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https://arxiv.org/abs/hep-ph/0101286 | hep-ph
(what is this?)
# Title: Reentrant violation of special relativity in the low-energy corner
Authors: G.E. Volovik
Abstract: In the effective relativistic quantum field theories the energy region, where the special relativity holds, can be sandwiched from both the high and low energies sides by domains where the special relativity is violated. An example is provided by 3He-A where the relativistic quantum field theory emerges as the effective theory. The reentrant violation of the special relativity in the ultralow energy corner is accompanied by the redistribution of the momentum-space topological charges between the fermionic flavors. At this ultralow energy an exotic massless fermion with the topological charge $N_3=2$ arises, whose energy spectrum mixes the classical and relativistic behavior. This effect can lead to neutrino oscillations if neutrino flavors are still massless at this energy scale.
Comments: RevTeX file, 5 pages, one figure, submitted to JETP Lett Subjects: High Energy Physics - Phenomenology (hep-ph); Condensed Matter (cond-mat); General Relativity and Quantum Cosmology (gr-qc) Journal reference: Pisma Zh.Eksp.Teor.Fiz. 73 (2001) 182-185; JETP Lett. 73 (2001) 162-165 DOI: 10.1134/1.1368706 Cite as: arXiv:hep-ph/0101286 (or arXiv:hep-ph/0101286v1 for this version)
## Submission history
From: Grigori Volovik [view email]
[v1] Thu, 25 Jan 2001 09:43:02 GMT (52kb) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7116305232048035, "perplexity": 3361.389441330586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511761.78/warc/CC-MAIN-20181018084742-20181018110242-00451.warc.gz"} |
http://umj.imath.kiev.ua/article/?lang=en&article=7552 | 2018
Том 70
№ 9
# Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials
Grigoryan A. L.
Abstract
We establish lower and upper bounds for the quantity $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ , where $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ , and D m (t) is the Dirichlet kernel, for the class W r of 2π-periodic functions, whose rth derivative satisfies the condition |f r (x)| ≤ 1.
English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 12, pp 1998-2006.
Citation Example: Grigoryan A. L. Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials // Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1691-1698.
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https://www.atlantis-press.com/proceedings/icwcsn-16/25870991 | # Fuzzy Roughness Measurement Model Based on Membership Effect
Authors
Fa-Chao Li, Li-Kun Wang
Corresponding Author
Fa-Chao Li
Available Online December 2016.
DOI
https://doi.org/10.2991/icwcsn-16.2017.87How to use a DOI?
Keywords
Fuzzy set; Roughness; Fuzzy rough set; Membership effect function; Attribute reduction
Abstract
Rough fuzzy set theory and fuzzy set theory are two commonly used tools in today's uncertain type of information processing. How to construct a fusion method for two kinds of uncertain information systematically has been a focus in both academic and applied fields. By analyzing the characteristics and shortcomings of the current fuzzy roughness sets, this paper puts forward concept of membership effect function and establishes a fuzzy roughness measurement model based on membership effect (denoted by FRM-BME, for short). And then, several necessary and sufficient conditions are given to reflect the value of FRM-BME. Finally, we propose an attribute reduction algorithm based on FRM-BME, and further analyze the characteristics and effectiveness of FRM-BME combined with specific cases. Theoretical analysis and experimental results show that, FRM-BME not only has good structural characteristics and interpretability, but also can simply integrate the fuzzy processing preference into roughness measurement system. To a certain extent, it not only enriches the existing related theories, but also can be widely used in artificial intelligence, resource management and many other fields.
Open Access
This is an open access article distributed under the CC BY-NC license.
Proceedings
Part of series
Advances in Computer Science Research
Publication Date
December 2016
ISBN
978-94-6252-302-9
ISSN
2352-538X
DOI
https://doi.org/10.2991/icwcsn-16.2017.87How to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.
TY - CONF
AU - Fa-Chao Li
AU - Li-Kun Wang
PY - 2016/12
DA - 2016/12
TI - Fuzzy Roughness Measurement Model Based on Membership Effect
PB - Atlantis Press
SP - 411
EP - 416
SN - 2352-538X
UR - https://doi.org/10.2991/icwcsn-16.2017.87
DO - https://doi.org/10.2991/icwcsn-16.2017.87
ID - Li2016/12
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https://scicomp.stackexchange.com/questions/20351/are-there-any-possible-applications-of-real-time-finite-element-analysis/20353 | # Are there any possible applications of real-time Finite Element Analysis?
FEA when applied to solid mechanics / structural engineering result in the solution to the equation $$F = K\cdot x$$ $F$ being the forces, $K$ the stiffness of the system and $x$ represents the resulting displacement.
Now if I were to analyse a live experiment where, lets say, I am applying a point load to a beam, gradually increasing in magnitude. As the load increases the deflection in the beam increases.
Using some sensors and data-acquisition system, if I were to read this load, into a FE program, then I would have, in the equation above, a scenario of load which updates, every few seconds, and this should result in new solution to the FE equation. Also, the actual displacement values can be read using a displacement sensor and fed to the FE program for validation / model updation.
This creates kind of a loop, and completes the definition of real-time FE solver.
Would such a system have any practical use?
• Some applications have been in the field of surgery. E.g., amses-journal.com/content/pdf/2213-7467-1-11.pdf Aug 1, 2015 at 17:07
• I seem to recall a guest lecture discussing wind farm blade control systems ... I will see if I can find the reference, Aug 4, 2015 at 8:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.72758948802948, "perplexity": 809.9990924537366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335004.95/warc/CC-MAIN-20220927100008-20220927130008-00096.warc.gz"} |
https://www.scholars.northwestern.edu/en/publications/the-impact-of-r-process-heating-on-the-dynamics-of-neutron-star-m | # The impact of r-process heating on the dynamics of neutron star merger accretion disc winds and their electromagnetic radiation
Hannah Klion*, Alexander Tchekhovskoy, Daniel Kasen, Adithan Kathirgamaraju, Eliot Quataert, Rodrigo Fernández
*Corresponding author for this work
Research output: Contribution to journalArticlepeer-review
## Abstract
Neutron star merger accretion discs can launch neutron-rich winds of >10-2M⊙. This ejecta is a prime site for r-process nucleosynthesis, which will produce a range of radioactive heavy nuclei. The decay of these nuclei releases enough energy to accelerate portions of the wind by ∼0.1c. Here, we investigate the effect of r-process heating on the dynamical evolution of disc winds. We extract the wind from a 3D general relativistic magnetohydrodynamic simulation of a disc from a post-merger system. This is used to create inner boundary conditions for 2D hydrodynamic simulations that continue the original 3D simulation. We perform two such simulations: one that includes the r-process heating, and another one that does not. We follow the hydrodynamic simulations until the winds reach homology (60 s). Using time-dependent multifrequency multidimensional Monte Carlo radiation transport simulations, we then calculate the kilonova light curves from the winds with and without dynamical r-process heating. We find that the r-process heating can substantially alter the velocity distribution of the wind, shifting the mass-weighted median velocity from 0.06c to 0.12c. The inclusion of the dynamical r-process heating makes the light curve brighter and bluer at $\sim 1\, \mathrm{d}$ post-merger. However, the high-velocity tail of the ejecta distribution and the early ($\lesssim 1\, \mathrm{d}$) light curves are largely unaffected.
Original language English (US) 2968-2979 12 Monthly Notices of the Royal Astronomical Society 510 2 https://doi.org/10.1093/mnras/stab3583 Published - Feb 2022
## Keywords
• neutron star mergers | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8451849222183228, "perplexity": 3849.1104975140247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103640328.37/warc/CC-MAIN-20220629150145-20220629180145-00202.warc.gz"} |
http://mathhelpforum.com/pre-calculus/199454-determining-lines-symmetry-scratch.html | # Math Help - Determining Lines of Symmetry from Scratch?
1. ## Determining Lines of Symmetry from Scratch?
Hi all ---
When I tried the following question in the black oval --- the answer is in red --- I was completely stuck on the logic/thought process/intuition in the first part. I'm fine with the algebra --- it's NOT what's troubling me for the past hours.
I know how to check for symmetry about the origin and y-axis. A function, f(x), is symmetric about the origin if $f(-x) = -f(x)$ and about y-axis if $f(-x) = f(x)$.
But here, how on earth would you even know to check for symmetry about $y = \pm x$? How would you even suspect or be aware of this in the first place?
My work: For $xy = v \Longleftrightarrow y = \frac{v}{x}$, I sort of see this because this is just the reciprocal function with the constant $v$.
So here, I'd expect symmetry about $y = \pm x$ because I've seen the graph of the reciprocal function before. But I'm completely lost for $x^4 + y^4 = u$?
Also, what would lead you to check for symmetry about x- and y-axis here? Question doesn't give any information that would lead you to do this?
In general, if I see this kind of question that only asks me to state all lines of symmetry (WITHOUT telling me what the lines are), how would I know
--- which lines I should check for symmetry?
--- how many lines there are?
Maybe because I don't understand it, but the answer looks like a bit of guesswork to me...
Thank you ---
2. ## Re: Determining Lines of Symmetry from Scratch?
The fact that the exponents on x and y are even so that $(-x)^4= x^4$ and $(-y)^4= y^4$ would lead me to say that the graph is symmetric about the axes: the point "symmetric" to (x, y) about the y-axis is (-x, y) and about the x-axis is (x, -y). The fact that the formula is "symmetric" with respect to x and y: replacing x with y and y with x changes $x^4+ y^4= u$ to $(y)^4+ (x^4)= u$ which is the same equation, tells me that it is symmetric about the lines y= x and y= -x.
3. ## Re: Determining Lines of Symmetry from Scratch?
Hi HallsofIvy ---
Thanks a lot for your answer. It really helped with why one'd think $y = \pm x$ and $x, y$ axes give symmetry for the functions.
Just one last question that was maybe unclear in my post --- How would you know/infer that there are no other lines of symmetry to check or test for either curve?
4. ## Re: Determining Lines of Symmetry from Scratch?
After thinking about this for the past week, I still do not see how and why there are no other lines of symmetry? Has someone gotten farther than me?
5. ## Re: Determining Lines of Symmetry from Scratch?
Hi all again --- I should point out that the red box is the solution, so we are NOT told or hinted that both graphs are symmetric about $y = \pm x$ and $x = 0, y = 0$.
After two more weeks of banging my head on this problem, I still do not see how you can tell or guess that there are no other lines of symmetry? Does anyone have more insight?
Thank you --- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5710878372192383, "perplexity": 381.14924083700674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1433195036613.12/warc/CC-MAIN-20150601214356-00032-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://scicomp.stackexchange.com/questions/8573/why-performance-is-given-in-gflop-s-rather-than-actual-time-in-seconds | # Why performance is given in Gflop/s rather than actual time in seconds
While reading many research-papers comparing parallel implementations of algorithms on different machines/architectures, I have noticed that the performance comparison is always listed in terms of GFlop/s and not the actual wall-clock time for the run in seconds. I am curious why this convention is used.
My only guess is that since every company advertises its device as having a certain peak flop-counts/second such research papers investigate how much of its "potential" has been achieved by listing the performance as "GFlop/s" for the particular application at hand.
Is this correct?
Also, say the performance of a $m$ x $n$ Matrix -- $n$ x $1$ Vector multiply has been stated as 4 GFlop/s. Is it reasonable to obtain the wall clock time in seconds by the following formula?
$$\frac{m(2n-1)}{4 * 10^9} \hspace{3mm} \text{seconds}$$ where $m(2n-1)$ is the number of floating point operations for the matrix-vector multiplication
Traditionally, I'd say that people more or less understood the number of floating-point operations required to solve a problem ($O(n^3)$ and the like), and so reporting performance as a rate had meaning. Readers could then obtain the run time via the method you describe, but they could also compare that to the theoretical peak performance of the hardware used to find the efficiency of the method.
Personally, I like to see run time and performance rate reported.
The only thing that might be unreasonable about your method for determining the run time is that you have to be sure you have the right operational complexity formula. If the authors are using scheme with a different operation count, then you might get the calculation very wrong. For example, what you give is probably OK for dense matrix-vector multiplication, but if this was a sparse example, some sort of approach where the zeros were not multiplied was probably used. If you use your approach to calculate the run time without trying to account for the sparsity, then you'll run into trouble.
• Thank you for the nice answer! Are there any standard software tools for measuring the flop rate of a code on a parallel / serial machines? – smilingbuddha Sep 22 '13 at 17:47
• Tons. Google HPCToolkit, PAPI, PerfExpert, and VTune. Though, I would say that if you are on a modern Intel platform (Sandbridge or later), your measurement will be wildly inaccurate. SNB counts floating-point instructions issued not floating-point instructions executed or retired. This means that if the core has to reissue an FP instruction because a value has arrived from memory or is otherwise unavailable, you will over count instructions. This is extremely common. – Bill Barth Sep 22 '13 at 17:52
• Yes I will be working on an Intel platform, and infact on a small Intel Xeon Phi cluster soon. If the Flop rate on modern Intel platforms is so inaccurate, are the figures in research papers documenting performance to be trusted? – smilingbuddha Sep 22 '13 at 18:02
• Depends. You can actually count the operations involved in your implementation and then divide by the time, so you don't have to rely on the hardware to measure them. People who do this are presenting legitimate, reliable results, most likely. If they try to measure them without validation, then they may be problematic. Take everything with a grain of salt. – Bill Barth Sep 22 '13 at 18:12
With the Linpack benchmark used in ranking supercomputers (and in many other situations where people are benchmarking high performance computing systems), it's common to scale the size of the problem and report the performance in Gflop/s for the particular value of the problem size $n$ at which the machine hits peak performance. In the case of the Linpack benchmark, the operation count is well known and standardized, so there's no question about how many floating point operations were actually performed in solving the benchmark problem at that size.
If two different machines have peak performance at very different problem sizes, then comparing the run times at these sizes (or even at one common problem size) wouldn't give a reasonable comparison.
On the other hand, if all you want is to solve a 100x100 system of linear equations, don't expect a machine that has a peak performance in the PetaFlop/s range to be much faster at solving your tiny system of equations than a good desktop machine! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40859872102737427, "perplexity": 606.0158754625537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141672314.55/warc/CC-MAIN-20201201074047-20201201104047-00500.warc.gz"} |
https://stats.stackexchange.com/questions/156519/whether-to-use-a-hierarchical-linear-model | # Whether to use a hierarchical linear model
I've been reading through Gelmans book: Data Analysis Using Regression and Multilevel/Hierarchical Models trying to learn more about how to implement hierarchal models. I have a dataset that I think is appropriate for this type of modeling however I want to get some other opinions. Basically the data I have is structured like this:
BRAND YEAR Y X1 X2 X3
company_1 2012 0.638042396 0.226787359 0.192104136 0.929220784
company_2 2012 0.983422117 0.308550049 0.527779594 0.106629747
company_n 2012 0.209276388 0.700314863 0.741787081 0.491451885
company_1 2013 0.833955686 0.735844101 0.518474158 0.117670754
company_2 2013 0.480778935 0.290739025 0.156177295 0.212643611
company_n 2013 0.69922326 0.188574282 0.448743735 0.609844836
company_1 2014 0.942147995 0.176500074 0.820207708 0.388313924
company_2 2014 0.503095705 0.987218933 0.834039587 0.42661805
company_n 2014 0.46569344 0.310693712 0.852694246 0.17574502
where I have about 15 different companies for each year. My thought was to have a model like this:
lmer(Y ~ X1 + X2 + X3 + (1 | BRAND) , h.data)
where I have a varying intercept for each company. So my question here is whether or not it makes sense to use a hierarchal model and if my data fits the archetype of hierarchal data? Also should I be including YEAR into the model somehow?
While I agree that multi-level modeling is an option for data with this structure, it's not the only option, especially given the lone time series dimension. Typically, the nestings within a heterarchical model are by category, e.g., students within classes or teachers, classes within schools, and so on, not ordinal dimensions like time.
Gelman and Hill's book is great, I agree. Perhaps even better is Singer and Willet's book Applied Longitudinal Data Analysis which, to one poster's point, goes into much greater depth than G&H on some topics, e.g., growth models, issues related to constructing an interpretable intercept, curvilinearity and survival analysis but S&W lack a Bayesian focus.
If you had an additional factor called "industry", then I would lean more strongly towards a heterarchical model. Given that you don't (i.e., you haven't posited "industry" as a factor. Do these companies belong to a single industry? How about by 6 or 8 digit SIC or NAIC codes?), another consideration would be pooled time series or event history analysis as it's called in sociology. Here, the advantages are that the models can be estimated in OLS, a more tractable functional form than multi-level models and that the industrial, organizational and economic literature has a long-standing history of published papers in this domain, beginning at least with F.M. Scherer but continuing up to the near-present with books like Wooldridge's Econometric Analysis of Cross Section and Panel Data.
My personal favorite in the field of PTS is Lee Cooper's ebook, Market Share Analysis, available on his UCLA website. Ignore the "share" part and even the "marketing" part. It's simply a great introduction to this class of models and it's quite accessible without sacrificing scientific rigor (he's an emeritus professor of mktg science). Not to mention that he develops different and carefully specified functional forms in terms of the data structure, elasticities, cross-elasticities and very practical advice on how to build these into your model. Depending on what your X factors are, this could be quite useful information.
• Great I will check out these resources. All the companies are from the same industry. Since that is the case you recommend a non-multilevel approach? – moku Jun 11 '15 at 17:42
• Yup. As noted, heterarchies are typically defined by categorical nestings, not time. – DJohnson Jun 11 '15 at 17:44
• Right. YEAR was a point of confusion for me I wasn't sure how to incorporate it into a multilevel model. Y = awareness(as a %) and my X variable are spend in various advertising channels (digital, tv, radio, print). Is there a specific package in R that accomplishes what Cooper talks about or are his suggestions more about model structure and transformations. Also I was curious maybe you know, when you standardize the data should I standardize within each year or overall among all three years? – moku Jun 11 '15 at 17:53
• Ah! Then you would want to use an MMM, marketing mix model, and PTS is definitely the right way to go, imho. Others might argue for much more rigorously econometric "state-space" approaches. Peter Cain is one of them, browse his profile and publications on LinkedIn for more...but he's selling his expertise as much as anything. Cooper was writing in 1989 when R wasn't around. If you follow his prescriptions, the R modules would be the typical OLS regression tools. Standardizing by year would erase variation across years, don't do it. Consider mktg vehicle interactions. There's a big lit on MMMs – DJohnson Jun 11 '15 at 18:24
I would say that using a hierarchical model is suitable in your case.
Following this guide, BRAND would be your Level-2-term and Year could be your Level-1-term, used as random slope.
You should also check the ICC afterwards to see whether hierarchical models give you any benefits over normal linear regression.
• Oh cool this guide is really nice! So you recommend this formula: $lmer(Y ~ X1 + X2 + X3 + YEAR + (1 | BRAND), data=data)$ – moku Jun 11 '15 at 17:16
• I must admit that I'm no super-expert, so I suggest waiting for another answer "validating" my proposal. You could also add YEAR as random slope, but I would in any case include YEAR as term as you did in your formula. – Daniel Jun 11 '15 at 17:30
Yes, you should probably use multilevel modeling, possibly with company at level 3 and year at level 2. This would be a multilevel growth curve approach. You could then analyze the different kinds of changes in x different types of companies had over time.
• Ok cool could you explain the levels a bit more what does each level correspond to? For instance level 1 = normal term, level 2= random intercepts, level 3 = random slopes. – moku Jun 11 '15 at 17:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22410640120506287, "perplexity": 1813.5550350400556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999130.50/warc/CC-MAIN-20190620004625-20190620030625-00555.warc.gz"} |
http://www.ats.ucla.edu/stat/stata/faq/outgraph.htm | Stata FAQ How do I copy Stata output and Stata graphs into Word?
It is rather easy to copy Stata output and Stata graphs into Microsoft Word. This page shows how you can copy your Stata output and graphs into Microsoft Word.
The Stata Results window holds the results of your Stata commands. We should point out two things about the Stata Results window that may surprise you. First, by default it holds only about 500 lines of output... anything after that is discarded. Second, you cannot use the pulldown menus to save the contents of the results window (i.e., you cannot go File Save to save the results). How can you save your output? Here are a couple of solutions.
Copy as text from the results window and paste into Word
You can use the mouse to scroll through the results window and mark an area that you want to save. You can then use the pulldown menu to choose Edit and then Copy. This is illustrated below.
You can then go to Microsoft Word and from its pulldown menu choose Edit then Paste. Most likely, the results will look lousy, that is, the text will be misaligned as shown below.
This happens because the output from Stata uses fixed space fonts, and most fonts in Microsoft Word are proportionally spaced fonts (for example, the text in the window above is Times New Roman). If you select the text in Word and choose a fixed space font like Courier the output will then look as it did in Stata, as shown below.
Copy as image from the results window and paste into Word
An alternative to copying the output as text from the Stata window is to copy the output as an imagine. Highlight the output you want to save, then use the pulldown menu to choose Edit and then Copy as Picture. This is illustrated below.
You can then go to Microsoft Word and from its pulldown menu choose Edit then Paste. When you do this, the table you see will probably be quite small. To make the output more readable, you can enlarge this table using the picture editor in Word as shown below.
Right-click on the output and select Format Picture... to edit the picture properties.
On the Size tab of the Format Picture box, change the scale of both the height and width. Finding the correct size for your output may take some trial and error, in this case, we'll start by increasing both the height and width from 100% to 130% and click OK.
Below is the output shown at 130%, in this case, the table is readable so no further adjustment is needed.
Copy .log file into Word
At the start of your session, you can start a log file, and then at the end of your session you can close the log file. This log contains the commands you typed and the output produced. For example,
log using d:\data\homework.log, text
<...you enter a bunch of commands...>
log close
Then, the file homework.log would contain the commands and output they produced. You can open this file in Microsoft Word and edit it, but there are a couple of tricks. First, you need to point Word to the folder (directory) where homework.log is located. By default, Word only shows you Word documents (i.e., files that end in .doc). For example, in the screen below we have pointed Word to the right folder, but we cannot see homework.log because it is only showing us Files of Type Word Documents (*.doc), see lower left of window.
If we change the Files of type to be All Files (*.*) we can now see our file called homework.log.
You can then edit this file. If you want to add comments, you may notice that text is in a fixed space font. It will probably look better if you convert that text to Times New Roman (mark your comments and then choose the Times New Roman font). You will want to choose File then Save As and save the file as a word document so it will contain all of the formatting you have created.
Copying graphs to Word
If you create a graph in Stata, you can copy that graph and then paste it into Microsoft Word. Below we show a graph we created in Stata and show how to copy the graph by pointing at the graph and right-clicking and choosing Copy
You can then open Word and then paste the graph by choosing from the Word pulldown menu Edit and then Paste. You will then see the Stata graph pasted into the Word. This covers the basics of copying Stata output and graphs into Microsoft Word.
The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4992607831954956, "perplexity": 1496.5799909899404}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368711005723/warc/CC-MAIN-20130516133005-00035-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://se.mathworks.com/help/stats/nlmefitsa.html | # nlmefitsa
Fit nonlinear mixed-effects model with stochastic EM algorithm
## Syntax
[BETA,PSI,STATS,B] = nlmefitsa(X,Y,GROUP,V,MODELFUN,BETA0) [BETA,PSI,STATS,B] = nlmefitsa(X,Y,GROUP,V,MODELFUN,BETA0,'Name',Value)
## Description
[BETA,PSI,STATS,B] = nlmefitsa(X,Y,GROUP,V,MODELFUN,BETA0) fits a nonlinear mixed-effects regression model and returns estimates of the fixed effects in BETA. By default, nlmefitsa fits a model where each model parameter is the sum of a corresponding fixed and random effect, and the covariance matrix of the random effects is diagonal, i.e., uncorrelated random effects.
The BETA, PSI, and other values this function returns are the result of a random (Monte Carlo) simulation designed to converge to the maximum likelihood estimates of the parameters. Because the results are random, it is advisable to examine the plot of simulation to results to be sure that the simulation has converged. It may also be helpful to run the function multiple times, using multiple starting values, or use the 'Replicates' parameter to perform multiple simulations.
[BETA,PSI,STATS,B] = nlmefitsa(X,Y,GROUP,V,MODELFUN,BETA0,'Name',Value) accepts one or more comma-separated parameter name/value pairs. Specify Name inside single quotes.
## Input Arguments
Definitions:
In the following list of arguments, the following variable definitions apply:
• n — number of observations
• h — number of predictor variables
• m — number of groups
• g — number of group-specific predictor variables
• p — number of parameters
• f — number of fixed effects
X An n-by-h matrix of n observations on h predictor variables. Y An n-by-1 vector of responses. GROUP A grouping variable indicating to which of m groups each observation belongs. GROUP can be a categorical variable, a numeric vector, a character matrix with rows for group names, a string array, or a cell array of character vectors. V An m-by-g matrix of g group-specific predictor variables for each of the m groups in the data. These are predictor values that take on the same value for all observations in a group. Rows of V are ordered according to GRP2IDX(GROUP). Use an m-by-g cell array for V if any of the group-specific predictor values vary in size across groups. Specify [] for V if there are no group predictors. MODELFUN A handle to a function that accepts predictor values and model parameters, and returns fitted values. MODELFUN has the form YFIT = MODELFUN(PHI,XFUN,VFUN) with input arguments PHI — A 1-by-p vector of model parameters. XFUN — An l-by-h array of predictor variables where l is 1 if XFUN is a single row of Xl is ni if XFUN contains the rows of X for a single group of size nil is n if XFUN contains all rows of X. VFUN — Either A 1-by-g vector of group-specific predictors for a single group, corresponding to a single row of VAn n-by-g matrix, where the k-th row of VFUN is V(i,:) if the k-th observation is in group i. If V is empty, nlmefitsa calls MODELFUN with only two inputs. MODELFUN returns an l-by-1 vector of fitted values YFIT. When either PHI or VFUN contains a single row, that one row corresponds to all rows in the other two input arguments. For improved performance, use the 'Vectorization' parameter name/value pair (described below) if MODELFUN can compute YFIT for more than one vector of model parameters in one call. BETA0 An f-by-1 vector with initial estimates for the f fixed effects. By default, f is equal to the number of model parameters p. BETA0 can also be an f-by-REPS matrix, and the estimation is repeated REPS times using each column of BETA0 as a set of starting values.
### Name-Value Arguments
By default, nlmefitsa fits a model where each model parameter is the sum of a corresponding fixed and random effect. Use the following parameter name/value pairs to fit a model with a different number of or dependence on fixed or random effects. Use at most one parameter name with an 'FE' prefix and one parameter name with an 'RE' prefix. Note that some choices change the way nlmefitsa calls MODELFUN, as described further below.
FEParamsSelect A vector specifying which elements of the model parameter vector PHI include a fixed effect, as a numeric vector with elements in 1:p, or as a 1-by-p logical vector. The model will include f fixed effects, where f is the specified number of elements. FEConstDesign A p-by-f design matrix ADESIGN, where ADESIGN*BETA are the fixed components of the p elements of PHI. FEGroupDesign A p-by-f-by-m array specifying a different p-by-f fixed effects design matrix for each of the m groups. REParamsSelect A vector specifying which elements of the model parameter vector PHI include a random effect, as a numeric vector with elements in 1:p, or as a 1-by-p logical vector. The model will include r random effects, where r is the specified number of elements. REConstDesign A p-by-r design matrix BDESIGN, where BDESIGN*B are the random components of the p elements of PHI. This matrix must consist of 0s and 1s, with at most one 1 per row.
The default model is equivalent to setting both FEConstDesign and REConstDesign to eye(p), or to setting both FEParamsSelect and REParamsSelect to 1:p.
Additional optional parameter name/value pairs control the iterative algorithm used to maximize the likelihood:
CovPattern Specifies an r-by-r logical or numeric matrix PAT that defines the pattern of the random effects covariance matrix PSI. nlmefitsa computes estimates for the variances along the diagonal of PSI as well as covariances that correspond to non-zeroes in the off-diagonal of PAT. nlmefitsa constrains the remaining covariances, i.e., those corresponding to off-diagonal zeroes in PAT, to be zero. PAT must be a row-column permutation of a block diagonal matrix, and nlmefitsa adds non-zero elements to PAT as needed to produce such a pattern. The default value of PAT is eye(r), corresponding to uncorrelated random effects. Alternatively, specify PAT as a 1-by-r vector containing values in 1:r. In this case, elements of PAT with equal values define groups of random effects, nlmefitsa estimates covariances only within groups, and constrains covariances across groups to be zero. Cov0 Initial value for the covariance matrix PSI. Must be an r-by-r positive definite matrix. If empty, the default value depends on the values of BETA0. ComputeStdErrors true to compute standard errors for the coefficient estimates and store them in the output STATS structure, or false (default) to omit this computation. ErrorModel A character vector or string scalar specifying the form of the error term. Default is 'constant'. Each model defines the error using a standard normal (Gaussian) variable e, the function value f, and one or two parameters a and b. Choices are 'constant' — y = f + a*e'proportional' — y = f + b*f*e'combined' — y = f + (a+b*f)*e'exponential' — y = f*exp(a*e), or equivalently log(y) = log(f) + a*e If this parameter is given, the output STATS.errorparam field has the value a for 'constant' and 'exponential' b for 'proportional' [a b] for 'combined' ErrorParameters A scalar or two-element vector specifying starting values for parameters of the error model. This specifies the a, b, or [a b] values depending on the ErrorModel parameter. LogLikMethod Specifies the method for approximating the loglikelihood. Choices are: 'is' — Importance sampling 'gq' — Gaussian quadrature'lin' — Linearization 'none' — Omit the loglikelihood approximation (default) NBurnIn Number of initial burn-in iterations during which the parameter estimates are not recomputed. Default is 5. NChains Number c of "chains" simulated. Default is 1. Setting c>1 causes c simulated coefficient vectors to be computed for each group during each iteration. Default depends on the data, and is chosen to provide about 100 groups across all chains. NIterations Number of iterations. This can be a scalar or a three-element vector. Controls how many iterations are performed for each of three phases of the algorithm: simulated annealingfull step size reduced step size Default is [150 150 100]. A scalar is distributed across the three phases in the same proportions as the default. NMCMCIterations Number of Markov Chain Monte Carlo (MCMC) iterations. This can be a scalar or a three-element vector. Controls how many of three different types of MCMC updates are performed during each phase of the main iteration: full multivariate updatesingle coordinate updatemultiple coordinate update Default is [2 2 2]. A scalar value is treated as a three-element vector with all elements equal to the scalar. OptimFun Optimization function for the estimation process that maximizes a likelihood function, specified as 'fminsearch' (default) or 'fminunc'. Use of 'fminunc' requires Optimization Toolbox™. The fminsearch function uses a direct search method that uses only function evaluations. The fminunc (Optimization Toolbox) function uses gradient methods and is generally more efficient for an optimization problem that maximizes the likelihood function. Options A structure created by a call to statset. nlmefitsa uses the following statset parameters: 'DerivStep' — Relative difference used in finite difference gradient calculation. May be a scalar, or a vector whose length is the number of model parameters p. The default is eps^(1/3).Display — Level of display during estimation. 'off' (default) — Displays no information'final' — Displays information after the final iteration of the estimation algorithm'iter' — Displays information at each iteration FunValCheck 'on' (default) — Check for invalid values (such as NaN or Inf) from MODELFUN'off' — Skip this check OutputFcn — Function handle specified using @, a cell array with function handles or an empty array. nlmefitsa calls all output functions after each iteration. See nlmefitoutputfcn.m (the default output function for nlmefitsa) for an example of an output function.TolX — Termination tolerance on the estimated fixed and random effects. The default is 1e-4. ParamTransform A vector of p-values specifying a transformation function f() for each of the p parameters: XB = ADESIGN*BETA + BDESIGN*B PHI = f(XB) Each element of the vector must be one of the following integer codes specifying the transformation for the corresponding value of PHI: 0: PHI = XB (default for all parameters) 1: log(PHI) = XB2: probit(PHI) = XB3: logit(PHI) = XB Replicates Number REPS of estimations to perform starting from the starting values in the vector BETA0. If BETA0 is a matrix, REPS must match the number of columns in BETA0. Default is the number of columns in BETA0. Vectorization Determines the possible sizes of the PHI, XFUN, and VFUN input arguments to MODELFUN. Possible values are: 'SinglePhi' — MODELFUN is a function (such as an ODE solver) that can only compute YFIT for a single set of model parameters at a time, i.e., PHI must be a single row vector in each call. nlmefitsa calls MODELFUN in a loop if necessary using a single PHI vector and with XFUN containing rows for a single observation or group at a time. VFUN may be a single row that applies to all rows of XFUN, or a matrix with rows corresponding to rows in XFUN.'SingleGroup' — MODELFUN can only accept inputs corresponding to a single group in the data, i.e., XFUN must contain rows of X from a single group in each call. Depending on the model, PHI is a single row that applies to the entire group, or a matrix with one row for each observation. VFUN is a single row. 'Full' — MODELFUN can accept inputs for multiple parameter vectors and multiple groups in the data. Either PHI or VFUN may be a single row that applies to all rows of XFUN, or a matrix with rows corresponding to rows in XFUN. Using this option can improve performance by reducing the number of calls to MODELFUN, but may require MODELFUN to perform singleton expansion on PHI or V. The default for 'Vectorization' is 'SinglePhi'. In all cases, if V is empty, nlmefitsa calls MODELFUN with only two inputs.
## Output Arguments
BETA Estimates of the fixed effects PSI An r-by-r estimated covariance matrix for the random effects. By default, r is equal to the number of model parameters p. STATS A structure with the following fields: logl — The maximized loglikelihood for the fitted model; empty if the LogLikMethod parameter has its default value of 'none' rmse — The square root of the estimated error variance (computed on the log scale for the exponential error model)errorparam — The estimated parameters of the error variance model aic — The Akaike information criterion (empty if logl is empty), calculated as aic = –2 * logl + 2 * numParam, where logl is the maximized loglikelihood.numParam is the number of fitting parameters, including the degree of freedom for covariance matrix of the random effects, the number of fixed effects and the number of parameters of the error model. bic — The Bayesian information criterion (empty if logl is empty), calculated as bic = -2*logl + log(M) * numParam M is the number of groups.logl and numParam are defined as in aic. Note that some literature suggests that the computation of bic should be , bic = -2*logl + log(N) * numParam, where N is the number of observations. To adjust the value of the output you can redefine bic as follows: bic = bic - numel(unique(group)) + numel(Y)sebeta — The standard errors for BETA (empty if the ComputeStdErrors parameter has its default value of false)covb — The estimated covariance of the parameter estimates (empty if ComputeStdErrors is false)dfe — The error degrees of freedompres — The population residuals (y-y_population), where y_population is the population predicted valuesires — The population residuals (y-y_population), where y_population is the individual predicted valuespwres — The population weighted residualscwres — The conditional weighted residualsiwres — The individual weighted residuals
## Examples
collapse all
load indomethacin
Fit a model to data on concentrations of the drug indomethacin in the bloodstream of six subjects over eight hours.
model = @(phi,t)(phi(:,1).*exp(-phi(:,2).*t)+phi(:,3).*exp(-phi(:,4).*t)); phi0 = [1 1 1 1]; xform = [0 1 0 1]; % log transform for 2nd and 4th parameters [beta,PSI,stats,br] = nlmefitsa(time,concentration, ... subject,[],model,phi0,'ParamTransform',xform)
beta = 4×1 0.8630 -0.7897 2.7762 1.0785
PSI = 4×4 0.0585 0 0 0 0 0.0248 0 0 0 0 0.5068 0 0 0 0 0.0139
stats = struct with fields: logl: [] aic: [] bic: [] sebeta: [] dfe: 57 covb: [] errorparam: 0.0811 rmse: 0.0772 ires: [66x1 double] pres: [66x1 double] iwres: [66x1 double] pwres: [66x1 double] cwres: [66x1 double]
br = 4×6 -0.2302 -0.0033 0.1625 0.1774 -0.3334 0.1129 0.0363 -0.1502 0.0071 0.0471 0.0068 -0.0481 -0.7631 -0.0553 0.8780 -0.8120 0.5429 0.1695 -0.0030 -0.0223 0.0192 -0.0830 0.0505 -0.0066
Plot the data along with an overall population fit.
figure phi = [beta(1),exp(beta(2)),beta(3),exp(beta(4))]; h = gscatter(time,concentration,subject); xlabel('Time (hours)') ylabel('Concentration (mcg/ml)') title('{\bf Indomethacin Elimination}') xx = linspace(0,8); line(xx,model(phi,xx),'linewidth',2,'color','k')
Plot individual curves based on random-effect estimates.
for j=1:6 phir = [beta(1)+br(1,j), exp(beta(2)+br(2,j)), ... beta(3)+br(3,j), exp(beta(4)+br(4,j))]; line(xx,model(phir,xx),'color',get(h(j),'color')) end
## Algorithms
In order to estimate the parameters of a nonlinear mixed effects model, we would like to choose the parameter values that maximize a likelihood function. These values are called the maximum likelihood estimates. The likelihood function can be written in the form
$p\left(y|\beta ,{\sigma }^{2},\Sigma \right)=\int p\left(y|\beta ,b,{\sigma }^{2}\right)p\left(b|\Sigma \right)db$
where
• y is the response data
• β is the vector of population coefficients
• σ2 is the residual variance
• ∑ is the covariance matrix for the random effects
• b is the set of unobserved random effects
Each p() function on the right-hand-side is a normal (Gaussian) likelihood function that may depend on covariates.
Since the integral does not have a closed form, it is difficult to find parameters that maximize it. Delyon, Lavielle, and Moulines [1] proposed to find the maximum likelihood estimates using an Expectation-Maximization (EM) algorithm in which the E step is replaced by a stochastic procedure. They called their algorithm SAEM, for Stochastic Approximation EM. They demonstrated that this algorithm has desirable theoretical properties, including convergence under practical conditions and convergence to a local maximum of the likelihood function. Their proposal involves three steps:
1. Simulation: Generate simulated values of the random effects b from the posterior density p(b|Σ) given the current parameter estimates.
2. Stochastic approximation: Update the expected value of the loglikelihood function by taking its value from the previous step, and moving part way toward the average value of the loglikelihood calculated from the simulated random effects.
3. Maximization step: Choose new parameter estimates to maximize the loglikelihood function given the simulated values of the random effects.
## References
[1] Delyon, B., M. Lavielle, and E. Moulines, "Convergence of a stochastic approximation version of the EM algorithm." Annals of Statistics, 27, 94-128, 1999.
[2] Mentré, F., and M. Lavielle, "Stochastic EM Algorithms in Population PKPD analyses." American Conference on Pharmacometrics, 2008.
## Version History
Introduced in R2010a | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6869871616363525, "perplexity": 2013.070421231152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334871.54/warc/CC-MAIN-20220926113251-20220926143251-00352.warc.gz"} |
http://math.stackexchange.com/questions/374940/intersection-of-sigma-algebras-and-set-theory | # Intersection of $\sigma$-algebras and set theory
Theorem: Given $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$, where each $E_\alpha$ is a $\sigma$-algebra on $X$. Then $E:=\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a $\sigma$-algebra.
Proof: Take $\{S_i\}_{i=1}^\infty$ with each $S_i \in E$. Then for each $\alpha \in \mathcal{A}$ we have $S_i \in E_\alpha$ and therefore $\bigcup_{i=1}^\infty S_i \in E_\alpha$. So $\bigcup_{i=1}^\infty S_i \in E$. And complement is same reasoning.
Now suppose we are given an arbitrary set $F$ and we say that $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$ is the set of all $\sigma$-algebras containing $F$. Then the above theorem would imply that there exists a unique smallest $\sigma$-algebra that contains $F$.
I don't know any set theory but I know that not all sets you can write down are legit (Russell's paradox), so I am a little uncomfortable with this. So I'd like to ask,
1. Suppose we have $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$. That is, we have a function $f:\mathcal{A} \rightarrow \mathcal{P}\left(\mathcal{P}\left(X\right)\right)$, where $E_\alpha := f(\alpha)$. Can we always say that $\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a legit set?
2. Is it easy to show that you can construct $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$? (I.e. the existence of an $f$ such that $f(\alpha)=E_\alpha$) Do we need to worry about this constructability in practice? Why?
-
One of the standard axioms of set theory says that if $X$ is a set, then any definable collection of elements of $X$ is also a set. This is the axiom of separation or comprehension. "Definable" means here "expressible by a first order formula, with parameters".
In particular, $\bigcap_{\alpha\in\mathcal A}E_\alpha$ is a set as long as $\mathcal A\ne\emptyset$: Fix $\alpha_0\in\mathcal A$, and note that $$\bigcap_{\alpha\in\mathcal A}E_\alpha=\{x\in E_{\alpha_0}\mid\forall \alpha\in\mathcal A\,(x\in E_\alpha)\}.$$
The second issue is also fine: Each $E_\alpha$ is a $\sigma$-algebra on $X$, so it is a collection of subsets of $X$, so it is a subset of $\mathcal P(X)$, so it belongs to $\mathcal P(\mathcal P(X))$. The function $f$ has domain a set, in this case $D=\mathcal A$, and range $R$ contained in a set, in this case the double power set of $X$. In set theory we code functions by their graphs, so $f$ is just a subset of $D\times R$ with some properties, which means (again) that it is a set by separation. That $D\times R$ itself is a set also follows from standard axioms as long as $D$ and $R$ are sets; the key is that we can code ordered pairs as sets.
In general, we do not encounter any foundational issues at this level: In order to obtain a collection that is not a set, for example by taking a union $\bigcup_{i\in I}A_i$ or a product $\prod_{i\in I}A_i$, we would need that either the index set $I$ or one of the $A_i$ is not a set. We simply do not encounter these issues in practice. The axioms of set theory are powerful enough that imply that we obtain sets if we begin with sets, and apply standard mathematical operations such as those obtained by iterating applications of pairing $x,y\mapsto\{x,y\}$, taking unions, power-sets, forming subcollections of given sets, or taking pointwise images of sets.
(On the other hand, it is easy to describe collections or objects that are not sets: The collection of all groups, or the collection of all $\sigma$-algebras.)
-
Note that Russell's paradox comes from trying to collect all objects with a given property (in this case, all objects $x$ such that $x\notin x$). Separation, on the other hand, is formulated precisely to avoid this issue, since it can only collect together all such objects that already belong to a set. This is the so-called Cantorian limitation of size, and is central to standard set theory, $\mathsf{ZF}$, see en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory. – Andrés Caicedo Apr 28 '13 at 6:18
Other set theories follow other conventions; if you are interested in this, you may want to look at this question: math.stackexchange.com/q/331567/462. – Andrés Caicedo Apr 28 '13 at 6:19
How do we know that $D=\mathcal{A}$ is a set? Do we just take $\mathcal{A} = \{ \text{the set of all } \sigma \text{-algebras containing } F \}$, and then define $f=\{ (x,x) : x\in D \}$? So why do we even use indexed sets, it seems we could just write $\bigcap_{C\in \mathcal{A}} C$? – StuartHa Apr 28 '13 at 16:26
@Stuart Sure, we could even write $\bigcap \mathcal A$. But we use indexed sets because sometimes it is more natural to use a different index set than the set itself we are considering. For example, we could have several families, all indexed by the same set, so even though it is not mathematically necessary, it tends to be notationally advantageous. Anyway, if $\mathcal A$ is not a set, the intersection is still a set, as it is still given by the separation as before. – Andrés Caicedo Apr 28 '13 at 17:53
(Of course, if $\mathcal A$ is not a set, then mentions of $\mathcal A$ are understood as abbreviations where $\mathcal A$ is replaced by a description of what it is. We call such collections $\mathcal A$ proper classes. That said, in your setting, since the collection of $\sigma$-algebras over a given set is a set anyway, then, even if $\mathcal A$ is a proper class, there is a set $\mathcal A'$ that we can use instead giving rise to precisely the same intersection.) – Andrés Caicedo Apr 28 '13 at 17:54
A $\sigma$-algebra on a set $X$ is a subset of $\mathcal{P} ( X )$, meaning that they are elements of $\mathcal{P} ( \mathcal{P} ( X ) )$. The usual axioms of set-theory imply that the collection $$\mathfrak{S} = \{ \mathcal{S} \in \mathcal{P} ( \mathcal{P} ( X ) ) : \mathcal{S}\text{ is a }\sigma\text{-algebra on }X \}$$ is a set, and so are any subcollections of this set.
The paradoxes of naive set theory came about mainly because the "size" of the collections considered were unbounded; e.g., the family of all sets, the family of all ordinal numbers, etc. As long as you restrict yourself to considering collections that are built from some specific set $X$ in a definite way (using, for example, power set and subset definition) no paradoxes have ever come around.
Note that if $\{ \mathcal{S}_i : i \in I \} \subseteq \mathfrak{S}$ is nonempty, then picking any $i_0 \in I$ we have that $$\bigcap_{i \in I} \mathcal{S}_i = \{ A \in \mathcal{S}_{i_0} : ( \forall i \in I ) ( A \in \mathcal{S}_i ) \}$$ and so this causes no problems since we are using a subset definition of a "good" set.
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http://math.stackexchange.com/users/50894/v4r?tab=questions | # v4r
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https://www.ideals.illinois.edu/handle/2142/14771/browse?type=subject&value=threonine | # Browse Dissertations and Theses - Animal Sciences by Subject "threonine"
• (2012-09-18)
Three experiments were conducted to determine the combined effects of purified dietary fiber and dietary threonine (Thr) on chick growth and intestinal health. The objectives of the first two experiments were to define the ...
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• (2015-02-04)
Five experiments were conducted to determine the ideal Thr:Lys ratio for 25 to 50 kg growing gilts and the effects of fiber on said ratio. In Exp. 1, the objectives were to determine the standardized ileal digestible (SID) ...
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http://www.coreboot.org/pipermail/coreboot/2009-April/047141.html | # [coreboot] [v2] r4193 - trunk/coreboot-v2/documentation
svn at coreboot.org svn at coreboot.org
Thu Apr 23 05:59:33 CEST 2009
Author: rminnich
Date: 2009-04-23 05:59:33 +0200 (Thu, 23 Apr 2009)
New Revision: 4193
Modified:
trunk/coreboot-v2/documentation/LinuxBIOS-AMD64.tex
Log:
This continues the doco attempt.
This also mentions some ideas on the new booting setup for v2.
The latest changes will remove all need for people to do math.
With Peter's corrections as well.
Signed-off-by: Ronald G. Minnich <rminnich at gmail.com>
Acked-by: Marc Jones <marcj303 at gmail.com>
Modified: trunk/coreboot-v2/documentation/LinuxBIOS-AMD64.tex
===================================================================
--- trunk/coreboot-v2/documentation/LinuxBIOS-AMD64.tex 2009-04-22 23:39:19 UTC (rev 4192)
+++ trunk/coreboot-v2/documentation/LinuxBIOS-AMD64.tex 2009-04-23 03:59:33 UTC (rev 4193)
@@ -1551,7 +1551,7 @@
\subsection{romcc images (from emulation/qemu)}
ROMCC images are so-called because C code for the ROM part is compiled with romcc. romcc is an optimizing C compiler which compiles one, and only
one file; to get more than one file, one must include the C code via include statements. The main ROM code .c file is usually called auto.c.
-\subsubsection{how it is built}
+\subsubsection{How it is built}
Romcc compiles auto.c to produce auto.inc. auto.inc is included in the main crt0.S, which is then preprocessed to produce crt0.s. The inclusion of files into crt0.S is controlled by the CRT0\_INCLUDES variable. crt0.s is then assembled.
File for the ram part are compiled in a conventional manner.
@@ -1630,7 +1630,7 @@
What does this mean? the non-fallback image has a 32-bit entry point; fallback has a 16-bit entry point. The reason for this is that some code from fallback always runs, so as to pick fallback or normal; but the normal is always called from 32-bit code.
CAR images in their simplest form are modified romcc images. The file is usually cache\_as\_ram\_auto.c. C inclusion is still used. The main difference is in the build sequence. The compiler command line is a very slight changed: instead of using romcc to generate an auto.inc include file, gcc us used. Then, two perl scripts are used to rename the .text and .data sections to .rom.text and .rom.data respectively.
-\subsubsection{how it is built}
+\subsubsection{How it is built}
The build is almost identical to the romcc build. Since the auto.inc file exists, it can be included as before. The crt0\_includes.h file has one addition: a file that enables CAR, in this case it is \textit{src/cpu/amd/model\_lx/cache\_as\_ram.inc}.
\subsubsection{layout}
No significant change from romcc code.
@@ -1640,7 +1640,7 @@
\subsection{car + CONFIG\_USE\_INIT images (new emulation/qemu}
This type of image makes more use of the C compiler. In this type of image, in fact,
seperate compilation is possible but is not always used. Oddly enough, this option is only used in PPC boards. That said, we need to move to this way of building. Including C code is poor style.
-\subsubsection{how it is built}
+\subsubsection{How it is built}
There is a make variable, INIT-OBJECTS, that for all our other targets is empty. In this type of build, INIT-OBJECTS is a list of C files that are created from the config tool initobject command. Again, with INIT-OBJECTS we can finally stop including .c files and go with seperate compilation.
\subsubsection{layout}
No significant change from romcc code.
@@ -1687,7 +1687,128 @@
\subsubsection{boot sequence}
No significant change from romcc code, except that the CAR code has to set up a stack.
\subsection{failover}
+Failover is the newest way to lay out a ROM. The choice of which image to run is removed from the fallback image and moved into a small, standalone piece of code. The code is simple enough to show here:
+\begin{verbatim}
+static unsigned long main(unsigned long bist)
+{
+ if (do_normal_boot())
+ goto normal_image;
+ else
+ goto fallback_image;
+normal_image:
+ __asm__ __volatile__("jmp __normal_image" : : "a" (bist) : );
+
+cpu_reset:
+ __asm__ __volatile__("jmp __cpu_reset" : : "a" (bist) : );
+
+fallback_image:
+ return bist;
+}
+
+\end{verbatim}
+Some motherboards have a more complex bus structure (e.g. Opteron). In those cases, the failover can be more complex, as it requires some hardware initialization to work correctly. As of this writing (April 2009), these boards have their own failover:
+\begin{quote}
+./src/mainboard/iei/nova4899r/failover.c
+./src/mainboard/emulation/qemu-x86/failover.c
+./src/mainboard/supermicro/x6dhr_ig/failover.c
+./src/mainboard/supermicro/x6dai_g/failover.c
+./src/mainboard/supermicro/x6dhe_g2/failover.c
+./src/mainboard/supermicro/x6dhr_ig2/failover.c
+./src/mainboard/supermicro/x6dhe_g/failover.c
+./src/mainboard/dell/s1850/failover.c
+./src/mainboard/intel/xe7501devkit/failover.c
+./src/mainboard/intel/jarrell/failover.c
+./src/mainboard/olpc/btest/failover.c
+./src/mainboard/olpc/rev_a/failover.c
+./src/mainboard/via/epia-m/failover.c
+\end{quote}
+Here is one of the more complicated ones:
+\begin{verbatim}
+static unsigned long main(unsigned long bist)
+{
+ /* Did just the cpu reset? */
+ if (memory_initialized()) {
+ if (last_boot_normal()) {
+ goto normal_image;
+ } else {
+ goto cpu_reset;
+ }
+ }
+
+ /* This is the primary cpu how should I boot? */
+ else if (do_normal_boot()) {
+ goto normal_image;
+ }
+ else {
+ goto fallback_image;
+ }
+ normal_image:
+ asm volatile ("jmp __normal_image"
+ : /* outputs */
+ : "a" (bist) /* inputs */
+ : /* clobbers */
+ );
+ cpu_reset:
+ asm volatile ("jmp __cpu_reset"
+ : /* outputs */
+ : "a"(bist) /* inputs */
+ : /* clobbers */
+ );
+ fallback_image:
+ return bist;
+}
+
+\end{verbatim}
+They're not that different, in fact. So why are there different copies all over the tree? Simple: code inclusion. Most of the failover.c are different because they include different bits of code. Here is a key reason for killing C code inclusion in the tree.
+\subsubsection{How it is built}
+\begin{itemize}
+\item HAVE\_FAILOVER\_IMAGE Has to be defined when certain files are included.
+\item USE\_FAILOVER\_IMAGE Enables the use of the failover image
+\end{itemize}
+Confusingly enough, almost all the uses of these two variables are either nested or both required to be set, e.g.
+The fallback and normal builds are the same. The target config has a new clause that looks like this:
+\begin{verbatim}
+romimage "failover"
+ option USE_FAILOVER_IMAGE=1
+ option USE_FALLBACK_IMAGE=0
+ option ROM_IMAGE_SIZE=FAILOVER_SIZE
+ option XIP_ROM_SIZE=FAILOVER_SIZE
+ option COREBOOT_EXTRA_VERSION="\\$(shell cat ../../VERSION)\_Failover"
+end
+\end{verbatim}
+This new section uses some constructs not yet discussed in detail. XIP\_ROM\_SIZE just refers to the
+fact that the failover code is eXecute In Place, i.e. not copied to RAM. Of course, the ROM part of normal/fallback is as well, so the usage of XIP here is somewhat confusing. Finally, the USE\_FAILOVER\_IMAGE variable is set, which changes code compilation in a few places. If we just consider non-mainbard files, there are:
+\begin{verbatim}
+src/cpu/amd/car/cache_as_ram.inc
+src/arch/i386/Config.lb
+\end{verbatim}
+For the cache\_as\_ram.inc file, the changes relate to the fact that failover code sets up CAR, so that fallback code need not.
+
+For the Config.lb, several aspects of build change.
+When USE\_FAILOVER\_IMAGE, entry into both normal and fallback bios images is via a 32-bit entry point (when not defined, entry into fallback is a 16-entry point at the power-on reset vector).
+\subsubsection{layout}
+Failover.c becomes the new bootblock at the top of memory. It calls either normal or fallback. The address of normal and fallback is determined by ldscript magic.
+\subsubsection{boot sequence}
+failover.c tests a few variables and the calls the normal or fallback payload depending on those variables; usually they are CMOS settings.
+\subsection{Proposed new image forat}
+The new image format will use seperate compilation -- no C code included! -- on all files.
+
+The new design has a few key goals:
+\begin{itemize}
+\item Always use a bootblock (currently called failover).
+The name failover.c, being utterly obscure, will not be used; instead, we will name the file bootblock.c. Instead of having a different copy for each mainboard, we can have just one copy.
+\item Always use seperate compilation
+\item Always use printk etc. in the ROM code
+\item (longer term) from the bootblock, always use cbfs to locate the normal/fallback etc. code. This code will be XIP.
+\end{itemize}
+
+\subsubsection{How it is built}
+For now, since we are still using the config tool, we'll need a new command: bootblockobject, which creates a list of files to be included in the bootblock. Not a lot else will have to change. We are going to move to using the v3 CAR code assembly code (one or two files at most, instead of many) and, instead of the thicket of little ldscript files, one ldscript file. This strategy is subject to modification as events dictate.
+\subsubsection{layout}
+Almost the same, for now, as the current failover code.
+\subsubsection{boot sequence}
%
% 14 Glossary
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https://www.groundai.com/project/recent-star-formation-in-high-redshift-early-type-galaxies-insights-from-the-rest-frame-uv/ | [
# [
[
###### Abstract
We combine deep photometry from the MUSYC survey with redshifts from the COMBO-17 survey to study the rest-frame ultraviolet () properties of 674 high-redshift () early-type galaxies, drawn from the Extended Chandra Deep Field South (E-CDFS). Galaxy morphologies are determined through visual inspection of Hubble Space Telescope (HST) images taken from the GEMS survey. We harness the sensitivity of the to young ( Gyrs old) stars to quantify the recent star formation history of the early-type population. We find compelling evidence that early-types of all luminosities form stars over the lifetime of the Universe, although the bulk of their star formation is already complete at high redshift. Luminous () early-types form 10-15 percent of their mass after , while their less luminous () counterparts form 30-60 percent of their mass in the same redshift range.
Keywords. Galaxies: elliptical and lenticular, cD, galaxies: evolution, galaxies: formation, galaxies: stellar contents, ultraviolet: galaxies
SFH of early-type galaxies at ] Recent star formation in high-redshift early-type galaxies: insights from the rest-frame UV S. Kaviraj et al.] S. Kavirajthanks: skaviraj@astro.ox.ac.uk, S. K. Yi, E. Gawiser, P. G. van Dokkum, S. Khochfar, K. Schawinski and J. Silk
## 1 Introduction
The vast majority of work on early-type galaxies in the past has focussed on optical spectro-photometric data. A significant drawback of optical photometry is its lack of sensitivity to moderate amounts of recent star formation (RSF). While red optical colours imply a high-redshift formation epoch for the bulk of the stellar population in early-type galaxies, the optical spectrum remains largely unaffected by the minority of stellar mass that forms in these systems at low and intermediate redshift. As a result it is virtually impossible to quantify early-type star formation histories (SFHs) over the last 8 billion years ().
RSF can be efficiently traced using the rest-frame ultraviolet () spectrum, which is sensitive to young, massive main sequence stars with ages less than 1 Gyr. Using a large sample of early-type galaxies detected by the GALEX space telescope, Kaviraj et al. (2006) have recently shown that, contrary to the expectations of traditional ‘monolithic collapse’ models (e.g. Bower et al. 1992), local early-types show widespread recent star formation (RSF) - at least 30 percent show blue -optical colours indicative of unambiguous RSF in these systems. The work presented here extends these results to high redshift by exploiting deep optical photometry to trace the rest-frame properties of early-type galaxies in the redshift range .
## 2 Estimation of recent star formation
The RSF in individual galaxies is calculated by comparing their multi-wavelength photometry with synthetic galaxy populations, generated in the framework of the standard model. Synthetic populations are generated using the semi-analytical model of Khochfar and Burkert (2003). A library of synthetic photometry is constructed by combining each of 15,000 model galaxies with a single metallicity in the range to , dust extinction parametrised by a value of in the range 0 to 0.5 and convolving them with the stellar models of Yi (2003). We construct an ‘average SFH’ for each galaxy, by combining the SFHs of all models in the library weighted by their individual likelihoods. The RSF, defined as the mass fraction of stars formed within the last Gyr, is calculated from this average SFH. Figure 1 shows the derived rest-frame colour-magnitude relation and the values of RSF in the E-CDFS galaxy population.
## 3 Summary and discussion
The early-type population as a whole exhibits a typical RSF between 5 and 13 percent in the redshift range , while the early-types on the broad ‘red sequence’, (), typically show RSF values less than 5 percent. The reddest early-types (which are also the most luminous) are virtually quiescent with RSF values of percent. Since the timescale of this study is Gyrs, a simple extrapolation (from RSF values in Figure 1) indicates that luminous () early-type galaxies typically form up to 10-15 percent of their mass after (with a tail to higher values), while less luminous early-types () form 30-60 percent of their mass after .
• Bower et al. (1992) Bower, R. G., J. R. Lucey, and R. Ellis (1992). MNRAS 254, 589.
• Kaviraj and GALEX Science Team (2006) Kaviraj, S. and GALEX Science Team (2006). ApJ GALEX dedicated issue, Dec 2007.
• Khochfar and Burkert (2003) Khochfar, S. and A. Burkert (2003). ApJ 597, L117–L120.
• Yi (2003) Yi, S. K. (2003). ApJ 582, 202–214.
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https://it.overleaf.com/latex/examples/footnote-citations-with-biblatex/sdnfkjhrnpkp | AbstractThis example shows how to create citations in footnotes using biblatex. Biblatex automatically formats references and citations, much like BibTeX, but biblatex is more robust and more powerful. You can (almost certainly) use your existing .bib databases with biblatex, it comes with a wide variety of styles built in, and it's much easier to write your own custom styles. In this example, we use the verbose-ibid style to generate footnotes with automatic “ibid.” abbreviations. For a full list of styles, see the user guide in the biblatex manual. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37155237793922424, "perplexity": 2662.988326680667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711013.11/warc/CC-MAIN-20221205064509-20221205094509-00454.warc.gz"} |
https://physics.stackexchange.com/questions/531098/why-do-batteries-need-a-loop-to-discharge | Why do batteries need a loop to discharge?
I'm an electrical engineer who has been taught transistors and silison and solid state theory and all that. But when I apply the theory to batteries and cathodic protection, my understanding seems to break down. Can someone explain how batteries and cathodic protection work, from the point of view of band diagrams?
I understand that when two metals touch, their outer electrons will not be of the same energy, so the electrons will move from one metal to the other, leaving both metals charged. I also understand that this charge will dissipate at the other surfaces of both metals, such that near the surface, the metals appear far less charged.
Then, if this combination is dunked in an ionic solution (such as lead and lead oxide in sulphuric acid, or zinc and steel in salt water) then ions of opposing charge will each gather at one of the metals, pulled in by the charge (I guess?), then react at the surface, causing .... the charge to build the other way? This part has me confused.
In a battery the metals are not touching, so the charge builds up and a voltage appears across the terminal, and this charge pushes away the ions, slowing the reaction and slowing the build up of charge. If a circuit is connected, then the metals are "touching" via this circuit and charge flows from one metal to the other, allowing the reaction with the disolved ions to proceed.
But then in cathodic protection, the metals are bolted to one another, and both will react with the ions, causing a build up of charge, which can dissipate to the other metal via the interface. I understand that then a big metal boat, with a big cathodic protection, would have a voltage (and current flow) across its hull, as the reaction with the steel of the hull will be spread across the surface of hull where it meets the water. But then why would this potential difference continue above the water, and if and anchor were tossed over to land, and the chain didn't touch the water, why would there be current through to ground? Why does it not all flow back through the bolt connecting the zinc and to the boat?
• This is a standard exercise in electrochemistry- specifically corrosion science. A textbook on corrosion will answer all your questions. – niels nielsen Feb 14 at 18:08
In a battery and electrochemical cells we typically explain the phenomenon using reduction potentials (these can be gotten from a standard reduction potential table). The Gibbs free energy dictates whether a reaction will happen and for electrochemical cells is given by the Nernst equation $$\Delta G=-nFE=-RT\ln K_\mathrm{eq}+RT\ln Q$$ so we need negative Gibbs free energy for a reaction to occur which means positive $$E$$ (cell potential). Also we can relate the Gibbs free energy to the chemical potential through thermodynamics $$\Delta G=\mu\,\mathrm dN$$ The reaction is between the two metals where one is oxidized (loses electrons) in this case the zinc anode and one is reduced (gains electrons) in this case the iron cathode. Oxidation happens at the anode and reduction happens at the cathode, thus cathodic protection as metal oxides are reduced back to their pure metals. When the metals are not touching as is typically the case in batteries(galvanic cells) the electrons travel through an outer circuit creating electrical energy and half cell redox reactions occur at each terminal(anode and cathode). The reactions cannot occur if the circuit is not closed as the cathodes sides reaction needs electrons and the anodes sides reaction expels electrons. The electrolyte serves to prevent charge buildup as during the oxidation reaction at the anode the positive metal ions are produced and need negative electrolyte ions to balance while at the cathode the positive metal ions are reduced leaving their negative counter parts that need to be balanced by positive electrolyte ions. If charge buildup occurs the reaction will stop and no current will flow (this is almost immediate). Cathodic protection can occur with the metals touching or not touching. When the metals are touching the full redox reaction occurs and no external circuit is necessary. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7614922523498535, "perplexity": 872.2982380571539}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400203096.42/warc/CC-MAIN-20200922031902-20200922061902-00044.warc.gz"} |
https://www.ihes.fr/en/library/ | The N. H. Kuiper Library collection focuses on research level materials in mathematics and pure physics. The library also offers various services.
The journal Les Publications mathématiques de l’IHES publishes two annual volumes totalling 400 pages and is a global reference in its field. Preprints collect works conducted at the Institute and often show the scientific collaborations between invited researchers and professors.
# The Nicolaas Hendrik Kuiper library
The new IHES library was inaugurated on 23 May 2003 and bears the name of the Institute’s second director, thus honouring the memory of Nicolaas Hendrik Kuiper. The library was created as soon as the Institute moved to Bures-sur-Yvette and was initially situated in the music pavilion in the Bois-Marie site. It is now located on the first floor of the science building and the view it offers of the surrounding woods is conducive to study and inspiration for its readers.
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# Les publications mathématiques de l’IHES
Published internationally in three languages, Les Publications mathématiques de l’IHES is one of the most prestigious journals in its field.
Les Publications mathématiques de l’IHES was created in 1959 under the leadership of Léon Motchane, and Jean Dieudonné, professor at the institute.
Issues were first published at irregular intervals and in four languages: French, English, German and Russian. (The latter language has now been abandoned). Gradually, IHES published two annual volumes totalling 400 pages. Since 2012, the journal has had a circulation of 320 printed copies. It is also available on line and on numdam.org.
Les Publications mathématiques de l’IHES is an international journal publishing papers of highest scientific level. Thanks to its worldwide distribution (it can be found in the libraries of the major mathematical institutions in the world), to its tradition of publishing landmark articles and to its broad coverage of the discipline, it has won international recognition.
Jean Dieudonné was the journal’s first editor-in-chief. Jacques Tits succeeded him in 1980 to 1998, followed by Etienne Ghys until 2009.
The editorial board has expanded over the years. Pierre Deligne, Dennis Sullivan, René Thom, all IHES professors, were the first to join the committee in 1980.
They were followed by Jean Bourgain in 1987, Mikhail Gromov in 1982, Maxim Kontsevich in 1996, Laurent Lafforgue in 1999 and Claire Voisin in 2007.
Les Publications mathématiques de l’IHES was managed in 2010 and 2011 by Claire Voisin and Sergiu Klainerman and has been managed by Claire Voisin since 1 January 2012.
## Current editorial board
This has now been extended to scientists outside the IHES
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Preprint Date PDF
Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak $\infty$-Groupoids for Spaces M/18/03
16/01/2018 PDF
Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces M/18/02
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The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area is recently called as "Dynamical Topology". For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we (mostly) survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". Slovak space is a dynamical analogue of the rigid space: a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness: all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.
08/01/2018 PDF
From Euler's play with infinite series to the anomalous magnetic moment P/18/01
05/01/2018 PDF
Introduction to Higher Cubical Operads. First Part: The Cubical Operad of Cubical Weak $\infty$-Categories M/17/19
21/12/2017 PDF
In memoriam: Cécile DeWitt-Morette M/17/15
08/12/2017 PDF
Ars combinatoria'' chez Gian-Carlo Rota ou le triomphe du symbolisme M/17/18
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Gian-Carlos Rota est l'inventeur d'une nouvelle science : la combinatoire algébrique qui n'était jusque-là qu'un ensemble de questions disparates, dont la solution, parfois ingénieuse, ne laissait entrevoir la méthode. Il s'agit ici de décrire la stratégie mise en oeuvre sous sa forme mathématique dont les enjeux philosophiques ne sont pas négligeables, en poursuivant le geste de Rota, pour sortir la combinatoire de son ghetto.
06/12/2017 PDF
Generalized conformal Hamiltonian dynamics and the pattern formation equations M/17/16
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We demonstrate the significance of the Jacobi last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator-inhibitor (AI) systems. We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems, and demonstrate that various subsystems of AI, depending on the choices of parameters, are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics are subclasses of another dynamics, known as Jacobi mechanics. Furthermore we show that for non Turing pattern formation, like the Gray-Scott model, may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians. Finally, we construct a locally defined dissipative Hamiltonian generating function \cite{HHG} of the original system. This generating function coincides with the free energy'' of the associated system if it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the method.
05/12/2017 PDF
Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy M/17/17
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The hierarchy of equations belonging to two different but related integrable systems, the Nonlinear Schrödinger and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not preserve the integrability, only asymptotically integrable, and non-holonomic deformation (N HD) that does. QID is carried out generically for the NLS hierarchy while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No QI anomaly is observed at the level of EOMs which suggests that at that level the QID may be identified as some integrable deformation. NHD is applied to the NLS hierarchy generally as well as with the specific focus on the NLS equation itself and the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN) and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently, different aspects of the results are discussed.
05/12/2017 PDF
Noncommutative Catalan Numbers M/17/14
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The goal of this paper is to introduce and study non-commutative Catalan numbers C_n which belong to the free Laurent polynomial algebra in n generators. Our non-commutative numbers admit interesting (commutative and non-commutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another to solving non-commutative quadratic equations. We also establish total positivity of the corresponding non-commutative Hankel matrices and introduce accompanying nonc-mmutative binomial coefficients.
25/08/2017 PDF
Theory of Morphogenesis M/17/13
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A model of morphogenesis is proposed based upon seven explicit postulates. The mathematical import and biological significance of the postulates are explored and discussed.
08/06/2017 PDF
Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription P/17/12
07/06/2017 PDF
Properties of soliton surfaces associated with integrable $\mathbb{C}P^{N-1}$ sigma models P/17/11
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We investigate certain properties of $\\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel\'fand have been discussed in detail. Explicit connections among these three surfaces are also established by purely analytical descriptions and, it is demonstrated that the three immersion formulas actually correspond to the single surface parametrized by some specific conditions.
06/06/2017 PDF
q-deformed quadrature operator and optical tomogram P/17/10
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In this paper, we define the homodyne q-deformed quadrature operator and find its eigenstates in terms of the deformed Fock states. We find the quadrature representation of q-deformed Fock states in the process. Furthermore, we calculate the explicit analytical expression for the optical tomogram of the q-deformed coherent states.
22/05/2017 PDF
SKEW PRODUCT SMALE ENDOMORPHISMS OVER COUNTABLE SHIFTS OF FINITE TYPE M/17/09
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We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi (EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we
14/05/2017 PDF
M-theoretic Lichnerowicz formula and supersymmetry P/17/06
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A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric operators to the effective action, the Bianchi identities for fluxes, and some equations of motion. Recently, such formulae have also been shown to underlie the (generalised) geometry of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that describes eleven-dimensional supergravity together with its higher-derivative couplings. The first corrections to the action appear at eight-derivative level, and the construction yields two different supersymmetric invariants, each with a free coefficient. We discuss the restriction of our construction to seven-dimensional internal spaces, and implications for compactifications on manifolds of G 2 holonomy. Inclusion of fluxes and computation of contributions with higher than eight derivatives are also discussed.
12/05/2017 PDF
Modification of Schrodinger-Newton equation due to braneworld models with minimal length P/17/07
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We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schrodinger-Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory.
12/05/2017 PDF
Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure P/17/08
12/05/2017 PDF
Software modules and computer-assisted proof schemes in the Kontsevich deformation quantization M/17/05
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The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the non-commutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter ħ. Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo o(ħ^4), for the newly built ⋆-product expansion.
11/05/2017 PDF
Galois equivariance of critical values of $L$-functions for unitary groups M/17/04
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The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic $L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture.
08/04/2017 PDF
Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra P/17/03
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We continue the study undertaken in [DV] of the exceptional Jordan algebra $J = J_3^8$ as (part of) the finite-dimensional quantum algebra in an almost classical space-time approach to particle physics. Along with reviewing known properties of $J$ and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel-Siebenthal theory of maximal subgroups of simple compact Lie groups.
30/03/2017 PDF
Examples of pre-CY structures, associated operads and cohomologies M/17/02
01/02/2017 PDF
On the four-loop static contribution to the gravitational interaction potential of two point masses P/17/01
19/01/2017 PDF
Three-body problem in 3D space: ground state, (quasi)-exact-solvability P/16/29
22/11/2016 PDF
Diffeomorphisms of quantum fields P/16/28
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We study field diffeomorphisms $\Phi(x)= F(\rho(x))=a_0\rho(x)+a_1\rho^2(x)+\ldots=\sum_{j+0}^\infty a_j \rho^{j+1},$ for free and interacting quantum fields $\Phi$. We find that the theory is invariant under such diffeomorphisms if and only if kinematic renormalization schemes are used.
07/10/2016 PDF
Sur la dualité des topos et de leurs présentations et ses applications : une introduction M/16/26
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Ce texte est une version écrite enrichie des notes d'un exposé donné à l'Université de Nantes le 1er avril 2016. Il a été rédigé par le second auteur, à partir de notes succinctes et d'expositions orales du premier auteur. Il peut servir d'introduction à la technique des topos comme ponts et à ses applications, en particulier pour les géomètres peu familiers de la logique catégorique et de la théorie des topos classifiants.
22/09/2016 PDF
Le principe de fonctorialité de Langlands comme un problème de généralisation de la loi d'addition M/16/27
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Ce texte représente les notes écrites d'un cours donné à l'Université de Nottingham en juin 2016. Dans la continuité des écrits précédents de l'auteur, il étudie le transfert automorphe de Langlands des groupes réductifs vers les groupes linéaires sous la forme équivalente de la définition d'opérateurs de transformation de Fourier locaux et globaux sur les groupes réductifs induits par les représentations de leurs groupes duaux, d'espaces fonctionnels locaux et globaux fixés par ces opérateurs et d'une fonctionnelle linéaire de Poisson globale qui serait invariante par transformation de Fourier. Se fondant sur une étude du cas des tores, il propose dans le cas des groupes réductifs non abéliens généraux une définition conjecturale des espaces fonctionnels recherchés et une caractérisation conjecturale de la fonctionnelle de Poisson associée. Il montre que la définition des opérateurs de transformations de Fourier et l'éventuelle vérification des propriétés attendues de ces espaces fonctionnels et de cette fonctionnelle de Poisson posent la question cruciale de la construction et de l'étude des opérateurs de convolution (transformés de Fourier de l'opérateur de multiplication point par point des fonctions) associés. Le texte propose pour ces opérateurs de convolution une conjecture d'algébricité qui semble plus raisonnable que la conjecture de "stabilité du tore par convolution" qui avait été faite dans la précédente pr
22/09/2016 PDF
Algebraic flows on Shimura varieties M/16/21
15/09/2016 PDF
Holomorphic curves in compact Shimura varieties M/16/22
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We prove a hyperbolic analogue of the We prove an analogue of the Bloch-Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties.
15/09/2016 PDF
Structures spéciales et problème de Zilber-Pink M/16/23
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The Manin-Mumford and the André-Oort conjectures as well as the one formulated by Zilber and Pink concern algebraic varieties (algebraic tori, Abelian or semi-Abelian varieties, pure or mixed Shimura varieties) endowed with a natural set of special points and special subvarieties. An axiomatisation, in the spirit of model theory, is presented for a description of algebraic varieties endowed with a natural set of special points and special subvarieties with an emphasis on the bi-algebraic nature of the question. The text also reviews recent results on these conjectures.
15/09/2016 PDF
Bi-algebraic geometry and the André-Oort conjecture M/16/24
15/09/2016 PDF
O-minimal flows on abelian varieties M/16/25
15/09/2016 PDF
Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory P/16/20
09/09/2016 PDF
Golod-Shafarevich type theorems and potential algebras M/16/19
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Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Groebner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras, and prove that they can not have dimension smaller than 8, using Groebner bases arguments, and arguing in terms of associated truncated algebra. We derive from the improved version of the Golod-Shafarevich theorem,that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree n>=3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class P_n of potential algebras with homogeneous potential of degree n+1>= 4, the minimal Hilbert series is H_n=1/1-2t+2t^n-t^{n+1}, so they are all infinite dimensional. Moreover, growth could be polynomial (but at least quadratic) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invaria
02/07/2016 PDF
Divisor Braids M/16/18
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We study a novel type of braid groups on a closed orientable surface Σ. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on Σ; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over Σ. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph Γ. In situations where there is more than one strand of each colour, we show that the corresponding braid group admits a metabelian presentation as a central extension of the free Abelian group H_1(Σ;Z)^r, where r is the number of colours, and describe its Abelian commutator. This computation relies crucially on producing a link invariant (of closed divisor braids) in the three-manifold S^1×Σ for each graph Γ. We also describe the von Neumann algebras associated to these groups in terms of rings that are familiar from noncommutative geometry.
25/05/2016 PDF
One question from the Polishchuk and Positselski book on Quadratic algebras M/16/16
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In the book 'Quadratic algebras' due to Polishchuk and Positselski algebras with small number of generators (n=2,3) is considered. For some number of relations r possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and three relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebra A with dim A_1=dim A_2=3 we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not.
05/05/2016 PDF
On the proof of the homology conjecture for monomial non-unital algebras M/16/15
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We consider the bar complex of a monomial non-unital associative algebra A=k / (w_1,...,w_t). For any fixed monomial w=x_1..x_n in A one can define certain subcomplex of the Bar complex of A. It was conjectured in [3] that homology of this complex is at most one. We prove here this conjecture, and describe the place where this nontrivial homology appears in terms of length of the Dyck path associated to a given word in A.
04/05/2016 PDF
Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy M/16/14
19/04/2016 PDF
Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism P/16/13
01/04/2016 PDF
Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors? M/16/12
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We examine two claims from the paper "Formality Conjecture" (Ascona 1996): specifically, that 1) a certain tetrahedral graph flow preserves the class of (real-analytic) Poisson structures, and that 2) another tetrahedral graph flow vanishes at every such Poisson structure. By using twelve Poisson structures with high-order polynomial coefficients as explicit counterexamples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at the Poisson bi-vectors. The counterexamples at hand themselves suggest a correction to the formula for the "exotic" flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance (1:6) for which the flow does preserve the space of Poisson bi-vectors.
