Summary

The concept of infinity is a complex and abstract one, which has challenged mathematicians and philosophers for centuries. In order to measure infinity, various mathematical methods and techniques have been developed over time. One approach is to use ensemble size, which involves creating a large number of models or simulations to get an accurate representation of the system being studied. Studies have shown that increasing the ensemble size can improve the accuracy of predictions, but this needs to be balanced with computational efficiency, particularly in the context of resolving higher resolution and larger ensembles. Another method involves the use of statistical consistency, where members are consistent draws from the same distribution as the observations. This has been shown to be effective in small ensembles of two to four members for standard evaluations. However, the appropriate ensemble size can vary depending on the needs of different users and the forecast range being considered, and research in this area continues. In addition to mathematical methods, the study of epistemological beliefs about infinity has shown that students can struggle to understand and reconcile the concept of infinity with their existing knowledge of finite numbers. Exposing students to new conceptualizations of infinity can provide a valuable strategy for understanding their epistemological beliefs about mathematics. Ultimately, measuring infinity requires a combination of mathematical expertise and philosophical inquiry, and further research is needed to continue advancing our understanding of this complex concept.

Consensus Meter

Then we derive a proof of the uniqueness of the invariant Radon measure for the Markov chain induced on Rd by the left random walk and prove a stronger property of divergence for the discrete time process on Rd induced by the right random walk. The potential kernels of the right and left random walk give the expected numbers of visits when the random walks start from a generic point g Aff(Rd ), and are respectively given by U r 1B = δg U and U l 1B = U δg. We observe that a(Rn ) = a(Ln ) = A1 An is a classical multiplicative random walk on R∗. We will suppose that the random walk is non-degenerate in the sense that there exists no y Rd fixed by X1 and that we are really dealing with affinities and not just with translations, i.e. ∀y Rd : P < 1 and P a(X1 ) = 1 < 1. The transience of the set Cs,t for the right random walk is thus a quite subtle phenomenon, and instead to prove it directly we will show, following the ideas of , that the right random walk can cross the border between Cs,t and its complement only a finite numbers of times. The transience of the random walk with law follows from the transience of random walk with law by duality.

Published By:

S Brofferio - Annales de l'Institut Henri Poincare (B) Probability and …, 2003 - Elsevier

Cited By:

47

Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely large and infinitely small quantities, A suitable extension is the superreal number system described here; an alternative extension is the hyperreal number field used in non-standard analysis which is also mentioned. The superreals In modern mathematics the number line is usually interpreted in terms of the real number system where each number has a decimal expansion a-m10m a-110 a0 a110-1 an10-n with the coefficients ai being digits between 0 and 9. The net result of all these cardinality computations is that, if we consider the number of infinitesimal indivisibles needed to cover a finite interval, then a superreal interval requires as many as there are real numbers in that interval, whilst a 'rational interval with infinitesimals' requires as many as there are rational numbers in that interval. For instance the consonance of intuitions of infinity with infinite measuring number rather than cardinal number in some children requires an answer from those psychologists who see the child's interpretation of number only within a cardinal paradigm.

Published By:

D Tall - Educational Studies in Mathematics, 1980 - Springer

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137

STUDENTS' EPISTEMOLOGIES OF MATHEMATICS Francisco makes a strong argument for the need of more studies investigating secondary school students' epistemological beliefs about mathematics and observes that many findings regarding school students are assumed to be true only because 3 - 131 2018. Umeå, Sweden: PME. Iannone, Rizza, & Thoma they are found to be true for university students and not because they originate from empirical research involving school-age students. The extract below is from one of the focus group interviews with the Italian students: Student 1: I think this is a contradiction - it is a concept which I cannot make mine because it is in contradiction to what I know Interviewer: contradictory because it has both characteristics of infinity and characteristics of finite numbers? Student 1: Yes ... Student 2: If you consider it as an infinite big number it is not contradictory because in the end this is not [the] infinity Student 3: It is one of the characteristics of grossone continuously increasing From this extract emerges a distinct sense of unease on the part of the students and especially of Student 1. We tested whether asking secondary school students to work through a worksheet introducing a new conceptualisation of infinity, unseen and somewhat incompatible with some of their existing knowledge, could provide a strategy suitable to expose secondary school students' epistemological beliefs.

