diffusion models

Application Of Fractional Calculus in Modeling and Reducing Road Accidents

Abstract Traditional mathematical models of traffic flow and driver behavior rely on integer-order calculus, which assumes localized, instantaneous changes. However, real-world traffic systems exhibit strong memory effects, non-local interactions, and anomalous diffusion. This paper explores the application of fractional calculus utilizing non-integer orders in reducing road accidents. By integrating fractional derivatives into traffic flow dynamics, viscoelastic tire-road friction models, and advanced driver assistance systems (ADAS), we demonstrate how capturing hereditary properties can optimize highway design, vehicular control, and active safety systems to actively prevent collisions.

Accelerating Materials Discovery: A Review of Machine Learning in X‐Ray Absorption Spectroscopy

X‐ray absorption spectroscopy (XAS) is a critical technique for probing the local structural and electronic properties of materials. Advanced synchrotron radiation facilities generate complex, high‐dimensional spectra, which pose significant challenges for traditional analysis methods while simultaneously offering unprecedented opportunities for machine learning (ML). This review systematically elaborates how ML models are driving the transformation of XAS data analysis. We not only cover supervised and unsupervised learning methods for spectral classification and clustering but also delve into cutting‐edge deep learning architectures. These include graph neural networks for precise “structure‐to‐spectra” mapping and diffusion models for generative tasks and structure prediction. We provide a comprehensive overview of the key challenges in data‐driven XAS, including the feature engineering of spectra and structures, strategies for solving the “spectra‐to‐structure” Inverse Problem, and Sim2Real methods for bridging the “Domain Gap” between simulated and experimental data. Furthermore, we emphasize the importance of model eXplainable artificial intelligence and uncertainty quantification for building trust in scientific research. Finally, this review looks ahead to the future of the field, driven by materials informatics and autonomous experiments. Active Learning techniques, represented by Bayesian optimization, are pioneering “self‐driving” smart XAS experiments, which will greatly accelerate the discovery and design of new materials.

The Cognitive Familiarity Supremacy Theory (CFST)

The Cognitive Familiarity Supremacy Theory (CFST) proposes that a substantial portion of human certainty, ideological attachment, collective identity formation, and perceived superiority emerges not primarily from objective rational evaluation, but from repeated familiarity encoding mechanisms operating within subconscious cognitive architectures.This framework argues that repeated environmental exposure, social conditioning, emotional reinforcement, identity fusion, symbolic repetition, and institutional amplification collectively construct familiarity-driven epistemic structures that are frequently mistaken for objective truth, rational certainty, or universal superiority. The theory synthesizes and mathematically formalizes principles from cognitive neuroscience, psychology, sociology, political theory, philosophy of mind, epistemology, systems theory, information theory, complexity science, behavioral economics, evolutionary biology, communication studies, artificial intelligence, anthropology, cybernetics, and cultural theory into a unified explanatory framework.CFST introduces a comprehensive causal chain model:Repeated Exposure \rightarrow Subconscious Encoding \rightarrow Identity Fusion \rightarrow Emotional Reinforcement \rightarrow Bias Formation \rightarrow Perceived Superiority.The theory proposes that human cognition operates through familiarity-weighted interpretive systems, where the subjective sensation of certainty often emerges from accumulated familiarity intensity rather than objective verification.The framework further integrates: Bayesian epistemology, predictive processing, Hebbian learning, social identity theory, information entropy, network propagation, algorithmic amplification, memetic evolution, cultural conditioning, political hegemony, and neurocognitive attractor-state dynamics.The theory also develops: formal mathematical models, belief topology equations, dynamic systems formulations, stochastic familiarity propagation systems, agent-based ideological simulations, network-theoretical belief diffusion structures, and computational cognitive equilibrium equations.At the civilizational level, CFST proposes that societies are partially constructed upon collectively reinforced familiarity architectures rather than purely objective truth systems. At the individual level, it explains ideological rigidity, nationalism, fanaticism, cultural supremacy perception, identity-protective cognition, and epistemic polarization.Finally, the theory proposes that genuine epistemic liberation requires conscious disruption of subconscious familiarity monopolies through critical reasoning, diversity exposure, meta-cognitive awareness, and reflective epistemological reconstruction.