31/03/2016 PDF
BADLY APPROXIMABLE VECTORS AND FRACTALS DEFINED BY CONFORMAL DYNAMICAL SYSTEMS M/16/10
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We prove that if J is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in J. The same is true if J is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of J that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author (’12) by showing that every hyperplane diffuse set supports an absolutely decaying measure.
22/03/2016 PDF
Real Analyticity for random dynamics of transcendental functions M/16/11
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Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh \cite{Rug08} leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge. We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.
22/03/2016 PDF
SU(N) transitions in M-theory on Calabi-Yau fourfolds and background fluxes P/16/07
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We study M-theory on a Calabi-Yau fourfold with a smooth surface S of AN−1 singularities. The resulting three-dimensional theory has a =2 SU(N) gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional =1 SU(N) gauge theory on the surface S. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial SU(N) principal bundle over S, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an AN−1 singularity in codimension two.
09/03/2016 PDF
New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole P/16/08
09/03/2016 PDF
High post-Newtonian order gravitational self-force analytical results for eccentric orbits around a Kerr black hole P/16/09
09/03/2016 PDF
Intermittency in the Hodgkin-Huxley model M/16/06
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We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.
26/02/2016 PDF
The Langlands-Shahidi method over function fields: the Ramanujan Conjecture and the Riemann Hypothesis for the unitary groups M/16/05
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On étudie la méthode de Langlands-Shahidi sur les corps de fonctions de caractéristique p. On prouve la fonctorialité de Langlands globale et locale des groupes unitaires vers les groupes linéaires pour les représentations génériques. Supposant connue la conjecture de Shahidi pour les L-paquets modérés, on donne une extension de la définition des fonctions L et des facteurs ε. Enfin, utilisant le travail de L. Lafforgue, on établit la conjecture de Ramanujan et on prouve que les fonctions L automorphes de Langlands-Shahidi satisfont l'hypothèse de Riemann.
11/02/2016 PDF
Class number problems and Lang conjectures M/16/04
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Given a square-free integer d we introduce an affine hypersurface whose integer points are in one-to-one correspondence with ideal classes of the quadratic number field Q(\sqrt{d}). Using this we relate class number problems of Gauss to Lang conjectures.
01/02/2016 PDF
On Emergent Geometry from Entanglement Entropy in Matrix Theory P/16/03
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Using Matrix theory, we compute the entanglement entropy between a supergravity probe and modes on a spherical membrane. We demonstrate that a membrane stretched between the probe and the sphere entangles these modes and leads to an expression for the entanglement entropy that encodes information about local gravitational geometry seen by the probe. We propose in particular that this entanglement entropy measures the rate of convergence of geodesics at the location of the probe.
29/01/2016 PDF
Perturbative quantum field theory meets number theory P/16/02
27/01/2016 PDF
On the conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity P/16/01
08/01/2016 PDF
Modular Graph Functions P/15/29
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We consider properties of modular graph functions, which are non-holomorphic modular functions associated with the Feynman graphs for a conformal scalar field theory on a two-dimensional torus. Such functions arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We demonstrate that these functions are sums, with rational coefficients, of special values of single-valued elliptic multiple polylogarithms, which will be introduced in this paper. This insight suggests the many interrelations between these modular graph functions (a few of which were motivated in an earlier paper) may be obtained as a consequence of identities involving elliptic polylogarithms.
22/12/2015 PDF
Cutkosky Rules and Outer Space P/15/34
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We derive Cutkosky’s theorem starting from Pham’s classical work. We emphasize structural relations to Outer Space.
08/12/2015 PDF
Higher Chern classes in Iwasawa theory M/15/33
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We begin a study of m-th Chern classes and m-th characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz p-adic L-functions.
05/12/2015 PDF
Confirming and improving post-Newtonian and effective-one-body results from self-force computations along eccentric orbits around a Schwarzschild black hole P/15/31
04/12/2015 PDF
Minisuperspace quantum supersymmetric cosmology (and its hidden hyperbolic Kac-Moody structures) P/15/32
04/12/2015 PDF
Persistent homology and string vacua P/15/30
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We use methods from topological data analysis to study the topological features of certain distributions of string vacua. Topological data analysis is a multi-scale approach used to analyze the topological features of a dataset by identifying which homological characteristics persist over a long range of scales. We apply these techniques in several contexts. We analyze N=2 vacua by focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg models. We then turn to flux compactifications and discuss how we can use topological data analysis to extract physical informations. Finally we apply these techniques to certain phenomenologically realistic heterotic models. We discuss the possibility of characterizing string vacua using the topological properties of their distributions.
03/12/2015 PDF
Catégories syntactiques pour les motifs de Nori M/15/26
13/11/2015 PDF
A new effective-one-body Hamiltonian with next-to-leading order spin-spin coupling P/15/27
13/11/2015 PDF
Spin-dependent two-body interactions from gravitational self-force computations P/15/28
13/11/2015 PDF
BPS spectra, barcodes and walls P/15/25
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BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper we approach this problem from the perspective of persistent homology. Persistent homology is at the base of topological data analysis, which aims at extracting topological features out of a set of points. We use these techniques to investigate the topological properties which characterize the spectra of several supersymmetric models in field and string theory. We discuss how such features change upon crossing walls of marginal stability in a few examples. Then we look at the topological properties of the distributions of BPS invariants in string compactifications on compact threefolds used to engineer black hole microstates. Finally we discuss the interplay between persistent homology and modularity by considering certain number theoretical functions used to count dyons in string compactifications and by studying equivariant elliptic genera in the context of the Mathieu moonshine.
04/11/2015 PDF
Moduli Spaces and Macromolecules M/15/24
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Techniques from moduli spaces are applied to biological macromolecules. The first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally. The second main result identifies up to homotopy the natural moduli space of several interacting RNA molecules with the Riemann moduli space of a surface with several boundary components in each fixed genus. Applications to RNA folding prediction are discussed. The mathematical and biological frameworks are surveyed and presented from first principles.
30/10/2015 PDF
Smoothness and classicality on eigenvarieties M/15/23
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Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimension by making crucial use of the "patched eigenvariety".
07/10/2015 PDF
A program for branching problems in the representation theory of real reductive groups M/15/22
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We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{a priori}} known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.
29/09/2015 PDF
Decorated super-Teichmueller space M/15/21
25/09/2015 PDF
La suite spectrale de Hodge-Tate M/15/20
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The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the étale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic étale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differential forms by a construction reminiscent of the Cartier isomorphism.
14/09/2015 PDF
On the modular structure of the genus-one Type II superstring low energy expansion P/15/04
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The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order $D^{10} \cR^4$ are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.
02/09/2015 PDF
Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus P/15/07
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The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight $w$, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given $w$, but different numbers of loops $\le w-1$. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight $w=4$ and expresses the three-loop modular function $D_4$ in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities.
02/09/2015 PDF
More Graviton Physics P/15/05
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The interactions of gravitons with spin-1 matter are calculated in parallel with the well known photon case. It is shown that graviton scattering amplitudes can be factorized into a product of familiar electromagnetic forms, and cross sections for various reactions are straightforwardly evaluated using helicity methods. Universality relations are identified. Extrapolation to zero mass yields scattering amplitudes for photon-graviton and graviton-graviton scattering.
02/09/2015 PDF
Fractal Tube Formulas and a Minkowski Measurability Criterion for Compact Subsets of Euclidean Spaces M/15/17
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We establish fractal tube formulas valid for a large class of compact subsets (and more generally, relative fractal drums, RFDs) in Euclidean spaces of any dimension. These formulas express the volume of the tubular neighborhoods of the fractal under consideration in terms of the residues of the associated fractal zeta functions and their poles (i.e., the complex dimensions). Under suitable assumptions, we also show that a compact subset of R^N is Minkowski measurable if and only if its only complex dimension of maximum real part is the Minkowski (or box) dimension D of the fractal (or RFD), and D is simple. These results extend to arbitrary dimensions N greater than 1 the corresponding ones obtained by Lapidus and van Frankenhuijsen for fractal strings (i.e., when N =1). We illustrate them by means of several examples.
27/07/2015 PDF
Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums M/15/14
15/07/2015 PDF
Distance and Tube Zeta Functions of Fractals and Arbitrary Compact Sets M/15/15
15/07/2015 PDF
Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory M/15/16
15/07/2015 PDF
Deformation approach to quantisation of field models M/15/13
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Associativity-preserving deformation quantisation via Kontsevich's summation over weighted graphs is lifted from the algebras of functions on finite-dimensional Poisson manifolds to the algebras of local functionals within the variational Poisson geometry of gauge fields over the space-time.
14/07/2015 PDF
The Sound of Fractal Strings and the Riemann Hypothesis M/15/11
07/07/2015 PDF
Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis M/15/12
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An asymmetric criterion for the Riemann hypothesis is provided, expressed in terms of the invertibility of the spectral operator. This criterion is asymmetric, in the sense that it is valid for all fractal dimensions c in (0,1/2) if and only if the Riemann hypothesis is true, but that (unconditionally) it fails to be true for any value of c in (1/2, 1), a fact which is closely connected to the universality of the Riemann zeta function.
07/07/2015 PDF
Energetics and phasing of nonprecessing spinning coalescing black hole binaries P/15/19
28/06/2015 PDF
Deformations of complex structures on Riemann surfaces and integrable structures of Whitham type hierarchies M/15/10
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We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.
12/06/2015 PDF
On the large-scale geometry of the $L^p$-metric on the symplectomorphism group of the two-sphere M/15/09
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We prove that the vector space $R^d$ of any finite dimension $d$ with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms of the two-sphere endowed with the $L^p$-metric for $p>2$. Along the way we show that the $L^p$-metric on this group is unbounded for $p>2$ by elementary methods.
10/06/2015 PDF
The Equivalence Principle in a Quantum World P/15/06
20/05/2015 PDF
Lectures on Regular and Irregular Holonomic D-modules M/15/08
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This is a survey paper based on lectures given by the authors at Ihes, February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.
18/05/2015 PDF
Fourth post-Newtonian effective one-body dynamics P/15/18
30/04/2015 PDF
Analytic determination of high-order post-Newtonian self-force contributions to gravitational spin precession P/15/17
04/03/2015 PDF
Detweiler's gauge-invariant redshift variable: analytic determination of the nine and nine-and-a-half post-Newtonian self-force contributions P/15/16
09/02/2015 PDF
Differential symmetry breaking operators I. General theory and F-method M/15/02
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We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case symmetry breaking operators are characterized by differential equations of second order via the F-method.
11/01/2015 PDF
Differential symmetry breaking operators. II. Rankin--Cohen operators for symmetric pairs M/15/03
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Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae, for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose we use a new method (F-method) developed in the first part of the series and based on the algebraic Fourier transform for generalized Verma modules. The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulae, of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.
11/01/2015 PDF
Global uniqueness of small representations M/15/01
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We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
04/01/2015 PDF
On the integral law of thermal radiation P/14/43
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The integral law of thermal radiation by finite size emitters is studied. Two geometrical characteristics of a radiating body or a cavity, its volume and its boundary area, define two terms in its radiance. The term defined by the volume corresponds to the Stefan-Boltzmann law. The term defined by the boundary area is proportional to the third power of temperature and inversely proportional to the emitter's effective size, which is defined as the ratio of its volume to its boundary area. This generalized law is valid for arbitrary temperature and effective size. It is shown that the cubic temperature contribution is observed in experiments. This term explains the intrinsic uncertainty of the NPL experiment on radiometric determination of the Stefan-Boltzmann constant. It is also quantitatively confirmed by data from the NIST calibration of cryogenic blackbodies. Its relevance to the size of source effect in optical radiometry is proposed and supported by the experiments on thermal emission from nano-heaters.
12/12/2014 PDF
On weight modules of algebras of twisted differential operators on the projective space M/14/42
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We classify blocks of categories of weight and generalized weight modules of algebras of twisted differential operators on P^n. Necessary and sufficient conditions for these blocks to be tame and proofs that some of the blocks are Koszul are provided. We also establish equivalences of categories between these blocks and categories of bounded and generalized bounded weight sl(n+1)-modules in the cases of nonintegral and singular central character.
11/12/2014 PDF
Topological invariants in magnetohydrodynamics and DNA supercoiling M/14/41
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We discuss the structure of topological invariants in two different media. The first example relates to the problem of reconnection in magnetohydrodynamics and the second one to the supercoiling of DNA. Despite the apparently different systems, the behavior of magnetic spread lines and supercoiling process in DNA display some common features based on the existence of topological invariants of Hopf's type.
10/12/2014 PDF
$r_\infty$-Matrices, triangular $L_\infty$-bialgebras, and quantum$_\infty$ groups M/14/40
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A homotopy analogue of the notion of a triangular Lie bialgebra is proposed with a goal of extending the basic notions of theory of quantum groups to the context of homotopy algebras and, in particular, introducing a homotopical generalization of the notion of a quantum group, or quantum$_\infty$-group.
09/12/2014 PDF
The calculus of multivectors on noncommutative jet spaces M/14/39
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The Leibniz rule for derivations is invariant under cyclic permutations of the co-multiples within the derivations' arguments. We now explore the implications of this fundamental principle, developing the calculus of variations on the infinite jet spaces for maps from sheaves of free associative algebras over commutative manifolds to the quotients of free associative algebras over the linear relation of equivalence under cyclic shifts. In the frames of such variational noncommutative symplectic geometry, we prove the main properties of the Batalin-Vilkovisky Laplacian and variational Schouten bracket. As a by-product of this intrinsically regularised picture, we show that the structures that arise in the classical variational Poisson geometry of infinite-dimensional integrable systems - such as the KdV, NLS, KP, or 2D Toda - do actually not refer to the graded commutativity assumption.
07/12/2014 PDF
Smooth approximation of plurisubharmonic functions on almost complex manifolds M/14/38
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This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth strictly J-plurisubharmonic functions point-wise decreasing down to u. In any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.
27/11/2014 PDF
Une interprétation modulaire de la variété trianguline M/14/37
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En utilisant le système de Taylor-Wiles-Kisin construit dans un travail récent de Caraiani, Emerton, Gee, Geraghty, Paškūnas et Shin, nous construisons un analogue de la variété de Hecke. Nous montrons que cette variété coïncide avec une union de composantes irréductibles de l'espace des représentations galoisiennes triangulines. Nous précisons les relations de cette construction avec les conjectures de modularité dans le cas cristallin ainsi qu'avec une conjecture de Breuil sur le socle des vecteurs localement analytiques de la cohomologie complétée. Nous donnons également une preuve d'une conjecture de Bellaïche et Chenevier sur l'anneau local complété en certains points des variétés de Hecke.
26/11/2014 PDF
The BV formalism for L$_\infty$-algebras M/14/36
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The notions of a BV$_\infty$-morphism and a category of BV$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain subcategory of the category of BV$_\infty$-algebras. This provides a Fourier-dual, BV alternative to the standard characterization of the category of L$_\infty$-algebras as a subcategory of the category of dg cocommutative coalgebras or formal pointed dg manifolds. The functor assigning to a BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint.
23/10/2014 PDF
Graviton-Photon Scattering P/14/32
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We use that the gravitational Compton scattering factorizes on the Abelian QED amplitudes to evaluate various gravitational Comp- ton processes. We examine both the QED and gravitational Compton scattering from a massive spin-1 system by the use of helicity am- plitudes. In the case of gravitational Compton scattering we show how the massless limit can be used to evaluate the cross-section for graviton-photon scattering and discuss the difference between photon interactions and the zero mass spin-1 limit. We show that the forward scattering cross-section for graviton photo-production has a very pecu- liar behaviour, differing from the standard Thomson and Rutherford cross-sections for a Coulomb-like potential.
16/10/2014 PDF
Three dimensional Sklyanin algebras and Groebner bases M/14/35
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We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field K with 3 generators x,y,z given by 3 relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin, Odesskii, and Artin, Tate, Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three cubes of p, q and r are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 variables. It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh.
14/10/2014 PDF
Geometry of Morphogenesis M/14/34
01/10/2014 PDF
On Koszulity for operads of Conformal Field Theory M/14/31
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We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Gr ̈bner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity (α(ab))c = a(α(bc)) turns out to be confluent. We introduce a new Hom structure, namely Hom–Gelfand-Dorfman algebras and study their basic properties.
24/09/2014 PDF
The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations M/14/30
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For an arbitrary associative unital ring R, let J1 and J2 be the following noncommutative birational partly defined involutions on the set M3 (R) of 3 × 3 matrices over R: J1 (M ) = M −1 (the usual matrix inverse) and J2 (M )jk = (Mkj )−1 (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich (1996) saying that (J2 ◦ J1 )3 −1 is the identity map modulo the DiagL × DiagR action (D1 , D2 )(M ) = D1 M D2 of pairs of invertible diagonal matrices. That is, we show that for each M in the domain where (J2 ◦J1 )3 is defined, there are invertible −1 diagonal 3 × 3 matrices D1 = D1 (M ) and D2 = D2 (M ) such that (J2 ◦ J1 )3 (M ) = D1 M D2.
24/09/2014 PDF
Topos-theoretic background M/14/27
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This text, which will form the first chapter of my book in preparation "Lattices of theories", is a self-contained introduction to topos theory, geometric logic and the 'bridge' technique.
23/09/2014 PDF
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective M/14/28
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We establish, generalizing Di Nola-Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.
23/09/2014 PDF
Quasi-exact-solvability of the $A_2$ elliptic model: Algebraic form, $sl(3)$ hidden algebra, polynomial eigenfunctions P/14/29
23/09/2014 PDF
Dimensional exactness of self-measures for random countable iterated function systems with overlaps. M/14/26
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We study projection measures for random countable (finite or infinite) conformal iterated function systems with arbitrary overlaps. In this setting we extend Feng's and Hu's result from [6] about deterministic finite alphabet iterated function systems. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist numerous differences between our case and the finite deterministic case. We give then applications and concrete estimates for pointwise dimensions of measures, with respect to various classes of random countable IFS with overlaps. Namely, we study several types of randomized systems related to Kahane-Salem sets; also, a random system related to a statistical problem of Sinai; and randomized infinite IFS in the plane for which the number of overlaps is uniformly bounded from above.
09/09/2014 PDF
Principe de fonctorialité et transformations de Fourier non linéaires : proposition de définitions et esquisse d'une possible (?) démonstration M/14/25
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Ce texte rassemble les notes écrites d'une série de quatre exposés donnés à l'IHES les 19 juin, 26 juin, 3 juillet et 8 juillet 2014. Il introduit une nouvelle approche pour une éventuelle démonstration - encore à vérifier - du transfert automorphe de Langlands sur les corps globaux. En attendant donc de vérifier soigneusement si cela marche ou bien non, d'abord dans le cas de GL(2) et des représentations de puissances symétriques de son dual. Le point le plus essentiel, sur lequel tout est fondé, est la propriété de stabilité du tore maximal par convolution (définie comme la transformée de Fourier de la multiplication point par point des fonctions) apparue dans la dernière partie de la partie III.
21/08/2014 PDF
Extensions of flat functors and theories of presheaf type M/14/23
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We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos.
26/06/2014 PDF
Cyclic theories M/14/22
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We describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos $[{\mathbb N}^{\ast}, \Set]$.
26/06/2014 PDF
A Feynman integral via higher normal functions P/14/06
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We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne's conjectures relating values of $L$-functions inside the critical strip to periods.
10/06/2014 PDF
Beltrami-Courant Differentials and $G_{\infty}$-algebras M/14/19
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Using the symmetry properties of two-dimensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the Einstein equations.
06/05/2014 PDF
2-CY-tilted algebras that are not Jacobian M/14/17
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Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials. In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2.
29/04/2014 PDF
Algebras of quasi-quaternion type M/14/18
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We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by Labardini to triangulations of closed surfaces with punctures on the other hand, hence it serves as a bridge between modular representation theory of finite groups and cluster algebras.
29/04/2014 PDF
SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction P/14/07
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The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine its exact $SL(2,\Z)$-invariant solution $f(condition as$y\to \infty$(the weak coupling limit). The solution is presented as a Fourier series with modes$\widehat{f}_n(y) e^{2\pi i n x}$, where the mode coefficients,$\widehat{f}_n(y)$are bilinear in$K$-Bessel functions. Invariance under$SL(2,\Z)$requires these modes to satisfy the nontrivial boundary condition$ \widehat{f}_n(y) =O(y^{-2})$for small$y$, which uniquely determines the solution. The large-$y$expansion of$f(ower-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.
09/04/2014 PDF
Algebraic rational cells, equivariant intersection theory, and Poincaré duality M/14/15
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We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of Q-fi ltrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any Q-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more general spherical varieties. In view of the localization theorem for equivariant operational Chow rings, we get some conditions for Pointcarré duality in this setting.
09/04/2014 PDF
Motivic Cohomology Spectral Sequence and Steenrod Algebra M/14/16
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For an odd prime number $p$, it is shown that differentials $d_n$ in the motivic cohomology spectral sequence with $p$-local coefficients vanish unless $p-1$ divides $n-1$. We obtain an explicit formula for the first non-trivial differential $d_p$, expressing it in terms of motivic Steenrod $p$-power operations and Bockstein homomorphisms. Finally, we construct examples of varieties, having non-trivial differentials $d_p$ in their motivic spectral sequences.
09/04/2014 PDF
Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1,2)$ M/14/14
01/04/2014 PDF
Calculabilité de la cohomologie étale modulo l M/14/13
20/03/2014 PDF
Scattering Equations and String Theory Amplitudes P/14/11
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Scattering equations for tree-level amplitudes are viewed in the context of string theory. As a result of the comparison we are led to define a new dual model which coincides with string theory in both the small and large $\alpha'$ limit, and whose solution is found algebraically on the surface of solutions to the scattering equations. Because it has support only on the scattering equations, it can be solved exactly, yielding a simple resummed model for $\alpha'$-corrections to all orders. We use the same idea to generalize scattering equations to amplitudes with fermions and any mixture of scalars, gluons and fermions. In all cases checked we find exact agreement with known results.
19/03/2014 PDF
Localization in equivariant operational K-theory and the Chang-Skjelbred property M/14/12
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We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifi cations of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.
18/03/2014 PDF
The physics of quantum gravity P/14/08
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Quantum gravity is still very mysterious and far from being well under- stood. In this text we review the motivations for the quantification of gravity, and some expected physical consequences. We discuss the remarkable rela- tions between scattering processes in quantum gravity and in Yang-Mills theory, and the role of string theory as an unifying theory.
17/03/2014 PDF
Polylogarithms and multizeta values in massless Feynman amplitudes P/14/10
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The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists' oriented survey of Francis Brown's work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz's graphical functions and to $x$-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT.
19/02/2014 PDF
Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability P/14/09
17/02/2014 PDF
The physics and the mixed Hodge structure of Feynman integrals P/14/04
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This expository text is an invitation to the relation between quantum field theory Feynman integrals and periods. We first describe the relation between the Feynman parametrization of loop amplitudes and world-line methods, by explaining that the first Symanzik polynomial is the determinant of the period matrix of the graph, and the second Symanzik polynomial is expressed in terms of world-line Green s functions. We then review the relation between Feynman graphs and variations of mixed Hodge structures. Finally, we provide an algorithm for generating the Picard-Fuchs equation satisfied by the all equal mass banana graphs in a two-dimensional space-time to all loop orders.
14/02/2014 PDF
Du transfert automorphe de Langlands aux formules de Poisson non linéaires M/14/05
17/01/2014 PDF
Non-Abelian Lie algebroids over jet spaces M/14/03
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We associate Hamiltonian homological evolutionary vector fields -- which are the non-Abelian variational Lie algebroids' differentials -- with Lie algebra-valued zero-curvature representations for partial differential equations.
07/01/2014 PDF
Une loi de réciprocité explicite pour le polylogarithme elliptique M/14/01
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On démontre une compatibilité entre la réalisation p-adique et la réalisation de de Rham des sections de torsion du profaisceau polylogarithme elliptique. La preuve utilise une variante pour H1 de la loi de réciprocité explicite de Kato pour le H2 des courbes modulaires.
06/01/2014 PDF
Le système d'Euler de Kato (II) M/14/02
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Ce texte est le deuxième article d’une série de trois articles sur une généralisation de système d’Euler de Kato. Il est consacré e d’Euler de Kato raffiné associé systèmes d’Euler de Kato.
06/01/2014 PDF
Studying Quantum Field Theory P/13/38
13/12/2013 PDF
Topological pressure and measure-theoretic degrees for non-expanding transformations M/13/39
11/12/2013 PDF
Density of potentially crystalline representations of fixed weight M/13/37
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Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology. The main hypothesis we need is the existence of a potentially diagonalizable lift, so that in the two-dimensional case, the result is unconditional.
15/11/2013 PDF
Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds M/13/36
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We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-s. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multi-way isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multi-way isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux result to the k-th eigenvalue and the -way isoperimetric constant. As a consequence the k-th eigenvalue of the weighted Laplacian and the k-way isoperimetric constant are equivalent up to polynomials of k on closed weighted manifolds of nonnegative Bakry-
21/10/2013 PDF
Allure of Quotations and Enchantment of Ideas M/13/35
02/10/2013 PDF
Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3 M/13/34
01/10/2013 PDF
Poisson varieties from Riemann surfaces M/13/33
30/09/2013 PDF
The elliptic dilogarithm for the sunset graph P/13/24
24/09/2013 PDF
The geometry of variations in Batalin-Vilkovisky formalism M/13/32
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We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "delta(0)=0" and "log delta(0)=0" within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Dirac's delta-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving -but not just "formally postulating"- the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation).
15/09/2013 PDF
On-shell Techniques and Universal Results in Quantum Gravity P/13/23
04/09/2013 PDF
Quantum supersymmetric cosmology and its hidden Kac--Moody structure P/13/29
04/09/2013 PDF
Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation P/13/30
04/09/2013 PDF
Merger states and final states of black hole coalescences: a numerical-relativity-assisted effective-one-body approach P/13/31
04/09/2013 PDF
The tensor hierarchy algebra P/13/27
02/09/2013 PDF
The tensor hierarchy simplified P/13/28
02/09/2013 PDF
The Nielsen and the Reidemeister Zeta Functions of maps on infra-solvmanifolds of type (R) M/13/26
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We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R).
30/08/2013 PDF
Anyon wave functions and probability distributions P/13/25
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The problem of determining the ground state energy for a quantum gas of anyons in two dimensions is considered. A recent approach to this problem by means of lower bounds is here refined to bring out the dependence on the n-particle probability distributions encoded in the wave functions. Furthermore, a class of states which has been proposed in the context of upper bounds for a related many-anyon problem, is here considered from the point of view of these refined lower bounds. A numerical approach to determining their corresponding probability distributions is employed for a limited number of particles.
06/08/2013 PDF
Ramification and nearby cycles for l-adic sheaves on relative curves M/13/22
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Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito’s ramification theory.
09/07/2013 PDF
Conformal and Einstein gravity in twistor space P/13/13
18/06/2013 PDF
Distributional Geometry of Squashed Cones P/13/21
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A regularization procedure developed in \cite{Fursaev:1995ef} for integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational $O(2)$ symmetry in a subspace orthogonal to a singular surface $\Sigma$ so that the surface is allowed to have extrinsic curvatures. A new feature of squashed conical singularities is that the surface terms in the integral invariants, in the limit of small angle deficit, now depend also on the extrinsic curvatures of $\Sigma$. A case of invariants which are quadratic polynomials of the Riemann curvature is elaborated in different dimensions and applied to several problems related to entanglement entropy. The results are in complete agreement with computations of the logarithmic terms in entanglement entropy of 4D conformal theories \cite{Solodukhin:2008dh}. Among other applications of the suggested method are logarithmic terms in entanglement entropy of non-conformal theories and a holographic formula of entanglement entropy for theories with gravity duals.
17/06/2013 PDF
Nonholonomic deformation of coupled and supersymmetric KdV equation and Euler-Poincaré-Suslov method M/13/15
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Recently Kupershmidt \cite{Kup} presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al \cite{KKK}. In this paper we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler-Poincarnsional construction to construct nonholonomic deformation of a wide class of coupled KdV equations, all these equations follow from the Euler-PoincarS^1) \bo C^{\infty}(S^1)}}$, where$Diff(S^1)$is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to two component Camassa-Holm equation. We also give a derivation of a nonholonomic deformation of the$N=1$supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler-Poincar 14/06/2013 PDF Application of Jacobi's last multiplier for construction of singular Hamiltonian of the activator-inhibitor model and conformal Hamiltonian dynamics M/13/16 See more > See less > The relationship between Jacobi's last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator and inhibitor model and these are connected to conformal Hamiltonian structure. 14/06/2013 PDF The role of the Jacobi Last Multiplier in Nonholonomic Systems and Almost Symplectic Structure M/13/17 See more > See less > The relationship between Jacobi's last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure is elucidated in this paper. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers, can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. 14/06/2013 PDF Contiguity relations for linearisable systems of Gambier type P/13/18 See more > See less > We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations of the solutions of the Gambier mapping. In all cases the resulting contiguity relation is a linearisable equation, involving free functions, and which can be related to the generic Gambier mapping. 14/06/2013 PDF Quantum aspects of the Liénard II equation and Jacobi's Last Multiplier-II P/13/19 See more > See less > This is a continuation of the paper [J. Phys. A: Math. Theor. 46 (2013) 165202], in which we mapped the Lis ordering technique. In this paper we present further results on the construction of three sets of exactly solvable potentials giving rise to bound-state solutions of the Schr 14/06/2013 PDF Chemotherapy in heterogeneous cultures of cancer cells with interconversion M/13/20 See more > See less > Recently, it has been observed the interconversion between differentiated and stem-like cancer cells. Here, we model the \textit{in vitro} growth of heterogeneous cell cultures in the presence of interconversion from differentiated cancer cells to cancer stem cells, showing that, targeting only cancer stem cells with cytotoxic agents, it is not always possible to eradicate cancer. We have determined the kinetic conditions under which cytotoxic agents in \textit{in vitro} heterogeneous cultures of cancer cells eradicate cancer. In particular, we have shown that the chemotherapeutic elimination of \textit{in vitro} cultures of heterogeneous cancer cells is effective if it targets all the cancer cell types, and if the induced death rates for the different subpopulations of cancer cell types are large enough. 14/06/2013 PDF Nouveaux développements sur les valeurs des caractères des groupes symétriques; méthodes combinatoires M/13/14 See more > See less > L’étude asymptotique des diagrammes de Young de grande taille a été entreprise cu dans le but de contrôler les algèbres d’opérateurs liées aux groupes libres ; elles ont servi ensuite tude des zéros de la fonction zeta de Riemann. Plus récemment , elles ont été appliquées par Biane aux propriétés asymptotiques des permutations. Nous insisterons surtout sur les formules exactes qui sous-tendent ces formules asymptotiques obtenues par les collaborateurs de Biane (Sniady, Féray), et qui développent de nouveaux domaines de la combinatoire (principalement cartes planaires). 05/06/2013 PDF Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function M/13/12 15/05/2013 PDF mRNA diffusion as a mechanism of morphogenesis in Drosophila early development M/13/11 See more > See less > In Drosophila early development, bicoid mRNA of maternal origin is deposited in one of the poles of the egg, determining the anterior tip of the embryo and the position of the head of larvae. The deposition of mRNAs is done during oogenesis by the mother ovary cells and is transported into the oocyte along microtubules. Initially, the oocyte has only one nucleus, but after fertilization and deposition of the egg, nuclear duplication by mitosis is initiated without the formation of cellular membranes. During the first 14 nuclear divisions of the developing embryo, bicoid mRNA of maternal origin is translated into protein in the ribosomes and accumulates near the external nuclear walls of the recently formed nuclei. Here, we show that mRNA diffusion is the main morphogenesis mechanism explaining consistently the establishment of Bicoid protein gradients. Moreover, we show that if diffusion for both bicoid mRNA and Bicoid protein were assumed, a steady distribution of Bicoid protein would result, with a constant concentration along the embryo, contradicting observations. 29/04/2013 PDF Equivariant operational Chow rings of spherical varieties and T-linear varieties. M/13/10 See more > See less > We stablish localization theorems for the equivariant operational Chow rings (or equivariant Chow cohomology) of singular spherical varieties and T-linear varieties. Our main results provide a GKM description of these rings in the case of spherical varieties admitting a BB-decomposition into algebraic rational cells. Our description extends certain topological results to intersection theory on singular varieties. 12/04/2013 PDF Dynamic trajectory control of gliders M/13/09 See more > See less > A new dynamic control algorithm in order to direct the trajectory of a glider to a pre-assigned target point is proposed. The algorithm runs iteratively and the approach to the target point is self-correcting. The algorithm is applicable to any non-powered lift-enabled vehicle (glider) travelling in planetary atmospheres. As a proof of concept, we have applied the new algorithm to the command and control of the trajectory of the Space Shuttle during the Terminal Area Energy Management (TAEM) phase. 04/04/2013 PDF Chemotaxis with directional sensing during Dictyostelium aggregation M/13/07 See more > See less > With an in silico analysis, we show that the chemotactic movements of colonies of the starving amoeba Dictyostelium discoideum are driven by a force that depends on both the direction of propagation (directional sensing) of reaction-diffusion chemotactic waves and on the gradient of the concentration of the chemoattractant. It is shown that the directional sensing of amoebae is due to the sensitivity of the cells to the time variation of the concentration of the chemoattractant combined with its gradient. It is also shown that chemotaxis exclusively driven by local concentration gradients leads to unstable local motion, preventing cells from aggregation. These facts show that the formation of mounds, which initiate multicellularity in Dictyostelium discoideum, is caused by the sensitivity of the amoebae to three factors, namely, to the direction of propagation of the chemoattractant, to its gradient, and to the spiral spatial topology of the propagating chemoattractant. 24/03/2013 PDF The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models M/13/08 See more > See less > To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deductible from the mass action law and, in the limit$n \to \infty$, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability. 24/03/2013 PDF Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires: Notes de cours M/13/06 18/02/2013 PDF On weakly group-theoretical non-degenerate braided fusion categories M/13/05 See more > See less > We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braided fusion category whose Frobenius-Perron dimension is a natural number less than 1800, or an odd natural number less than 33075, is weakly group-theoretical. 01/02/2013 PDF Special Functions in Minimal Representations M/13/04 See more > See less > Minimal representations of a real reductive group G are the smallest' irreducible unitary representations of G. We discuss special functions that arise in the analysis of L^2-model of minimal representations. 22/01/2013 PDF Rankin-Cohen Operators for Symmetric Pairs M/13/03 10/01/2013 PDF Sur la correspondance de Simpson p-adique. II : aspects globaux M/13/02 See more > See less > We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This second article is devoted to the global aspects of the theory. 09/01/2013 PDF Sur la correspondance de Simpson p-adique. 0 : une vue d'ensemble M/13/01 See more > See less > We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. The aim of this article is to give an extensive overview of the theory that has been developped in two articles, the first one (arXiv:1102.5466) devoted to the local aspects and the second one (arXiv:1301.0904) to the global aspects. 09/01/2013 PDF F-method for constructing equivariant differential operators M/12/36 See more > See less > Using an algebraic Fourier transform of operators, we develop a method (F-method) to obtain explicit highest weight vectors in the branching laws by differential equations. This article gives a brief explanation of the F-method and its applications to a concrete construction of some natural equivariant operators that arise in parabolic geometry and in automorphic forms. 31/12/2012 PDF Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring P/12/35 10/12/2012 PDF Gravitational radiation reaction along general orbits in the effective one-body formalism P/12/34 10/12/2012 PDF Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators M/12/33 04/12/2012 PDF Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets M/12/32 02/12/2012 PDF The Decimation Method for Laplacians on Fractals: Spectra and Complex Dynamics M/12/31 27/11/2012 PDF Towards an axiomatic geometry of fundamental interactions in noncommutative space-time at Planck scale P/12/30 See more > See less > We outline an axiomatic quantum picture unifying the four fundamental interactions; this is done by exploring a possible physical meaning of the notions, structures, and logic in a class of noncommutative geometries which has been introduced in [IHES/M-12-13]. We try to recognise a mathematical formalisation of such phenomena of Nature as the oriented space-time, gravity (here, dark matter and vacuum energy), Hubble's law, inflation, formation and structure of sub-atomic particles, antimatter, annihilation, CP-symmetry violation, mass and mass endowment mechanism, three lepton-neutrino matchings, spin, helicity and chirality, electric charge and electromagnetism, as well as the weak and strong interaction between particles, admissible transition mechanisms (e.g., muon to muon neutrino, electron, and electron antineutrino), and decays (e.g., neutron to proton, electron, and electron antineutrino). Our approach is based on the understanding of Physics as text which is written in the language of affine Lie algebras and associated homeo-class noncommutative structures over the space-time. 12/11/2012 PDF Temperedness of Reductive Homogeneous Spaces M/12/29 See more > See less > Let G be a semisimple algebraic Lie group and H a reductive subgroup. We compute geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^p. As an application, we give a criterion which detects whether this representation is tempered. 05/11/2012 PDF Noyaux du transfert automorphe de Langlands et formules de Poisson non lin\'eaires M/12/28 See more > See less > On montre qu'un certain type de formules de Poisson non lin'eaires explicites, qui est impliqu'e par le principe de fonctorialit'e de Langlands, permet de construire des noyaux'' du transfert automorphe. Il y a donc 'equivalence entre le principe de fonctorialit'e et ces formules de Poisson non lin'eaires. 29/10/2012 PDF The Current State of Fractal Billiards M/12/27 16/10/2012 PDF Varna Lecture on$L^2$-Analysis of Minimal Representations M/12/26 See more > See less > Minimal representations of a real reductive group G are the 'smallest' irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: 'small' representation of a group = 'large' symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with emphasis on the Schroedinger model. 12/10/2012 PDF Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator M/12/25 30/09/2012 PDF Discrete Spectrum for non-Riemannian Locally Symmetric Spaces --- I. Construction and Stability M/12/24 See more > See less > We study the discrete spectrum of the Laplacian on certain pseudo-Riemannian manifolds M which are quotients of reductive symmetric spaces X by discrete groups of isometries acting properly discontinuously. Assuming that X admits a maximal compact subsymmetric space of full rank, we construct L^2-eigenfunctions on M for an infinite set of eigenvalues. In contrast to the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface, we prove that this infinite set of eigenvalues is stable under any small deformation of discrete groups, for a large class of groups. We actually construct joint L2-eigenfunctions for the whole commutative algebra of invariant differential operators on M. 19/09/2012 PDF Minkowski Measurability and Exact Fractal Tube Formulas for p-Adic Self-Similar Strings M/12/23 14/08/2012 PDF One-Loop four-graviton amplitudes in N=4 supergravity models P/12/21 13/08/2012 PDF From Global to Local M/12/18 27/07/2012 PDF The zero locus of the infinitesimal invariant M/12/20 See more > See less > Let$\nu$be a normal function on a complex manifold$X$. The infinitesimal invariant of$\nu$has a well-defined zero locus inside the tangent bundle$TX$. When$X$is quasi-projective, and$\nu$is admissible, we show that this zero locus is constructible in the Zariski topology. 27/07/2012 PDF Box-Counting Fractal Strings, Zeta Functions, and Equivalent Forms of Minkowski Dimension M/12/22 27/07/2012 PDF Multifractal Analysis via Scaling Zeta Functions and Recursive Structure of Lattice Strings M/12/19 26/07/2012 PDF On dimension growth of groups M/12/17 See more > See less > The (asymptotic) dimension growth functions of groups were introduced by Gromov in 1999. In this paper, we show connections between dimension growth and expansion properties of graphs, Ramsey theory and the Kolmogorov-Ostrand dimension of groups and prove that all solvable subgroups of the R.Thompson group F have polynomial dimension growth. We introduce controlled dimension growth function and prove that the exponentially controlled dimension growth is exponential for the Thompson group F and some solvable of class 3 groups. The paper contains many open questions. 24/07/2012 PDF Sequences of Compatible Periodic Hybrid Orbits of Prefractal Koch Snowflake Billiards M/12/16 18/07/2012 PDF Partition Zeta Functions, Multifractal Spectra, and Tapestries of Complex Dimensions M/12/15 11/07/2012 PDF The twelve lectures in the (non)commutative geometry of differential equations M/12/13 See more > See less > These notes follow the twelve-lecture course in the geometry of nonlinear partial differential equations of mathematical physics. Briefly yet systematically, we outline the geometric and algebraic structures associated with such equations and study the properties of these structures and their inter-relations. The lectures cover the standard material about the infinite jet bundles, systems of differential equations (e.g., Lagrangian or Hamiltonian), their symmetries and conservation laws (together with the First and Second Noether Theorems), and the construction of the nonlocalities. Besides, in the lectures we introduce the calculus of variational multivectors --- in terms of the Schouten bracket, or the antibracket --- on the (non)commutative jet spaces and proceed with its applications to the variational Poisson formalism and the BRST- or BV-approach to the gauge systems. The course differs from other texts on the subject by its greater emphasis on the physics that motivates the model geometries. Simultaneously, the course attests to the applicability of the algebraic techniques in the analysis of the geometry of fundamental interactions. These lectures could be a precursor to the study of the (quantum) field and string theory. 02/07/2012 PDF Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals M/12/14 02/07/2012 PDF Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description P/12/10 29/06/2012 PDF Theoretical aspects of the equivalence principle P/12/11 29/06/2012 PDF Measurability of the tidal polarizability of neutron stars in late-inspiral gravitational-wave signals P/12/12 29/06/2012 PDF Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings M/12/09 09/06/2012 PDF Gravity, strings, modular and quasimodular forms P/12/08 09/05/2012 PDF On the Tutte-Krushkal-Renardy polynomial for cell complexes M/12/07 See more > See less > Recently V.Krushkal and D.Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A.~Duval, C.~Klivans, and J.~Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial. 16/04/2012 PDF Hopf Galois (Co)Extensions In Noncommutative Geometry M/12/06 See more > See less > We introduce an alternative proof, with the use of tools and notions for Hopf algebras, to show that Hopf Galois coextensions of coalgebras are the sources of stable anti Yetter-Drinfeld modules. Furthermore we show that two natural cohomology theories related to a Hopf Galois coextension are isomorphic. 10/04/2012 PDF Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems P/12/05 03/04/2012 PDF On Generalizations of Connes-Moscovici Characteristic Map M/12/04 See more > See less > In this paper we generalize the Connes-Moscovici characteristic map for cyclic cohomology of extended version of Hopf algebras called x-Hopf algebras. To do this, we define a pairing for cyclic cohomology of module algebras and module coalgebras under the symmetry of a x-Hopf algebra. We introduce more examples of similar generalized characteristic maps for quantum algebraic torus and enveloping algebras. 12/03/2012 PDF Algebras for Amplitudes P/12/03 See more > See less > Tree-level amplitudes of gauge theories are expressed in a basis of auxiliary amplitudes with only cubic vertices. The vertices in this formalism are explicitly factorized in color and kinematics, clarifying the color-kinematics duality in gauge theory amplitudes. The basis is constructed making use of the KK and BCJ relations, thereby showing precisely how these relations underlie the color-kinematics duality. We express gravity amplitudes in terms of a related basis of color-dressed gauge theory amplitudes, with basis coefficients which are permutation symmetric. 05/03/2012 PDF A$R^4$non-renormalisation theorem in${\mathcal N} = 4$supergravity P/12/02 17/02/2012 PDF Quantization is a mystery P/12/01 See more > See less > Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students who have had a first course in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called {\it prequantization} thus preparing the ground for an outline of {\it geometric quantization} (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of {\it quantization of K"ahler manifolds}. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of {\it quantum mechanics in phase space}, ending with a brief survey of the modern development of {\it deformation quantization}. Sect. 5 provides a review of {\it second quantization} and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the theory of quantized matter waves''. 31/01/2012 PDF Spatial variation of fundamental couplings and Lunar Laser Ranging P/11/30 22/12/2011 PDF Energy versus Angular Momentum in Black Hole Binaries P/11/31 22/12/2011 PDF History of Mathematics from a working mathematician's view M/11/29 12/12/2011 PDF SMALL REPRESENTATIONS, STRING INSTANTONS, AND FOURIER MODES OF EISENSTEIN SERIES P/11/25 21/11/2011 PDF A Juzvinski\u{i} Addition Theorem for Finitely Generated Free Groups Actions M/11/28 See more > See less > The classical Juzvinski\u{i} Addition Theorem states that the entropy of an automorphism of a compact group decomposes along invariant subgroups. Thomas generalized the theorem to a skew-product setting. Using L. Bowen's f-invariant we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or finitely dimensional Lie Groups and discuss examples. 17/11/2011 PDF Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models M/11/26 16/11/2011 PDF Symplectic structures on moduli spaces of framed sheaves on surfaces M/11/27 See more > See less > We provide generalizations of the notions of Atiyah class and the Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we de ne a symplectic structure on the moduli spaces of framed sheaves on the second Hirzebruch surface. This generalizes a result of Bottacin for the locally free case. 15/11/2011 PDF N=4 SYM Regge Amplitudes and Minimal Surfaces in AdS/CFT Correspondence P/11/24 See more > See less > The high-energy behavior of N=4 SYM elastic amplitudes at strong coupling is studied by means of the AdS/CFT correspondence. For massless gluon-gluon scattering, we consider the amplitude found by Alday and Maldacena using a minimal surface in$AdS_5$momentum space. For elastic scattering of massive quarks, we reconsider the eikonal method proposed by Janik and one of the authors of this paper, where the relevant minimal surface is a "generalized helicoid" in hyperbolic space ("Euclidean AdS_5"), from which the physical amplitude is obtained after an appropriate analytic continuation. Exploiting a conformal transformation, we are able to show that the quark-quark amplitude is dominated by the same cusp contribution already found in the gluon-gluon case. Both amplitudes are shown to be of Regge type at high energy, with the same logarithmic Regge trajectory, in agreement with the expected universality property of Regge amplitudes. The subleading constant term in the trajectory is known for gluon-gluon scattering, but it is regularization-scheme dependent. Hence, the full content of Regge universality remains an open problem. 17/10/2011 PDF The growth rate of symplectic Floer homology M/11/23 See more > See less > The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. 14/09/2011 PDF Collapsing of Abelian Fibred Calabi-Yau Manifolds M/11/22 04/08/2011 PDF Topos co-évanescents et généralisations M/11/20 See more > See less > Cet article est consacré à l'étude d'un topos introduit par Faltings pour les besoins de la théorie de Hodge$p$-adique. Nous en présentons une nouvelle approche basée sur une généralisation des topos co-évanescents de Deligne. Chemin faisant, nous corrigeons la définition originelle de Faltings. This article is devoted to studying a topos introduced by Faltings for the purpose of$p$-adic Hodge theory. We present a new approach based on a generalisation of Deligne's co-vanishing topos. Along the way, we correct Faltings' original definition. 13/07/2011 PDF On the double zeta values M/11/21 13/07/2011 PDF Surjectivity and equidistribution of the word$x^ay^b$on$PSL(2,q)$and$SL(2,q)$M/11/19 See more > See less > We determine the positive integers a,b and the prime powers q for which the word map w(x,y)=x^ay^b is surjective on the group PSL(2,q) (and SL(2,q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2,q) (and SL(2,q)). Our proof is based on the investigation of the trace map of positive words. 29/06/2011 PDF From Traditional Set Theory -- that of Cantor, Hilbert, G\"odel, Cohen -- to Its Necessary Quantum Extension M/11/18 17/06/2011 PDF The vanishing volume of D=4 superspace P/11/14 07/06/2011 PDF Darboux coordinates, Yang-Yang functional, and gauge theory P/11/16 07/06/2011 PDF Tempered automorphic representations of the unitary group M/11/17 See more > See less > Following Arthur's study of the representations of the orthogonal and symplectic groups, we prove many cases of both the local and global Arthur conjectures for tempered representations of the unitary group. This completes the proof of Arthur's description of the discrete series representations of the quasi-split$p$-adic unitary group, and Arthur's description of the tempered discrete automorphic representations of the unitary group, satisfying certain technical conditions. 04/06/2011 PDF How to take advantage of the blur between the finite and the infinite M/11/15 04/05/2011 PDF Derived equivalences for cluster-tilted algebras of Dynkin type D M/11/11 See more > See less > We provide a far reaching derived equivalence classification of cluster-tilted algebras of Dynkin type D. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with normal forms. We also suggest normal forms for the derived equivalence classes, but some subtle questions in the derived equivalence classification remain open. 28/04/2011 PDF Mutation classes of certain quivers with potentials as derived equivalence classes M/11/12 See more > See less > We characterize the marked bordered unpunctured oriented surfaces with the property that all the Jacobian algebras of the quivers with potentials arising from their triangulations are derived equivalent. These are either surfaces of genus g with b boundary components and one marked point on each component, or the disc with 4 or 5 points on its boundary. We show that for each such marked surface, all the quivers in the mutation class have the same number of arrows, and the corresponding Jacobian algebras constitute a complete derived equivalence class of finite-dimensional algebras whose members are connected by sequences of Brenner-Butler tilts. In addition, we provide explicit quivers for each of these classes. We consider also 10 of the 11 exceptional finite mutation classes of quivers not arising from triangulations of marked surfaces excluding the one of the quiver X_7, and show that all the finite-dimensional Jacobian algebras in such class (for suitable choice of potentials) are derived equivalent only for the classes of the quivers E_6^(1,1) and X_6. 28/04/2011 PDF Which mutation classes of quivers have constant number of arrows? M/11/13 See more > See less > We classify the connected quivers with the property that all the quivers in their mutation class have the same number of arrows. These are the ones having at most two vertices, or the ones arising from triangulations of marked bordered oriented surfaces of two kinds: either surfaces with non-empty boundary having exactly one marked point on each boundary component and no punctures, or surfaces without boundary having exactly one puncture. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is a naturally associated potential such that the Jacobian algebras of all the QP in its mutation class are derived equivalent. 28/04/2011 PDF Multiscale analysis of biological functions: the example of biofilms P/11/10 See more > See less > Biological functions involve processes at different scales. This statement is obviously true for organismic processes like development. It is already relevant for a bacterial colony, the example on which we shall more specifically focus here. Understanding biological functions thus requires to integrate knowledge and data of different natures, available at different levels, and described within different frameworks, from quantum mechanics (for elementary intracellular processes e.g. light transduction) to stochastic kinetics to deterministic rate equations and continuous medium theory (e.g. elasticity theory or hydrodynamics). Beyond the epistemic issue of capturing a real process in descriptions and measurements prescribed by our own abilities and limitations, biological functions and their regulation present a greater challenge: they are intrinsically and irreducibly multiscale processes. Indeed regulation of a biological function has to bridge the overall state of the cells as well as some surroundings features with the basic ingredients and mechanisms at the atomic or molecular scale, in an adaptive and interrelated way. A bacterial cell itself has to perform a multiscale integration. Accordingly our analysis and modeling should follow the same line. For these two main reasons, multiscale approaches play an essential role in the way towards the integrated understanding of biological functions, and all the more of biological systems. 17/04/2011 PDF Configuration Space Renormalization of Massless QFT as an Extension Problem for Associate Homogeneous Distributions P/11/07 12/04/2011 PDF Quantum Einstein-Dirac Bianchi Universes P/11/08 12/04/2011 PDF Accurate numerical simulations of inspiralling binary neutron stars and their comparison with effective-one-body analytical models P/11/09 12/04/2011 PDF Absolute algebra III-the saturated spectrum M/11/06 15/03/2011 PDF Sur la correspondance de Simpson p-adique. I : étude locale M/11/05 See more > See less > Nous développons une nouvelle approche pour la correspondance de Simpson p-adique, intimement liée à l'approche originelle de Faltings, mais aussi inspirée du travail d'Ogus et Vologodsky sur un analogue en caractéristique p>0. Ce premier article est consacré aux aspects locaux de la théorie. \\ We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This first article is devoted to the local aspects of the theory. 01/03/2011 PDF On Vassiliev invariants of braid groups of the sphere M/11/03 See more > See less > We construct a universal Vassiliev invariant for braid groups of the sphere and the mapping class groups of the sphere with$n$punctures. The case of a sphere is different from the classical braid groups or braids of oriented surfaces of genus strictly greater than zero, since Vassiliev invariants in a group without 2-torsion do not distinguish elements of braid group of a sphere. 31/01/2011 PDF Shannon entropy: a rigorous mathematical notion at the crossroads between probability, information theory, dynamical systems and statistical physics M/11/04 See more > See less > Statistical entropy was introduced by Shannon as a basic concept in information theory, measuring the average missing information on a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. I here present how statistical entropy and entropy rate relate to other notions of entropy, relevant either to probability theory (entropy of a discrete probability distribution measuring its unevenness), computer sciences (algorithmic complexity), the ergodic theory of dynamical systems (Kolmogorov-Sinai or metric entropy), or statistical physics (Boltzmann entropy). Their mathematical foundations and correlates (entropy concentration, Sanov, Shannon-McMillan-Breiman, Lempel-Ziv and Pesin theorems) clarify their interpretation and offer a rigorous basis to maximum entropy principles. Although often ignored, these mathematical perspectives give a central position to entropy and relative entropy in statistical laws describing generic collective behaviors. They provide insights into the notions of randomness, typicality and disorder. The relevance of entropy outside the realm of physics, for living systems and ecosystems, is yet to be demonstrated. 31/01/2011 PDF On finite arithmetic groups M/11/02 See more > See less > In this paper we study representations of finite groups stable under Galois operation over arithmetic rings in local and global fields. 21/01/2011 PDF On localization in holomorphic equivariant cohomology M/11/01 See more > See less > We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. localization formula. 18/01/2011 PDF Statistical Properties of Cosmological Billiards P/10/16 16/12/2010 PDF On Effective Action of Multiple M5-branes and ABJM Action P/10/45 See more > See less > We calculate the fluctuations from the classical multiple M5-brane solution of ABJM action which we found in the previous paper. We obtain D4-brane-like action but the gauge coupling constant depends on the spacetime coordinate. This is consistent with the expected proporties of M5-brane action, although we will need to take into account the monopole operators in order to fully understand M5-branes. We also see that the Nambu-Poisson bracket is hidden in the solution. 15/12/2010 PDF Cyclic structures in algebraic (co)homology theories M/10/44 See more > See less > This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd~modules. 06/12/2010 PDF Generalized matrix models and AGT correspondence at all genera P/10/43 See more > See less > We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the$\CN=2$gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera. 24/11/2010 PDF Incompressibility of generic orthogonal grassmannians M/10/42 23/11/2010 PDF A uniqueness theorem for meromorphic mappings with two families of hyperplanes M/10/41 See more > See less > In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate. 07/11/2010 PDF TENSOR STRUCTURE FROM SCALAR FEYNMAN MATROIDS P/10/40 See more > See less > We show how to interpret the scalar Feynman integrals which appear when reducing tensor integrals as scalar Feynman integrals coming from certain nice matroids. 04/11/2010 PDF Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism P/10/37 27/10/2010 PDF Analytic modelling of tidal effects in the relativistic inspiral of binary neutron stars P/10/38 27/10/2010 PDF Accuracy and effectualness of closed-form, frequency-domain waveforms for non-spinning black hole binaries P/10/39 27/10/2010 PDF Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces M/10/36 See more > See less > We compute the open Gromov-Witten invariants for every compact semi-Fano toric surface, i.e. a toric surface$X$with nef anticanonical bundle. Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As an application, an explicit expression of the superpotential$W$for the mirror of$X$is obtained, which in turn gives an explicit ring presentation of the small quantum cohomology of$X$. We also give a computational verification of the natural ring isomorphism between the small quantum cohomology of$X$and the Jacobian ring of$W$. 25/10/2010 PDF The momentum kernel of gauge and gravity theories P/10/32 See more > See less > We derive an explicit formula for factorizing an n-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory. 20/10/2010 PDF Highly Transitive Actions of Out(Fn) M/10/35 See more > See less > An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n>3 the group Out(Fn) = Aut(Fn)/Inn(Fn) admits a faithful highly transitive action on a countable set. 05/10/2010 PDF Seiberg-Witten curve via generalized matrix model P/10/34 See more > See less > We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N=2 SU(2)^n gauge theory with circular quiver diagram by the AGT relation. We first verify that it is obtained from the perturbative calculation of the Liouville correlation function. We derive the Seiberg-Witten curve for N=2 gauge theory as a spectral curve of the generalized matrix model. 28/09/2010 PDF Geometric analysis on small representations of GL(N,R) M/10/33 See more > See less > The most degenerate unitary principal series representations \pi_{i\lambda,\delta} (\lambda \in R, \delta \in Z/2 Z) of G = GL(N,R) attain the minimum of the Gelfand--Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction \pi_{i\lambda,\delta}|_H (\textit{branching law}) with respect to all symmetric pairs (G,H). For N=2n with n \ge 2, the restriction \pi_{i\lambda,\delta}|_H remains irreducible for H=Sp(n,R) if \lambda\ne0 and splits into two irreducible representations if \lambda=0. The branching law of the restriction \pi_{i\lambda,\delta}|_H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) \times GL(q,R) (p+q=N), and contains both discrete and continuous spectra for H=O(p,q) (p>q \ge 1). Our emphasis is laid on geometric analysis, which arises from the restriction of small representations' to various subgroups. 27/09/2010 PDF On the Surjectivity of Engel Words on PSL(2,q) M/10/31 See more > See less > We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q). 23/09/2010 PDF a-Maximization in N=1 Supersymmetric Spin(10) Gauge Theories P/10/30 See more > See less > A summary is reported on our previous publications about four dimensional N=1 supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of \amax, we exlpored decoupling operators in the infared and the renormalization flow of the theory. \ We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory. This is a review article invited to International Journal of Modern Physics A. 17/09/2010 PDF Topological invariants and moduli spaces of Gorenstein quasi-homogeneous surface singularities. M/10/29 See more > See less > We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of Rd by a discrete group. 15/09/2010 PDF Restrictions of generalized Verma modules to symmetric pairs M/10/28 See more > See less > We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs (g, g'). In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple (g, g', p) such that the restriction X|_g' always contains simple g'-modules for any g-module X lying in the parabolic BGG category O^p attached to a parabolic subalgebra p of g. Formulas are derived for the Gelfand–Kirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction X|_g' is generically multiplicity-free for any p and any X\in O^p if and only if (g, g') is isomorphic to (A_n,A_n−1), (B_n,D_n), or (D_n+1,B_n). Explicit branching laws are also presented. 26/08/2010 PDF Ramification and cleanliness M/10/27 See more > See less > This article is devoted to studying the ramification of Galois torsors and of$\ell$-adic sheaves in characteristic$p>0$(with$\ell\not=p$). Let$k$be a perfect field of characteristic$p>0$,$X$be a smooth, separated and quasi-compact$k$-scheme,$D$be a simple normal crossing divisor on$X$,$U=X-D$,$\Lambda$be a finite local${\mathbb Z}_\ell$-algebra,$F$be a locally constant constructible sheaf of$\Lambda$-modules on$U$. We introduce a boundedness condition on the ramification of$F$along$D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of$F$. The cleanliness condition extends the one introduced by Kato for rank one sheaves. Roughly speaking, it means that the ramification of$F$along$D$is controlled by its ramification at the generic points of$D$. Under this condition, we propose a conjectural Riemann-Roch type formula for$F$. Some cases of this formula have been previously proved by Kato and by the second author (T.S.). 23/08/2010 PDF Noncommutative Toda Chains, Hankel quasideterminants and Painlev\'e II equation M/10/25 See more > See less > We construct solutions of an infinite Toda system and an analogue of Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices. 27/07/2010 PDF Global Stringy Orbifold Cohomology, K-theory and de Rham Theory M/10/26 See more > See less > There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. This has been used to define a global stringy extension of the functors$K_0,K^{top}, A^*,H^*$. The second one is given by first pushing forward and then pulling back. This has been used in the cyclic case and in particular for singularities with symmetries and for symmetric products. For Abelian quotients Chen and Hudiscussed such a construction in the de Rham setting. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull--push formalism has a solution by the push--pull equationsin the setting of cyclictwisted sectors. In the general case, we introduce ring extensions that allow us to treat all the stringy multiplications mentioned above. The first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull-push solution while fractional Euler classes give a trivialization of the co--cycles of the pull-push formalism using the presentation of the obstruction bundle of Jarvis--Kaufmann--Kimura This trivialization can be interpreted as defining twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau--Ginzburg models. 27/07/2010 PDF Phenomenology of the Equivalence Principle with Light Scalars P/10/23 22/07/2010 PDF Equivalence Principle Violations and Couplings of a Light Dilaton P/10/24 22/07/2010 PDF Categorification of the Jones-Wenzl Projectors M/10/22 See more > See less > The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes, whose graded Euler characteristic is the classical''projector in the Temperley-Lieb algebra.We show that the projectors are homotopy idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of quantum su(2), and a categorification of quantum spin networks. We introduce 6j-symbols in this context. 14/07/2010 PDF FEYNMAN AMPLITUDES AND LANDAU SINGULARITIES FOR 1-LOOP GRAPHS M/10/20 See more > See less > We use mixed Hodge structures to investigate Feynman amplitudes as functions of external momenta and masses. 02/07/2010 PDF Algebraic Structures in local QFT P/10/21 See more > See less > A review of the Hodge and Hopf-algebraic approach to QFT. 02/07/2010 PDF Instantons on Gravitons P/10/19 See more > See less > Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data is formulated in terms of matrix-valued functions of a single variable, that are organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions.The Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spacesof instantons on ALF spaces. 23/06/2010 PDF Remarks on some locally Qp-analytic representations of GL2(F) in the crystalline case M/10/18 01/06/2010 PDF The emerging p-adic Langlands programme M/10/17 20/05/2010 PDF Classical analog of quantum Schwarzschild black hole: local vs global, and the mystery of log 3 P/10/15 See more > See less > The model is built in which the main global properties of classical and quasi-classical black holes become local. These are the event horizon, no-hair, temperature and entropy. Our construction is based on the features of a quantum collapse, discovered when studying some quantum black hole models. But our model is purely classical, and this allows to use selfconsistently the Einstein equations and classical (local) thermodynamics and explain in this way the log 3-puzzle. 20/04/2010 PDF A Non-differentiable Noether's theorem M/10/14 See more > See less > In the framework of the non-differentiable embedding of Lagrangian systems, defined by Cresson and Greff, we prove a Noether's theorem based on the lifting of one-parameter groups of diffeomorphisms. 14/04/2010 PDF The critical ultraviolet behaviour of N=8 supergravity amplitudes P/10/13 09/04/2010 PDF Eisenstein series for higher-rank groups and string theory amplitudes P/10/10 06/04/2010 PDF Kramers-Wannier Duality for Non-Abelian Lattice Spin Systems and Hecke Surfaces M/10/12 See more > See less > We discuss two themes: 1.Duality transformation for generalized Potts models and Hecke surfaces and$K$-regular graphs 01/04/2010 PDF Pseudogroupes de Lie et théorie de Galois différentielle M/10/11 26/03/2010 PDF Intersecting D4-branes Model of Holographic QCD and Tachyon Condensation P/10/09 See more > See less > We consider the intersecting D4-brane and anti-D4-brane model of holographic QCD, motivated by the model that has recently been suggested by Van Raamsdonk and Whyte. We analyze such D4-branes by the use of the tachyonic Dirac-Born-Infeld action, so that we find the classical solutions describing the intersecting D4-branes and the U-shaped D4-branes. We show that the bi-fundamental tachyon'' field in the bulk theory provides a current quark mass and a quark condensate to the dual gauge theory and thatthe lowest modes of mesons obtain mass via tachyon condensation. Then evaluating the properties of a pion, one can reproduce Gell-Mann-Oakes-Renner relation. 15/03/2010 PDF Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes P/10/08 See more > See less > We discuss monodromy relations between different color-ordered amplitudes in gauge theories. We show that Jacobi-like relations of Bern, Carrasco and Johansson can be introduced in a manner that is compatible with these monodromy relations. The Jacobi-like relations are not the most general set of equations that satisfy this criterion. Applications to supergravity amplitudes follow straightforwardly through the KLT-relations. We explicitly show how the tree-level relations give rise to non-trivialidentities at loop level. 12/03/2010 PDF Monodromy and Kawai-Lewellen-Tye Relations for Gravity Amplitudes P/10/07 08/03/2010 PDF A matrix model for the topological string I: deriving the matrix model P/10/06 04/03/2010 PDF String theory dualities and supergravity divergences P/10/05 22/02/2010 PDF Single-Lifting Macaulay-Type Formulae of Generalized Unmixed Toric Resultants M/10/04 28/01/2010 PDF Minimal representations and reductive dual pairs in conformal field theory P/10/03 15/01/2010 PDF Automorphic properties of low energy string amplitudes in various dimensions P/10/01 14/01/2010 PDF On the ultraviolet behaviour of N=8 supergravity amplitudes P/10/02 14/01/2010 PDF Sugawara-type constraints in hyperbolic coset models P/09/47 22/12/2009 PDF$E_{7(7)}$invariant Lagrangian of$d=4{\mathcal N} = 8$supergravity P/09/52 22/12/2009 PDF Crystal melting on toric surfaces P/09/53 02/12/2009 PDF Almost etale resolution of foliations M/09/51 25/11/2009 PDF La cosmologie: un laboratoire pour la théorie des cordes P/09/50 17/11/2009 PDF Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism P/09/49 03/11/2009 PDF Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries P/09/48 03/11/2009 PDF Sur un problème de compatibilité local-global modulo p pour GL2 M/09/46 05/10/2009 PDF Three \'Etudes in QFT P/09/45 23/09/2009 PDF Construire un noyau de la fonctorialité~? \\ Le cas de l'induction automorphe \\ sans ramification de${\rm GL}_1$à${\rm GL}_2$M/09/42 See more > See less > Le but de cet article (à paraître aux Annales de l'Institut Fourier) est de présenter une nouvelle méthode purement adélique pour réaliser le principe de fonctorialité de Langlands dans le cas de l'induction automorphe sans ramification de GL(1) à GL(2) sur les corps de fonctions. On construit sur le produit des groupes adéliques GL(1) et GL(2) un noyau de la fonctorialité. C’est une version « en famille » et locale de la construction par les modèles de Whittaker globaux, utilisée classiquement dans les « théorèmes réciproques » de Weil et Piatetski-Shapiro. La plus grande partie de la construction et des vérifications nécessaires est locale, c’est-à-dire se fait place par place. Il s’agit de prouver que deux fonctions, dont chacune est définie localement, deviennent égales après sommation sur les éléments rationnels de certains groupes. Cela résulte de la formule de Poisson, sur le modèle de la thèse de Tate, dès lors que l’on comprend comment nos deux fonctions se déduisent localement l’une de l’autre par une certaine transformation de Fourier. 18/09/2009 PDF Construire des noyaux de la fonctorialité~? \\ Définition générale, \\ cas de l'identité de${\rm GL}_2$\\ et construction générale conjecturale \\ de leurs coefficients de Fourier M/09/43 18/09/2009 PDF We are all your students, Mr. Cartan'' M/09/44 18/09/2009 PDF Notes sur l'histoire et la philosophie des mathématiques V : le problème de l'espace M/09/41 16/09/2009 PDF Noncommutative$\mathbf{K}$-correspondence categories, simplicial sets and pro$C^*$-algebras M/09/40 28/08/2009 PDF A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle M/09/39 27/08/2009 PDF Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory P/09/35 20/08/2009 PDF Living in a contradictory world: categories vs sets? M/09/37 20/08/2009 PDF RENORMALIZATION AND RESOLUTION OF SINGULARITIES P/09/36 07/08/2009 PDF Two-Dimensional Topological Strings Revisited P/09/17 03/08/2009 PDF Minimal Basis for Gauge Theory Amplitudes P/09/33 15/07/2009 PDF Vinberg Algebras and Combinatorics M/09/34 See more > See less > Vinberg algebras are usually called pre-Lie algebras and were introduced long ago by Gerstenhaber. We propose to follow a different route by motivating these algebras by problems coming from differential geometry, and first studied in depth by Vinberg. We shall recall how the Lie bracket of vector fields can be obtained by skewsymmetriz- ing a more fundamental product. We shall then develop a combinatorial method for the higher order differential operators, quite similar to the procedure used in study- ing Runge–Kutta methods. We shall then move to nilpotent (or pronilpotent) Lie groups. In the last part of these lectures, I shall apply the previous methods in the renormalization theory of quantum fields (à la Connes–Kreimer). 26/06/2009 PDF On the gravitational polarizability of black holes P/09/28 17/06/2009 PDF Fermionic Kac-Moody Billiards and Supergravity P/09/31 17/06/2009 PDF The Equivalence Principle and the Constants of Nature P/09/32 17/06/2009 PDF Supersymmetric Vacua and Bethe Ansatz P/09/09 16/06/2009 PDF The Effective One Body description of the Two-Body problem P/09/27 16/06/2009 PDF Relativistic tidal properties of neutron stars P/09/29 16/06/2009 PDF An improved analytical description of inspiralling and coalescing black-hole binaries P/09/30 16/06/2009 PDF Simplicity of Amplitudes in Gravity and Yang-Mills Theories P/09/26 15/06/2009 PDF Algebras for quantum fields P/09/24 See more > See less > We give an account of the current state of the approach to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic structures here inone place which are either scattered over the literature, or only implicit in previous writings. In particular we point out mathematical structures which can be helpful to farther develop our mathematical understanding of quantum fields. 1.1. 09/06/2009 PDF The QCD$\beta$-function from global solutions to Dyson-Schwinger equations P/09/25 See more > See less > We study quantum chromodynamics from the viewpoint of untruncated Dyson–Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson– Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and alsoinvestigatethe low energy regime and the possibility for a mass gap in the asymptotically free theory. 09/06/2009 PDF The Big Bang as the Ultimate Traffic Jam P/09/23 See more > See less > We present a novel solution to the nature and formation of the initial state of the Universe. It derives from the physics of a generally covariant extension of Matrix theory. We focus on the dynamical state space of this background independent quantum theory of gravity and matter, an infinite dimensional, complex non-linear Grassmannian. When this space is endowed with a Fubini–Study-like metric, the associated geodesic distance between any two of its points is zero. This striking mathematical result translates into a physical description of a hot, zero entropy Big Bang. The latter is then seen as a far from equilibrium, large fluctuation driven, metastable ordered transition, a “freezing by heating” jamming transition. Moreover, the subsequent unjamming transition could provide a mechanism for inflation while rejamming may model a Big Crunch, the final state of gravitational collapse. 18/05/2009 PDF Surprising simplicity of N=8 supergravity P/09/22 15/05/2009 PDF An infinite family of solvable and integrable quantum systems on a plane P/09/21 See more > See less > An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of thisfamily. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvableand integrable generalization of the family is found. 13/05/2009 PDF Quantum Groups and Braid Group Statistics in Conformal Current Algebra Models P/09/18 24/04/2009 PDF Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence M/09/20 See more > See less > We discuss derived categories of the first kind for DG-modules, DG-comodules, and DG-contramodules, derived categories of the second kind for CDG-modules, CDG-comodules, and CDG-contramodules. For the latter two, the comodule-contramodule correspondence is constructed. Nonhomogeneous Koszul duality (equivalence of exotic derived categories) is obtained in conilpotent and nonconilpotent versions. A_\infty structures are also considered. 24/04/2009 PDF Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles M/09/19 23/04/2009 PDF Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariable Polynomials. II.$E_7$case P/09/15 08/04/2009 PDF Discrete Minimal Surface Algebras M/09/14 See more > See less > We consider Discrete Minimal Surface Algebras (DMSA) as noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in the context of Membrane Theory, where sequences of their representations are used as a regularization of the theory. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the universal enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras. 27/03/2009 PDF Higher-loop amplitudes in the non-minimal pure spinor formalism P/09/13 23/03/2009 PDF Recursive relations in the core Hopf algebra P/09/12 See more > See less > We study co-ideals in the core Hopf algebra underlying a quantum field theory. 17/03/2009 PDF Integrable Systems in Noncommutative Spaces P/09/11 See more > See less > We discuss extension of soliton theories and integrable systems into non-commutative (NC) spaces. In the framework of NC integrable hierarchy, we give infinite conserved quantities and exact soliton solutions for many NC integrable equations,which are represented in terms of Strachan's products and quasi-determinants, respectively. We also present a relation to an NC Anti-Self-Dual Yang-Mills equation, and make comments on how integrability'' should be considered in noncommutative spaces. 13/03/2009 PDF Scattering Amplitudes and BCFW Recursion in Twistor Space P/09/10 See more > See less > A number of recent advances in our understanding of scattering amplitudes have been inspired by ideas from twistor theory. While there has been much work studying the twistor space support of scattering amplitudes, this has largely been done by examiningthe amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. The main reasons for doing so are to seek a formulation of scattering amplitudes in N=4 super Yang-Mills in which superconformal symmetry ismanifest, and to use the progress in on-shell methods in momentum space to build our understanding of how to construct quantum field theory in twistor space. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistorformulation that, together with the three-point seed amplitudes, allow us in principle to recursively construct general tree amplitudes in twistor space. The twistor space BCFW recursion is tractable, and we obtain explicit formulae for n-particle MHV and NMHV amplitudes, together with their CPT conjugates (whose representations are distinct in our chiral framework). The amplitudes are a set of purely geometric, superconformally invariant delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in N=4 super Yang-Mills, and we trace their origin to a topological property of split signature spacetime. Our work is related via a twistor transform to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan. 12/03/2009 PDF An accurate few-parameter ground state wave function for the Lithium atom P/09/08 25/02/2009 PDF On two-dimensional quantum gravity and quasiclassical integrable hierarchies P/09/06 24/02/2009 PDF Essential hyperbolic Coxeter polytopes M/09/07 See more > See less > We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least$6$which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, and realize it in four-dimensional case. 24/02/2009 PDF The core Hopf algebra P/09/05 See more > See less > We study the core Hopf algebra underlying the renormalization Hopf algebra. 13/02/2009 PDF On a class of hamiltonian fiber bundles M/09/04 See more > See less > We study an interesting class of hamiltonian fiber bundles whose fibers are compact homogeneous symplectic manifolds. Applications to the cohomology of their symplectomorphism group are given. 04/02/2009 PDF Exploiting N=2 in consistent coset reductions of type IIA P/09/03 27/01/2009 PDF Generalized E(7(7)) coset dynamics and D=11 supergravity P/09/02 See more > See less > The hidden on-shell E(7(7)) symmetry of maximal supergravity is usually discussed in a truncation from D=11 to four dimensions. In this article, we reverse the logic and start from a theory with manifest off-shell E(7(7)) symmetry inspired by West's coset construction. Following de Wit's and Nicolai's idea that a 4+56 dimensional exceptional geometry'' underlies maximal supergravity, we construct the corresponding Lagrangian and the supersymmetry variations for the 56 dimensional subsector. We prove that both the dynamics and the supersymmetry coincide with D=11 supergravity in a truncation to d=7 in the expected way. 11/01/2009 PDF Un pays dont on ne connaîtrait que le nom (Grothendieck et les 'motifs') M/09/01 08/01/2009 PDF Topological String on${\mathcal S}^2$Revisited P/08/18 31/12/2008 PDF Théories de Galois géométriques M/08/62 See more > See less > Nous présentons de manière conceptuelle les idées de Riemann sur les singularités et la monodromie, et nous l'illustrons par l'étude des fonctions algébriques, et des solutions des équations différentielles. Cela fournit une approche unifiée aux diverses théories de Galois. 29/12/2008 PDF Yet Another Poincaré's Polyhedron Theorem M/08/64 See more > See less > This work contains a new version of Poincare's Polyhedron Theorem that also suits geometries of nonconstant curvature lacking concepts of convexity. Most conditions of the theorem, being as local as possible, are easy to verify in practice. 22/12/2008 PDF Notion de spectre M/08/61 See more > See less > La notion de spectre est au départ une notion physique. Elle a pris progressivement une signification de plus en plus large en mathématique, sa signification mathématique la plus importante lui ayant été donnée par Grothendieck dans sa théorie des schémas. Nous nous promènerons donc de la physique à la géométrie algébrique. 19/12/2008 PDF Entropy estimation of symbolic sequences: How short is a short sequence? P/08/63 See more > See less > While entropy per unit time is a meaningful index to quantify the dynamic features of experimental time series, its estimation is often hampered by the finite length of the data. We here investigate the performance of entropy estimation procedures, relying either on block entropies or Lempel-Ziv complexity, when only {\it very short symbolic sequences} are available. Heuristic analytical arguments point at the influence of temporal correlations on the bias and statistical fluctuations, and put forward a reduced effective sequence length suitable for error estimation. Numerical studies are conducted using, as benchmarks, the wealth of different dynamic regimes generated by the family of logistic maps and stochastic evolutions generated by a Markov chain of tunable correlation time. Practical guidelines and validity criteria are proposed, based on the result that the quality of entropy estimation is sensitive to the sequence temporal correlation hence self-consistently depends on the entropy value itself. 19/12/2008 PDF Quantum Integrability and Supersymmetric Vacua P/08/59 02/12/2008 PDF Phase space polarization and the topological string: a case study P/08/60 02/12/2008 PDF Instanton Partition Functions and M-Theory P/08/16 27/11/2008 PDF About Time P/08/58 24/11/2008 PDF Simplicity in the Structure of QED and Gravity Amplitudes P/08/54 21/11/2008 PDF Dimensional Regularization of the Gravitational Interaction of Point Masses in the ADM Formalism P/08/55 21/11/2008 PDF What is Missing from Minkowski s Raum und Zeit Lecture P/08/56 21/11/2008 PDF Improved Resummation of Post-Newtonian Multipolar Waveforms from Circularized Compact Binaries P/08/57 21/11/2008 PDF Two Dimensional Topological Strings and Gauge Theory P/08/17 18/11/2008 PDF Multi-valued hyperelliptic continued fractions of generalized Halphen type M/08/53 See more > See less > We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperel- liptic Halphen (HH) element p X2g+2 ¡ p Y2g+2 x ¡ y ; depending on parameter y, where X2g+2 is a polynomial of degree 2g + 2 and Y2g+2 = X2g+2(y). We study regular and irregular HH elements, their continued fraction developments and some basic properties of such developments such as: even and odd symmetry and periodicity. There is a 2$ g + 1 dynamics which lies in the basis of the developed continued fractions theory. We give two geometric realizations of this dynamics. The ¯rst one deals with nets of polynomials and with polygons circumscribed about a conic K. The dynamics is realized as a path ofpolygons of g + 1 sides inscribed in a curve B of degree 2g and circumscribed about the conic K obtained by successive moves, so called { °ips along edges. The second geometric realization leads to a new interpretation of generalized Jacobians of hyperelliptic curves. 1
18/11/2008 PDF
Partition Functions of Matrix Models as the First Special Functions of String Theory II. Kontsevich Model P/08/52
07/11/2008 PDF
Diagrammes de Diamond et $(\varphi,\Gamma)$-modules M/08/51
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Piecewise principal comodule algebras M/08/50
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A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P -> P_i whose kernels intersect to zero, P is principal if and only if all P_i's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P -> P_i and such that the comodule algebra of global sections is P.
