Pith. sign in

REVIEW 4 major objections 4 minor 33 references

The Monte Carlo Method and New Device and Architectural Techniques for Accelerating It

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read An uncertainty-tracking microarchitecture stores probability distributions in processor state and computes the output distribution of a program in one deterministic pass — measured here as 51–114× faster than Monte Carlo at matched accuracy

desk verdict A vendor-run benchmark claiming 51-114x speedups for uncertainty-tracking hardware over Monte Carlo, but the timing comparison is not apples-to-apples and the headline numbers are not yet established. read the letter →

arxiv 2508.07457 v1 pith:YCCS2KOD submitted 2025-08-10 cs.AR

Janith Petangoda , Chatura Samarakoon , James Meech , Divya Thekke Kanapram , Hamid Toshani , Nathaniel Tye , Vasileios Tsoutsouras , Phillip Stanley-Marbell This is my paper · ORCID
classification cs.AR
keywords MonteCarlomethoduncertaintypropagationuncertainty-trackingmicroarchitectureprobabilitydistributionrepresentationrandomvariategenerationWassersteindistanceconvergence-obliviouscomputingphysics-basedgenerators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the Monte Carlo method — sampling, evaluation, and post-processing — can be accelerated at every stage, and that one stage can be eliminated entirely. It presents physics-based programmable random variate generators (Spot and Grappa) that make the sampling step dramatically faster, then presents Signaloid's UxHw, a microarchitecture that stores a compact representation of a probability distribution in its own state and updates that distribution deterministically as arithmetic executes, so the output density emerges from one pass over the program. Across two benchmarks — a Gaussian-mixture input pushed through a sigmoid, and Poiseuille's law for blood transfusion — the paper measures that UxHw at representation size $r=32$ reaches a Wasserstein distance near $10^{-3}$, while the Monte Carlo runs matching that accuracy (32,000 and 128,000 samples) take 113.85× and 51.53× longer. The significance is architectural: if distributional state lives in the processor the way IEEE-754 floating-point numbers live in registers, uncertainty propagation stops being a sample-count game and becomes a deterministic, convergence-oblivious computation with an accuracy ceiling set by representation size. The paper concedes the boundary: when substantially higher accuracy is required, Monte Carlo remains the method of choice.

What carries the argument

The carrying object is Signaloid's UxHw, an uncertainty-tracking microarchitecture that stores a compact, fixed-bit-width representation of an entire probability distribution — the Telescopic Torques Representation (TTR) — in microarchitectural state and updates it deterministically under each arithmetic operation, transparently to the running program. Its role is to distributions what IEEE-754 is to real numbers: an approximate finite representation whose size $r$ stands in for precision. The analytic machinery is the change-of-variables formula for densities, which shows what an exact single-pass computation of $p_Y$ requires. The paper's organizing frame is the three-step decomposition of

What would settle it

Run both methods on identical hardware — the same two benchmarks on a UxHw core (or its cycle-accurate model) and the same C code on the same machine — and include in UxHw's clock the time to draw samples from its output representation whenever the deliverable is sample-based. If the Wasserstein-matched speedup at $r=32$ falls far below 51×–114× once scoring samples are paid for or silicon is shared, convergence-obliviousness may still hold but the headline factors do not. Separately, check whether increasing $r$ keeps shrinking the Wasserstein distance to the ground truth; if it saturates whi

Watch

Extended reading notes

Core claim

The mathematical spine is the change-of-variables formula: for invertible, differentiable $f$, the density of $Y=f(X)$ is $p_Y(y)=p_X(f^{-1}(y))\,|\det\nabla f^{-1}(y)|$; a machine that tracks $p_X$ through arithmetic can emit $p_Y$ without sampling. UxHw implements this as its fixed-bit-width Telescopic Torques Representation (TTR), where $r$ controls accuracy. Scored by Wasserstein distance against a one-million-sample ground truth, UxHw at $r=32$ gives 0.00167 and 0.00033; Monte Carlo needs 32,000 and 128,000 samples to match — 113.85× and 51.53× more run time. The paper's framing: UxHw is to Monte Carlo what floating-point is to real arithmetic — a bounded, deterministic, convergence-obl

Load-bearing premise

The load-bearing premise is that the timing comparison is apples-to-apples: UxHw's run times come from the vendor's cloud engine (release 2.6), while Monte Carlo runs natively on an Apple M1 Pro, and the million samples drawn from UxHw's output only for scoring are excluded from its clock, while Monte Carlo's clock includes generating its own output samples; the authors also concede the result assumes the accuracy at $r=32$ is "sufficient."

