survival. Our conclusion is nonetheless tempered by several assumptions and limitations of our speculative exercise, which ultimately does not affirm the existence of an ‘afterlife’ but rather highlights the need for measurements with greater precision and/ or a more comprehensive set of quantifiable variables. Therefore, we discuss how our probabilistic approach provides important heuristics to guide future research in this highly controversial domain that touches both parapsychology and transpersonal psychology.

An analytical framework for modeling ride pooling efficiency and minimum fleet size

Ride pooling (RP) is a transport mode using on-demand buses to combine the trips of multiple users into one vehicle. Its required fleet size and carbon emissions are quantified by the system’s efficiency. Due to the complex interplay between street network, buses, users and dispatch algorithm, efficiency case studies are available but bottom-up predictions are not. Here we close this gap using probabilistic and geometric arguments in an analytical model framework. Its modular design allows for adaptation to specific usage scenarios and provides an over-arching view of them. In a showcase on Euclidean spaces, our model quantifies how RP outperforms private cars as user demand increases. Its predicted power-law scaling is verified using a custom simulation framework, which further reveals improved scaling on real street networks and graphs with hierarchical structures. Henceforth, our work may help to identify street networks well-suited for RP, and predict key performance indicators analytically.

Subspace coverings with multiplicities

AbstractWe study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$ , while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

A class of supercritical/critical singular stochastic PDEs: Existence, non-uniqueness, non-Gaussianity, non-unique ergodicity

We study the surface quasi-geostrophic equation with an irregular spatial perturbation∂tθ+u⋅∇θ=−ν(−Δ)γ/2θ+ζ,u=∇⊥(−Δ)−1θ, on [0,∞)×T2, with ν⩾0, γ∈[0,3/2) and ζ∈B∞,∞−2+κ(T2) for some κ>0. This covers the case of ζ=(−Δ)α/2ξ for α<1 and ξ a spatial white noise on T2. Depending on the relation between γ and α, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures [38]. Based on purely analytical tools from convex integration and without the need of any probabilistic arguments including renormalization, we prove existence of infinitely many analytically weak solutions in Llocp(0,∞;B∞,1−1/2)∩Cb([0,∞);B∞,1−1/2−δ)∩Cb1([0,∞);B∞,1−3/2−δ) for all p∈[1,∞) and δ>0. We are able to prescribe an initial as well as a terminal condition at a finite time T>0, and to construct steady state, i.e. time independent, solutions. In all cases, the solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times. Moreover, a coming down from infinity with respect to the perturbation and the initial condition holds. Finally, we show that our solutions generate statistically stationary solutions as limits of ergodic averages, and we obtain existence of infinitely many non-Gaussian time dependent ergodic stationary solutions. We also extend our results to a more general class of singular SPDEs.

Reliability Assessment and Safety Arguments for Machine Learning Components in System Assurance

The increasing use of Machine Learning (ML) components embedded in autonomous systems—so-called Learning-Enabled Systems (LESs)—has resulted in the pressing need to assure their functional safety. As for traditional functional safety, the emerging consensus within both, industry and academia, is to use assurance cases for this purpose. Typically assurance cases support claims of reliability in support of safety, and can be viewed as a structured way of organising arguments and evidence generated from safety analysis and reliability modelling activities. While such assurance activities are traditionally guided by consensus-based standards developed from vast engineering experience, LESs pose new challenges in safety-critical application due to the characteristics and design of ML models. In this article, we first present an overall assurance framework for LESs with an emphasis on quantitative aspects, e.g., breaking down system-level safety targets to component-level requirements and supporting claims stated in reliability metrics. We then introduce a novel model-agnostic Reliability Assessment Model (RAM) for ML classifiers that utilises the operational profile and robustness verification evidence. We discuss the model assumptions and the inherent challenges of assessing ML reliability uncovered by our RAM and propose solutions to practical use. Probabilistic safety argument templates at the lower ML component-level are also developed based on the RAM. Finally, to evaluate and demonstrate our methods, we not only conduct experiments on synthetic/benchmark datasets but also scope our methods with case studies on simulated Autonomous Underwater Vehicles and physical Unmanned Ground Vehicles.