26/08/2008 PDF
Tamagawa defect of Euler systems M/08/48
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As remarked by Mazur an Rubin (2004, {\em Mem. Amer. Math. Soc.\/}, 168(799)) one does not expect the Kolyvagin system obtained from an Euler system for a $p$-adic Galois representation $T$ to be \emph{primitive} (in the sense of \emph{loc. cit.}) if$p$ divides a Tamagawa number at a prime $\ell\neq p$; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of $T$, refining a result of \emph{loc. cit.}. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.
19/08/2008 PDF
Stickelberger elements and Kolyvagin systems M/08/49
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In this paper we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Stark elements. We show how to apply this construction to prove results on the \emph{odd} parts ofthe ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Stark elements) on how to study \emph{Kolyvagin systems of core rank $r>1$} (in the sense of Mazur and Rubin). The analogous (and in some sense complementary) results for \emph{even} parts of the ideal class groups and main conjectures for totally even characters of totally real number fields have been previously obtained by the author using similar ideas.
19/08/2008 PDF
Analytic subvarieties with many rational points M/08/46
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Dyson's Theorem for curves M/08/47
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Vertex algebroids over Veronese rings M/08/45
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We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.
29/07/2008 PDF
LOCAL STABILITY OF A QUASI-LINEAR AGE-SIZE STRUCTURED POPULATION DYNAMICS MODEL M/08/44
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The local stability of a quasi-linear age-size structured population model studied in Tchuenche (2007) is analysed. If a certain threshold parameter known as the basic reproductive rate is less than unity, then the trivial steady state is locally asymptotically stable. Also, it is shown that if the only real root of the equation ${\cal R}(m')=1$ is negative, then, the non trivial steady state is locally exponentially asymptotically stable.
03/07/2008 PDF
Pure Spinor Partition Function and the Massive Superstring Spectrum P/08/31
03/07/2008 PDF
One-loop $\beta$ functions of a translation-invariant renormalizable noncommutative scalar model P/08/43
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Recently, a new type of renormalizable $\phi^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(\theta^2p^2)$ term. We calculate here the$\beta$ and $\gamma$ functions at one-loop level for this model. The coupling constant $\beta_\lambda$ function is proved to have the same behaviour as the one of the $\phi^4$ model on the commutative $\mathbb{R}^4$. The $\beta_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.
24/06/2008 PDF
SELF-SIMILAR P-ADIC FRACTAL STRINGS AND THEIR COMPLEX DIMENSIONS M/08/42
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We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geo- metric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings.
11/06/2008 PDF
Regularization, renormalization, and renormalization groups: relationships and epistemological aspects P/08/41
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This paper confronts renormalization used in quantum field theory and that used in critical phenomena studies in statistical mechanics or dynamical systems theory. Regularization that cures spurious divergences is distinguished from renormalization transformations allowing to compute actual physical divergences. The former generates a group, and is also encountered in singular perturbation analyses in nonlinear physics. The latter generates a semi-group, and is implemented as a flow in a space of models; its analysis, focusing on fixed points and their neighborhood, allows to determine asymptotic scaling behavior, to delineate universality classes and to assess model structural stability (or instability, i.e. crossovers). The renormalization group can be seen as a symmetry group and a general covariant formulation is proposed. Aspects presented here show that renormalization theory has emulated a shift of focus from the investigation of outcomes of a given model to the analysis of models themselves, by relating models of the same system at different scales or grouping models of different systems exhibiting the same large-scale behavior. So doing, not only (subjective and partial) models are distinguished from underlying physical systems, but also intrinsic physical features can be derived from model comparison and classification.
10/06/2008 PDF
Open-Closed Moduli Spaces and Related Algebraic Structures M/08/40
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We set up a Batalin-Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered $J$-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L_infty and A_infty algebraic structures associated to it.
06/06/2008 PDF
A trace on fractal graphs and the Ihara zeta function M/08/36
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Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.
29/05/2008 PDF
Ihara's zeta function for periodic graphs and its approximation in the amenable case M/08/37
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In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in \cite{GrZu} by Grigorchuk and $\dot{\text{Z}}$uk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.
29/05/2008 PDF
Bartholdi Zeta Functions for Periodic Simple Graphs M/08/38
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The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von~Neumann algebras studied by the authors in \cite{GILa03}, a determinant formula and functional equations are obtained for this zeta function.
29/05/2008 PDF
Ihara zeta functions for periodic simple graphs M/08/39
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The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.
29/05/2008 PDF
Toward zeta functions and Complex Dimensions of Multifractals M/08/34
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Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this text as a spring board to define similaar tools for the study of multifractals such as the binomial measure. The goal of this work is to shine light on new ideas and perspectives rather than to summarize a coherent theory. Progress has been made which connects these new perspectives to and expands upon classical results, leading to a healthy variety of natural and interesting questions for further investigation and elaboration.
28/05/2008 PDF
Turbulence and Holography P/08/35
28/05/2008 PDF
Sutherland-type trigonometric models, trigonometric invariants, and multivariate polynomials P/08/32
16/05/2008 PDF
The QED $\beta$-function from global solutions to Dyson-Schwinger equations P/08/33
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We discuss the structure of beta functions as determined by the recursive nature of Dyson-Schwinger equations turned into an analysis of ordinary differential equations, with particular emphasis given to quantum electrodynamics. In particular we determine when a separatrix for solutions to such ODEs exists and clarify the existence of Landau poles beyond perturbation theory. Both are determined in terms of explicit conditions on the asymptotics for the growth of skeleton graphs.
15/05/2008 PDF
Tube Formulas and complex Dimensions of Self-Similar tilings M/08/27
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We use the self-similar tilings constructed by Erin Pearse to define a generating function for the geometry of a self-similar set in Euclidean space. This geometric zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the geometric zeta function and hence develop a tube formula for self-similar tilings in Rd. The resulting power series in epsilon is a fractal extension of Steiner's classical tube formula for convex bodies K of Rd. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,...,d-1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term complex dimension''. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained by Michel Lapidus and Machiel van Frankenhuisjen.
02/05/2008 PDF
Tube Formulas for Self-Similar Fractals M/08/28
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Tube formulas (by which we mean an explicit formula for the volume of an (inner) epsilon-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl's celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, thetube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which may be too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here.
02/05/2008 PDF
Nonarchimedean Cantor Set and String M/08/29
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We construct a nonarchimedean (or p-adic) analogue of the classical ternary set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterize C3 as the subset of 3-adic integers whose elements contain only 0's and 2's in their 3-adic expansions and prove that C3 is naturally homeomorphic to C. Finally, from the point of view of fractal strings and their complex dimensions (see the books by Lapidus and van Frankhuijsen), the corresponding non archimedean Cantor string resembles the standard archimedean (or real) Cantor string perfectly.
02/05/2008 PDF
MIXED HODGE STRUCTURES AND RENORMALIZATION IN PHYSICS M/08/30
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We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs.
02/05/2008 PDF
Recycling the Independent Field Approximation argument in the far field P/08/26
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The Independent Field Approximation for the entropy production of Laplacian mild diffusional fields is rigorously introduced and discussed. Some new results due to super-convergent algorithms are presented and the meaning of the active zone concept is enlightened.
23/04/2008 PDF
On the origin of time and the Universe P/08/25
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Fractal structure of the block-complexity function M/08/24
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We demonstrate that the block-complexity function for words from 3-letter and 4-letter alphabets exhibits a fractal structure. The resulting fractals have dimensions approximately equal to 1.892 and 1.953 respectively. We visualize approximations of the corresponding fractals using sequences of length 6 and 5 respectively. We note that a similar fractal structure has been established recently for the block-complexity function for words from a 2-letter alphabet, using a different terminology. In this case, the resulting fractal has dimension approximately equal to 1.584.
18/04/2008 PDF
Infinite Dimensional Lie Algebras in 4D Conformal Field Theory P/08/23
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On a class of holonomic D-modules on symmetric matrices attached to the general linear group M/08/22
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We give a classification of regular holonomic D-modules on complex symmetric matrices whose characteristic variety is the union of conormal bundles to the orbits of the general linear group
02/04/2008 PDF
Accurate Effective-One-Body waveforms of inspiralling and coalescing black-hole binaries P/08/21
01/04/2008 PDF
SQCD: A Geometric Apercu P/08/04
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We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.
01/04/2008 PDF
Groupoides de Lie et leurs alg\'ebroides M/08/20
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Résumé : Ce texte d’un exposé prochain au Séminaire Bourbaki est une revue des notions de géométrie différentielle liées aux variétés symplectiques et de Poisson, aux groupoides de Lie et aux algébroides de Lie. Ces notions ont des liens multiples,et la correspondance groupoide de Lie $rightleftarrows$ algébroide de Lie généralise la correspondance bien connue groupe de Lie $rightleftarrows$ algèbre de Lie. Les groupoides de Lie jouent un rôle analogue aux champs algébriques et permettent, grâce à l’équivalence de Morita entre groupoides de Lie, de définir des variétés-quotients généralisées (par exemple, espace des feuilles d’un feuilletage). L’exposé s’achève par une présentation élémentaire de la théorie de Galois des équations différentielles (Picard-Vessiot-Ritt-Kolchin). Abstract: The purpose of this coming lecture at the Bourbaki Seminar is a review of a set of important notions in differential geometry: symplectic and Poisson varieties, Lie groupoids and Lie algebroids. All these notions are strongly connected and the classical correspondence Lie groups $rightleftarrows$ Lie algebras extends to a correspondence Lie groupoids $rightleftarrows$ Lie algebroids. One can considers Lie groupoids up to an equivalencemodelled after the Morita equivalence in algebra. They form the ob jects of a category, closely related to stacks in algebraic geometry, and representing generalized varieties like the space of leaves of a foliation. We end up with a new presentation of the differential Galois theory.
25/03/2008 PDF
Instantons beyond topological theory II P/08/15
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On the conjecture of Kevin Walker M/08/19
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In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relativelengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in the 3-dimensional space. We also prove that for planar polygon spaces the conjecture holds is several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze.
21/03/2008 PDF
Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling P/08/14
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String theory, gravity and experiment P/08/08
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Introductory lectures on the Effective One Body formalism P/08/09
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Faithful Effective One Body waveforms of equal-mass coalescing black-hole binaries P/08/10
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Constraints on the variability of quark masses from nuclear binding P/08/11
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Comparing Effective One Body gravitational waveforms to accurate numerical data P/08/12
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Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling P/08/13
04/03/2008 PDF
On the Brownian gas: a field theory with a Poissonian ground state P/08/05
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As a first step towards a successful field theory of Brownian particles in interaction, we study exactly the non-interacting case, its combinatorics and its nonlinear time-reversal symmetry. Even though the particles do not interact, the field theory contains an interaction term: the vertex is the hallmark of the original particle nature of the gas and it enforces the constraint of a strictly positive density field, as opposed to a Gaussian free field. We compute exactly all the n-point density correlation functions, determine non-perturbatively the Poissonian nature of the ground state and emphasize the futility of any coarse-graining assumption for the derivation of the field theory. We finally verify explicitly, on the n-point functions, the fluctuation-dissipation theorem implied by the time-reversal symmetry of the action.
27/02/2008 PDF
NOT SO NON-RENORMALIZABLE GRAVITY P/08/06
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We review recent ideas [1] how gravity might turn out to be a renormalizable theory after all.
27/02/2008 PDF
Quantum Groups and Braid Group Statistics in Conformal Field Theory P/08/07
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A quantum universal enveloping algebra $U_q$ and the braid group on $n$ strands ${\mathcal B}_n$ mutually commute when acting on the $n$-fold tensor product of a $U_q$-module. Their combined action is applied to low dimensional systems -- the only ones that admit a nontrivial monodromy and hence a braid group (rather than a permutation group) statistics. The lectures introduce the notions of braid group and Hopf algebra and apply them to examples of 2-dimensional (rational) conformal field theory. The case of the $su(2)$ current algebra model, for which the deformation parameter $q$ is an even root of unity, is considered in some detail. In particular, the solution to the Schwarz problem for the $su(2)$ Knizhnik-Zamolodchikov equation is reviewed.
27/02/2008 PDF
Classical and quantum integrability M/08/03
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Twistor Strings, Gauge Theory and Gravity P/08/02
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Cofiniteness conditions, projective covers and the logarithmic tensor product theory M/08/01
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We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C_{1}-cofinite in the sense of Li. 2. There existsa positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_{N}(V) is finite dimensional. This result shows that the category of grading-restricted generalized $V$-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V satisfies in addition Condition 3, that irreducible V-modules are $\R$-graded and C_{1}-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under the P(z)-tensor product operation for $z\in C^{\times}. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C_{2}-cofinite, Conditions 1--3 are satisfied and thus all the conclusions hold. 09/01/2008 PDF On the Projective Hull of Certain Curves in$C^2$M/07/39 See more > See less > The projective hull X^ of a compact set X in projective n-space P^n is an analogue of the classical polynomial hull of a set in C^n. In the special case that X lies in an affine chart C^n in P^n, the part of X^ lying in C^n can be defined as the set of points x in C^n for which there exists a constant M(x) so that |p(x)| < M(x)^d sup_X|p| for all polynomials p of degree less than or equal to d, and any d > 0. Let X^(M) denote the set of points x where M(x) < M. Using an argument of E. Bishop, we show that if g is a compact real analytic curve in C^2 (not necessarily connected), then for any linear projection p:C^2 --> C^1, that part of g^(M) which lies above z is finite for almost all z in C^1. It is then shown that for anycompact stable real-analytic curve g in P^n, the set g^ - g is a 1-dimensional complex analytic subvariety of P^n - g. 14/12/2007 PDF Operator-Valued Involutive Distributions of Evolutionary Vector Fields and their Affine Geometry M/07/38 04/12/2007 PDF Describing general cosmological singularities in Iwasawa variables P/07/36 31/10/2007 PDF When steric hindrance facilitates processivity: polymerase activity within chromatin P/07/37 See more > See less > During eukaryotic transcription, polymerase activity generates torsional stress in DNA, having a negative impact in polymerase processivity. Using our previous studies of the chromatin fiber structure and conformational transitions, we suggest that this torsional stress can be alleviated thanks to a balance between fiber twist and a nucleosome conformational transition into a reversome state. Our model enlightens the origin of polymerase pauses, and leads to the counter-intuitive conclusion that chromatin organized compaction might facilitate polymerase processivity. Indeed, in a compact and well-structured chromatin loop, steric hindrance between nucleosomes enforce sequential transitions, thus ensuring that the polymerase always meets a permissive nucleosomal state. 31/10/2007 PDF Quelques remarques sur le principe de fonctorialit\'e M/07/31 28/10/2007 PDF Dirichlet Duality and the Nonlinear Dirichlet Problem M/07/34 See more > See less > We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet problem.This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of theSpecial Lagrangian potential equation. 23/10/2007 PDF Scalar curvature on lightlike hypersurfaces. M/07/35 See more > See less > In a recent paper by K. Duggal, the concept of induced scalar curvature of lightlike hypersurfaces is introduced, restricting on a specific class of the latter. This paper removes some of these constraints and construct this scalar quantity by an approach that is consistent with the well-known nondegenerate theory. Basic calculations supported by examples are provided. 23/10/2007 PDF Deforming, revolving and resolving - New paths in the string theory landscape P/07/33 See more > See less > In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau with h^{1,1}=86 and h^{2,1}=2. We then show how to construct infinite series of continuously connected minima to the mirror quintic potential by moving into this larger moduli space, applying its monodromies, and moving back. We provide an example of such series, and discuss their implications for the string theory landscape. 04/10/2007 PDF L'universalisme math\'ematique M/07/29 20/09/2007 PDF The diffeomorphism group of a$K3$surface and Nielsen realization M/07/32 See more > See less > We use moduli spaces of various geometric structures on a manifold M to probe the cohomology of the diffeomorphism group and mapping class group of M. The general principle is that existence of a moduli problem for which the Teichmuller space resembles apoint implies that the homomorphism from the diffeomorphism group (or the mapping class group) to an appropriate discrete group resembles a retraction after applying the classifying space functor. Our main application of this idea is for K^4 a K3 surface;here the maps BDiff(K) --> BAut(H^2(K;Z)) and B\pi_0 Diff(K) --> BAut(H^2(K;Z)) are injective on real cohomology in degrees * < 10. The work of Borel and Matsushima determines the real cohomology of BAut(H^2(K;Z)) in these degrees. Using the above injections, the Borel classes provide cohomological obstructions to a generalized Nielsen realization problem which asks when subgroups of the mapping class group can be lifted to the diffeomorphism group. We conclude that the homomorphism Diff(M) --> \pi_0 Diff(M) does not admit a section if M contains a K3 surface as a connected summand. 20/09/2007 PDF Constraints and the$E_{10}$Coset Model P/07/30 See more > See less > We continue the study of the one-dimensional$E_{10}$coset model (massless spinning particlemotion on$E_{10}/K(E_{10}))$whose dynamics at low levels is known to coincide with the equations of motion of maximal supergravity theories in appropriate truncations. We show that the coset dynamics (truncated at levels$\ell \leq 3$) can be consistently restricted by requiring the vanishing of a set of constraints which are in one-to-one correspondence with the canonical constraints of supergravity. Hence,the resulting constrained$\sigma$-model dynamics captures the full (constrained) supergravity dynamics in this truncation. Remarkably, the bosonic constraints are found to be expressible in a Sugawara-like (current$\times$current) form in terms of the conserved$E_{10}$Noether current, and transform covariantly under an upper parabolic subgroup$E_{10}^+ \subset E_{10}$. We discuss the possible implications of this result, and in particular exhibit a tantalising link with the usual affine Sugawara construction in the truncation of$E_{10}$to its affine subgroup$E_9$. 18/09/2007 PDF Restricted set addition: The exceptional case of the Erdos-Heilbronn conjecture M/07/28 See more > See less > Let A,B be different nonempty subsets of the group of integers modulo a prime p. If p is not smaller than |A|+|B|-2, then at least this many residue classes can be represented as a+b, where a and b are different elements of A and B, respectively. This result complements the solution of a problem of Erdos and Heilbronn obtained by Alon, Nathanson, and Ruzsa. 08/08/2007 PDF Tubular Neighborhoods of Nodal Sets and Diophantine Approximation M/07/27 See more > See less > We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers. 31/07/2007 PDF Hochschild cohomology, the characteristic morphism and derived deformations M/07/26 See more > See less > A notion of Hochschild cohomology of an abelian category was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism from the Hochschild cohomology into the graded centre of the (bounded) derived category. An element in the second Hochschild cohomology corresponds to a first order deformation of the abelian category (Lowen and Van den Bergh, 2006). The problem of deforming single objects of the bounded derived category was treated by Lowen (2005). In this paper we show that the image of the Hochschild cohomology element under the characteristic morphism encodes precisely the obstructions to deforming single objects of the bounded derived category to the bounded derived category of the deformed abelian category. Hence this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a derived deformation theory''. 24/07/2007 PDF Towards a modulo$p$Langlands correspondence for GL$_2$M/07/25 04/07/2007 PDF Subgroup separability in residually free groups M/07/24 See more > See less > We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type$FP_\infty$are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups. 29/06/2007 PDF Perverse coherent sheaves and the geometry of special pieces in the unipotent variety M/07/23 See more > See less > Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subscheme whose complement has codimension at least 2. We extend the theory of perverse coherent sheaves, due to Deligne and disseminated by Bezrukavnikov, by showing that a coherent middle extension functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities that has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S_2-extension") in terms of perverse coherent sheaves of algebras on Xthat takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S_2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S_2-extension to give a uniform construction of the desired variety. 28/06/2007 PDF On the derived category of 1-motives, I M/07/22 See more > See less > We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an 'etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding almost" has a left adjoint, that we call LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Ro\v\i tman type theorems (in characteristic 0). 14/06/2007 PDF A remark on quantum gravity P/07/20 See more > See less > We discuss the structure of Dyson--Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson--Schwinger equations. 29/05/2007 PDF Wormholes as Black Hole Foils P/07/19 See more > See less > We study to what extent wormholes can mimic the observational features of black holes. It is surprisingly found that many features that could be thought of as 'characteristic' of a black hole (endowed with an event horizon) can be closely mimicked by aglobally static wormhole, having no event horizon. This is the case for: the apparently irreversible accretion of matter down a hole, no-hair properties, quasi-normal-mode ringing, and even the dissipative properties of black hole horizons, such as a finite surface resistivity equal to 377 Ohms. The only way to distinguish the two geometries on an observationally reasonable time scale would be through the detection of Hawking's radiation, which is, however, too weak to be of practical relevance for astrophysical black holes. We point out the existence of an interesting spectrum of quantum microstates trapped in the throat of a wormhole which could be relevant for storing the information 'lost' during a gravitational collapse. 04/05/2007 PDF Cosmological Singularities and a Conjectured Gravity/Coset Correspondence P/07/15 See more > See less > We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like "chaotic" structure of generic cosmological singularities in eleven-dimensional supergravity and the "last" hyperbolic Kac-Moody algebra$E_{10}$. This intriguing connection suggests the existence of a hidden "correspondence" between supergravity (or even$M$-theory) and null geodesic motion on the infinite-dimensional coset space$E_{10}/K(E_{10})$. If true, this gravity/coset correspondence would offer a new view of the (quantum) fate of space (and matter) at cosmological singularities. 13/04/2007 PDF Binary Systems as Test-beds of Gravity Theories P/07/16 See more > See less > We review the general relativistic theory of the motion, and of the timing, of binary systems containing compact objects (neutron stars or black holes). Then we indicate the various ways one can use binary pulsar data to test the strong-field and/or radiative aspects of General Relativity, and of general classes of alternative theories of relativistic gravity. 13/04/2007 PDF General Relativity Today P/07/17 See more > See less > After recalling the conceptual foundations and the basic structure of general relativity, we review some of its main modern developments (apart from cosmology): (i) the post-Newtonian limit and weak-field tests in the solar system, (ii) strong-gravitational fields and black holes, (iii) strong-field and radiative tests in binary pulsar observations, (iv) gravitational waves, (v) general relativity and quantum theory. 13/04/2007 PDF Chaos and Symmetry in String Cosmology P/07/18 See more > See less > We review the recently discovered interplay between chaos and symmetry in the general inhomogeneous solution of many string-related Einstein-matter systems in the vicinity of a cosmological singularity. The Belinsky-Khalatnikov-Lifshitz-type chaotic behaviour is found, for many Einstein-matter models (notably those related to the low-energy limit of superstring theory and$M$-theory), to be connected with certain (infinite-dimensional) hyperbolic Kac-Moody algebras. In particular, the billiard chambers describing the asymptotic cosmological behaviour of pure Einstein gravity in spacetime dimension$d+1$, or the metric-three-form system of$11$-dimensional supergravity, are found to be identical to the Weyl chambers of the Lorentzian Kac-Moody algebras$AE_d$, or$E_{10}$, respectively. This suggests that these Kac-Moody algebras are hidden symmetries of the corresponding models. There even exists some evidence of a hidden equivalence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space$E_{10} / K (E_{10})$, where$K (E_{10})$is the maximal compact subgroup of$E_{10}$. 13/04/2007 PDF Local Asymmetry and the Inner Radius of Nodal Domains M/07/14 See more > See less > Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue lambda. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/lambda^n. Weapply this result to prove that each nodal domain contains a ball of radius > C/lambda^n. 12/04/2007 PDF B$_{\rm cris}^{\varphi = 1}$-représentations et$(\varphi , \Gamma)$-modules M/07/13 12/04/2007 PDF The cyclomatic number of connected graphs without solvable orbits M/07/12 See more > See less > A graph is without solvable orbits if its group of automorphisms acts on each of its orbits through a non-solvable quotient. We prove that there is a connected graph without solvable orbits of cyclomatic number c if and only if c is equal to 6, 8, 10, 11, 15, 16, 19, 20, 21, 22, or is at least 24, and briefly discuss the geometric consequences. 05/04/2007 PDF The Geometry of Small Causal Diamonds P/07/11 See more > See less > The geometry of causal diamonds or Alexandrov open sets whose initial and final events$p$and$q$respectively have a proper-time separation$ au$small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in$ au$whose coefficients involve the curvature at the centre of the diamond. We give formulae for the total 4-volume$V$of the diamond, the area$A$of the intersection the future light cone of$p$with the past light coneof$q$and the 3-volume of the hyper-surface of largest 3-volume bounded by this intersection valid to${cal O } ( au ^4) $. The formula for the 4-volume agrees with a previous result of Myrheim. Remarkably, the iso-perimetric ratio${3V_3 øver 4 pi }/({ A øver 4 pi } ) ^{3 øver 2} $depends only on the energy density at the centre and is bigger than unity if the energy density is positive. These results are also shown to hold in all spacetime dimensions. Formulae are also given, valid to nextnon-trvial order, for causal domains in two spacetime dimensions. We suggest a number of applications, for instance, the directional dependence of the volume allows one to regard the volumes of causal diamonds as an observable providing a measurement of the Ricci tensor. 14/03/2007 PDF An$A_{\infty}$-structure for lines in a plane M/07/10 See more > See less > As an explicit example of an$A_{\infty}$-structure associated to geometry, we construct an$A_{\infty}$-structure for a Fukaya category of finitely many lines (Lagrangians) in$R^2$, ie., we define also non-transversal$A_{\infty}$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where$R^2$is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory (HPT) as discussed by Kontsevich and Soibelman in cite{KoSo}, where we introduce a special DG category which is a key idea of our construction. 06/03/2007 PDF A vanishing theorem in positive characteristic and tilting equivalences M/07/09 02/03/2007 PDF Notes on instantons in topological field theory and beyond P/07/08 21/02/2007 PDF Renormalisation of non-commutative field theories P/07/07 See more > See less > The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject. 13/02/2007 PDF Familles de représentations de De Rham et monodromie p-adique M/07/06 08/02/2007 PDF One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model P/07/05 See more > See less > We compute at the one-loop order the beta-functions for a renormalisable non-commutative analog of the Gross-Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The beta-function for the non-commutative counterpart of the Thirring model is found to be non vanishing. 02/02/2007 PDF A counterexample to Premet's and Joseph's conjectures M/07/01 See more > See less > Let$g$be a finite-dimensional reductive Lie algebra of rank$l$over an algebraically closed field of characteristic zero. Given an element$x$of$g$, we denote by$g_x$the centraliser of$x$in$g$. It was conjectured by Premet, that the algebra$S(g_x)^{g_x}$of the symmetrci$mathfrak g_x$-invariants is a graded polynomial algebra in$l$variables. In this note, we show that this conjecture does not hold for the minimal nilpotent orbit in the simple Lie algebra of type$E_8$. As a consequence, a conjecture of Joseph on the semi-invariants of (bi)parabolics is not true either. 26/01/2007 PDF Lifetime of a massive particle in a de Sitter universe P/07/02 See more > See less > We study particle decay in de Sitter space-time as given by first order perturbation theory in an interacting quantum field theory. We show that for fields with masses above a critical mass$m_c$there is no such thing as particle stability, so thatdecays forbidden in flat space-time do occur there. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with lower mass are even stranger: The masses of their decay products must obey quantification rules, and their lifetime is zero. 26/01/2007 PDF Semi-classical open string corrections and symmetric Wilson loops P/07/03 See more > See less > In the AdS/CFT correspondence, an AdS_2 x S^2 D3-brane with electric flux in AdS_5 x S^5 spacetime corresponds to a circular Wilson loop in the symmetric representation or a multiply wound one in N=4 super Yang-Mills theory. In order to distinguish the symmetric loop and the multiply wound loop, one should see an exponentially small correction in large 't Hooft coupling. We study semi-classically the disk open string attached to the D3-brane. We obtain the exponent of the term and it agrees with the result of the matrix model calculation of the symmetric Wilson loop. 26/01/2007 PDF Geometry of differential equations M/07/04 See more > See less > This is a review of classical and modern methods of geometric-algebraic approach to (overdeteremined) systems of partial differential equations. 26/01/2007 PDF Recursion and growth estimates in renormalizable quantum field theory P/06/62 See more > See less > In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be${ m min}{ ho,1/b_1}$, where$ ho$is the bound on the convergent ones, the instanton radius, and$b_1$the first coefficient of the$eta$-function. 19/12/2006 PDF Comparaison des théories de Deitmar et de Zhu M/06/63 19/12/2006 PDF The next-to-ladder approximation for Dyson-Schwinger equations P/06/64 See more > See less > We solve the linear Dyson Schwinger equation for a massless vertex in Yukawa theory, iterating the first two primitive graphs. 19/12/2006 PDF Extended Seiberg-Witten Theory and Integrable Hierarchy P/06/43 08/12/2006 PDF Sur un corps de caractéristique 1' (d'après Zhu) M/06/61 See more > See less > Nous exposons la théorie de Zhu concernant un analogue formel du corps${mathbf F}_{p}$, pour $p=1$'. 06/12/2006 PDF Homotopy graph-complex for configuration and knot spaces M/06/58 See more > See less > In the paper we prove that the primitive part of the Sinha homology spectral sequence E2-term for the space of long knots is rationally isomorphic to the homotopy E2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces. 05/12/2006 PDF Modified Hochschild and Periodic Cyclic Homology M/06/59 See more > See less > The Hochschild and (periodic) cyclic homology of the algebra of continuous functions on a smooth manifold are trivial. In this paper we create an analogue of the Hochschild and periodic cyclic homology which gives the right result when applied onto the algebra of continuous functions on smooth manifolds ($Z_{2}$-graded de Rham co-homology of the manifold). This is realized by replacing the Connes periodic bi-complex (b, B) by the bi-complex$( ilde{b}, d),$where the operator$ ilde{b}$is obtained by blending the Hochschild boundary$b$with the Alexander-Spanier boundary$d$; the operator$ ilde{b}$anti-commutes with the operator$d$. In order to reach this objective, as in the classical case, one has to consider the Alexander-Spanier complex of germs. As the notion of germ is a locality notion, our procedure will apply to topological algebras. The problem of producing a tool able to extract the correct homology from the algebra of continuous functions was addressed before byM. Puschnigg. 05/12/2006 PDF A generalization of residual finiteness M/06/60 See more > See less > The concept of residual finiteness with respect to automorphic equivalence, a property generalizing residual finiteness and conjugacy separability is introduced. A sufficient condition for a group G to be residually finite with respect to automorphic equivalence is proven (Theorem). It is then used to give some examples of automorphic equivalent residually finite groups. 05/12/2006 PDF Noncommutative Geometry Approach to Principal and Associated Bundles M/06/65 01/12/2006 PDF The map from the cyclohedron to the associahedron is left cofinal M/06/66 See more > See less > Two natural projections from the cyclohedron to the associahedron are defined. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disc, but are still contractible. We briefly explain anapplication of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss embedding calculus. 01/12/2006 PDF Injectivity Radius of Lorentzian Manifolds M/06/67 01/12/2006 PDF Reduction Theorems for characteristic functors on finite$p$-groups and applications to$p$-nilpotence criteria M/06/55 22/11/2006 PDF Strongly homotopy Lie bialgebras and Lie quasi-bialgebras M/06/57 See more > See less > Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras. Cohomology theories of all these structures are described in a concise way using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an$L_infty$-(quasi)bialgebra (strong homotopy Lie (quasi)bialgebra). We recover an$L_infty$-algebra structure as a particularcase of our construction. The formal geometry interpretation leads to a definition of an$L_infty$(quasi)bialgebra structure on$V$as a differential operator$Q$on$V,$self-commuting with respect to the Poisson bracket. Finally, we establish an$L_infty$-version of a Manin (quasi) triple and get a correspondence theorem with$L_infty$-(quasi) bialgebras. 21/11/2006 PDF Cardy condition for Open-closed field algebras M/06/56 See more > See less > Let$V$be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that$mathcal{C}_V$, the category of$V$-modules, is a modular tensor category. We study open-closed field algebras over$V$equipped with nondegenerateinvariant bilinear forms for both open and closed sectors. We show that they give algebras over certain$C$-extension of the Swiss-cheese partial dioperad, and we obtain Ishibashi states easily in such algebras. We formulate Cardy condition algebraically in terms of the action of the modular transformation$S: au mapsto -frac{1}{ au}$on the space of intertwining operators. We then derive a graphical representation of$S$in the modular tensor category$mathcal{C}_V$. This result enablesus to give a categorical formulation of Cardy condition and modular invariant conformal full field algebra over$Vøtimes V$. Then we incorporate the modular invariance condition for genus-one closed theory, Cardy condition and the axioms for open-closed field algebra over$V$equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called Cardy$mathcal{C}_V|mathcal{C}_{Vøtimes V}$-algebra. We also give a categorical construction of Cardy$mathcal{C}_V|mathcal{C}_{Vøtimes V}$-algebra in Cardy case. 15/11/2006 PDF Field rotation parameters and limit cycle bifurcations M/06/54 See more > See less > In this paper, the global qualitative analysis of planar polynomial dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem on the maximum number and relative position of their limit cycles in two special cases of such systems is suggested. First, using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles ina quadratic system is equal to four and the only possible their distribution is (3:1). Then, by means of the same geometric approach, we solve the Problem for Li'{e}nard's polynomial system (in this special case, it is considered as Smale's Thirteenth Problem). Besides, generalizing the obtained results, we present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles surrounding a singular point for an arbitrary polynomial system and, applying the Wintner-Perko terminationprinciple for multiple limit cycles, we develop an alternative approach to solving the Problem. By means of this approach, for example, we give another proof of the main theorem for a quadratic system and complete the global qualitative analysis ofa generalized Li'{e}nard's cubic system with three finite singularities. We discuss also some different approaches to the Problem. 08/11/2006 PDF Correlator of Fundamental and Anti-symmetric Wilson loops in AdS/CFT Correspondence P/06/53 27/10/2006 PDF Instantons beyond topological theory I P/06/42 See more > See less > Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories. 17/10/2006 PDF Open-closed field algebras, operads and tensor categories M/06/51 See more > See less > We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra$V$. In the case that$V$satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over$V$canonically gives an algebra over a$C$-extension of the Swiss-cheese partial operad. We also give a tensor categorical formulation and categorical constructions of open-closed field algebras over$V$. 10/10/2006 PDF Classification of low energy sign-changing solutions of an almost critical problem M/06/52 See more > See less > In this paper we make the analysis of the blow-up of low energy sign-changing solutions of a semi-linear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball. 10/10/2006 PDF Remarks on compact shrinking Ricci solitons of dimension four M/06/50 03/10/2006 PDF A primer of Hopf algebras M/06/40 See more > See less > In this paper, we review a number of basic results about so-called Hopf algebras. We begin by giving a historical account of the results obtained in the 1930's and 1940's about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with Milnor-Moore's theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions andthecorresponding multiple zeta values. 26/09/2006 PDF Rigidity theorem for expanding Gradient Ricci Solitons M/06/49 See more > See less > In this paper, we study the rigidity problem for expanding gradient Ricci soliton equation on a complete conformally compact Riemannian manifold. We show that under a natural condition on the Ricci curvature and the scalar curvature, the expanding Ricci soliton is Poincare-Einstein. 21/09/2006 PDF Linear dependence in Mordell-Weil groups M/06/48 See more > See less > We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety defined over a number field. 19/09/2006 PDF A Lie theoretic approach to renormalization P/06/46 See more > See less > Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on the fine properties of Hopf algebras and their associated descent algebras. Besides leading very directly to proofs of the main combinatorial properties of the renormalization procedures, the new techniques do not depend on the geometry underlying the particular case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme. 13/09/2006 PDF Le Calcul des Probabilités de Poincaré M/06/47 See more > See less > Résumé: Le présent texte est extrait d'un ouvrage à paraître sur L'héritage mathématique de Poincaré' (éditions Belin). Il s'agit d'une analyse du livre de Poincaré sur les probabilités, qui reprend l'un de ses cours, et son texte fameux sur le hasard. Le plus intéressant est la manière dont Poincaré traite des problèmes de théorie cinétique des gaz et des cheminements aléatoires. Abstract: We analyse the classical work of Poincaré about probability theory. This report is part of a forthcoming book on the Mathematical Heritage of Poincaré'. We focus on the relevance of the book for the development of statistical physics and random walks. 13/09/2006 PDF Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism P/06/41 See more > See less > Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes fromregularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the ten-dimensional superspace action which do not involve integration over the maximum number of$ heta$'s. 06/09/2006 PDF Dessins d'enfants: Solving equations determining Belyi pairs M/06/44 See more > See less > This paper deals with the Grothendieck dessins d'enfants, that is tamely embedded graphs on surfaces. We investigate combinatorics of systems of equations determining corresponding Belyi pair, that is a rational function with at most 3 critical values onan algebraic curve, such that its preimage is the dessin under consideration. Several properties of extra, or so-called parasitic, solutions of such systems are described. 05/09/2006 PDF Dyson Schwinger Equations: From Hopf algebras to Number Theory P/06/45 See more > See less > We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive non-perturbative results for the short-distance singular sector of a renormalizable quantum field theory. We focus on the short-distance behaviour and thus discuss renormalized Green functions$G_R(alpha,L)$which depend on a single scale$L=ln q^2/mu^2$. 05/09/2006 PDF Séries hypergéométriques multiples et polyzêtas M/06/36 See more > See less > Nous décrivons un algorithme théorique et effectif permettant de démontrer que des séries et intégrales hypergéométriques multiples relativement générales se décomposent en combinaisons linéaires à coefficients rationnels de polyzêtas. 29/06/2006 PDF Phénomènes de symétries dans des formes linéaires en polyzêtas M/06/37 See more > See less > On donne deux généralisations, en profondeur quelconque, du phénomène de symétrie utilisé par Ball-Rivoal pour démontrer qu'une infinité de valeurs de la fonction zeta de Riemann aux entiers impairs sont irrationnelles. Ces généralisations concernent des séries multiples de type hypergéométrique qui s'écrivent comme formes linéaires en certains polyzêtas. La preuve utilise notamment la régularisation des polyzêtas à divergence logarithmique. 29/06/2006 PDF Tensor gauge fields in arbitrary representations of$GL(D,{\bf R})$: II. Quadratic actions P/06/33 See more > See less > Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreduciblein$D$-dimensional Minkowski space are explicitly written in a compact form by making use of Levi--Civita tensors. The field equations derivedfrom these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously.Moreover, these actions allow a frame-like reformulation \a la MacDowell--Mansouri, without any trace constraint in the tangent indices. 29/06/2006 PDF On matrix differential equations in the Hopf algebra of renormalization P/06/38 29/06/2006 PDF Bounds for the dimensions of$p$-adic multiple$L$-value spaces M/06/39 See more > See less > First, we will define$p$-adic multiple$L$-values ($p$-adic MLV's), which are generalizations of Furusho's$p$-adic multiple zeta values ($p$-adic MZV's) in Section 2. Next, we prove bounds for the dimensions of$p$-adic MLV-spaces in Section 3, assuming results in Section 4. The bounds come from the rank of$K$-groups of ring of$S$-integers of cyclotomic fields, and these are$p$-adic analogues of Deligne-Goncharov's bounds for the dimensions of (complex) MLV-spaces. In the case of MLV-spaces, the gap between the dimensions and the bounds is related to spaces of modular forms similarly as the complex case. In Section 4, we define the crystalline realization of mixed Tate motives and show a comparison isomorphism, by uing$p$-adic Hodge theory. 29/06/2006 PDF Quantum effects in gravitational wave signals from cuspy superstrings P/06/35 See more > See less > We study the gravitational emission, in Superstring Theory, from fundamental strings exhibiting cusps. The classical computation of the gravitational radiation signal from cuspy strings features strong bursts in the special null directions associated to the cusps. We perform a quantum computation of the gravitational radiation signal from a cuspy string, as measured in a gravitational wave detector using matched filtering and located in the special null direction associated to the cusp. We study the quantum statistics (expectation value and variance) of the measured filtered signal and find that it is very sharply peaked around the classical prediction. Ultimately, this result follows from the fact that the detector is a low-pass filter which is blind tothe violent high-frequency quantum fluctuations of both the string worldsheet, and the incoming gravitational field. 23/06/2006 PDF Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I M/06/34 See more > See less > We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an abstract'' Calabi-Yau manifold). 21/06/2006 PDF$K(E_{10})$, Supergravity and Fermions P/06/28 See more > See less > We study the fermionic extension of the E10/K(E10) coset model and its relation to eleven-dimensional supergravity. Finite-dimensional spinor representations of the compact subgroup K(E10) of E(10,R) are studied and the supergravity equations are rewritten using the resulting algebraic variables. The canonical bosonic and fermionic constraints are also analysed in this way, and the compatibility of supersymmetry with local K(E10) is investigated. We find that all structures involving A9 levels 0,1 and 2 nicely agree with expectations, and provide many non-trivial consistency checks of the existence of a supersymmetric extension of the E10/K(E10) coset model, as well as a new derivation of the bosonic dictionary' between supergravity and coset variables.However, there are also definite discrepancies in some terms involving level 3, which suggest the need for an extension of the model to infinite-dimensional faithful representations of the fermionic degrees of freedom. 15/06/2006 PDF Coherent algebras and noncommutative projective lines M/06/32 See more > See less > A well-known conjecture says that every one-relator discrete group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show every that Gorenstein algebra$A$of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line$PP1$as a noncommutative scheme based on the coherent noncommutative spectrum$cohp A$of such an algebra$A$, that is, the category of coherent$A$-modulesmodulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent sheaves on$PP1$. In this way, we obtain a sequence$PP1_n $($nge 2$) of pairwise non-isomorphic noncommutative schemes which generalize the scheme$PP1 = PP1_2$. 13/06/2006 PDF About the Trimmed and the Poincaré-Dulac normal form of diffeomorphisms M/06/29 See more > See less > We study two particular continuous prenormal forms as defined by Jean Ecalle and Bruno Vallet for local analytic diffeomorphism: the Trimmed form and the Poincare-Dulac normal form. We first give a self-contain introduction to the mould formalism of Jean Ecalle. We provide a dictionary between moulds and the classical Lie algebraic formalism using non-commutative formal power series. We then give full proofs and details for results announced by J. Ecalle and B. Vallet about the Trimmed form of diffeomorphisms. We then discuss a mould approach to the classical Poincare-Dulac normal form of diffeomorphisms. We discuss the universal character of moulds taking place in normalization problems. 07/06/2006 PDF Fractional embedding of differential operators and Lagrangian systems M/06/30 See more > See less > This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provide a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e. that the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of F. Riewe, we obtain a fractional Ostrogradski formalism which allows us to derive non-conservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equation and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss as well as the fractional diffusion equation. 07/06/2006 PDF Mode conversion in the cochlea? linear analysis P/06/31 See more > See less > It is suggested that the frequency selectivity of the ear may be based on the phenomenon of mode conversion rather than critical layer resonance. The distinction is explained and supporting evidence discussed. 07/06/2006 PDF Stochastic Embedding of Dynamical Systems M/06/27 19/05/2006 PDF Théorème de Noether stochastique M/06/25 17/05/2006 PDF Non-differentiable deformations of$R^nM/06/26 See more > See less > Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian spaceR^n$which can be viewed as the basic bricks to construct irregular objets. They are obtain by taking the topological product of$n$-graphs of nowhere differentiable real valued functions. Our point of view is to replace the study of a non-differentiable function by the dynamical study of a one-parameter family of smooth regularization of this function. In particular, this allows us to construct a one parameter family of smooth coordinates systems on non-differentiable deformations of$R^n$whichdepend on the smoothing parameter via an explicit differential equation called a scale-law. Deformations of$R^n$are examples of a new class of geometrical objects called scale manifolds which are defined in this paper. As an application, we derive rigorously the main results of the scale relativity theory developed by L. Nottale in the framework of a Scale space-time manifold. 17/05/2006 PDF Etude for linear Dyson-Schwinger Equations P/06/23 See more > See less > We discuss properties of linear Dyson-Schwinger equations. 10/05/2006 PDF An Etude in non-linear Dyson-Schwinger equations P/06/24 See more > See less > We show how to use the Hopf algebra structure of quantum field theory to derive nonperturbative results for the short-distance singular sector of a renormalizable quantum field theory in a simple but generic example. We discuss renormalized Green functions$G_R(alpha,L)$in such circumstances which depend on a single scale$L=ln q^2/mu^2$and start from an expansion in the scale$G_R(alpha,L)=1+sum_k gamma_k(alpha)L^k$. We derive recursion relations between the$gamma_k$which make full use of the renormalization group. We then show how to determine the Green function by the use of a Mellin transform on suitable integral kernels. We exhibit our approach in an example for which we find a functional equation relating weak and strong coupling expansions. 10/05/2006 PDF Constructing conformal field theory models P/06/05 09/05/2006 PDF Calcul Moulien M/06/22 09/05/2006 PDF Curvature corrections and Kac-Moody compatibility conditions P/06/21 See more > See less > We study possible restrictions on the structure of curvature corrections to gravitational theories in the context of their corresponding Kac--Moody algebras, following the initial work on E10 in Class. Quant. Grav. 22 (2005) 2849. We first emphasize thatthe leading quantum corrections of M-theory can be naturally interpreted in terms of (non-gravity) fundamental weights of E10. We then heuristically explore the extent to which this remark can be generalized to all over-extended algebras by determining which curvature corrections are compatible with their weight structure, and by comparing these curvature terms with known results on the quantum corrections for the corresponding gravitational theories. 26/04/2006 PDF Evolution in random environment and structural instability M/06/20 See more > See less > We consider stability and evolution of complex biological systems in particular, genetic networks. We focus our attention on supporting of homeostasis in these systems with respect to fluctuations of an external medium (the problem is posed by M. Gromov, A.Carbone cite{Gr}). Using a measure of stochastic stability we show that a generic system with fixed parameters is unstable, i.e., the probability to support homeostasis converges to zero as time$T o infty$. However, if we consider a population of unstable systems, which are capable to evolve (change their parameters), then such a population can be stable as$T o infty$. This means that the probability to survive may be non-zero as$T o infty$. Evolution algorithms, that provide stability of populations, are not trivial. We show that the mathematical results on evolution algorithms are consistent with experimental data on genetic evolution. 13/04/2006 PDF L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld au niveau des points M/06/19 11/04/2006 PDF Wilson Loops of Anti-symmetric Representation and D5-branes P/06/18 See more > See less > We use a D5-brane with electric flux in AdS_5 x S^5 background to calculate the circular Wilson loop of anti-symmetric representation in N=4 super Yang-Mills theory in 4 dimensions. The result agrees with the Gaussian matrix model calculation. 30/03/2006 PDF Nonequilibrium statistical mechanics and entropy production in a classical infinite system of rotators. P/06/17 29/03/2006 PDF Matrix Representation of Renormalization in Perturbative Quantum Field Theory P/06/15 See more > See less > We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota-Baxter anti-homomorphism from general regularized functionals on the Feynmangraph Hopf algebra to triangular matrices with entries in a Rota--Baxter algebra. For characters mapping to the group of unipotent triangular matrices we derive the algebraic Birkhoff decomposition for matrices using Spitzer's identity. This simple matrix factorization is applied to characterize and calculate perturbative renormalization. 28/03/2006 PDF L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld : Décomposition cellulaire de la tour de Lubin-Tate M/06/16 See more > See less > Cet article est le premier d'une série visant à construire un isomorphisme entre les tours p-adiques de Lubin-Tate et de Drinfeld, décrire cet isomorphisme et en donner des applications. Nous-y construisons un modèle entier p-adique équivariant en niveauinfini de la tour de Lubin-Tate. Ce schéma formel p-adique sera comparé plus tard à un autre associé à la tour de Drinfeld. 28/03/2006 PDF Birkhoff type decompositions and the Baker--Campbell--Hausdorff recursion P/06/14 See more > See less > We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decompositionthat was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer canbe obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations. 23/03/2006 PDF The structure of double groupoids M/06/13 See more > See less > We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goesas follows: in a first step we prove that every double groupoid is obtained as an extension of its pith groupoid, which is an abelian group bundle, by its frame double groupoid. The frame satisfies that every box is determined by its edges, and thusis called a 'thin' double groupoid. In a second, independent, step we prove that every thin double groupoid with filling condition is completely determined by a factorization of a certain canonically defined 'diagonal' groupoid. 22/03/2006 PDF First steps towards p-adic Langlands functoriality M/06/12 See more > See less > By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups. 21/03/2006 PDF Existence of closed$G_2$-structures on 7-manifolds M/06/11 See more > See less > In this note we propose a new way of constructing compact 7-manifolds with a closed$G_2$-structure. As a result we find a first example of a closed$G_2$-structure on$S^3 imes S^4$. We also prove that any integral closed$G_2$-structure on a compact7-manifold$M^7$can be obtained by embedding$M^7$to a universal space$(W ^{3(80 +8 cdot C^3 _8)}, h)$. 08/03/2006 PDF Existence d'immeubles triangulaires quasi-périodiques M/06/08 03/03/2006 PDF Ergodic pumping: a mechanism to drive biomolecular conformation changes P/06/09 See more > See less > We propose that a significant contribution to the power stroke of myosin and similar conformation changes in other biomolecules is the pressure of a single molecule (e.g. a phosphate ion) expanding a trap, a mechanism we call ergodic pumping''. We demonstrate the principle with a toy computer model and discuss the mathematics governing the evolution of slow degrees of freedom in large Hamiltonian systems. We indicate in detail how the mechanism could fit with known features of the myosin cycle. Manyother biomolecular conformation changes could be driven in part by ergodic pumping. We suggest the use of ergodic pumping as a design principle in nanobiotechnology. 03/03/2006 PDF Interaction of two charges in a uniform magnetic field I: planar problem P/06/10 See more > See less > The interaction of two charges moving in$R^2$in a magnetic field${f B}$can be formulated as a hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reducethis Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitableregime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb-type interaction. Explicit knowledge of the reconstruction map and adynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types fo dynamical behaviours in this system: from periodic ro quasiperiodic and chaotic orbits, from bounded to unbounded motion. 03/03/2006 PDF Cerbelli and Giona's map is pseudo-Anosov and 9 consequences M/06/06 See more > See less > It is shown that a piecewise affine area-preserving homeomorphism of the 2-torus studied by Cerbelli and Giona is pseudo-Anosov. This enables one to prove various of their conjectures, quantify the multifractality of its "w-measures", calculate many other quantities for its dynamics, and construct an exact area-preserving tilt map of the cylinder with proved diffusive behaviour. 10/02/2006 PDF Discommensuration Theory and Shadowing in Frenkel-Kontorova Models P/06/07 See more > See less > We prove that if the minimum energy advancing discommensuration of mean spacing$p/q$for a Frenkel-Kontorova chain is unique up to translations and has phonon gap then all minimum energy states with mean spacing$\omega$just above$p/q$are approximated exponentially well in$q\omega - p$by concatenations of advancing$p/q$discommensuration. 10/02/2006 PDF Dirichlet forms and Markov semigroups on non-associative vector bundles M/06/04 01/02/2006 PDF Sur quelques représentations potentiellement cristallines de$GL_2(Q_p)$M/06/03 24/01/2006 PDF Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces M/06/02 20/01/2006 PDF Bubbling Geometries for Half BPS Wilson lines P/06/01 18/01/2006 PDF Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces M/05/51 21/12/2005 PDF On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces M/05/56 21/12/2005 PDF Universal enveloping algebras and some applications in physics P/05/26 See more > See less > These notes are intended to provide a self-contained and pedagogical introduction to the universal enveloping algebras and some of their uses in mathematical physics. After reviewing their abstract definitions and properties, the focus is put on their relevance in Weyl calculus, in representation theory and their appearance as higher symmetries of physical systems. 20/12/2005 PDF Semi-global invariants of piecewise smooth Lagrangian fibrations M/05/55 See more > See less > We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the information on the non-smoothness into certain invariants consisting, roughly, of a sequence of closed 1-forms on a torus. The main motivation for this work is given by the piecewise smooth Lagrangian fibrations previously constructed by the authors, which topologicallycoincide with the local models used by Gross in "Topological Mirror Symmetry". 20/12/2005 PDF Hidden symmetries and the fermionic sector of eleven-dimensional supergravity P/05/53 See more > See less > We study the hidden symmetries of the fermionic sector of D=11 supergravity, and the role of K(E10) as a generalised R symmetry'. We find a consistent model of a massless spinning particle on an E10/K(E10) coset manifold whose dynamics can be mapped onto the fermionic and bosonic dynamics of D=11 supergravity in the near space-like singularity limit. This E10-invariant superparticle dynamics might provide the basis of a new definition of M-theory, and might describe the de-emergence' of space-time near a cosmological singularity. 15/12/2005 PDF On Killing tensors and cubic vertices in higher-spin gauge theories P/05/54 See more > See less > The problem of determining all consistent non-Abelian local interactions is reviewed in flat space-time. The antifield-BRST formulation of the free theory is an efficient tool to address this problem. Firstly, it allows to compute all on-shell local Killing tensor fields, which are important because of their deep relationship with higher-spin algebras. Secondly, under the sole assumptions of locality and Poincar?e invariance, all non-trivial consistent deformations of a sum of spin-three quadratic actions deforming the Abelian gauge algebra were determined. They are compared with lower-spin cases. 15/12/2005 PDF Relative Seiberg-Witten and Ozsvath-Szabo invariants for surfaces in 4-manifolds M/05/57 See more > See less > We study the invariants of surfaces in 4-manifolds extracted from the Seiberg-Witten and the Ozsvath-Szabo invariants of their fiber sums with auxiliary Lefschetz fibrations. Such invariants involve relative Spin_c structures and can be treated as refinements of the usual Seiberg-Witten and Ozsvath-Szabo invariants. We prove several properties of the relative invariants, for instance the adjunction inequality for membranes, which estimates their genus, and the product formula. 01/12/2005 PDF Topological strings and two dimensional electrons P/05/34 01/12/2005 PDF Collapsed 5-manifolds with pinched sectional curvature M/05/52 See more > See less > Let$M$be a closed$5$-manifold of pinched curvature$0<\delta\le \text{sec}_M\le 1$. We prove that$M$is homeomorphic to a spherical space form if$M$satisfies one of the following conditions: (i)$\delta =1/4$and the fundamental groupis a non-cyclic group of order$\ge C$, a constant. (ii) The center of the fundamental group has index$\ge w(\delta)$, a constant depending on$\delta$. (iii) The ratio of the volume and the maximal injectivity radius is$<\epsilon(\delta)$. (iv) The volume is less than$\epsilon(\delta)$and the fundamental group$\pi_1(M)$has a center of index at least$w$, a universal constant, and$\pi_1(M)$is either isomorphic to a spherical$5$-space group or has an odd order. 23/11/2005 PDF A Chiral Perturbation Expansion for Gravity P/05/48 22/11/2005 PDF Filtration de monodromie et cycles évanescents formels M/05/50 16/11/2005 PDF Refined Analytic Torsion as an Element of the Determinant Line M/05/49 See more > See less > We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E. We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on$E$, we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual. 08/11/2005 PDF Lectures on curved beta-gamma systems, pure spinors, and anomalies P/05/35 See more > See less > The curved beta-gamma system is the chiral sector of a certain infinite radius limit of the non-linear sigma model with complex target space. Naively it only depends on the complex structures on the worldsheet and the target space. It may suffer from theworldsheet and target space diffeomorphism anomalies. We analyze the curved beta-gamma system on the space of pure spinors, aiming to verify the consistency of Berkovits covariant superstring quantization. We demonstrate that under certain conditions bothanomalies can be cancelled for the pure spinor sigma model, in which case one reproduces the old construction of B.Feigin and E.Frenkel. 02/11/2005 PDF On motives associated to graph polynomials M/05/46 See more > See less > The appearance of multiple zeta values in anomalous dimensions and$\beta$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. 11/10/2005 PDF On obstructions to asphericity of certain crossed modules M/05/45 11/10/2005 PDF Noncritical osp($1 vert 2$,R) M-theory matrix model with an arbitrary time dependent cosmological constant P/05/42 See more > See less > Dimensional reduction of the D=2 minimal super Yang-Mills to the D=1 matrix quantum mechanics is shown to double the number of dynamical supersymmetries, from N=1 to N=2. We analyze the most general supersymmetric deformation of the latter, in order to construct the noncritical 3D M-theory matrix model on generic supersymmetric backgrounds. It amounts to adding a harmonic oscillator potential with an arbitrary time dependent coefficient or cosmological constant,' \Lambda(t). The resulting matrix modelenjoys, irrespective of \Lambda(t), two dynamical supersymmetries which further reveal three hidden so(1,2) symmetries. All together they form the supersymmetry algebra, osp(1|2,R). Each so(1,2) multiplet in the Hilbert space visualizes a dynamics constrained on either Euclidean or Minkowskian dS_{2}/AdS_{2} space, depending on its Casimir. In particular, all the unitary as well as BPS multiplets have the Euclidean dS_{2}/AdS_{2} geometry. We conjecture that the matrix model provides holographic duals tothe 2D superstring theories on various backgrounds having the spacetime signature Minkowskian if \Lambda(t)>0, or Euclidean if \Lambda(t)<0. In particular, we argue that the choice of the negative constant \Lambda corresponds to the N=2 super Liouvilletheory. 11/10/2005 PDF Anatomy of a gauge theory P/05/40 See more > See less > Resumé: We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the Slavnov-Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams. 07/10/2005 PDF Théorie d'Iwasawa des représentations cristallines II M/05/41 07/10/2005 PDF Représentations modulaires de$mathrm{GL}_2(mathbf{Q}_p)$et représentations galoisiennes de dimension$2$M/05/44 See more > See less > On montre la conjecture de Breuil concernant la r\'eduction modulo$p$des repr\'esentations triangulines$V$et des repr\'esentations$\Pi(V)$de$\mathrm{GL}_2(\mathbf{Q}_p)$qui leur sont associ\'ees par la correspondance de Langlands$p$-adique. L'ingr\'edient principal de la d\'emonstration est l'\'etude de certaines repr\'esentations lisses irr\'eductibles de$\mathrm{B}(\mathbf{Q}_p)$via des mod\eles construits en utilisant les$(\varphi,\Gamma)$-modules. 05/10/2005 PDF Kähler flat manifolds of low dimensions M/05/43 See more > See less > We give a list of six dimensional flat K\"ahler manifolds. Moreover, we present an example of eight dimensional flat K\"ahler manifold M with finite Out(\pi_1(M)) group. 04/10/2005 PDF The Hopf algebra structure of renormalizable Quantum Field Theory P/05/39 See more > See less > Review for the Encyclopedia of Mathematical Physics. 20/09/2005 PDF Quaternion Landau-Ginsburg models and noncommutative Frobenius manifolds M/05/37 See more > See less > We extend topological Landau-Ginsburg models with boundaries to Quaternion Landau-Ginsburg models that satisfy the axioms for open-closed topological field theories. Later we prove that moduli spaces of Quaternion Landau-Ginsburg models are non-commutative Frobenius manifolds in means of [J. Geom. Phys, 51 (2003),387-403.]. 14/09/2005 PDF IR Free or Interacting? A Proposed Diagnostic P/05/38 14/09/2005 PDF The continuous spin limit of higher spin equations of motion P/05/36 See more > See less > We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the spin to infinity with their product being fixed. The result allows to transform the Wigner equations to a gauge invariant Fronsdal-like form. We also give the generalisation of the Wigner equations to higher dimensions with fields belonging to arbitrary representations of the massless little group. 13/09/2005 PDF On non-commutative analytic spaces over non-archimedean fields M/05/33 See more > See less > We discuss various examples of non-commutative spaces which can be called non-commutative rigid analytic spaces 30/08/2005 PDF On the torsion of optimal elliptic curves over function fields M/05/32 26/08/2005 PDF Torsion as a function on the space of representations M/05/30 16/08/2005 PDF Dynamics, Laplace transform and Spectral geometry M/05/31 16/08/2005 PDF Jean Dieudonné (1906-1992) mathematician M/05/28 See more > See less > Jean Dieudonn\'e has been one of the most influential French mathematicians during the 20$^{\rm th}$century, especially through his association -- even identification -- with the Bourbaki group. An excellent biography has been written by his friend P. Dugac, a historian of science [4]. We shall retrace here his long and distinguished career. 09/08/2005 PDF An algebraic proof of a cancellation theorem for surfaces M/05/29 See more > See less > Let$\K$be an algebraically closed field of arbitrary characteristic. We give a short self-contained algebraic proof of the following statement: If the cylinder over an affine surface (i. e. the product of our surface and an affine line) over$\K$is (isomorphic to) an affine space then the surface is (isomorphic to) an affine plane. 09/08/2005 PDF Spin three gauge theory revisited P/05/25 See more > See less > We study the problem of consistent interactions for spin-$3$gauge fields in flat spacetime of arbitrary dimension$n > 3$. Under the sole assumptions of Poincar\'e and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total ofeither three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions$n>4$and for completely antisymmetric structure constant tensor, a new covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed. 05/08/2005 PDF A group of diffeomorphisms of the interval with intermediate growth M/05/27 04/08/2005 PDF Sur la dynamique unidimensionnelle en régularité intermédiaire M/05/24 02/08/2005 PDF Représentations p-adiques ordinaires de GL2(Qp) et compatibilité local-global M/05/22 See more > See less > On définit les représentations p-adiques de GL2(Qp) "correspondant" aux représentations potentiellement cristallines réductibles (et éventuellement scindées) de Gal(Qpbar/Qp) de dimension 2 et on montre qu'elles apparaissent naturellement dans la cohomologie étale complétée de la tour en p des courbes modulaires. 21/07/2005 PDF Représentations semi-stables de$GL_2 ({mathbb Q}_p)$, demi-plan$p$-adique et réduction modulo$p$M/05/23 See more > See less > On calcule par voie cohomologique la r\'eduction modulo$p$de certaines repr\'esentations$p$-adiques semi-stables de${\rm GL}_2({\mathbb Q}_p)$. Les calculs exploitent la g\'eom\'etrie du demi-plan$p$-adique. Ils permettent de retrouver certaines formules de la r\'eduction modulo$p$de repr\'esentations$p$-adiques semi-stables de${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$. 20/07/2005 PDF Cauchy-Davenport theorem in group extensions M/05/21 See more > See less > Let A and B be nonempty subsets of a finite group G in which the order of the smallest nontrivial subgroup is not smaller than d=|A|+|B|-1. Then the product set AB has at least d elements. This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper's inverse theorem in the same spirit, giving a complete description of the critical pairs. The proofs depend on the structure of group extensions. 13/07/2005 PDF Sur la conjecture faible de Greenberg dans le cas abélien$p$-décomposé M/05/20 See more > See less > Let$p$be an odd prime. For any CM number field$K$containing a primitive$p^{\rm th}$-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality$\lambda^+ \leq \lambda^-$between the plus'' andminus'' parts of the$\lambda$-invariant of$K$. Greenberg's classical conjecture predicts the vanishing of$\lambda^+$. We propose a weak form of this conjecture:$\lambda^+ = \lambda^-$if and only if$\lambda^+ = \lambda^- = 0$, and we proveit when$K^+$is abelian,$p$is totally split in$K^+$, and certain (mild) conditions on the cohomology of circular units are satisfied. 13/07/2005 PDF On the Riemann zeta-function and analytic characteristic functions M/05/18 See more > See less > Set$f(s) :=1/(\sin(\pi s/4)q(1/2 + s))$with$q(s) := \pi^{-s/2} \, 2 \Gamma (1 + s/2)(s-1) \zeta (s)$. The Riemann hypothesis, RH, and the simple zeros conjecture, SZC, together with conjectures advanced by the author are used to show that$f(s)$on each vertical strip$V_{4n}$of$s$with$4n < {\rm Re} \, (s) < 4(n+1)$provides an analytic characteristic function,$(-1)^n \cdot f(s) = \int_R (dy) e^{sy} P_{4n} (y)$with$P_{4n} (y)$positive. The essential case with$n = 0$implies RH. A formula is obtained for$P_{4n} (y)$, which for$y$negative involves the critical zeros. An alternative formula is obtained for$P_{4n} (y)$, without relying on RH, SZC or other unproven conjectures. It does not involve the critical zeros. Analogous resultsfor the cases of the Dirichlet$L$-functions and the Ramanujan tau Dirichlet$L$-function are conjectured. 28/06/2005 PDF Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology P/05/19 See more > See less > In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. 24/06/2005 PDF Semi-stable extensions on arithmetic surfaces M/05/17 See more > See less > On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles. 07/06/2005 PDF Estimates from below for the spectral function and for the remainder in local Weyl's law M/05/16 See more > See less > We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. 19/05/2005 PDF Formal Lagrangian Operad M/05/14 See more > See less > Given a symplectic manifold$M$, we may define an operad structure on the the spaces$\op^k$of the Lagrangian submanifolds of$(\bar{M})^k\times M$via symplectic reduction. If$M$is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semi-classical part of Kontsevich's deformation of$C^\infty(\R^d)$is adeformation of the trivial symplectic groupoid structure of$T^*\R^d$. 18/05/2005 PDF Gauge invariants and Killing tensors in higher-spin gauge theories P/05/15 See more > See less > In free completely symmetric tensor gauge field theories on Minkowski space-time, all gauge invariant functions and Killing tensor fields are computed, both on-shell and off-shell. These problems are addressed in the metric-like formalisms. 18/05/2005 PDF Partial wave expansion and Wightman positivity in conformal field theory P/05/13 See more > See less > A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a non-perturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space (GCI). Bilocal fields arising in the harmonic decomposition of the operator product expansion (OPE) prove to be a powerful instrument in exploring the field content. In particular,in the theory of a field$\LL$of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part fromthe nontrivial part. 26/04/2005 PDF Sur la réduction des représentations cristallines de dimension 2 en poids moyens M/05/12 See more > See less > On calcule la r\'eduction modulo$p$des repr\'esentations cristallines de dimension$2$dont les poids de Hodge-Tate sont$0$et$k-1$avec$k \in \{p+2,\cdots,2p-1\}$. 20/04/2005 PDF On the Hochschild-Kostant-Rosenberg map for graded manifolds M/05/10 See more > See less > We show that the Hochschild--Kostant--Rosenberg map from the space of multivector fields on a graded manifold$N$(endowed with a Berezinian volume) to the cohomology of the algebra of multidifferential operators on$N$(as a subalgebra of the Hochschild complex of$C^\infty(N)$) is an isomorphism of Batalin--Vilkovisky algebras. Moreover, with an example inspired by string topology, we prove that in general the inclusion of multidifferential operators in the Hochschild complex is not a quasi-isomorphism. 15/04/2005 PDF Rota-Baxter Algebras, Dendriform Algebras and Poincare-Birkhoff-Witt Theorem M/05/11 See more > See less > Rota-Baxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative Rota-Baxter algebras relate to double shuffle relations for multiple zeta values. The interest in the non-commutative setting arose in connection with the work of Connes and Kreimer on the Birkhoff decomposition in renormalization theory in perturbative quantum field theory. We construct free non-commutative Rota-Baxter algebras and apply the construction to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We also prove an analog of the Poincare-Birkhoff-Witt theorem for universal enveloping algebra in the context of dendriform trialgebras. In particular, every dendriform dialgebra and trialgebra is a subalgebra of a Rota-Baxter algebra. We explicitly show that the free dendriform dialgebras and trialgebras, as represented by planar trees, are canonical subalgebras of free Rota-Baxter algebras. 15/04/2005 PDF The character of pure spinors P/05/01 14/04/2005 PDF Dynamics of higher spin fields and tensorial space P/05/06 14/04/2005 PDF Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions M/05/08 See more > See less > We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalizedintegrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladikand Volterra lattices. We establish corresponding Christoffel-Darboux formul\ae . For all these classes of polynomials a$2\times 2$system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudomeasures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted T\"oplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) T\"oplitz (H\"ankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic taufunction. 14/04/2005 PDF On the Distribution of the Wave Function for Systems in Thermal Equilibrium P/05/09 See more > See less > A density matrix that is not pure can arise, via averaging, from many different distributions of the wave function. This raises the question, which distribution of the wave function, if any, should be regarded as corresponding to systems in thermal equilibrium as represented, for example, by the density matrix$\rho_\beta = (1/Z) \exp(- \beta H)$of the canonical ensemble. To answer this question, we construct, for any given density matrix$\rho$, a measure on the unit sphere in Hilbert space, denoted GAP($\rho$), using the Gaussian measure on Hilbert space with covariance$\rho$. We argue that GAP($\rho_\beta$) corresponds to the canonical ensemble. 14/04/2005 PDF An invariant for non simply connected manifolds M/05/07 See more > See less > For a closed manifold$M$we introduce the set of co-Euler structures and we define the modified Ray-Singer torsion, a positive real number associated to$M,$a co-Euler structure and an acyclic representation$\rho$of the fundamental group of$M$with$H^\ast(M;\rho)=0.$If the co-Euler structure is integral we show that the modified Ray--Singer torsion, regarded as a positive (real valued) function on the variety of some complex representations, is the absolute value of a (complex valued) rational function which carries interesting topological information about the manifold. This rational function is the invariant in the title. If the co-Euler structure is arbitrary one obtains a more general object, a holomorphic 1-cocycle. Interesting rational functions in topology appear in this way.The argument of this rational function when defined, is an interesting and apparently unexplored invariant which reminds the Atiyah--Patodi--Singer eta invariant. 02/02/2005 PDF Uniform uniformisation M/05/03 01/02/2005 PDF Factorization Conjecture and the Open/Closed String Correspondence P/05/04 01/02/2005 PDF Analyticity of the susceptibility function for unimodal Markovian maps of the interval P/05/05 01/02/2005 PDF Semi-stable reduction of fiolations M/05/02 26/01/2005 PDF Gromov-Witten invariants and pseudo symplectic capacities M/04/60 See more > See less > We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate itin terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that the Hofer-Zehnder capacity is finite on a small neighborhood of a rational connected closed symplectic submanifold of codimension two in a symplectic manifold. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants. 24/12/2004 PDF A LA RECHERCHE DE LA m-THÉORIE PERDUE Z-THEORY: CHASING m/f THEORY P/04/52 23/12/2004 PDF LECTURES ON NONPERTURBATIVE ASPECTS OF SUPERSYMMETRIC GAUGE THEORIES P/04/53 23/12/2004 PDF FROM SUPERSTRINGS TO QUANTUM FOAM USING SUPERSYMMETRY P/04/54 23/12/2004 PDF GROMOV-WITTEN THEORY AND DONALDSON-THOMAS THEORY, II M/04/55 23/12/2004 PDF Global geometric deformations of current algebras as Krichever-Novikov type algebras M/04/56 See more > See less > We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. 23/12/2004 PDF Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence M/04/57 See more > See less > We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginzburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces. 23/12/2004 PDF On the Galois cohomology of unipotent algebraic groups over local and global function fields M/04/58 See more > See less > We discuss some results on the triviality and finiteness for Galois cohomology of connected unipotent groups over local and global function fields, and their relation with the closedness of orbits. As application, we show that a separable additive polynomial over a global field$k$of characteristic$p>0$in two variables is universal over$k$if and only if it is so over all completions$k_v$of$k$. 05/12/2004 PDF Nonlinear Higher Spin Theories in Various Dimensions P/04/47 01/12/2004 PDF Some Rationality Properties of Observable Groups and Related Questions M/04/62 01/12/2004 PDF Chiral polyhedra in ordinary space, II M/04/61 See more > See less > A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary Euclidean 3-space with finite skew faces and finite skew vertex-figures. Part II completes the enumeration of all discrete chiral polyhedra in 3-space. There exist several families of chiral polyhedra with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I. 01/12/2004 PDF EXPLICIT MUMFORD ISOMORPHISM FOR HYPERELLIPTIC CURVES M/04/51 See more > See less > We give an explicit version of the Mumford isomorphism on the moduli stack of hyperelliptic curves of any given genus. 16/11/2004 PDF LA MONODROMIE HAMILTONIENNE DES CYCLES ÉVANESCENTS M/04/50 See more > See less > Nous montrons sous des hypoth\eses pr\'ecis\'ees dans l'\'enonc\'e que le premier groupe d'homologie \'evanescente d'une fibration lagrangienne singuli\ere est librement engendr\'e par les cycles \'evanescents. Nous en d\'eduisons que l'op\'erateur de variation associ\'e est un isomorphisme. 05/11/2004 PDF Four basic symmetry types in the universal 7-cluster structure of 143 complete bacterial genomic sequences M/04/49 28/10/2004 PDF Fermions in the harmonic potential and string theory P/04/41 See more > See less > We explicitly derive collective field theory description for the system of fermions in the harmonic potential. This field theory appears to be a coupled system of free scalar and (modified) Liouville field. This theory should be considered as an exact bosonization of the system of non-relativistic fermions in the harmonic potential. Being surprisingly similar to the world-sheet formulation of c=1 string theory, this theory has quite different physical features and it is conjectured to give space-time description of the string theory, dual to the fermions in the harmonic potential. A vertex operator in this theory is shown to be a field theoretical representation of the local fermion operator, thus describing a D0 brane in the string language. Possible generalization of this result and its derivation for the case of c=1 string theory (fermions in the inverse harmonic potential) is discussed. 28/10/2004 PDF Creation of Toy Universe P/04/48 See more > See less > General ideas of gauge/gravity duality allow for the possibility of time dependent solutions that interpolate between a perturbative gauge theory phase and a weakly curved string/gravity phase. Such a scenario applied to cosmology would exhibit a non-geometric phase before the big bang. We investigate a toy model for such a cosmology, whose endpoint is the classical limit of the two-dimensional non-critical string. We discuss the basic dynamics of this model, in particular how it evolves toward the double scaling limit required for stringy dynamics. We further comment on the physics that will determine the fluctuation spectrum of the scalar tachyon. Finally, we discuss various features of this model, and what relevance they might have for a more realistic, higher dimensional scenario. 18/10/2004 PDF REPRÉSENTATIONS CRISTALLINES IRRÉDUCTIBLES DE${ M/04/46
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Nous démontrons certaines conjectures (non nullit\'e, irr\'eductibilit\'e topologique, admissibilit\'e) dues au second auteur concernant des représentations unitaires de GL2(Qp) sur des espaces de Banach p-adiques associées aux représentations cristallines irréductibles de dimension 2 de Gal(Qpbar/Qp). Pour cela, nous réinterprétons ces espaces de Banach comme espaces de fonctions sur Qp d'un certain type, puis nous utilisons la théorie des (phi,Gamma)-modules associés aux représentations cristallines correspondantes.
05/10/2004 PDF
Boundary Liouville theory at $c=1$ P/04/42
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The c=1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c=1 limit of unitary minimal models. Here we extend the analysis of the c=1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c=1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of finite width. We propose explicit formulas for the boundary 2-point function and for the bulk-boundary operator product expansion in the c=1 boundary Liouville model. As a by-product of our analysis we also provide a nice geometric interpretation for ZZ branes and their relation with FZZT branes in the c=1 theory.
29/09/2004 PDF
On the Landau Background Gauge Fixing and the IR Properties of YM Green Functions P/04/23
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We analyse the complete algebraic structure of the background field method for Yang--Mills theory in the Landau gauge and show several structural simplifications within this approach. In particular we present a new way to study the IR behavior of Green functions in the Landau gauge and show that there exists a unique Green function whose IR behaviour controls the IR properties of the gluon and the ghost propagators.
29/09/2004 PDF
Towards Integrability of Topological Strings I: Three-forms on Calabi-Yau manifolds P/04/43
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The precise relation between Kodaira-Spencer path integral and a particular wave function in seven dimensional quadratic field theory is established. The special properties of three-forms in 6d, as well as Hitchin's action functional, play an important role. The latter defines a quantum field theory similar to Polyakov's formulation of 2d gravity; the curious analogy with world-sheet action of bosonic string is also pointed out.