Editorial extensions

If this is right

  • Uncertainty propagation becomes a single deterministic pass: the paper names Gaussian-process posterior prediction with uncertain inputs as a direct target, and its own Poiseuille benchmark is a medical flow model — tens of microseconds per computation where Monte Carlo needs milliseconds.
  • The accuracy-versus-cost trade-off changes shape: Monte Carlo error falls only as $1/\sqrt{n}$ with sample count (halving variance requires quadrupling samples), while UxHw error falls with representation size $r$; on both benchmarks the Pareto-optimal point is $r=32$, so the architecture wins at moderate accuracy.
  • Outputs are reproducible: because the computation is deterministic, run-to-run scatter comes only from the million-sample scoring draws, not from the computation itself — a property the paper ties to consistency and safety in repeated use.
  • The gap widens as Monte Carlo is pushed: matching UxHw at $r=32$ costs 113.85× and 51.53× more time; beating its accuracy by one or two standard deviations costs 411×–732× more on the first benchmark and 160× on the second — until UxHw's own accuracy ceiling binds.
  • The deliverable is a full distribution, not a summary: UxHw's representation carries the output density's shape, whereas the paper notes its Monte Carlo comparison computes only mean and variance, which cannot capture non-Gaussian structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper does not run: put both sides on identical hardware and charge UxHw for drawing samples from its output representation whenever the deliverable is itself a sample set or histogram; the methods section (Appendix C.1) references supplementary material with 'further measures to ensure that our results were meaningful,' but that material is not included. The 51×–114× fact
  • The Appendix F histograms show UxHw putting more density at the mode and less in the tails than the ground truth, even at large $r$. If that bias belongs to the representation rather than to these two benchmarks, the general claim to test is bounded representation error, and fat-tailed or strongly multi-modal inputs are the natural stress case.
  • The paper's IEEE-754 analogy invites a research program it opens but does not start: a theory of representation-error bounds for composed operations on distributional state, analogous to floating-point error analysis, so users would know in advance how much accuracy a given $r$ buys.
  • A competitor the paper does not benchmark: quasi-Monte Carlo and other low-discrepancy methods also attack the $1/\sqrt{n}$ variance wall while staying inside the sampling paradigm. Comparing UxHw against them would isolate how much of the measured gain comes from the architecture and how much from abandoning sampling altogether.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper decomposes the Monte Carlo method into sampling, evaluation, and post-processing; reviews two earlier physics-based programmable non-uniform random variate generators (Spot and Grappa) from the same group; and describes Signaloid's UxHw, a microarchitecture that represents and propagates probability distributions in processor state using a Telescopic Torques Representation (TTR). The empirical section compares UxHw (commercial cloud engine, release 2.6) with a C implementation of traditional Monte Carlo running on an Apple M1 Pro for two applications: a Gaussian-mixture/sigmoid convergence challenge and Poiseuille's law for blood transfusion. At representation size r=32, UxHw is reported to reach Wasserstein distances 0.00167 and 0.00033 in 0.020 ms and 0.173 ms, respectively, while Monte Carlo reaches similar distances with 32,000 and 128,000 samples in 2.277 ms and 8.914 ms, giving claimed speedups of 113.85x and 51.53x. The authors conclude that UxHw is convergence-oblivious and can replace Monte Carlo when the achieved accuracy is sufficient.