Beating the Probabilistic Lower Bound on q-Perfect Hashing

In this paper, we investigate the achievable rate of perfect hash code. For an integer $$q\ge 2$$ , a perfect q-hash code C is a block code over $$[q]:=\{1,\ldots ,q\}$$ of length n in which every subset $$\{{\textbf{c}}_1,{\textbf{c}}_2,\dots ,{\textbf{c}}_q\}$$ of q elements is separated, i.e., there exists $$i\in [n]$$ such that $$\{\textrm{proj}_i({\textbf{c}}_1),\dots ,\textrm{proj}_i({\textbf{c}}_q)\}=[q]$$ , where $$\textrm{proj}_i({\textbf{c}}_j)$$ denotes the ith position of $${\textbf{c}}_j$$ . Finding the maximum size M(n, q) of perfect q-hash codes of length n, for given q and n, is a fundamental problem in combinatorics, information theory, and computer science. In this paper, we are interested in asymptotic behavior of this problem. Precisely speaking, we will focus on the quantity $$R_q:=\limsup _{n\rightarrow \infty }\frac{\log _2 M(n,q)}{n}$$ . A well-known probabilistic argument shows an existence lower bound on $$R_q$$ , namely $$R_q\ge \frac{1}{q-1}\log _2\left( \frac{1}{1-q!/q^q}\right) $$ (Fredman and Komlós, SIAM J Algebr Discrete Methods 5(1):61–68, 1984; Körner, SIAM J Algebr Discrete Methods 7(4):560–570, 1986). This is still the best-known lower bound till now except for the case $$q=3$$ (Körner and Marton, Eur J Comb 9(6):523–530, 1988). The improved lower bound of $$R_3$$ was discovered in 1988 and there has been no progress on the lower bound of $$R_q$$ for more than 30 years. In this paper we show that this probabilistic lower bound can be improved for q from 4 to 15 and all odd integers between 17 and 25, and all sufficiently large q.

Axiology and the Problem of Evil

Abstract In this article, I contend that (a) disagreement over the definition of ‘good person’ indicates a challenge for the probabilistic argument from evil (PAE) and (b) the debate between value monism and value pluralism exposes obstacles for the PAE. I also highlight areas for further axiological inquiry with respect to the problem of evil and related problems. My goal is not to argue that the PAE fails, but to examine the axiology of the argument, to investigate some of its vulnerabilities, and to motivate novel evaluations of it by reframing it as an axiological rather than moral issue.

Wong-Zakai type approximations of rough random dynamical systems by smooth noise

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path ω with Hurst index H∈(13,12]. We first construct the approximation ωδ of ω by probabilistic arguments, and then use the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and the approximative system generate a continuous random dynamical systems φ and φδ. As a consequence of the Wong-Zakai approximation of the solution, φδ converges to φ as δ→0.

Probabilistic Truth, Relativism, and Objective Chance

AbstractIn Probabilistic Knowledge Sarah Moss proposes that our credences and subjective probability judgments (SPJs) can constitute knowledge. Mossean probabilistic knowledge is grounded in probabilistic beliefs that are justified, true, and unGettiered. In this paper I aim to address and solve two challenges that arise in the vicinity of the factivity condition for probabilistic knowledge: the factivity challenge and the challenge from probabilistic arguments from ignorance (probabilistic AIs). I argue that while Moss's deflationary solution to the factivity challenge formally works, it leaves us ill-equipped to handle probabilistic AIs. An account of probabilistic knowledge that cannot overcome probabilistic AIs makes knowledge of thoroughly probabilistic contents a rare and unstable phenomenon, at best, or, at worst, impossible. I hold that establishing a metaphysically enriched account of probabilistic truth is therefore mandatory. I go on to develop a truth-conditional approach on probabilistic factivity that is relativistic in its nature and centers on objective chances. I show that while the approach is still compatible with Moss's overall semantics for probabilistic knowledge, it provides us with a simple but forceful answer to probabilistic AIs.

Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity.

The Yang-Mills heat flow with random distributional initial data

We construct local solutions to the Yang–Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato–Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way toward the construction of 3D Yang–Mills measures.

A converse to the neo-classical inequality with an application to the Mittag-Leffler function

We prove two inequalities for the Mittag-Leffler function, namely that the function log E_alpha (x^alpha ) is sub-additive for 0<alpha <1, and super-additive for alpha >1. These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull London Math Soc 2010). For 0<alpha <2, we also show that E_alpha (x^alpha ) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.

Random subcomplexes of finite buildings, and fibering of commutator subgroups of right‐angled Coxeter groups

AbstractThe main theme of this paper is higher virtual algebraic fibering properties of right‐angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type but not , and examples where the RACG is hyperbolic and the kernel is finitely generated and non‐hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz–Norin–Wise involving Bestvina–Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

Ответ Р. Суинберна на критику статьи «Моральная благость Бога»