29/09/2004 PDF
THE K-THEORY OF HEEGAARD-TYPE QUANTUM 3-SPHERES M/04/44
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We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.
29/09/2004 PDF
Lattice Geometry P/04/45
29/09/2004 PDF
The Structure of the Ladder Insertion-Elimination Lie Algebra M/04/40
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We continue our investigation into the insertion-elimination Lie algebra of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson-Schwinger equations. We work out the relation of this Lie algebra to some classical infinite dimensional Lie algebra and we determine its cohomology.
01/09/2004 PDF
Saddle point equations in Seiberg-Witten theory P/04/38
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N=2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for (almost) all models allowed by the asymptotic freedom the 1-instanton corrections which follows from these equations agree with the direct computations and with known results.
24/08/2004 PDF
What is the trouble with Dyson--Schwinger equations? P/04/37
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We discuss similarities and differences between Green Functions in Quantum Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint equations which originate from an underlying Hopf algebra structure. Typically, the equation is linear for the polylog, and non-linear for Green Functions. We argue though that the crucial difference lies not in the non-linearity of the latter, but in the appearance of non-trivial representation theory related to transcendental extensions of the number field which governs the linear solution. An example is studied to illuminate this point.
11/08/2004 PDF
Spitzer's identity and the algebraic Birkhoff decomposition in pQFT P/04/39
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In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.
11/08/2004 PDF
Approximation by analytic operator functions. Factorizations and very badly approximable functions M/04/36
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This is a continuation of our earlier paper. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $\T$ and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly approximable operator functions. We obtain algebraic and geometric characterizations of badly approximable and very badly approximable operator functions. Note that there is an important difference between the case of finite matrix functions and the case of operator functions. Our criteria for a function to be very badly approximable in the case of finite matrix functions also guarantee that the zero function is the only superoptimal approximant. However in the case of operator functions this is not true.
11/08/2004 PDF
AN EXTENSION OF THE KOPLIENKO-NEIDHARDT TRACE FORMULAE M/04/35
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Koplienko found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class $\bS_2$. A similar formula in the case of unitary operators was obtained by Neidhardt. In this paper we improve their results and obtain sharp conditions under which the Koplienko--Neidhardt trace formulae hold.
30/07/2004 PDF
Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model M/04/34
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We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes.
29/07/2004 PDF
Differentiating the absolutely continuous invariant measure of an interval map $f$ with respect to $f$ M/04/32
23/07/2004 PDF
Differentiation of SRB states for hyperbolic flows M/04/33
23/07/2004 PDF
ABCD of instantons P/04/19
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We solve N=2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do thiswithout resolving singularities of the compactified instanton moduli spaces.
21/07/2004 PDF
Bernoulli Number Identities from Quantum Field Theory P/04/31
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We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families ofconvolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki's identity due to Gessel and obtain a new similar identity generalizing the FPZ identity. The generalization of the method to the derivation of convolution identities of arbitrary order is outlined. We also describe an extension to identities which relate convolutions of Euler and Bernoulli numbers.
29/06/2004 PDF
Holomorphically Covariant Matrix Models P/04/30
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We present a method to construct matrix models on arbitrary simply connected oriented real two dimensional Riemannian manifolds. The actions and the path integral measure are invariant under holomorphic transformations of matrix coordinates.
28/06/2004 PDF
Distance Function, Linear quasi-Connections and Chern Character. M/04/27
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In Sect.4 we show how the Chern character of the tangent bundle of a smooth manifold may be extracted from the geodesic distance function by means of cyclic homology. In Sect.5 we introduce the notion of coarse linear connection in vector bundles, notion which generalizes the notion of linear connection. We show next that the algebraic procedure for constructing the Chern character, discussed in Sect.4, applies also in the case of coarse linear connections. Looking in retrospect, the constructions we present here represent the non commutative counterpart of a geometric construction of the Chern character, see Teleman N.[Tn] and Teleman C.[Tc]. The arguments discussed here may be formulated within the language of groupoids. In a subsequent paper we are going to improve some of the considerations presented here and extend their field of application to more singular situations.
25/06/2004 PDF
Optimal Trade-Off for Merkle Tree Traversals M/04/29
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We prove tight upper and lower bounds for computing Merkle tree traversals, and display optimal trade-offs between time and space complexity of that problem.
17/06/2004 PDF
Gravitational radiation from inspiralling compact binaries completed at the third post-Newtonian order P/04/25
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The gravitational radiation from point particle binaries is computed at the third post-Newtonian (3PN) approximation of general relativity. Three previously introduced ambiguity parameters, coming from the Hadamard self-field regularization of the 3PN source-type mass quadrupole moment, are consistently determined by means of dimensional regularization, and proved to have the values xi = -9871/9240, kappa = 0 and zeta = -7/33. These results complete the derivation of the general relativistic prediction for compact binary inspiral up to 3.5PN order, and should be of use for searching and deciphering the signals in the current network of gravitational wave detectors.
15/06/2004 PDF
Mathematical models in population dynamics and ecology M/04/26
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We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals. The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short- and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of theMcKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.
15/06/2004 PDF
Approximation Hardness of Short Symmetric Instances of MAX-3SAT M/04/28
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We prove approximation hardness of short symmetric instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3. We display also an explicit approximation lower bound for that problem. The bound two on the number of occurrences of literals in symmetric MAX-3SAT is thus the smallest possible bound for the MAX-3SAT hardness gap property to exist and making the instances hard to approximate.
15/06/2004 PDF
Deformation of outer representations of Galois group M/04/24
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To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, weintroduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite dimensional graded components to serve our purpose.
02/06/2004 PDF
Phasing of gravitational waves from inspiralling eccentric binaries P/04/22
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We provide a method for analytically constructing high-accuracy templates for the gravitational wave signals emitted by compact binaries moving in inspiralling eccentric orbits. By contrast to the simpler problem of modeling the gravitational wave signals emitted by inspiralling {\it circular} orbits, which contain only two different time scales, namely those associated with the orbital motion and the radiation reaction, the case of {\it inspiralling eccentric} orbits involves {\it three different time scales}: orbital period, periastron precession and radiation-reaction time scales. By using an improved method of variation of constants', we show how to combine these three time scales, without making the usual approximation of treating the radiativetime scale as an adiabatic process. We explicitly implement our method at the 2.5PN post-Newtonian accuracy. Our final results can be viewed as computing new post-adiabatic' short period contributions to the orbital phasing, or equivalently, new short-period contributions to the gravitational wave polarizations, $h_{+,\times}$, that should be explicitly added to the post-Newtonian' expansion for $h_{+,\times}$, if one treats radiative effects on the orbital phasing of the latter in the usual adiabatic approximation. Our results should be of importance both for the LIGO/VIRGO/GEO network of ground based interferometric gravitational wave detectors (especially if Kozai oscillations turn out to be significant in globular cluster triplets), and for thefuture space-based interferometer LISA.
27/05/2004 PDF
Conformal invariance and rationality in an even dimensional quantum field theory P/04/21
21/05/2004 PDF
Vertex Operators for Closed Superstrings P/04/20
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We construct an iterative procedure to compute the vertex operators of the closed superstring in the covariant formalism given a solution of IIA/IIB supergravity. The manifest supersymmetry allows us to construct vertex operators for any generic background in presence of Ramond-Ramond (RR) fields. We extend the procedure to all massive states of open and closed superstrings and we identify two new nilpotent charges which are used to impose the gauge fixing on the physical states. We solve iteratively theequations of the vertex for linear x-dependent RR field strengths. This vertex plays a role in studying non-constant C-deformations of superspace. Finally, we construct an action for the free massless sector of closed strings, and we propose a form for the kinetic term for closed string field theory in the pure spinor formalism.
13/05/2004 PDF
Self-similar Fractals in Arithmetic M/04/18
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We define a notion of self-similarity on algebraic varieties by considering algebraic endomorphisms as "similarity" maps. Self-similar objects are called fractals, for which we present several examples and define a notion of dimension in many differentcontexts. We also present a strong version of Roth's theorem for algebraic points on a variety approximated by fractal elements. Fractals provide a framework in which one can unite several important conjectures in Diophantine geometry.
30/04/2004 PDF
The residues of quantum field theory - numbers we should know M/04/17
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We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory.
20/04/2004 PDF
Noncentral extension of the $AdS_{5} \times S^{5}$ superalgebra : supermultiplet of brane charges P/04/15
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We propose an extension of the su(2, 2|4) superalgebra to incorporate the F1/D1 string charges in type IIB string theory on the $AdS_{5} \times S^{5}$ background, or the electro-magnetic charges in the dual super Yang-Mills theory. With the charges introduced, the superalgebra inevitably undergoes a noncentral extension, as noted recently in [1]. After developing a group theoretical method of obtaining the noncentral extension, we show that the charges form a certain nonunitary representation of the original unextended superalgebra, subject to some constraints. We solve the constraints completely and show that, apart from the su(2,2|4) generators, there exist 899 complex brane charges in the extended algebra. Explicitly we present all the super-commutation relations among them.
18/04/2004 PDF
D-brane charges on SO(3) P/04/14
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In this letter we discuss charges of D-branes on the group manifold SO(3). Our discussion will be based on a conformal field theory analysis of boundary states in a Z_2-orbifold of SU(2). This orbifold differs from the one recently discussed by Gaberdieland Gannon in its action on the fermions and leads to a drastically different charge group. We shall consider maximally symmetric branes as well as branes with less symmetry, and find perfect agreement with a recent computation of the corresponding K-theory groups.
12/04/2004 PDF
Weak Bézout inequality for D-modules M/04/16
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A bound is obtained on the leading coefficient of the Hilbert-Kolchin polynomial of a D-module in terms of the degrees of its generators.
10/04/2004 PDF
The Hopf algebra of rooted trees in Epstein-Glaser Renormalization P/04/12
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We show how the Hopf algebra of rooted trees, in a somewhat modified presentation, encodes the combinatorics of Epstein-Glaser renormalization and position space renormalization in general. In particular we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions, and by twisting the antipode with a renormalization character, which formally solves the Bogoliubov recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator $B_+$.
01/04/2004 PDF
S-duality and Topological Strings P/04/09
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In this paper we show how S-duality of type IIB superstrings leads to an S-duality relating A and B model topological strings on the same Calabi-Yau as had been conjectured recently: D-instantons of the B-model correspond to A-model perturbative amplitudes and D-instantons of the A-model capture perturbative B-model amplitudes. Moreover this confirms the existence of new branes in the two models. As an application we explain the recent results concerning A-model topological strings on Calabi-Yau and its equivalence to the statistical mechanical model of melting crystal.
31/03/2004 PDF
Lusternik-Schnirelman theory and dynamics, II M/04/13
30/03/2004 PDF
An introduction to arithmetic groups M/04/11
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Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by generators and relations. The next topic will be: are all subgroups of finite index given by congruence conditions? Finally, we shall discuss rigidity properties of arithmetic groups.
20/03/2004 PDF
Integrable Renormalization II: the general case P/04/08
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We extend the results we obtained in an earlier. The cocommutative case of rooted ladder graphs is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.
17/03/2004 PDF
Torsion cohomology classes and algebraic cycles on complex projective manifolds M/04/10
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Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes are small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration on Chow groups.
10/03/2004 PDF
Systèmes de Taylor-Wiles pour ${\rm GSp}_4$ M/04/07
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On montre qu'une représentation galoisienne symplectique de degré quatre congrue modulo p à une représentation galoisienne provenant d'une forme de Siegel de genre deux provient elle-même d'une telle forme de Siegel. On suppose pour cela que l'image de Galois est grosse modulo p et que des hypothèses locales sont satisfaites (minimalité en dehors de p et ordinarité cristalline en p, de poids de Hodge-Tate adéquats). On utilise pour cela la technique des systèmes de Taylor-Wiles.
07/03/2004 PDF
07/03/2004 PDF
An inverse theorem for the restricted set addition in Abelian groups M/04/06
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Let $A$ be a set of $k\ge 5$ elements of an Abelian group $G$ in which the order of the smallest nontrivial subgroup is larger than $2k-3$. Extending a result of Dias da Silva and Hamidoune (Bull. London Math. Soc. 26 (1994) 140-146), in a recent paperwe proved that the number of different elements of $G$ that can be written in the form $a+a'$, where $a, a'\in A$, $a\ne a'$, is at least $2k-3$. Here we prove that the bound is attained if and only if the elements of $A$ form an arithmetic progression in $G$, thus completing the solution of a problem of Erd\H os and Heilbronn. The proof is based on the so-called Combinatorial Nullstellensatz'.
25/02/2004 PDF
Spectral Asymmetry, Zeta Functions and the Noncommutative Residue M/04/02
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In this paper, motivated by an approach developped by Wodzicki, we look at the spectral asymmetry of elliptic PsiDO's in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic PsiDO'sand of geometric Dirac operators. In particular, we show that the eta function of a selfadjoint elliptic odd PsiDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of Branson-Gilkey for Dirac operators), and we relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemmanian geometric data, which yields a new spectral interpretation of the Einstein action from gravity. We also obtain a large class of examples of elliptic PsiDO's for which the regular values at the origin of the (local) zeta functions can easily be seen to be independent of the spectral cut. On the other hand, we simplify the proofs of two well-known and difficult results of Wodzicki: (i) The independence with respect to the spectral cut of the regular value at the origin of the zeta function of an elliptic PsiDO; (ii) The vanishing of the noncommutative residue of a zero'th order PsiDO projector. These results were proved by Wodzickiusing a quite difficult and involved characterization of local invariants of spectral asymmetry, which we can bypass here. Finally, in an appendix we give a new proof of the aforementioned asymmetry formulas of Wodzicki.
30/01/2004 PDF
The entropy of black holes: a primer P/04/04
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After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the "correspondence" between massive string states and non-supersymmetric Schwarzschild black holes.
21/01/2004 PDF
Cayley-Hamilton Decomposition and Spectral Asymmetry M/04/01
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In this paper we derive Cayley-Hamilton decompositions, along some of their consequences, for compact operators and closed operators with compact resolvent on a (separable) Hilbert space. In particular, we make use these decompositions to give a spectralinterpretation of a projector found by Wodzicki to encode the spectral asymmetry of elliptic PsiDO's on a compact manifold. As another application we get a convenient definition of the partial inverse of a closed operator with compact resolvent. Finally,we work out the results of this paper in the examples of an elliptic PsiDO on a compact manifold and of an elliptic PsiDO on a spectral triple (i.e. on a noncommutative manifold in the sense of Connes's noncommutative geometry).
20/01/2004 PDF
Unraveling the Fourier Law for Hamiltonian Systems P/04/03
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We exhibit simple Hamiltonian and stochastic models of heat transport in non-equilibrium problems. Theoretical arguments are given to show that, for a wide class of models, the temperature profile obeys a universal law depending on a parameter $\alpha$. When $\alpha=1$, the law is linear, but, depending on the nature of the energy exchange mechanism by tracer particles, we find that $\alpha$ is, in many cases, different from $1$. When $\alpha \neq 1$, the temperature profile is not linear, although translation invariance and, in some models, local thermal equilibrium, hold.
14/01/2004 PDF | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8860054016113281, "perplexity": 1225.876635571693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813571.24/warc/CC-MAIN-20180221063956-20180221083956-00305.warc.gz"} |
http://www.gradesaver.com/tuesdays-with-morrie/q-and-a/who-is-morries-immediate-family-272779 | # who is morries immediate family
Who is Morrie's Immediate family?
##### Answers 1
I think it is just his wife. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9873419404029846, "perplexity": 25829.358108511562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607242.32/warc/CC-MAIN-20170522230356-20170523010356-00575.warc.gz"} |
https://www.physicsforums.com/threads/electrostatics-quick-question-help.247303/ | # Electrostatics quick question help
1. Jul 28, 2008
1. The problem statement, all variables and given/known data
I have a series of about a dozen questions, my son is taking a course that was outlined by his school " as a great way to get ready for Grade 12" it is mostly multiple choice so if someone could help us choose the right answer with a little bit of an explanation. He has done all the questions himself i just need help marking them.
Question 1.
One of the ways in which Newton's law of universal gravitation differs from Coulomb's law is that gravitational force can only attract, whereas the electric force can only repel.
True Or False
2. Relevant equations
3. The attempt at a solution
I think it is True because no where does it state in electrostatics that the electric force Can repel.
2. Jul 28, 2008
### nicksauce
Well look at Coulomb's law: F = k*q1q2/r^2. Note F ~ q1q2. Charges can be either positive (like protons) or negative (like electrons).
(Edited for clarity). If q1 and q2 are of the opposite sign then F is negative and attractive, whereas if q1 and q2 are of the same sign then F is positive and repulsive.
Now compare to the Gravitational force: F ~ m1m2. Since mass is only positive, F can only have one sign, and so it is always attractive.
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https://rupress.org/jcb/article/109/2/529/55702/Structural-changes-induced-in-Ca2-regulated-myosin | We have used electron microscopy and proteolytic susceptibility to study the structural basis of myosin-linked regulation in synthetic filaments of scallop striated muscle myosin. Using papain as a probe of the structure of the head-rod junction, we find that this region of myosin is approximately five times more susceptible to proteolytic attack under activating (ATP/high Ca2+) or rigor (no ATP) conditions than under relaxing conditions (ATP/low Ca2+). A similar result was obtained with native myosin filaments in a crude homogenate of scallop muscle. Proteolytic susceptibility under conditions in which ADP or adenosine 5'-(beta, gamma-imidotriphosphate) (AMPPNP) replaced ATP was similar to that in the absence of nucleotide. Synthetic myosin filaments negatively stained under relaxing conditions showed a compact structure, in which the myosin cross-bridges were close to the filament backbone and well ordered, with a clear 14.5-nm axial repeat. Under activating or rigor conditions, the cross-bridges became clumped and disordered and frequently projected further from the filament backbone, as has been found with native filaments; when ADP or AMPPNP replaced ATP, the cross-bridges were also disordered. We conclude (a) that Ca2+ and ATP affect the affinity of the myosin cross-bridges for the filament backbone or for each other; (b) that the changes observed in the myosin filaments reflect a property of the myosin molecules alone, and are unlikely to be an artifact of negative staining; and (c) that the ordered structure occurs only in the relaxed state, requiring both the presence of hydrolyzed ATP on the myosin heads and the absence of Ca2+.
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http://www.edri.gov.et/documents/total-factor-productivity-and-technical-efficiency-in-the-ethiopian-manufacturing-sector/ | # EDRI Working Paper 010: Total Factor Productivity and Technical Efficiency in the Ethiopian Manufacturing Sector
## Abstract
In this paper a Stochastic Frontier Model is used to examine the technical efficiency and total factor productivity (TFP) growth in the Ethiopian manufacturing sector over the period 1996– 2009. TFP growth is decomposed into technical change (progress), technical efficiency change, and scale effect. With a firm level unbalanced panel data collected by CSA, individual estimations are made for each industrial group categorized by two digit ISIC except a three digit ISIC for food and beverage industrial groups. The empirical results indicate existence of large inefficiencies, inefficiency that explains at least 14 percent of output variation among firms. TFP has shown better progress after 2001/02 and the growth is largely explained by technical change which is a shift in production frontier. The effect of efficiency change is very small as most industrial groups have time invariant efficiency. In addition, the scale effect is zero or very small because most industrial groups have constant returns to scale or small deviation from constant returns to scale. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8107220530509949, "perplexity": 3469.5084083191523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997731.69/warc/CC-MAIN-20190616042701-20190616064701-00175.warc.gz"} |
http://www.qspace-cost.eu/publications/?publication_category=6 | # Publications
A list of the publications of the several Working Groups and (software) tools developed by participants. Publications by project members should be sent to the following website representatives:
## WG1
• Feature article on CQG+: Bouncing a cosmic brew
• We consider implications of the microscopic dynamics of spacetime for the evolution of cosmological models. We argue that quantum geometry effects may lead to stochastic fluctuations of the gravitational constant, which is thus considered as a macroscopic effective dynamical quantity. Consistency with Riemannian geometry entails the presence of a time-dependent dark energy term in the […]
• We show that in the perturbative regime defined by the coupling constant, the θ-exact Seiberg-Witten map applied to the noncommutative U(N) Yang-Mills — with or without Supersymmetry — gives an ordinary gauge theory which is, at the quantum level, dual to the former. We do so by using the on-shell DeWitt effective action and dimensional […]
• The equivalence of the noncommutative U(N) quantum field theories related by the θ-exact Seiberg- Witten maps is, in this paper, proven to all orders in the perturbation theory with respect to the coupling constant. We show that this holds for super Yang-Mills theories with N 1⁄4 0, 1, 2, 4 supersymmetry. A direct check of […]
• In this paper, we discuss two features of the noncommutative geometry and spectral action approach to the Standard Model: the fact that the model is inherently Euclidean, and that it requires a quadrupling of the fermionic degrees of freedom. We show how the two issues are intimately related. We give a precise prescription for the […]
• Abstract: We compute the one-loop 1PI contributions to all the propagators of the noncommutative (NC) N = 1, 2, 4 super Yang-Mills (SYM) U(1) theories defined by the means of the θ-exact Seiberg-Witten (SW) map in the Wess-Zumino gauge. Then we extract the UV divergent contributions and the noncommutative IR divergences. We show that all […]
• I make some considerations on the quantization of spacetime from a spectral point of view. The considerations range from the renormalization flow, to the standard model, to a new phase of spacetime.
• Is it possible that the fundamental Planck constant has stochastic fluctuations? We entertain this possibility in this talk, motived by a possible quantum structure of spacetime. We describe in which sense Panck’s constant is inconstant (a nontrivial affair for fundamental dimensionful quantities. In our scheme h is time dependent, given by random fluctuations around zero, […]
• One examines putative corrections to the Bell operator due to the noncommutativity in the phase-space. Starting from a Gaussian squeezed envelop whose time evolution is driven by commutative (standard quantum mechanics) and noncommutative dynamics respectively, one concludes that, although the time evolving covariance matrix in the noncommutative case is different from the standard case, the […]
• Motivated by the Dirac idea that fundamental constants are dynamical variables and by conjectures on quantum structure of space–time at small distances, we consider the possibility that Planck constant ℏ is a time depending quantity, undergoing random Gaussian fluctuations around its measured constant mean value, with variance σ2 and a typical correlation timescale Δt. We […]
• We calculate the Green functions for a scalar field theory with quartic interactions for which the fields are multiplied with a generic translation invariant star product. Our analysis involves both noncommutative products, for which there is the canonical commutation relation among coordinates, and nonlocal commutative products. We give explicit expressions for the one-loop corrections to […]
• We analyze the running at one-loop of the gauge couplings in the spectral Pati-Salam model that was derived in the framework of noncommutative geometry. There are a few different scenarios for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different […]
• We deduce an evolution equation for an arbitrary hybrid Seiberg-Witten map for compact gauge groups by using the antifield formalism. We show how this evolution equation can be used to obtain the hybrid Seiberg-Witten map as an expansion, which is θ-exact, in the number of ordinary fields. We compute explicitly this expansion up to order […]
• We consider the spectral action within the context of a 4-dimensional manifold with torsion and show that, in the vacuum case, the equations of motion reduce to Einstein’s equations, securing the linear stability of the theory. To subsequently investigate the nonvacuum case, we consider the spectral action of an almost commutative torsion geometry and show […]
## WG2
• We introduce ABJM quantum field theory in the noncommutative spacetime by using the component formalism and show that it is N = 6 supersymmetric. For the U(1)κ × U(1)−κ case, we compute all one-loop 1PI two and three point functions in the Landau gauge and show that they are UV finite and have well-defined commutative […]
• For any number h such that hbar:=h/(2\pi) is irrational, let A_{g,h} be the corresponding Weyl *-algebra over Z^{2g} and consider the ergodic group of *-automorphisms of A_{g,h} induced by the action of Sp(2g,Z) on Z^{2g}. We show that the only Sp(2g,Z)-invariant state on A_{g,h} is the trace state.
• We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms.
• This note describes a quantization of the relational symplectic groupoid (RSG) for a constant Poisson structure using the BV-BFV formalism and shows how this induces the Moyal deformation quantization for the underlying Poisson manifold.
• It was pointed out by Shifman and Yung that the critical superstring on $X^{10}={\mathbb R}^4\times Y^6$, where $Y^6$ is the resolved conifold, appears as an effective theory for a U(2) Yang-Mills-Higgs system with four fundamental Higgs scalars defined on $\Sigma_2\times{\mathbb R}^2$, where $\Sigma_2$ is a two-dimensional Lorentzian manifold. Their Yang-Mills model supports semilocal vortices on […]
• We present the minimal realization of the ℓ-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex Pais-Uhlenbeck oscillator equations. We introduce a minimal deformation of the ℓ=1/2 conformal Galilei (a.k.a. Schr\”odinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d=1 conformal […]
• We discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the discussion in [arXiv:0706.3164], where a gauge fixing defined by a compatible complex structure was introduced, by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss […]
• Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1,1|2) in mechanics. Remarking that SU(1,1|2) is a particular member of a chain of supergroups SU(1,1|n) parametrized by an integer n, here we begin a systematic study of SU(1,1|n) multi-particle mechanics. A representation of the superconformal algebra su(1,1|n) is constructed […]
• We consider Spin(4)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M^d \times T^{1,1}$, where $M^d$ is a smooth manifold and $T^{1,1}$ is a five-dimensional Sasaki-Einstein manifold Spin(4)/U(1). We obtain new quiver gauge theories on $M^d$ extending those induced via reduction over the leaf spaces $\mathbb{C}P^1 \times \mathbb{C}P^1$ in $T^{1,1}$. We describe the […]
• We extract the leading-order entropy of a four-dimensional extremal black hole in ${\cal N}{=}2$ ungauged supergravity by formulating the CFT$_1$ that is holographically dual to its near-horizon AdS$_2$ geometry, in terms of a rational Calogero model with a known counting formula for the degeneracy of states in its Hilbert space.
• In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker’s polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information […]
• The spherical reduction of the rational Calogero model (of type A_{n−1} and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at […]
• We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge […]
• We recall the emergence of a generalized gauge theory from a noncommutative Riemannian spin manifold, viz. a real spectral triple (A,H,D;J). This includes a gauge group determined by the unitaries in the *-algebra A and gauge fields arising from a so-called perturbation semigroup which is associated to A. Our main new result is the interpretation […]
• Deformation theory refers to an apparatus in many parts of math and physics for going from an infinitesimal (= first order) deformation to a full deformation, either formal or convergent appropriately. If the algebra being deformed is that of observables, the result is deformation quantization, independent of any realization in terms of Hilbert space operators. […]
• We derive an explicit expression for the star product reproducing the κ-Minkowski Lie algebra in any dimension n. The result is obtained by suitably reducing the Wick-Voros star product defined on ℂdθ with n=d+1. It is thus shown that the new star product can be obtained from a Jordanian twist.
• This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.
• The goal of this note is to give a brief overview of the BV-BFV formalism developed by the first two authors and Reshetikhin in [arXiv:1201.0290], [arXiv:1507.01221] in order to perform perturbative quantisation of Lagrangian field theories on manifolds with boundary, and present a special case of Chern-Simons theory as a new example.
• Over many decades, the word “double” has appeared in various contexts, at times seemingly unrelated. Several have some relation to mathematical physics. Recently, this has become particularly strking in DFT (double field theory). Two ‘doubles’ that are particularly relevant are double vector bundles and Drinfel’d doubles. The original Drinfel’d double occurred in the contexts of […]
• We consider linear star products on Rd of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with […]
• This is a survey of our program of perturbative quantization of gauge theories on manifolds with boundary compatible with cutting/pasting and with gauge symmetry treated by means of a cohomological resolution (Batalin-Vilkovisky) formalism. We also give two explicit quantum examples -- abelian BF theory and the Poisson sigma model. This exposition is based on a […]
• We formulate an algebraic criterion for the presence of global anomalies on globally hyperbolic space-times in the framework of locally covariant field theory. We discuss some consequences and check that it reproduces the well-known global SU(2) anomaly in four space-time dimensions.
## WG3
• The space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise […]
• We discuss a (3+1)-dimensional covariant quantum space-time describing a FLRW cosmology with Big Bounce, obtained by a projection of the fuzzy hyperboloid H4n. This provides a background solution of the IKKT matrix model with mass term. We characterize the bosonic fluctuation spectrum, which consists of a tower of higher-spin modes, truncated at n. The modes […]
• The eight nonisomorphic Drinfel’d double (DD) structures for the Poincaré Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincaré group are also identified and constructed, while in (3+1) dimensions no Poincaré DD structure does exist. Each of […]
• We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on […]
• We consider the SO(4,1)-covariant fuzzy hyperboloid H4n as a solution of Yang-Mills matrix models, and study the resulting higher-spin gauge theory. The degrees of freedom can be identified with functions on classical H4 taking values in a higher-spin algebra associated to (4,1), truncated at spin n. We develop a suitable calculus to classify the higher-spin […]
• Curved momentum spaces associated to the $\kappa$-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the $\kappa$-deformation with non-vanishing cosmological constant. The $\kappa$-de Sitter and $\kappa$-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, […]
• Geometrically the phase space of a mechanical system involves the co-tangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In […]
• For a field theory that is invariant under diffeomorphisms there is a subtle interplay between symmetries, conservation laws and the phase space of the theory. The natural language for describing these ideas is that of differential forms and both differential forms on space-time and differential forms on the infinite dimensional space of solutions of the […]
• We consider SU(N) N=4 super Yang-Mills with cubic and quadratic soft SUSY breaking potential, such that the global SU(4)R is broken to SU(3) or further. As shown recently, this set-up supports a rich set of non-trivial vacua with the geometry of self-intersecting SU(3) branes in 6 extra dimensions. The zero modes on these branes can […]
• We compute the lowest order gravitational UV divergent radiative corrections to the S matrix element of the fermion + fermion → fermion + fermion scattering process in the massive Yukawa theory, coupled either to Unimodular Gravity or to General Relativity. We show that both Unimodular Gravity and General Relativity give rise to the same UV […]
• We obtain massive deformations of Type IIA supergravity theory through duality twisted reductions of Double Field Theory (DFT) of massless Type II strings. The mass deformation is induced through the reduction of the DFT of the RR sector. Such reductions are determined by a twist element belonging to Spin+(10,10), which is the duality group of […]
• We work out the one-loop and order κ2 m2 UV divergent contributions, coming from Unimodular Gravity and General Relativity, to the S matrix element of the scattering process + → + in a λ 4 theory with mass m. We show that both Unimodular Gravity and General Relativity give rise to the same UV divergent contributions in Dimensional Regularization. […]
• The $\kappa$-deformation of the (2+1)D anti-de Sitter, Poincaré and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson-Lie structure underlying the $\kappa$-deformation are explicitly given, and the three quantum kinematical groups are obtained […]
• We present simple solutions of IKKT-type matrix models describing a quantized homogeneous and isotropic cosmology with k=−1, finite density of microstates and a resolved Big Bang. At late times, a linear coasting cosmology a(t)∝t is obtained, which is remarkably close to observation. The solution consists of two sheets with opposite intrinsic chiralities, which are connected […]
• We present simple solutions of IKKT-type matrix models describing quantized homogeneous and isotropic cosmologies, with finite density of microstates and a regular Big Bang (BB). The BB arises from a signature change of the effective metric on a fuzzy brane embedded in Lorentzian target space, in the presence of a quantized 4-volume form. The Hubble […]
• We study duality twisted reductions of the Double Field Theory (DFT) of the RR sector of massless Type II theory, with twists belonging to the duality group Spin+(10,10). We determine the action and the gauge algebra of the resulting theory and determine the conditions for consistency. In doing this, we work with the DFT action […]
• We study the beta functions of the quartic and Yukawa couplings of General Relativity and Unimodular Gravity coupled to the λφ^4 and Yukawa theories with masses. We show that the General Relativity corrections to those beta functions as obtained from the 1PI functional by using the standard MS multiplicative renormalization scheme of Dimensional Regularization are […]
• The correspondence between Poisson homogeneous spaces over a Poisson-Lie group $G$ and Lagrangian Lie subalgebras of the classical double $D({\mathfrak g})$ is revisited and explored in detail for the case in which ${\mathfrak g}=D(\mathfrak a)$ is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous […]
• We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a […]
• We work out the one-loop contribution to the lepton anomalous mag- netic moment coming from Unimodular Gravity. We use Dimen- sional Regularization and Dimensional Reduction to carry out the com- putations. In either case, we find that Unimodular Gravity gives rise to the same one-loop correction as that of General Relativity.
• We examine in detail the higher spin fields which arise on the basic fuzzy sphere S4N in the semi-classical limit. The space of functions can be identified with functions on classical S4 taking values in a higher spin algebra associated to (5). Yang-Mills matrix models naturally provide an action formulation for higher spin gauge theory […]
• We study quantum tunnelling in Dante’s Inferno model of large field inflation. Such a tunnelling process, which will terminate inflation, becomes problematic if the tunnelling rate is rapid compared to the Hubble time scale at the time of inflation. Consequently, we constrain the parameter space of Dante’s Inferno model by demanding a suppressed tunnelling rate […]
• We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as SO(5)-equivariant projections of quantized coadjoint orbits of SO(6). We show that they arise as solutions in Yang-Mills matrix models, which naturally leads to higher-spin gauge theories on S4. Several types of embeddings in matrix models are found, […]
• We show that the non-Abelian nature of geometric fluxes - the corner-stone in the definition of quantum geometry in the framework of loop quantum gravity (LQG) - follows directly form the continuum canonical commutations relations of gravity in connection variables and the validity of the Gauss law. The present treatment simplifies previous formulations and thus […]
• This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely the classically-equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method.
• The maximally helicity violating tree-level scattering amplitudes involving three, four or five gravitons are worked out in Unimodular Gravity. They are found to coincide with the corresponding amplitudes in General Relativity. This a remarkable result, insofar as both the propagators and the vertices are quite different in the two theories.
• We prove that Kitaev’s lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev-Viro and […]
• The symmetries of unimodular gravity are clarified somewhat.
• We study perturbations of the 4-dimensional fuzzy sphere as a background in the IKKT or IIB matrix model. The linearized 4-dimensional Einstein equations are shown to arise from the classical matrix model action, without adding an Einstein-Hilbert term. The excitation modes with lowest spin are identified as gauge fields, metric and connection fields. In addition […]
• Refining previous work by Iso, Kawai and Kitazawa, we discuss bi-local string states as a tool for loop computations in noncommutative field theory and matrix models. Defined in terms of coherent states, they exhibit the stringy features of noncommutative field theory. This leads to a closed form for the 1-loop effective action in position space, […]
• The equation of state associated with ${\cal N}=4$ supersymmetric Yang-Mills in 4 dimensions, for $SU(N)$ in the large $N$ limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang-Mills theory on the boundary in which the cosmological constant is equivalent to a volume. […]
• We use Hamiltonian reduction to simplify Falqui and Mencattini’s recent proof of Sklyanin’s expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of Falqui and Mencattini, and to obtain Sklyanin’s formula as a corollary.
• When quantizing conformal dilaton gravity, there is a conformal anomaly which starts at two-loop order. This anomaly stems from evanescent operators on the divergent parts of the effective action. The general form of the finite counterterm, which is necessary in order to insure cancellation of the Weyl anomaly to every order in perturbation theory, has […]
• We initiate a systematic study of 3-dimensional `defect’ topologi- cal quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categori- cation of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it […]
• We consider a modified gravity plus single scalar-field model, where the scalar Lagrangian couples symmetrically both to the standard Riemannian volume-form (spacetime integration measure density) given by the square-root of the determinant of the Riemannian metric, as well as to another non-Riemannian volume-form in terms of an auxiliary maximal-rank antisymmetric tensor gauge field. The pertinent […]
• Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is […]
• We develop a systematic approach to determine and measure numerically the geometry of generic quantum or “fuzzy” geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in ℝd including the well-known examples of fuzzy spaces, but it applies much more generally. […]
• The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any d+1≠2 (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered […]
• Hopf algebra gauge theory on a ribbon graph We generalise the notion of a group gauge theory on a graph embedded into an oriented surface to finite-dimensional ribbon Hopf algebras. By linearising the corresponding structures for groups, we obtain axioms that encode the notions of connections, the algebra of functions on connections, gauge transformations and […]
• We describe a stabilization mechanism for fuzzy $S^4_N$ in the Euclidean IIB matrix model in the presence of a positive mass term. The one-loop effective potential for the radius contains an attractive contribution attributed to supergravity, while the mass term induces a repulsive contribution for small radius due to SUSY breaking. This leads to a […]
•
• n this paper we study the structure of the phase space in noncommutative geometry in the presence of a nontrivial frame. Our basic assumptions are that the underlying space is a symplectic and parallelizable manifold. Furthermore, we assume the validity of the Leibniz rule and the Jacobi identities. We consider noncommutative spaces due to the […]
• Non-abelian gauge theories in the context of generalized complex geometry are discussed. The generalized connection naturally contains standard gauge and scalar fields, unified in a purely geometric way. We define the corresponding Yang-Mills theory on particular subbundles of a Courant algebroid, known as Dirac structures, where the generalized curvature is a tensor. Different Dirac structures […]
• It is shown that in the noncommutative version of QED (NCQED) Gribov copies induced by the noncommutativity of space-time appear in the Landau gauge. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.
• The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any d+1≠2 (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered […]
• It is shown that a matrix model with SO(d,d) global symmetry is derived from a generalized Yang-Mills theory on the standard Courant algebroid. This model keeps all the positive features of the well-studied type IIB matrix model, and it has many additional welcome properties. We show that it does not only capture the dynamics of […]
• The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the […]
• Target space duality is one of the most profound properties of string theory. However it customarily requires that the background fields satisfy certain invariance conditions in order to perform it consistently; for instance the vector fields along the directions that T-duality is performed have to generate isometries. In the present paper we examine in detail […]
• Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter $z$ is related to a Planck scale, and the cosmological constant $\Lambda$ plays […]
• All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and […] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93917316198349, "perplexity": 592.643764439329}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039546945.85/warc/CC-MAIN-20210421161025-20210421191025-00079.warc.gz"} |
https://kb.osu.edu/dspace/handle/1811/17184 | # VELOCITY MODULATION INFRARED LASER SPECTROSCOPY OF NEGATIVE IONS: THE $\nu_{3}, \nu_{3}+v_{1}-\nu_{1}, \nu_{3}+\nu_{2}-\nu_{2}$ AND $\nu_{3}+2\nu_{2}-2\nu_{2}$ BANDS OF CYANATE ($NCO^{-}$)
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/17184
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Title: VELOCITY MODULATION INFRARED LASER SPECTROSCOPY OF NEGATIVE IONS: THE $\nu_{3}, \nu_{3}+v_{1}-\nu_{1}, \nu_{3}+\nu_{2}-\nu_{2}$ AND $\nu_{3}+2\nu_{2}-2\nu_{2}$ BANDS OF CYANATE ($NCO^{-}$) Creators: Gruebele, M.; Polak, M.; Saykally, R. J. Issue Date: 1987 Publisher: Ohio State University Abstract: 132 transitions in the $\nu_{3}$ (CN stretching) fundamental and the corresponding bending and stretching hotbands of the cyanate anion, ranging from P(52) to R(66), have been measured with a tunable diode laser using the velocity modulation technique. The spectra were fit to the standard linear triatomic molecule rotation-vibration Hamiltonian, yielding effective molecular parameters for the $(00^{0}0), (01^{1}0), (02^{0}0), (10^{0}0), (00^{0}1), (01^{1}1), (02^{0}1)$ and $(10^{0}1)$ states. The equilibrium rotational constant was determined to be $B_{e} = 0.385933(116) cm^{-1}$. A comparison with condensed phase results is presented. As previously observed for $N_{3}^{-}$, cyanate is detected with high vibrational excitation in the bending mode, most likely as a consequence of the formation mechanism. Description: Author Institution: Department of Chemistry, University of California URI: http://hdl.handle.net/1811/17184 Other Identifiers: 1987-ME-11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8554540872573853, "perplexity": 5451.5830510030755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246656168.61/warc/CC-MAIN-20150417045736-00147-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://brilliant.org/problems/a-number-theory-problem-by-dewita-sonya/ | # A number theory problem by dewita sonya
Let $$N=\dfrac{60^{2014}}{7}$$. What is the sum of the first 2014 digits before the decimal point of $$N$$?
×
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Set Loading... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7693501114845276, "perplexity": 1379.1434552610708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607702.58/warc/CC-MAIN-20170523202350-20170523222350-00482.warc.gz"} |
https://collaborate.princeton.edu/en/publications/characterizing-the-nature-of-the-unresolved-point-sources-in-the- | # Characterizing the nature of the unresolved point sources in the Galactic Center: An assessment of systematic uncertainties
Laura J. Chang, Siddharth Mishra-Sharma, Mariangela Lisanti, Malte Buschmann, Nicholas L. Rodd, Benjamin R. Safdi
Research output: Contribution to journalArticlepeer-review
27 Scopus citations
## Abstract
The Galactic Center excess (GCE) of GeV gamma rays can be explained as a signal of annihilating dark matter or of emission from unresolved astrophysical sources, such as millisecond pulsars. Evidence for the latter is provided by a statistical procedure - referred to as non-Poissonian template fitting (NPTF) - that distinguishes the smooth distribution of photons expected for dark matter annihilation from a "clumpy" photon distribution expected for point sources. In this paper, we perform an extensive study of the NPTF on simulated data, exploring its ability to recover the flux and luminosity function of unresolved sources at the Galactic Center. When astrophysical background emission is perfectly modeled, we find that the NPTF successfully distinguishes between the dark matter and point source hypotheses when either component makes up the entirety of the GCE. When the GCE is a mixture of dark matter and point sources, the NPTF may fail to reconstruct the correct contribution of each component. These results are related to the fact that in the ultrafaint limit, a population of unresolved point sources is exactly degenerate with Poissonian emission. We further study the impact of mismodeling the Galactic diffuse backgrounds, finding that while a dark matter signal could be attributed to point sources in some outlying cases for the scenarios we consider, the significance of a true point source signal remains robust. Our work enables us to comment on a recent study by Leane and Slatyer (2019) that questions prior NPTF conclusions because the method does not recover an artificial dark matter signal injected on actual Fermi data. We demonstrate that the failure of the NPTF to extract an artificial dark matter signal can be natural when point sources are present in the data - with the effect further exacerbated by the presence of diffuse mismodeling - and does not on its own invalidate the conclusions of the NPTF analysis in the Inner Galaxy.
Original language English (US) 023014 Physical Review D 101 2 https://doi.org/10.1103/PhysRevD.101.023014 Published - Jan 27 2020
## All Science Journal Classification (ASJC) codes
• Physics and Astronomy (miscellaneous)
## Fingerprint
Dive into the research topics of 'Characterizing the nature of the unresolved point sources in the Galactic Center: An assessment of systematic uncertainties'. Together they form a unique fingerprint. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9254763126373291, "perplexity": 1904.22576234105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104432674.76/warc/CC-MAIN-20220704141714-20220704171714-00168.warc.gz"} |
https://dev.goldbook.iupac.org/terms/view/S05695/plain | ## singlet molecular oxygen (singlet molecular dioxygen)
https://doi.org/10.1351/goldbook.S05695
Oxygen molecule (dioxygen), O2, in an excited singlet state. The ground state of O2 is a triplet (3Σg-). The two metastable singlet states derived from the ground-state configuration are 1Δg and 1Σg+ (the latter with the higher energy).
Notes:
1. Use of the term singlet oxygen alone, without mention of the chemical species is discouraged since it can also refer to an oxygen atom in a $$^{1}\text{S}$$ or $$^{1}\text{D}$$ @E02257@. The oxygen atom @G02704@ is a triplet $$^{3}\text{P}$$ state and the $$^{1}\text{S}$$ and $$^{1}\text{D}$$ states are also derived from the ground-state configuration.
2. @T06503-1@ @Q05007@ by triplet dioxygen (i.e., a process of @P04652@) is the most common procedure for the production of singlet molecular dioxygen in solution. For many chemical species, the efficiency with which the @T06503-2@ is quenched by triplet dioxygen and, independently, the efficiency with singlet molecular dioxygen is produced, is controlled by the spin-@R05175@.
Source:
PAC, 2007, 79, 293. 'Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)' on page 420 (https://doi.org/10.1351/pac200779030293) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7581087350845337, "perplexity": 2839.8689297693136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056890.28/warc/CC-MAIN-20210919125659-20210919155659-00680.warc.gz"} |
https://api.philpapers.org/rec/CARAOE-2 | # Aristotle on Earlier Greek Psychology: The Science of Soul
Cambridge University Press (2019)
Authors Jason W. Carter Ludwig Maximilians Universität, München Abstract This volume is the first in English to provide a full, systematic investigation into Aristotle's criticisms of earlier Greek theories of the soul from the perspective of his theory of scientific explanation. Some interpreters of the De Anima have seen Aristotle's criticisms of Presocratic, Platonic, and other views about the soul as unfair or dialectical, but Jason W. Carter argues that Aristotle's criticisms are in fact a justified attempt to test the adequacy of earlier theories in terms of the theory of scientific knowledge he advances in the Posterior Analytics. Carter proposes a new interpretation of Aristotle's confrontations with earlier psychology, showing how his reception of other Greek philosophers shaped his own hylomorphic psychology and led him to adopt a novel dualist theory of the soul–body relation. His book will be important for students and scholars of Aristotle, ancient Greek psychology, and the history of the mind–body problem. Keywords Aristotle psychology soul science Categories (categorize this paper) Reprint years 2021 Buy this book $32.99 new$36.83 used Amazon page ISBN(s) 9781108481076 1108481078 110870378X 9781108703789 Options Mark as duplicate Export citation Request removal from index
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## References found in this work BETA
No references found.
## Citations of this work BETA
Aristotle on the Soul as Harmony.Melpomeni Vogiatzi - 2020 - Elenchos: Rivista di Studi Sul Pensiero Antico 41 (2):245-268.
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https://tex.stackexchange.com/questions/58848/ap%C3%A9ndices-appendix-spanish-accent/58883 | # Apéndices (Appendix spanish accent)
Dear friends: there is a problem with appendix and ápendice.
I have a scrbook document and I use appendix package,
\usepackage[title, titletoc]{appendix}
because I like the word "Apéndice" in the toc
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
However, when I compile with latex there is an error in file.toc (maybe in file.aux before). The reason is that in the file.aux there is a line like this:
\@writefile{toc}{\contentsline {chapter}{Ap\IeC {\'e}ndice \numberline {A}Sobre el \LaTeX }{27}{ApÈndice.1.Alph1}}
and in the second compilation the character "È" produces and error. If I compile with lualatex I have not any error.
I have tried to modify \indexname, \@chapapp but I can not find any solution.
Yes, that is a solution. I am going to clarify the problem with a MWE. In my opinion the problem is motivated by UTF-8 codify and inputenc package, as it is possible to see with this code:
\documentclass[paper=a4,10pt, twoside]{scrbook}%
\usepackage{etoolbox}
\usepackage{tgheros} % (Una versión libre de Helvetica)
\renewcommand{\sfdefault}{qhv} %qhv
%:Fuente para todo lo demás (Una versión libre de Times)
\renewcommand{\rmdefault}{qtm}
\usepackage{textcomp}
\normalfont
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} %para LaTex
%:Paquetes relacionados con el idioma y codificación
\usepackage[english, spanish, es-nosectiondot, es-noindentfirst, es-nolists, activeacute]{babel}
\spanishdecimal{.}
%:Para poner las referencias y los nombres de figuras y tablas en castellano
{%
\def\figurename{Figura}%
\def\tablename{Tabla}%
\def\listfigurename{Índice de Figuras}
\def\listtablename{Índice de Tablas}
\def\contentsname{Índice}%
\def\chaptername{Capítulo}
\def\Prefacename{Prefacio}%
\def\refname{Referencias}%
\def\abstractname{Resumen}%
\def\bibname{{Bibliografía}}
\def\appendixname{Apéndice}
\def\miapendice{\appendixname}
\def\glossaryname{Glosario}%
\def\indexname{{Índice Alfabético}}
}
%:Para definir APÉNDICES
\usepackage[title, titletoc]{appendix}
\usepackage{hyperref}
\begin{document}
\cleardoublepage
\phantomsection
\tableofcontents
%:Empieza el contenido del libro
\mainmatter
\chapter{Primer capítulo}
\section{Primera sección, primer capítulo}
\chapter{Segundo capítulo}
\section{Primera sección, segundo capítulo}
\begin{appendices}
\chapter{Primer apéndice}
\section{Primera sección, primer apéndice}
\chapter{Segundo apéndice}
\section{Primera sección, segundo apéndice}
\end{appendices}
\end{document}
If you compile, you obtain this error message:
Package inputenc Error: Unicode char \u8:énd not set up for use with La- TeXSee the inputenc package documentation for explanation.Your command was ignored.Type I to replace it with another command,or to continue without it.ice.alph1.Alph11ice.alph1.Alph20P
But the solution proposed it is perfect. Thank you
• Hi Javier, Welcome to TeX.SE! I edited your code using the {} and backticks- it just makes it a bit easier to distinguish code from text. It would be best if you could post a complete MWE to help those who might be able to work on your problem. Welcome to the group! – cmhughes Jun 6 '12 at 23:56
• Please edit your question and add a minimal, yet complete, version of your document allowing us to reproduce the problem. – Gonzalo Medina Jun 7 '12 at 0:06
• Javier: I could answer just because this reminded me about another question. Please, always try to present a minimal complete example; I add one to my answer so you can adjust it if something doesn't work as advertised. – egreg Jun 7 '12 at 8:31
This seems very similar to How to make appendix and hyperref packages work together with cyrillic (non ASCII) characters? and indeed a part of that answer should make your day.
Add the following after loading appendix (it's needed only when hyperref is used)
\usepackage{etoolbox}
\makeatletter
\appto{\appendices}{\def\Hy@chapapp{Appendix}}
\makeatother
For some reasons, hyperref defines \Hy@chapapp as \appendixname, which in turn becomes (with Spanish babel and appendix) \spanishappendixname that again becomes Ap\'{e}ndice. However, it seems that \Hy@chapapp is used only for building the anchor name for hyperlinks, so it should be immaterial what string is used, as long as it is unique. Since the expansion of \Hy@chapapp is subject to complete expansion, it's better for it to be composed by characters only.
Here's a minimal document
\documentclass{scrbook}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[spanish]{babel}
\usepackage[title, titletoc]{appendix}
\usepackage{etoolbox} % we load it for the workaround
\usepackage{hyperref}
% workaround
\makeatletter
\appto{\appendices}{\def\Hy@chapapp{Appendix}}
\makeatother
\begin{document}
\frontmatter
\tableofcontents
\mainmatter
\begin{appendices}
\chapter{Sobre el \LaTeX}
\end{appendices}
\end{document} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8600561618804932, "perplexity": 6417.423100790151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703517966.39/warc/CC-MAIN-20210119042046-20210119072046-00016.warc.gz"} |
http://mathhelpforum.com/algebra/50433-quadradic-equation-condition.html | x^2+2px+(3p+4)=0 , where p is a positive constant, has equal roots
a) Find the value of p.
b) for this value of p, solve the equation.
2. Greetings,
First of all, think when quadratic equations have equal roots.
From there you get two values for p, one of which is negative and solving the equation with the other p is not a problem, I shall assume.
3. Originally Posted by randomised
x^2+2px+(3p+4)=0 , where p is a positive constant, has equal roots
a) Find the value of p.
b) for this value of p, solve the equation.
An equation has repeating [equal] roots when the discriminant is equal to zero.
$\displaystyle \Delta=b^2-4ac\implies \Delta=(2p)^2-4(1)(3p+4)=4p^2-12p-16$
Thus, $\displaystyle \Delta=0$ when $\displaystyle 4p^2-12p-16=0\implies p^2-3p-4=0$
Can you take it from here and solve for $\displaystyle p$?
--Chris
4. Originally Posted by randomised
x^2+2px+(3p+4)=0 , where p is a positive constant, has equal roots
a) Find the value of p.
b) for this value of p, solve the equation.
Set the discriminant to zero for a double root.
$\displaystyle b^2-4ac=0$
a = 1
b = 2p
c = 3p + 4
Geeze! Way too slow today!
5. Perhaps D = k^2 - ac, k = b/2 would shorten the path.
,
,
,
,
,
# the set of values of p for which the expression x^2-2px 3p 4 is negative for at least one real x
Click on a term to search for related topics. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9719390869140625, "perplexity": 851.071316709012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946199.72/warc/CC-MAIN-20180423203935-20180423223935-00266.warc.gz"} |
https://www.vernier.com/experiments/bio-i/11/fermentation_with_yeast/ | Vernier Software & Technology
# Fermentation with Yeast
## Introduction
Yeast can metabolize sugar in two ways, aerobically, with the aid of oxygen, or anaerobically, without oxygen. When yeast metabolizes a sugar under anaerobic conditions, ethanol (CH3CH2OH) and carbon dioxide (CO2) gas are produced. An equation for the fermentation of the simple sugar glucose (C6H12O6) is:
The metabolic activity of yeast can be determined by the measurement of gas pressure inside the fermentation vessel.
## Objectives
In this Preliminary Activity, you will use a Gas Pressure Sensor to monitor the pressure inside a conical tube as yeast metabolizes glucose anaerobically. When data collection is complete, you will perform a linear fit on the resultant graph to determine the fermentation rate.
After completing the Preliminary Activity, you will first use reference sources to find out more about sugar fermentation by yeast before you choose and investigate a researchable question dealing with fermentation. Some topics to consider in your reference search are:
• sugars
• glucose
• monosaccharides
• disaccharides
• fermentation
• anaerobic respiration
• aerobic respiration
• yeast
• enzyme
• substrate
• enzyme inhibitor
## Sensors and Equipment
This investigation features the following Vernier sensors and equipment.
### Option 2
You may also need an interface and software for data collection. What do I need for data collection?
## Investigating Biology through Inquiry
See other experiments from the lab book.
1 Buffer Investigations 2 Diffusion 3 Investigating Osmosis 4 Chemistry of Membranes 5 Investigating Protein 6A Testing Catalase Activity (O2 Gas Sensor) 6B Testing Catalase Activity (Gas Pressure) 6C Testing Enzyme Activity (Spectrometer) 7 Introduction to Biofuels: Enzyme Action 8 Analysis of Enzymes using Tyrosinase 9 Cellular Respiration 10A Sugar Metabolism with Yeast (Carbon Dioxide Gas) 10B Sugar Metabolism with Yeast (Ethanol) 11 Fermentation with Yeast 12 Photosynthesis by Chloroplasts 13 Transpiration of Plants 14 Plant Pigments 15 Heart Rate 16 Investigating Dissolved Oxygen 17 Investigating Primary Productivity 18 Modeling Population Dynamics 19 Water Monitoring 20 Evolution of Cellobiase in Fungi 21 Introduction to Molecular Evolution 22 Evolution of Yeast
### Investigation 11 from Investigating Biology through Inquiry Lab Book
#### Included in the Lab Book
Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30474287271499634, "perplexity": 17319.53276644181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665985.40/warc/CC-MAIN-20191113035916-20191113063916-00122.warc.gz"} |
https://wikibump.com/what-is-the-coriolis-force-equation/ | # What is the Coriolis force equation?
The Coriolis force has a very simple mathematical form, ∝ 2Ω × V , where Ω is the rotation vector and V is the velocity observed from the rotating frame.
## How do you calculate the Coriolis effect?
where Fc = Coriolis effect, V = wind velocity, A = angular velocity of the earth, and l = latitude. Based on this equation, magnitude of the Coriolis effect at the poles “is equal to twice the product of wind velocity, V, and the angular velocity of the earth, A” (Battan, 1961, p. 46).
## How Coriolis force is formed?
Noun. the result of Earth’s rotation on weather patterns and ocean currents. The Coriolis effect makes storms swirl clockwise in the Southern hemisphere and counterclockwise in the Northern Hemisphere.
## What is called Coriolis force?
Coriolis Force an artifact of the earth’s rotation. Once air has been set in motion by the pressure gradient force, it undergoes an apparent deflection from its path, as seen by an observer on the earth. This apparent deflection is called the Coriolis force and is a result of the earth’s rotation.
## What are Coriolis and centrifugal forces?
The Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to the square of the rotation rate. … The centrifugal force acts outwards in the radial direction and is proportional to the distance of the body from the axis of the rotating frame.
## What is Coriolis effect in simple terms?
In simple terms, the Coriolis Effect makes things (like planes or currents of air) traveling long distances around the Earth appear to move at a curve as opposed to a straight line. It’s a pretty weird phenomenon, but the cause is simple: Different parts of the Earth move at different speeds.
## What is the Coriolis effect for dummies?
As air moves away from the mid-latitudes toward the poles and toward the equator, it doesn’t move in a straight line relative to the earth’s surface. Rather, it moves in a slightly curved direction as a result of the Earth’s rotation. This phenomenon is called the Coriolis effect.
## What is Coriolis law?
Coriolis effect is an inertial force described by the French mathematician Gustave-Gaspard Coriolis. Based on Newton’s laws of motion Coriolis determined that a “mass moving in a rotating system experiences a force acting perpendicular to the direction of motion and to the axis of rotation” 1.
## What is Coriolis force example?
Cyclones are an example of the influence of the Coriolis effect. A cyclone is a large air mass that rotates around a center. As they rotate, cyclones suck air into their center, or eye. The air currents are pulled in from all directions. In the Northern Hemisphere, they are then deflected to the right.
## Does the Earth spin clockwise?
Its rotation direction is prograde, or west to east, which appears counterclockwise when viewed from above the North Pole, and it is common to all the planets in our solar system except Venus and Uranus, according to NASA.
## How do you calculate Coriolis acceleration?
The Coriolis acceleration is then given by: du/dt = f v, \; dv/dt = – fu, \; f=2 \Omega \sin \phi . The Coriolis acceleration can be derived in a more intuitive way if we consider fluid velocities (u,v) much smaller than the earth’s surface rotation velocity U = R \Omega \cos \phi.
## What is Coriolis force and Ferrel’s law?
The key difference between Coriolis effect and Ferrel’s law is that the Coriolis effect is the deflection that occurs due to the Coriolis force, whereas Ferrel’s law is the tendency of rising warm air to pull in air from more equatorial and warmer regions and transport it poleward.
## How is centrifugal force calculated?
Use the centrifugal force equation: F = m v² / r . In our example, it will be equal to (10 kg) * (5 m/s)² / (2 m) = 125 kg⋅m/s² = 125 N .
## How does Coriolis force work?
But because the Earth rotates, circulating air is deflected. Instead of circulating in a straight pattern, the air deflects toward the right in the Northern Hemisphere and toward the left in the Southern Hemisphere, resulting in curved paths. This deflection is called the Coriolis effect.
## What are examples of centrifugal forces?
When a car in motion takes a sudden turn towards the left, passengers in a car experience an outward push. This is due to the centrifugal force acting on passengers.
## What are 3 things that are affected by the Coriolis effect?
The Coriolis effect occurs because of Earth’s rotation and the fact that the atmosphere and oceans are not connected to the solid part of the planet.
• Atmospheric Circulation Patterns. Earth rotates eastward. …
• Oceanic Circulation Patterns. …
• Flight Paths.
## How does Coriolis effect wind?
What is the Coriolis effect? The Earth’s rotation means that we experience an apparent force known as the Coriolis force. This deflects the direction of the wind to the right in the northern hemisphere and to the left in the southern hemisphere.
## Why does Coriolis effect happen?
The observed Coriolis effect arises because the Earth is rotating, and in non- equatorial locations, is actually turning underneath as a horizontally and freely moving object travels forward. Because the motion is being measured relative to Earth, the motion appears to be along a curved path.
## Do toilets actually flush backwards Australia?
Australian Toilets Don’t Flush Backwards Because of the Coriolis Effect. … The real cause of backwards-flushing toilets is just that the water jets point in the opposite direction.
## What is Coriolis effect geography class 9?
Due to the earth’s rotation, the direction of the winds and ocean currents get changed. As the Earth rotates, the winds and the ocean currents are deflected towards the right in the Northern Hemisphere and towards the left in the Southern Hemisphere. This is known as Coriolis Effect.
## What is Coriolis force Class 8?
What is Coriolis Force? Coriolis force is the invisible force that appears to deflect the objects. The Coriolis force is determined by the object’s rate of rotation and mass of the object.
## What is Coriolis force and its applications?
The Coriolis force is a force which acts upon any moving body in an independently rotating system. The most well known application of the Coriolis force is for the movement or flow of air across the Earth. … Moving air undergoes an apparent deflection from its path, as seen by an observer on the Earth.
## Which statement is true about the Coriolis force?
The correct answer is It deflects the wind to the right direction: in the southern hemisphere. The Coriolis effect describes the pattern of deflection taken by objects not firmly connected to the ground as they travel long distances around Earth.
## What is Ferrel’s law class 9?
Ferrel’s Law states that as a result of the Earth’s rotation on its axis from west to east, wind or any other moving object in the Northern Hemisphere is deflected to the right and in the Southern Hemisphere, it is deflected to the left of its course. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8765523433685303, "perplexity": 584.7916729740062}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531779.10/warc/CC-MAIN-20220520093441-20220520123441-00698.warc.gz"} |
https://physics.stackexchange.com/questions/504428/why-are-worldsheets-of-strings-holomorphic/505370 | # Why are worldsheets of strings _holomorphic_?
Disclaimer. I am a mathematician (algebraic geometer) who knows nothing about physics. Even worse, I might have major misconceptions about the objects I'll ask about. The level of the question is pop-science, but perhaps just a bit too specific for something like Quora. I apologize in advance for my ignorance.
According to my understanding, a worldsheet is the 2D surface traced out by a vibrating string. We can think of this as (the image of) a map from a Riemann surface into spacetime. However, my understanding is that this map ought to be holomorphic. [Here I am already a bit confused, though this is not the main point of the question: do we only consider maps into a (6D) Calabi-Yau fiber of spacetime (are these called instantons?), or does this somehow make sense for maps whose image isn't contained in one of these fibers?]
My question: why does it have to be holomorphic?
How to come to terms (technically or philosophically) with this very strong hypothesis/property?
• I don't think the map needs to be holomorphic in general. Do you have a reference that says this? – d_b Sep 24 '19 at 7:27
• Maybe what you mean is that the worldsheet equations of motion separate solutions in holomorphic and antiholomorphic parts? (See e.g. arxiv.org/abs/hep-th/9709062) – Toffomat Sep 24 '19 at 10:58
I suggest learning the basics of conformal field theory before diving into string theory. To illustrate where holomorphicity comes in, let's consider the free bosonic string (the simplest example of a CFT). One starts with the worldsheet embedding $$\phi(z,\overline{z}): \mathbb{C}P^1 \to \mathbb{C}$$ and derives the equations of motion $$\partial_{\overline{z}} \partial_z \phi(z,\overline{z}) = 0$$. This implies the existence of the fields $$$$\partial \phi(z) = \sum_{n \in \mathbb{Z}} a_n z^{-n-1} \ \ \ \ \ \ \ \ \ \ \overline{\partial} \phi(\overline{z}) = \sum_{n \in \mathbb{Z}} \overline{a}_n \overline{z}^{-n-1},$$$$ which are holomorphic and antiholomorphic respectively. So one answer to your question is that the fields of interest are holomorphic/antiholomorphic as a result of the equations of motion (which in the case of the free boson are derived from the action of the theory).
There is much more to be said here. To obtain a quantum theory, the coefficients $$a_n$$ are promoted to elements of a Lie algebra. This is known as canonical quantisation. In the case of the free boson, the Lie algebra is generated by the elements $$\{a_n\}_{n \in \mathbb{Z}}$$ and $$\{k\}$$, with Lie bracket defined by $$$$[a_n, a_m] = \delta_{n, -m}k \ \ \ \ \ [a_n, k] = 0.$$$$ There is another Lie algebra of interest known as the Virasoro algebra. The Virasoro algebra appears in 2d CFT as a result of conformal invariance of the theory. The Virasoro algebra has generators $$L_n$$ and one obtains another holomorphic field $$$$T(z) = \sum_{n \in \mathbb{Z}} L_n z^{-n-2}$$$$ known as the energy-momentum tensor. There is also an antiholomorphic analogue of the energy-momentum tensor. For this reason, we say the free boson is composed of holomorphic and antiholomorphic "sectors". | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9748210906982422, "perplexity": 172.00920070311022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251690095.81/warc/CC-MAIN-20200126165718-20200126195718-00017.warc.gz"} |
https://www.physicsforums.com/threads/localization-of-states-and-elementary-vs-composite-in-qm.796090/ | Localization of states and elementary vs composite in QM
1. Feb 5, 2015
TrickyDicky
While browsing Wikipedia I bumped into this sentence that seemed partially wrong to me but maybe I didn't understand what it is referring to so would like for some expert to help me elucidate it: "Even if an elementary particle has a delocalized wavepacket, the wavepacket is in fact a quantum superposition of quantum states wherein the particle is exactly localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states."
There are two parts: If the wavepacket is a superposition of plane waves, in what sense are they "quantum states wherein the particle is exactly localized"?
And if this was so, why in the bound state wavepacket case this can never be so?
2. Feb 5, 2015
kith
You could write
$$e^{-ipx'} \propto \int_{-\infty}^{+\infty} \delta(x-x') e^{-ip(x-x')} dx,$$
so in this sense, a plane wave can be viewed as a formal superposition of delta functions just like a Gaussian wavepacket can be viewed as a formal superposition of plane waves. But as you probably know, delta functions are not square integrable so they are not proper quantum states but idealizations.
This probably refers to entanglement. If the composite state is not a product state, you cannot assign definite wavefunctions to the single particles at all.
3. Feb 5, 2015
TrickyDicky
Thanks, kith.
Yes, they are idealizations, with certain validity as approximations for many different situations(like for VP propagators) but not to undermine the very foundation of QM, the HUP. It looks like in particle physics they take seriously this idealization, the paragraph I quoted goes on to say: "It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero."
Since in particle physics this assumption about the internal structure of elementary particles is essential to their interpretation of scattering experiments of i.e.electrons as probes of the internal structure of i.e. protons I wonder if some inconsistency can be derived from this de facto dispensing with the HUP.
I see what you mean, you cannot assign them states since when considering the isolated composite particle they are acting as virtual, so this is by definition. What I find ironic here is that the composite state itself is mathematically equivalent to elementary particle one, even if physically by definition we cannot assign it the same meaning as in the elementary particle case.
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http://docmadhattan.fieldofscience.com/2012/11/looking-through-opaque-material.html | ### Looking through an opaque material
I am proud to publish a feature article about the research word of a Wikipedia's friend like Jacopo Bertolotti. The work was finally published on "Nature", that decided also to honor the paper with the cover.
I admit to have received the paper a couple of weeks ago, so I hope to have made a good service to Jacopo and all of his colleagues.
Recently one of my students asked me why glass is transparent while other materials are not. Its transparency is substantially due to the interaction between the electrons of the glass and the incident light, and so from the interaction between the photons (or electromagnetic radiation) and matter. A photon, when it interacts with matter, can be absorbed, reflected or continue on its way without change. These different behaviors are due to the energy levels occupied by the electrons of the atoms that constitute matter, in particular by the energy difference between these levels. We know, thanks to the photoelectric effect and the explanation given to it by Einstein, that the electrons are excited by the incident photons only if the energy of these photons is equal to (or greater than) the energy required to jump to another level. This means that if the light does not excite the electrons of the material, this is transparent to its passage, just as occurs in the glass: the visible light, in fact, has not enough energy to excite the electrons of the glass, which is therefore transparent to its passage, despite the rays of light are reflected.
Now, if we take a sheet of glass and do hit by light from one of its ends, a part of the rays will be reflected and then detected by a device (such as our eyes) placed for example at the opposite end. Not all reflected rays, however, follow an equal path and therefore not all the rays reach the eyes at the same time.
Something similar also occurs when light passes through the glass of our windows: the light rays don't travel all at the same speed.
In order to enable to all the rays of light to reach the detection point at the same time one can construct a structure which gradually tapering as it approaches the ends, i.e. it builds a lens.
One of the ways to use a lens is for example for the magnification of objects, but not all magnifications can be made using the lenses that take advantage of the visible light. If, for example, we have some objects that are at a scale of less than 200 nm, the usual optical lenses are not able to resolve their details. Unless you build a HIRES(1) (High Resolution Enhancement by Scattering Index) lens.
This lens, developed by the group of van Putten and Bertolotti (our Jacopo!) consists of:
(...) of a homogeneous slab of high-index material on top of a strongly disordered scattering layer. The disordered layer breaks the translational invariance of the interface, which enables incident light to be coupled to all propagating angles inside the highrefractive-index material as shown in figure.
Yet multiple scattering also scrambles the wavefront creating a speckle-like pattern on the object plane that itself cannot be used for imaging. Therefore we manipulate the incident wavefront in order to force constructive interference of the scattered light at a position in the object plane of our HIRES-lens.(1)
The lens developed by Dutch laboratories COPS is able to detect gold nanoparticles of the order of 100 nm or less, as you can see in the picture below that compares the observation of these particles with a magnifying glass commercial high quality (left) and with the HIRES(1) (right):
But if we want to observe an object in the sky hidden by the clouds, we can enlarge the portion of the sky until we want, but we will never take more than a few details. In fact JR Fienup, thinking about atmospheric turbulence, developed a modified version of the Gerchberg-Saxton algorithm(2).
The idea is simple: every object, even hidden, sends light information that we are able to detect, but may not be sufficient to give us an idea of the shape and details that characterize it. Then it is possible to derive this information extracting them with a suitable iterative algorithm starting from the Fourier transforms of the scattered light(2).
The method, already tested by Fienup, has recently been refined and adapted again at the COPS for the detection of objects in the order of micrometers(3) or even nanometer, like Allard Mosk stated in the official press release:
This will be very useful in nanotechnology. We would like to bring structures to light that are hidden inside a complex environments like computer chips.
Let suppose you have a fluorescent object completely hidden by an opaque screen with respect to the scattering of light. One way to reconstruct the hidden object is
(...) to separate the small amount of light that did not change direction owing to random scattering (ballistic light) from the scattered background using a gated technique such as optical coherence tomography.(3)
In this way it is possible to recover the information within a semi-transparent means, but it is very difficult, if not impossible, to use non-invasive methods with increasing diffusion. You could use diffuse optical tomography, but this only allows the location of the object but not the reconstruction of the details. The idea then becomes to retrieve the details thanks to the correlations in scattered light.
In order to retrieve this information you must first hit the object, which in this case was modeled in the form of $\pi$, with the laser light that pass through the opaque surface. What we get from scattered light, however, is something quite incomprehensible, if we observe results with naked eyes:
At this point the iterative algorithm is used to retrieve the correlations between the points of the previous figure and from these correlations researchers retrieve the details of the object itself:
The method, however, has enormous potentials: it was also tested with a complex biological structure as the stem of Convallaria majalis (lily of the valley) and the results are very encouraging:
If the main limitation of the method is dued
(...) by the background signal, which can distort the measured autocorrelation(3)
possible future developments of the technique are therefore the detection of small subcutaneous structures, and possible subsequent medical applications.
The research, finally, also received the honor of the cover of Nature released two days ago and the draft title of this post was A Wikipedian on the cover of Nature. When I talked to Jacopo, he timidly remmbered me that he was no longer an active Wikipedian, but as far as I'm concerned, paraphrasing the famous Avengers,
Once a wikipedian, always a wikipedian!
P.S.: the first three images are from the book QED by Richard Feynman
Video: Why is glass transparent?
(1) van Putten, E., Akbulut, D., Bertolotti, J., Vos, W., Lagendijk, A., & Mosk, A. (2011). Scattering Lens Resolves Sub-100 nm Structures with Visible Light Physical Review Letters, 106 (19) DOI: 10.1103/PhysRevLett.106.193905 (arXiv)
(2) Fienup, J. (1978). Reconstruction of an object from the modulus of its Fourier transform Optics Letters, 3 (1) DOI: 10.1364/OL.3.000027 (pdf)
(3) Bertolotti J., van Putten E.G., Blum C., Lagendijk A., Vos W.L. & Mosk A.P. (2012). Non-invasive imaging through opaque scattering layers, Nature, 491 (7423) 232-234. DOI: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.595139741897583, "perplexity": 706.2899136489127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038119532.50/warc/CC-MAIN-20210417102129-20210417132129-00033.warc.gz"} |
https://www.physicsforums.com/threads/chain-rule-or-substitution-rule.572792/ | Chain rule or Substitution rule?
1. Jan 31, 2012
cris(c)
Chain rule or Substitution rule???
1. The problem statement, all variables and given/known data
It appears that a standard result in ODE is the following: if $f(x,t)$ is smooth enough, then the solution $\psi(t)$ to the initial value problem:
$x(t)=x_{0}+\int_{0}^{t}f(x(s),s)ds$
$x(0)=x_{0}$
is continuously differentiable with respect to its initial condition (or a parameter in the problem), and that this derivative satisfies the following first variational equation:
$\frac{\partial \psi(t)}{\partial x}=c +\int_{0}^{t}\frac{\partial f(\psi(s,x),s)}{\partial x}\frac{\partial \psi}{\partial x}ds$
2. Relevant equations
To obtain the second equation above one makes use of the chain rule applied to $f(\psi(t,x),t)$. However, I am not certain we can always use this rule. Consider the following `f' function:
$f(\psi(s),s)=\biggl(\frac{1-\int_{0}^{s}G(\psi(\xi))d\xi}{F(t)F(\psi(t))+\int_{0}^{t}G(\psi(\xi))d\xi}\biggl)^N$
where $G$ is a sufficiently continuously differentiable function, and $N>2$. The question is how could I apply the chain rule to obtain a first variational equation as the one I presented above?
3. The attempt at a solution
No sure yet. I truly need help here as I've trying to understand this for a long time without any success...Thanks!
Can you offer guidance or do you also need help?
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https://math.hawaii.edu/wordpress/category/undergrad-posts/computing-posts/ | # Courses and Course Materials with Online content
##### University online courses
Links for Math History
##### Outside online courses
Thinkwell Instruction on a CD; tests online. Has "live" lectures.
Aleks Highly interactive lessions. It even has graphical rulers and pencil the students can use to draw graphs. Algebra, basic math.
eGrade Wiley's eGrade interactive math testing program. Good, allows fill-in-the-blank questions.
eCollege Well-packaged course module to which professors may add content.
Online Plato Well-illustrated courses with audio. Low math/word ratio.
Cogito Online courses in Statistics, Precal, Calculus
ElementK Computer courses, no math
Hungry Minds Go to "Mathematics", then "Trigonometry" and you'll find Math 140, Honolulu Community College, by Wayne Lewis, online for $117 MindEdge Offers online courses from various colleges, typically$200-\$900/student, no math.
I can learn Claims to have interactive courses, no demos.
Houghton-Mifflin Larson's Precalculus on CD-ROM (to be published).
McGrawhill Barnett's Precalculus.
Virtual High School Online high school courses.
Other online material
Project Mathematics Tom Apostle's famous video tape series.
Calculus OnLine
MegaMath
WebMath
Platonic Realms
Math Topics
Interactive CDs
Thinkwell Instruction on a CD; tests online. Has "live" lectures.
Math Deluxe (available at CompUSA). A Mathematica-type environment with instructional worksheets.
MathPert Assistant. Student is given problems, student chooses what to do at each step (subtract from both sides, factor, ... ).
Calculus WIZ. A Mathematica based Calculus tutorial and problem solver.
Course development tools
Laulima UH uses this system.
WBT TopClass.
Blackboard
LearningSpace
Click2Learn (Asymmetrix) ToolBook.
Information about online resources
PEW - successors of Virginia Tech Math Emporium
# Computer lab
The Department has a computer lab in Physical Science Building 208, equipped with Matlab, Mathematica, etc., for the benefit of students and faculty. Access is either by numerical code or by permission of the Lab Monitor.
When space is available, other students can use the lab during Math 140 exam times. However, the last Math 140 computer exam is the last day to withdraw. After that, the lab will be closed during these times.
#### Lab Monitor Schedule
The schedule shows when the various Lab Monitors will be on duty.
Terms of Acceptable Use
Each student must agree to the Terms of Acceptable Use statement for the UH Math Dept Computer Lab. The students agree to obey all of the terms of use and conditions set forth in the UH Information Technology Resources Policy. Their policy is here:
The policy for the math lab will be the same, but adds in particular that
Students agree not to:
1) Download, install software or run any outside programs or software packages.
2) Download or upload work protected by copyright or files not related to class work.
3) Attempt to gain unauthorized access to remote computer systems.
In addition you acknowledge that:
4) There is no expectation of privacy in the Math Lab and your activities may be monitored.
Each lab account can use a portion of the network hard for storing files. You save your files to the H: drive and not the local hard disk or C: drive. In Derive you would typically save your lab by using the Save As ... option on the File tab and then saving you file with the name H:\lab1. To open a file you would use the Open option on the File tab and enter H: <cr> to see a list of your files.
In order to keep others from using your account you need to keep your password secret. For example, don't write it down next to your account number in your notebook where others can see it.