A new integrated conditional positive definite Gaussian kernel with application in the solution of the Bates jump-diffusion model

TriM-D: Accelerating 2-D/3-D Diffusion Generation Via Triple-Mixed Quantization and Sparsity-Aware Dual-Branched Computation

Diffusion models have shown marked advancements in 2-D generation and have also become focal points in 3-D generation via consistent multiview image generation. However, the computation and memory demands hinder their real-time deployment on mobile and edge devices. Moreover, the reduction of diffusion timesteps leads to a decrease in interstep similarity, thereby making previous methods ineffective in optimizing computation. The varied layer quantization sensitivity and optimal format in diffusion models also present challenges for traditional quantization methods to achieve efficient memory compression. To address these issues, we first introduce <inline-formula> <tex-math notation="LaTeX">${M}^{3}$ </tex-math></inline-formula> quantization, a mixed-precision, mixed-format, and mixed-granularity quantization framework that allocates optimal precision and format to different data with a unified FP8 computation flow for efficient implementation. Building upon <inline-formula> <tex-math notation="LaTeX">${M}^{3}$ </tex-math></inline-formula> quantization, we present TriM-D, a hardware accelerator designed to optimize computation. It features sparsity-aware dual-branched FP8-MAC units that introduce a dual-branch path and an efficient approximate adder to fully utilize bit-level sparsity of <inline-formula> <tex-math notation="LaTeX">${M}^{3}$ </tex-math></inline-formula>-quantized data. Such an FP8-MAC scheme preserves both efficiency and numerical accuracy. Our experiments demonstrate that TriM-D significantly outperforms the state-of-the-art (SOTA) diffusion accelerator Cambricon-D (Camb-D), achieving an average reduction of 56.5% in memory access and 80.6% in computation cost. In addition, TriM-D provides a <inline-formula> <tex-math notation="LaTeX">$55\times $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$6.72\times $ </tex-math></inline-formula> improvement in energy efficiency and <inline-formula> <tex-math notation="LaTeX">$3.69\times $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$2.67\times $ </tex-math></inline-formula> speedups over NVIDIA A100 and Camb-D.

A reduced-order Galerkin-projected flame transfer model for coupled dual jet diffusion flames based on mixture fraction

Adsorption of Ciprofloxacin onto Lotus Stem-Derived Biochar: Effects of Solution pH, Temperature, and Coexisting Cations

The discharge of ciprofloxacin (CFX) into aquatic environments poses a threat to ecosystem health. In this study, biochar derived from lotus stem (Nelumbo nucifera) was evaluated as a low-cost adsorbent for CFX removal from aqueous solution. The biochar was characterized by SEM/EDX, BET surface area, FTIR, and point of zero charge (pHpzc). Batch sorption experiments were conducted to examine the effects of solution pH (3, 7, and 10), temperature (20, 26, and 32 °C), and coexisting cations. The kinetics of sorption could be explained using a two-stage intraparticle diffusion model, and the equilibrium isotherms were fitted to the Langmuir model. The maximum adsorption capacity (Qmax) was 22.96 mmol kg⁻¹ at pH 7 and 26 °C. Solution pH influenced the sorption capacity, yet substantial sorption was maintained across the entire pH range studied. Thermodynamic analysis revealed an endothermic, entropy-driven and spontaneous process at all temperatures. Divalent Ca²⁺ suppressed Qmax by 49.0% through competition with CFX for adsorption sites. The sorption process under pH 7 was facilitated by electrostatic attraction, hydrogen bonding, and π–π stacking interactions. Overall, unmodified lotus stem biochar proved to be a promising and cost-effective adsorbent for CFX removal from water.

Trustworthy RAG: An Evaluation Agent for Detecting Misinformation and Knowledge Poisoning in Generative AI Systems