This paper is a response to the critiques in Filosophia Religii 2022 and 2023 of my paper, “God’s Moral Goodness” (in Filosophia Religii 2022). I argue for the preferability of Aquinas’s way of arguing to God’s nature from the most general observable features of the world, over Anselm’s way adopted by Mark Murphy of drawing out the consequences of God being a “most perfect being”. I claimed against Vladimir Shokhin, that my probabilistic argument leads to the simplest and so most probable kind of God. To Igor Gasparov’s claim that an even simpler kind of God would be one who has no un-realised potentialities, and so one in which his one property of divinity includes all his activity, I argued that on the contrary, such a God would have no free will, and so not be truly omnipotent. To Mark Wynn’s claim that Christianity holds that God does not choose arbitrarily between equally good alternatives, but has preferences which do not merely track the relative goodness states of affairs, I answer that to postulate a God with such preferences simply moves the arbitrariness of God from his actions to his nature, and so postulates a less simple God than I postulate. I argue also against Vladimir Shokhin’s claim that humans do not have the power to discover God’s reasons for permitting horrendous evils, by drawing attention to some features of my own theodicy, which might begin to make it plausible that we do have this power. I reject Mark Wynn’s claim that on my theodicy no possible balance of evil over good observable by humans on earth could constitute evidence against the existence of God, and that in consequence nothing observable by such humans could constitute evidence for the existence of God. Kirill Karpov pointed out that there are vast disagreements about which world states are good and which are evil, and so we are in no position to judge whether my requirement for a cogent theodicy that every evil state is a logically necessary condition for some comparable good state, is satisfied in our world. I argued against him, that the mere existence of moral disagreement does not show that none of us can discern moral truths; and that on the contrary, many moral disagreements are resoluble by the method of “reflective equilibrium”. To Mark Murphy’s claim that God does not have “requiring reasons” to conform to the “familiar welfare orientated” morality, which humans are obliged to follow, I reply that the same very general principles of morality apply both to God and to humans, but their application varies with the degree of power, knowledge and freedom possessed, and the causal power exercised, by humans and by God. To Murphy’s claim that God as “the ultimate source” of the goodness of created beings cannot be constrained by what constitutes their goodness, I object that what constitutes moral goodness is determined in part by paradigm examples of moral goodness, including promoting, and preventing setbacks to, the well-being of created beings. To Vladimir Shokhin’s endeavor to support his view that God is too great for humans to understand his ways, by various quotations from Scripture, I adduce quotations from Scripture which suggest the contrary. None of the contributors expressed a view on my claim in the earlier paper that God is not merely a “best acting” God, but also a “best feeling” God, except for Vladimir Shokhin who agreed with me; and so I welcome this conclusion to our symposium.

Lattice packings through division algebras

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh’s lattice packing result Venkatesh (Int Math Res Notices 2013(7): 1628–1642, 2013). In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. We employ a probabilistic argument to show the existence of lattices in certain dimensions with good packing densities. The approach improves the best known lower bounds on the lattice packing problem for certain dimensions.We work with a moduli space of lattices that are invariant under the action of a finite group, one that can be embedded inside a division algebra. To obtain our existence result, we prove a division algebra variant of the Siegel’s mean value theorem. In order to establish this, we describe a useful description of the Haar measure on our moduli space and a coarse fundamental domain to perform the integration.

Polymer dynamics via cliques: New conditions for approximations

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition—the clique dynamics condition—, which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least 21/α for the hard-core model on bipartite α-expanders.

The M/M/k with Deterministic Setup Times

Capacity management, whether it involves servers in a data center, or human staff in a call center, or doctors in a hospital, is largely about balancing a resource-delay tradeoff. On the one hand, one would like to turn off servers when not in use (or send home staff that are idle) to save on resources. On the other hand, one wants to avoid the considerable setup time required to turn an ''off'' server back ''on.'' This paper aims to understand the delay component of this tradeoff, namely, what is the effect of setup time on average delay in a multi-server system? Surprisingly little is known about the effect of setup times on delay. While there has been some work on studying the M/M/k with Exponentially-distributed setup times, these works provide only iterative methods for computing mean delay, giving little insight as to how delay is affected by k , by load, and by the setup time. Furthermore, setup time in practice is much better modeled by a Deterministic random variable, and, as this paper shows, the scaling effect of a Deterministic setup time is nothing like that of an Exponentially-distributed setup time. This paper provides the first analysis of the M/M/k with Deterministic setup times. We prove a lower bound on the effect of setup on delay, where our bound is highly accurate for the common case where the setup time is much higher than the job service time. Our result is a relatively simple algebraic formula which provides insights on how delay scales with the input parameters. Our proof uses a combination of renewal theory, martingale arguments and novel probabilistic arguments, providing strong intuition on the transient behavior of a system that turns servers on and off.

New upper bounds for the Erdős-Gyárfás problem on generalized Ramsey numbers

AbstractA $(p,q)$ -colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$ -clique. The problem of determining the minimum number of colours, $f(n,p,q)$ , needed to give a $(p,q)$ -colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$ . The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$ , each of which was proved by giving a deterministic construction which combined a $(p,p-1)$ -colouring using few colours with an algebraic colouring.In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$ -cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$ -colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$ -colourings, which would otherwise make this problem intractable for large values of $p$ . In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$ .

Exact solution of an integrable non-equilibrium particle system

We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the non-equilibrium steady-state can be written in the closed form for each N. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: (i) the introduction of a dual absorbing process reducing the problem to a finite number of particles and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Long-range correlations are computed in the finite-volume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.