### Graphing and Calculus Software
#### wxMaxima
You can download this free software which is available for Window, MacOS, Linux and UNIX computers. This software is installed on the lab PCs and is used in the calculus classes for graphing and symbolic calculus calculations. In addition, the math department has many sample sessions available on their website.
### Other Free Graphing Software:
Online Grapher: (click to try)
Try out this online grapher. Be sure to use the * character for multiplication and you zoom in on a point by dragging your cursor to form a rectangle around the point.
Winplot is a small but useful plotting program that can be downloaded and used on any Windows PC. It is available on the desktop in the PC lab in addition to Derive. Give it a try!
Mac users with OS Version 7.5 or later already have a plotting program on their Apple menu. Check it out.
### Other Free Software:
Just for fun ... read about an important math/computer problem:
The famous problem "P versus NP" problem is decades old and there is a million dollar prize to solve it. Now a mathematician in England has shown that the problem is equivalent to solving a larger version of the well know Minesweeper game that is included with the Windows operating system. Read the news story, Scientific American article or just play the game.
# Online grapher
Skip instructions
This program was one of John Orr's Java Projects. His homepage has several other interesting projects.
Press "Add Function" button and type in a formula to be graphed in standard mathematical notation (eg "e^(-x/10)sin(x)"). The only variable the formula can use is x. Use Pi for pi; for example, sin(Pi/2 x). You can add as many functions as you like, and "Delete Function" deletes the last one you entered.
Rescaling
This brings up a panel which you can use to edit the scale of the graph. The x variable runs between xMin and xMax, and the y values run between yMin and yMax. Use xScale and yScale to set the spacing of tick marks on the axes. If xScale or yScale is zero, the graph will have no tickmarks on that axis.
Zooming
You can change the scale of your picture quickly with "Zoom In", "Zoom Out" and "Zoom to Box". The first two zoom in or out around the center of the graph. If you click and drag on the graphing window, you can draw out a box, and Zoom To Box will zoom in to make that box fill the whole window.
# Octave
The PC Lab in PSB 208 has the computer software Octave installed. Octave is a clone of the popular mathematical software system MatLab. The MatLab system is also available in our computer lab and there is a student version available on the web for a moderate fee. The MatLab system is used in many engineering and scientific computing laboratories. Octave is a free system available online by one of the creators of MatLab at the University of Wisconsin. It is licensed under the General Public License or GPL (see Wikipedia for more information). Its abilities include matrix computations, linear algebra, plotting and structured programming.
When used as a programming tool the program code is edited in an external editor supplied with the system and then programs are run by entering commands in the Octave command window. With a little care it is possible to use core elements of the MatLab/Octave programming language so that the program code will work with either system.
#### Installing Octave On Your Computer
1) Download the installation file by clicking the link below:
Window: octave-3.0.1-setup.exe
Mac-386: octave-3.0.1-i386.dmg
Either save it to a temporary location and then execute it or else execute the program immediately during the download process.
Note: I recommend choosing the non-Java version during the installation since it seems to be more stable.
2) The program will be installed in your C: drive and there will be startup folder containing a startup icon and PDF file version of the manual. The startup folder can be opened by clicking Start/All Programs/GNU Octave.
You start the program by clicking the startup icon and waiting for the prompt to appear. You then can just start entering statements into Octave by typing expressions and then clicking the enter key:
octave:1> 2 + 3
ans = 5
octave:2> 4^(1/2)
ans = 2
octave:3> 2*sin(pi/2)
ans = 2
octave:4> sin(2)
ans = 0.90930
octave:5> exit
That last command "exit" will close the Octave window and end the program.
3) By default, Octave will look for your program code in the folder
C:\Program Files\Octave
You want to create a better location for your files. A good solution is to create a folder "Octave Programs" in your "My Documents" folder. Next you need to make this folder the current working directory in Octave. This can be done at startup by using the change directory command cd as follows for Windows XP:
octave:1> cd "C:/Documents and Settings/Dave/My Documents/Octave Programs"
octave:2> pwd
ans = C:\Documents and Settings\Dave\My Documents\Octave Programs
For Windows Vista this would be:
octave:1> cd "C:/Users/Dave/Documents/Octave Programs"
octave:2> pwd
ans = C:\Users\Dave\Documents\Octave Programs
The best method is to have the system automatically make this change for you by editing the properties of the desktop startup icon. To do this right click the startup icon, click "Properties" and then edit the "Start In:" folder to be your code folder.
The next time you start Octave the working directory will be the one you entered in the "Start In:" field. You can check whether this is the case by using the pwd command as above.
4) The default text editor for writing octave code is the SciTE program and you can copy a startup link to your desktop also. It is located at
C:\Program Files\GNU Octave 3.0.1\tools\wscite\SciTE.exe
This last step may not be necessary since the Windows system will associate this editor with text file with the m-extension. Thus, if you create an empty plain text file in your code folder (Step 2) with the name test.m then double clicking this file icon will open the editor. At that point you can use the File/Save As option to rename it.
5) To test your system create the file test.m in Step 4 and enter the single statement:
disp('Hello World!')
and then Save your changes. Next double-click the Octave startup on your desktop and enter the statement 'test' followed by the Enter key. Here's what should happen:
octave:1> test
Hello World!
octave:2>
Alternately, you can just enter statements directly into Octave by typing say:
octave:3> disp('Hi!')
Hi!
octave:4> for i=1:5, disp(i^2), end
1
4
9
16
25
The preferred method of writing code is to use a source file (as above) and then test the code in Octave. When something is not right you should modify the source file, save the changes and repeat the testing step. Nevertheless, any code statement can be entered directly into Octave for testing purposes or for doing very simple examples.
6) To get started with programming read over the introductory material in the Octave Manual starting at page 33. Another way to get started is to skim over the MatLab Getting Started Manual starting with Chap 3 and Chap 5. The Matlab system has a completely different interface than Octave but the MatLab Command window is very similar to the Octave system language:
Octave Manual (378 page PDF)
Getting Started with MatLab (138 page PDF)
Octave and MatLab are quite compatible and with a little care one can write programs that will simultaneously work with either system. In addition, MatLab has an extensive website to help you learn their programming language, see
7) It is possible to paste code into the command line and to copy portions of the command window for use in text files. Here's how you do that:
1. Enlarge the octave window so that all computations are showing.
2. Right-click in the top window strip to reveal the edit menu. Choose Edit and select Mark.
3. With your mouse select button held down select the portion of the window you want to copy. It should highlight as you move your mouse. When the material is selected press the Enter-key. The selection is now on the clipboard and can be pasted into a text editor.
The reverse process of pasting a command into Octave is done by first copying the command to the clipboard from a file or web page. Then, in the octave window right-click the top window strip and select Edit then Paste.
These direction may vary slightly for Mac or Linux computers. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2494361251592636, "perplexity": 3152.453215272111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105341.69/warc/CC-MAIN-20170819105009-20170819125009-00661.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=562993&page=5 | by Opus_723
Tags: ddwfttw
P: 206
Quote by kmarinas86 If so, then some of this phenomenon could be related even to General Relativity, as it appears that the fundamental microscopic non-inertial, vibratory/rotational motions involved must some how have changed course to some small degree (in this non-relativistic example) as a result of the force interactions involved.
Nothing fancy -- simple straightforward fluid mechanics involved and it's not even anything new, having been done for hundreds (thousands?) of years. A simple flip of the environment (viewed from the perspective of a fish for example) easily shows that the keel of the very first boat to ever tack its way upwind was achieving a downfluid VMG faster than the fluid, absolutely steady state.
JB
P: 4,191
Quote by kmarinas86 I will now concede that in this there is a source of energy that would make for the illusion that the apparent wind can accelerate a sail craft in the opposite direction that it is blowing
It is not "an illusion". It is a well verified empirical fact and in full agreement with Newtonian physics. The "source of energy" is the velocity difference between the air & surface which is always being reduced.
Quote by kmarinas86 ...matter's energetic motions were somehow deflected internally ... some of this phenomenon could be related even to General Relativity...fundamental microscopic non-inertial, vibratory/rotational motions involved ...
LOL. It's just simple mechanics as the vectors diagrams show.
P: 1,011
Quote by ThinAirDesign Nothing fancy -- simple straightforward fluid mechanics involved and it's not even anything new, having been done for hundreds (thousands?) of years. A simple flip of the environment (viewed from the perspective of a fish for example) easily shows that the keel of the very first boat to ever tack its way upwind was achieving a downfluid VMG faster than the fluid, absolutely steady state. JB
Quote by A.T. It is not "an illusion". It is a well verified empirical fact and in full agreement with Newtonian physics. The "source of energy" is the velocity difference between the air & surface which is always being reduced. LOL. It's just simple mechanics as the vectors diagrams show.
Without positing pre-existing "hidden" momentum inside the mass of the apparent wind and/or the craft itself, I cannot at all see how something that blows at you can pull you forward. It seems to make more sense to imagine that the apparent wind is simply allowing this "hidden" momentum (which we know exists in the form of the $\vec{p}$ in the equation $(m_{whole}c^2)^2$$=E_{whole}^2=(m_{parts}c^2)^2+\left\|\sum \vec{p}\ c \right\|^2$ to appear visible to a human observer, than it is to believe that apparent wind would be gaining energy by doing work on the sail. After all, if the apparent wind is moving to the left and pushes the sail to the right, this would mean that the apparent wind would have to accelerate both itself and the sail, which would violate the conservation of energy, if it were not for this hidden momentum. It is no coincidence to me that so many think incorrectly that the DDTFTTW craft is impossible. They see the apparent wind as a source of energy, and they cannot imagine how it would increase speed relative to the craft by being thrusted by the propeller tailwards while at the same time having that power of the propeller being explained by the same incoming headwind. The reason why this is so prevalent is that there is something wrong with apparent wind being able to do that with the cart, if you don't accept that there is a hidden source of energy! Yes, the forces can explain conservation of momentum, but from the inertial frame of the craft at time $t$, it is very clear that without a hidden source of energy, we cannot explain why the head wind and the craft with increase respect to the frame once $t$ has passed. In this case, it is not hidden because of deception, but rather, it is hidden because the energy is that of atoms and molecules. So the skeptics of DDTFTTW are not entirely wrong in their skepticism. There must be a hidden energy source (It's nature's energy!). :)
P: 4,191
Quote by kmarinas86 I cannot at all see how something that blows at you can pull you forward.
Sailors use this of ages.
Quote by kmarinas86 ...hidden momentum...
Very creative, but there is no need for such obfuscatory nonsense. All the momentum is clearly visible all the time and is being conserved.
P: 206
Quote by kmarinas86 I cannot at all see how something that blows at you can pull you forward.
Before I respond I really must clarify something.
Sailboats sail upwind all the time. They can leave a point downwind and readily arrive at a point directly upwind of where they were by simply sailing towards a point situation to the right (or left) of the upwind goal, and then once halfway there, they turn and sail directly towards that point.
A: If taken literally and without context, your above quoted statement would make it seem as though you don't see how a boat can sail upwind such as the above. I'm pretty sure that's not what you mean but I do want to ask the question.
B: If you believe boats can make upwind progress by sailing at an angle to the wind, but are having difficulty believing that *anything* wind powered can make steady state progress directly into the wind using basic Newtonian physics then I need to know that.
A or B or other?
Thanks
JB
P: 1,011
Quote by A.T. Sailors use this of ages. Very creative, but there is no need for such obfuscatory nonsense. All the momentum is clearly visible all the time and is being conserved.
The net momentum is indeed conserved.
2 + (-2) = (3) + (-3)... etc.
That makes it look like there is no mystery.
What does not make sense to many still-skeptical skeptics is how would the Blackbird DDTWFTTW sand yacht conserve energy. Neither you, nor them, seem to have the explanation.
Note that:
(2)^2 + (-2)^2 is not (3)^2 + (-3)^2... etc.
Would you mind explaining where the energy comes from to allow the wind to do work on the DDWFTTW vehicle (in the time between $t$ and $t+\epsilon$) at the same time the DDWFTTW vehicle accelerates, with respect to the initial inertial frame of the vehicle at time $t$? The work is done in opposite directions, conserving momentum even macroscopically, but not the kinetic energies of both (both increase as far as the initial inertial frame is concerned). My "very creative" resolution addresses this problem by bringing up the point about the true and factual existence of the below-macroscopic energy of atoms and molecules as being the entity that accounts for this apparent gap.
My explanation is not that of a hidden net momentum, but a hidden set of vector momenta which sums to zero in the frame being evaluated (i.e. the momentum whose energy is identical to the rest mass of a body, as evaluated from the system frame in question, times the speed of light squared). I have a hunch that somehow this is either the static P-V energy that was already present in the air mass prior to vehicle operation, and/or the vibrational and rotational energy of the vehicle's particle makeup. Probably both.
P: 1,011
Quote by ThinAirDesign Before I respond I really must clarify something. Sailboats sail upwind all the time. They can leave a point downwind and readily arrive at a point directly upwind of where they were by simply sailing towards a point situation to the right (or left) of the upwind goal, and then once halfway there, they turn and sail directly towards that point. A: If taken literally and without context, your above quoted statement would make it seem as though you don't see how a boat can sail upwind such as the above. I'm pretty sure that's not what you mean but I do want to ask the question. B: If you believe boats can make upwind progress by sailing at an angle to the wind, but are having difficulty believing that *anything* wind powered can make steady state progress directly into the wind using basic Newtonian physics then I need to know that. A or B or other? Thanks JB
Neither. Potential energy must be extracted from the system. Either the static P-V of the wind and/or the energy from the particle make-up of vehicle (and/or even that of the ground, if need be). A correct explanation cannot be found by trying to conserve "kinetic energy+heat" while ignoring potential energy.
The quote (taken out from a sentence after a comma) is taken too far out of context. It doesn't even relate to what I am saying.
P: 206
Quote by kmarinas86 Neither. The quote (taken out from a sentence after a comma) is taken too far out of context. It doesn't even relate to what I am saying.
Ok, got it. I was pretty sure from your previous posts that "A" wasn't what you meant, but didn't want to move forward without confirmation.
Thanks.
It appears that what you are saying the examples I have in that post can and do happen, you just don't believe they can be explained through simple Newtonian physics.
Would that be a fair representation of your position?
JB
HW Helper
P: 7,166
Quote by kmarinas86 Potential energy must be extracted from the system.
For a fluid or gas, potential energy is used to describe the gravitational potential energy of a gas or fluid. For a wind driven vehicle, the energy extracted from the air affected by the wind driven vehicle corresponds to the reduction in kinetic energy (wrt ground) of the affected air. (Pressure effects are short term and only exist in the immediate vincinity of the propeller.)
P: 1,011
Quote by ThinAirDesign Ok, got it. I was pretty sure from your previous posts that "A" wasn't what you meant, but didn't want to move forward without confirmation. Thanks. It appears that what you are saying the examples I have in that post can and do happen, you just don't believe they can be explained through simple Newtonian physics. Would that be a fair representation of your position? JB
I guess that depends on what you call simple, what you call Newtonian, and what you mean by explain. You can explain things using forces without highlighting the apparent non-conservation of "kinetic energy+heat". That is simple (to me), but ignoring where this kinetic energy comes from doesn't do it for me, so I would disagree that it is somehow an adequate explanation. (If a claimed-to-be explanation is inadequate, does it really explain what needs to be explained?) Potential energy latent inside matter isn't exactly something that I would call part of "Newtonian" physics. It's not included in most of the (simple enough to be convincing to most) analyses that have been offered to explain the Blackbird. Certainly it can be explained using classical physics.
The lack of significant mention of potential energy when discussing how tacking can allow sails to move ahead of the wind, in addition to the absolute absence of this point in many of the videos that I have seen that try to explain DDTWFTTW, has (I bet) contributed much confusion for people (including skeptics and naysayers) who wonder where the energy comes from and who, like myself, have for a time not been able to see how tacking would be of any benefit to it.
P: 206
Quote by kmarinas86 The lack of significant mention of potential energy when discussing how tacking can allow sails to move ahead of the wind, in addition to the absolute absence of this point in many of the videos that I have seen that try to explain DDTWFTTW, has (I bet) contributed much confusion for people (including skeptics and naysayers) who wonder where the energy comes from and who, like myself, have for a time not been able to see how tacking would be of any benefit to it.
If by "potential energy" you mean something other than the mass of one fluid moving relative to another (or surface) and the kinetic energy contained thus, then the reason it isn't used in an explanation or video is that it would be flat wrong. There is NO other energy involved in accelerating the craft.
If by "potential energy" you mean the mass of one fluid moving relative to another (or surface) and the kinetic energy contained thus - meaning the power of the wind, I can't imagine how you have missed such explanations.
JB
HW Helper
P: 7,166
Quote by kmarinas86 where this kinetic energy comes from
You could consider the source of energy for the true wind to be the heat from the sun.
Quote by kmarinas86 Potential energy latent inside matter isn't exactly something that I would call part of "Newtonian" physics.
I don't recall any mention of potential energy in the descriptions of how wind driven vehicles operates (sail boats, DDWFTTW vehicles, DUW vehicles, ... ). Extracting potential energy within matter involves a chemical or nuclear reaction, which doesn't occur with the wind powered vehicles being discussed here.
Quote by kmarinas86 The lack of significant mention of potential energy when discussing how tacking can allow sails to move ahead of the wind. ... where the energy comes from
I'm not sure what you mean by potential energy. Wind driven vehicles extract kinetic energy from the wind (using a ground or water based frame of reference).
Quote by kmarinas86 how tacking would be of any benefit to it.
Tacking isn't required for a DDWFTTW vehicle. A DDWFTTW vehicle could connect the wheels to a treadmill geared so the upper surface of the treadmill moves upwind at some fraction of the vehicles speed, for example 1/2 of the vehicles speed (an advance ratio of .5). The treadmill could pull parachutes along the upper surface and then collapse them (perhaps pull them through a tube) along the lower surface. It wouldn't be as efficient as a propeller, but if the losses could be reduced enough, it would work.
A sail can't generate thrust from an apparent headwind component, so it needs an apparent crosswind component which it diverts to aft of the boat's heading to generate thrust, which is why a sail boat needs to tack in order to acheive vmg downwind greater than true wind.
P: 1,011
Quote by rcgldr I'm not sure what you mean by potential energy. Wind driven vehicles extract pressure energy and kinetic energy from the wind (using a ground or water based frame of reference).
Pressure energy to me is a form of potential energy, though I tend to look at it from a "molecular" perspective where electric forces reign supreme over gravitational ones.
P: 203
Quote by kmarinas86 Pressure energy to me is a form of potential energy, though I tend to look at it from a "molecular" perspective where electric forces reign supreme over gravitational ones.
Even pressure energy doesn't come into the equation. The vehicle simply slows down the air relative to the ground beneath it. That's all. For pressure energy to benefit the cart it would have to leave a volume of air in its wake that has been expanded to greater volume and lower pressure. It doesn't do this.
HW Helper
P: 7,166
Quote by spork Even pressure energy doesn't come into the equation.
There's a pressure differential in the immediate vincinity of the propeller, but eventually the affected air's pressure returns to ambient and it's velocity is changed. From a DDWFTTW vehicle's frame of reference, the pressure and flow near the propeller corresponds to the description in this NASA article:
http://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html
From a ground frame of reference, the pressure differential at the propeller results in the affected air from the true wind to be slowed down.
P: 203
Quote by rcgldr There's a pressure differential in the immediate vincinity of the propeller...
Of course there is. But you said: "Wind driven vehicles extract pressure energy and kinetic energy from the wind"
And this isn't the case. Sure they create a lower pressure in front of the disk and a high pressure behind the disk, but ultimately, they don't take any "pressure energy" from the wind. That would be equivalent to saying that an airplane's exploits the "pressure energy" of the air to stay aloft. The fact that it has a local effect (both positive and negative) on air pressure does not imply that it extracts pressure energy from the wind.
HW Helper
P: 7,166
Quote by spork But you said: "Wind driven vehicles extract pressure energy and kinetic energy from the wind"
Yeah I worded that badly and corrected my previous posts. The pressure effects are short term and only exist in the immediate vicinity of the propeller.
P: 203
Quote by rcgldr Yeah I worded that badly. The pressure effects are short term and only exists in the immediate vicinity of the propeller.
I figured it was a simple mis-statement. That's one of the reasons I didn't mention it when you posted it, but only when kmarinas86 talked about it as potential energy. I got the distinct idea he saw it as something it's not.
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https://www.nist.gov/publications/accurate-and-precise-laser-spectroscopy-emerging-technology-study-atmospheric | # Accurate and Precise Laser Spectroscopy: Emerging Technology for the Study of Atmospheric Constituents, Radiative Transfer, and Planetary Dynamics
Published: May 21, 2018
### Author(s)
Adam J. Fleisher, David A. Long, Joseph T. Hodges, Gerd A. Wagner, David F. Plusquellic
Proceedings Title: National Academies of Science Workshop on the Future of Atmospheric Boundary Layer Observations
Conference Dates: October 23-26, 2017
Conference Location: Warrenton, VA
Pub Type: Conferences | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.906466543674469, "perplexity": 15631.43944654134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584445118.99/warc/CC-MAIN-20190124014810-20190124040810-00161.warc.gz"} |
http://semantic-domain.blogspot.com/2014/06/g-ontological-argument.html | ## Thursday, June 19, 2014
### Gödel's Ontological Argument
In his paper Jokes and their Relation to the Cognitive Unconscious: Marvin Minsky argued that a sense of humor would be an essential component of a rational agent. His argument went a bit like this: an AI would use need to use logic to deduce consequences from their data about the world. However, any sufficiently large database of facts would inevitably contain inconsistencies, and a blind application of ex falso quodlibet would lead to disaster.
The bus timetable, which you believe to be correct, says the bus arrives at 3:35, but it has actually arrived at 3:37. This is a contradiction, and so by ex falso, you think it's a good idea to give me all your money!
Clearly, the correct response to such an argument is to laugh at it and move on. Whence the sense of humor, according to Minksy.
When I first read Anselm's ontological argument for the existence of God, I had the correct Minskyan reaction — I laughed at it and moved on. However, one of the curiosities of mathematical logic is that Kurt Gödel did not laugh at it. He found all of the gaps in Anselm's reasoning, and then — and here we see what being one of the greatest logicians of all time gets you — he proceeded to repair all the holes in the proof.
That is: Gödel has given a proof of the existence of God. So let's look at the proof!
We begin by axiomatizing a predicate
$\Good : (i \to \prop) \to \prop$
To do this, we first give the following auxilliary definitions.
• $\Godlike : i \to \prop$
$\Godlike(x) = \forall \phi:i \to \prop.\; \Good(\phi) \Rightarrow \phi(x)$
• $\EssenceOf : i \to (i \to \prop) \to \prop$
$\EssenceOf(x, \phi) = \phi(x) \wedge \forall \psi:i \to \prop.\; \psi(x) \Rightarrow \Box(\forall x.\; \phi(x) \Rightarrow \psi(x))$
• $\Essential : i \to \prop$
$\Essential(x) = \forall \phi:i \to \prop.\; \EssenceOf(x,\phi) \Rightarrow \Box(\exists y.\phi(y))$
So something is godlike if it has all good properties. Something $x$'s essence is a property $\phi$, if $x$ has property $\phi$, and furthermore every property $x$ has is implied by $\phi$. Something $x$ is essential, if something with its essence necessarily exists. (This sounds like medieval theology already, doesn't it?)
Now, we can give the axioms describing the $\Good$ predicate.
1. $\forall \phi:i \to \prop.\; \Good(\lambda x.\lnot \phi(x)) \iff \lnot \Good(\phi)$
2. $\forall \phi, \psi:i \to \prop.\; (\forall x:i.\; (\phi(x) \Rightarrow \psi(x))) \Rightarrow \Good(\phi) \Rightarrow \Good(\psi)$
3. $\Good(\Godlike)$
4. $\Good(\phi) \Rightarrow \Box(\Good(\phi))$
5. $\Good(\Essential)$
The first axiom says that every property is exactly good or not good (bad). The second says that if a property is a logical consequence of a good property, it is also a good property. The third says that being godlike is good. The third property says that every good property is necessarily good. Finally, being essential (as opposed to accidental) is good. I feel like I could argue with any of these axioms (except maybe 2), but honestly I'm more interested in the proof itself.
Below, I give Goedel's result, rearranged a bit and put into a natural deduction style. Loosely speaking, I'm using the natural deduction format from Davies and Pfenning's A Judgmental Reconstruction of Modal Logic. In my proof, I use the phrase “We have necessarily $P$” to mean that $P$ goes into the valid context, and I use “Necessarily:” with an indented block following it to indicate that I'm proving a box'd proposition. The rule is that within the scope of a “Necessarily:”, I can only use “necessarily $P$” results from outside of it.
This style is occasionally inconvenient when proving lemmas, so I'll also use the “necessitation rule” from traditional modal logic, which says that if you have a closed proof of $P$, you can conclude $\Box(P)$. (This rule is implied by the Davies and Pfenning rules, but it's handy in informal proofs.)
Now, their proof system is basically for S4, and the ontological argument uses S5. So I'll also make use (as an axiom) of the fact that $\Diamond(\Box(P))$ implies $\Box(P)$ — that is, that possibly necessary things are actually necessary. This saves having to mention world variables in the proof.
Overall, the proof system needed for the proof is a fully impredicative second-order classical S5 — a constructivist may find this a harder lift than the existence of God! (Can God create a consistent axiomatic system so powerful She cannot believe it?) Jokes aside, it's an interesting proof nonetheless.
We begin by showing that any Godlike entity only has necessarily good properties.
Lemma 1. (God is good) $\forall x:i.\; \Godlike(x) \Rightarrow \forall \phi:i \to \prop.\; \phi(x) \Rightarrow \Box(\Good(\phi))$
Proof.
• Assume $x$ and $\Godlike(x)$, $\phi$ and $\phi(x)$.
• For a contradiction, suppose $\lnot \Good(\phi)$.
• Then by axiom 1, $\Good(\lambda x.\lnot \phi(x))$.
• Unfolding $\Godlike$ and instantiating with $\lambda x.\;\lnot \phi(x), \Good(\lambda x.\lnot \phi(x)) \Rightarrow \lnot \phi(x)$.
• We know $\Good(\lambda x.\lnot \phi(x))$.
• Hence $\lnot \phi(x)$
• This contradicts $\phi(x)$.
• Therefore $\Good(\phi)$.
• By Axiom 4, $\Box(\Good(\phi))$.
Then, we show that all of a Godlike entity's properties are entailed by being Godlike. I was initially tempted to dub this the “I am that I am” lemma, but decided that “God has no hair” was funnier. The name comes from the theorem in physics that “black holes have no hair” — they are completely characterized by their mass, charge and angular momentum. Similarly, here being Godlike completely characterizes Godlike entities.
If you accept Leibniz's principle, this implies monotheism, as well. Hindus and Buddhists will disagree, because they often deny that things are characterized by their properties. (For differing reasons, Jean-Paul Sartre might say the same, too, as would Per Martin-Löf!)
Lemma 2. (God has no hair): $\forall x:i.\; \Godlike(x) \Rightarrow \EssenceOf(x, \Godlike)$
Proof.
• Assume $x$ and $\Godlike(x)$.
• By definition, $\EssenceOf(x, \Godlike)$ = $\Godlike(x) \wedge \forall \psi:i \to \prop.\; \psi(x) \Rightarrow \Box(\forall x.\; \Godlike(x) \Rightarrow \psi(x))$
1. $\Godlike(x)$ holds by assumption.
2. Assume $\psi:i \to \prop$ and $\psi(x)$.
• By Lemma (God is good), $\Box(\Good(\psi))$.
• So necessarily $\Good(\psi)$.
• Necessarily:
• Assume $x$ and $\Godlike(x)$.
• By definition of $\Godlike$, $\Good(\psi) \Rightarrow \psi(x)$.
• But necessarily $\Good(\psi)$.
• Hence $\psi(x)$
• Therefore $\forall x.\; \Godlike(x) \Rightarrow \psi(x)$
• Therefore $\Box(\forall x.\; \Godlike(x) \Rightarrow \psi(x))$
• Therefore $\forall \psi:i \to \prop.\; \psi(x) \Rightarrow \Box(\forall x.\; \Godlike(x) \Rightarrow \psi(x))$
• Therefore $\EssenceOf(x, \Godlike)$
Next, we show that if a Godlike object could exist, then a Godlike entity necessarily exists.
Lemma 3. (Necessary Existence): $(\exists x:i.\; \Godlike(x)) \Rightarrow \Box(\exists y:i.\; \Godlike(y))$
Proof.
• Assume there is an $x$ such that $\Godlike(x)$.
• By axiom 5, $\Good(\Essential)$.
• By the definition of $\Godlike$, we have $\Essential(x)$.
• Unfolding $\Essential$, we get $\forall \phi:i \to \prop. \EssenceOf(x,\phi) \Rightarrow \Box(\exists y.\phi(y))$
• By Lemma (God has no hair), We know $\EssenceOf(x, \Godlike)$.
• Hence $\Box(\exists y:i.\; \Godlike(y))$
We can now show that the above implication is itself necessary. We could have stuck all of the above theorems inside a “Necessarily:” block, but that would have been an annoying amount of indentation. So I used the necessitation principle, which is admissible in Davies/Pfenning.
Lemma 4. (Necessary Necessary Existence): $\Box(\exists x:i.\; \Godlike(x)) \Rightarrow \Box(\exists y:i.\; \Godlike(y))$
Proof. By inlining the proofs of all the earlier lemmas, we can give a closed proof of the necessary existence lemma (i.e., using nothing but the axioms and definitions).
So by necessitation, we get $\Box(\exists x:i.\; \Godlike(x)) \Rightarrow \Box(\exists y:i.\; \Godlike(y))$.
Now, we‘ll show that it's possible that God exists. I would have called it the “no atheists in foxholes” lemma, except that the US military atheists’ association maintains a list of atheists in foxholes.
Lemma 5. (God is possible): $\Diamond (\exists x.\Godlike(x))$
Proof.
• For a contradiction, suppose $\Box(\forall y:i.\lnot \Godlike(y))$.
• Assume $y$ and $\Godlike(y)$.
• Instantiating the contradiction hypothesis, $\lnot \Godlike(y)$.
• Therefore $\forall y:i.\; \Godlike(y) \Rightarrow \lnot \Godlike(y)$.
• By Axiom 2, $\Good(\lambda y.\lnot \Godlike(y))$
• By Axiom 1, $\lnot \Good(\Godlike)$.
• By Axiom 3, $\Good(\Godlike)$.
• Therefore $\lnot \Box(\forall y.\lnot \Godlike(y))$.
• By quantifier twiddling, $\lnot \Box(\lnot \exists y.\Godlike(y))$
• By clasical definition of possibility $\Diamond (\exists y.\Godlike(y))$
Now we can finish off Gödel's argument!
Theorem 1. (God necessarily exists): $\Box(\exists y:i.\; \Godlike(y))$
Proof.
• By Lemma (Necessary Necessary Existence), $\Box[(\exists x:i.\; \Godlike(x)) \Rightarrow \Box(\exists y:i.\; \Godlike(y))]$
• Necessarily $(\exists x:i.\; \Godlike(x)) \Rightarrow \Box(\exists y:i.\; \Godlike(y))$.
• By Lemma (God is Possible), $\Diamond ((\exists x:i.\; \Godlike(x)))$.
• Stipulating $\exists x:i.\; \Godlike(x)$:
• By using the implication, $\Box(\exists y:i.\; \Godlike(y))$.
• Therefore $\Diamond \Box(\exists y:i.\; \Godlike(y))$.
• In S5, $\Diamond \Box P$ implies $\Box P$ for any $P$.
• Therefore $\Box(\exists y:i.\; \Godlike(y))$
When I set out to understand this proof, I wanted to see where the proof was nonclassical — I initially thought it would be funny to constructivize his argument and announce the existence of a computer program which was a realizer for God. It turns out that this is impossible, since this is a deeply classical argument. However, there might still be some computational content in this proof, since Jean-Louis Krivine has looked at realizability interpretations of very powerful classical systems such as ZF set theory. AFAICT, the tricky part will be finding a specific predicate satisfying the axioms of $\Good$.
However, as far as understanding the proof goes, the use of excluded middle actually turns out to not bother me overmuch. After all, proving that it is not the case that God doesn't exist, sort of makes a good case for apophatic theology, which is pretty congenial to me as a cultural (albeit firmly atheist) Hindu!
Instead, the part I find most difficult to swallow is the proof of the possible existence of God, because it runs afoul of relevance. Specifically, the proof that $\Godlike(y) \Rightarrow \lnot \Godlike(y)$ does not use its hypothesis. As a result, I find myself quite dubious that being $\Godlike$ is a $\Good$ property (Axiom 3). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9813246130943298, "perplexity": 1145.605849083907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215843.55/warc/CC-MAIN-20180820042723-20180820062723-00473.warc.gz"} |
https://theculture.sg/2016/06/inequality-question/ | So lets see how to resolve inequalities that a bit longer than usual.
$\frac{x+3}{x-1} \textless x \textless \frac{1}{2}$
$\frac{x+3}{x-1} \textless x$ and $x \textless \frac{1}{2}$
$\frac{x+3}{x-1} -x \textless 0$
$\frac{x+3-x(x-1)}{x-1} \textless 0$
$\frac{x+3-x^2+x}{x-1} \textless 0$
$\frac{-x^2+2x+3}{x-1} \textless 0$
$\frac{(-x+3)(x+1)}{x-1} \textless 0$
$\Rightarrow -1 \textless x \textless 1$ or $x >3$ and $x \textless \frac{1}{2}$.
Thus, $-1 \textless x \textless \frac{1}{2}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 12, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7249654531478882, "perplexity": 579.6989002932246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806419.21/warc/CC-MAIN-20171121151133-20171121171133-00082.warc.gz"} |
https://blog.mbedded.ninja/programming/operating-systems/linux/programs/ps/ | PROGRAMS
# ps
## Overview
The ps command (a.k.a process status) provides a snapshot of the current processes running on the system.
For repetitive updating of the process information (e.g. updates on how much CPU and memory each process is using, similar to Task Manager in Windows) see the top command instead.
## Default Implementation
By default, ps only prints processes owned by the current user, AND are processes that are associated with the terminal that called ps. Normally this will result in quite a small amount of output, perhaps only 2-5 processes.
gbmhunter@ubuntu:~$ps PID TTY TIME CMD 71811 pts/3 00:00:00 bash 73771 pts/3 00:00:00 ps gbmhunter@ubuntu:~$
You will always be guaranteed these two above when running from a bash terminal, obviously bash is running, and so is the ps command (it includes itself).
## More Complete Information
More complete information on the running processes of the system can be found by providing arguments, such as the UNIX-style ps -e:
[ghunter@ubuntu ~]$ps -e PID TTY TIME CMD 1 ? 01:15:50 systemd 2 ? 00:00:26 kthreadd 3 ? 00:02:24 ksoftirqd/0 8 ? 00:01:55 migration/0 9 ? 00:00:00 rcu_bh 10 ? 06:22:40 rcu_sched 11 ? 00:00:51 watchdog/0 12 ? 00:01:00 watchdog/1 ... If you want more detail on the command that started each process, use ps -ef. [ghunter@ubuntu ~]$ ps -ef
UID PID PPID C STIME TTY TIME CMD
root 1 0 0 Jan02 ? 01:15:51 /usr/lib/systemd/systemd --system --deserialize 22
root 2 0 0 Jan02 ? 00:00:26 [kthreadd]
root 45147 1 0 Jun06 ? 00:00:19 /usr/sbin/crond -n
root 45697 1 0 Jun06 ? 00:06:39 /usr/sbin/irqbalance --foreground
root 45718 1 0 Jun06 ? 00:00:00 /usr/sbin/atd -f
ntp 45742 1 0 Jun06 ? 00:00:18 /usr/sbin/ntpd -u ntp:ntp -g
Remember that PID is the process ID and PPID is the parent process ID.
If you use BSD style options (no dash), ps will print the command and the provided options for each process, rather than the executable name:
gbmhunter@ubuntu:~$ps -aux USER PID %CPU %MEM VSZ RSS TTY STAT START TIME COMMAND root 1 0.0 0.0 33896 4020 ? Ss Jan27 0:02 /sbin/init root 2 0.0 0.0 0 0 ? S Jan27 0:00 [kthreadd] root 3 0.0 0.0 0 0 ? S Jan27 0:01 [ksoftirqd/0] root 5 0.0 0.0 0 0 ? S< Jan27 0:00 [kworker/0:0H] root 7 0.0 0.0 0 0 ? S Jan27 0:30 [rcu_sched] root 8 0.0 0.0 0 0 ? S Jan27 0:00 [rcu_bh] root 9 0.0 0.0 0 0 ? S Jan27 0:00 [migration/0] root 10 0.0 0.0 0 0 ? S Jan27 0:00 [watchdog/0] root 11 0.0 0.0 0 0 ? S Jan27 0:00 [watchdog/1] root 12 0.0 0.0 0 0 ? S Jan27 0:00 [migration/1] root 13 0.0 0.0 0 0 ? S Jan27 0:01 [ksoftirqd/1] root 15 0.0 0.0 0 0 ? S< Jan27 0:00 [kworker/1:0H] ... This is only a snapshot of the total number of processes it will print! ## Supported Options ps can support a large and confusing amount of different option styles, including UNIX options (one dash), BSD options (no dash) and GNU long options (two dashes). ## ps With grep The output of ps can be piped to grep to filter the results. For example, if you wanted to only look for processes with the word hocus_pocus in it: $ ps -aux | grep hocus_pocus
Note: grep will match anything on the line printed by ps -aux. That means that hocus_pocus will be matched against the username column and any paths in the process name.
However, aside from having to use two commands, there are other disadvantages to using ps with grep. A completely new program, pgrep was built to try and provide a better process-searching tool. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20418184995651245, "perplexity": 8510.654731741048}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363072.47/warc/CC-MAIN-20210301212939-20210302002939-00404.warc.gz"} |
http://gplv3.fsf.org/gpl3-dd4-rationale.tex/view?searchterm=None | # GPLv3
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# GPLv3 - Last call rationale (LaTeX)
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\documentclass[11pt]{article}
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\huge{GPLv3 Final Discussion Draft Rationale}
\end{center}
\begin{center}
\Large{Free Software Foundation}
\end{center}
\vspace{300pt}
\noindent
Fifth Floor, Boston, MA 02110-1301, USA\\
\noindent
Verbatim copying and distribution of this entire article are permitted
worldwide, without royalty, in any medium, provided this notice is
preserved.
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\subsection*{Foreword}
This two-part document states the rationale for the changes in the final
discussion draft of GPLv3. Part I provides a discussion of the most
significant changes we have made since the release of the previous
draft, organized by subject matter. Part II is an annotated markup
version of the final draft, with \sout{strikeout} indicating text
present in the third discussion draft that we have removed and
\textbf{bold} indicating text we have added. The annotations state the
reasons for specific changes; some annotations refer the reader to Part
I. We refer to the third and final discussion drafts of GPLv3 as
Draft 3'' and the Final Draft,'' respectively.
A public comment period lasting 29 days will begin on the date of our
release of the Final Draft. At the end of that period we will formally
promulgate the final version of GPLv3.
%\clearpage
\dblnew
\part{Discussion of Principal Changes}
\dblnew
We are pleased to report that the Final Draft makes the Apache License,
version 2.0, fully compatible with GPLv3. We are grateful to the Apache
Software Foundation for working with us to achieve this long-sought
goal.
The concerns we stated in the Draft 3 Rationale were based on varying
literal readings of section 9 of the Apache license that differed from
the interpretation of section 9 held by the ASF itself. During the
course of productive discussions with the ASF following the release of
Draft 3, we ascertained that, to the ASF, the words by reason of'' in
the section 9 upstream indemnification clause meant nothing broader or
vaguer than directly as a result of.'' Read in this light, section 9
seems to us a reasonable and fair approach to protecting upstream
developers, even though we do not wish to adopt such a provision in our
The Final Draft makes the Apache indemnification clause compatible with
GPLv3 by adding a new category of additional conditions in section 7
that may be applied, with appropriate copyright authorization, to
material added to a covered work. Subsection 7f allows terms that
require indemnification of upstream licensors and authors of the
material by a downstream distributor who conveys with contractual
assumptions of liability to the recipient, for any liability that such
assumptions directly impose on those upstream parties.
Another change we have made to section 7 should make clearer that the
trademark clause in section 6 of the Apache license is compatible with
GPLv3. The new subsection 7e permits addition of terms that decline to
grant rights under trademark law for use of trademarks. We have no
objection to such terms, since they do not limit the rights of users
beyond what applicable trademark law would itself require. However,
mandatorily-worded trademark clauses that purport to use the power of
copyright or contract to affirmatively prohibit users from exercising
rights otherwise available under trademark law continue to be
incompatible with the GPL.
The GPLv3 compatibility of the Apache license patent termination clause
was accomplished in Draft 3 by the second specific example of an
impermissible further restriction given in the third paragraph of
section 10. We revised this clause in the Final Draft in order to avoid
the use of the terms contributor'' and contribution'' (now
contributor version''), which we wished to confine to section 11 for
greater readability. Given the way in which we defined contribution''
in Draft 3, the parenthetical wording in the Draft 3 version of the
section 10 clause was actually broader than necessary to achieve Apache
license compatibility. The Final Draft therefore replaces the words
the Program (or the contribution of any contributor)'' with the
Program or any portion of it'' without affecting Apache license
compatibility. We confirmed this conclusion in discussions with the
ASF.
\section{Cut-off Date in Section 11, Paragraph 7}\label{11par7}
In the Final Draft we have removed the square brackets surrounding the
cut-off date at the end of section 11, paragraph 7 (corresponding to
section 11, paragraph 5 in Draft 3). That is to say, the Final Draft
limits the effect of this provision to deals involving discriminatory
patent promises that do not predate the release of Draft 3.
The main reason for this is tactical. We believe we can do more to
protect the community by allowing Novell to use software under GPL
version 3 than by forbidding it to do so. This is because of
paragraph 6 of section 11 (corresponding to paragraph 4 in Draft 3).
It will apply, under the Microsoft/Novell deal, because of the coupons
that Microsoft has acquired that essentially commit it to participate
in the distribution of the Novell SLES GNU/Linux system.
Microsoft is scrambling to dispose of as many Novell SLES coupons as
possible prior to the adoption of GPLv3. Unfortunately for Microsoft,
those coupons bear no expiration date, and paragraph 6 has no cut-off
date. Through its ongoing distribution of coupons, Microsoft will
have procured the distribution of GPLv3-covered programs as soon as
they are included in Novell SLES distributions, thereby extending
patent defenses to all downstream recipients of that software by
operation of paragraph 6.
A secondary reason is to avoid affecting other kinds of agreements for
other kinds of activities. We have tried to take care in paragraph 7
to distinguish pernicious deals of the Microsoft/Novell type from
business conduct that is not particularly harmful, but we cannot be
sure we have entirely succeeded. There remains some risk that other
unchangeable past agreements could fall within its scope.
In future deals, distributors engaging in ordinary business practices
can structure the agreements so that they do not fall under paragraph
7. However, it will block Microsoft and other patent aggressors from
further such attempts to subvert parts of our community.
\section{User Products}\label{userprod}
Section 6 of Draft 3 contained a bracketed reference to the
Magnuson-Moss Warranty Act, stating that interpretation of the term
consumer product'' in that United States statute would control
interpretation of the corresponding term in section 6. Many readers of
Draft 3, particularly those outside the U.S., objected to the inclusion
of a U.S. statutory reference in the GPL. This reference was merely
intended to guide interpretation of the User Product definition in
difficult cases, but we agree that it is better for a license designed
for international use to avoid country-specific legal references. We
recognize also that the interpretive history of a U.S. statute will be
relatively inaccessible even to users in the U.S., let alone those in
other countries.
In the Final Draft, therefore, we have replaced the Magnuson-Moss
reference with three sentences that encapsulate the judicial and
United States concerning the Magnuson-Moss consumer product definition.
First, we state that doubtful cases are resolved in favor of coverage
under the definition. Second, we indicate that the words normally
used'' in the consumer product definition refer to a typical or common
use of a class of product, and not the status of a particular user or
expected or actual uses by a particular user. Third, we make clear that
the existence of substantial non-consumer uses of a product does not
negate a determination that it is a consumer product, unless such
non-consumer uses represent the only significant mode of use of that
product.
It should be clear from these added sentences that it is the general
mode of use of a product that determines objectively whether or not it
is a consumer product. One could not escape the effects of the User
Products provisions by labeling what is demonstrably a consumer product
in ways that suggest it is for professionals,'' for example, contrary
to what some critics of Draft 3 have suggested.
We have made one additional change to the User Products provisions of
section 6. In Draft 3 we made clear that the requirement to provide
Installation Information implies no requirement to provide warranty or
support for a work that has been modified or installed on a User
Product. The Final Draft adds that there is similarly no requirement to
provide warranty or support for the User Product itself.
\section{Conveying to Outside Contractors}\label{contractor}
Large enterprise users of free software often contract with non-employee
developers, often working offsite, to make modifications intended for
the user's private or internal use, and often arrange with other
companies to operate their data centers. Whether GPLv2 permits these
activities is not clear and may depend on variations in copyright law.
The practices seem basically harmless, so we have decided to make it
clear they are permitted.
GPLv3 now gives an explicit permission for a client to provide a copy of
its modified software to a contractor exclusively for that contractor to
modify it further, or run it, on behalf of the client. However, the
client can only exercise this control over its own copyrighted changes
to the GPL-covered program. The parts of the program it obtained from
other contributors must be provided to the contractor with the usual GPL
freedoms.
This permission is stated in section 2. It permits a user to convey
covered works to contractors operating exclusively on the user's behalf,
under the user's direction and control, and to require the contractors
to keep the user's copyrighted changes confidential, but only if the
contractor is limited to acting on the user's behalf, just as the user's
employees would have to act.
The strict conditions in this provision are needed so that it cannot be
twisted to fit other activities, such as making a program available to
users or customers. By making the limits on this provision very narrow,
we ensure that in all other cases the contractor gets the full freedoms
of the GPL.
\clearpage
\part{Annotated Markup of Final Discussion Draft}
\dblnew
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\maketitle
\vspace{-.6in}
\begin{center}
{\parindent 0in
\textbf{Final} Discussion Draft \sout{3} of Version 3, \sout{28 March}
\textbf{31 May} 2007
\bigskip
THIS IS A DRAFT, NOT A PUBLISHED VERSION OF THE \\ GNU GENERAL PUBLIC
\bigskip
Inc. (http://fsf.org) \\ 51 Franklin Street, Fifth Floor,
Boston, MA 02110-1301 USA \\ Everyone is permitted to copy and
distribute verbatim copies of this license document, but changing it is
not allowed.}
\end{center}
\vspace{.3in}
\section*{Preamble}
The GNU General Public License is a free, copyleft license for software
and other kinds of works.
The licenses for most software and other practical works are designed
to take away your freedom to share and change the works. By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change \sout{free software} \textbf{all versions of a
program}---to make sure \sout{the software is} \textbf{it remains} free
\textbf{software} for all its users. We, the Free Software Foundation,
use the GNU General Public License for most of our software; it applies
also to any other \sout{program whose} \textbf{work released this way by
its} authors \sout{commit to using it}. You can apply it to your
programs, too.
When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
\sout{this service} \textbf{them} if you wish), that you receive source
code or can get it if you want it, that you can change the software or
use pieces of it in new free programs, and that you know you can do
these things.
To protect your rights, we need to \sout{make requirements that forbid
anyone to deny} \textbf{prevent others from denying} you these rights or
you have certain responsibilities if you distribute copies of the
software, or if you modify it\textbf{: responsibilities to respect the
freedom of others}.
For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received. You must make sure that they, too,
receive or can get the source code. And you must show them these
terms so they know their rights.
Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
\sout{which gives} \textbf{giving} you legal permission to copy,
distribute and/or modify \sout{the software} \textbf{it}.
For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software. For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be \sout{associated}
\textbf{attributed} erroneously \sout{with the} \textbf{to authors of}
previous versions.
Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so. This is fundamentally incompatible with the \sout{purpose}
\textbf{aim} of \sout{the GPL, which is to protect} \textbf{protecting}
users' freedom to change the software \sout{where changes are possible}.
The systematic pattern of such abuse occurs in the area of products for
individuals to use, which is precisely where it is most unacceptable.
Therefore, we have designed this version of the GPL to prohibit the
practice for those products. If such problems arise substantially in
other domains, we stand ready to extend this provision to those domains
in future versions of the GPL, as needed to protect the freedom of
users.
Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in \sout{places where they}
\textbf{those that} do, we wish to avoid the special danger that patents
applied to a free program could make it effectively proprietary. To
prevent this, the GPL assures that patents cannot be used to render the
program non-free.
The precise terms and conditions for copying, distribution and
modification follow.
\vspace{.2in}
\centerline{TERMS AND CONDITIONS}
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}
\ordsec{Definitions.}
This License'' refers to version 3 of the GNU General Public License.
Copyright'' also means copyright-like laws that apply to other kinds
of works, such as semiconductor masks.
The Program'' refers to any copyrightable work licensed under this
License. Each licensee is addressed as you.'' Licensees'' and
recipients'' may be individuals or organizations.
To modify'' a work means to copy from or adapt all or part of the
work in a fashion requiring copyright permission, other than the making
of \sout{a verbatim} \textbf{an exact} copy. The resulting work is
called a modified version'' of the earlier work or a work based on''
the earlier work. A covered work'' means either the unmodified
Program or a work based on the Program.
\sout{A contributor'' is a party who licenses under this License a
work on which the Program is based. Such a work is called the party's
contribution.''}\footnote{We moved the definition of contributor and
contributor version (the replacement for the term contribution'') to
section 11, since in the Final Draft those terms are used only in that
section. See n.~\ref{contrib-n}.\label{contrib-o}}
To propagate'' a work means to do \sout{(or cause others to do)}
anything with it that\textbf{,} \sout{requires} \textbf{without}
permission\textbf{, would make you directly or secondarily liable for
infringement} under applicable copyright law,\footnote{As we noted in
the Draft 3 Rationale, the parenthetical wording or cause others to
do'' introduced in Draft 3 was intended to explicitly incorporate
concepts of secondary copyright liability into the definition of
propagation. However, the wording we chose in Draft 3 was broader than
necessary to do that job. Some readers pointed out that, if taken
literally, it could make a customer the propagator of the work the
customer receives. The revised definition of propagate achieves the
intended result more precisely.} except executing it on a
computer or making modifications that you do not share. Propagation
includes copying, distribution (with or without modification), making
available to the public, and in some countries other activities as well.
To convey'' a work means any kind of propagation that enables other
parties to make or receive copies, excluding sublicensing. Mere
interaction with a user through a computer network, with no transfer of
a copy, is not conveying.
\sout{A party's essential patent claims'' in a work are all patent
claims owned or controlled by the party, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted by
this License, of making, using, or selling the work, but do not include
claims that would be infringed only as a consequence of further
modification of the work. For purposes of this definition, control''
includes the right to grant sublicenses in a manner consistent with the
requirements of this License.}\footnote{We moved the definition of
essential patent claims to section 11, since it is used only in that
section. See n.~\ref{epc-n}.\label{epc-o}}
\textbf{An interactive user interface displays Appropriate Legal
Notices'' to the extent that it includes a convenient and prominently
visible feature that (1) displays an appropriate copyright notice, and
(2) tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the work
under this License, and how to view a copy of this License. If the
interface presents a list of user commands or options, such as a menu, a
prominent item in the list meets this criterion.}\footnote{We factored
this definition out of subsection 5d. In doing so, we have tempered the
5d requirement somewhat in response to arguments by several readers that
it was too restrictive of the right of modification. The term
Appropriate Legal Notices is now also used in subsection 7b.\label{aln}}
\ordsec{Source Code.}
The source code'' for a work means the preferred form of the work for
making modifications to it. Object code'' means any non-source form
of a work.
A Standard Interface'' means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that is
widely used among developers working in that language.
The System Libraries'' of an executable work include anything, other
than the work as a whole, that (a) is \sout{normally} included in the
\textbf{normal form} \sout{distribution} of \textbf{packaging} a Major
Component,\footnote{We think that speaking of the packaging'' of a
System Library with a Major Component is somewhat clearer than the
notion of its being included in a distribution'' of such a component.}
but which is not part of that Major Component, and (b) serves only to
enable use of the work with that Major Component, or to implement a
Standard Interface for which an implementation is available to the
public in source code form. A Major Component'', in this context,
means a major essential component (kernel, window system, and so on) of
the specific operating system (if any) on which the executable work
runs, or a compiler used to produce the work, or an object code
interpreter used to run it.
The Corresponding Source'' for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities. However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work. For example, Corresponding Source
includes interface definition files associated with source files for the
work, and the source code for shared libraries and dynamically linked
subprograms that the work is specifically designed to require, such as
by intimate data communication or control flow between those subprograms
and other parts of the work.
The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.
The Corresponding Source for a work in source code form is that
same work.
\ordsec{Basic Permissions.}
All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
permission to run the unmodified Program. The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work. This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.
\sout{Propagation of} \textbf{You may make, run and propagate} covered
works that you do not convey, \sout{and making modified versions of the
Program that you do not convey, are permitted} without
\textbf{You may convey covered works to others for the sole purpose of
having them make modifications exclusively for you, or provide you with
facilities for running those works, provided that you comply with the
terms of this License in conveying all material for which you do not
hold copyright. Those thus making or running the covered works for you
must do so exclusively on your behalf, under your direction and control,
on terms that prohibit them from making any copies of your copyrighted
material outside their relationship with you.}\footnote{See Part I, \S\
\ref{contractor}.}
Conveying \textbf{under any other circumstances} is permitted
\textbf{solely} under the conditions stated below. Sublicensing is not
allowed; section 10 makes it unnecessary.
\ordsec{\sout{No Denying} \textbf{Protecting} Users' \textbf{Legal}
Rights \sout{through Technical Measures} \textbf{From Anti-Circumvention
Law}.\protect\footnote{This section shields users from being subjected
to liability under anti-circumvention law for exercising their rights
under the GPL, so far as the GPL can do so. Some readers seem to have
assumed that this provision contains a prohibition on DRM; it does not
(no part of GPLv3 forbids DRM). We think that the title of section 3
may have contributed to this misunderstanding, and so we have replaced
it with a more descriptive one.}}
No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article 11
of the WIPO copyright treaty adopted on 20 December 1996, or similar
laws prohibiting or restricting circumvention of such measures.
When you convey a covered work, you waive any legal power to forbid
circumvention of technical measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technical measures.
\bracketsec{Conveying Verbatim Copies.}
You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.
You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.
\bracketsec{Conveying Modified Source Versions.}
You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4 above, provided that you also meet all of these
conditions:
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item
The work must carry prominent notices stating that you
modified it, and giving a relevant date.
\item
The work must carry prominent notices stating that it is released
requirement modifies the requirement in section 4 to keep intact
all notices''.
\item
You must license the entire work, as a whole, under this License to
anyone who comes into possession of a copy. This License will
therefore apply, \sout{unmodified except as permitted by}
\textbf{along with any applicable} section 7 \textbf{additional
terms},\footnote{The revised wording concerning section 7 expresses
more clearly what was stated by the previous wording.} to the whole
of the work, and all its parts, regardless of how they are packaged.
This License gives no permission to license the work in any other
way, but it does not invalidate such permission if you have
\item
If the work has interactive user interfaces, each must \sout{include
a convenient feature that displays an appropriate copyright notice,
and tells the user that there is no warranty for the work (unless
you provide a warranty), that licensees may convey the work under
this License, and how to view a copy of this License. Specifically,
if the interface presents a list of user commands or options, such
as a menu, a command to display this information must be prominent
in the list; otherwise, the work must display this information at
startup.} \textbf{display Appropriate Legal Notices;} \sout{However}
\textbf{however}, if the Program has interactive interfaces that do
not \sout{comply with this subsection} \textbf{display Appropriate
Legal Notices}, your work need not make them \sout{comply}
\textbf{do so}.\footnote{See n.~\ref{aln}.}
\end{enumerate}
A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work, in
or on a volume of a storage or distribution medium, is called an
aggregate'' if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work in
an aggregate does not cause this License to apply to the other parts of
the aggregate.
\bracketsec{Conveying Non-Source Forms.}
You may convey a covered work in object code form under the terms of
sections 4 and 5, provided that you also convey the machine-readable
Corresponding Source under the terms of this License, in one of these
ways:
\begin{enumerate}
\renewcommand{\labelenumi}{\alph{enumi})}
\item
Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by the
Corresponding Source fixed on a durable physical medium
customarily used for software interchange.
\item
Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by a written
offer, valid for at least three years and valid for as long as you
offer spare parts or customer support for that product model, either
(1) to give anyone who possesses the object code a copy of the
Corresponding Source for all the software in the product that is
covered by this License, on a durable physical medium customarily
used for software interchange, for a price no more than your
reasonable cost of physically performing this conveying of source,
or (2) to provide access to copy the Corresponding Source from a
network server at no charge.
\item
Convey individual copies of the object code with a copy of the
written offer to provide the Corresponding Source. This alternative
is allowed only occasionally and noncommercially, and only if you
received the object code with such an offer, in accord with
subsection 6b.
\item
Convey the object code by offering access from a designated place
Corresponding Source in the same way through the same place at no
further charge. You need not require recipients to copy the
Corresponding Source along with the object code. If the place to
copy the object code is a network server, the Corresponding Source
may be on a different server (operated by you or a third party) that
supports equivalent copying facilities, provided you maintain clear
directions next to the object code saying where to find the
Corresponding Source. Regardless of what server hosts the
Corresponding Source, you remain obligated to ensure that it is
available for as long as needed to satisfy these requirements.
\item
Convey the object code using peer-to-peer transmission, provided you
inform other peers where the object code and Corresponding Source of
the work are being offered to the general public at no charge under
subsection 6d.
\end{enumerate}
A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.
A User Product'' is either (1) a consumer
product\textbf{,}''\sout{,} which means any tangible personal property
which is normally used for personal, family, or household purposes, or
(2) anything designed or sold for incorporation into a dwelling.
\sout{[In cases of doubt concerning whether an item is a consumer
product'', the interpretation of the Magnuson-Moss Warranty Act, 15
U.S.C. 2301 et seq., shall provide the basis for interpretation,
regardless of the choice of law determination for this License as a
whole.]} \textbf{In determining whether a product is a consumer
product, doubtful cases shall be resolved in favor of coverage. For a
particular product received by a particular user, normally used''
refers to a typical or common use of that class of product, regardless
of the status of the particular user or of the way in which the
particular user actually uses, or expects or is expected to use, the
product. A product is a consumer product regardless of whether the
product has substantial commercial, industrial or non-consumer uses,
unless such uses represent the only significant mode of use of the
product.}\footnote{See Part I, \S\ \ref{userprod}.}
Installation Information'' for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product
from a modified version of its Corresponding Source. The information
must suffice to ensure that the continued functioning of the modified
object code is in no case prevented or interfered with solely because
If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information. But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).
The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient\textbf{,
or for the User Product in which it has been modified or
installed}.\footnote{See Part I, \S\ \ref{userprod}.} Network access
may be denied when the modification itself materially and adversely
affects the operation of the network or violates the rules and protocols
for communication across the network.
Corresponding Source conveyed, and Installation Information provided, in
accord with this section must be in a format that is publicly
documented\sout{,} \textbf{(and} with an implementation available to the
public in source code form\textbf{)}, and must require no special
Additional permissions'' are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law. If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by this
When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.) You may place
for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, \textbf{for
material you add to a covered work,} you may \textbf{(if authorized by
the copyright holders of that material)} supplement the terms of this
License with terms \sout{effective under, or drafted for compatibility
with, local law}:\footnote{The limitation stated in Draft 3 to terms
effective under, or drafted for compatibility with, local law'' was
more confusing than useful. It highlighted one possible reason for
desire for compatibility with local legal requirements, but the more
common use of this provision will be to permit combination of
GPLv3-covered code with code covered by other separately written
added terms must be effective under applicable local law to be
effectively placed on a GPL-covered work is obviously true. We
therefore have deleted this wording.
The wording we added in the Final Draft makes clearer than before that
a covered work, and only if use of those terms is authorized by the
copyright holders of that material. This also enables some
simplification of the wording of the listed categories.}
\renewcommand{\labelenumi}{\alph{enumi}.}
\begin{enumerate}
\item
disclaiming warranty or limiting liability differently from the
terms of section\textbf{s} 15 \textbf{and 16} of this
\item
requiring preservation of specified reasonable legal notices or
author attributions in \sout{source or object code forms of}
\textbf{that} material \sout{added by you to a covered work}
\textbf{or in the Appropriate Legal Notices displayed by works
containing it};\footnote{The typical licensing term made
GPLv3-compatible by the first part of this clause is one which
requires preservation of notices in source code itself or in
materials accompanying the distribution of object code. We have
broadened the clause slightly to also permit terms that require
preservation of such notices in the Appropriate Legal Notices
displayed by interactive user interfaces that comply with the
requirements of subsection 5d.} or
\item
prohibiting misrepresentation of the origin of \textbf{that}
material \sout{added by you to a covered work}, or requiring that
modified versions of such material be marked in reasonable ways as
different from the original version; or
\item
limiting the use for publicity purposes of \sout{specified} names
of licensors or authors \textbf{of the material;}\sout{,} or
the extent otherwise permitted by law}
\renewcommand{\labelenumi}{\textbf{\alph{enumi}.}}
\item
\textbf{declining to grant rights under trademark law for use of
I, \S\ \ref{apache}.} \textbf{or}
\item
\textbf{requiring indemnification of licensors and authors of that
material by anyone who conveys the material (or modified versions
of it) with contractual assumptions of liability to the recipient,
for any liability that these contractual assumptions directly
impose on those licensors and authors}.\footnote{See Part I, \S\
\ref{apache}.}
\end{enumerate}
All other non-permissive additional terms are considered further
restrictions'' within the meaning of section 10. If the Program as you
received it, or any part of it, purports to be governed by this License,
supplemented by a term that is a further restriction, you may remove
that term. If a license document contains a further restriction but
permits relicensing or conveying under this License, you may add to a
covered work material governed by the terms of that license document,
provided that the further restriction does not survive such relicensing
or conveying.
If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the additional
terms that apply to those files, or a notice indicating where to find
the applicable terms.
\textbf{Additional terms, permissive or non-permissive, may be stated
in the form of a separately written license, or stated as exceptions;
the above requirements apply either way.}\footnote{Section 7 applies
regardless of whether an additional term is contained within a formal
terms of the GPL.}
\bracketsec{Termination.}
You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void. If you violate this License, any copyright holder
\textbf{of the work} may put you on notice by notifying you of the
violation, by any reasonable means, provided 60 days have not elapsed
since the most recent violation. Having put you on notice, the
copyright holder may, at any time, terminate the rights (including any
patent rights) that the copyright holder has granted to you under this
However, if this is \sout{your} \textbf{the} first \textbf{time you
software)} with respect to a given copyright holder, and you cure the
violation within 30 days following your receipt of the notice, then your
first and second paragraphs of section 8 are clarifications of the
previous wording. It is a copyright holder of the work being violated
who can notify the violator of the violation and terminate the
violator's rights. The cure opportunity is specifically available for
violators who have not previously received notice of a GPLv3 violation
from a particular copyright holder with respect to any GPLv3-covered
software.}
In the event that your rights are terminated under this section,
parties who have received copies, or rights, from you under this License
will not have their licenses terminated so long as they remain in full
compliance. \textbf{Unless and until your rights are restored by the
material under section 10.}\footnote{This point is already implicit in
the GPL. A violator cannot simply avoid termination of rights for
certain material by obtaining a new copy of the material and claiming
the benefit of automatic licensing under the first paragraph of section
10.}
\bracketsec{Acceptance Not Required for Having Copies.}
You are not required to accept this License in order to receive or run
a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission to
receive a copy likewise does not require acceptance. However, nothing
other than this License grants you permission to propagate or modify any
covered work. These actions infringe copyright if you do not accept
this License. Therefore, by modifying or propagating a covered work,
\bracketsec{Automatic Licensing of Downstream Recipients.}
Each time you convey a covered work, the recipient automatically
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.
An entity transaction'' is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that transaction
work the party's predecessor in interest had or could give under the
previous paragraph, plus a right to possession of the Corresponding
Source of the work from the predecessor in interest\textbf{, if the
predecessor has it or can get it with reasonable efforts}.\footnote{It
is necessary to state this qualification because the predecessor in
interest might not be able to obtain the Corresponding Source. For
example, as one of the comments submitted to \texttt{gplv3.fsf.org}
pointed out, a three-year written offer to provide the Corresponding
Source might have expired by the time of the entity transaction.}
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may not
impose a license fee, royalty, or other charge for exercise of rights
granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that any
patent claim is infringed by making, using, selling, offering for sale,
or importing the Program \sout{(}or \sout{the contribution of any
contributor)} \textbf{any portion of it}.\footnote{See Part I, \S\
\ref{apache}.}
\ordsec{Patents.}
\textbf{A contributor'' is a copyright holder who authorizes use
under this License of the Program or a work on which the Program is
based. The work thus licensed is called the contributor's contributor
version.''}\footnote{This paragraph was relocated from section 0 and
revised for clarity. See n.~\ref{contrib-o}. The new definition of
contributor makes clearer that contributors are a subset of the
copyright holders of the Program. The definition corrects the failure
of the previous definition to include copyright holders who make the
Program itself available, rather than a work further upstream on which
the Program is based. A comment submitted to \texttt{gplv3.fsf.org}
We agree with those who argued that our use of the term contribution''
to mean the entire licensed work (and not just the contributor's
copyrighted portions of it) was confusing. In the Final Draft we have
replaced it with contributor version,'' a term used with approximately
the same meaning in the Mozilla Public License and its
progeny.\label{contrib-n}}
\textbf{A contributor's essential patent claims'' are all patent
claims owned or controlled by the contributor, whether already acquired
or hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a consequence
of further modification of the contributor version. For purposes of
this definition, control'' includes the right to grant patent
sublicenses in a manner consistent with the requirements of this
License.}\footnote{This paragraph was relocated from section 0. See
n.~\ref{epc-o}. The definition was revised to make it specific to
contributors and contributor versions, since the term essential patent
claims is used in no other context.\label{epc-n}}
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims \sout{in
its contribution}, to make, use, sell, offer for sale, import and
otherwise run, modify and propagate the \textbf{contents of its}
\sout{contribution} \textbf{contributor version}.\footnote{We specify
that the patent license covers making, using and selling the contents of
the contributor version to make clear that the permissions under the
patent license extend to the entirety of the contributor version and any
part of it.}
For purposes of the following three paragraphs, a patent license''
\sout{means a patent license, a covenant not to bring suit for patent
infringement, or} \textbf{is} any \sout{other} express agreement or
commitment, however denominated, not to enforce a patent\textbf{, and to
grant'' a patent license to a party means to make such an agreement or
commitment not to enforce a patent against the party}.\footnote{The
definition of patent license has been made more concise, and we have
clarified what it means to grant'' such a patent license, as used in
the sixth and seventh paragraphs of section 11, given that patent
non-assertion promises falling under the definition of patent license
might not be stated as a formal grant of patent rights.}
If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone to
copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) disclaim the patent license for this particular work,
or (3) arrange, in a manner consistent with the requirements of this
recipients. Knowingly relying'' means you have actual knowledge that,
but for the patent license, your conveying the covered work in a
country, or your recipient's use of the covered work in a country, would
infringe one or more identifiable patents in that country that you have
reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license \sout{providing freedom}
\textbf{to some of the parties receiving the covered work authorizing
them} to use, propagate, modify or convey a specific copy of the covered
work \sout{to any of the parties receiving the covered work}, then the
patent license you grant is automatically extended to all recipients of
the covered work and works based on it.\footnote{We made minor changes
to the wording of this provision for clarity.}
\textbf{A patent license is discriminatory'' if it does not
include within the scope of its coverage, prohibits the exercise of, or
is conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License.} You may not convey a covered
work if you are a party to an arrangement with a third party that is in
the business of distributing software, under which you make payment to
the third party based on the extent of your activity of conveying the
work, and under which the third party grants, to any of the parties who
would receive the covered work from you, a \textbf{discriminatory}
patent license (a) in connection with copies of the covered work
conveyed by you\sout{, and/or} \textbf{(or} copies made from those
\textbf{copies)}, or (b) primarily for and in connection with specific
products or compilations that contain the covered work, \sout{which
license does not cover, prohibits the exercise of, or is conditioned on
the non-exercise of any of the rights that are specifically granted to
recipients of the covered work under this License[,} unless you entered
into that arrangement, or that patent license was granted, prior to
\textbf{28} March \sout{28,} 2007\sout{]}.\footnote{We factored out a
definition of discriminatory patent license'' to permit some
simplification and clarification of the provision. For a discussion of
our removal of the brackets from the cut-off date, see Part I, \S\
\ref{11par7}.}
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may otherwise
be available to you under applicable patent law.
\bracketsec{No Surrender of Others' Freedom.}
If conditions are imposed on you (whether by court order, agreement or
excuse you from the conditions of this License. If you cannot convey
the Program, or other covered work, so as to satisfy simultaneously your
obligations under this License and any other pertinent obligations, then
as a consequence you may not convey it at all. For example, if you
agree to terms that obligate you to collect a royalty for further
conveying from those to whom you convey the Program, the only way you
could satisfy both those terms and this License would be to refrain
entirely from conveying the Program.
\ordsec{Use with the \textbf{GNU} Affero General Public License.}
Notwithstanding any other provision of this License, you have
permission to link any covered work with a work licensed under version
Foundation)} of the \textbf{GNU} Affero General Public License, and to
convey the resulting combination. The terms of this License will
continue to apply to your covered work but will not apply to the work
with which it is linked, which will remain governed by the \textbf{GNU}
Affero General Public License.\footnote{We accompany the release of the
Final Draft with a discussion draft of version 3 of the GNU Affero
General Public License. The GNU AGPLv3 is identical in its terms and
conditions to GPLv3 except for its section 13. The first paragraph of
section 13 states the additional requirement for modified versions that
support remote network interactive use:
\begin{quote}
Notwithstanding any other provision of this License, if you modify the
Program, your modified version must give all users interacting with it
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https://www.hepdata.net/search/?q=cmenergies%3A%5B1.3+TO+1.4%7D&author=Eichon%2C+A.D.&page=1&phrases=E%2B+E-+Scattering | Showing 10 of 11 results
#### rivet Analysis Measurements of $e^{+} e^{-} \to K^{+} K^{-} \eta$, $K^{+} K^{-} \pi^0$ and $K^0_{s} K^\pm \pi^\mp$ cross- sections using initial state radiation events
The collaboration Aubert, Bernard ; Bona, M. ; Boutigny, D. ; et al.
Phys.Rev.D 77 (2008) 092002, 2008.
Inspire Record 765258
This paper reports measurements of processes: e+e- -> gamma KsK+pi-, e+e- -> gamma K+K-pi0, e+e- -> gamma phi eta, and e+e- -> gamma phi pi0. The initial state radiated photon allows to cover the hadronic final state in the energy range from thresholds up to ~4.6 GeV. The overall size of the data sample analyzed is 232 fb-1, collected by the BaBar detector running at the PEP-II e+e- storage ring. From the Dalitz plot analysis of the KsK+pi- final state, moduli and relative phase of the isoscalar and the isovector components of the e+e- -> K K*(892) cross section are determined. Parameters of phi and rho recurrences are also measured, using a global fitting procedure which exploits the interconnection among amplitudes, moduli and phases of the e+e- -> KsK+pi-, K+K-pi0, phi eta final states. The cross section for the OZI-forbidden process e+e- -> phi pi0, and the J/psi branching fractions to KK*(892) and K+K-eta are also measured.
3 data tables match query
The cross section for E+ E- --> K0S K+ PI- + CC with statistical errors only.
The cross section for E+ E- --> K+ K- PI0 with statistical errors only.
The cross section for E+ E- --> PHI PI0 with statistical errors only.
#### Study of the process e+e-\to \mu+\mu- in the energy region \sqrt{s}=980, 1040 -- 1380 MeV
Achasov, M.N. ; Aulchenko, V.M. ; Beloborodov, K.I. ; et al.
Phys.Rev.D 79 (2009) 112012, 2009.
Inspire Record 798415
The cross section of the process e+e-\to\mu+\mu- was measured in the SND experiment at the VEPP-2M e+e- collider in the energy region \sqrt{s}=980, 1040 -- 1380 MeV. The event numbers of the process e+e-\to\mu+\mu- were normalized to the integrated luminosity measured using e+e-\to e+e- and e+e-\to\gamma\gamma processes. The ratio of the measured cross section to the theoretically predicted value is 1.006\pm 0.007 \pm 0.016 and 1.005 \pm 0.007 \pm 0.018 in the first and second case respectively. Using results of the measurements, the electromagnetic running coupling constant \alpha in the energy region \sqrt{s}=1040 -- 1380 MeV was obtained <1/\alpha> = 134.1\pm 0.5 \pm 1.2 and this is in agreement with theoretical expectation.
2 data tables match query
The E+ E- --> MU+ MU- cross section obtained using the (GAMMA GAMMA) luminosity measurement.
The measured E+ E- --> E+ E- cross section in the electron angle 30 to 150 degrees.
#### rivet Analysis The e+ e- ---> 2(pi+ pi-) pi0, 2(pi+ pi-) eta, K+ K- pi+ pi- pi0 and K+ K- pi+ pi- eta Cross Sections Measured with Initial-State Radiation
The collaboration Aubert, Bernard ; Bona, M. ; Boutigny, D. ; et al.
Phys.Rev.D 76 (2007) 092005, 2007.
Inspire Record 758568
We study the processes $e^+ e^-\to 2(\pi^+\pi^-)\pi^0\gamma$, $2(\pi^+\pi^-)\eta\gamma$, $K^+ K^-\pi^+\pi^-\pi^0\gamma$ and $K^+ K^-\pi^+\pi^-\eta\gamma$ with the hard photon radiated from the initial state. About 20000, 4300, 5500 and 375 fully reconstructed events, respectively, are selected from 232 fb$^{-1}$ of BaBar data. The invariant mass of the hadronic final state defines the effective $e^+ e^-$ center-of-mass energy, so that the obtained cross sections from the threshold to about 5 GeV can be compared with corresponding direct \epem measurements, currently available only for the $\eta\pi^+\pi^-$ and $\omega\pi^+\pi^-$ submodes of the $e^+ e^-\to 2(\pi^+\pi^-)\pi^0$ channel. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from $e^+ e^-\to\omega(782)\pi^+\pi^-$ and study the $\omega(1420)$ and $\omega(1650)$ resonances. In the charmonium region, we observe the $J/\psi$ in all these final states and several intermediate states, as well as the $\psi(2S)$ in some modes, and we measure the corresponding branching fractions.
4 data tables match query
Measured cross section for E+ E- --> 2(PI+ PI-) PI0 with statistical errorsonly.
Measured cross section for E+ E- --> ETA PI+ PI- with statistical errors only.
Measured cross section for E+ E- --> OMEGA PI+ PI- with statistical errors only.
More…
#### Experiments at VEPP-2M with SND detector
The collaboration Achasov, M.N. ; Aulchenko, V.M. ; Baru, S.E. ; et al.
1998.
Inspire Record 476386
Short overview of experiments with SND detector at VEPP-2M e^+e^- collider in the energy range 2E = 400 - 1400 MeV and preliminary results of data analysis are presented.
4 data tables match query
No description provided.
No description provided.
No description provided.
More…
#### rivet Analysis Measurement of the e+e- -> K+K- process cross-section in the energy range s**(1/2) = 1.04 - 1.38 GeV with the SND detector in the experiment at VEPP-2M e+e- collider
Achasov, M.N. ; Beloborodov, K.I. ; Berdyugin, A.V. ; et al.
Phys.Rev.D 76 (2007) 072012, 2007.
Inspire Record 755881
The cross section for the process $e^+e^- \to K^+K^-$ was measured in the energy range $\sqrt{s}$ = 1.04--1.38 GeV in the SND experiment at VEPP-2M $e^+e^-$ collider. The measured cross section is described by the Vector Meson Dominance model with contributions from the light vector mesons $\rho$, $\omega$, $\phi$ and their lowest excitations. The mean statistical accuracy of the measurement is 4.4 %, and the systematic uncertainty is 5.2 %.
1 data table match query
The measured cross section and charged kaon form factor.
#### rivet Analysis Inclusive Lambda(c)+ Production in e+ e- Annihilations at s**(1/2) = 10.54-GeV and in Upsilon(4S) Decays
The collaboration Aubert, Bernard ; Bona, M. ; Boutigny, D. ; et al.
Phys.Rev.D 75 (2007) 012003, 2007.