Significance. If the reported speedups were reproducible, uncertainty-tracking microarchitectures would be a significant advance: distributional state in the datapath removes the sample-count dependence of Monte Carlo and gives deterministic, converged outputs at a fixed representation size. The paper's expository framework, the three-step decomposition, and the change-of-variables derivation in Appendix A are useful, and the authors are candid that UxHw has an accuracy ceiling. However, the quantitative evidence presented here does not yet support the claimed one-to-two-order-of-magnitude speedups. The timing comparison mixes platforms and runtime-accounting conventions, the r=32 anchor is chosen post hoc from the same measured frontier, and the referenced supplementary material is absent. Independent validation on identical hardware, with complete end-to-end accounting and pre-specified accuracy criteria, is needed.

major comments (4)
  1. [Appendix C.3 / Table 1] The headline speedup figures are based on UxHw runs on Signaloid's commercial cloud engine (release 2.6) and Monte Carlo runs natively on an Apple M1 Pro. The manuscript gives no CPU, clock, memory, or virtualization details for the cloud engine. Because the claim is architectural, the 113.85x and 51.53x ratios in Table 1 are not established as UxHw-vs-Monte Carlo speedups; they are confounded by hardware and environment differences. The comparison should be repeated on the same machine or at least on the same cloud instance type with the same compiler and library stack for both methods.
  2. [Appendix C.1 / C.3] Runtime accounting is asymmetric. Monte Carlo's reported time includes generating the n output samples, while UxHw's reported time excludes the 1,000,000-sample generation from the TTR representation that is used only to compute the Wasserstein distance. For r=32, the reported UxHw times are 0.020 ms and 0.173 ms; drawing 1,000,000 samples is likely to dominate these figures. The paper argues the sampling is 'not part of a typical use case,' but if an application requires concrete samples, that cost cannot be excluded. Please report both UxHw times with and without output sampling and provide an end-to-end comparison.
  3. [Section 6 / Appendix F.1.1] The r=32 anchor is selected post hoc. Appendix F.1.1 says 'we chose r = 32 to be the best overall configuration' because r>32 improves Wasserstein distance less than it worsens runtime, and F.1.2 makes a similar choice. This selection is made from the same measured frontier that generates the headline speedup, and no quantitative objective (e.g., a Wasserstein-runtime cost function or a target accuracy) is defined in advance. Different anchors, such as r=16 or r=64, would produce different speedup ratios. The selection rule should be specified before seeing the results, or the full Pareto frontier should be treated as the primary result.
  4. [Appendix C.1] The manuscript states: 'We took further measures to ensure that our results were meaningful; these are detailed in the supplementary material.' The supplementary material is not present in the submission. Because the timing methodology is already a central concern, this missing reference blocks verification of the experimental protocol. Please provide the supplementary material, or remove the reference and include the measures in the paper itself.
minor comments (4)
  1. [Section 4] In the derivation of E[f(X)], the middle equality writes y pY(f^{-1}(y)) |det(gradient f^{-1}(y))| dy. As written this is false; the integrand should be y pX(f^{-1}(y)) |det(gradient f^{-1}(y))| dy, with pY(y) appearing only after applying Theorem 1. Please correct the subscript.
  2. [Appendix F, Table 2] The table caption and header refer to 'the factor increase in dynamic instructions required,' but the table contains no dynamic-instruction counts. Either add the missing column or remove the phrase from the header/caption.
  3. [Figure 2 caption] The caption lists the subfigures as '8 (b), 256 (b), 1024 (c), and 128,000 (d)'; the subfigure labels should be sequential, so the second occurrence of (b) is a typo and the 128,000-sample subfigure is not labeled consistently.
  4. [Section 5 / Appendix E.1] The Gaussian-mixture sampling procedure is not fully specified. The paper says the mixture has weights 0.6 and 0.4, but it does not state how this mixture is sampled in the C implementation (e.g., component selection, seed management, and whether the ziggurat generator is used for each component). Please provide the code or a precise algorithmic description.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the headline speedup comparison is an empirical measurement, not a derivation whose conclusion is equivalent to its inputs.

full rationale

The paper's claimed derivation chain is: (1) Monte Carlo is decomposed into sampling, evaluation, and post-processing; (2) the change-of-variables formula gives the density pY of Y=f(X); (3) UxHw is presented as a microarchitecture that tracks distributional state and, by prior work [5,6], computes such densities; (4) runtime and Wasserstein accuracy are measured for UxHw and for traditional Monte Carlo on two applications. No step defines a quantity in terms of the target conclusion. The self-citations to Spot [9], Grappa [10], and UxHw [5,6] are background capability claims, not the source of the new speedup numbers: the speedups in Table 1 and Section 6 are measured with a stated protocol (30 repetitions, gettimeofday, same C code, no parallelization), and the r=32 configuration is selected from the reported Pareto data, not fitted to produce the speedup. The main concerns are experimental fairness, not circularity: UxHw runs on Signaloid's commercial cloud engine while Monte Carlo runs on an Apple M1 Pro (Section C.3), and the 1,000,000-sample scoring time for UxHw is excluded (Section C.3). These affect external validity of the absolute speedup factors, but they do not make the comparison equivalent to its inputs by construction. The missing supplementary material referenced in C.1 is also a completeness limitation, not a circular derivation. Under the rules requiring explicit reduction of a result to its own inputs, no circular step is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