Retrieval-Augmented Generation (RAG) grounds Large Language Model (LLM) outputs in external knowledge, but RAG systems usually trust whatever they retrieve, creating a Security-Reliability Gap: high semantic relevance does not guarantee factual truth. Adversaries exploit this through knowledge poisoning, inserting malicious documents to cause targeted misinformation. We propose an Evaluation Agent, middleware that combines Natural Language Inference (NLI) factual verification, a five-signal poison detector with relevance-weighted aggregation, and a Trust Index T = 0.4 F + 0.35 C + 0.25 (1 - P ) with a non-linear dampener for high-contamination contexts. On TruthfulQA with Llama 3.3 70B, the agent reaches 91% accuracy and 100% precision, with 100% recall on instruction injection, while in-place edits, such as entity swaps, remain hard to detect. Across three LLMs the Trust Index stays discriminative, with a Receiver Operating Characteristic Area Under the Curve (ROC-AUC) of 0.73 to 0.81; generation style matters more than model size, and per-LLM threshold calibration restores baseline competitive accuracy, whereas a weaker FEVER result shows that cross-dataset generalization requires domain-specific calibration. In a software-engineering use case, a secure-coding assistant over guidance from the Open Worldwide Application Security Project (OWASP) Top 10 and the Common Weakness Enumeration (CWE), the agent reliably blocks instruction injection of unsafe advice (F1 92%), while contradiction and subtle semantic weakening remain hard. Throughout, the agent measures detection of poisoned context before generation, not whether the LLM adopts the injected misinformation. We release the proposed approach, attack generator, and experimental artifacts at the link: https://github.com/GPT-Laboratory/TrustworthyRAG.

The Economics of Artificial Intelligence: Scaling, Verification, Assignment, Capital, Growth, and Value

This paper develops the economics of artificial intelligence as a single connected structure, from the physics of the production function to the aggregate growth constraint and the valuation of the firms building and adopting it. Part I derives the cost of capability from scaling laws, shows why deployed models are systematically overtrained, and estimates the task-success slope directly from 23,235 public evaluation runs: $\hat\beta=0.83$ with no detectable release-date trend. Part II treats market structure: minimum efficient scale, the two-tier equilibrium in which open weights contest the trailing edge but never the frontier, and inference as a capacity-constrained short-run market that rations rather than prices. Part III is the core. We replace the standard automation assignment rule with one that prices reliability, obtaining an automation calendar $t_{\mathrm{aut}}=t_{1/2}+(\tau/\beta)\log_2\gamma$ in which verification cost, not task difficulty, sets the date; derive optimal checkpoint spacing $k^\star\approx\sqrt{v_{\mathrm{ver}}/\lambda}$; and prove the exact best-of-$k$ result. Against a sound verifier, sampling divides the reliability lag by $k$ in the small-$k$ regime and does better outside it; against an unsound verifier, it leaves an error floor that no amount of sampling removes. Part IV aggregates: diffusion inherits its time dispersion from verification costs, and revenue growth is governed by the density of tasks at the current threshold. Part V proves a Baumol bound --- with elasticity of substitution below one, aggregate growth converges to that of the least automatable essential input --- and states three jointly necessary conditions for explosive growth. Part VI treats measurement, policy, and financial markets. Part VII states the investment bridge: technological importance, industry profit, and security return are distinct objects, and a coherent valuation must respect the automation calendar, rent migration, capital consumption, and expectations already in price. The full valuation architecture is reserved for a separate companion paper. Part VIII states eighteen open problems.

A Coupled LBM‐FDEM Framework for Simulating Solute Transport and Mineral Dissolution Under H‐M‐C Conditions

Abstract The injection of CO 2 into geological reservoirs can acidify the reservoir environment, leading to the dissolution of mineral grains and increasing the risk of CO 2 leakage. A deeper understanding of these micro‐scale dissolution mechanisms is therefore essential for the safety of geological CO 2 sequestration. To address this challenge, we develop a multi‐field coupled simulation framework that integrates the Lattice Boltzmann Method (LBM) with the Finite‐Discrete Element Method (FDEM) to model solute transport and mineral micro‐structure dissolution. The framework incorporates a geometric mapping detection scheme and an improved momentum exchange algorithm to capture fluid‐solid interactions. Furthermore, a novel updating scheme for dissolution rates and solute concentrations is proposed, explicitly accounting for mass conservation and complex mineral morphology. A dynamic geometric topology updating algorithm is also proposed to track the morphological evolution of mineral grains. The accuracy and capability of the LBM‐FDEM framework are validated through several numerical benchmarks, including particle settling, solute release from a moving particle, dissolution of polymorphic particles, and reactive transport within a non‐uniform channel, yielding excellent agreement with existing solutions. Finally, this proposed method is applied to investigate the spatiotemporal evolution of fracture dissolution under varying Péclet numbers. The numerical results reveal that under diffusion‐dominated low Péclet regimes, product accumulation restricts dissolution to the upstream inlet, yielding a typical face dissolution pattern. Conversely, strong advection under high Péclet conditions drives rapid reactant penetration, triggering a uniform channelized widening along the entire fracture. Ultimately, this framework demonstrates significant potential for investigating complex hydro‐mechanical‐chemical couplings in subsurface engineering.