Inspire Record 725377
We present measurements of the total production rates and momentum distributions of the charmed baryon $\Lambda_c^+$ in $e^+e^- \to$ hadrons at a center-of-mass energy of 10.54 GeV and in $\Upsilon(4S)$ decays. In hadronic events at 10.54 GeV, charmed hadrons are almost exclusively leading particles in $e^+e^- \to c\bar{c}$ events, allowing direct studies of $c$-quark fragmentation. We measure a momentum distribution for $\Lambda_c^+$ baryons that differs significantly from those measured previously for charmed mesons. Comparing with a number of models, we find none that can describe the distribution completely. We measure an average scaled momentum of $\left< x_p \right> = 0.574\pm$0.009 and a total rate of $N_{\Lambda c}^{q\bar{q}} = 0.057\pm$0.002(exp.)$\pm$0.015(BF) $\Lambda_c^+$ per hadronic event, where the experimental error is much smaller than that due to the branching fraction into the reconstructed decay mode, $pK^-\pi^+$. In $\Upsilon (4S)$ decays we measure a total rate of $N_{\Lambda c}^{\Upsilon} = 0.091\pm$0.006(exp.)$\pm$0.024(BF) per $\Upsilon(4S)$ decay, and find a much softer momentum distribution than expected from B decays into a $\Lambda_c^+$ plus an antinucleon and one to three pions.
1 data table match query
The integrated number of LAMBDA/C+'s per hadronic event for the continuum at cm energy 10.54 GeV.
#### Pion Form-Factor Measurement by e+ e- --> pi+ pi- in the Energy Range 2 e from 0.78-GeV Up to 1.34-GeV
Bukin, A.D. ; Vasserman, I.B. ; Koop, I.A. ; et al.
Phys.Lett.B 73 (1978) 226-228, 1978.
Inspire Record 134220
The cross section of the reaction e + e − → π + π − has been measured in the energy range 2 E from 0.78 up to 1.34 GeV. The obtained experimental values of the pion form factor exceed notably the ρ -meson tail.
1 data table match query
No description provided.
#### rivet Analysis The $e^+e^- \to 3(\pi^+ \pi^-), 2(\pi^+ \pi^- \pi^0)$ and $K^+ K^- 2(\pi^+ \pi^-)$ cross sections at center-of-mass energies from production threshold to 4.5-GeV measured with initial-state radiation
The collaboration Aubert, Bernard ; Barate, R. ; Boutigny, D. ; et al.
Phys.Rev.D 73 (2006) 052003, 2006.
Inspire Record 709730
We study the processes e+ e- --> 3(pi+pi-)gamma, 2(pi+pi-pi0)gamma and K+ K- 2(pi+pi-)gamma, with the photon radiated from the initial state. About 20,000, 33,000 and 4,000 fully reconstructed events, respectively, have been selected from 232 fb-1 of BaBar data. The invariant mass of the hadronic final state defines the effective e+e- center-of-mass energy, so that these data can be compared with the corresponding direct e+e- measurements. From the 3(pi+pi-), 2(pi+pi-pi0) and K+ K- 2(pi+pi-) mass spectra, the cross sections for the processes e+ e- --> 3(pi+pi-), e+ e- --> 2(pi+pi-pi0) and e+ e- --> K+ K- 2(pi+pi-) are measured for center-of-mass energies from production threshold to 4.5 GeV. The uncertainty in the cross section measurement is typically 6-15%. We observe the J/psi in all these final states and measure the corresponding branching fractions.
2 data tables match query
The cross section for E+ E- --> 3PI+ 3PI- as measured with the ISR data. Errors are statistical only.
The cross section for E+ E- --> 2PI+ 2PI- 2PI0 as measured with the ISR data. Errors are statistical only.
#### rivet Analysis Study of the process e+ e- ---> pi+ pi- pi0 in the energy region s**(1/2) from 0.98-GeV to 1.38-GeV
Achasov, M.N. ; Aulchenko, V.M. ; Beloborodov, K.I. ; et al.
Phys.Rev.D 66 (2002) 032001, 2002.
Inspire Record 582183
The cross section of the process $e^+e^-\to \pi^+\pi^-\pi^0$ was measured in the Spherical Neutral Detector experiment at the VEPP-2M collider in the energy region $\sqrt[]{s} = 980 \div 1380$ MeV. The measured cross section, together with the $e^+e^-\to \pi^+\pi^-\pi^0$ and $\omega\pi^+\pi^-$ cross sections obtained in other experiments, was analyzed in the framework of the generalized vector meson dominance model. It was found that the experimental data can be described by a sum of $\omega$, $\phi$ mesons and two $\omega^\prime$ and $\omega^{\prime\prime}$ resonances contributions, with masses $m_{\omega^\prime}\sim 1490$,$m_{\omega^{\prime\prime}}\sim 1790$ MeV and widths $\Gamma_{\omega^\prime}\sim 1210$, $\Gamma_{\omega^{\prime\prime}}\sim 560$ MeV. The analysis of the $\pi^+\pi^-$ invariant mass spectra in the energy region $\sqrt[]{s}$ from 1100 to 1380 MeV has shown that for their descriptionone should take into account the $e^+e^-\to\omega\pi^0\to\pi^+\pi^-\pi^0$ mechanism also. The phase between the amplitudes corresponding to the $e^+e^-\to\omega\pi$ and $e^+e^-\to\rho\pi$ intermediate states was measured for the first time. The value of the phase is close to zero and depends on energy.
1 data table match query
The measured E+ E- --> PI+ PI- PI0 cross section.
#### INVESTIGATION OF THE PROCESS E+ E- ---> E+ E- GAMMA IN THE ENERGY REGION 0.64-GEV - 1.40-GEV. (IN RUSSIAN)
Bukin, A.D. ; Kurdadze, L.M. ; Lelchuk, M.Yu. ; et al. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9844460487365723, "perplexity": 5040.208074251214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488269939.53/warc/CC-MAIN-20210621085922-20210621115922-00133.warc.gz"} |
http://mathhelpforum.com/differential-geometry/119533-uniform-convergence-measures.html | # Math Help - Uniform convergence in measures
1. ## Uniform convergence in measures
Just stuck on a Measure theory question right now:
Suppose $\{f_n\}\subset {L^1} and {f_n}\longrightarrow f$ uniformly. Show that if $\mu(X)<\infty , then f\in {L^1}$ and $\int f_n\longrightarrow \int f$. Find an example that shows that the result is not necessarily true when $\mu(X)= \infty$
Little bit stumped on how to start. Initially I thought I could use the dominated convergence theorem, but not fully sure. We do have uniform convergence and the dominated convergence theorem claims if we can bound a function by g then the $\int f =lim_{n\rightarrow\infty} \int {f_n}$.
If not I'm thinking of using chebychev's inequality.
Any help would be great.
I also apologise for any problems with the LaTex, it's my first time using it.
Thanks
2. Since $f_n \rightarrow f$ uniformly we have that for all $x\in X$ there exists an $N$ such that $m,n\geq N$ implies $\vert f_n(x)\vert - \vert f_m(x) \vert \leq \vert f_m(x)-f_n(x) \vert \leq 1$ now taking $m=N$ we get $\vert f_n(x) \vert \leq 1+ \vert f_N(x)\vert$ for all $n\geq N$ and this last one is integrable since $\int_{X} d\mu =\mu (X)<\infty$ so picking $g(x)=\max \{ \vert f_1(x) \vert ,...,\vert f_{N-1}(x) \vert , 1+ \vert f_N(x) \vert \}$ we get our dominating function.
When $\mu (X)= \infty$ see here
3. ## uniform convergence
looks like what u used cauchy sequence rather than uniformly convergence. can you still do what you did with uniform convergent sequence?
4. $|\int_Xf_n-f|\le\int_X|f_n-f|\le\mu(X)\cdot\sup|f_n-f|<\epsilon\mu(X)$.
For a counter example consider $f_n=\frac{1}{n}\chi_{[0,n]}$. Then $f_n\to 0$ but $\int_\mathbb{R}f_n=1$ for all $n$.
5. ## L1 space
can you show me how to show that $f$ is in L1, please?
6. I think that the following version of the triangle inequality should help: $|a|-|b|\le |a-b|$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 23, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.985261082649231, "perplexity": 211.30458308209123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657133564.63/warc/CC-MAIN-20140914011213-00138-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://aps.arxiv.org/list/cs.NA/pastweek?skip=15&show=25 | # Numerical Analysis
## Authors and titles for recent submissions, skipping first 15
[ total of 91 entries: 1-25 | 16-40 | 41-65 | 66-90 | 91 ]
[ showing 25 entries per page: fewer | more | all ]
### Tue, 7 Feb 2023 (continued, showing last 15 of 30 entries)
[16]
Title: Convergence Analysis of the Deep Galerkin Method for Weak Solutions
Comments: arXiv admin note: substantial text overlap with arXiv:2107.14478
Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
[17]
Title: Error analysis of the implicit variable-step BDF2 method for the molecular beam epitaxial model with slope selection
Subjects: Numerical Analysis (math.NA)
[18]
Title: High order computation of optimal transport, mean field planning, and mean field games
Subjects: Numerical Analysis (math.NA)
[19]
Title: VEM discretization allowing small edges for the reaction-convection-diffusion equation: source and spectral problems
Subjects: Numerical Analysis (math.NA)
[20]
Title: Recursive Analytical Quadrature of Laplace and Helmholtz Layer Potentials in $\mathbb{R}^3$
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
[21]
Title: Analysis of a fully discretized FDM-FEM scheme for solving thermo-elastic-damage coupled nonlinear PDE systems
Subjects: Numerical Analysis (math.NA)
[22]
Title: Construction of Hierarchically Semi-Separable matrix Representation using Adaptive Johnson-Lindenstrauss Sketching
Subjects: Numerical Analysis (math.NA)
[23] arXiv:2302.02894 (cross-list from math.SP) [pdf, ps, other]
Title: Bounds on the eigenvalues of matrix rational functions
Subjects: Spectral Theory (math.SP); Numerical Analysis (math.NA)
[24] arXiv:2302.02795 (cross-list from cs.MS) [pdf, other]
Title: Trimpack: Unstructured Triangular Mesh Generation Library
Subjects: Mathematical Software (cs.MS); Numerical Analysis (math.NA)
[25] arXiv:2302.02671 (cross-list from math.DS) [pdf, ps, other]
Title: Solving fractional Hantavirus model: A new approach
Subjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
[26] arXiv:2302.02665 (cross-list from math.AP) [pdf, other]
Title: Adjoint Method in PDE-based Image Compression
Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
[27] arXiv:2302.02640 (cross-list from math-ph) [pdf, ps, other]
Title: A simple formula of the magnetic potential and of the stray field energy induced by a given magnetization
Authors: Tahar Zamene Boulmezaoud (UVSQ, UVIC)
Subjects: Mathematical Physics (math-ph); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
[28] arXiv:2302.02548 (cross-list from cs.LG) [pdf, other]
Title: Learning Trees of $\ell_0$-Minimization Problems
Authors: G. Welper
Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
[29] arXiv:2302.02464 (cross-list from math.OC) [pdf, other]
Title: On the numerical stability of discretised Optimal Control Problems
Subjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
[30] arXiv:2302.02214 (cross-list from cs.CV) [pdf, other]
Title: Variational multichannel multiclass segmentation\endgraf using unsupervised lifting with CNNs
Comments: 20th INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS
Subjects: Computer Vision and Pattern Recognition (cs.CV); Functional Analysis (math.FA); Numerical Analysis (math.NA)
### Mon, 6 Feb 2023 (showing first 10 of 14 entries)
[31]
Title: A coupling generalized multiscale finite element method for coupled thermomechanical problems
Subjects: Numerical Analysis (math.NA)
[32]
Title: A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations
Comments: 62 pages, 5 figures
Subjects: Numerical Analysis (math.NA)
[33]
Title: Error analysis for a Crouzeix-Raviart approximation of the obstacle problem
Comments: 31 pages, 6 figures, 2 tables. arXiv admin note: text overlap with arXiv:2210.12116
Subjects: Numerical Analysis (math.NA)
[34]
Title: An Equilibrated Error Estimator for the 2D/1D MSFEM T-Formulation of the Eddy Current Problem
Subjects: Numerical Analysis (math.NA)
[35]
Title: Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation
Subjects: Numerical Analysis (math.NA); Machine Learning (stat.ML)
[36]
Title: Improving Recommendation Relevance by simulating User Interest
Subjects: Numerical Analysis (math.NA); Information Retrieval (cs.IR); Machine Learning (cs.LG); Applications (stat.AP)
[37]
Title: Numerical solutions to an inverse problem for a non-linear Helmholtz equation
Comments: 10 pages, 2 figures
Subjects: Numerical Analysis (math.NA)
[38]
Title: Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier-Stokes equations as a paradigm
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Probability (math.PR)
[39]
Title: Analysis of view aliasing for the generalized Radon transform in $\mathbb R^2$
Subjects: Numerical Analysis (math.NA)
[40]
Title: Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computations
Comments: arXiv admin note: text overlap with arXiv:2207.10978
Subjects: Numerical Analysis (math.NA)
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Links to: arXiv, form interface, find, cs, new, 2302, contact, help (Access key information) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6224077343940735, "perplexity": 18837.258528768183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00166.warc.gz"} |
https://brilliant.org/problems/chomp-again/ | Chomp again
Refer to Chomp for the rules of this tasty game.
Knowing that the game on rectangular boards is rigged, the players are now playing with irregular boards as depicted on the picture. With optimal play, who wins each game? Answer $$1$$ if the first player wins and $$2$$ if the second player wins. String all five digits together. Thus, if the first player wins on all boards but the M, your answer should be $$11121$$.
Remember that each player must select a piece; they cannot select an empty square! So on the P, it's illegal to move on $$(1,1)$$ to capture the three pieces to the right and below it, as $$(1,1)$$ is not a piece.
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http://mathoverflow.net/tags/topological-groups/new | # Tag Info
0
Assuming the Hilbert-Smith conjecture, $G$ is a Lie group. For sufficiently large $n$ (I think $n=5$ will do) any $n$-connected Lie group is unipotent. But $G$ is not unipotent, because it contains $GL_m$. So no such $G$ can exist.
3
Here is a partial positive answer, going in the opposite direction of Terry´s comment. Let $X$ be an infinite topological space. Suppose $X$ is first countable at $p \in X$ and for every open neighborhood $U$ of $p$ there is a smaller neighborhood $V \subseteq U$ such that $|U \setminus V|=|U|$. Then there is a group structure on $X$ with identity $p$ such ...
5
Having slept on it, I realise that the answer is No (even if we omit the condition on the inverse). The first example of infinite countable non-topologizable group due to Olshanskii (1980) is actually locally non-topologizable. Indeed, Olshanskii's group $G$ has exponent $p^2$ and the cyclic centre of order $p$. In addition, $g^p\ne1$ for any non-central ...
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http://mathematica.stackexchange.com/questions/13387/define-product-derivative/13388 | # Define product derivative
How do I define the $n$th product derivative of a function in Mathematica?
The first product derivative $f^\ast$ of a function $f$ is $$f^\ast(x)=\exp\left(\frac{f^\prime(x)}{f(x)}\right)$$ The $n$th product derivative is the result of applying this operator $n$ times.
This is my attempt at a recursive definition:
In[111]:= Clear[prodd]
prodd[f_, n_] := prodd[e^(f'/f), n - 1]; prodd[f_, 0] = f
Out[112]= f
In[113]:= prodd[E^x, 1](* Should print E *)
Out[113]= e^(E^-x Derivative[1][(E^x)])
In[114]:= prodd[E^E^x, 1](* Should print E^E^x *)
Out[114]= e^(E^-E^x Derivative[1][(E^E^x)])
In[114]:= prodd[E^x, 2](* Should print 1 *)
-
As an aside, what is the product derivative used for? – rcollyer Oct 22 '12 at 3:29
@rcollyer: see the preprint by Mike Spivey that I linked to. See also this, this, and this. – Guess who it is. Oct 22 '12 at 11:17
@J.M. the preprint is awesome. I like the exponential approximations, and it makes me wonder what else I've missed in calculus, or was ignored ... – rcollyer Oct 22 '12 at 20:19
From Corollary 1 of this preprint by Mike Spivey, there is a simple nonrecursive definition for the product derivative:
ProductD[f_, x_] := ProductD[f, {x, 1}];
ProductD[f_, {x_, k_Integer?NonNegative}] := Exp[D[Log[f], {x, k}]]
Verify a few identities:
ProductD[f[x] g[x], x] == ProductD[f[x], x] ProductD[g[x], x] // Simplify
True
ProductD[x^a, x] == Exp[a/x]
True
ProductD[Exp[Exp[x]], x] == Exp[Exp[x]]
True
ProductD[x^x, x] == E x
True
Here's the corresponding multiplicative calculus analog of Derivative[]:
ProductDerivative[0] = Identity;
ProductDerivative[k_Integer?Positive][f_] :=
Derivative[k][Composition[Log, f]] /. Function[ff_] :> Function[Evaluate[E^ff]]
To use the example given by the OP in the comments:
ProductDerivative[1][Sin][4]
E^Cot[4]
ProductDerivative[1][Sin[#] &][4]
E^Cot[4]
ProductDerivative[1][Function[u, Sin[u]]][4]
E^Cot[4]
-
Thanks. I guess I should have tried to find an explicit definition first. – Navin Oct 21 '12 at 19:42
Is there a way to make this return a function? For example, ProductD[Sin[x], {x, 1}][4] should evaluate to E^Cot[4]. – Navin Oct 21 '12 at 19:53
I've added an implementation of ProductDerivative[] now. – Guess who it is. Oct 22 '12 at 2:19
e is not the same as E, and f' expects f to be a function but E^x is just an expression, so I use D[f, x] instead:
Clear[prodd]
prodd[f_, n_] := prodd[E^(D[f, x]/f), n - 1];
prodd[f_, 0] := f
prodd[E^x, 1](*Should print E*)
prodd[E^E^x, 1](*Should print E^E^x*)
prodd[E^x, 2](*Should print 1*)
E
E^E^x
1
You could also write:
Clear[prodd]
prodd[f_, n_] := Nest[E^(D[#, x]/#) &, f, n]
If you are going to use x in this manner I recommend using \[FormalX] instead to prevent failure if you accidentally assign a value to x.
You might also consider something like this:
Clear[prodd]
prodd[f_, n_] := Nest[E^(#'[\[FormalX]]/#[\[FormalX]]) &, f, n]
prodd[E^# &, 1]
prodd[E^E^# &, 1]
prodd[E^# &, 2]
E
E^E^x
1
-
Thanks, this seems to work. – Navin Oct 21 '12 at 19:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3859172463417053, "perplexity": 6088.21171024508}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988924.75/warc/CC-MAIN-20150728002308-00214-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://iwaponline.com/wst/article/79/11/2145/68070/An-electrode-assisted-anaerobic-digestion-process | ## ABBREVIATIONS
Anaerobic digestion
•
• COD
Chemical oxygen demand
•
• MEC
Microbial electrolysis cell
•
• OCV
Open circuit voltage
•
• PCoA
Principal coordinates analysis
•
• TS
Total solids
•
• VFA
Volatile fatty acids
•
• VS
Volatile solids
## INTRODUCTION
Anaerobic digestion (AD) is a microbial process that occurs in oxygen-free environments by which organic waste is decomposed by complex metabolic pathways, eventually leading to the formation of biogas, a mixture of carbon dioxide and methane (CO2 + CH4), that is considered as a renewable energy source that can potentially replace fossil fuels (Chen et al. 2008). The production of biogas by AD has been implemented in the treatment of waste for over a century, and in recent years a substantial increase in the number of installations of industrial-scale reactors for wastewater treatment has been observed (Appels et al. 2011). In wastewater treatment, AD offers several advantages compared with other technologies, including lower sludge production and lower energy requirements along with a high potential for energy recovery. Additionally, the produced biogas of the existing anaerobic system will be more economically valuable when meeting the quality of natural gas to be utilized as either vehicle fuel or to be connected to the natural gas infrastructure. Nevertheless, AD is not yet widely applied world-wide because it often has low operational stability (Chen et al. 2008), including activated biomass washout, lengthy periods of acclimation, low quality of the produced biogas, in addition to sensitivity to xenobiotics and micropollutants that may inhibit process stability and performance (Ruel et al. 2002; Rajagopal et al. 2013).
Although biogas is considered a potentially important energy resource, it has limited application and utilization, since it typically consists of 60%–70% CH4 and 30%–40% CO2, and often contains additional undesirable gases, (i.e., traces of water vapor, H2S and H2, and other contaminants). Therefore, although biogas can be used for heating or to generate power, the large volume of CO2 reduces the heating value of the biogas, increases compression and transportation costs and limits its economic feasibility (Ruel et al. 2002; Zhao et al. 2010; Muñoz Torre et al. 2015). Thus, low-quality biogas cannot be used as a vehicle fuel or integrated into the natural gas energy system before removing substantial amounts of the CO2 component. In this regard, upgrading it to natural-gas quality (CH4 content of 90% or higher) will enable broad applications, substantially increasing its value.
Most of the methods for biogas upgrading are based on chemical and physical techniques that remove CO2 with minimal loss of CH4 (Niesner et al. 2013), yet most of them are not cost-effective or sustainable in terms of capital investment and operational costs, and require expensive rare elements for chemical catalysis (Guebitz et al. 2015; Angelidaki et al. 2018). About 200 biogas upgrading plants operated world-wide make use of five main technologies for bio-methane processes including those able to produce bio-methane to the required purity (Niesner et al. 2013). In contrast, biogas upgrading can also be accomplished by biologically based methods, where hydrogenotrophic methanogens utilize H2 (from an external source) as an electron donor to reduce CO2 to CH4 (Luo et al. 2012; Rachbauer et al. 2016; Kougias et al. 2017). This leads to higher CH4 concentrations than those achieved by the methanogenic microbial community in a conventional AD reactor during wastewater treatment. This approach, which mitigates CO2 that is present in the system, can be implemented under mild operational conditions, and thus is considered a sustainable process for both biogas upgrading and, on top of that, reducing anthropogenic greenhouse gas emissions from the wastewater treatment process. While many studies chose to inject pure hydrogen to experimentally investigate biological biogas upgrading (Bassani et al. 2015; Agneessens et al. 2017; Wahid et al. 2019), microbial electrolysis cells (MECs) can be used as an alternative low-cost approach for hydrogen production (Logan et al. 2008; Yu et al. 2018).
In this study, a novel method for treating wastewater while simultaneously producing biomethane (upgraded biogas) by an integrated three-chamber AD-MEC-AD system is presented. The method has the significant advantage of avoiding the necessity of pre-harvesting the H2 that is essentially produced from the same source of treated wastewater. The objectives of this research were two-fold: (1) to design an innovative biological approach combining a process for CO2 reduction to CH4 with a hybrid high-rate anaerobic–microbial electrolysis cell (MEC) system for low-cost hydrogen production; and (2) as the upgrading process is biologically based on the activity of hydrogenotrophic methanogens, it was aimed to enhance the stability of methanogenesis through the utilization of the hybrid system, and to better understand the prokaryotic community dynamics that potentially govern the biogas upgrade capacity in the hybrid high-rate anaerobic–microbial electrolysis cell system. Unlike recent applications in which electrodes were implanted directly into AD reactors (in situ electrode-configuration) (e.g. Bo et al. 2014; Liu et al. 2017), here, an ex situ electrode-configuration was used (in(ex) situ electrode-configuration and in situ bio-methanation are distinguished).
## MATERIALS AND METHODS
### Anaerobic bioreactor inoculation and operation conditions
Two laboratory-scale anaerobic reactors (AD1 and AD2) were inoculated separately, using an anaerobic granular biomass that was collected from a well operated, up-flow anaerobic sludge blanket (UASB) bio-reactor used to treat the wastewater of a citrus-based soft drink factory (PRIGAT) at Kibbutz Givat Haim, Israel. The volume ratio between the two reactors was 75% for the first stage (AD1) and 25% for the second tank (AD2), as illustrated in Supplementary Figure S1 (available with the online version of this paper). The second reactor operated under conditions to enrich primarily hydrogenotrophic methanogen activity, as described below. The active volumes of the two bioreactors were 2,300 mL and 800 mL, respectively. Both reactors were equipped with cylindrical double jackets maintaining a constant temperature of 37 °C by circulating heated water. The system was fed with synthetic wastewater (kept at 4 °C), where the effluent of the first unit (AD1) was fed into the subsequent reactor, and the produced biogas of the first methanogenic stage was mixed with hydrogen from an external source (gas bag) at a ratio of 4:1 (H2:CO2), and the entire mixture was fed into the second-stage tank (AD2). In addition, a circulation of 1.5–2 times the biogas flow rate was applied to the second AD reactor in order to increase the solubility of hydrogen, and hence, to enhance its bioavailability and allow the enrichment of the hydrogenotrophic microorganisms.
### Microbial electrolysis cell (MEC)
The MEC (active volume 2,100 mL) had a spirally wound design comprised of a carbon cloth electrode for the anode attached to another carbon cloth electrode for the cathode. Each electrode surface area was 900 cm2. The electrodes (with a flow spacer and an ion permeable separator) were connected to a potentiostat (IviumStat, Ivium Technologies, The Netherlands) under an applied voltage of 1.0 V. After an acclimation period, the MEC was coupled to the combined-AD reactor. Polarization assays were performed once the current stabilized after every increase in organic loading rate by applying various external voltages (OCV, 1,500, 1,300, 1,100, 900, 700 and 500 mV), with each voltage being applied for 25 minutes, and the current was recorded by the potentiostat. The electrodes' potential measurements (vs Ag/AgCl) were manually recorded routinely, as well as for every applied voltage during polarization assays.
### Operation conditions
The integrated AD1-MEC-AD2 system operated continuously, in series by implementing two consecutive methanogenesis steps: the effluent of the first AD unit was fed into the subsequent reactor (MEC); and the produced biogas of the first methanogenic stage was also fed to the second AD system, as illustrated in Supplementary Figure S1. The system was continuously fed with synthetic wastewater containing glucose as a carbon source. The used synthetic wastewater was prepared based on three stock solutions. All chemicals, of analytical grade, were obtained from Sigma-Aldrich and used as received without any further purification (concentrations of the chemicals given below are in g L−1 in distilled water):
• A. NH4Cl, 100; MgCl2·6H2O, 10; NaCl, 10; CaCl2·2H2O, 5; (NH4)2HPO4, 40.
• B. K2HPO4·3H2O, 152.6.
• C. Trace-metal and saline solution: FeCl2·4H2O, 2; H3BO3, 0.05; ZnCl2, 0.05; CuCl2·2H2O, 0.038; MnCl2·4H2O, 0.05; (NH4)6Mo7O244H2O, 0.05; AlCl3, 0.05; CoCl2·6H2O, 0.05; NiCl2·6H2O, 0.092; ethylenediamine tetra acetate, 0.5; concentrated HCl, 1 mL; Na2SeO35H2O, 0.1.
To a volume of 900 mL of distilled water, 10 mL of stock solution (A), 2 mL of stock solution (B) and 1 mL of stock solution (C) were added, also 0.2 g yeast extract, 0.33 g peptone, 1.5 g glucose, 2.6 g NaHCO3 and 0.25 g Na2S·9H2O were added.
### Analytical methods and calculations
The produced biogas in the anaerobic system was collected in 3 L Tedlar Air® sampling bags. Collected biogas volume was measured daily and the CH4 and CO2 contents were analyzed with special CH4 sensors (Guardian Plus, model 97,460, Edinburgh Sensors and Guardian NG Edinburgh Sensors respectively). H2 was analyzed using an (F-12D ATI) ATI hydrogen sensor by using a gas transmitter. Chemical oxygen demand (COD), total solids (TS) and volatile solids (VS) were measured according to the 18th edition of Standard Methods for the Examination of Water and Wastewater (APHA 2005). The concentration of volatile fatty acids (VFA) was analyzed using a combination of the potentiometric titration methods for acidity (Method 2310B) and alkalinity (Method 2320B) with sludge sample preparation techniques according to the 18th edition of Standard Methods (APHA 2005). The total alkalinity was calculated using the amount of acid needed to titrate the sample from the starting pH to pH 4; the volatile acids were calculated using the amount of hydroxide needed to titrate the sample from pH 4 back to pH 7 (APHA et al. 2005).
To evaluate the contribution of combining the MEC with an enriched hydrogenotrophic reactor (AD2) for upgrading the produced CH4 from a fixed amount of organic substrate, methane production yield (mL CH4/g COD removed) was calculated based on biogas and COD removal measurements taken for three different operation conditions of the systems. The three different operation conditions were: AD alone (without MEC), AD-MEC without circulation (AD-MEC, without circulation) and AD-MEC under applied circulation between MEC-AD2 ×9 the flow rate (AD-MEC, with circulation). As described by Speece (1983) and Jiménez et al. (2018), 1 g of COD removed typically corresponds to the production of 350 mL CH4 when the COD is used only by methanogens under standard conditions. However, this volume can be recalculated to 337.5 mL under NTP conditions (20 °C) while taking into account that 10% of the substrate was assimilated by microorganisms for cell growth.
CH4 production yield (mL/g COD removed) was calculated based on biogas and COD measurements taken from AD2, using the following equation:
### DNA extraction
Water samples from each of the three-reactor systems were collected at different time points for bacterial and archaeal community analysis. Total DNA was extracted from suspended cells in the water samples using the PowerWater DNA isolation kit (DNeasy PowerWater, QIAGEN, Israel), according to the manufacturer's instructions. For cell suspension, 20 mL of each of the three-reactor systems were filtered through a 0.22 μm sterile filter paper. The filter paper was inserted into the Power Water® bead tube to extract microbial DNA according to the manufacturer's instructions (DNeasy PowerWater, QIAGEN, Israel). Additionally, metagenomic DNA was extracted from another AD reactor as a control for AD community composition without an integrated MEC. Quality and quantity of the genomic DNA were determined using NanoDrop (Thermo-Fisher Scientific, Waltham, MA) and agarose gel electrophoresis.
### Analysis of the 16S rRNA gene sequences
For community analysis, fragments of the bacterial and archaeal 16S rRNA genes were separately amplified (for archaea using the Arc344F/Arc806R primer set and for bacteria using the universal Bac515F/Bac806R primer set) while incorporating linkers required for subsequent sequencing with the MiSeq 500 sequencer platform (Illumina). PCR amplification, quality control and sequencing were performed at Hy Laboratories Ltd (Rehovot, Israel). Data analysis is described in the Supplementary Methods (available online).
## RESULTS AND DISCUSSION
### Applying circulation between AD and MEC enhances hydrogen solubility and increases its availability for the hydrogenotrophic community
Figure 1
CH4 production yield (±SD) produced by the AD2 reactor in the different operation phases (calculations are described in Materials and methods section). Dashed line denotes the theoretical CH4 production yield obtained from the removal of 1 g of COD.
Figure 1
CH4 production yield (±SD) produced by the AD2 reactor in the different operation phases (calculations are described in Materials and methods section). Dashed line denotes the theoretical CH4 production yield obtained from the removal of 1 g of COD.
On the other hand, a significant increase in the observed CH4 production yield was noted following the application of the circulation of the effluent between AD2 and MEC (Figure 1). These results clearly indicate that circulation enhances hydrogen solubility and, therefore, increases the bioavailability of hydrogen for the hydrogenotrophic methanogenic community. Other studies have reported that coupling AD and MEC reactors increased methane production. Bo et al. (2014) showed a 2.3-fold increase in methane yield when the coupled AD-MEC reactor was applied with 1 V. Liu et al. (2017) found that only a two-chamber configuration enabled methane enrichment in biogas compared with a single-chamber AD-MEC comprised of an in situ electrode-configuration. In essence, most of the studies combining the AD reactor with MEC have directly applied the electrodes within the AD reactor (Bo et al. 2014; Park et al. 2018; Yu et al. 2018). However, Xu et al. (2014) compared between an in situ and an ex situ electrode-configuration in which an external digester was fed with glucose. In contrast to results presented by Liu et al. (2017), Xu et al. (2014) found that the in situ biogas upgrading system was more effective than the ex situ one, however, no circulation was applied in both studies. Therefore, here is the first time that methane production was improved through increasing hydrogen solubility by applying the electrodes in a separate MEC reactor combined with high-rate circulation of the hydrogen-rich effluent flowing into the hydrogenotrophic-enriched AD reactor. The results show that circulation was instrumental in substantially increasing methane production yields relative to the AD reactor without MEC, as well as the combined AD-MEC without circulation.
Importantly, it was shown that while the CH4 production yield of AD without MEC was identical to the theoretical value, the combined system showed 1.17 and 1.45 increases for AD-MEC without and with circulation, respectively. These results correlate with other studies that reported little improvement of methane production after combining an AD-MEC relative to theoretical maximum yields (Park et al. 2018). Overall, the results emphasize the potential of the combined AD-MEC system with circulation to substantially increase CH4 production yields beyond the theoretical expected values.
The observed current density of the MEC reactor in the different operation phases is shown in Figure 2. The current density is produced as a result of organic matter oxidation in the anode surface by electrochemically active microorganisms that transfer electrons to the anode and secrete protons to the liquid media. The electrons travel through the external circuit and react on the cathode surface with the protons that diffuse toward it from the anode. Since an additional voltage is applied to the circuit, hydrogen formation becomes thermodynamically possible (Logan et al. 2008). Figure 2 shows that current density was not affected by the increased circulation. This result indicates that MEC performance was robust under the different conditions without any lack of organic matter, especially VFA, on the anode. The observed current density was lower than that reported (1 A m−2) for an in situ electrode-configuration fed with glucose (Xu et al. 2014), but similar to that obtained by the ex situ electrode-configuration in the same study (0.4 A m−2). Yet, current densities were substantially higher compared with other combined AD-MEC reactor systems fed with glucose (Gajaraj et al. 2017). The differences in current densities may have resulted from gas–liquid transfer limitations that occurred in the different systems and differences in overall MEC internal resistance.
Figure 2
Current density produced by the MEC reactor in the different operation phases.
Figure 2
Current density produced by the MEC reactor in the different operation phases.
### MEC performance is highly dependent on the organic load influent
The MEC was connected to the AD reactors starting on day 117 and until the last day of the experiment (255 days). The average current density throughout the experiment was relatively constant, and predicatively it was highly dependent on the VFA concentration of the inlet (Figure 3). There were two time-points (day 143 and day 180) in which the current was considerably altered due to a sudden drop in VFA concentrations (hereafter starvation period, days 143 and 180). Subsequently, at these time-points the anode and cathode potentials became more positive (on day 143: −93 mV, −293 mV, respectively, and on day 180: −153 mV, −293 mV, respectively), as shown in Figure 4.
Figure 3
VFA concentration in the inlet into the MEC and current density as a function of time.
Figure 3
VFA concentration in the inlet into the MEC and current density as a function of time.
Figure 4
Anode and cathode potentials (vs Ag/AgCl).
Figure 4
Anode and cathode potentials (vs Ag/AgCl).
Once current density reached steady-state values following the starvation period (Figure 3), the circulation ratio was increased to nine times the feed flow rate between AD2 and MEC. Comparing the polarization curve for MEC performance on day 187 (pre-circulation) and on day 215 (post-circulation), it was observed that MEC performance was considerably improved for the entire range of the applied voltage (−500–1,500 mV, Supplementary Figure S2, available with the online version of this paper). The maximal current and power density for pre- and post-circulation were 0.50 and 1.01 A m−2, respectively. Maximal power density was 744.6 and 1,517.4 mW m−2, respectively.
### Microbial community analysis of the hybrid AD-MEC
A principal coordinates analysis (PCoA) was plotted for both the bacterial and archaeal samples (Figure 5). The plots differ in the degree of separation between the samples taken from the combined and non-combined systems. The bacterial plot (Figure 5(a)) shows a slightly stronger separation between the AD-MEC samples and control samples (ANOSIM p = 0.0005, R = 44%) relative to the archaea (ANOSIM p = 0.0002, R = 37%). Additionally, the bacterial plot shows a stronger separation between the combined system-derived samples and the sludge samples, while the archaeal samples separated less well from the sludge sample used for inoculation (Supplementary Tables 1 and 2, available online). This indicates that there were bacterial community members appearing in higher abundances in the combined system (i.e. electroactive species known to be enriched primarily in microbial electrochemical systems).
Figure 5
Principal coordinates analysis (PCoA) based on the Bray Curtis distance metric between (a) bacterial microbial communities and (b) archaeal microbial communities of the samples from the different reactors, as indicated by the different colors. PC: principal coordinate.
Figure 5
Principal coordinates analysis (PCoA) based on the Bray Curtis distance metric between (a) bacterial microbial communities and (b) archaeal microbial communities of the samples from the different reactors, as indicated by the different colors. PC: principal coordinate.
As expected, known electroactive species (e.g. Geobacter and Desulfuromonas spp.) that are typically enriched primarily in microbial electrochemical systems (Yu et al. 2018) were more abundant in the combined system relative to control AD reactors and the sludge samples (Figure 6).
Figure 6
Relative abundance of the electroactive spp. (sum of Geobacter and Desulfuromonas relative to the sum of the Deltaproteobacteria class) in the control reactor, the combined system reactors and sludge samples. Both Geobacteraceae and Desulfuromonadaceae are known as electroactive bacterial groups and they are closely related both phylogenetically and functionally.
Figure 6
Relative abundance of the electroactive spp. (sum of Geobacter and Desulfuromonas relative to the sum of the Deltaproteobacteria class) in the control reactor, the combined system reactors and sludge samples. Both Geobacteraceae and Desulfuromonadaceae are known as electroactive bacterial groups and they are closely related both phylogenetically and functionally.
Figure 7 shows the composition of the different methanogenic orders in the reactors at the different time-points sampled. All reactors were composed primarily of three orders: Methanobacteriales, Methanomicrobiales and Methanosarcinales, in agreement with previous reports on AD communities (Yu et al. 2006). While these three groups were generally evenly distributed in the control reactors and sludge samples (Figure 7(a) and 7(b)), in the combined system, Methanomicrobiales predominated in both reactors (Figure 7(c) and 7(d)). Interestingly, while the control reactors were relatively stable with Methanosarcinales relative abundance throughout their operation, an increase in this group's abundance in the combined system was correlated with time-points at which there was a technical problem. Essentially, there was a switch between the Methanomicrobiales and Methanosarcinales groups on day 150 in both of the combined reactors (AD and MEC, Figure 7(c) and 7(d)). Yet, this switch did not persist, as the Methanomicrobiales relative abundance almost immediately increased and stabilized for the rest of the operation until the last time-point at which an additional technical failure occurred. Thus, it is assumed that in such systems, the appearance of Methanosarcinales can be defined as a marker for system failure in such enriched hydrogenotrophic reactors, since these methanogens can utilize a variety of non-hydrogen substrates. Under steady-state conditions when hydrogen is bioavailable in the system, Methanomicrobiales dominate the system and outcompete other methanogens (mainly acetoclastic). However, a rapid shift from Methanomicrobiales to Methanosarcinales could predict a technical or possible system failure probably reflecting a reduction in hydrogen availability in the bioreactor.
Figure 7
Relative abundance of archaeal orders belonging to the Euryarchaeota phylum in the different reactors: (a) samples collected from the control AD reactor not connected to an MEC; (b) samples collected from six inoculum sludge samples; (c) samples collected from the MEC; (d) samples collected from the AD2 (the combined system). Each color on the graph represents a different order of methanogenic archaea.
Figure 7
Relative abundance of archaeal orders belonging to the Euryarchaeota phylum in the different reactors: (a) samples collected from the control AD reactor not connected to an MEC; (b) samples collected from six inoculum sludge samples; (c) samples collected from the MEC; (d) samples collected from the AD2 (the combined system). Each color on the graph represents a different order of methanogenic archaea.
Key changes of both bacterial and archaeal family-level groups were followed in AD2 relative to the control AD reactor to further examine the dynamics of the above-mentioned switch (around day 150 of the experiment), as presented in Figure 8. Key groups were chosen based on significant fold-changes of relative abundance compared with their average abundance observed in the sludge samples used for initial inoculation of the reactor, as well as compared with the relative abundance in the control AD reactor that was not connected to the MEC. According to control AD (Figure 8(a)), while the order Methanobacteriales was relatively stable throughout the time, Methanomicrobiales and Methanosarcinales were highly dynamic and their abundance oscillated antagonistically to each other. Similar observations were made for the combined system (Figure 8(b)). Family-level investigation revealed that while Methanosaetaceae was the more dynamic group in the control AD, Methanosarcinaceae belonging to the Methanosarcinales order was highly stable through time (Figure 8(c)). On the other hand, Methanosarcinaceae abundance in the AD-MEC system substantially increased on day 150 and Methanosaetaceae was less affected by the technical problem that occurred around that day. Conversely, the abundance of the hydrogenotrophic methanogen family, Methanospirillaceae, substantially decreased at this time-point which presumably allowed the rise of the acetoclastic methanogen Methanosarcinaceae (Figure 8(d)). Interestingly, the abundance of the other highly abundant groups, Methanobacteriaceae, Methanoregulaceae (Methanomicrobiales) and Methanosaetaceae (Methanosarcinales) were dynamically consistent with each other in the combined reactor. The hydrogenotrophic Methanospirillaceae was previously shown to be the key CO2 reducer when applying sufficient voltage for H2 production in other combined AD-MEC systems (Bo et al. 2014). The other archaeal groups were also reported previously (Bassani et al. 2015). Thus, Methanospirillaceae is suggested to be the primary methanogen family responsible for the biogas upgrading in our system, while Methanoculleus was also recently identified as a dominant hydrogenotroph in such systems (Treu et al. 2018).
Figure 8
(a) Relative abundance of most abundant archaeal orders belonging to the Euryarchaeota phylum in control AD; (b) relative abundance of most abundant archaeal orders in AD2 (the combined system); (c) relative abundance of most abundant archaeal families in control AD; (d) relative abundance of most abundant archaeal families in AD2; (e) relative abundance of most abundant bacterial families in control AD; (f) relative abundance of most abundant bacterial families in AD2.
Figure 8
(a) Relative abundance of most abundant archaeal orders belonging to the Euryarchaeota phylum in control AD; (b) relative abundance of most abundant archaeal orders in AD2 (the combined system); (c) relative abundance of most abundant archaeal families in control AD; (d) relative abundance of most abundant archaeal families in AD2; (e) relative abundance of most abundant bacterial families in control AD; (f) relative abundance of most abundant bacterial families in AD2.
In accordance with bacterial communities reported in previous studies, Bacteroidetes, Proteobacteria, and Firmicutes were the most abundant groups in the system, and were likely involved in substrate degradation (Nelson et al. 2011; Yi et al. 2016). Methanogens, and especially hydrogenotrophs, generally metabolically interact with different bacterial community members (Stams et al. 2003; Treu et al. 2018). Thermodynamically, hydrogenotrophic methanogens are crucial for keeping a low H2 concentration that maintains a favorable reaction rate of acetogenic reactions that channel acetic acid into H2 and CO2 (Stams et al. 2003), but are often outcompeted by other microbes. These methanogens are the key players in ensuring complete conversion of the organic waste to methane. Thus, the purpose of separating the AD reactor into AD1 and AD2 was to overcome this thermodynamic challenge, allowing a function-based subdivision between the following: (1) AD1 – the primary steps of biomass degradation are hydrolysis, fermentation and acetogenesis. The latter includes bacterial members that are highly sensitive to high hydrogen concentration. (2) AD2 – the biogas upgrading process that requires an external input of hydrogen and CO2 that leads to the enrichment of hydrogenotrophic methanogens.
Interestingly, the bacterial community composition in the control reactor exhibited a dramatic shift on day 396. While all other time-points had a stable hydrogen consumption rate, on day 396 a reduced consumption rate was observed (data not shown), leading to a temporary, short-term higher concentration of hydrogen in the system. Although the archaeal community composition was not affected by this change (Figure 7(a)), the bacterial community revealed an increase in Bacteroidetes and a reduction in Proteobacteria (Figure S3). In contrast to the combined reactor's critical time-point (around day 150) at which, due to the technical failure, the MEC directed a lower portion of hydrogen-enriched liquid into AD2, the same bacterial community groups were affected in an opposite manner (Figure 8(e) and 8(f)). While the abundance of the same specific proteobacterial family (Campylobacteraceae, Epsilonproteobacteria) was affected similarly in both cases, the Bacteroidetes families were different (Lentimicrobiaceae belonging to the order Sphingobacteriales in the control reactor and Porphyromonadaceae belonging to the order Bacteroidales in AD2 of the AD-MEC system). Bassani et al. (2015) reported that an unclassified member of Sphingobacteriaceae was found to decrease after the addition of H2. Here, these closely related members of Bacteroidetes responded differently to hydrogen exposures. Remarkably, Lentimicrobiaceae was found to grow syntrophically with the hydrogen-consuming methanogen Methanospirillum hungatei (Sun et al. 2016), supporting its enhanced growth once hydrogen availability increased at this time-point. Essentially, the bacterial composition dynamics indicated specific groups that flourish under high and low hydrogen concentrations, as well as important bacterial–archaeal interactions that occurred under those conditions.
## CONCLUSION
The results show that the combination of MEC with AD significantly improved the performance of an anaerobic treatment system, while substantially upgrading the potential for a higher-quality biogas in a cost-effective manner. Yet, the successful upgrading process was achieved only when circulation of the hydrogen-enriched liquid was applied between the MEC and the second AD reactor, leading to a higher hydrogen bioavailability for hydrogenotrophic methanogens. Accordingly, microbial community analysis confirmed that the process was enriched with electroactive species in the MEC-based enhanced hydrogen production module of the system. It was also observed that while in the control reactor there was a relatively even distribution of acetoclastic and hydrogenotrophic methanogens, in the combined system hydrogenotrophic methanogens were more abundant at most of the time-points. However, at critical time-points when failure occurred, the dominance of acetoclastic methanogens suddenly increased, as well as other bacterial members that thrive under hydrogen-limiting conditions, clearly indicating an association between such combined anaerobic system performance and an increased abundance of specific microbial groups. Consequently, the microbial results show proof of concept that upgrading the produced biogas by using soluble hydrogen directly from a microbial electrochemical device is plausible, and thus has great potential for biogas upgrading without the need for direct hydrogen harvesting. Nevertheless, this process, undoubtedly, requires further optimization from the engineering and microbial perspectives, to facilitate a more practical upscaled application for commercial wastewater treatment systems.
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https://www.physicsforums.com/threads/volume-of-solids.67280/ | # Volume of Solids
1. Mar 14, 2005
### kreil
The base of the solid is the disc x2+y2=1. The cross-section by planes perpendicular to the y-axis between y=-1 and y=1 are isosceles right triangles with 1 leg in the disc. Find the volume of the solid.
I did the problem, but I am not sure if I did it correctly, and if I did I really just need the problem explained. I understand most of it, I think:
$$A(y)=\frac{1}{2}bh=(\frac{1}{2})(2\sqrt{1-y^2})(2\sqrt{1-y^2})=2(1-y^2)$$
$$V_s=\int_{-1}^1A(y)dy=2\int_{-1}^1(1-y^2)dy=2(y-\frac{1}{3}y^3]_{-1}^1)=2(\frac{4}{3})=\frac{8}{3}$$
Thanks
Can you help with the solution or looking for help too?
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Similar Discussions: Volume of Solids | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8533340692520142, "perplexity": 658.7509790379563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281450.93/warc/CC-MAIN-20170116095121-00225-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.openwetware.org/index.php?title=6.021/Notes/2006-10-16&oldid=80427 | 6.021/Notes/2006-10-16
(diff) ←Older revision | Current revision (diff) | Newer revision→ (diff)
• $V_m^o = \sum_n \frac{G_n}{G_m}V_n$
• Rest is not equal to equilibrium
• eventually all ions would equilibrate inside and outside and there would be no membrane potential
• For this to not happen, must have pumps (primary active transport)
• Pumps allow us to go from rest to equilibrium
• treat as current source
• $J_n^a+J_n^p=0$ for all n where $J_n^a$ is the active current (pump) and $J_n^p$ is the passive current (electro-diffusion).
• Define quasi-equilibrium: no net flux but requires energy
• $J_m = 0 = \sum_n G_n(V_m^o-V_n) + \sum_n J_n^a$
• $V_m^o = \sum_n \frac{G_n}{G_m}V_n - \frac{1}{G_m}\sum_n J_n^a$
• The first term is the "indirect effect" whereas the second term is the "direct effect" of the pump
• Both terms depend on the pump as without the pump, both would be 0
• All pumps have indirect effect but only some pumps have direct effect
• A non-electrogenic pump has no net charge change (e.g. trade one sodium ion for one potassium ion)
• An electrogeneic pump such as 3 sodium for 2 potassium would have net active current
• Active transport gets energy from glucose metabolism (namely ATP)
• Experiment shows that sodium and potassium transport are linked
• Sodium is needed for potassium transport and vice-versa
• Sodium is pumped out, potassium pumped in
• The direct effect on membrane potential is negative | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7560263872146606, "perplexity": 5216.397353465434}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988722951.82/warc/CC-MAIN-20161020183842-00110-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Group_testing | # Group testing
An animation of the false coin problem being solved for 10 coins, where the goal is to find the single coin that is lighter than the others. Many fewer than 10 tests are needed since at each stage, at least half of the 'good' coins are eliminated.
In combinatorial mathematics, group testing refers to any procedure which breaks up the task of locating elements of a set which have certain properties into tests on groups of items, rather than on individual elements. A familiar example of this type of technique is the false coin problem of recreational mathematics. In this problem there are n coins and one of them is false, weighing less than a real coin. The objective is to find the false coin, using a balance scale, in the fewest number of weighings. By repeatedly dividing the coins in half and comparing the two halves, the false coin can be found quickly as it is always in the lighter half.[a]
Schemes for carrying out such group testing can be simple or complex and the tests involved at each stage may be different. Schemes in which the tests for the next stage depend on the results of the previous stages are called adaptive procedures, while schemes designed so that all the tests are known beforehand are called non-adaptive procedures. The structure of the scheme of the tests involved in a non-adaptive procedure is known as a pooling design.
## Background
The field of (Combinatorial) Group Testing was introduced by Robert Dorfman in 1943. The motivation arose during the Second World War when the United States Public Health Service and the Selective service embarked upon a large scale project to weed out all syphilitic men called up for induction. Testing an individual for syphilis involves drawing a blood sample from them and then analysing the sample to determine the presence or absence of syphilis. However, at the time, performing this test was expensive, and testing every soldier individually would have been very cost heavy and inefficient.
Supposing there are ${\displaystyle n}$ soldiers, this method of testing leads to ${\displaystyle n}$ separate tests. If we have 70-75% of the people infected then this method would be reasonable. Our goal however, is to achieve effective testing in the more likely scenario where it does not make sense to test 100,000 people to get (say) 10 positives.
The feasibility of a more effective testing scheme hinges on the following property. We can combine blood samples and test a combined sample together to check if at least one soldier has syphilis. This is the central idea behind group testing. If one or more of the soldiers in this group has syphilis, then a test is wasted (more tests need to be performed to find which soldier(s) it was). On the other hand, if no one in the pool has syphilis then many tests are saved.
Modern interest in these testing schemes has been rekindled by the Human Genome Project.[1]
## Types of group-testing algorithm
As well as adaptivity, all group testing algorithms are either combinatorial or probabilistic. A combinatorial algorithm finds all the defectives with certainty. In contrast, a probabilistic algorithm has some non-zero probability of making a mistake (i.e. deciding a defective item is non-defective or vice versa). It is known that zero-error algorithms require significantly more tests asymptotically (in the number of defective items) than algorithms that allow asymptotically small probabilities of error.[4]
Another class of algorithms are so-called noisy algorithms. These deal with the situation that with some non-zero probability, ${\displaystyle q}$, the result of a group test is erroneous (e.g. comes out positive when the test contained no defectives). A noisy algorithm will always have a non-zero probability of making a mistake.
## Formalization of the problem
We now formalize the group-testing problem abstractly.
Let the total number of items to be tested be ${\displaystyle n}$, and an upper bound on the number of defective items be ${\displaystyle d}$. The (unknown) information about which items are defective is described as a vector, ${\displaystyle \mathbf {x} =(x_{1},x_{2},...,x_{n})}$, where ${\displaystyle x_{i}=1}$ if i-th item is defective and ${\displaystyle x_{i}=0}$ otherwise. ${\displaystyle \mathbf {x} }$ is called the input vector.
The Hamming Weight of ${\displaystyle \mathbf {x} }$ is defined as the number of ${\displaystyle 1}$'s in ${\displaystyle \mathbf {x} }$. Hence, ${\displaystyle |\mathbf {x} |\leq d}$ where ${\displaystyle |\mathbf {x} |}$ is the Hamming weight. The vector ${\displaystyle \mathbf {x} }$ is an implicit input since we do not know the positions of the ${\displaystyle 1}$'s. The only way to find out is to run the tests.
### Formal notion of a test
A query/test ${\displaystyle S}$ is a subset of ${\displaystyle [n]}$. The answer to the query ${\displaystyle S\subseteq [n]}$ is defined as follows:
${\displaystyle A(S)={\begin{cases}1,{\mbox{ if }}\displaystyle \sum _{k\in S}x_{k}\geq 1\\0,{\mbox{ otherwise.}}\end{cases}}}$
Note that the addition operation used by the summation is the logical OR, i.e.
${\displaystyle A(S)=\displaystyle \bigvee _{i\in S}x_{i}}$.
### Goal
The goal of group testing is to compute or estimate ${\displaystyle \mathbf {x} }$ and minimize the number of tests required to do so. Here, we have no direct questions or answers. Any piece of information can only be obtained using an indirect query. Group testing is ultimately a question of combinatorial searching. The major issue with such problems is that the solution can grow exponentially in the size of the input.
### Definitions and initial bounds
${\displaystyle t(d,n)}$ is defined as the minimum number of non-adaptive tests that one would have to make to detect all of ${\displaystyle d}$ defective items with a total of ${\displaystyle n}$ items. Similarly, we use ${\displaystyle t^{a}(d,n)}$ to denote the number of adaptive tests that one would have to make to detect all the defective items.
Consider the case when only one person in the group will test positive. Then if we tested in the naive way, in the best case we would at least have to test the first person to find out if he/she is infected. However, in the worst case one might have to end up testing the entire group and only the last person we test will turn out to really be the one who was infected. Hence, ${\displaystyle 1\leq t(d,n)\leq n}$. We also have that ${\displaystyle t^{a}(d,n)\leq t(d,n)}$ due to the fact that any non-adaptive test can be performed by an adaptive test by running all of the tests in the first step of the adaptive test. Adaptive tests can be more efficient than non-adaptive tests since the test can be changed after certain things are discovered.
In summery, ${\displaystyle 1\leq t^{a}(d,n)\leq t(d,n)\leq n}$.
### Mathematical representation of non-adaptive algorithms
A typical group testing setup. A non-adaptive algorithm first chooses the matrix ${\displaystyle M}$, and is then given the vector y. The problem is then to find an estimate for x.
Algorithms for non-adaptive group testing consist of two distinct phases. First, it is decided how many tests to perform and which items to include in each test. This is usually encoded in an ${\displaystyle t\times n}$ binary matrix, ${\displaystyle M}$, where ${\displaystyle t}$ is the number of tests and ${\displaystyle n}$ is the total number of items. Each column of ${\displaystyle M}$ represents an item and each row represents a test, with a ${\displaystyle 1}$ in the (i,j)th entry indicating that the i-th test included the j-th item and a ${\displaystyle 0}$ indicating otherwise. In the second phase, often called the decoding step, the results of each group test are analysed to determine which items are likely to be defective.
As well as the vector ${\displaystyle \mathbf {x} }$ (of length ${\displaystyle n}$) that describes the (unknown) defective set, we introduce the vector ${\displaystyle \mathbf {y} }$, of length ${\displaystyle t}$, describing the results of each test. A ${\displaystyle 1}$ in the j-th entry of ${\displaystyle \mathbf {y} }$ indicates that the j-th test was positive (i.e. contained at least one defective). With these vectors the problem can be reframed as follows: first we choose some testing matrix, ${\displaystyle M}$, after which we are given ${\displaystyle \mathbf {y} }$. Then the problem is to analyse ${\displaystyle \mathbf {y} }$ to find some estimate for ${\displaystyle \mathbf {x} }$.
These notions can be expressed more formally as follows. Let ${\displaystyle S\subseteq [n]}$ be a test and define ${\displaystyle \chi _{i}\in \{0,1\}^{n}}$ such that ${\displaystyle i\in S\Leftrightarrow \chi _{s}(i)=1}$, so the k-th test is described by ${\displaystyle \chi _{k}}$. Let ${\displaystyle M}$ be the ${\displaystyle t\times n}$ matrix with rows ${\displaystyle \chi _{i}}$, so that ${\displaystyle (M)_{i,j}=\chi _{i}(j)}$. This construction is based on the grounds that non-adaptive testing with ${\displaystyle t}$ tests is represented by a ${\displaystyle t}$-subset ${\displaystyle S_{i}\subseteq [n]}$ where ${\displaystyle 1\leq i\leq t}$.
Under this setup, ${\displaystyle \mathbf {y} }$ is defined by the matrix multiplication relation: ${\displaystyle M\mathbf {x} =\mathbf {y} }$, where multiplication is logical AND (${\displaystyle \wedge }$) and addition is logical OR (${\displaystyle \vee }$). Here, ${\displaystyle \mathbf {y} }$ will have a ${\displaystyle 1}$ in position ${\displaystyle i}$ if and only if ${\displaystyle (M)_{i,j}}$ and ${\displaystyle \mathbf {x} _{j}}$ are both ${\displaystyle 1}$ for any ${\displaystyle j}$. That is, if and only if at least one defective item was included in the i-th test.
## Generalized binary splitting algorithm
The generalized binary splitting algorithm is an essentially-optimal adaptive group-testing algorithm that proceeds as follows:[5][6]
1. If ${\displaystyle n\leq 2d-2}$, test the ${\displaystyle n}$ items individually. Otherwise, set ${\displaystyle l=n-d+1}$ and ${\displaystyle \alpha =\lfloor \log {l/d}\rfloor }$.
2. Test a group of size ${\displaystyle 2^{\alpha }}$. If the outcome is negative, every item in the group is declared to be non-defective; set ${\displaystyle n:=n-2^{\alpha }}$ and go to step 1. Otherwise, use a binary search to identify one defective and an unspecified number, called ${\displaystyle x}$, of non-defective items; set ${\displaystyle n:=n-1-x}$ and ${\displaystyle d:=d-1}$. Go to step 1.
The generalized binary splitting algorithm requires no more than ${\displaystyle T}$ tests where ${\displaystyle T={\begin{cases}n&n\leq 2d-2\\(\alpha +2)d+p-1&n\geq 2d-1\end{cases}}}$.
Non-adaptive group-testing algorithms tend to assume that the number of defectives, or at least a good upper bound on them, is known.[7] We will denote this quantity ${\displaystyle d}$. If no bounds are known, there are non-adaptive algorithms with low query complexity that can help estimate ${\displaystyle d}$.[8]
### Combinatorial Orthogonal Matching Pursuit (COMP)
An illustration of the COMP algorithm. COMP identifies item a as being defective and item b as being non-defective. However, it incorrectly labels c as a defective, since it is “hidden” by defective items in every test in which it appears.
Combinatorial Orthogonal Matching Pursuit, or COMP, is a simple non-adaptive group-testing algorithm that forms the basis for the more complicated algorithms that follow in this section.
First, each entry of the testing matrix is chosen i.i.d. to be ${\displaystyle 1}$ with probability ${\displaystyle 1/d}$ and ${\displaystyle 0}$ otherwise.
The decoding step proceeds column-wise (i.e. by item). If every test in which an item appears is positive, then the item is declared defective; otherwise the item is assumed to be non-defective. Or equivalently, if an item appears in any test whose outcome is negative, the item is declared non-defective; otherwise the item is assumed to be defective. Of particular note here is that this algorithm never creates false negatives, though a false positive occurs when all locations with ones in the j-th column of ${\displaystyle M}$ (corresponding to a non-defective item j) are “hidden” by the ones of other columns corresponding to defective items.
The COMP algorithm requires no more than ${\displaystyle ed(1+\delta )\ln(n)}$ tests to have an error probability less than or equal to ${\displaystyle n^{-\delta }}$.[9]
In the noisy case, we relax the requirement in the original COMP algorithm that the set of locations of ones in any column of ${\displaystyle M}$ corresponding to a positive item be entirely contained in the set of locations of ones in the result vector. Instead, we allow for a certain number of “mismatches” – this number of mismatches depends on both the number of ones in each column, and also the noise parameter, ${\displaystyle q}$. This noisy COMP algorithm requires no requires no more than ${\displaystyle 4.36({\sqrt {\delta }}+{\sqrt {1+\delta }})^{2}(1-2q)^{-2}d\log {n}}$ tests to achieve an error probability at most ${\displaystyle n^{-\delta }}$.[10]
### Definite Defectives (DD)
The definite defectives method is an extension of the COMP algorithm that attempts to remove any false positives. Performance guarantees for DD have been shown to strictly exceed those of COMP.[11]
The decoding step uses a useful property of the COMP algorithm: that every item that COMP declares non-defective is certainly non-defective (that is, there are no false negatives). It proceeds as follows:
1. We first run the COMP algorithm, and remove any non-defectives that it detects. All remaining items are now “possibly defective”
2. Next the algorithm looks at all the positive tests. If an item appears as the only “possible defective” in a test, then it must be defective, so the algorithm declares it to be defective.
3. All other items are assumed to be non-defective. The justification for this last step comes from the assumption that the number of defectives is much smaller than the total number of items.
Note that steps 1 and 2 never make a mistake, so the algorithm can only make a mistake if it declares a defective item to be non-defective. Thus the DD algorithm can only create false negatives.
### Sequential COMP (SCOMP)
SCOMP is an algorithm that makes use of the fact that DD makes no mistakes until the last step, were we assume remaining items to be defective. Let the set of declared defectives be ${\displaystyle K}$. We say that a positive test is explained by ${\displaystyle K}$ if it contains at least one item in ${\displaystyle K}$. The key observation with SCOMP is that the set of defectives found by DD may not explain every positive test, and that every unexplained test must contain a hidden defective.
The algorithm proceeds as follows:
1. First carry out steps 1 and 2 of the DD algorithm to obtain ${\displaystyle K}$, an initial estimate for the set of defectives.
2. If ${\displaystyle K}$ explains every positive test, terminate the algorithm: ${\displaystyle K}$ is our final estimate for the set of defectives.
3. If there are any unexplained tests, find the “possible defective” that appears in the largest number of unexplained tests, and declare it to be defective (that is, add it to the set ${\displaystyle K}$). Go to step 2.
In simulations, SCOMP has been shown to perform close to optimally.[12]
## Bounds
In the combinatorial setup, there are a number of upper and lower bounds on t(d,n) (and t^a(d,n)), the minimum number of tests required to detect all defectives.
### Lower bound on ${\displaystyle t^{a}(d,n)}$
Fix a valid group testing scheme with ${\displaystyle t}$ tests. Now, for two distinct input vectors ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {x} ^{\prime }}$ where ${\displaystyle |\mathbf {x} |,|\mathbf {x} ^{\prime }|\leq d}$, the resulting vectors will not be the same i.e. ${\displaystyle \mathbf {y(x)} \neq \mathbf {y(x^{\prime })} }$, were ${\displaystyle \mathbf {y(x)} }$ is the resultant for input vector ${\displaystyle \mathbf {x} }$. This is because two valid inputs will never give us the same result. If this ever happened, then we would always have an error in finding both ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {x} ^{\prime }}$. This gives us that the total number of distinct results is the volume of a Hamming Ball of radius ${\displaystyle d}$, centered about ${\displaystyle n}$ i.e. ${\displaystyle Vol_{2}(d,n)}$. However, for ${\displaystyle t}$ bits, the total number of possible distinct vectors is ${\displaystyle 2^{t}}$. Hence, ${\displaystyle 2^{t}\geq Vol_{2}(d,n)}$. Taking the ${\displaystyle \log }$ on both sides gives us ${\displaystyle t\geq \log\{Vol_{2}(d,n)\}}$.
Now, ${\displaystyle Vol_{2}(d,n)\geq {n \choose d}\geq \left({\frac {n}{d}}\right)^{d}}$. Therefore, we will end up having to perform a minimum of ${\displaystyle d\log {\frac {n}{d}}}$ tests.
Thus we have proved that ${\displaystyle t^{a}(d,n)\geq d\log {\frac {n}{d}}}$
### Upper bound on ${\displaystyle t^{a}(d,n)}$
${\displaystyle t^{a}(d,n)\leq O(d\log {n})}$.
Since we know that the upper bound on the number of positives is ${\displaystyle d}$, we run a binary search at most ${\displaystyle d}$ times or until there are no more values to be found. To simplify the problem we try to give a testing scheme that uses ${\displaystyle O(\log {n})}$ adaptive tests to figure out a ${\displaystyle i}$ such that ${\displaystyle x_{i}=1}$. The related problem is solved by splitting ${\displaystyle [n]}$ in two halves and querying to find a ${\displaystyle 1}$ in one of those and then proceeding recursively to find the exact position in the half where the query returned a ${\displaystyle 1}$. This will take ${\displaystyle 2\lceil \log {n}\rceil }$ time or if the first query is performed on the whole set, it will take ${\displaystyle \lceil \log {n}\rceil +1}$. Once a ${\displaystyle 1}$ is found, the search is then repeated after removing the ${\displaystyle i^{th}}$ co-ordinate. This can be done at most ${\displaystyle d}$ times. This justifies the running time of ${\displaystyle O(d\log {n})}$. For a full proof and an algorithm for the problem refer to: CSE545 at the University at Buffalo
### Upper bound on ${\displaystyle t(1,n)}$
${\displaystyle t(1,n)\leq \lceil \log {n}\rceil }$ This upper bound is for the special case where ${\displaystyle d=1}$ i.e. there is a maximum of 1 positive. In this case, the matrix multiplication gets simplified and the resultant ${\displaystyle \mathbf {y} }$ represents the binary representation of ${\displaystyle i}$ for test ${\displaystyle i}$. This gives a lower bound of ${\displaystyle \lceil \log {n}\rceil }$. Note that decoding becomes trivial because the binary representation of ${\displaystyle i}$ gives us the location directly. The group test matrix here is just the parity check matrix ${\displaystyle H_{m}}$ for the ${\displaystyle [2^{m}-1,2^{m}-m-1,3]}$ Hamming code.
Thus as the upper and lower bounds are the same, we have a tight bound for ${\displaystyle t(d,n)}$ when ${\displaystyle d=1}$. Such tight bounds are not known for general ${\displaystyle d}$.
### Upper bounds for non-adaptive group testing
For non-adaptive group testing upper bounds we shift focus toward disjunct matrices, which have been used for many of the bounds because of their nice properties. It has been shown that ${\displaystyle \Omega \left({\frac {d^{2}}{\log {d}}}\log {n}\right)\leq t(d,n)}$. Also for upper bounds we currently have ${\displaystyle t(d,n)\leq {\mathcal {O}}(d^{2}\log {n})}$ with a strongly explicit construction. So the smallest upper bound and biggest lower bound they are only off by a factor of ${\displaystyle 1/\log {d}}$, which is fairly small.
Separately, we can see that the current known lower bound for ${\displaystyle t(d,n)}$ is already a factor of ${\displaystyle d/\log {d}}$ larger than the upper bound for ${\displaystyle t^{a}(d,n)}$.
## Notes
1. ^ A bit more precisely – if there are an odd number of coins to be weighed, pick one to put aside and divide the rest into two equal piles. If the two piles have equal weight, the bad coin is the one put aside, otherwise the one put aside was good and no longer has to be tested.
## References
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8
• Atia, George Kamal; Saligrama, Venkatesh (2009). "Boolean Compressed Sensing and Noisy Group Testing". IEEE Transactions on Information Theory. 58 (3): 1880–1901. arXiv:. doi:10.1109/TIT.2011.2178156.
• Ding-Zhu, Du; Hwang, Frank K. (1993). Combinatorial group testing and its applications. Singapore: World Scientific. ISBN 9810212933.
• Hwang, Frank K. (September 1972). "A Method for Detecting All Defective Members in a Population by Group Testing". Journal of the American Statistical Association. 67 (339): 605–608. doi:10.2307/2284447. JSTOR 2284447.
• Chun Lam Chan; Pak Hou Che; Jaggi, Sidharth; Saligrama, Venkatesh (2011), Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms, arXiv: [cs.IT]
• Sobel, Milton; Elashoff, R. M. (1975). "Group testing with a new goal, estimation". Biometrika. 62 (1): 181–193. doi:10.1093/biomet/62.1.181.
• Aldridge, Matthew; Baldassini, Leonardo; Johnson, Oliver (June 2014). "Group Testing Algorithms: Bounds and Simulations". IEEE Transactions on Information Theory. 60 (6): 3671–3687. doi:10.1109/TIT.2014.2314472.
• Atri Rudra's course on Error Correcting Codes: Combinatorics, Algorithms, and Applications (Spring 2007), Lectures 7.
• Atri Rudra's course on Error Correcting Codes: Combinatorics, Algorithms, and Applications (Spring 2010), Lectures 10, 11, 28, 29
• Dorfman, R. (December 1943), "The Detection of Defective Members of Large Populations", The Annals of Mathematical Statistics, 14 (4): 436–440, JSTOR 2235930
• Du, D., & Hwang, F. (2006). Pooling Designs and Nonadaptive Group Testing. Boston: Twayne Publishers.
• Ely Porat, Amir Rothschild: Explicit Non-adaptive Combinatorial Group Testing Schemes. ICALP (1) 2008: 748-759
• Kagan, Eugene; Ben-gal, Irad (2014), "A group testing algorithm with online informational learning", IIE Transactions, 46: 164–184, doi:10.1080/0740817X.2013.803639, ISSN 0740-817X | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 159, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8720727562904358, "perplexity": 408.6184631114271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542712.12/warc/CC-MAIN-20161202170902-00264-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://mersenneforum.org/showthread.php?s=6611e5a07a0787c83e681835aa86f410&p=486302 | mersenneforum.org > YAFU "too few cycles, matrix probably cannot build"
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2018-04-26, 16:02 #1 EdH "Ed Hall" Dec 2009 Adirondack Mtns 71508 Posts "too few cycles, matrix probably cannot build" I'm running several instances of a variant of yoyo's yafu.pl and have discovered it's failing regularly on many 100 dd composites. I've started gathering examples. Here are five: Code: 1574511877475876335316372499005658025399466737139649097909807522789878189373960844795718204018822451 1800564888014950938521459017065544154608289307498073557047579694832535418806830055691078264692681971 1799069283919885736773557773320655283053593370861452162980412937307504604322920717391232110473296877 1941652481212430800771664311330077572073638734507074252601030360303069469052917861590445467265513617 1600108174771308311440054791086668943407407343625259307020396274538801026760430257008769115560700427 I've traced the trouble down to YAFU failing to create a matrix and simply quitting. Here's the nfs.log from one of the above numbers: Code: Thu Apr 26 11:18:34 2018 Thu Apr 26 11:18:34 2018 commencing relation filtering Thu Apr 26 11:18:34 2018 estimated available RAM is 7867.5 MB Thu Apr 26 11:18:34 2018 commencing duplicate removal, pass 1 Thu Apr 26 11:19:06 2018 found 408823 hash collisions in 4600431 relations Thu Apr 26 11:19:10 2018 added 155071 free relations Thu Apr 26 11:19:10 2018 commencing duplicate removal, pass 2 Thu Apr 26 11:19:11 2018 found 159049 duplicates and 4596453 unique relations Thu Apr 26 11:19:11 2018 memory use: 19.6 MB Thu Apr 26 11:19:11 2018 reading ideals above 100000 Thu Apr 26 11:19:11 2018 commencing singleton removal, initial pass Thu Apr 26 11:19:40 2018 memory use: 172.2 MB Thu Apr 26 11:19:40 2018 reading all ideals from disk Thu Apr 26 11:19:40 2018 memory use: 142.6 MB Thu Apr 26 11:19:40 2018 keeping 5297083 ideals with weight <= 200, target excess is 41917 Thu Apr 26 11:19:40 2018 commencing in-memory singleton removal Thu Apr 26 11:19:40 2018 begin with 4596453 relations and 5297083 unique ideals Thu Apr 26 11:19:42 2018 reduce to 1321099 relations and 1271455 ideals in 21 passes Thu Apr 26 11:19:42 2018 max relations containing the same ideal: 83 Thu Apr 26 11:19:42 2018 relations with 0 large ideals: 593 Thu Apr 26 11:19:42 2018 relations with 1 large ideals: 5128 Thu Apr 26 11:19:42 2018 relations with 2 large ideals: 34697 Thu Apr 26 11:19:42 2018 relations with 3 large ideals: 144841 Thu Apr 26 11:19:42 2018 relations with 4 large ideals: 331827 Thu Apr 26 11:19:42 2018 relations with 5 large ideals: 425278 Thu Apr 26 11:19:42 2018 relations with 6 large ideals: 270876 Thu Apr 26 11:19:42 2018 relations with 7+ large ideals: 107859 Thu Apr 26 11:19:42 2018 commencing 2-way merge Thu Apr 26 11:19:43 2018 reduce to 698957 relation sets and 649742 unique ideals Thu Apr 26 11:19:43 2018 ignored 429 oversize relation sets Thu Apr 26 11:19:43 2018 commencing full merge Thu Apr 26 11:19:51 2018 memory use: 33.7 MB Thu Apr 26 11:19:51 2018 found 70082 cycles, need 163940 Thu Apr 26 11:19:51 2018 too few cycles, matrix probably cannot build This is what I see in a terminal running YAFU directly: Code: ... nfs: commencing algebraic side lattice sieving over range: 1160000 - 1162500 nfs: commencing algebraic side lattice sieving over range: 1162500 - 1165000 nfs: commencing algebraic side lattice sieving over range: 1177500 - 1180000 nfs: commencing algebraic side lattice sieving over range: 1165000 - 1167500 nfs: commencing algebraic side lattice sieving over range: 1167500 - 1170000 nfs: commencing algebraic side lattice sieving over range: 1172500 - 1175000 nfs: commencing algebraic side lattice sieving over range: 1170000 - 1172500 nfs: commencing algebraic side lattice sieving over range: 1175000 - 1177500 total yield: 39005, q=1177507 (0.00138 sec/rel) total yield: 41529, q=1175003 (0.00138 sec/rel) total yield: 41540, q=1172503 (0.00138 sec/rel) total yield: 42503, q=1162501 (0.00138 sec/rel) total yield: 44463, q=1167539 (0.00133 sec/rel) total yield: 45453, q=1170007 (0.00135 sec/rel) total yield: 46028, q=1165001 (0.00133 sec/rel) total yield: 47561, q=1180009 (0.00133 sec/rel) nfs: commencing msieve filtering 1800564888014950938521459017065544154608289307498073557047579694832535418806830055691078264692681971 ~/Math/yafu\$ This is what I see from the perl script: Code: ... nfs: commencing algebraic side lattice sieving over range: 1167500 - 1170000 nfs: commencing msieve filtering 1600108174771308311440054791086668943407407343625259307020396274538801026760430257008769115560700427 Error, no result found This has put a big hurdle in my factordb composite work, since my machines continually try to clear these without success. Here's the YAFU header info from one of the machines: Code: 04/26/18 11:51:19 v1.35-beta @ math94, System/Build Info: Using GMP-ECM 7.0.4, Powered by GMP 6.1.2 detected Intel(R) Core(TM) i7-2600 CPU @ 3.40GHz detected L1 = 32768 bytes, L2 = 8388608 bytes, CL = 64 bytes measured cpu frequency ~= 3392.244760 using 1 random witnesses for Rabin-Miller PRP checks =============================================================== ======= Welcome to YAFU (Yet Another Factoring Utility) ======= ======= bbuhrow@gmail.com ======= ======= Type help at any time, or quit to quit ======= =============================================================== cached 78498 primes. pmax = 999983 One last observation: When YAFU quits when run via: Code: ./yafu "factor()" I can no longer type within the terminal that ran YAFU. CTRL+C functions properly, but standard letters and numbers do not appear. Thanks for any help.
2018-04-26, 17:32 #2 VBCurtis "Curtis" Feb 2005 Riverside, CA 112018 Posts The script is supposed to interpret that msieve outcome as "we need another round of sieving". I don't use the perl script, but in the python script I can edit the number of relations needed to attempt filtering; if you adjust that upward by 20% or so the error should disappear (or be 5% as common). Of course, that avoids the problem rather than cures it, as there's a bug preventing a resumption of sieving when a matrix cannot be built. Note that "filtering needs 1000000 more relations" usually means you're not close to building a matrix, while "too few cycles" usually means you're right on the cusp of a usable matrix. That's why I suggest 15-20% more relations as a way to avoid the bug.
2018-04-26, 17:33 #3 bsquared "Ben" Feb 2007 22×23×37 Posts The joys of working with branch code . The "matrix probably cannot build" message is printed by the msieve library, just prior to that library calling exit(0). In the pre-packaged binaries that I used to provide I removed all of these (yes, there are others for a variety of abnormal conditions) exit calls before linking in the library. If you are building the branch wip version of yafu then your msieve probably still has the exits. That particular one, at least in the msieve code I currently have, occurs on line 627 of merge.c. You'd have to comment out the exit and rebuild msieve, then rebuild yafu. The command line bit is something I'm working on where the yafu shell in linux will recognize backspaces to delete text, and up arrows and down arrows for command browsing. But, if yafu exits abnormally (ctrl-c, or exit()), then you get the effect you observed. The temporary fix (I hope, this is untested) would be to: 1) edit line 121 to : #if 0 //defined(__unix__) 2) edit line 355 to: #if 1 //!defined(__unix__) 3) edit line 337 to: #if 0 //defined(__unix__) Sorry! Thanks for the feedback!