Everything load-bearing in this paper traces to the authors' own prior work (Spot, Grappa, UxHw) and to assumptions about their product's fidelity and timing. The benchmark adds three hand-chosen scales (tested representation sizes, hand-built mixture/sigmoid task, 10^6-sample ground truth). No new physical or mathematical entity is introduced by this paper; UxHw is presented as established prior work with a commercial implementation.

free parameters (3)
  • UxHw representation size r = 32 (best overall per measured frontier; range 16-2048)
    The paper tests r in {16..2048} and then selects r=32 as the best accuracy-time configuration; every headline speedup factor is anchored to this post hoc choice (Section 6, F.1.1, F.1.2).
  • Convergence Challenge benchmark parameters = mixture weights 0.6/0.4, component means 2 and -1, standard deviations 0.5 and 1.0, sigmoid shift (x-1)
    Chosen by hand (Equations 2-3, Section 5.1) to create a bimodal input whose modes map through a sigmoid, deliberately producing the small-sample modal-bias failure the paper wants to show.
  • Ground-truth and scoring sample scale = 1,000,000 Monte Carlo samples
    Both the ground-truth distribution and the Wasserstein scoring of UxHw outputs use 10^6 samples (Sections 6, C.2). The reported Wasserstein estimates include the noise of this choice.
assumptions (5)
  • standard math Change-of-variables formula for probability densities (Theorem 1)
    Used in Section 4 to argue that access to pY makes E[f(X)] computable without Monte Carlo. Requires f invertible and differentiable, which holds for the two benchmark transforms.
  • domain assumption UxHw's TTR representation tracks the true output distribution within its representation budget
    The central premise of Section 4: distributional microarchitectural state evolves 'transparently to the applications'. Grounded only in the authors' prior papers [5,6] and the commercial implementation [23]; no independent validation is offered in this paper.
  • domain assumption The chosen MC baseline (C rand + GSL ziggurat, single-threaded, -O3) is a representative baseline
    Section 5. Variance-reduced MC schemes (quasi-MC, stratified sampling, antithetic variates) would close part of the measured gap; the speedup factors are specific to this vanilla baseline.
  • domain assumption Cloud-based UxHw run times are comparable to native local MC run times
    Section 5 and C.3: UxHw runs on a commercial cloud engine (release 2.6) while MC runs on an Apple M1 Pro; UxHw's 10^6-sample output scoring is excluded from its run time while MC's run time includes generating its output samples. Neither assumption is validated, and the promised supplementary timing detail (C.1) is absent.
  • domain assumption 1,000,000-sample Monte Carlo is a valid ground truth for Wasserstein scoring
    Section C.2. The ground truth is itself a sampled estimate; Wasserstein distances below about 10^-4 approach the noise floor of this reference.
invented entities (1)
  • Signaloid UxHw (uncertainty-tracking microarchitecture with TTR distribution representation) independent evidence
    purpose: Stores a probability distribution as microarchitectural state and propagates it through arithmetic operations in one pass, so the output density pY is produced without Monte Carlo sampling (Section 4).
    It has falsifiable handles: the prior Laplace/UxHw publications [5,6], the commercial cloud engine [23], and measurable output distributions that can be compared with analytic propagation. Within this paper, however, all evidence is produced by the authors' own group and company for their own product.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Monte Carlo Method and New Device and Architectural Techniques for Accelerating It." pith.science (2026). https://pith.science/paper/YCCS2KOD

@misc{pith2026250807457,
  author       = {Pith},
  title        = {Pith review of: The Monte Carlo Method and New Device and Architectural Techniques for Accelerating It},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCCS2KOD}},
  note         = {Machine review of arXiv:2508.07457}
}
read the original abstract