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D Rizza, P Iannone, A Thoma - 2018 - ueaeprints.uea.ac.uk

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12

If there had been only a finite number of rooms, the guest in the last room would have had to move out into interstellar space. Because the hotel had infinitely many rooms, there was space for all, and I was able to move in without depriving any of the cosmic zoologists of his room. "Half the rooms are empty. We won't fulfill the financial plan." Actually, I was not quite sure what financial plan he was talking about; after all, he was getting the fee for an infinite number of rooms, but I nevertheless gave him some advice: "Well, why don't you move the guests closer together; move them around so as to fill all the rooms." This turned out to be easy to do. More generally, if the guest occupies room n in the mth hotel, then if n m he will occupy number2 m, and if n < m, number m2 - n 1." The proposed project was recognized to be the best - all the guests from all hotels would find a place in our hotel, and not even one room would be empty.

Published By:

NY Vilenkin - 2013 - books.google.com

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72

M. Leutbecher Ensemble size: How suboptimal is less than infinity? 13 September 2017 3 Buizza and Palmer Comparison of 2, 4, 8, 16, 32 members Verified against analysis Verified against member Z500 in NH; T63L19 model, initial uncertainty represented with singular vectors, no representation of model uncertainty Careful conclusions that do not rule out increases in skill beyond 32 members. Impact of resolution and ensemble size In December 1996, resolution was increased from T63 to TL159 and ensemble size was increased from 32 to 50 members. In Richardson show that the Brier score also satisfies BSM = BS∞. M. Leutbecher Ensemble size: How suboptimal is less than infinity? 13 September 2017 15 Analytic result for statistically consistent ensembles When members are statistically consistent draws from same distribution as observation, the CRPS for an m-member ensemble satisfies M −1 1 0 CRPSM = 1 E x −x = 1 CRPS∞ 2M M Eqns. Two to four members may be enough for standard evaluations M. Leutbecher Ensemble size: How suboptimal is less than infinity? 13 September 2017 26 Discussion How can we increase ensemble size when we need to increase resolution too? Different users will have different needs, how to obtain a good compromise for all of them? How to increase ensemble size in a computationally efficient way for all forecast ranges from medium-range to extended-range? What is an adequate ensemble size for the reforecasts? Which other proper scores permit the construction of an associated fair score? M. Leutbecher Ensemble size: How suboptimal is less than infinity? 13 September 2017 27..

Published By:

M Leutbecher - Quarterly Journal of the Royal Meteorological …, 2019 - Wiley Online Library

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70

The Proceedings of the Aristotelian Society is an online journal that publishes philosophical papers read at the meetings of the Aristotelian Society, as well as discussion notes on those papers. The journal features contributions from global sources, discussing issues across a wide range of philosophical traditions, including those of current interest. The hardback volume of the journal is published every October, while the online journal is issued quarterly in March, June, and September. The membership lists of the Aristotelian Society are not available for display until 75 years after their initial publication, in compliance with the UK's Data Protection Act of 1998. Oxford University Press is the parent publisher of the journal, and it publishes over 6,000 publications annually, with offices in around 50 countries and a global workforce of over 5,500 people.

Published By:

J Lear - Proceedings of the Aristotelian Society, 1979 - JSTOR

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104

What is the bias introduced by these algorithmic choices for neural networks? What ensures generalization in neural networks? What is the relevant notion of complexity or capacity control? As mentioned above, simply accounting for complexity in terms of the number of parameters, or any measure which is uniform across all functions representable by a given architecture, is not sufficient to explain the generalization ability of neural networks trained in practice. Suggested different norms of network parameters to measure the capacity of neural networks. Dividing all the weights by the same number will scale down the output of the network but does not change the 0/1 loss, and hence it is possible to get a network with arbitrary small norm and the same 0/1 loss. For networks with more hidden units, `2 norm and `1 -path norm increase with the size of the network.

Published By:

B Neyshabur, S Bhojanapalli… - Advances in neural …, 2017 - proceedings.neurips.cc

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996

In the last decades a new challenge took hold: the academic and industrial focus, in the most different fields, extended from developing new technologies to acquire data and more accurate ways to process it, to also developing software environments to properly handle and process the big amount and large variety of available data. As acquiring data becomes easier and more efficient, and data acquisition automation is also finding its way in construction processes, the amount of available data is therefore growing over time, and the biggest challenge is properly handling the acquired data, fusing different data sources together, processing them, and getting valuable and accurate information to be used in the construction monitoring. The goal for the Infinity product is to bring the different sensor data to a geodetically correct and traceable workflow, working in a common reference frame; it uses data dependencies for simplifying the visualization of the various data to the work. Each point of the point cloud has a confidence value as other data source, such as a total station and GNSS observation, and working with all data sources in a project and being able to visualize the relation to the source of the data is removing the uncertainties for trusting the results.