Multiscale physicochemical evolution of mined coal links structure transport and reaction kinetics during thermochemical conversion

Global reliance on coal for energy generation and chemical production necessitates physics-based frameworks that move beyond empirical bulk characterization toward predictive and sustainable utilization strategies. Coal is a heterogeneous, reactive porous medium whose structure, physicochemical properties, and reactivity evolve continuously during thermochemical conversion. Coal mining conditions and associated geological and mechanical disturbances may influence pore–fracture architecture, mineral exposure, and transport accessibility, thereby affecting subsequent conversion behavior. This review develops a unified structure–transport–reaction framework that links coal’s multiscale architecture to transport behavior, reaction regimes, and process-scale performance. Particular emphasis is placed on how thermochemical loading governs pore–fracture evolution, mineral redistribution, and interfacial functionality, thereby controlling heat and mass transfer, adsorption–desorption behavior, and heterogeneous reaction kinetics. Unlike conventional constant-property approaches, the framework highlights conversion-dependent transport properties, spatially heterogeneous reactivity, and dynamic transitions between kinetic- and diffusion-controlled regimes. Coal exhibits a hierarchical pore–fracture system spanning micropores (< 2 nm), mesopores (2–50 nm), macropores, and cleat–fracture networks, with permeability typically ranging from 10⁻²¹ to 10⁻¹⁵ m² and thermal conductivity varying between 0.15 and 0.60 W m⁻¹ K⁻¹ depending on coal rank and conversion state. Under thermochemical conditions of 800–1800 °C, continuous structural evolution modifies pore connectivity, mineral distribution, and surface chemistry, leading to significant changes in transport pathways and reaction accessibility. Major conversion routes, including combustion, gasification, liquefaction, and coal-derived carbon-material production, are critically evaluated in terms of transport limitations, reactivity evolution, ash behavior, and mechanical stability. Advances in operando characterization, image-informed pore-scale simulations, multiscale multiphysics modeling, and physics-informed data-driven approaches are assessed for their ability to bridge scales and improve predictive capability. Overall, this review establishes a mechanistic foundation for understanding mined coal as an evolving porous medium and provides a roadmap for integrating multiscale characterization, transport physics, and reactor-scale modeling to advance next-generation coal conversion technologies.

The Four Pi That Catches Molecules Is the Four Pi That Spreads a Field ── Steady Diffusion Is Laplace's Equation: the Flux Through Every Enclosing Sphere Is 12.566370614359, and in Two Dimensions No Steady Absorber Exists ── [Paper 249]