Computing systems interacting with real-world processes must safely and reliably process uncertain data. The Monte Carlo method is a popular approach for computing with such uncertain values. This article introduces a framework for describing the Monte Carlo method and highlights two advances in the domain of physics-based non-uniform random variate generators (PPRVGs) to overcome common limitations of traditional Monte Carlo sampling. This article also highlights recent advances in architectural techniques that eliminate the need to use the Monte Carlo method by leveraging distributional microarchitectural state to natively compute on probability distributions. Unlike Monte Carlo methods, uncertainty-tracking processor architectures can be said to be convergence-oblivious.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 31 canonical work pages

  1. [1]

    Rogers and M

    S. Rogers and M. Girolami, A first course in machine learning. Chapman and Hall/CRC, 2016

    work page 2016

  2. [2]

    C. M. Bishop and N. M. Nasrabadi, Pattern recognition and machine learning, vol. 4. Springer, 2006

    work page 2006

  3. [3]

    Introduction to monte carlo methods,

    S. Weinzierl, “Introduction to monte carlo methods,” arXiv preprint hep-ph/0006269, 2000

    arXiv 2000

  4. [4]

    Introduction to monte carlo simulation,

    R. L. Harrison, “Introduction to monte carlo simulation,” in AIP conference proceedings , vol. 1204, pp. 17–21, American Institute of Physics, 2010

    work page 2010

  5. [5]

    The laplace microarchitecture for tracking data uncertainty and its implementation in a risc-v processor,

    V . Tsoutsouras, O. Kaparounakis, B. Bilgin, C. Samarakoon, J. Meech, J. Heck, and P. Stanley- Marbell, “The laplace microarchitecture for tracking data uncertainty and its implementation in a risc-v processor,” in MICRO-54: 54th Annual IEEE/ACM International Symposium on Microarchitecture, pp. 1254–1269, 2021

    work page 2021

  6. [6]

    The signaloid’s uxhw microarchitecture for tracking data uncertainty,

    V . Tsoutsouras, O. Kaparounakis, C. Samarakoon, B. Bilgin, J. Meech, J. Heck, and P. Stanley- Marbell, “The signaloid’s uxhw microarchitecture for tracking data uncertainty,”IEEE Micro, vol. 42, no. 4, pp. 78–86, 2022

    work page 2022

  7. [7]

    A note on the generation of random normal deviates,

    G. E. Box, “A note on the generation of random normal deviates,”Ann. Math. Statist., vol. 29, pp. 610–611, 1958

    work page 1958

  8. [8]

    Univariate distribution relationships,

    L. M. Leemis and J. T. McQueston, “Univariate distribution relationships,” The American Statistician, vol. 62, no. 1, pp. 45–53, 2008

    work page 2008

Show all 33 references
  1. [9]

    Efficient programmable random variate generation ac- celerator from sensor noise,

    J. T. Meech and P. Stanley-Marbell, “Efficient programmable random variate generation ac- celerator from sensor noise,” IEEE Embedded Systems Letters , vol. 13, no. 3, pp. 73–76, 2020

    2020

  2. [10]

    A system for generating non- uniform random variates using graphene field-effect transistors,

    N. J. Tye, J. T. Meech, B. A. Bilgin, and P. Stanley-Marbell, “A system for generating non- uniform random variates using graphene field-effect transistors,” in2020 IEEE 31st International Conference on Application-specific Systems, Architectures and Processors (ASAP), pp. 101...

    2020

  3. [11]

    J. T. Meech, Electron Tunneling Noise Based Programmable Random Variate Accelerator. PhD thesis, Apollo - University of Cambridge Repository, 2023

    2023

  4. [12]

    Tye, Analog-Domain Machine Learning Computations Exploiting 2D Materials Properties

    N. Tye, Analog-Domain Machine Learning Computations Exploiting 2D Materials Properties. PhD thesis, Apollo - University of Cambridge Repository, 2022

    2022

  5. [13]

    Ern and J.-L

    A. Ern and J.-L. Guermond, Theory and practice of finite elements, vol. 159. Springer Science & Business Media, 2013

    2013

  6. [14]

    M. P. Deisenroth, A. A. Faisal, and C. S. Ong, Mathematics for machine learning. Cambridge University Press, 2020

    2020

  7. [15]