Published By:

M Di Rita, K Hanson - International Archives of the …, 2022 - search.ebscohost.com

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0

The text discusses the problem of disturbance attenuation using measurement feedback for an affine nonlinear system, with a focus on internal stability. Some simplifying assumptions in earlier analyses of this problem are shown to be unnecessary, and it is demonstrated that the solution is linked to the existence of solutions of Hamilton-Jacobi inequalities. The paper was presented at a conference in 1992 and published by the IEEE. In conclusion, the research extends and clarifies previous work on disturbance attenuation, contributing to the body of knowledge on system control and stability. The findings may have practical applications in fields such as robotics, aerospace engineering, and industrial automation.

Published By:

A Isidori - [1992] Proceedings of the 31st IEEE Conference on …, 1992 - ieeexplore.ieee.org

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49

Governance good governance is not an outcome of itself. Governance governance is perhaps, simply the answer to this question. Governance "How well are both your organisation, and the people within your organisation, 'being steered' and how well are they 'steering'?". Measuring most attempts to measure the quality of corporate governance focus on compliance related issues.

Published By:

P Carey - 2020 - comsuregroup.com

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0

We draw out consequences for the resource divergences that delineate the structural hierarchy of ergodic processes and for processes that are themselves hierarchical. Edu [11, 12]. Even if each individual trial is a realization generated by a stationary process with finite memory and exponentially decaying correlations, the resulting process over many trials can be infinitary [3-5]. Why can the past-future mutual information of Bandit processes diverge? The answer is remarkably simple: Bandit processes are nonergodic. Intrinsic complexity characterizations have been most constructively and thoroughly developed for finitememory, finite-randomness processes, despite the fact that many important natural processes are infinitary. What is known about information divergences in ergodic processes? An information divergence hints at a structural level in the space of ergodic processes; a space that is itself highly organized.

Published By:

JP Crutchfield, S Marzen - Physical Review E, 2015 - APS

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19

The Philosophical Review is a journal edited by the faculty of the Sage School of Philosophy at Cornell University. It has been in continuous publication since 1892 and aims to provide an impartial and catholic tone and spirit that is not influenced by any institution, sect, or interest. Duke University Press publishes approximately one hundred books per year and thirty journals, primarily in the humanities and social sciences. The publisher has developed a strong reputation in the broad and interdisciplinary area of theory and history of cultural production and is known for taking chances with nontraditional and interdisciplinary publications. Overall, The Philosophical Review and Duke University Press are dedicated to advancing knowledge in philosophy and the humanities while maintaining editorial independence and freedom of thought.

Published By:

N Rescher - The Philosophical Review, 1955 - JSTOR

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12

We choose a vertex c 1 in G 1 and a vertex c 2 in G 2 and construct a new graph G by merging c 1 and c 2 into the so-called common vertex c. We speak of the graph G generated by joining two graphs G 1 and G 2 by means of a common vertex c. Note that the new graph G is a simple connected undirected graph of order n = n 1 n 2 1. Let us take for the graph G 1 the graph K 3 and for the graph G 2 the path graph P 4. Clearly the degree of the terminal vertices A 1 and A k 1 in G is equal to 1 because our path is the largest simple path in the graph G. If the vertex A 2 has degree 2 we take the next vertex A 3 and so on till we get the vertex A i of degree greater than 2 in the graph G. If there is no such vertex in our path then our graph is clearly isomorphic to P n. So let a vertex A i be the first vertex of degree greater than 2 in the graph G. It means that all vertices A 2 , A 3 , A i −1 have degree 2 in the graph G and there is a vertex B which is adjacent to the vertex A i and does not belong to our largest path. More precisely, we can ask for graph indices that cannot be reconstructed by sequences of type but require, for example, more than just two graphs per fusion, graphs which are at the same time more complex than fully connected graphs and path graphs.

Published By:

T Lokot, O Abramov, A Mehler - Plos one, 2021 - journals.plos.org

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0