2018-04-26, 17:52 #4 EdH "Ed Hall" Dec 2009 Adirondack Mtns 71508 Posts Thanks for all the help from both of you. I will look into the modifications. I do have a bit of interesting additional info, though. All the failures I referenced were from i7 2600 machines, but I tried a Core2 and it finished properly. There are three differences, (ECM, GMP, CPU): Code: 04/26/18 13:33:11 v1.35-beta @ math44, System/Build Info: Using GMP-ECM 7.0.3, Powered by GMP 6.1.1 detected Intel(R) Core(TM)2 Quad CPU Q8400 @ 2.66GHz detected L1 = 32768 bytes, L2 = 2097152 bytes, CL = 64 bytes measured cpu frequency ~= 2660.013600 using 1 random witnesses for Rabin-Miller PRP checks =============================================================== ======= Welcome to YAFU (Yet Another Factoring Utility) ======= ======= bbuhrow@gmail.com ======= ======= Type help at any time, or quit to quit ======= =============================================================== cached 78498 primes. pmax = 999983 The tail end of the run: Code: ... nfs: commencing msieve filtering 1600108174771308311440054791086668943407407343625259307020396274538801026760430257008769115560700427 nfs: commencing algebraic side lattice sieving over range: 1190000 - 1200000 nfs: commencing algebraic side lattice sieving over range: 1180000 - 1190000 total yield: 167185, q=1200007 (0.00134 sec/rel) total yield: 173738, q=1190011 (0.00133 sec/rel) nfs: commencing msieve filtering 1600108174771308311440054791086668943407407343625259307020396274538801026760430257008769115560700427 nfs: commencing msieve linear algebra linear algebra completed 238825 of 240167 dimensions (99.4%, ETA 0h 0m) nfs: commencing msieve sqrt NFS elapsed time = 4628.5065 seconds. Total factoring time = 5027.3603 seconds ***factors found*** P50 = 44073246161242488859802237851252243217008990661099 P50 = 36305657380382052089741809788137222014836299339873 1 Thanks again. Off to try the changes, as soon as I get through some other of life's delays...
2018-04-26, 18:25 #5 bsquared "Ben" Feb 2007 D4C16 Posts A thought, RE: different outcomes on different machines, is that with different thread counts, the job may proceed through different ranges of q. Since the error is related to matrices "on the cusp" of being created, small difference in ranges of q is all that may be needed to have one work and another fail.
2018-04-26, 19:04 #6
chris2be8
Sep 2009
111111011002 Posts
Quote:
Originally Posted by EdH One last observation: When YAFU quits when run via: Code: ./yafu "factor()" I can no longer type within the terminal that ran YAFU. CTRL+C functions properly, but standard letters and numbers do not appear. Thanks for any help.
Try "stty echo" (you will need to type it blind), next time you get into that state.
If that fails "stty sane" might fix it.
Chris
2018-04-26, 21:04 #7 jasonp Tribal Bullet Oct 2004 33·131 Posts The calls to exit() were a holdover from when the library was only used in one application and not the complete factoring frameworks we have now. Getting rid of them is easy but they need to be replaced with actual return paths that e.g. close any open files and propagate error information up. Not having this makes embedding libmsieve within application much harder.
2018-04-26, 22:36 #8 EdH "Ed Hall" Dec 2009 Adirondack Mtns 23×461 Posts Well, commenting out exit(0) didn't solve it. It just delayed it: Code: Thu Apr 26 14:31:22 2018 commencing linear algebra Thu Apr 26 14:31:22 2018 read 71308 cycles Thu Apr 26 14:31:22 2018 cycles contain 388962 unique relations Thu Apr 26 14:31:24 2018 read 388962 relations Thu Apr 26 14:31:24 2018 using 20 quadratic characters above 4294917295 Thu Apr 26 14:31:26 2018 building initial matrix Thu Apr 26 14:31:28 2018 memory use: 49.8 MB Thu Apr 26 14:31:28 2018 read 71308 cycles Thu Apr 26 14:31:28 2018 matrix is 161626 x 71308 (34.5 MB) with weight 10709416 (150.19/col) Thu Apr 26 14:31:28 2018 sparse part has weight 8195436 (114.93/col) Thu Apr 26 14:31:29 2018 filtering completed in 2 passes Thu Apr 26 14:31:29 2018 matrix is 148675 x 58795 (29.1 MB) with weight 8958009 (152.36/col) Thu Apr 26 14:31:29 2018 sparse part has weight 6934564 (117.94/col) Thu Apr 26 14:31:29 2018 matrix starts at (0, 0) Thu Apr 26 14:31:29 2018 matrix is 148675 x 58795 (29.1 MB) with weight 8958009 (152.36/col) Thu Apr 26 14:31:29 2018 sparse part has weight 6934564 (117.94/col) Thu Apr 26 14:31:29 2018 matrix needs more columns than rows; try adding 2-3% more relations As jasonp referred to, removing the exit(-1) in the "matrix needs more columns" conditional statement had some really "bad" effects: Code: nfs: commencing msieve linear algebra *** Error in ./yafu': free(): invalid next size (normal): 0x0000000002208d50 *** ======= Backtrace: ========= /lib/x86_64-linux-gnu/libc.so.6(+0x777e5)[0x7f57bf1a37e5] /lib/x86_64-linux-gnu/libc.so.6(+0x8037a)[0x7f57bf1ac37a] /lib/x86_64-linux-gnu/libc.so.6(cfree+0x4c)[0x7f57bf1b053c] ./yafu[0x499653] ./yafu[0x491403] ./yafu[0x448113] ./yafu[0x426e18] ./yafu[0x4179d6] ./yafu[0x4195ed] ./yafu[0x4198d8] ./yafu[0x40569d] /lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf0)[0x7f57bf14c830] ./yafu[0x406279] ======= Memory map: ======== 00400000-00561000 r-xp 00000000 08:01 2627508 /home/math94/Math/yafu/yafu 00760000-00761000 r-xp 00160000 08:01 2627508 /home/math94/Math/yafu/yafu 00761000-00771000 rwxp 00161000 08:01 2627508 /home/math94/Math/yafu/yafu 00771000-00779000 rwxp 00000000 00:00 0 00d6a000-0316f000 rwxp 00000000 00:00 0 [heap] 7f5788000000-7f5788189000 rwxp 00000000 00:00 0 7f5788189000-7f578c000000 ---p 00000000 00:00 0 7f5790000000-7f579018b000 rwxp 00000000 00:00 0 7f579018b000-7f5794000000 ---p 00000000 00:00 0 7f5798000000-7f579818d000 rwxp 00000000 00:00 0 7f579818d000-7f579c000000 ---p 00000000 00:00 0 7f57a0000000-7f57a0189000 rwxp 00000000 00:00 0 7f57a0189000-7f57a4000000 ---p 00000000 00:00 0 7f57a4000000-7f57a418d000 rwxp 00000000 00:00 0 7f57a418d000-7f57a8000000 ---p 00000000 00:00 0 7f57a8000000-7f57a818b000 rwxp 00000000 00:00 0 7f57a818b000-7f57ac000000 ---p 00000000 00:00 0 7f57b0000000-7f57b018b000 rwxp 00000000 00:00 0 7f57b018b000-7f57b4000000 ---p 00000000 00:00 0 7f57b77ff000-7f57b7800000 ---p 00000000 00:00 0 7f57b7800000-7f57b8000000 rwxp 00000000 00:00 0 7f57b8000000-7f57b8189000 rwxp 00000000 00:00 0 7f57b8189000-7f57bc000000 ---p 00000000 00:00 0 7f57bc126000-7f57bc127000 ---p 00000000 00:00 0 7f57bc127000-7f57bc927000 rwxp 00000000 00:00 0 7f57bc927000-7f57bc928000 ---p 00000000 00:00 0 7f57bc928000-7f57bd128000 rwxp 00000000 00:00 0 7f57be715000-7f57be72b000 r-xp 00000000 08:01 2888878 /lib/x86_64-linux-gnu/libgcc_s.so.1 7f57be72b000-7f57be92a000 ---p 00016000 08:01 2888878 /lib/x86_64-linux-gnu/libgcc_s.so.1 7f57be92a000-7f57be92b000 rwxp 00015000 08:01 2888878 /lib/x86_64-linux-gnu/libgcc_s.so.1 7f57be92b000-7f57be92c000 ---p 00000000 00:00 0 7f57be92c000-7f57bf12c000 rwxp 00000000 00:00 0 7f57bf12c000-7f57bf2ec000 r-xp 00000000 08:01 2888840 /lib/x86_64-linux-gnu/libc-2.23.so 7f57bf2ec000-7f57bf4ec000 ---p 001c0000 08:01 2888840 /lib/x86_64-linux-gnu/libc-2.23.so 7f57bf4ec000-7f57bf4f0000 r-xp 001c0000 08:01 2888840 /lib/x86_64-linux-gnu/libc-2.23.so 7f57bf4f0000-7f57bf4f2000 rwxp 001c4000 08:01 2888840 /lib/x86_64-linux-gnu/libc-2.23.so 7f57bf4f2000-7f57bf4f6000 rwxp 00000000 00:00 0 7f57bf4f6000-7f57bf4f9000 r-xp 00000000 08:01 2888864 /lib/x86_64-linux-gnu/libdl-2.23.so 7f57bf4f9000-7f57bf6f8000 ---p 00003000 08:01 2888864 /lib/x86_64-linux-gnu/libdl-2.23.so 7f57bf6f8000-7f57bf6f9000 r-xp 00002000 08:01 2888864 /lib/x86_64-linux-gnu/libdl-2.23.so 7f57bf6f9000-7f57bf6fa000 rwxp 00003000 08:01 2888864 /lib/x86_64-linux-gnu/libdl-2.23.so 7f57bf6fa000-7f57bf802000 r-xp 00000000 08:01 2888910 /lib/x86_64-linux-gnu/libm-2.23.so 7f57bf802000-7f57bfa01000 ---p 00108000 08:01 2888910 /lib/x86_64-linux-gnu/libm-2.23.so 7f57bfa01000-7f57bfa02000 r-xp 00107000 08:01 2888910 /lib/x86_64-linux-gnu/libm-2.23.so 7f57bfa02000-7f57bfa03000 rwxp 00108000 08:01 2888910 /lib/x86_64-linux-gnu/libm-2.23.so 7f57bfa03000-7f57bfa1b000 r-xp 00000000 08:01 2888986 /lib/x86_64-linux-gnu/libpthread-2.23.so 7f57bfa1b000-7f57bfc1a000 ---p 00018000 08:01 2888986 /lib/x86_64-linux-gnu/libpthread-2.23.so 7f57bfc1a000-7f57bfc1b000 r-xp 00017000 08:01 2888986 /lib/x86_64-linux-gnu/libpthread-2.23.so 7f57bfc1b000-7f57bfc1c000 rwxp 00018000 08:01 2888986 /lib/x86_64-linux-gnu/libpthread-2.23.so 7f57bfc1c000-7f57bfc20000 rwxp 00000000 00:00 0 7f57bfc20000-7f57bfc96000 r-xp 00000000 08:01 2635156 /usr/local/lib/libgmp.so.10.3.2 7f57bfc96000-7f57bfe96000 ---p 00076000 08:01 2635156 /usr/local/lib/libgmp.so.10.3.2 7f57bfe96000-7f57bfe97000 r-xp 00076000 08:01 2635156 /usr/local/lib/libgmp.so.10.3.2 7f57bfe97000-7f57bfe98000 rwxp 00077000 08:01 2635156 /usr/local/lib/libgmp.so.10.3.2 7f57bfe98000-7f57bfebe000 r-xp 00000000 08:01 2888812 /lib/x86_64-linux-gnu/ld-2.23.so 7f57bffa3000-7f57c00a6000 rwxp 00000000 00:00 0 7f57c00bc000-7f57c00bd000 rwxp 00000000 00:00 0 7f57c00bd000-7f57c00be000 r-xp 00025000 08:01 2888812 /lib/x86_64-linux-gnu/ld-2.23.so 7f57c00be000-7f57c00bf000 rwxp 00026000 08:01 2888812 /lib/x86_64-linux-gnu/ld-2.23.so 7f57c00bf000-7f57c00c0000 rwxp 00000000 00:00 0 7ffd4d1d1000-7ffd4d1f2000 rwxp 00000000 00:00 0 [stack] 7ffd4d1fb000-7ffd4d1fe000 r--p 00000000 00:00 0 [vvar] 7ffd4d1fe000-7ffd4d200000 r-xp 00000000 00:00 0 [vdso] ffffffffff600000-ffffffffff601000 r-xp 00000000 00:00 0 [vsyscall] Aborted (core dumped) However, trying the same number with 4 threads instead of 8 on the same machine worked. I'll try the trunk version for my 8 thread machines? Thanks everyone.
2018-04-27, 02:42 #9 EdH "Ed Hall" Dec 2009 Adirondack Mtns 23×461 Posts The /trunk svn only made it a little further: Code: Thu Apr 26 18:23:22 2018 commencing linear algebra Thu Apr 26 18:23:22 2018 read 71308 cycles Thu Apr 26 18:23:22 2018 cycles contain 388962 unique relations Thu Apr 26 18:23:24 2018 read 388962 relations Thu Apr 26 18:23:24 2018 using 20 quadratic characters above 4294917295 Thu Apr 26 18:23:26 2018 building initial matrix Thu Apr 26 18:23:29 2018 memory use: 49.8 MB Thu Apr 26 18:23:29 2018 read 71308 cycles Thu Apr 26 18:23:29 2018 matrix is 161626 x 71308 (34.5 MB) with weight 10709416 (150.19/col) Thu Apr 26 18:23:29 2018 sparse part has weight 8195436 (114.93/col) Thu Apr 26 18:23:29 2018 filtering completed in 2 passes Thu Apr 26 18:23:29 2018 matrix is 148675 x 58795 (29.1 MB) with weight 8958009 (152.36/col) Thu Apr 26 18:23:29 2018 sparse part has weight 6934564 (117.94/col) Thu Apr 26 18:23:29 2018 matrix starts at (0, 0) Thu Apr 26 18:23:29 2018 matrix is 148675 x 58795 (29.1 MB) with weight 8958009 (152.36/col) Thu Apr 26 18:23:29 2018 sparse part has weight 6934564 (117.94/col) Thu Apr 26 18:23:29 2018 matrix needs more columns than rows; try adding 2-3% more relations Thu Apr 26 18:23:29 2018 saving the first 48 matrix rows for later Thu Apr 26 18:23:29 2018 matrix includes 64 packed rows Thu Apr 26 18:23:29 2018 matrix is 148627 x 58795 (28.3 MB) with weight 7735194 (131.56/col) Thu Apr 26 18:23:29 2018 sparse part has weight 6835511 (116.26/col) Thu Apr 26 18:23:29 2018 using block size 8192 and superblock size 786432 for processor cache size 8192 kB Thu Apr 26 18:23:30 2018 commencing Lanczos iteration Thu Apr 26 18:23:30 2018 memory use: 21.9 MB Thu Apr 26 18:23:30 2018 lanczos error: submatrix is not invertible Thu Apr 26 18:23:30 2018 lanczos halted after 2 iterations (dim = 63) hard/software info: Code: 04/26/18 22:29:46 v1.34.5 @ math94, System/Build Info: Using GMP-ECM 7.0.4, Powered by GMP 6.1.2 detected Intel(R) Core(TM) i7-2600 CPU @ 3.40GHz detected L1 = 32768 bytes, L2 = 8388608 bytes, CL = 64 bytes measured cpu frequency ~= 3392.234920` Ok, I'm at a loss as to which way I go with these machines. I hate to cripple all of them to 4 cores. But, I also think I missed a lot of composites that had been processed by these and returned errors.
2018-04-27, 03:30 #10
EdH
"Ed Hall"
Dec 2009
23×461 Posts
Quote:
Originally Posted by chris2be8 Try "stty echo" (you will need to type it blind), next time you get into that state. If that fails "stty sane" might fix it. Chris
stty echo worked great. Thanks!
2018-04-27, 04:31 #11 Dubslow Basketry That Evening! "Bunslow the Bold" Jun 2011 40
Similar Threads Thread Thread Starter Forum Replies Last Post wildrabbitt Miscellaneous Math 11 2015-03-06 08:17 Andi47 Msieve 2 2010-01-11 08:50 mdettweiler Msieve 10 2009-03-17 02:38 nitai1999 Software 7 2004-08-26 18:12
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Sat Apr 17 17:09:01 UTC 2021 up 9 days, 11:49, 0 users, load averages: 1.27, 1.35, 1.47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32403722405433655, "perplexity": 28275.751133386555}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038461619.53/warc/CC-MAIN-20210417162353-20210417192353-00275.warc.gz"} |
http://mathhelpforum.com/pre-calculus/149443-need-help-finding-how-man-solutions-there-please.html | 1. ## need help finding how man solutions there are please.
how many solutions are there on the interval [0,2pi]?
cos(x)sin(x)+3cos(x)+sin(x)+3=0
Im looking at this problem and Im just lost.
2. Here is a hint
Note that it can be rewritten as
$\cos(x)\sin(x)+\sin(x)+3\cos(x)+3=0$
Now factor by grouping to get
$\sin(x)[\cos(x)+1]+3[\cos(x)+1]=(\cos(x)+1)(\sin(x)+3)=0$
Now use the zero factor principle to solve. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6404088735580444, "perplexity": 879.1654678975528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170925.44/warc/CC-MAIN-20170219104610-00328-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://www.math.gatech.edu/seminars-colloquia/series/aco-seminar/christian-borgs-20131126 | ## Convergence of sparse graphs as a problem at the intersection of graph theory, statistical physics and probability
Series:
ACO Seminar
Tuesday, November 26, 2013 - 14:05
1 hour (actually 50 minutes)
Location:
Skiles 005
,
Microsoft Research (New England), Cambridge, MA
,
Organizer:
Many real-world graphs are large and growing. This naturally raises the question of a suitable concept of graph convergence. For graphs with average degree proportional to the number of vertices (dense graphs), this question is by now quite well-studied. But real world graphs tend to be sparse, in the sense that the average or even the maximal degree is bounded by some reasonably small constant. In this talk, I study several notions of convergence for graphs of bounded degree and show that, in contrast to dense graphs, where various a priori different notions of convergence are equivalent, the corresponding notions are not equivalent for sparse graphs. I then describe a notion of convergence formulated in terms of a large deviation principle which implies all previous notions of convergence. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9261384010314941, "perplexity": 630.763117981873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864648.30/warc/CC-MAIN-20180522073245-20180522093245-00296.warc.gz"} |
https://www.nature.com/articles/s41467-019-10713-z?error=cookies_not_supported | ## Introduction
The behaviour of nanoparticles at elevated temperatures is interesting from a fundamental perspective (thermodynamics on the nanoscale) and is also relevant to functionalities such as catalysis. Specifically, it is now well-known that gold nanoparticles exhibit catalytic activity1. Understanding their morphology and atomic structure under relevant reaction conditions, such as high temperatures, may ultimately prove useful in the design of catalyst materials. The suppression of the melting point at the nanoscale2 is a longstanding issue. First predicted by Pawlow in 19083 and detected experimentally by Takagi in 1954 via shape changes in the diffraction rings of a nanoparticle ensemble4, a $$1/r$$ dependence of melting temperature at the nanoscale seems to hold true for all except few materials5,6. Early experimental observations of melting point suppression in supported Au nanoparticles were reported in the seminal electron diffraction study by Buffat and Borel7, as well as in a transmission electron microscope (TEM) evaporation rate investigation by Sambles8. Subsequently, there have been several experimental studies of the high temperature behaviour of gold nanoparticles. However, at present there is only very limited single particle time-resolved data on melting, obtained by evaporation at fixed temperature9, with no data below 5 nm—the catalytic size regime10. In addition, because previously reported experimental studies of Au nanoparticle melting do not track individual particles in real space as the temperature is increased (instead they use static temperature evaporation or ensemble diffraction methods), the exact mechanism by which melting occurs remains unresolved, such as whether a surface liquid layer is formed.
There are several theoretical models for melting point suppression at the nanoscale. Pawlow’s theory is a thermodynamic model based on the triple point equilibrium of spherical solid and liquid particles of the same material and equal mass surrounded by their vapour. The liquid shell model, liquid nucleation and growth model, numerous molecular dynamics (MD) studies of Au11,12,13,14,15 and other metal16,17,18 nanoparticles predict the formation of a liquid shell as a mechanism for nanoparticle melting. The liquid shell model, first suggested by Reiss and Wilson19 and developed by others20,21, including Sambles8, is a thermodynamic model that assumes a solid core surrounded by a liquid shell of constant thickness in the proximity of the melting temperature. The liquid nucleation and growth model, proposed by Couchman and Jesser22, is based on nucleation theory and describes melting as the nucleation of a liquid shell on the surface of the nanoparticle which then spreads into the core until a critical radius is reached and the whole particle melts. Thus, the model predicts a melting region in which there is solid–liquid coexistence. There has been no experimental observation of the existence of a liquid shell prior to melting for Au nanoparticles. Young et al.23 reported surface roughening (amorphous regions) in 10.2 nm diameter Au particles at 600 °C, however, a liquid shell was not observed. Such a solid core, liquid shell structure and thus solid–liquid coexistence have been reported in electron microscopy investigations of embedded lead24, and polymer-capped platinum particles25, but without atomic resolution, and most recently for large gallium nanoparticles at fixed (room) temperature26.
Here we observe the behaviour of individual, supported, size-selected Au nanoclusters (≤5 nm) in real space with atomic resolution as their temperature is increased from 20 °C up to 1000 °C. We employ an in situ heating stage in the aberration-corrected scanning transmission electron microscope (ac-STEM). Single particle measurements of the surface melting temperature on amorphous carbon supports are made by a unique method in which shape changes of the nanoparticles are observed as the temperature is ramped. The results show an approximately $$1/r$$ dependence on the surface melting temperature, in agreement with Pawlow’s triple point model and the liquid nucleation and growth model. However, the cluster core-melting temperature is much higher than predicted by the models of melting point suppression. The ac-STEM images further reveal the formation of a quasi-liquid shell that persists over a range of temperatures. Large-scale ab initio MD simulations of cuboctahedral 561-atom nanoclusters at temperatures in the experimental range address the previously neglected effect of the support on the melting of the nanoparticles. The theoretical results support the experimental observations of solid core-liquid shell coexistence and agree with the measured surface and core-melting points if the surface is understood to constrain the facet bound to it.
## Results
### Experimental
Figure 1a shows a Au561 particle that is heated incrementally from 550 to 857 °C. The frames shown in the figure are taken from a series of 22 high-angle annular dark-field (HAADF) STEM images of this particle. The shape of the particle is first changed at 657 °C, where there is a protrusion from the cluster surface, which we take to be indicative of surface melting. For this particle the surface melting temperature is recorded as 654 ± 4 °C. Similarly, Fig. 1b shows a Au2530 particle formed by aggregation heated from 556 to 1000 °C. In this case the change in shape first appears at 801 °C, and the surface melting temperature is recorded as 800 ± 1 °C (i.e. between images recorded at 799 and 801 °C). The errors stated here do not include the potential systematic error of <5% arising from the heating chip calibration.
There are at least two ways in which the melting temperature of a nanoparticle could be measured using in situ heating in the STEM. The first, as employed conventionally, would be to use the loss of atomic structure in the images as an indication of melting9. The second, proposed here, is to use changes in shape—a method that arises directly from the quality of data collected here (although shape changes have been previously used to describe melting in field emission microscopy experiments27). In our study the criterion of loss of atomic structure is not suitable due to the time resolution of the experiments (each image taking 5.4 s to record). The phenomenon of quasi-melting below the melting point, where the particle structure rapidly fluctuates, has been reported in both experimental and theoretical studies28,29. If this were to occur, the rapidly fluctuating particle would likely appear amorphous in the recorded STEM image and could be misinterpreted as melted. The shape change method attributes changes in shape in the projected images to diffusion of atoms in the molten surface layers. MD simulations of cluster surface melting show that diffusion of atoms results in the formation of anisotropic cluster shapes11,13,15, such as those seen in our experimental STEM images. The association between shape changes and peripheral melting in these papers is notable, and consistent with our interpretation of our observations. At still more elevated temperatures more dramatic shape changes are sometimes found.
Figure 2 shows the results of our single particle analysis of the surface melting temperature of the Au clusters as a function of the reciprocal radius. A clear inverse correlation is seen. Where possible, cluster core (i.e., complete) melting temperatures are also shown, measured by observing the loss of core atomic structure. The scatter points are measurements of size-selected Au309 and Au561 particles and aggregated Au1110, Au2530, and Au3390 particles. The error on the surface melting temperature includes both the error arising from the temperature window and the 5% heating chip calibration error, as described in the Methods section. The particle radius was calculated by making four separate measurements of the average particle diameter, then taking the mean value; the error is given by the maximum and minimum of these values, shown by error bars in the figure.
Also shown in Fig. 2 are Pawlow’s triple point model3, the liquid shell model8 and the liquid nucleation and growth (LNG) model22 are also plotted for comparison (for details see Supplementary Methods and Supplementary Fig. 1). For the liquid shell model, the smallest possible shell thickness of 2.7 Å (the atomic diameter) is used. Pawlow’s model is for complete (i.e. including core) particle melting; thus, it is reasonable to assume that our surface melting measurements should fall below those predicted by this model. This is true for all but one particle (Au1110). However, the experimental core-melting temperatures are much higher than predicted. The liquid shell model predicts significantly lower whole particle melting temperatures than Pawlow across our size range, even more so for small particles. The melting temperatures predicted are much lower than the observed core and even the observed surface melting temperatures. Moreover, additional ac-STEM images (see below) show a liquid shell thickness which varies between particles and is generally greater than the value of 1 atomic layer embedded in the theory curve. Increasing the shell thickness in the liquid shell model amplifies the melting point suppression for small particles sizes, which is not mirrored by the experimental data. Indeed, the data are well fitted by a straight line ($$T_m = - 904/r + 1518$$, R2 = 0.9). The liquid nucleation and growth model predicts a melting region; the lower boundary is the onset of surface melting and the upper boundary is the point by which the complete particle melts. Within this region solid and liquid phases coexist. Again, our surface and core-melting temperature measurements are higher than those predicted, but this LNG model is in the best qualitative agreement with our data. It captures the experimental result that surface melting precedes core melting, so there is a solid–liquid coexistence which occurs over a range of temperatures. The size of our solid–liquid coexistence region is in very good agreement with the coexistence region predicted by the LNG model, deviating by ~20 °C. The different melting temperatures observed for the different 561-atom nanoclusters is a noteworthy result of our paper. The differences between the core-melting temperatures in Fig. 2 could be due to specific structural characteristics of each examined nanoparticle, whether intrinsic (e.g. we sometimes observe twin structures as shown by the power spectrum fast Fourier transform (FFT) image in Supplementary Fig. 2, and also observable by eye) or extrinsic (such as different orientations of the cluster on the support).
To check the surface melting temperatures obtained from the observed onset of the cluster shape changes, we developed a numerical approach, based on tracking the curvature along the perimeter of the nanoclusters. This is described in detail in the Supplementary Methods and shown in Supplementary Fig. 3 for a 561- and a 1110-atom nanocluster. This approach is in good agreement with the observations by eye, yielding results which are either identical or deviate by only a small margin. The results of the two methods for the 561-, 1110-, 2530- and 3390-atom nanoclusters are shown in Table 1. The average of the melting temperature for the 561-atom nanoclusters obtained by observation is 619 °C, while the average by the numerical method is 603 °C, a difference of only 16 °C.
Furthermore, the coexistence of an amorphous melted surface with a solid crystalline core can be verified by power spectrum FFT images of multiple areas of each nanoparticle. In Fig. 3, experimental images of a nanoparticle at 600 and 657 °C are shown with the power spectra obtained from the marked regions. The additional spots in Fig. 3a power spectra indicate a degree of crystallinity in the shell. In Fig. 3b, it is obvious that the central part of the nanoparticle is still crystalline, while the surface areas are not (lack of peripheral spots).
Figure 4 shows two series of HAADF STEM images, which further demonstrate the formation of a quasi-liquid non-uniform layer at the cluster surface and the coexistence of solid and liquid phases at high temperatures. Figure 4a, for a Au561 particle above its surface melting point, exhibits a surface protrusion at 650 °C indicating the surface is molten. At 800 °C, a complete liquid shell-solid core structure is visible. Figure 4b shows a cuboctahedral Au1110 particle which is solid at 704 °C and exhibits dramatic shape change at 801 °C. Again, we see a solid core-uneven liquid shell structure. The corresponding profile plots in Fig. 4c, d indicate the fact in atom density at the edge of the solid core. In both clusters the core atomic structure persists after surface melting. The larger Au1110 particle shows some face-centred cubic (fcc) structure in the shell too. We found no evidence of a rapidly fluctuating quasi-molten state23,28,29,30 nor an icosahedral or icosahedral solid–liquid coexistence state prior to melting12,31,32.
The degree of formation of the solid core-liquid shell structure (as seen in Fig. 4) does vary from cluster to cluster, but in all cases melting initiates at the surface. The notable delay between cluster surface and cluster core melting is consistent with the liquid nucleation and growth model, however, a comparison of our experimentally measured critical core radii (Supplementary Fig. 1) with the model show some significant differences. Experimentally, we obtain a notably lower limit on the critical core radius and a much higher upper limit on the core-melting temperature (see Fig. 2).
### Theoretical
An intriguing issue raised by the mismatch between the experimental and conventional models of nanoscale melting is the influence of the substrate. In order to explore the effect of the carbon support on the melting of the clusters, we adopt an ab initio computational approach with two sets of density functional theory molecular dynamics (DFT-MD) simulations: one where all the atoms of the nanocluster are free to move and one where the atoms of one (100) facet of the nanocluster are fixed in place during the simulations. We proceed to extract the degree of crystallinity of the nanoclusters during the last steps of the simulations. The results for all the investigated cases and analysis algorithms applied are shown in Fig. 5. The calculations give a sizable difference in the behaviour and melting temperatures for the free nanocluster and the nanocluster with a frozen facet. The two types of nanoclusters are almost identical at room temperature but display major differences at elevated temperatures. The freezing of the facet, i.e. the simulated influence of the support, is found to delay the melting of the nanocluster considerably.
We use the criterion that surface melting occurs when the count of the atoms recognised as fcc and their 12 nearest neighbours is <309, i.e. one atomic layer is melted. This critical point is identified at a temperature of 500 °C for the free and 586 °C for the frozen-facet Au561 clusters, with the average of the experimental measurements at 619 °C lying closer to the frozen-facet case. The surface melting temperature difference between the free and frozen-facet nanoclusters is 86 °C. If we proceed further and check where the aforementioned count is lower than 147 atoms, we get temperatures of 680 and 822 °C for the free and frozen-facet clusters, respectively. For cluster core melting we use the criterion that the number of atoms that belong to an fcc arrangement is lower than 13, which corresponds to the smallest possible crystalline magic-number nanocluster. Using this threshold, we observe cluster core melting at 807 and 951 °C for the free and frozen-facet nanoclusters, respectively, a temperature difference of 144 °C. These results are in good agreement with the average core-melting temperature for 561-atom nanoclusters shown in Fig. 2, which is 878 °C, between the core-melting temperatures of the free and frozen-facet nanoclusters.
We also note that a solid core is observed in the nanoclusters after the melting process, located at the centre of the nanocluster for the free nanoclusters, as shown in Fig. 6a. The core is of larger size and displaced towards the frozen facet in the other nanoclusters, as shown in Fig. 6b, revealing another effect of the support on the melting process. The solid–liquid coexistence interval is found to be 307 and 365 °C for the free and frozen-facet cases, respectively, higher than the 259 °C found by the experimental observations shown in Fig. 2. This is possibly due to the lower surface melting temperatures obtained by the simulations compared with the experiments.
## Discussion
We have employed in situ heating experiments on sub-5 nm size-selected Au nanoparticles supported on a-C in the aberration-corrected STEM, demonstrating melting initiated at the cluster surface as well as size-dependent melting point suppression. The surface melting temperature of the nanoclusters was measured using a method—the observation of nanoparticle shape changes at high temperatures. We have observed single particle surface melting in real space, identified the formation of a solid core-liquid shell structure at elevated temperatures and proved the coexistence of solid and liquid phases. The mechanism is broadly consistent with the liquid nucleation and growth model, with good agreement for the solid–liquid coexistence interval, but notably higher surface and cluster core-melting temperatures.
The influence of the support is apparent from large-scale ab initio simulations that mimic the effect of cluster adsorption by a frozen-facet approach. Quantitative agreement between experiment and theory is obtained when the surface is introduced in this way. The theoretical results confirm and illuminate our interpretation of the experimental results. Specifically, the melting point suppression and the coexistence of a melted shell and a crystalline core at elevated temperatures are both confirmed. The introduction of the frozen-facet approximation to cluster adsorption leads to core-melting temperatures for both the free and frozen-facet clusters which are notably higher than the standard models but in good agreement with the experimental results. Moreover, we find that the solid core in the coexistence region is enlarged due to the effect of the support, and displaced towards the surface. We therefore conclude that the main origin of the deviations of the experimental work from the previous models lies in the effect of the cluster-support interaction and is well captured by the frozen-facet model. Since the Au particles studied here lie in the catalytically active size regime, the observations also raise the question of whether the liquid surface affects the catalytic activity at high temperatures.
## Methods
### Experimental
The size-selected Au nanoclusters were prepared using a magnetron sputtering, gas aggregation cluster source33 with lateral time of flight mass filter (M/∆M = 22)34. Clusters containing either 309 ± 7 or 561 ± 13 Au atoms were deposited onto amorphous carbon membranes of the heating chips, using low deposition energy (soft landing35) to preserve their original structures. STEM imaging was performed using a 200 keV JEOL 2100F instrument with spherical aberration corrector (CEOS). A HAADF detector, with inner collection angle of 62 mrad, as well as a bright field detector were employed for imaging. Use of the HAADF detector enabled any aggregate particles to be sized accurately by using the size-selected clusters as ‘mass-standards’36. Imaging was performed at an electron dose of $$2.5 \times 10^4\,e^ - \cdot {\mathrm{{\AA}}}^{ - 2} \cdot {\mathrm{frame}}^{ - 1}$$ for the 309-atom nanoclusters and $$4.8 \times 10^4\,e^ - \cdot {\mathrm{{\AA}}}^{ - 2} \cdot {\mathrm{frame}}^{ - 1}$$ for the 561-, 1110-, 2530- and 3390-atom nanoclusters.
For in situ heating a high temperature heating holder (DENS Solutions Wildfire) was used in conjunction with MEMS-based heating chips. The chips featured 5-nm-thick amorphous carbon membranes, on which the Au nanoparticles were deposited. They were heated by applying a current to a metal heater coil embedded in the chip. Heating experiments were performed by STEM imaging of individual particles at incrementally increasing temperatures. The surface melting temperature of the nanoparticles was determined by observing the onset of shape changes (shape changes due to rotations of the particle were excluded by comparison against simulation atlases37 showing particles at different angles of orientation). When a shape change was observed, the surface melting temperature for that particle was recorded as the average between the temperatures of the last observed particle of the original-shape and that of the first changed shape. In order to check the assignments of the surface melting temperatures obtained from visual inspection of the images, a numerical method was developed to assess protrusions and dents on the nanoclusters. It tracks the curvature along the perimeter of the nanoclusters. The method identifies surface alterations which show significant deviation from quasi-spherical shape. The occurrence of such deviations is associated with surface melting of the nanoclusters (further details on the method are given in the Supplementary Methods).
The precision of the measured surface melting temperature is a function of the temperature window in which the shape change occurs and the temperature stability of the MEMS heating chip, which is essentially negligible in comparison (<0.1 °C). Another source of error arises from the calibration of the heating chip; there may be a systematic error—an offset of up to 5% on the stated temperature. This systematic error effects the accuracy of all temperature measurements. However, so long as measurements are made using the same heating chip the general trend (melting temperature vs particle size) should not be affected.
### Theoretical
The DFT-MD calculations were performed using the Vienna Ab initio Simulation Package (VASP)38,39 under the local density approximation (LDA)40 with projector-augmented wave (PAW) pseudopotentials41,42. A canonical ensemble under Born–Oppenheimer MD was simulated with a time step of 2 fs to integrate the equations of motion using the algorithm of Nosé43,44,45, with a Nosé mass equal to 40 time steps. The energy cut-off of the plane-wave basis set was 240 eV and the tolerance for self-consistency for the electronic steps was set at 10−6 eV. The lattice constant of the 8-atom Au unit cell was found to be equal to 4.057 Å. The Brillouin zone was sampled using a Γ-centred 5 × 5 × 5 mesh for the unit cell, scaled accordingly for the large-scale calculations.
Magic-number cuboctahedral nanoclusters, chosen on the basis that they have been proven to be of the greatest abundance and lower energy amongst the three magic-number structural isomers of nanoparticles46, that consisted of 561 atoms were constructed for the simulations. After an initial structural relaxation to obtain the ground-state configuration for the nanoclusters at T = 0 K, separate NVT simulations of the temporal behaviour were performed at temperatures from 300 to 1200 K at 150 K intervals. The total duration of each simulation was 2.4 ps. The optimized width of the vacuum surrounding the nanoclusters was equal to 14 Å in all directions, which ensured that there was no interaction between the nanocluster and its nearest image and took into account the thermal expansion identified during the simulations.
The simulation approach we followed in this work was based on two sets of DFT-MD simulations. In one set all the atoms of the nanocluster were free to move (free clusters). In the other the atoms of one (100) facet of the nanocluster, consisting of 36 atoms for the 561-atom nanocluster, were fixed in place during the simulations to capture the effect of the surface on the melting of the clusters. A Common Neighbour Analysis (CNA)47 using a fixed cut-off radius equal to 3.4629 Å was performed with the OVITO software48 in order to extract the degree of crystallinity of the nanocluster as a qualitative measure of the melting process. Statistical averages were collected over the last 600 steps of the simulations. It should be noted that the atoms belonging to the outer layer of the nanocluster, i.e. 252 atoms for the 561-atom nanocluster, were always recognized as non-crystalline by the CNA. In order to assess the surface melting in a better way, the atom selection was expanded to include the atoms recognised to be in an fcc crystalline arrangement by the CNA algorithm and each of their 12 nearest neighbours (each atom was counted only once). In this way the initial count was equal to 561. Thus, the modified CNA approach was used to extract the surface melting temperatures, whereas the CNA approach was used to extract the cluster core-melting temperatures. We adopt the criterion that surface melting occurs when the count of the atoms obtained by the modified CNA approach is <309, i.e. one atomic layer is melted, while for cluster core melting we adopt the criterion that the count of atoms obtained by the CNA approach is lower than 13.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8420872092247009, "perplexity": 1286.9414486431103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335276.85/warc/CC-MAIN-20220928180732-20220928210732-00076.warc.gz"} |
http://mathhelpforum.com/calculus/134195-y-sqrt2x-2sqrt2x.html | # Thread: y= sqrt2x + 2sqrt2x
1. ## y= sqrt2x + 2sqrt2x
y= sqrt2x + 2sqrt2x
I need to find dy/dx,
When i differentiate each part I get the same result which I know is not correct,
y = 2x^-1/2 + 2(x^-1/2)
dy/dx = 1/2sqrtx + 1/2sqrtx
any help would be good
2. Originally Posted by drb11
y= sqrt2x + 2sqrt2x
I need to find dy/dx,
When i differentiate each part I get the same result which I know is not correct,
y = 2x^-1/2 + 2(x^-1/2) $\displaystyle \longleftarrow$ should be $\displaystyle y = (2x)^{1/2} + 2(2x)^{1/2}$
. . .
3. well using the power rule:
$\displaystyle f'(x)=nx^{n-1}$
We see in your problem:
$\displaystyle y= \sqrt{2x} + 2\sqrt{2x}$
1.
$\displaystyle y= (2x)^{1/2}+2(2x)^{1/2}$
2.
$\displaystyle y= (x)^{-1/2}+2(x)^{-1/2}$
3.
$\displaystyle y= x^{-1/2}+2x^{-1/2}$
4. [quote=Zanderist;475964]well using the power rule:
$\displaystyle f'(x)=nx^{n-1}$
We see in your problem:
$\displaystyle y= \sqrt{2x} + 2\sqrt{2x}$
$\displaystyle y= (2x)^{1/2}+2(2x)^{1/2}$
$\displaystyle y'= (x)^{-1/2}+2(x)^{-1/2}$ $\displaystyle \longleftarrow$ should be $\displaystyle y'= (2x)^{-1/2}+2(2x)^{-1/2}$
[\quote]
. . .
5. You are aware that $\displaystyle \sqrt{2x}+ 2\sqrt{2x}= 3\sqrt{2x}$ aren't you? Its derivative is $\displaystyle \frac{3}{2}\sqrt{2}x^{-1/2}= \frac{3}{2}\sqrt{\frac{2}{x}}$.
6. oops, it seems I made a mistake when posting the initial problem,
it was supposed to be:
y = sqrt2x + 2sqrtx
sorry about that
I know what the answer is but I don't know how to get there....
7. Originally Posted by drb11
oops, it seems I made a mistake when posting the initial problem,
it was supposed to be:
y = sqrt2x + 2sqrtx
sorry about that
I know what the answer is but I don't know how to get there....
$\displaystyle \sqrt{2x}+2\sqrt{x}=(\sqrt{2}+2)\sqrt{x}$
8. Sweet, thanks
9. Originally Posted by drb11
I know what the answer is but I don't know how to get there....
We see people writing this all the time. However you do not know what the answer is all you know is what it says at the back of the book. Which is not the same thing at all.
CB
10. Sorry captainblack sparrow. The reason I asked the question; for help on the process on how to get to the correct solution.I thought this was a math help forum ,obviously not, oh well, please revoke my membership. Feel free to feed upon my ego
11. Originally Posted by drb11
Sorry captainblack sparrow. The reason I asked the question; for help on the process on how to get to the correct solution.I thought this was a math help forum ,obviously not, oh well, please revoke my membership. Feel free to feed upon my ego
You had your help.
But it seems that if we comment on your sloppy use of language you take your balls back and run off. Now who is the loser, you want help, we give it (or rather DeDust, Halls, Zanderist and Drexel28 give it), but you won't play any more because I wish to correct a ridiculous use of language that I see all the time.
Ohhh.. well, my loss I suppose. Goodbye
CB | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7967819571495056, "perplexity": 1369.048465928015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867417.71/warc/CC-MAIN-20180526112331-20180526132331-00434.warc.gz"} |
http://cds.cern.ch/collection/SPS%20and%20CNGS?ln=fr | # SPS and CNGS
Derniers ajouts:
2018-09-19
13:49
Effets chromatiques et passage de la transition à grand $\Delta.p/p$ / Faugeras, P E ; Faugier, A ; Gareyte, J CERN-SPS-Improvement-Report-130. - 1978. - 12 p. Full text
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09:31
Effective spill times, RF and LQE servo / Actis, P ; Burnod, L ; d'Amico, E ; Ferioli, G ; Mann, J ; de Vries, J CERN-SPS-Improvement-Report-126. - 1978. - 8 p. Full text
2018-09-11
11:17
Vacuum tubes for kickers in emittance measure line : Visit to Ferranti Ltd (Manchester) / Pearce, N CERN-SI-Mi-ME-70-02. - 1970. - Mult. p.
2018-09-08
06:01
Status and Prospects for the AWAKE Experiment / Turner, Marlene (CERN) /AWAKE Collaboration The AWAKE Collaboration is pursuing a demonstration of proton-driven plasma wakefield acceleration of electrons. The AWAKE experiment uses a \SI{400}{GeV/c} proton bunch from the CERN SPS, with a rms bunch length of $6$-\SI{15}{cm}, to drive wakefields in a \SI{10}{m} long rubidium plasma with an electron density of $10^{14}-10^{15}$cm$^{-3}$ [...] arXiv:1809.01813.- 2018-06 - 4 p. - Published in : 10.18429/JACoW-IPAC2018-TUXGBE1 Fulltext: PDF; Preprint: PDF; In : 9th International Particle Accelerator Conference, Vancouver, Canada, 29 Apr - 4 May 2018, pp.TUXGBE1
2018-09-01
06:30
Review of hydrodynamic tunneling issues in high power particle accelerators / Tahir, N A (Darmstadt, GSI) ; Burkart, F (CERN) ; Schmidt, R (CERN) ; Shutov, A (IPCP, Chernogolovka) ; Piriz, A R (Castilla-La Mancha U.) Full impact of one Large Hadron Collider (LHC) 7 TeV proton beam on solid targets made of different materials including copper and carbon, was simulated using an energy deposition code, FLUKA and a two-dimensional hydrodynamic code, BIG2, iteratively. These studies showed that the penetration depth of the entire beam comprised of 2808 proton bunches significantly increases due to a phenomenon named hydrodynamic tunneling of the protons and the shower. [...] 2018 - 17 p. - Published in : Nucl. Instrum. Methods Phys. Res., B 427 (2018) 70-86
2018-08-28
08:12
A method for radiological characterization based on fluence conversion coefficients / Froeschl, Robert (CERN) Radiological characterization of components in accelerator environments is often required to ensure adequate radiation protection during maintenance, transport and handling as well as for the selection of the proper disposal pathway. The relevant quantities are typical the weighted sums of specific activities with radionuclide-specific weighting coefficients. [...] 2018 - 10 p. - Published in : J. Phys. : Conf. Ser. 1046 (2018) 012006 Fulltext: pdf - PDF; 10.1088_1742-6596_1046_1_012006 - PDF; In : 4th International Workshop on Accelerator Radiation Induced Activation, Lund, Sweden, 22 - 24 May 2017, pp.012006
2018-08-28
08:12
Measurements and FLUKA Simulations of Bismuth, Aluminium and Indium Activation at the upgraded CERN Shielding Benchmark Facility (CSBF) / Iliopoulou, E (CERN) ; Bamidis, P ; Brugger, M (CERN) ; Froeschl, R ; Infantino, A ; Kajimoto, T ; Nakao, N ; Roesler, S ; Sanami, T ; Siountas, A et al. The CERN High energy AcceleRator Mixed field (CHARM) facility is situated in the CERN Proton Synchrotron (PS) East Experimental Area. The facility receives a pulsed proton beam from the CERN PS with a beam momentum of 24 GeV/c with $5 \cdot 10^{11}$ protons per pulse with a pulse length of 350 ms and with a maximum average beam intensity of $6.7 \cdot 10^{10}$ protons per second. [...] 2018 - 14 p. - Published in : J. Phys. : Conf. Ser. 1046 (2018) 012004 Fulltext: 10.1088_1742-6596_1046_1_012004 - PDF; pdf - PDF; In : 4th International Workshop on Accelerator Radiation Induced Activation, Lund, Sweden, 22 - 24 May 2017, pp.012004
2018-08-23
18:25
HIE-ISOLDE : technical design report for the energy upgrade / Kadi, Y ; Fraser, M A (ed.) ; Papageorgiou-Koufidou, A (ed.) The Isotope mass Separator On-Line facility (ISOLDE) at CERN occupies a leading position in the field of radioactive ion beams research, as it can produce the largest range of isotopes worldwide —over 1000 isotopes of more than 70 elements [...] CERN-2018-002-M. - Geneva : CERN, 2018. - 240 p. (CERN Yellow Reports: Monographs ; 1/2018)
10.23731/CYRM-2018-001
Full text
2018-08-23
17:41
First magnetic measurements of fast-ramping dipole DHPTB102 of BTF upgraded beam-lines / Valente, Paolo (INFN Roma) ; Armenti, Gustavo (INFN LNF) ; Bolli, Bruno (INFN LNF) ; Casamatta, Alberto (INFN LNF) ; Di Pasquale, Enrico (INFN LNF) ; Iungo, Franco (INFN LNF) ; Lauciani, Stefano (INFN LNF) ; Martelli, Stefano (INFN LNF) ; Paris, Marco (INFN LNF) ; Pellegrino, Luigi (INFN LNF) et al. AIDA-2020-NOTE-2018-003; INFN–18/08/LNF.- Geneva : CERN, 2018 Fulltext: PDF;
2018-08-23
16:18
Analysis of images of a self-modulated proton bunch exiting a plasma in AWAKE / Andreini, Paul (2018 CERN Summer Student) ; Muggli, Patric (AWAKE Scientist) We have analyzed data recorded by the streak cameras at the CERN AWAKE experiment on 10 September 2017. [...] CERN-STUDENTS-Note-2018-063. - 2018. Access to fulltext | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8006538152694702, "perplexity": 21046.759189223798}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159359.58/warc/CC-MAIN-20180923114712-20180923135112-00498.warc.gz"} |
http://docs.duckietown.org/DT19/downloads/AIDO/docs-AIDO/branch/jzilly001/AIDO/out/amod_baselines.html | Home
build details
# Agent Template for AMOD
Modified 2019-04-17 by Liam Paull
If you have not done this yet, study the general introduction to the task and the explanation of the performance objectives on these two pages:
Please briefly study the our client-server protocol which is described in our github repository (link below). It details the communication between the Aido-Host container in which the autonomous mobility-on-demand (AMoD) simulation is running and the Aido-Guest container in which you can implement your fleet operational policy:
aido-client-protocol.md
Next, you can use one of the baselines that we provided to understand how an Aido-Guest to manage an AMoD task is actually implemented. We provide one baseline in JAVA and one in Python. Both do not represent well-performing policies but are simply demonstrations of how an Aido-Guest could be implemented, therefore you should rapidly be able to beat their performance.
No questions found. You can ask a question on the website. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37181928753852844, "perplexity": 1740.9650018532507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527396.78/warc/CC-MAIN-20190721225759-20190722011759-00252.warc.gz"} |
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http://kenbenoit.net/assets/courses/dmslTCD2015/Days5-6_Exercise.html | ** Due: April 3, 2015**
### Working with Textual Data
This exercise is designed to get you working with quanteda. The focus will be on exploring the package and getting some texts into the corpus object format. quanteda package has several functions for creating a corpus of texts which we will use in this exercise.
1. Getting Started.
You can use R or Rstudio for these exercises. You will first need to install the package,
using the commands below. Also see the instructions for installation from the dev branch page of http://github.com/kbenoit/quanteda.
# needs the devtools package for this to work
if (!require(devtools)) install.packages("devtools", dependencies=TRUE)
# be sure to install the latest version from GitHub, using dev branch:
# if this fails remove the ref="dev" part
devtools::install_github("quantedaData", username="kbenoit")
2. Exploring quanteda functions.
You can try running demo(quanteda), and also use the example() function for any function in the package, to run the examples and see how the function works. Of course you should also browse the documentation, especially ?corpus to see the structure and operations of how to construct a corpus.
3. Making a corpus and corpus structure
1. From a vector of texts already in memory.
The simplest way to create a corpus is to use a vector of texts already present in R’s global environment. Some text and corpus objects are built into the package, for example inaugTexts is the UTF-8 encoded set of 57 presidential inaugural addresses. Try using corpus() on this set of texts to create a corpus.
Once you have constructed this corpus, use the summary() method to see a brief description of the corpus. The names of the character vector inaugTexts should have become the document names.
2. From a directory of text files.
The corpus() function can take as its main argument the name of a directory, if you wrap the path to the directory within a directory() call. (See ?directory for an example.) If you call directory() with no arguments, then it should allow you to choose the directory interactively (you will need to have installed the tcltk2 package first though.)
Here you are encouraged to select any directory of plain text files of your own.
How did it work? Try using docvars() to assign a set of document-level variables.
Note that if you document level metadata in your filenames, then this can be automatically parsed by corpus.directory() into docvars.
require(quanteda)
## Loading required package: quanteda
mydir <- textfile("~/Dropbox/QUANTESS/corpora/ukManRenamed/*.txt")
mycorpus <- corpus(mydir)
summary(mycorpus, 5)
## Corpus consisting of 101 documents, showing 5 documents.
##
## Text Types Tokens Sentences
## UK_natl_1945_en_Con.txt 1578 6095 275
## UK_natl_1945_en_Lab.txt 1258 4975 241
## UK_natl_1945_en_Lib.txt 1061 3377 158
## UK_natl_1950_en_Con.txt 1806 7411 381
## UK_natl_1950_en_Lab.txt 1342 4879 275
##
## Source: /Users/kbenoit/Dropbox/Classes/Trinity/Data Mining 2015/Notes/Day 6 - Text/* on x86_64 by kbenoit.
## Created: Mon Mar 23 22:11:21 2015.
## Notes: .
3. From .csv or .json files — see the documentation with ?textfile.
4. Explore some phrases in the text.
You can do this using the kwic (for “key-words-in-context”) to explore a specific word or phrase.
kwic(inaugCorpus, "terror", 3)
## preword word
## [1797-Adams, 1183] or violence, by terror,
## [1933-Roosevelt, 100] nameless, unreasoning, unjustified terror
## [1941-Roosevelt, 252] by a fatalistic terror,
## [1961-Kennedy, 763] uncertain balance of terror
## [1961-Kennedy, 872] instead of its terrors.
## [1981-Reagan, 691] Americans from the terror
## [1981-Reagan, 1891] those who practice terrorism
## [1997-Clinton, 929] the fanaticism of terror.
## [1997-Clinton, 1462] strong defense against terror
## [2009-Obama, 1433] aims by inducing terror
## postword
## [1797-Adams, 1183] intrigue, or venality,
## [1933-Roosevelt, 100] which paralyzes needed
## [1941-Roosevelt, 252] we proved that
## [1961-Kennedy, 763] that stays the
## [1961-Kennedy, 872] Together let us
## [1981-Reagan, 691] of runaway living
## [1981-Reagan, 1891] and prey upon
## [1997-Clinton, 929] And they torment
## [1997-Clinton, 1462] and destruction. Our
## [2009-Obama, 1433] and slaughtering innocents,
Try substituting your own search terms, or working with your own corpus.
5. Create a document-feature matrix, using dfm. First, read the documentation using ?dfm to see the available options.
mydfm <- dfm(inaugCorpus, ignoredFeatures=stopwords("english"))
## Creating a dfm from a corpus ...
## ... indexing 57 documents
## ... tokenizing texts, found 134,142 total tokens
## ... cleaning the tokens, 461 removed entirely
## ... ignoring 174 feature types, discarding 69,005 total features (51.6%)
## ... summing tokens by document
## ... indexing 9,085 feature types
## ... building sparse matrix
## ... created a 57 x 9085 sparse dfm
## ... complete. Elapsed time: 3.366 seconds.
dim(mydfm)
## [1] 57 9085
topfeatures(mydfm, 20)
## will people government us can upon
## 871 564 561 476 470 371
## must may great states shall world
## 363 338 334 331 314 305
## country every nation peace one new
## 294 291 287 253 244 241
## power public
## 232 223
Experiment with different dfm options, such as stem=TRUE. The function trim() allows you to reduce the size of the dfm following its construction.
Grouping on a variable is an excellent feature of dfm(), in fact one of my favorites.
For instance, if you want to aggregate all speeches by presidential name, you can execute
mydfm <- dfm(inaugCorpus, groups="President")
## Creating a dfm from a corpus ...
## ... grouping texts by variable: President
## ... indexing 34 documents
## ... tokenizing texts, found 134,142 total tokens
## ... cleaning the tokens, 461 removed entirely
## ... summing tokens by document
## ... indexing 9,208 feature types
## ... building sparse matrix
## ... created a 34 x 9208 sparse dfm
## ... complete. Elapsed time: 1.246 seconds.
dim(mydfm)
## [1] 34 9208
docnames(mydfm)
## [1] "Adams" "Buchanan" "Bush" "Carter" "Cleveland"
## [6] "Clinton" "Coolidge" "Eisenhower" "Garfield" "Grant"
## [11] "Harding" "Harrison" "Hayes" "Hoover" "Jackson"
## [16] "Jefferson" "Johnson" "Kennedy" "Lincoln" "Madison"
## [21] "McKinley" "Monroe" "Nixon" "Obama" "Pierce"
## [26] "Polk" "Reagan" "Roosevelt" "Taft" "Taylor"
## [31] "Truman" "VanBuren" "Washington" "Wilson"
Note that this groups Theodore and Franklin D. Roosevelt together – to separate them we would have needed to add a firstname variable using docvars() and grouped on that as well.
6. Explore the ability to subset a corpus.
There is a subset() method defined for a corpus, which works just like R’s normal subset() command. This provides an easy method to send specific documents to downstream functions, like dfm(), which will be useful workaround until I implement a full set of subsetting and indexing features for the dfm class object.
For instance if you want a wordcloud of just Obama’s two inagural addresses, you would need to subset the corpus first:
obamadfm <- dfm(subset(inaugCorpus, President=="Obama"), stopwords=TRUE)
## Creating a dfm from a corpus ...
## ... indexing 2 documents
## ... tokenizing texts, found 4,525 total tokens
## ... cleaning the tokens, 43 removed entirely
## ... summing tokens by document
## ... indexing 1,333 feature types
## ... building sparse matrix
## ... created a 2 x 1333 sparse dfm
## ... complete. Elapsed time: 0.05 seconds.
plot(obamadfm)
7. Preparing and pre-processing texts
1. “Cleaning”" texts
It is common to “clean” texts before processing, usually by removing punctuation, digits, and converting to lower case. Look at the documentation for quanteda}’s clean command ?clean and use the command on the exampleString text (you can load this from quantedaData using data(exampleString). Can you think of cases where cleaning could introduce homonymy?
2. Tokenizing texts
In order to count word frequencies, we first need to split the text into words through a process known as tokenization. Look at the documentation for quanteda’s tokenize command using the built in help function (? before any object/command). Use the tokenize command on exampleString, and examine the results. Are there cases where it is unclear where the boundary between two words lies? You can experiment with the options to tokenize.
Try reshaping the sentences from exampleString into sentences, using segmentSentence. What sort of object is returned if you tokenize the segmented sentence object?
3. Stemming.
Stemming removes the suffixes using the Porter stemmer, found in the SnowballC library. The quanteda function to invoke the stemmer is wordstem. Apply stemming to the exampleString and examine the results. Why does it not appear to work, and what do you need to do to make it work? How would you apply this to the sentence-segmented vector?
4. Applying pre-processing to the creation of a dfm.
quanteda’s dfm() function makes it wasy to pass the cleaning arguments to clean, which are executed as part of the tokenization implemented by dfm(). Compare the steps required in a similar text preparation package, tm:
require(tm)
## Loading required package: tm
##
## Attaching package: 'tm'
##
## The following objects are masked from 'package:quanteda':
##
## as.DocumentTermMatrix, stopwords
data("crude")
crude <- tm_map(crude, content_transformer(tolower))
crude <- tm_map(crude, removePunctuation)
crude <- tm_map(crude, removeNumbers)
crude <- tm_map(crude, stemDocument)
tdm <- TermDocumentMatrix(crude)
# same in quanteda
require(quanteda)
crudeCorpus <- corpus(crude)
crudeDfm <- dfm(crudeCorpus)
## Creating a dfm from a corpus ...
## ... indexing 20 documents
## ... tokenizing texts, found 3,863 total tokens
## ... cleaning the tokens, 0 removed entirely
## ... summing tokens by document
## ... indexing 973 feature types
## ... building sparse matrix
## ... created a 20 x 973 sparse dfm
## ... complete. Elapsed time: 0.047 seconds.
Inspect the dimensions of the resulting objects, including the names of the words extracted as features. It is also worth comparing the structure of the document-feature matrixes returned by each package. tm uses the slam simple triplet matrix format for representing a sparse matrix.
It is also – in fact almost always – useful to inspect the structure of this object:
str(tdm)
## List of 6
## $i : int [1:1954] 49 86 110 148 166 167 178 183 184 195 ... ##$ j : int [1:1954] 1 1 1 1 1 1 1 1 1 1 ...
## $v : num [1:1954] 1 2 1 1 1 1 2 1 1 2 ... ##$ nrow : int 943
## $ncol : int 20 ##$ dimnames:List of 2
## ..$Terms: chr [1:943] "abdulaziz" "abil" "abl" "about" ... ## ..$ Docs : chr [1:20] "127" "144" "191" "194" ...
## - attr(*, "class")= chr [1:2] "TermDocumentMatrix" "simple_triplet_matrix"
## - attr(*, "weighting")= chr [1:2] "term frequency" "tf"
THis indicates that we can extract the names of the words from the tm TermDocumentMatrix object by getting the rownames from inspecting the tdm:
head(tdm$dimnames$Terms, 20)
## [1] "abdulaziz" "abil" "abl" "about" "abov"
## [6] "above" "abroad" "accept" "accord" "across"
## [16] "adher" "advantag" "advisers" "after" "again"
Compare this to the results of the same operations from quanteda. To get the “words” from a quanteda object, you can use the features() function:
features_quanteda <- features(crudeDfm)
head(features_quanteda, 20)
## [1] "a" "abdulaziz" "abil" "abl" "about"
## [6] "abov" "above" "abroad" "accept" "accord"
## [16] "added" "address" "adher" "advantag" "advisers"
str(crudeDfm)
## Formal class 'dfmSparse' [package "quanteda"] with 9 slots
## ..@ settings :List of 1
## .. ..$: NULL ## ..@ weighting: chr "frequency" ## ..@ smooth : num 0 ## ..@ Dim : int [1:2] 20 973 ## ..@ Dimnames :List of 2 ## .. ..$ docs : chr [1:20] "reut-00001.xml" "reut-00002.xml" "reut-00004.xml" "reut-00005.xml" ...
## .. ..\$ features: chr [1:973] "a" "abdulaziz" "abil" "abl" ...
## ..@ i : int [1:2172] 0 1 2 3 4 5 6 7 8 9 ...
## ..@ p : int [1:974] 0 18 19 21 23 31 35 37 38 39 ...
## ..@ x : num [1:2172] 5 7 2 3 1 8 10 2 4 4 ...
## ..@ factors : list()
What proportion of the crudeDfm are zeros? Compare the sizes of tdm and crudeDfm using the object.size() function.
8. Keywords-in-context
1. quanteda provides a keyword-in-context function that is easily usable and configurable to explore texts in a descriptive way. Type ?kwic to view the documentation.
2. Load the Irish budget debate speeches for the year 2010 using
require(quantedaData)
## Loading required package: quantedaData
data(ie2010Corpus)
and experiment with the kwic function, following the syntax specified on the help page for kwic. kwic can be used either on a character vector or a corpus object. What class of object is returned? Try assigning the return value from kwic to a new object and then examine the object by clicking on it in the environment pane in RStudio (or using the inspection method of your choice).
3. Use the kwic function to discover the context of the word “clean”. Is this associated with environmental policy?
4. Examine the context of words related to “disaster”. Hint: you can use the stem of the word along with setting the regex argument to TRUE.
9. Descriptive statistics
1. We can extract basic descriptive statistics from a corpus from its document feature matrix. Make a dfm from the 2010 Irish budget speeches corpus.
2. Examine the most frequent word features using topfeatures. What are the five most frequent word in the corpus?
3. summary quanteda provides a function to count syllables in a word — syllables. Try the function at the prompt. The code below will apply this function to all the words in the corpus, to give you a count of the total syllables in the corpus.
# count syllables from texts in the 2010 speech corpus
textSyls <- syllables(texts(ie2010Corpus))
# sum the syllable counts
totalSyls <- sum(textSyls)
10. Lexical Diversity over Time
1. We can plot the type-token ratio of the Irish budget speeches over time. To do this, begin by extracting a subset of iebudgets that contains only the first speaker from each year:
data(iebudgetsCorpus, package="quantedaData")
finMins <- subset(iebudgetsCorpus, number=="01")
tokeninfo <- summary(finMins)
## Corpus consisting of 6 documents.
##
## Text Types Tokens Sentences year debate
## 2008_BUDGET_01_Brian_Cowen_FF 1705 8659 417 2008 BUDGET
## 2009_BUDGET_01_Brian_Lenihan_FF 1653 7593 418 2009 BUDGET
## 2009_BUDGETSUP_01_Brian_Lenihan_FF 1639 7500 410 2009 BUDGETSUP
## 2010_BUDGET_01_Brian_Lenihan_FF 1649 7719 390 2010 BUDGET
## 2011_BUDGET_01_Brian_Lenihan_FF 1539 7049 371 2011 BUDGET
## 2012_BUDGET_01_Michael_Noonan_FG 1521 6412 294 2012 BUDGET
## number namefirst namelast party
## 01 Brian Cowen FF
## 01 Brian Lenihan FF
## 01 Brian Lenihan FF
## 01 Brian Lenihan FF
## 01 Brian Lenihan FF
## 01 Michael Noonan FG
##
## Source: /Users/kbenoit/Dropbox/QUANTESS/quantedaData_kenlocal_gh/* on x86_64 by kbenoit.
## Created: Sat Nov 15 18:32:54 2014.
## Notes: .
Note the quotation marks around the value for number. Why are these required here?
2. Get the type-token ratio for each text from this subset, and plot the resulting vector of TTRs as a function of the year.
3. Now compare the results from the lexdiv function applied to the texts. Are the results the same?
11. Document and word associations
1. Load the presidential inauguration corpus selecting from 1900-1050, and create a dfm from this corpus.
2. Measure the document similarities using similarity(). Compare the results for Euclidean, Euclidean on the term frequency standardized dfm, cosine, and Jaccard.
3. Measure the term similarities for the following words: economy, health, women.
12. Working with dictionaries
1. Creating a simple dictionary.
Dictionaries are named lists, consisting of a “key” and a set of entries defining the equivalence class for the given key. To create a simple dictionary of parts of speech, for instance we could define a dictionary consisting of articles and conjunctions, using:
posDict <- dictionary(list(articles = c("the", "a", "and"),
conjunctions = c("and", "but", "or", "nor", "for", "yet", "so")))
To let this define a set of features, we can use this dictionary when we create a dfm, for instance:
posDfm <- dfm(inaugCorpus, dictionary=posDict)
## Creating a dfm from a corpus ...
## ... indexing 57 documents
## ... tokenizing texts, found 134,142 total tokens
## ... cleaning the tokens, 461 removed entirely
## ... applying a dictionary consisting of 2 key entries
## ... created a 57 x 3 sparse dfm
## ... complete. Elapsed time: 1.847 seconds.
posDfm[1:10,]
## Document-feature matrix of: 10 documents, 3 features.
## 10 x 3 sparse Matrix of class "dfmSparse"
## articles conjunctions Non_Dictionary
## 1789-Washington 178 73 1178
## 1793-Washington 15 4 116
## 1801-Jefferson 232 109 1385
## 1805-Jefferson 256 126 1784
## 1817-Monroe 458 174 2738
## 1821-Monroe 577 195 3685
## 1825-Adams 448 150 2317
Weight the posDfm by term frequency using tf(), and plot the values of articles and conjunctions (actually, here just the coordinating conjunctions) across the speeches. (Hint: you can use docvars(inaugCorpus, "Year")) for the x-axis.)
Is the distribution of normalized articles and conjunctions relatively constant across years, as you would expect?
2. Replicating a published dictionary analysis
Here we will create and implement the populism dictionary from Rooduijn, Matthijs, and Teun Pauwels. 2011. “Measuring Populism: Comparing Two Methods of Content Analysis.” West European Politics 34(6): 1272–83. Appendix B of that paper provides the term entries for a dictionary key for the concept populism. Implement this as a dictionary, and apply it to the same UK manifestos as in the article.
Hint: You can get a corpus of the UK manifestos from their article using the following:
data(ukManifestos, package="quantedaData")
ukPopCorpus <- subset(ukManifestos, (Year %in% c(1992, 2001, 2005) &
(Party %in% c("Lab", "LD", "Con", "BNP", "UKIP"))))
summary(ukPopCorpus)
## Corpus consisting of 11 documents.
##
## Text Types Tokens Sentences Country Type Year Language
## UK_natl_1992_en_Con 3886 29560 1605 UK natl 1992 en
## UK_natl_1992_en_Lab 2313 11355 623 UK natl 1992 en
## UK_natl_1992_en_LD 3004 17381 939 UK natl 1992 en
## UK_natl_2001_en_Con 2517 13196 721 UK natl 2001 en
## UK_natl_2001_en_Lab 3600 28704 1602 UK natl 2001 en
## UK_natl_2001_en_LD 3291 21174 1232 UK natl 2001 en
## UK_natl_2005_en_BNP 4444 25112 1058 UK natl 2005 en
## UK_natl_2005_en_Con 1860 7685 420 UK natl 2005 en
## UK_natl_2005_en_Lab 3579 23800 1557 UK natl 2005 en
## UK_natl_2005_en_LD 2859 16081 840 UK natl 2005 en
## UK_natl_2005_en_UKIP 2185 8856 425 UK natl 2005 en
## Party
## Con
## Lab
## LD
## Con
## Lab
## LD
## BNP
## Con
## Lab
## LD
## UKIP
##
## Source: /Users/kbenoit/Dropbox/QUANTESS/quantedaData_kenlocal_gh/* on x86_64 by kbenoit.
## Created: Sat Nov 15 18:43:36 2014.
## Notes: .
Create a dfm of the populism dictionary on the UK manifestos. Use this dfm to reproduce the x-axis for the UK-based parties from Figure 1 in the article. Suggestion: Use dotchart(). You will need to normalize the values first by term frequency within document. Hint: Use weight(x, "relFreq") on the dfm.
You can explore some of these terms within the corpus to see whether you think they are appropriate measures of populism. How can you search the corpus for the regular expression politici* as a “keyword in context”? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2824886441230774, "perplexity": 14354.948677726246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813322.19/warc/CC-MAIN-20180221024420-20180221044420-00314.warc.gz"} |
http://math.stackexchange.com/questions/643020/cumulative-distribution-function-proof | # Cumulative Distribution Function Proof
So I have this proof formula for the Cumulative Distribution Function and I understand it up till the point where we see square brackets, how did those end up there and how did division ended up there????
-
It's simply an application of the formula $$\sum_{j=0}^{n-1}r^j=\frac{1-r^n}{1-r}$$ for $r\neq 1$ and $n\geqslant 1$ an integer.
Indeed, $(1-r)\sum\limits_{j=0}^{n-1}r^j=\sum\limits_{j=0}^{n-1}r^j-r^{j+1}$ is a telescopic sum.
-
I somewhat understand but still feel confused :( – Chris Dobkowski Jan 18 '14 at 21:20
What is confusing you? – Davide Giraudo Jan 18 '14 at 21:21
where did that formula came from? – Chris Dobkowski Jan 18 '14 at 22:03
See edit. ${}{}$ – Davide Giraudo Jan 18 '14 at 22:07
Doesn't ring a bell :( – Chris Dobkowski Jan 18 '14 at 22:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8509621620178223, "perplexity": 1102.3247798782602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257827077.13/warc/CC-MAIN-20160723071027-00172-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://physics.anu.edu.au/contact/people/profile.php?ID=327 | Position Fellow Department of Theoretical Physics (02) 612 59475 Javascript Required Oliphant 4 08 http://tpsrv.anu.edu.au/Members/man...
## arXiv pre-prints
Boundary matrices for the higher spin six vertex model
arXiv:1903.00274v2
,
Density and current profiles in \$U_q(A^{(1)}_2)\$ zero range process
Nuclear Physics B 922 (2017) 148-185
arXiv:1705.10979v2
, ,
Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday
Journal of Physics A: Mathematical and Theoretical 50 (2017) 6pp
arXiv:1608.04899v2
Construction of \$R\$-matrices for symmetric tensor representations related to \$U_{q}(\widehat{sl_{n}})\$
arXiv:1607.07968v1
, , ,
Stochastic \$R\$ matrix for \$U_q(A^{(1)}_n)\$
Nuclear Physics B 913 (2016) 248-277
arXiv:1604.08304v4
, , ,
On the Yang-Baxter equation for the six-vertex model
Nuclear Physics B 882 (2014) 70-96
arXiv:1401.6494v2
, , ,
Q-operators in the six-vertex model
Nuclear Physics B 886 (2014) 166-184
arXiv:1406.0662v1
, , , , , , , , ,
An integrable 3D lattice model with positive Boltzmann weights
Journal of Physics A: Mathematical and Theoretical 46 (2013)
arXiv:1308.4773v2
, , , ,
Scaling and universality in the 2D Ising model with a magnetic field
Physical Review E-Statistical, Nonlinear and Soft Matter Physics 81 (2010) 4
arXiv:1002.4234v2
,
Form factor expansions in the 2D Ising model and Painlev\'e VI
Nuclear Physics B 838 (2010) 391-412
arXiv:1002.2480v2
Picard solution of Painlev\'e VI and related tau-functions
arXiv:1002.2327v1
,
Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results
Journal of Physics A: Mathematical and Theoretical 43 (2010) 16
arXiv:0912.2163v2
, ,
Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
XVIth International Congress on Mathematical Physics 2009 (2010) 23-44
arXiv:0911.3693v1
, , , ,
Variational approach to the scaling function of the 2D Ising model in a magnetic field
Journal of Physics A: Mathematical and Theoretical 42 (2009) 10
arXiv:0811.3271v2
, ,
Quantum geometry of 3-dimensional lattices
Journal of Statistical Mechanics: Theory and Experiment July 2008 (2008) 27p
arXiv:0801.0129v3
, ,
Exact solution of the Faddeev-Volkov model
Physics Letters A 372 (2008) 1547-1550
arXiv:0706.3077v2
, ,
Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
Nuclear Physics B 784 (2007) 234-258
arXiv:hep-th/0703041v2
An Analytic Formula for the A_2 Jack Polynomials
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 3 (2007) 014(1-11)
arXiv:math/0701677v1
,
Analytic theory of the eight-vertex model
Nuclear Physics B 775 (2007) 225-282
arXiv:hep-th/0609153v3
,
The eight-vertex model and Painleve VI
Journal of Physics A: Mathematical and General 39 (2006) 12235-12243
arXiv:hep-th/0602122v1
,
Eight-vertex model and non-stationary Lame equation
Journal of Physics A: Mathematical and General 38 (2005) L145-L153
arXiv:hep-th/0411094v2
, ,
Q-operator and factorised separation chain for Jack polynomials
Indagationes Mathematicae 14 (2003) 451-482
arXiv:math/0306242v2
Separation of variables for the A_3 elliptic Calogero-Moser system
Journal of Physics A: Mathematical and General 34 (2001) 4183-4195
arXiv:nlin/0105005v1
,
Some exact results for the three-layer Zamolodchikov model
Nuclear Physics B 592 (2001) 597-626
arXiv:hep-th/0008085v1
,
Bethe ansatz for the three-layer Zamolodchikov model
Journal of Physics A: Mathematical and General 32 (1999) 5285-5298
arXiv:solv-int/9903010v1
,
Functional relations and nested Bethe ansatz for sl(3) chiral Potts model at q^2=-1
Journal of Physics A: Mathematical and General 32 (1999) 3041-3054
arXiv:solv-int/9903001v1
V. V. Mangazeev
A_2 Macdonald polynomials: a separation of variables
arXiv:q-alg/9512003v1
S. M. Sergeev, V. V. Bazhanov, V. V. Mangazeev
Quantum Dilogarithm and Tetrahedron Equation
arXiv:q-alg/9511015v2
S. M. Sergeev, H. E. Boos, V. V Mangazeev, Yu. G. Stroganov
\$Psi\$ - Vectors for Three Dimensional Models
arXiv:q-alg/9509019v1
S. M. Sergeev, V. V. Mangazeev, Yu. G. Stroganov
The vertex formulation of the Bazhanov-Baxter Model
arXiv:hep-th/9504035v1
V. V. Mangazeev, S. M. Sergeev, Yu. G. Stroganov
New solution of vertex type tetrahedron equations
arXiv:hep-th/9410049v1
H. E. Boos, V. V. Mangazeev, S. M. Sergeev
Modified Tetrahedron Equations and Related 3D Integrable Models
arXiv:hep-th/9407146v1
V. V. Mangazeev, Yu. G. Stroganov
Elliptic solution for modified tetrahedron equations
arXiv:hep-th/9305145v1
## Publications
### Book chapter
, ,
QUANTUM GEOMETRY OF 3-DIMENSIONAL LATTICES AND TETRAHEDRON EQUATION
XVIth International Congress on Mathematical Physics 2009
World Scientific Publishing Company, Prague (2010) 23-44
(1 publications)
### Journal article
, , ,
Stochastic R matrix for U-q(A(n)((1)))
Nuclear Physics B 913() 248-277
,
Construction of R-matrices for symmetric tensor representations related to U-q ((sl(n))over-bar)over cap>)
Journal of Physics A: Mathematical and Theoretical 49, 49() 1-19
, , , , , , , , ,
An integrable 3D lattice model with positive Boltzmann weights
Journal of Physics A: Mathematical and Theoretical 46, 46()
, , , , , , , , , , ,
An elliptic parameterisation of the Zamolodchikov model
Nuclear Physics B 871, 1() 127-144
,
Form factor expansions in the 2D Ising model and Painleve VI
Nuclear Physics B 838, 3() 391-412
, , , ,
Scaling and universality in the two-dimensional Ising model with a magnetic field
Physical Review E-Statistical, Nonlinear and Soft Matter Physics 81, 6() 4
,
The eight-vertex model and Painleve VI equation II: eigenvector results
Journal of Physics A: Mathematical and Theoretical 43, 8() 16
, , , ,
Variational approach to the scaling function of the 2D Ising model in a magnetic field
Journal of Physics A: Mathematical and Theoretical 42, 042005() 10
, ,
Exact solution of the Faddeev-Volkov model
Physics Letters A 372, 10() 1547-1550
, ,
Quantum geometry of three-dimensional lattices
Journal of Statistical Mechanics: Theory and Experiment July 2008() 27p
An Analytic Formula for the A2 Jack Polynomials
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 3() 0141-11
,
The Eight-Vertex Model and Painlevé VI
Journal of Physics A: Mathematical and General 39() 12235-12243
,
Eight-vertex Model and Non-stationary Lame Equation
Journal of Physics A: Mathematical and General 38() L145-L153
, ,
Q-operator and factorized separation chain for Jack polynomials
Indagationes Mathematicae 14, 3-4() 451-482
Separation of variables for the A3 elliptic Calogero-Moser system
Journal of Physics A: Mathematical and General 34, 19() 4183-4195
,
Some exact results for the three-layer Zamolodchikov model
Nuclear Physics B 592, 3() 597-626
Separation of variables for the A3 elliptic Calegro-Moser system
Journal of Physics A: Mathematical and General 34() 4183-4195
,
Continuum limit of the triple tau-function model
Theoretical and Mathematical Physics 129() 1573-1580
,
Functional relations and nested Bethe ansatz for sl(3) chiral Potts model at q2 = -1
Journal of Physics A: Mathematical and General 32, 16() 3041-3054
,
Bethe ansatz for the three-layer Zamolodchikov model
Journal of Physics A: Mathematical and General 32, 28() 5285-5298
,
Functional relations and nested Bethe ansatz (3) chiral Potts model at q\^2=-1
Journal of Physics A: Mathematical and General 32, 16() 3041-3054
(29 publications)
### Journal short contribution (non refereed)
(1 publications)
Updated: 15 January 2019/ Responsible Officer: Director, RSPE/ Page Contact: Physics Webmaster | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9477941393852234, "perplexity": 7569.557028263813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997533.62/warc/CC-MAIN-20190616022644-20190616044644-00381.warc.gz"} |
http://stats.stackexchange.com/questions/41784/causation-implication | # Causation implication
I recently read an article about how you can increase longevity by sleeping less. This article, like many others I've read, references a statistical study and implies that causation was found between two events(sleeping less and living longer). Well I'm not a statistician, but I'm aware of the common fallacies about causation and I find it very hard to accept the information given in this article and other similar ones. Even further, I can almost never know from an article if the original study found causation or not, without taking the time and digging into it.