    IEEE standard for floating-point arithmetic,

    IEEE, “IEEE standard for floating-point arithmetic,” IEEE Std 754-2019 (Revision of IEEE 754-2008), pp. 1–84, 2019

    2019

  8. [16]

    What every computer scientist should know about floating-point arithmetic,

    D. Goldberg, “What every computer scientist should know about floating-point arithmetic,” ACM computing surveys (CSUR), vol. 23, no. 1, pp. 5–48, 1991

    1991

  9. [17]

    M. P. Deisenroth, Efficient reinforcement learning using Gaussian processes , vol. 9. KIT Scientific Publishing, 2010

    2010

  10. [18]

    Pilco: A model-based and data-efficient approach to policy search,

    M. Deisenroth and C. E. Rasmussen, “Pilco: A model-based and data-efficient approach to policy search,” in Proceedings of the 28th International Conference on machine learning (ICML-11), pp. 465–472, 2011

    2011

  11. [19]

    The gnu c library reference manual: for version 2.38,

    S. Loosemore, R. Stallman, R. McGrath, A. Oram, and U. Drepper, “The gnu c library reference manual: for version 2.38,” Free Software Foundation, 2022

    2022

  12. [20]

    Noise generators for the simulation of digital communi- cation systems,

    M. F. Schollmeyer and W. H. Tranter, “Noise generators for the simulation of digital communi- cation systems,” ACM SIGSIM Simulation Digest, vol. 21, no. 3, pp. 264–275, 1991

    1991

  13. [21]

    Galassi, J

    M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi, and R. Ulerich, GNU scientific library. Network Theory Limited Godalming, third ed., 2009

    2009

  14. [22]

    Llvm: a compilation framework for lifelong program analysis & transformation,

    C. Lattner and V . Adve, “Llvm: a compilation framework for lifelong program analysis & transformation,” in International Symposium on Code Generation and Optimization, 2004. CGO 2004., pp. 75–86, 2004. 7

    2004

  15. [23]

    Signaloid cloud compute engine

    “Signaloid cloud compute engine.” https://signaloid.com, 2024

    2024

  16. [24]

    Mathematical methods of organizing and planning production,

    L. V . Kantorovich, “Mathematical methods of organizing and planning production,”Manage- ment science, vol. 6, no. 4, pp. 366–422, 1960

    1960

  17. [25]

    J. L. M. Poiseuille, Recherches sur les causes du mouvement du sang dans les vaisseaux capillaires, vol. 7. Impr. royale, 1839

  18. [26]

    B. S. Massey and J. Ward-Smith, Mechanics of fluids, vol. 1. Crc Press, 1998

    1998

  19. [27]

    Principles of physics in surgery: the laws of flow dynamics physics for surgeons—part 1,

    A. Srivastava, A. Sood, S. P. Joy, and J. Woodcock, “Principles of physics in surgery: the laws of flow dynamics physics for surgeons—part 1,” Indian Journal of Surgery, vol. 71, pp. 182–187, 2009

    2009

  20. [28]

    Blood rheology: key parameters, impact on blood flow, role in sickle cell disease and effects of exercise,

    E. Nader, S. Skinner, M. Romana, R. Fort, N. Lemonne, N. Guillot, A. Gauthier, S. Antoine- Jonville, C. Renoux, M.-D. Hardy-Dessources, et al., “Blood rheology: key parameters, impact on blood flow, role in sickle cell disease and effects of exercise,”Frontiers in physiology, ...

    2019

  21. [29]

    Measuring real-time blood viscosity with a ventricular assist device,

    W. Hijikata, T. Maruyama, Y . Suzumori, and T. Shinshi, “Measuring real-time blood viscosity with a ventricular assist device,” Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, vol. 233, no. 5, pp. 562–569, 2019

    2019

  22. [30]

    A geometric derivation of the cantor distribution,

    B. Presnell, “A geometric derivation of the cantor distribution,” The American Statistician , vol. 76, no. 1, pp. 73–77, 2022

    2022

  23. [31]

    V . I. Bogachev and M. A. S. Ruas,Measure theory, vol. 1. Springer, 2007

    2007

  24. [32]

    Casella and R

    G. Casella and R. L. Berger, Statistical inference. Cengage Learning, 2021

    2021

  25. [33]

    SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y . Fen...

    2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.