When four pi turns up in a biological setting, is it a coincidence of numbers or the same root? This paper treats exactly one case: the four pi in the rate at which a cell captures molecules is the same solid angle as in Gauss's law. Not by analogy, but because it is the same equation. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): no new mathematical theorem and no new law is claimed. The Smoluchowski capture rate, the receptor count of Berg and Purcell, and the reduction of steady diffusion to Laplace's equation are all standard. No measured value is cited; every number is computed from a definition, and no measurement on a real cell is used. No physiology is discussed; nothing is claimed about the actual size, number or arrangement of receptors. The ratio of one to a thousand in Section 7 is a value put there for the sake of the calculation and belongs to no particular cell. No biological conclusion is drawn; nothing is said about why cells are the size they are or about how evolution acted. The reaction-limited case is not treated; only the diffusion-limited steady state appears, with no binding or unbinding rates. No general account of four pi in biology is attempted; one case is treated. Joining things because their numbers agree is what this paper most wants to avoid: the claim of a shared root rests on the equation being the same, not on the value being the same. The relation to earlier papers. Paper 2 treated the pure solid angle appearing in an inverse-square field; this paper shows the same solid angle in diffusion, from the identity of the equations. Paper 1 established that the exponent is the dimension minus one; the failure in two dimensions in Section 6 is where that exponent becomes zero. Paper 156 traced the flatness inside a spherical shell to harmonicity; this paper uses the same harmonicity. Papers 36 and 37 dissected the six pi of Stokes drag; Section 8 places that six pi beside this four pi as a different root. Paper 44 showed that the power of four pi is a mass dimension and that fixing it is a convention of electromagnetic units; Section 8 puts that conventional four pi third. The setting. Let a sphere of radius a absorb on contact the molecules drifting around it. With a distant concentration and a diffusion coefficient, the amount captured per unit time in the steady state is four pi times the diffusion coefficient times the radius times the concentration (Smoluchowski, 1917). The question is what that four pi is. First, steady diffusion is Laplace's equation. The diffusion equation relates the change of concentration in time to the Laplacian, and in the steady state the time derivative vanishes, leaving only the vanishing of the Laplacian. That is the same equation as electrostatics, so the solutions take the same form and one over r appears in spherical symmetry. The four pi enters when the flux through a sphere is counted, for exactly the reason it appears in Gauss's law: the full solid angle of a sphere is four pi. The harmonicity to which Paper 156 traced the flatness inside a shell is the same harmonicity used here. Second, the flux is the same through every enclosing sphere. Setting the diffusion coefficient, the concentration and the radius to one and measuring the flux from the steady solution, moving the radius from one and a half times to a thousand times gives 12.566370614359172 throughout, the six values spreading by 3.55 times ten to the minus fifteenth. And four pi is 12.566370614359172. That the flux does not depend on the radius is Gauss's law itself; this is not a resemblance but the same consequence of the same equation. Third, solving numerically without using the form of the solution gives the same value. Integrating the radial equation over two million points with the outer boundary at four hundred times the radius gives 12.597865194, which differs from the infinite formula 12.566370614 by 3.1 times ten to the minus second. That is not an error: the ratio is 1.002506259, agreeing with R over R minus a, namely 1.002506266, to 6.7 times ten to the minus ninth. What differs is not the method but the outer boundary being finite, and stretching the boundary tenfold shrinks the gap to exactly a tenth. When a number and a formula disagree, first ask what each of them assumed. Fourth, this is the core of the paper. Capture is proportional to the radius and not to the area. Moving the radius from a tenth to ten, the capture rate grows a hundredfold from 1.256637061 to 125.663706144 while the surface grows ten thousandfold from 0.125663706 to 1256.637061436. Capture per unit area falls from 10.000000000 to 0.100000000, by a factor of a hundred. One naturally expects a wider absorbing surface to catch more, but that is surface-limited thinking. In the diffusion-limited case what decides is not the surface but the rate at which molecules arrive from far away, which is proportional to the radius, since a gradient of one over r multiplied by an area of r squared leaves one power of r. That multiplication is exactly the exponent of Paper 1. Fifth, in two dimensions no steady absorber exists. The same calculation in d dimensions gives a solution containing r to the power two minus d, and at d equal to two that power is zero, the power-law solution disappears and a logarithm takes its place. With the outer boundary at ten times the radius the flux is 0.434294482, at a thousand times 0.144764827, at a million times 0.072382414, at ten to the twelfth 0.036191207, and even at ten to the hundredth 0.004342945 remains. It falls as one over the logarithm and reaches zero only in the limit, yet in that limit the steady solution itself does not exist. A creature in a flat world cannot gather food by diffusion alone; more precisely, it needs a wall at a finite distance. This is where the exponent of Paper 1 becomes one at two dimensions and a logarithm replaces one over r. Sixth, the surface may be left almost bare. The whole surface need not absorb: with small receptors scattered over the sphere, the capture rate is a fraction of that of a perfect absorber (Berg and Purcell, 1977). Putting the relative radius of a receptor at one thousandth, 3142 receptors reach half the rate of a perfect absorber while covering 0.078540 percent of the surface. Ninety percent needs 28275 receptors covering 0.706858 percent, and ninety-nine percent needs 311018 covering 7.775442 percent. Zero point zero eight percent of the surface gives half the rate, and more than ninety-nine percent may be left bare, because a molecule strikes the surface many times by diffusion: missing once, it wanders off and comes back. The ratio of one thousandth is a value put there for the calculation and is not a measurement on any cell. Seventh, three kinds of pi appear in this neighbourhood. The four pi of diffusion capture, of the volume of a sphere and of Gauss's law are all solid angles. In the six pi of Stokes drag the pi is a solid angle but the six has another source (Papers 36 and 37), and the six pi in the Einstein relation is inherited from it. The four pi in Poisson's equation in Gaussian units is a convention of units (Paper 44). The same characters sometimes mean a solid angle and sometimes a convention, and six pi is not one and a half times four pi. The only case in which a shared root is claimed here is diffusion capture, and the claim rests on the same Laplace equation, not on the values agreeing. Closing. The four pi in the rate at which a cell captures molecules is the same solid angle as in Gauss's law. Not because the values agree, but because steady diffusion is Laplace's equation. The same equation gives the same one over r, multiplied by the same spherical area, leaving the same four pi, and the check is that the flux does not depend on the radius. Two things followed: capture scales with the radius and not the area, and no steady absorber exists in two dimensions. Last, the surface may be left almost bare, zero point zero eight percent giving half the rate, because a molecule, by diffusion, comes back again and again. The separator is whether it comes from the same equation; that two values agree shows nothing about a shared root. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 生物の話に 4π が出てきたとき、それは数字の一致なのか、同じ根なのか。本稿が扱うのは一つだけである——細胞が分子を捕らえる速さに現れる 4π は、ガウスの法則の 4π と同じ立体角である。類推ではなく、同じ方程式だからである。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない。スモルコフスキーの拡散捕捉率、ベルクとパーセルによる受容体の勘定、定常拡散がラプラス方程式に帰着することは、いずれも標準的である。測定値を引かない——本稿の数はすべて定義から計算したものであり、実在の細胞の測定値は一つも使っていない。生理学を論じない——受容体の実際の大きさ・数・分布については何も主張しない。第7節の 1000 分の 1 という比は仮に置いた値であって、特定の細胞のものではない。生物学的な結論を引き出さない——細胞がなぜその大きさなのか、進化がどう働いたかについては何も述べない。反応律速の場合を扱わない——本稿が扱うのは拡散律速の定常状態だけであり、結合速度や解離は入っていない。生物に現れる 4π を一般に論じない——扱うのは拡散捕捉の一件だけである。数字が一致していることを根拠に何かを結ぶことは、本稿がもっとも避けたいことである——本稿が同根だと言えるのは、同じ方程式から出ているからであって、値が同じだからではない。 既刊との関係。論文2 は逆二乗場に純粋な立体角が現れることを扱った——本稿は同じ立体角が拡散にも現れることを、方程式の同一性から示す。論文1 は n = d−1 を示した——第6節の二次元の破れはその指数がゼロになる場所である。論文156 は球殻の内部が平らな理由を調和性に帰した——本稿が使うのは同じ調和性である。論文36・37 はストークス抵抗の 6π を解剖した——第8節はその 6π と本稿の 4π が別根であることを並べる。論文44 は 4π の冪が質量次元であり、a = 4π を固定するのが電磁単位の規約だと示した——第8節はその規約としての 4π を三つ目に置く。 設定。半径 a の球が、周囲にただよう分子を触れた瞬間に吸収するとする。遠方の濃度をC0、拡散係数を D とし、定常状態で単位時間に捕らえる量を求めると、答は 4πDaC0 である(スモルコフスキー 1917)。問いは、この 4π が何かである。 第一に、定常拡散はラプラス方程式である。拡散方程式は濃度の時間変化を D 掛ける濃度のラプラシアンと結ぶが、定常状態では時間変化が消えるので、残るのはラプラシアンがゼロという式だけである。これは静電場の方程式と同じ式であり、したがって解も同じ形になり、球対称なら 1 / r が出る。4π が現れるのは球面を通る流束を数えるときで、ガウスの法則で4π が出るのとまったく同じ理由——球の全立体角が 4π だからである。論文156 が球殻の内部の平らさを帰した調和性は、ここで使っている調和性と同じものである。 第二に、どの半径で測っても流束は変わらない。D も C0 も a も 1 と置き、定常解 C(r) = C0(1 − a/r) から半径 r の球面を通る流束を測ると、r を a の 1.5 倍から 1000 倍まで動かして 12.56637061435917