Although I'd really like to, I'm not going to go into any depth bringing examples of how the causation might be wrong in the given article, because I would most likely be preaching to the choir. Luckily though, to my relief, the original article in NY Times only talks about correlation and even has a disclaimer in the end that basically says that the study doesn't imply causation.
Also, I now read that, in accordance with my intuition, it is actually really hard to find and/or test for causation. Reading through the wiki page on the subject I saw two models which seemed to be the most common ones used to find actual links in causation: Granger causality test and convergent cross mapping. But I see, that both of these have some gaps in them.
Now, I have a few questions regarding all this.
Q1: Are there any models that can, with high accuracy, find actual causation in such widespread studies? If so what are the most common ones?
Q2: How often do these widespread studies actually test for causation? E.g. if I read somewhere that a very broad study implies that A causes B, do I have any reason whatsoever to believe that there's something more than correlation behind those claims.
Q3: Is there a good technique that can be used by a non-statistician to filter out good statistics from the bad ones in everyday life.
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Regarding Q1: Are you only referring to observational studies? If not the easiest way to test for a causal effect is to conduct a experiment in which you e.g. apply food-restriction to one group of mice but not to another group and then test if the starving mice live longer. – jokel Nov 5 '12 at 7:54
Well I'm referring to observational ones, yes. I understand, that blind (or even double-blind) experiments are quite good at finding/testing for causality. – Deiwin Nov 5 '12 at 9:28
Starting with Q2: convincing readers that a causal effect has been found is often the overriding concern of research articles in social sciences, psychology and medicine. However this usually cannot be done through a simple test but by proposing a causal theory, presenting evidence consistent with the theory in the form of correlations and other statistical patterns, pre-empting alternative explanations, and presenting evidence that is not consistent with the alternative explanations.
(Q1) Regression analysis allowing the researcher to examine the conditional correlation between an explanatory variable and a dependent variable while controlling for other possible explanatory variables is still probably the most common method. This does not by itself demonstrate causation between the variables of interest, but helps to isolate and reject competing causal explanations. In particular, it is difficult to argue that there is a causal relationship when there is no correlation after controlling for other variables (although when the explanatory variables are strongly correlated with each other, even this can be questioned).
Some other common methods that try to improve on basic regression analysis for eliminating competing explanations of hypothesised causal effects include
• 'natural experiments', for instance where one region of a country has benefited from a new policy but another, otherwise similar, region has not
• matching techniques such as propensity score matching, which try to control specifically for the fact that the 'treatment' group (e.g. people who sleep less) in an observational study may differ systematically from the 'control' group (people who sleep more) in ways that affect the outcome of interest (longevity)
(Q3) Arguably there is no simple technique for filtering out good statistics; it depends as much on knowledge of the field as knowledge of the statistical methods. Assuming an article shows a correlation between two variables, consider what the alternative explanations for that correlation could be (e.g. an unobserved variable influencing both, or reverse causation) and whether the article at least tries to eliminate those alternative explanations. There will nearly always be some form of regression or matching analysis to control for other variables; if nothing like that is mentioned in the abstract then a causal conclusion is likely to be unsound (but such articles would rarely get published in academic journals).
Also consider the level of generality claimed for the causal effect. E.g. the claim that less sleep leads to more longevity in abstract terms sounds like it is suggesting a universal effect, applicable to humans in all places and times. To make a general claim like that plausible would depend on subject knowledge (e.g. prior evidence that the determinants of longevity are similar across countries) and a strong argument that the sample is representative (e.g. the article describes how they were selected and discusses how potential sample bias was avoided).
For me the most convincing evidence from observational studies is when a series of studies by different researchers, using different datasets and techniques, have consistently found evidence that does not contradict the proffered causal explanation, and progressively eliminated alternative explanations.
-
Thanks! Really helpful. – Deiwin Nov 5 '12 at 12:44
Q2: As far as I know there are 2 good ways to test for causality. It is done by the design of the study. Two designs work here:
• In a situation where you can control ALL independent variables (usually in experiments). This makes sure that A is causing B, and there is no C causing the effect (and also not the reverse effect; B causing A).
• In longitudinal studies.
Q1: Rubin's Causal Model: http://en.wikipedia.org/wiki/Rubin_causal_model
Q3: There is no "trick" to seeing whether the statistics of one single article are good. I find the most important questions you should ask yourself:
• Who did the research?
• What is to be gained by the researcher with a favorable outcome?
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On bullet 1: Controlling all independent variable can also be wrong. Simple example: B is caused by A and C, but A does not cause C. Now, if you control for B and you'll be wrong about the causal impact of A on C (you'll think there is one). Also, in experiments that randomise and control, it is the randomisation that warrants the causal inference; control 'just' makes the inference more precise. – conjugateprior Nov 5 '12 at 12:45
You might find Winship and Morgan's overview article helpful. – conjugateprior Nov 5 '12 at 12:53
Controlling all other independent variables, while randomizing the variable expected to have a causal relation? – PascalvKooten Nov 5 '12 at 12:57
Even then. Try it: assume A and C are normalised test scores unrelated in the population: dd <- data.frame(A=rnorm(100), C=rnorm(100)) # A is randomised and unrelated to C. Now make B, say college admission, depend on them: transform(dd, B=ifelse(A+C > 1, 1, 0)) ## score over 1 get in. The correct causal inference is given by not conditioning on B: summary(lm(C ~ A, data=dd)) # no effect. Conditioning on B: summary(lm(C ~ A + B, data=dd)) makes it look like A causes C to be go down, but it doesn't (they're unrelated by construction). This is not specific to regression, btw. – conjugateprior Nov 5 '12 at 13:19
You might be thinking 'but wait, that B really is a dependent variable', and you'd be right, in a sense -- this effect is closely related to the hazards of 'selecting on the dependent variable'. In this case it's just not your dependent variable (that would be C in the example.) Nevertheless, if you didn't know A and C caused B, or more generally you didn't know that B had A and C somewhere way back in its causal chain, then you'd list it as an independent variable, condition on / control for it and things would go wrong, as they do above. – conjugateprior Nov 5 '12 at 13:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5993650555610657, "perplexity": 1094.059421856394}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246643283.35/warc/CC-MAIN-20150417045723-00164-ip-10-235-10-82.ec2.internal.warc.gz"} |
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# 442
Change yourself - September 19, 2011
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Do this is a derivative? I'm only in year 8 but I'm learning this stuff from my teacher
Yes, and in this case it can be interpreted as "you should always be growing."
im i year 8 too! :D
Or put on weight!
Change yourself - be more positive!
Sadly, the second derivative is smaller than zero.
Life is full of unattainable goals.
Yeah, but as I am a grad student, $0 < \frac{d(me)}{dt} < \episilon$
Oops. Apparently my bad LaTeX joke got cut off.
Should be fixed now :D
Hey hey just asking what is dt means?
dx/dt is the math way of saying "the change of x with respect to time" (it's called a derivative.) If the change is greater than zero, that means x (or, in this case, the person) is increasing.
I like the informal definition "differential (smallest possible change) in time". So the derivative becomes the smallest possible change in the top bit per every smallest possible change in the bottom. It also stands for "delerium tremens", but clearly this little man's over it.
:) This is great. I'd buy a T-shirt with this print.
Monotonic but not monotonous!
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Apps | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6384103298187256, "perplexity": 4615.244812630361}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190134.67/warc/CC-MAIN-20170322212950-00285-ip-10-233-31-227.ec2.internal.warc.gz"} |
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## AngelDreamGurl Group Title 5/6 - 1/3 PLEASE HELP 11 months ago 11 months ago Edit Question Delete Cancel Submit
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1. Florcelda Group Title
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You can add or subtract fractions if they have the same denominator Find the least common denominator or the least common multiple of 3 and 6
• 11 months ago
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What is the common denominator for 6 and 3?
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1/2
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It is the least number that can both be divided by 6 And 3
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\[5-2\div6=1\div2\]
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Or you can make the denominator of 1/3 to be 6, by multiplying both num and dem by 2 that will give you 2/6 Now that the denominators are the same you can now subtract the numerators 5/6 - 2/6 = ? Then simplify to its lowest term
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9967344403266907, "perplexity": 8305.75928332983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510268660.14/warc/CC-MAIN-20140728011748-00115-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-simplify-square-roots/e/simplifying_radicals | # Simplify square roots
For example, simplify √18 as 3√2.
### Problem
Simplify.
Remove all perfect squares from inside the square root.
root, start index, end index, square root of, 20, end square root, equals | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 13, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7927311062812805, "perplexity": 14817.505449098879}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170875.20/warc/CC-MAIN-20170219104610-00187-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/97031/how-to-modify-apa-biblatex-style-to-look-like-elsarticle-harv-style | # How to modify APA Biblatex style to look like “elsarticle-harv” style?
I have been using the \bibliographystyle{elsarticle-harv} style for a long time, since I newly swtiched to biblatex and it is only for natbib, so how to modify the default biblatex style or specifically the APA style, so that making the citation style same as elsarticle-harv.
The elsarticle-harv author-year citation style:
\cite{LathamVOF:2012} -----> Latham (2012)
\citep{LathamDEM:2012} -----> (Latham,2012)
\cite{LathamM:2010} -----> Latham et al.(2010) % multi-author
\cite{LathamMeer:2011} -----> Latham and Meer (2011) % 2 authors
\cite{LathamVOF:2012,LathamDEM:2012} -----> Latham (2012a, b)
The new setting of biblatex is given:
\usepackage{csquotes}
\usepackage[
style=apa,
backend=biber,
refsection=chapter,
natbib=true ] {biblatex}
\DeclareLanguageMapping{british}{british-apa}
However, the APA sytle was like:
\cite{LathamVOF:2012} -----> Latham, 2012
\citep{LathamDEM:2012} -----> (Latham,2012) % <--- same
\cite{LathamM:2010} -----> Latham, A & B, 2010 % <--- too many cited authors!
\cite{LathamMeer:2011} -----> Latham & Meer, 2011 % & instead of and!
\cite{LathamVOF:2012,LathamDEM:2012} -----> Latham, 2012a,2012b
So how to make the year in () when using \cite{} and lessen the authors (if>3) as FirstAuthor et al.
Edit1:
using \LetLtxMacro{\cite}{\citet} makes year in (), however, the setting maxcitenames=2 seems not working. Besides, how to change the & to and?
MWE codes:
\documentclass[12pt, a4paper]{book}
\usepackage{hyperref}
\usepackage{color}
\usepackage[ ]{libertine}
\usepackage[ T1 ]{fontenc}
\definecolor{DarkBlue}{RGB}{0,51,153}
\hypersetup{
citecolor=DarkBlue,%
filecolor=red,%
urlcolor=green,
\usepackage[UKenglish]{babel}
\usepackage{csquotes}
\usepackage[
style=apa,
backend=biber,
refsection=chapter,
maxcitenames=2,
natbib=true ] {biblatex}
\DeclareLanguageMapping{british}{british-apa}
\LetLtxMacro{\cite}{\citet} % year in ()
\begin{document}
\frontmatter
\tableofcontents
\mainmatter
\chapter{Introduction}
\citep[e.g.][]{Tsuji:2008, TsujiDEM:2008}
\cite{TsujiDEM2D:2008, TsujiDEM:2008}
\cite{Shirgaonkar:2009}.
\citep{Sorensen:2006}
\end{document}
and reffile.bib
@ARTICLE{Tsuji:2008,
author = {Tsuji, T. and Yabumoto, K. and Tanaka, T. and Peter, B.},
title = {Spontaneous structures in three-dimensional bubbling gas-fluidized bed by paralled {DEM-CFD} coupling simulation},
journal = { Powder Technology },
year = { 2008 },
volume = { 184 },
pages = { 132-140 },
number = {5 }
}
@ARTICLE{TsujiDEM:2008,
author = {Tsuji, T. },
title = {Spontaneous structures in three-dimensional bubbling},
journal = { Powder Technology },
year = { 2008 },
volume = { 11 },
pages = { 11-17 },
number = {2 }
}
@ARTICLE{TsujiDEM2D:2008,
author = {Tsuji, T. },
title = {Spontaneous structures in 2-{D} bubbling },
journal = { Powder Technology },
year = { 2008 },
volume = { 10 },
pages = { 13-19 },
number = { 7}
}
@ARTICLE{Shirgaonkar:2009,
author = {Shirgaonkar, A. A. and Maclver, M. A. and Patankar, N. A.},
title = {A new mathematical formulation and fast algorithm for fully resovled simulation of self-propulsion},
journal = {Journal of Computional Physics},
year = { 2009 },
volume = { 228 },
pages = { 2366-2390 },
number = {11 }
}
@ARTICLE{Haeri:2012,
author = {Haeri, S. and Shrimpton, J. S.},
title = {On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows},
journal = {International Journal of Multiphase Flow},
year = { 2012 },
volume = { 40 },
pages = { 38-55 },
number = { 18}
}
@BOOK{Sorensen:2006,
author = {Sorensen, R. M.},
title = { {Basic Coastal Engineering} },
publisher = {Springer Science, New York},
edition = {Third},
year = { 2006 },
}
Output: Displaying 3-5 authors in citation looks really odd :(, but maxcitenames is not available in APA
The printed references style: Journal name and volume is italic, how to make it as normal font; or just make the volume number non-italic, because the combination 18(5) (where 5 is the number) looks weird.
For comparison, the original natbib output is also attached:
I think the issue with the maxcitenames is that APA style requires that all the names be listed the first time the work is cited. This is controlled by the labelname format in apa.cbx. I modified
\DeclareNameFormat{labelname}{%
...
{\ifboolexpr{test {\ifnumcomp{\value{listtotal}}{>}{5}}
or test {\ifciteseen}}
...
}
to be
\DeclareNameFormat{labelname}{%
...
{\ifboolexpr{test {\ifnumcomp{\value{listtotal}}{>}{\value{maxnames}}}}
...
}
I believe you can just copy the original and paste a modified version into your preamble or you can define your own citation style.
As for dealing with & and "and", I do
\renewcommand*{\finalnamedelim}{\addspace{}\bibstring{and}\space}
I cannot remember if this handles changes in language properly.
• Unfortunately, it did not work :(, when I put it in preamble. – KOF Feb 6 '13 at 17:15
• I forgot to add makeletter...., it did work – KOF Feb 7 '13 at 20:15
The biblatex-apa package is carefully tailored to meet the standards of the APA style. Your requirements are relatively simple and likely have more in common with features of the standard authoryear-comp style.
Most of the changes you would need to make to authoryear-comp have already been addressed by other posts on this site.
The following example should get you most of the way there. I used BibTeX as the backend here as I can't see why your style needs biber. If you wish to use biber, load biblatex with:
\usepackage[backend=biber,style=authoryear-comp,natbib,uniquename=false,
uniquelist=false,maxcitenames=2,firstinits,dashed=false]{biblatex}
This will disable name and name list disambiguation. The latter overrides maxcitenames=2 to ensure that name lists in citations are unique. Note that in switching between backends you should delete auxiliary files (*.aux, *.bbl, *.bcf, etc.) before re-compiling.
\documentclass{article}
\usepackage[backend=bibtex,style=authoryear-comp,natbib,maxcitenames=2,
firstinits,dashed=false]{biblatex}
% \cite like \textcite
\renewrobustcmd*{\cite}{\citet}
\renewrobustcmd*{\Cite}{\Citet}
\renewrobustcmd*{\cites}{\textcites}
% all titles upright, no quotes
\DeclareFieldFormat{journaltitle}{#1}
\DeclareFieldFormat{issuetitle}{#1}
\DeclareFieldFormat{maintitle}{#1}
\DeclareFieldFormat{title}{#1}
\DeclareFieldFormat
[article,inbook,incollection,inproceedings,patent,thesis,unpublished]
{title}{#1}
% no "In: " preceding journal titles
\renewbibmacro{in:}{%
\ifentrytype{article}{}{\printtext{\bibstring{in}\intitlepunct}}}
% volume (number)
\renewbibmacro*{volume+number+eid}{%
\printfield{volume}%
\ifentrytype{article}
\printfield{number}%
\printfield{eid}}
\DeclareFieldFormat[article]{number}{\mkbibparens{#1}}
% no page prefix
\DeclareFieldFormat{pages}{#1}
% no parentheses around dates
\renewbibmacro*{date+extrayear}{%
\iffieldundef{year}{}{\printtext{\printdateextra}}}%
% no and before last name in list
\AtBeginBibliography{%
% all names last-first
\DeclareNameAlias{sortname}{last-first}
\begin{document}
\noindent
\cite[e.g.][10--15]{glashow}, \citet[e.g.][10--15]{glashow} \\
\cite{companion}, \citet{companion}, \citep{companion} \\
\cite*{companion}, \citet*{companion} \\
\cites[e.g.][10--15]{bertram}[10--14]{knuth:ct:a} \\
\cite{glashow,bertram,knuth:ct:a,knuth:ct:b,knuth:ct:c} \\
\citet{glashow,bertram,knuth:ct:a,knuth:ct:b,knuth:ct:c}
\printbibliography
\end{document}
• Great, now everything with use of this style is under control. – KOF Feb 7 '13 at 0:09
To add the parenthesis to the citation produced with \cite, you can use the package letltxmacro to define \cite as \citet
\LetLtxMacro{\cite}{\citet}
The number of authors to be printed before using et. al. is controlled by the parameter maxcitenames as one of the options of biblatex, i.e.
\usepackage[maxcitenames=2]{biblatex}
• The first command worked, but the option maxcitenames=2 did not change the maximum citation names. – KOF Feb 6 '13 at 14:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.624786913394928, "perplexity": 11651.716644054006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986666959.47/warc/CC-MAIN-20191016090425-20191016113925-00378.warc.gz"} |
http://www.ck12.org/physics/Universal-Gas-Law/quiz/Universal-Gas-Law-Quiz/r1/ | <meta http-equiv="refresh" content="1; url=/nojavascript/"> Universal Gas Law ( Assessments ) | Physics | CK-12 Foundation
# Universal Gas Law
%
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Universal Gas Law Quiz
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Reviews the difference between the combined gas law and the universal gas law and calculations involving the universal gas law.
# Reviews
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Well done! You've successfully verified the email address . | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9546735882759094, "perplexity": 23668.259118201124}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663167.8/warc/CC-MAIN-20140930004103-00141-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://scicomp.stackexchange.com/questions/30783/radiation-heat-transfer-between-surfaces | # Radiation heat transfer between surfaces
I'm trying to model the temperature distribution over a curved surface. Apart from the heat equation, I need to take into account the energy emission/absorption through electromagnetic radiation. The surface is divided into small triangles that will exchange energy.
If we focus on the radiation part, one way to start is to compute form factors that take into account how much one surface sees another one. Then, it is easy to derive a formula for the energy exchange between the triangles. However, this formula implies calculating powers of a matrix $$n\times n$$ where n is the number of triangles. This can be done in $$n^{2.8}$$. Besides that, form factors are also expensive to calculate.
This is quite expensive. For a mesh divided in 10,000 triangles, which is not a lot, it's more than $$10^{11}$$ calculations.
Does anyone know any reference to some algorithm that solves the same problem with better complexity? Note that game-like algorithms aren't likely to work well because here there are as many light sources and as many observers as there are triangles...
If you're not aware of any good deterministic algorithm but know about heuristic ones it can also be useful.
I think you are looking for a so-called fast algorithm for heat transfer problems. One family worth considering is a family of multipole algorithms. Fast Multipole Method (FMM) is extremely popular in the modeling of electromagnetic phenomena. But I've seen its application to heat transfer problems as well. FMM, in general, allows reducing the complexity to $$\mathcal O(N\log N)$$ (subject to the problem statement, preconditioning, etc.) There probably are certain limitations/details in applying FMM to heat transfer, but I would consider this approach very viable. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8491496443748474, "perplexity": 312.88130664702106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358973.70/warc/CC-MAIN-20211130110936-20211130140936-00063.warc.gz"} |
https://www.springerprofessional.de/study-on-cutting-force-cutting-temperature-and-machining-residua/18231414?fulltextView=true | main-content
Weitere Artikel dieser Ausgabe durch Wischen aufrufen
01.12.2020 | Original Article | Ausgabe 1/2020 Open Access
Study on Cutting Force, Cutting Temperature and Machining Residual Stress in Precision Turning of Pure Iron with Different Grain Sizes
Zeitschrift:
Chinese Journal of Mechanical Engineering > Ausgabe 1/2020
Autoren:
Yinfei Yang, Lu Jin, Jinpeng Zhu, Jinxing Kong, Liang Li
1 Introduction
Pure iron has the characteristics of soft texture, high plasticity and toughness, and excellent electromagnetic properties. It is mostly used in the manufacturing of precision instruments and components in the national defense and military industry. The thin-walled parts made of pure iron are crucial parts of a physical experiment with high machining accuracy. However, due to its poor rigidity and overall size, thin-walled parts deform owing to cutting force, cutting heat, residual stress, which made its accuracy challenging to meet the design requirements [ 1]. Therefore, it is a crucial and challenging point in the current manufacturing field to explore the mechanism of relevant factors on the machining deformation of thin-walled parts and providing the corresponding theoretical support of processing technology.
In the actual machining process, the above factors lead to the final formation of residual stress. Generally, the mechanical load plays a leading role in the generation of residual stress at low-speed cutting and low cutting temperature. After the cutting speed increases gradually, and the cutting temperature also rises, the thermal load on the workpiece surface will play a leading role. When the cutting temperature exceeds a specific value, the phase transition of workpiece material will take place. It can, that in the cutting process, the generation of residual stress is extremely complex, which is closely related to the thermal coupling in the material removal process [ 215].
Since the beginning of the last century, numerous experts and scholars have conducted preliminary studies on residual stresses applying many sophisticated testing methods, classified into a) destructive mechanical release measurement b) non-destructive physical measurement [ 16]. The residual stress of machining is a non-destructive physical method by the X-ray diffraction method and stripping method. Zhang et al. [ 17] selected the X-ray diffraction method based on a two-dimensional surface survey to measure GH4169 high-temperature alloy and found that two-dimensional surface survey technology could quickly obtain the complete Debye ring of the material, reflecting the crystallographic characteristics of the material. Moreover, it shows that the residual stress on the processed surface of the high-temperature alloy studied was uneven, and the microstructure was obvious. Xia et al. [ 18] measured the residual surface stress of the TP340 steel pipe by using an X-ray diffraction method combined with electrolytic polishing, which provided a reference for the study on reducing the stress corrosion cracking of this kind of steel pipe. Abboud et al. [ 19] carried out an experimental study on the residual stress in the finishing process of two types of aerospace titanium alloys Ti-64 and Ti-6246, believed that the residual stress is closely related to the cutting parameters. Increasing the material removal rate and reducing the radius of tooltip arc can increase the residual compressive stress on the surface and improve fatigue life. Mohammadpour [ 14] thought that the residual stress caused by material removal is the result of the multiple cutting. In order to explore the accumulated strain field and temperature field of the effect of residual stress by using multiple-step analysis using finite element software ABAQUS simulation. The authors have analyzed the milling of AISI4140 ultra high strength steel, and the highest temperature had influenced the residual stresses. The accuracy of the model has been proved by experimentation.
The influence of cutting parameters on the cutting force, heat, and machining residual stress has been focused on the turning process. Nevertheless, the influence of grain size on them has not been investigated comprehensively. In order to explore the effects of grain size, two kinds of grain size were selected to analyze their effect on cutting force, heat, and machining residual stress under precision turning of pure iron. The influence of grain size under different feed, cutting speed and depth of cut is analyzed and discussed in Section 3. Some conclusions are given in Section 4.
2 Experimental Details
2.1 Material Preparation
The pure iron bar (DT4E) material procured from Taiyuan Iron and Steel Company. The chemical composition of the workpiece is provided in Table 1. Its hardness is in the range of 50‒80 HB, the ultimate tensile strength of 313 MPa, and the impact toughness up to 234 J/cm 2.
Table 1
Chemical composition of electrical pure iron (DT4E)
Element
C
Ni
Mn
Si
Cr
S
Cu
P
Al
Content(≤ %)
0.025
0.20
0.30
0.020
0.1
0.020
0.10
0.020
0.15‒0.50
DT4E pure iron bar of initial grain size 0-grade was prepared, as shown in Figure 1(a). The cold forging process has been used to forge the Φ155 × 45 mm pure iron raw materials by 50% of the compression ratio at 700 °C temperature, maintained for two hours, realizing the grain refinement. Then grain size can be controlled in 5 to 6 level, as shown in Figure 1(b).
This section gives a brief description of the experimental setup, machine tool, workpiece material, applied cooling techniques, input and output parameters.
2.2 Sample Preparation and Selection
The cutting tool is DCGT11T302 HP sharp carbide blade produced by KENNAMETAL Company. The number of the blade is KC5010, PVD AlTiN coating material. The cutting experiments were conducted on a self-developed vertical lathe. The machine unit has spindle speed from 400‒12000 r/min with a spindle power of 5.7 kW. The workpiece clamping and cutting tools as shown in Figure 2(a). The workpiece was installed the handle for SDJCL2020K11, with installation tool cutting edge angle was 93°, chose to cut the lubricating way of dry cutting in processing disc face. According to the requirements of the cutting platform and the clamping, two kinds of grain size of pure black-smith respectively processed into Φ79 × 20 mm and Φ123 × 25 mm disk with a hole in the middle provided in Figure 2(a).
Cutting force was measured by Kistler 9119AA2 type high-precision three-way force sensor. The cutting temperature was measured by an infrared thermal imager produced by the German InfraTec Company. The model was Image IR 5300, and the temperature range was selected from 150‒300 °C. The Phantom series high-speed camera of American VRI Company was used to shoot the precision turning process. The model is VISION V2511, and the highest shooting rate is 1000000 f/s. The focus area was selected as the middle position of the end face to be processed.
Residual stress was measured by X-ray residual stress measurement instrument made in PULSTEC Japanese company, as shown in Figure 2(b). The main composition of pure iron is ferrite materials, so we chose the target of measurement for Cr in inclination degrees of 35° and crystal surface of 211 with working voltage and current are respectively 30 kV and 1 mA, its features of transmission line for Cr-Kɑ. We can get 125 points by incident only once on residual stress data fitting; the residual stress value is calculated based on the Debye diffraction ring. According to the tool feed direction in the process of turning the end face, the measurement point and direction of residual stress are defined as shown in Figure 3 below.
2.3 Infrared Thermal Imager Emissivity Check
Infrared Thermal Imager can convert invisible surface heat radiated by objects above absolute zero into visible images, to generate a thermal image and temperature value [ 20]. The instrument can not only obtain the thermal field distribution diagram of the tested sample surface but also obtain more accurate temperature data.
The precise measurement of the data is closely related to the emissivity, solar radiation, ambient temperature, and the corresponding emissivity of the different materials is substantially different. That’s why determining the emissivity of the tested surface is of considerable significance through the infrared thermal image temperature measurement technology. The heating furnace was used to heat the coarse and fine grain pure iron workpiece from room temperature to 400 °C, respectively, as shown in Figure 4.
In the heating process, the thermocouple method and infrared thermograph were selected to measure the processed surface temperature of the workpiece simultaneously. The machining parameters of the workpiece were within the range of experimental design parameters. Based on the real-time measurement results of the thermocouple, the radiance is adjusted in the configuration software of the infrared thermometer to keep the temperature consistent. The emissivity of the Infrared Thermal Imager for two kinds of pure iron materials with grain size in different temperature ranges was obtained, as shown in Table 2.
Table 2
Temperature range
Coarse crystal pure iron
Fine crystal pure iron
T < 150 °C
0.79
0.78
150 °C < T < 200 °C
0.71
0.72
200 °C < T < 250 °C
0.58
0.61
250 °C < T < 300 °C
0.55
0.57
300 °C < T 350 °C
0.52
0.53
350 °C < T
0.49
0.51
2.4 Experiment Parameter
Based on the range of the cutting parameter of the actual finishing pure iron, the test scheme was designed to carry out the single factor cutting test of coarse and fine grain size to measure the cutting force and cutting temperature during the cutting process to analyze the influence degree of cutting force and heat-affected by grain size. Cutting parameters are shown in Table 3.
Table 3
Single-factor cutting test parameters table
Feed (mm/r)
Speed of cut (m/min)
Depth of cut (mm)
0.02
160
0.01, 0.03, 0.05, 0.07, 0.09
0.01, 0.02, 0.03, 0.04, 0.05
120
0.04
0.05
80, 120, 160, 200, 240
0.04
Orthogonal cutting tests of pure iron with three factors and four levels were carried out as shown in Table 4, and groups’ results of tests are given in Table 5. The performance measures for this study were the surface residual stresses to analyze the influence of the residual stress on the surface of pure iron at different cutting conditions and grain sizes.
Table 4
Orthogonal test factors and their corresponding levels
No.
Feed (mm/r)
Speed of cut (m/min)
Depth of cut (mm)
1
0.02
80
0.01
2
0.05
120
0.04
3
0.08
160
0.07
4
0.11
200
0.10
Table 5
Groups’ results of orthogonal tests
No.
Feed (mm/r)
Speed of cut (m/min)
Depth of cut (mm)
1
0.02
80
0.01
2
0.02
120
0.04
3
0.02
160
0.07
4
0.02
200
0.10
5
0.05
80
0.04
6
0.05
120
0.01
7
0.05
160
0.10
8
0.05
200
0.07
9
0.08
80
0.07
10
0.08
120
0.10
11
0.08
160
0.01
12
0.08
200
0.04
13
0.11
80
0.10
14
0.11
120
0.07
15
0.11
160
0.04
16
0.11
200
0.01
Since the cutting platform used in the test does not have the function of maintaining a constant linear speed with infinitely variable speed, the constant speed when cutting the end face is calculated based on the linear speed designed in the test and the diameter of the middle position of the end face of the disk. The middle diameter of the end face of the fine-grain workpiece is 65 mm, at the mean diameter of the three end faces of the coarse grain workpiece is 55 mm, 83 mm, and 111 mm from the inside out. In the final collection of cutting force, cutting temperature data, we selected the middle data for analysis to ensure that the corresponding linear speed and the test design was consistent. When measuring the residual stress on the machined surface of the workpiece, since the collimator aperture of the X-ray diffractometer is 2 mm, it can be considered that as long as the measuring point is at the middle position of the end face of the disc, the measured value is the corresponding linear velocity in the test scheme.
3 Results and Discussion
3.1 Influence of Grain Size on Cutting Force and Heat
Similarly, Hou et al. [ 21] measured the milling force and surface roughness of the workpiece after milling K4169 superalloy with different grain sizes, and the results showed that smaller grain size was conducive to improving the machinability of the material. Cutting force is one of the underlying physical parameters in the cutting process, which directly affects the generation of cutting heat and then the generation of residual surface stress. Many factors are affecting cutting force, including cutting parameters, tool wear, tool coating, material state of the workpiece and so on. Figure 5 shows the comparison of three-way cutting force and cutting temperature of two kinds of pure iron materials with grain size in different feed rates when the speed of cut is 120 m/min, and the depth of cut is 0.04 mm.
It can be seen from Figure 4, the cutting force and cutting temperature of the two kinds of grain increase gradually with the increase of feed, and the value of the main cutting force is the largest, and the value of axial thrust force is the smallest. The figure also shows that the main cutting force increases with the feed the fastest, and the main cutting force of coarse grain is above the fine grain. Within the range of experimental parameters, the three-way cutting forces measured in both coarse and fine grain states is small, and the difference percentage is within 20%. However, as can be seen from Figure 4, the cutting temperature in the fine-grain state is significantly lower than that in the coarse grain state, and the difference percentage is between 6% and 30%.
Figure 6 shows the comparison of three-way cutting force and cutting temperature of two kinds of grain size in different cutting depths when the feed rate is 0.02 mm/r and speed of cut is 160 m/min. With the increase of depth of cut, the cutting force and cutting temperature of two kinds of grain increase gradually. Moreover, the value of the main cutting force is the largest, and the value of axial thrust force is the smallest, and the main cutting force increases the fastest. Within the range of experimental design parameters, the percentage of three-way cutting force difference between coarse and fine grain materials is within 14%. As can be seen from Figure 5, except for the depth of cut of 0.05 mm (may measure error), the cutting temperatures in the fine-grain state are significantly lower than those in the coarse grain state, and the difference percentage is between 10% and 26%.
Figure 7 shows the three-way cutting force and cutting temperature of two-grain states processed at different cutting speeds when the feed rate is 0.05 mm/r, and the depth of cut is 0.04 mm. With the increase of speed of cut, the cutting force gradually decreases. The figure shows that the main cutting force is the largest and the most apparent reduction, and the axial thrust force value is the minimum, while the main cutting force is significantly higher than fine-grain except for a few groups of coarse crystal pure iron (may measure error). The cutting temperature increases with the increase in cutting speed. The cutting temperature of pure iron in the fine-grain material state is generally smaller than that of pure iron in the coarse grain material state. The numerical deviation is 7.2% to 21%.
Figure 8 shows the chip outflow process when cutting the pure iron end face at different cutting speeds recorded by the high-speed camera.
Due to the high plasticity and high toughness of pure iron, when the cutting speed is low, the chip deformation is large, and it is easy to cold weld with rake face and bond on the tool surface, and then increases the cutting force, as shown in Figure 8(a). After further increasing the speed of cut, the friction between the bottom of the chip and rake face becomes more intense. The friction heat increases and the flow performance of the chip and rake face increases, which makes the chip slides out along the front surface more smoothly, as shown in Figure 8(b), the retention phenomenon of chip decreases or even disappears, thus reducing the cutting force.
3.2 Influence of Grain Size on Surface Residual Stress
A total of 16 groups of tests with three factors and four levels were used in the orthogonal test, aiming to analyze the influence of the residual stress on the machined surface of pure iron material in precision turning with the state of grain material under different cutting parameters and indicate the direction of cut and feed respectively, as shown in Figure 4. The test results are shown in Table 5.
In order to facilitate the comparison, the calculation formula of residual stress difference percentage w(1) and w(2) is defined as follows:
$$w_{{\sigma_{c} }} = \frac{{\sigma_{c} ({\text{Fine grain}}) - \sigma_{c} \text{(Coarse grain)}}}{{\sigma_{c} \text{(Coarse grain)}}} \times 100\% ,$$
(1)
$$w_{{\sigma_{f} }} = \frac{{\sigma_{f} ({\text{Fine grain}}) - \sigma_{f} \text{(Coarse grain)}}}{{\sigma_{f} \text{(Coarse grain)}}} \times 100\% .$$
(2)
According to the analysis results in Table 6, compared with coarse grain pure iron, the average difference percentage is − 14.66%, and the average difference percentage is − 12.50%, indicating that the residual surface stress of fine grain pure iron is less than that of coarse grain pure iron.
Table 6
Measurement results of residual stresses on the surface of coarse and fine grain pure iron
Number
Coarse grain
$$\sigma_{c}$$(MPa)
Fine grain
$$\sigma_{c}$$(MPa)
$$w_{{\sigma_{c} }} \%$$
Coarse grain
$$\sigma_{f}$$ (MPa)
Fine grain
$$\sigma_{f}$$(MPa)
$$w_{{\sigma_{f} }} \%$$
1
214
219
↓− 73.89
− 45
− 12
↑2.22
2
249
270
↑18.00
75
89
↑8.75
3
309
312
↑16.54
127
148
↑0.81
4
381
234
↓− 21.30
100
79
↓− 38.58
5
356
359
↓− 2.80
143
139
↑0.84
6
403
314
↑18.75
224
266
↓− 22.19
7
438
425
↑37.11
113
154
↓− 3.08
8
439
365
↓− 36.02
220
141
↓− 16.91
9
409
321
↓− 30.18
164
115
↓− 21.56
10
461
386
↑26.39
144
182
↓− 16.37
11
401
396
↓− 57.41
270
115
↓− 1.43
12
396
431
↑8.23
216
234
↑8.71
13
394
432
↓− 13.25
145
126
↑9.72
14
414
413
↑36.44
159
216
↓− 0.18
15
428
430
↑23.61
229
283
↑0.41
16
384
439
↑9.78
276
303
↑14.26
The average main effect diagram reflects how each factor affects response characteristics. When different levels of factors have different effects on response characteristics, there is a main effect [ 22]. For different levels corresponding to the surface, residual stress should calculate the average, and then draw a graph line. By comparing the slope, the difference of the relative influence of cutting parameters on the two kinds of pure iron grain size can be analyzed.
As shown in Figure 9, on the main cutting force direction, surface residual stress of two kinds of state grain is most affected by the feed. The depth of cut has greater effect on residual surface stress of coarse grain than the fine grain. With cutting speed and cutting depth increases, the residual surface stress of fine-grain appears certain fluctuation, but do not increase or decrease dramatically. In the direction of feed, the feed has the most significant influence on the residual surface stress of two-grain size. When the depth of cut increases, the residual stress fluctuates to different degrees, and cutting parameters have a smaller effect on surface residual of fine-grain than coarse grain.
4 Conclusions
This work presents an endeavor to obtain better machinability of pure iron. This experimental study has put forward a clear comparison of two-grain sizes, such as a level 0 and 5‒6 levels of the two kinds of grain size. Essential output responses, namely, cutting force, temperature, and machining residual stress were evaluated on the precision turning of pure iron. The following conclusions have been derived based on obtained results.
(1)
The cutting force results show that the effect of cutting parameter on the main cutting force is the greatest, and on the base of the influence of grain size is also the greatest. The cutting force of fine grain is less 2 N than coarse grain, but in general, the influence of cutting force of grain size in the finish turning process is not obvious.
(2)
The cutting temperature results show that the cutting temperature of fine-grain was smaller 50 °C than the coarse grain. This indicates that fine grain of pure iron in finishing can get smaller the cutting temperature.
(3)
Surface residual stress results show that every cutting parameter of fine-grain overall less surface residual stress than coarse grain. Under the same parameters after processing, the residual surface stress of fine grain of pure iron is less than the coarse grain, which obtains better cutting machinability.
Acknowledgements
The authors sincerely thank to Ning He and Muhammad Jamil of College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics for their critical discussion and reading during manuscript preparation.
Authors’ Information
Yinfei Yang, born in 1982, is currently the associate professor at College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China.
Lu Jin, born in 1994, is currently a master candidate at College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China.
Jinpeng Zhu, born in 1993, is a master candidate at College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China.
Jinxing Kong, born in 1987, is doctoral candidate at College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China.
Liang Li, born in 1959, is currently a professor at College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, China.
Competing Interests
The authors declare no competing financial interests.
Literatur
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https://tex.stackexchange.com/questions/20895/difference-between-footcite-and-cite-in-biblatex/20936 | # Difference between \footcite and \cite in BibLaTeX
Since recently, I am having some trouble with the short/ibidem and full citations in BibLaTeX. When I use a \cite command inside a footnote, it behaves differently than \footcite: it seems each keeps its own track of whether a citation has been used before. This simple example shows the ibid tracker works differently inside and outside a footnote:
\documentclass[a4paper]{memoir}
\usepackage[russian, english, UKenglish]{babel}
\usepackage[style=british,english=british]{csquotes}
\selectlanguage{UKenglish}
\usepackage[backend=biber,style=verbose-ibid]{biblatex}
\bibliography{test}
\begin{document}
Check.\footcite{Book1}
Check.\footcite{Book1}
Check.\footnote{\cite{Book1}.}
Check.\footcite{Book1}
Check.\footcite{Book2}
Check.\footnote{\cite{Book2}.}
\end{document}
This produces the following output:
• footnote 1: full citation 1
• footnote 2: ibid.
• footnote 3: full citation 1
• footnote 4: ibid.
• footnote 5: full citation 2
• footnote 6: full citation 2
I should think footnotes 3 and 6 should also be ibid., right? Or at least I would want them to be.
Another example more likely to occur in my writing practice is the following, in which I use cite and parencite in short footnote comments:
\documentclass[a4paper]{memoir}
\usepackage[english, UKenglish]{babel}
\usepackage[style=british,english=british]{csquotes}
\selectlanguage{UKenglish}
\usepackage[backend=biber,style=verbose-ibid]{biblatex}
\bibliography{test}
\begin{document}
Check 1.\footcite{Book1}
Check 2.\footcite{Book2}
Check 3.\footcite{Book1}
Check 4.%
\footnote{As the above author put it, life is just a box of chocolates'
\parencite{Book1}. This is pure nonsense, of course.}
Check 5.%
\footnote{There are various opinions on this matter. A very silly one is given
by author number one \parencite{Book1}. A much more clever approach has
recently been suggested in \cite{Book3}. I propose to follow this second
course.}
Check 6.\footcite{Book1}.
Check 7.\footcite{Book3}.
\end{document}
The first results are:
• footnote 1: full citation Book1
• footnote 2: full citation Book2
• footnote 3: short citation Book1
This is exactly how it should be. But then, when I use citation Book1 as a parencite within footnote 4, it is again a full citation, which is not what I want, since the full citation has appeared above. Likewise, in footnote 7, I get a full citation of Book3, which has already appeared in full in footnote 5. Finally, footnote 6 says Ibid., which is confusing, since it refers to the footcite command of footnote 3, not to what happened in footnote 5.
The last problem, with the ididtracker can be solved manually; I other ones I find more difficult. The situation as a whole, I'm afraid, is a mess. How to improve it? Am I doing something wrong here? I am now using Biber 0.9.2, MikTeX 2.9, and BibLaTeX 1.4.
• Huh. That's interesting. Why are you using both? Why not stick to one or the other. With \footcite[Text before cite][text after]{cite} you can put text either side of a citation in a footnote. – Seamus Jun 16 '11 at 10:25
This is quite interesting -- you seem to have stumbled upon a conflict between the memoir class and biblatex. Using the book class correctly yields "ibid." for citations no. 3 and 6 in your first example, but using the memoir class doesn't.
% Using the "book" class correctly yields "ibid." for citations no. 3 and 6
\documentclass{book}
% But using the "memoir" class doesn't
% \documentclass{memoir}
\usepackage[style=verbose-ibid]{biblatex}
\begin{filecontents}{\jobname.bib}
@misc{Book1,
author = {Author, A.},
year = {2001},
title = {Alpha},
}
@misc{Book2,
author = {Buthor, B.},
year = {2002},
title = {Bravo},
}
\end{filecontents}
\begin{document}
Check.\footcite{Book1}
Check.\footcite{Book1}
Check.\footnote{\cite{Book1}.}
Check.\footcite{Book1}
Check.\footcite{Book2}
Check.\footnote{\cite{Book2}.}
\end{document}
• Interesting, where does those \abx@aux@X come from in the aux file? In memoir they are not the same as in book – daleif Jun 17 '11 at 8:44
• many thanks for your perceptive analysis. memoir seems to have caused the problem. This conflict was way too complicated for me to handle, but it seems everything works fine again in BibLaTeX 1.5 and Biber 0.9.3 (which were not yet packaged by MikTeX). Let's hope that this will remain so in subsequent versions. My apologies for not using the very latest versions. – Rutger Jun 20 '11 at 6:36
In my opinion you are doing things wrong somehow.
Don't use \cite inside a \footnote when you can use a \footcite, since \footcite IS \cite inside a \footnote! \footcite allows you to put the citation, pre and post notes (EDIT: like Seamus demonstrated), and even to do multiple citations (with the \footcites command)... I don't see any reason why you could be interested in doing a \cite inside a \footnote and not a \footcite.
If you need to add footnotes not related to any citation, I strongly advise you to add a \mancite command inside your \footnote, like this:
\footnote{\mancite Bla bla}
`
It will reset ibid tracker, for they are usually tracking only footcites citations.
• and @Seamus: Sorry, I must have made my example a bit too simplified. In my opinion, footcites is not a full equivalent of placing various cites into a footnote. And it seems the developers of BibLaTeX think so, too, for they have created \smartcite, which is designed to behave differently outside and inside a footnote. What I want to do, every now and then, is give a brief discussion of various sources within a footnote; and if possible I would like to avoid squeezing this discussion in a messy collection of pre- and postnotes. I'll try to make a better example. – Rutger Jun 16 '11 at 15:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.768964946269989, "perplexity": 2143.997450357181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540508599.52/warc/CC-MAIN-20191208095535-20191208123535-00068.warc.gz"} |
http://debruijnsequence.org/db/home | # De Bruijn Sequence and Universal Cycle Constructions
A de Bruijn (DB) sequence is a circular string of length $k^n$ where every string of length $n$ over an alphabet of size $k$ appears exactly once as a substring.
Example $n=4$ and $k=2$ 0000101101001111 is a DB sequence where the 16 unique substrings of length 4 visited in order are: 0000 0001 0010 0101 1011 0110 1101 1010 0100 1001 0011 0111 1111 1110 1100 1000
The following different construction methods are presented:
• Greedy methods
• Shift rules (binary)
• Shift rules ($k$-ary)
• Concatenation approaches
• Euler cycle approach
• LFSR and NFSR
Additionally, there is an algorithm to decode the lexicographically smallest DB sequence (the Granddaddy), an implementation of the cutting down variant , and information on the de Bruijn torus .
More generally, a universal cycle for a set $\mathbf{S}$ of length $n$ strings, is a circular string of length $|\mathbf{S}|$ where every string in $\mathbf{S}$ appears exactly once as a substring. Universal cycle constructions are provided for:
• Permutations (shorthand)
• Weak orders
• Weight range $k$-ary strings and relatives | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4908948242664337, "perplexity": 802.8525308314053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314638.49/warc/CC-MAIN-20190819011034-20190819033034-00530.warc.gz"} |
http://mathhelpforum.com/math-topics/110998-problem.html | # Math Help - problem!!
1. ## problem!!
There is a trip between A to B of 100km.
It is divided into 5 parts of 20 km each
A man needs to cover this distance
He can cover 20 km in each day
At A unlimited quantity of food is there
For every 20 km the man needs to take 1 food
the man can carry a maximum of three foods
he can drop the food in between the path wherever he likes
how will he reach the destination
2. The way I understand what you're saying (which isn't too clear),
all he has to do is have 3 "foods" at the 40 point (end of day2);
then he takes all 3, eats one 40-60, one 60-80 and one 80-100.
3. Originally Posted by jashansinghal
There is a trip between A to B of 100km.
It is divided into 5 parts of 20 km each
A man needs to cover this distance
He can cover 20 km in each day
At A unlimited quantity of food is there
For every 20 km the man needs to take 1 food
the man can carry a maximum of three foods
he can drop the food in between the path wherever he likes
how will he reach the destination
4. yes i know the answer. ....
and i dont know what are you taking this question as....
5. Ok then, since you KNOW the answer:
I think your problem can be re-worded this way:
Jack needs to get from A to B, a distance of 100 km.
Jack can only walk 20 km each day.
At A is an unlimited number of water bottles.
Jack needs to drink one bottle of water for each 20 km.
And Jack can carry a maximum of only 3 bottles.
Jack may leave bottles at any spots from A to B.
How can Jack reach B ?
My solution:
1: take 3 bottles, walk 20 km, leave one at 20 km, return : 2 days
2: repeat 3 more times: total 8 days (4 bottles at 20 km)
3: go to 20 km : total 9 days (6 bottles at 20 km)
4: repeat above for 3 days: total 12 days (3 bottles at 40 km)
5: walk the remaining 60 km, drinking 1 bottle each day: total 15 days.
That's a "quick" solution, since you did not specify MINIMUM.
6. Originally Posted by jashansinghal
yes i know the answer. ....
and i dont know what are you taking this question as....
MrF means: "do you know how to solve this?" and not "is the answer at the back of the book?".
If you do know how to solve this and you have posted this as a challenge to other member then this is where the question belongs.
However if you do not know how to solve this and are look for help in solving it then it is not a challenge problem as defined in this forums title.
The question is being asked of this problem because to the staff it looks like the latter rather than the former
CB
7. okk ....i am sorry and the last answer posted is correct
8. Originally Posted by jashansinghal
okk ....i am sorry and the last answer posted is correct
WHY is my answer (15 days) "correct"?
Can you PROVE it's the minimum ?
9. no I CANT | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8557891249656677, "perplexity": 1413.1598272774304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462898.92/warc/CC-MAIN-20150226074102-00028-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://www.ck12.org/book/Basic-Speller-Student-Materials/r1/section/8.4/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: Basic Speller Student Materials Go to the latest version.
8.4: Sometimes [t] is Spelled < ed >
Difficulty Level: At Grade Created by: CK-12
Sometimes [t] is Spelled <ed>
1. Look at these sentences and fill in the blank:
He coughs a lot.
Last night he coughed all night long.
When you want to add the meaning “in the past” to a verb, usually you add the suffix _______.
2. The suffix -ed sometimes sounds like [d], sometimes like [id], and sometimes like [t]. Say each of the following words carefully and sort them into the three groups:
$&\text{addressed} && \text{approached} && \text{struggled} && \text{shoveled}\\&\text{adopted} && \text{collected} && \text{enjoyed} && \text{attached}\\&\text{accomplished} && \text{allowed} && \text{taxed} && \text{announced}\\&\text{murmured} && \text{assigned} && \text{attended} && \text{avoided}\\&\text{attacked} && \text{approved} && \text{coughed} && \text{telephoned}$
3. Sometimes the [t] at the end of a verb that has the meaning “in the past” is the suffix ______.
4. So far you have worked with three different spellings of [t]. They are ______, ______, and ______.
Word Scrambles. This Scrambles contains words that all contain the sound [t]. We have given you a start by filling in the three spellings of [t].
Subjects:
1 , 2 , 3 , 4 , 5
Feb 23, 2012
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https://iacr.org/cryptodb/data/author.php?authorkey=2570 | ## CryptoDB
### Kim Nguyen
#### Publications
Year
Venue
Title
2007
CHES
2003
JOFC
2001
EPRINT
In many cases, the security of a cryptographic scheme based on Diffie--Hellman does in fact rely on the hardness of the Diffie--Hellman Decision problem. In this paper, we show that the hardness of Decision Diffie--Hellman is a much stronger hypothesis than the hardness of the regular Diffie--Hellman problem. Indeed, we describe a reasonably looking cryptographic group where Decision Diffie--Hellman is easy while Diffie--Hellman is equivalent to a -- presumably hard -- Discrete Logarithm Problem. This shows that care should be taken when dealing with Decision Diffie--Hellman, since its security cannot be taken for granted.
Antoine Joux (2) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8386105298995972, "perplexity": 1318.3855047221323}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00376.warc.gz"} |
https://www.lessonplanet.com/teachers/density-review | # Density Review
In this density worksheet, students review how to calculate density and how to apply Archimedes' Principle, Boyle's Law, and Charles' Law. This worksheet has 10 matching, 13 short answer, and 6 fill in the blank questions.
Concepts
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https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-4-number-representation-and-calculation-4-3-computation-in-positional-systems-exercise-set-4-3-page-234/6 | # Chapter 4 - Number Representation and Calculation - 4.3 Computation in Positional Systems - Exercise Set 4.3: 6
$1244_{five}$
#### Work Step by Step
Rightmost column: $3+1=4=4\times1=4_{five}$ record the 4, no carry . .$\$3 2 $3_{five}$ $+$4 2 $1_{five}$ $\_ \_ \_ \_ \_$ $\qquad 4_{five}$ Next column: $2+2=4=4\times1=4_{five}$ record the 4, no carry . .$\ \$3 2 $3_{five}$ $+$4 2 $1_{five}$ $\_ \_ \_ \_ \_$ $\quad \ \ 4 \ 4_{five}$ Next column: $3+4=7=1\times 5+2\times1=12_{five}$ record the 12 $\ \$3 2 $3_{five}$ $+$4 2 $1_{five}$ $\_ \_ \_ \_ \_$ $1\ 2\ 4 \ 4_{five}$ Check in base 10: $3\times 25+2\times 5+3=88$ $4\times 25+2\times 5+1=111$ $1\times 125+2\times 25+4\times 5+4=199$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2726280987262726, "perplexity": 2199.517447624797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864848.47/warc/CC-MAIN-20180623000334-20180623020334-00530.warc.gz"} |
http://mathoverflow.net/questions/86404/quantum-e6-e7-knot-polynomials/86518 | # Quantum E6/E7 knot polynomials
Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8?
I suspect these haven't been worked out, but if anybody knows of a reference containing these or even just discussing them, please let me know.
-
It is certainly possible to evaluate these for links which can be obtained as the closure of a braid with at most three strings. – Bruce Westbury Jan 22 '12 at 22:17
I'm pretty sure that the quantum groups Mathematica package which is part of the Knot Atlas package (katlas.org/wiki/Main_Page) can do this. It'll probably be pretty slow. You can ask Scott Morrison if you want more info. – Noah Snyder Jan 22 '12 at 22:17
I spoke to Scott a couple of days ago. His package cannot compute quantum invariants for F4 or En due to the complexity of some intermediate computations. He noted that one could avoid these issues by, e.g. writing down the quantum positive roots in terms of the PBW bases... working this out is [currently] a bit beyond my primitive background in QA. However, if anybody knows of a good source for such things, that would be equally useful. – Ross Elliot Jan 22 '12 at 22:38
Ok, my bad. I'd used it for some more basic quantum En calculations before, but I guess they were all a lot simpler than what you need. – Noah Snyder Jan 23 '12 at 0:15
My recollection from conversations with Scott is that the main obstacle is that arithmetic with rational functions is slow. For the 3-strand braid group you're already talking about composing matrices that are roughly 20,000 by 20,000. – Noah Snyder Jan 23 '12 at 16:59
$(E7, 56)$ belongs to a series for which I computed some skein relations in my phd thesis (see p55-56 of http://web.univ-ubs.fr/lmam/patureau/articles/these.ps.gz with $Y=X^{19}$). Unfortunately the set of skein relations is not complete but you can use it to compute the quantum invariants of small knots.
-
Dear Bertrand & Ross - could you check the following? I neither speak French nor Math :-) but I computed the same skein relations with magic and trickery (so my results are unproven, of course). I get 11+6+2*8+2*3=39 basis 3+3 tangles (which I here already split into the D6h symmetry classes) and under the E7 family polynome 10+5+2*7+2*3=35 are independent. The linear rest should be your skein relations. So, are there exactly 4 (including symmetry-transformed!) of them, i.e. your Figs. 2/3? Since they don't look symmetric under rotation, it's hard to check for me.
(BTW, for 2+2 tangles, I "know" an even generalized result (of your Fig. 1) for 20 years :-)
Maybe going to 4+4 tangles could solve the problem of having a complete reducing skein set (or at least give an unproven solution) but since that might need hundreds of basis tangles you a) either have to crowdsource the calculation or b) do it with birdtracks or c) use math (I'm out then :-)
EDIT: Now that I'm back at my PC with access to the literature, 4+4 doesn't look too promising either. Kuperbergs G2 paper says there are 455 crossingless freeways with 8 endpoints. Uck. (And under E7 instead, a lot of them are inaccessible.) So if one could do the following: 1. Prove that: IF you can reduce 1-gons and 2-gons and two adjacent 3-gons (works for all the E7 family) AND an adjacent 3- and 4-gon pair AND two 3-gons touching on a corner (five crossings; i.e. simpler diagrams now span the 8-endpoint vector space...or so I hope!) THEN you could reduce any link diagram, and: 2. the compatibility rules of this reducing set are finite (I don't have much hope for that either!)... the whole E7 family polynomial proof would reduce to skein diagram manipulation (of gargantuan size, of course). Yup, it's probably better to use math instead :-)
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8122326135635376, "perplexity": 715.1685658560888}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701152130.53/warc/CC-MAIN-20160205193912-00243-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/89807-exponential-function-derivative.html | # Thread: exponential function derivative
1. ## exponential function derivative
Can anyone help me find the derivative of y=2^|x|?
2. Originally Posted by nejikun
Can anyone help me find the derivative of y=2^|x|?
If $f(x) = a^x$ then $f'(x) = a^xln(a)$
Spoiler:
Take the log of both sides:
$ln(y) = ln(2^{|x|}) = |x|ln(2)$
Differentiate both sides:
$\frac{1}{y} \cdot \frac{dy}{dx} = ln(2)$
Isolate dy/dx:
$\frac{dy}{dx} = yln(2) = 2^{|x|}ln(2)$
3. Originally Posted by nejikun
Can anyone help me find the derivative of y=2^|x|?
Hi nejikun.
$y\ =\ 2^{|x|}\ =\ e^{(\ln2)|x|}\ =\ e^u$ where $u=(\ln2)|x|$
So, using the chain rule, $\frac{dy}{dx}\ =\ \frac{dy}{du}\,\frac{du}{dx}\ =\ e^u\cdot(\ln2)\frac{|x|}x\ =\ \frac{(\ln2)|x|}x2^{|x|}.$
Note that $\frac{d\left(|x|\right)}{dx}=\frac{|x|}x.$
4. Originally Posted by e^(i*pi)
$ln(y) = ln(2^{|x|}) = |x|ln(2)$
Differentiate both sides:
$\frac{1}{y} \cdot \frac{dy}{dx} = ln(2)$
The derivative of $|x|$ is 1 if $x>0$ and $-1$ if $x<0;$ i.e. $\frac{d(|x|)}{dx}=\frac{|x|}x.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9891014099121094, "perplexity": 1105.1447549373777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823360.1/warc/CC-MAIN-20171019175016-20171019195016-00340.warc.gz"} |
https://quant.stackexchange.com/questions/21734/call-and-put-prices-equal-at-forward-price-why | # Call and Put Prices Equal at Forward Price - Why?
Consider a European call and put with values $C_t$ and $P_t$, respectively, under the Black-Scholes model. By put-call parity, $$C_t - P_t = S_t - Ke^{-r(T-t)}$$ for expiration time $T$. Note if $K = S_te^{r(T-t)}$ we get $$C_t = P_t. \qquad (1)$$ Of course, $S_te^{r(T-t)}$ is the time $T$-forward price of the stock at time $t$, which is arrived at from a no arbitrage argument and not just taking expectations. That is, $$S_te^{r(T-t)} \neq E_P(S_T \mid S_t) = S_te^{\mu(T-t)},$$ where $P$ is the physical measure and $\mu$ the drift rate.
I see that (1) holds simply by put-call parity, but I'm seeking a deeper understanding. Is it that, the call and put prices are equal because, under the risk-neutral measure $Q$, the expected value of the stock is the strike price (which is the forward price in this case)? That is, $$S_te^{r(T-t)} = E_Q(S_T \mid S_t) = K,$$ and hence the stock is equally likely to finish above or below the strike? Or, is there something deeper going on, like a no arbitrage argument?
Certainly, you must agree that $$C_{T}-P_{T}=\left(S_{T}-K\right)^{+}-\left(K-S_{T}\right)^{+}=S_{T}-K.$$ Therefore, since $$C_{t}=e^{-r\left(T-t\right)}E_{Q}\left[C_{T}\right]\text{ and }P_{t}=e^{-r\left(T-t\right)}E_{Q}\left[P_{T}\right]$$ it follows by the linearity of $E$ that $$C_{t}-P_{t}=e^{-r\left(T-t\right)}E_{Q}\left[C_{T}-P_{T}\mid \mathcal{F}_{t}\right]=e^{-r\left(T-t\right)}E_{Q}\left[S_{T}-K\mid \mathcal{F}_{t}\right]=e^{-r\left(T-t\right)}\left(e^{r(T-t)}S_{t}-K\right).$$ The put-call parity follows.
The Call-Put parity $$C_{T}-P_{T}=\left(S_{T}-K\right)^{+}-\left(K-S_{T}\right)^{+}=S_{T}-K$$ will exist under any condition, since it is pretty much a mathematical fact.
However, I think the straight answer to your question $$K = S_te^{r(T-t)} => C_{t} = P_{t}$$ is : not always, but only under an efficient market i.e. no-arbitrage, frictionless and complete market.
You can think of a not-so-rare market scenario when a very popular company issues its first stocks. Market expectations from the stock are high. It is not unusual to find call options to be highly overpriced compared to puts. Even when $$K = S_te^{r(T-t)}$$, you may find that $$C_{t} >> P_{t}$$. This may seem like a case of transient arbitrage, but think of an opposite scenario in case of a distressed stock - lack of liquidity may be the problem there. The efficient market hypothesis is not valid in these cases.
Fundamentally I think the definition of the forward price is the one that leads to call and put prices being equal, and not the other way round, i.e. $$C_{t} = P_{t} => K = E\left[S_{T} \right]$$ This is converse to your question, but there is a subtle difference here which is important.
This forward price is the expected future price of the asset under some measure. It so happens that only under efficient market hypothesis a risk-neutral measure $$Q$$ can be shown to exist, and you can use this to prove that this forward price is equal to the expected future price $$K = E_{Q}\left[S_{T} | S_{t} \right] = S_te^{r(T-t)}$$.
A speculator can define an alternate probability measure $$P$$ under which $$E_{P}\left[S_{T} | S_{t} \right] > S_te^{r(T-t)}$$. He will speculate different Call and Put prices under this measure, but the Call-Put Parity will still hold. $$C_{t}^{P} - P_{t}^{P} = E_{P}\left[(S_{T} - K)^{+}| S_{t}\right] - E_{P}\left[(K - S_{T})^{+} | S_{t}\right] = E_{P}\left[S_{T} | S_{t}\right] - K > 0$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9367477297782898, "perplexity": 316.78404025429245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987828425.99/warc/CC-MAIN-20191023015841-20191023043341-00063.warc.gz"} |
https://doc.cgal.org/5.4-beta1/Number_types/structCGAL_1_1Mpzf.html | CGAL 5.4 - Number Types
CGAL::Mpzf Struct Reference
#include <CGAL/Mpzf.h>
## Definition
An object of the class Mpzf is a multiple-precision floating-point number which can represent numbers of the form $$m*2^e$$, where $$m$$ is an arbitrary precision integer based on the GMP library, and $$e$$ is of type int.
This type can be considered exact, even if the exponent is not a multiple-precision number. A Mpzf constructed from an integer or a normalized finite floating point number represents exactly that number. This number type offers functionality very similar to MP_Float and Gmpzf but is faster.
Is Model Of:
IntegralDomainWithoutDivision
RealEmbeddable
Implementation
This class is only available on platforms on which GMP uses 64 bit limbs and the endianness is either big-endian or little-endian. The macro CGAL_HAS_MPZF will be defined when including CGAL/Mpzf.h if this is true. This class makes the assumption that the representation of a double in memory follows IEEE 754.
Currently, an Mpzf contains an array of a few limbs, in which it stores the data as long as it fits. If it does not fit, it dynamically allocates memory with new, and de-allocates it when it is done. Code to recycle the allocated memory (per thread) is included but disabled at the moment.
## Related Functions
(Note that these are not member functions.)
std::ostream & operator<< (std::ostream &out, const Mpzf &f)
writes a double approximation of f to the ostream out.
std::istream & operator>> (std::istream &in, Mpzf &f)
reads a double from in, then converts it to a Mpzf.
## Creation
Mpzf ()
creates a Mpzf initialized with 0.
Mpzf (int i)
creates a Mpzf initialized with i.
Mpzf (long int l)
creates a Mpzf initialized with l.
Mpzf (const Gmpz &i)
creates a Mpzf initialized with i.
Mpzf (const mpz_class &i)
creates a Mpzf initialized with i.
Mpzf (double d)
creates a Mpzf initialized with d.
## Conversion
operator Gmpq () const
creates a Gmpq initialized with *this.
operator mpq_class () const
creates a mpq_class initialized with *this. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2858659625053406, "perplexity": 8924.327467439285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00682.warc.gz"} |
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0069301 | Research Article
# A Strategy to Model Nonmonotonic Dose-Response Curve and Estimate IC50
• hui.zhang@stjude.org
Affiliation: Department of Biostatistics, St. Jude Children's Research Hospital, Memphis, Tennessee, United States of America
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• Affiliation: Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, New York, United States of America
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• Affiliation: Department of Medicine and Microbiology and Immunology, University of Rochester Medical Center, Rochester, New York, United States of America
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• Affiliation: Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, New York, United States of America
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• Published: August 01, 2013
• DOI: 10.1371/journal.pone.0069301
## Abstract
The half-maximal inhibitory concentration IC is an important pharmacodynamic index of drug effectiveness. To estimate this value, the dose response relationship needs to be established, which is generally achieved by fitting monotonic sigmoidal models. However, recent studies on Human Immunodeficiency Virus (HIV) mutants developing resistance to antiviral drugs show that the dose response curve may not be monotonic. Traditional models can fail for nonmonotonic data and ignore observations that may be of biologic significance. Therefore, we propose a nonparametric model to describe the dose response relationship and fit the curve using local polynomial regression. The nonparametric approach is shown to be promising especially for estimating the IC of some HIV inhibitory drugs, in which there is a dose-dependent stimulation of response for mutant strains. This model strategy may be applicable to general pharmacologic, toxicologic, or other biomedical data that exhibits a nonmonotonic dose response relationship for which traditional parametric models fail.
### Motivation
Drugs that inhibit the reverse transcriptase (RT) activity of Human Immunodeficiency Virus (HIV) are widely used to treat HIV infection. RT is an ideal target for antiviral HIV therapy because it is the key required for HIV replication. Non-nucleoside reverse transcriptase inhibitors (NNRTIs) inhibit RT activity by selectively binding RT at a hydrophobic binding pocket adjacent to the polymerase active site. Efavirenz (EFV) is an commonly used NNRTI to treat HIV infection [1][3] but patients can develop resistance to this drug because of the development of mutations in the NNRTI binding cite which in turn inhibits NNRTI binding [4] and can lead to resistance mutations such as K101E, K103N, Y188C, G190S, G190A [5], [6], and L100I [7].
Understanding the pharmacodynamic properties associated with the development of NNRTI resistant mutations is vital for devising treatment strategies for HIV. In pharmacodynamics, the drug-target interaction can be modeled by:
where denotes the drug and the target (usually enzyme). Drug efficiency is primarily determined by the drug target binding affinity. In pharmacodynamic studies, the drug target affinity is usually assessed by comparing dose response curves: the stronger the drug binds target, the steeper the curve is.
Therefore, a critical index of the dose response curve, the half-maximal inhibitory concentration (IC), is commonly used to compare the binding affinities of drugs to the same target. IC represents the concentration of a drug that is required for 50% of maximal inhibition in vitro. IC and IC are the concentrations corresponding to 25% and 75% inhibition, respectively. The dose response curve usually has the steepest portion in the middle. Thus, using IC, rather than IC or IC, minimizes the random error for estimation, making IC the preferred measure of drug affinity.
To estimate the IC value, several nonlinear functions have been commonly used, for example,(1)
where is the drug concentration, is the percentage of inhibition at this concentration, and is a shape parameter. Other parametric models include the complementary log-log model for asymmetric quantal response data, and the two-parameter Weibull model for carcinogenic experiments [8].
An important feature of the function (1) is that as the value of increases from to infinity, increases from to , reflecting that a drug's inhibitory potential changes from none to full inhibition as the drug concentration increases (Fig. 1A). As per (1), a steeper dose response curve corresponds to a smaller IC; for a given IC, the curve shape is depicted by (Fig. 1B). In addition, this function curve has the steepest portion in the middle, which is a characteristic of a sigmoidal dose response relationship.
The advantages of this function are that (a) it is symmetrically about the IC; (b) it is monotonic, which equals that the target protein has an inhibitory binding site only, and the drug has an inhibitory effect only; and (c) IC and can be easily estimated under certain conditions. On the other hand, these advantages become restrictive when some conditions fail, for instance, if observations are not monotonic. As a consequence, the models can produce remarkably biased estimates or not even fit the observations. For example, Bliss's beetle data show that symmetry is not a required feature of a dose response curve [9]. Another example is that a viral mutation occurs before the drug concentration reaches a certain level, such as in the following example.
A recent study on HIV mutations conferring resistance to NNRTI found that the monotonicity relation does not always hold [10]. The dose response was determined as proportion reduction in HIV replication at a given NNRTI dose relative to viral replication in the absence of drug. As seen in Fig. 2, the study shows that replication of HIV mutation M230L was promoted when the concentration of EFV is lower than 70 nM. Similarly, our data example shows that increasing the concentration of EFV stimulates the replication of HIV K101E+G190S mutant strain when EFV concentrations are below 2000 nM (Fig. 2). It has also been reported that EFV stimulation of the K101E+G190S double mutant strain can be abolished by the presence of additional M41L+T215Y mutation [11]. A potential explanation for this nonsigmoidal dose response relationship is that dimerization is essential for a fully functional RT. For the double mutant K101E+G190S strain, at low concentrations EFV can enhance the dimerization of the two subunits of RT without interfering with the binding of the incoming nucleotide during DNA polymerization.
As the real dose response relationship is nonmonotonic in our example, our preliminary analysis indicates that traditional estimation methods for the sigmoidal model (1) fitting lead to results that either do not converge or are remarkably biased, which may reach an erroneous conclusion. Thus, appropriate estimation of IC for this type of dose response relationship poses statistical challenges. If we use model (1) to fit data when the dose response pattern is nonmonotonic, the fit is poor, and the estimated IC values are not reliable because the fact that lower EFV concentrations can enhance replication of an HIV mutant strain is ignored.
Thus, to appropriately estimate the pattern of observations and then estimate IC, we developed a robust modeling strategy to test whether: (i) our model fitting is comparable to monotonic parametric models such as model (1) when the observed data are monotonic; and (ii) our model fitting yields reasonable estimates when the data pattern is nonmonotonic and monotonic parametric models, such as model (1), does not work.
The rest of this paper is organized as follows. Section 2 briefly introduces monotonicity testing, our model, estimation, and test methods. Section 3 gives simulation results, including p-values of the monotonicity test. Section 4 presents extensive analysis of our real data example, including estimated ICs using the proposed model and model (1) when it is appropriate.
### Methods
We propose that the inhibition percentage and concentration are related in the form(2)
where is the measurement error with mean zero and finite variance, is a mathematical function, but no restrictions are applied on the form of (i.e., no specified as quadratic, parametric, or increasing in , etc.) Since the structure of the model is not fixed, it is called a nonparametric method. Hence, we perform an empirical analysis of the data to estimate . We use the observations to estimate , denoted as , by appropriate statistical techniques. The IC may be estimated as the point that satisfies . As noted above, we first need to determine whether the function is monotonic.
#### Subsection 1 Monotonicity test
Testing the monotonicity of a dose response relationship is of practical interest and has been studies previously. Several parametric and nonparametric methods have been proposed in the statistical literature. For example, Ramsay [12] studied the use of monotone splines to model a dose response function. Bowman et al. [13] developed a monotonicity test by using local linear estimation of the curve, followed by a critical bandwidth test. Hall and Heckman [14] proposed an alternative approach that focuses on “ running gradient” estimation over very short intervals. The method of Hall and Heckman is more effective in estimating the flat part of the curve and is also more sensitive to small dips in the curve.
For our study, we adopt the method of Hall and Heckman, which is based on the following principle. Let be integers and be constants. For each pair of , define the estimators of and by and , as the arguments of the following objective function:
Define
Let , where satisfies . Hall and Heckman [14] stated that a large indicates that the null hypothesis, being monotonic, should be rejected. To obtain the on the basis of , they suggested the following procedure. First, data should be fit with the nonparametric model . An estimation of should be obtained by a consistent estimator of , such as the local linear estimator. Assuming that a constant function is the most difficult nondecreasing form to be tested, Hall and Heckman used to obtain the . Specifically, using the estimated , they resampled and obtained a new dataset , by which they obtained . Repeated sampling n times resulted in a set of size . By taking the th ordered s as the critical value for , that is, when obtained from the real data is greater than this , we claim that the function is nonmonotonic at the level.
#### Subsection 2 Nonparametric fitting
We used local linear regression [15] to fit the dose response curve. Assuming that has bounded, continuous second partial derivatives, by Taylor expansion, , in a neighborhood of , can be approximated as: The estimator of at is the solution of by minimizing subject to (), where is a bandwidth controlling the size of the local neighborhood, , with being a kernel function assigning weights to each data point.
### Simulation
To investigate how the Hall and Heckman [14] test performed for small and moderate sample sizes, we conducted a simulation study. Let and or , for . Consider 2 cases: (a) ; and (b) , where .
Case (a) reflects that and have a monotonic relationship, whereas case (b) indicates that the monotonicity is violated. Case (b) is based on the relationship between the inhibition of HIV mutant K101E+G190S strain and EFV concentration from our real data example, and is meant to show the performance power of the test. Figure 3 depicts the patterns of and , with showing a pronounced dip around .
The error follows a normal distribution , with . Our data are generated from the model (2) with or , and . For each case, we considered 3 sample sizes , and generated independent datasets for each of the 3 error variances.
-value was determined as the probability that is greater than . Table 1 shows the -values of different simulation cases. As is a nondecreasing function, the -values should be greater than , which was set at ; in contrast, -values for should be lower than . All test results based on gave -values greater than , although there was a slight trend of decreasing -values as increases. Thus, on the basis of simulations, the monotonicity test method did not reject the null hypothesis at and correctly concluded that the relationship is monotonic. In contrast, when , all test results based on showed -values lower than , indicating that the null hypothesis (monotonicity) would be correctly rejected even when the sample size is very small. When was increased to and the sample size was as small as 20, smaller values, i.e. higher dip sizes (Fig. 3), still gave -values lower than . However, greater values showed -values slightly higher than 0.05. When the sample size was increased to 50, all -values were lower than 0.05. When the sample size was further increased to 100, all -values were lower than . These results indicate that even with a high noise level, the monotonicity test is still reliable, particularly for large sample sizes. These results show that the monotonicity test is reliable and robust.
### Real Data Analysis
We performed a monotonicity test for all HIV mutation dose response curves for percent viral replication compared to no drug (Fig. 2). For each mutant strain, we repeated the monotonicity test 100 times and have reported the average -value in Table 2.
The null hypothesis of monotonicity in HIV mutants K101E+G190S, M230L, and K101E+G190S(D10) was rejected with (Table 2). This result is consistent with the observed shape of the dose response curves (Fig. 2). Surprisingly, the test failed to reject the monotonicity hypothesis for M41L+K101E+G190S+T215Y(D10) data, probably because the local polynomial fitting of this dataset still gives a non decreasing curve.
We then used the traditional sigmoidal model to fit the data and found that this method did not converge for HIV mutants K101E+G190S, M230L, and K101E+G190S(D10) because of the nonmonotonicity while the model did converge for datasets G190S, K101E, L74V+K101E+G190S, M41L+K101E+G190S+T215Y, M41L+G190S+.
T215Y, M41L+K101E+G190S+T215Y(D10), and M41L+G190S+T215Y(D10). We then fit all datasets again, using the local polynomial regression method (Section 2). Figure 4 shows the fitted curves for datasets K101E+G190S, L74V+K101E+G190S, M230L, and M41L+K101E+G190S+.
T215Y. Fitted curves for datasets L74V+K101E+G190S and M41L+K101E+G190S+T215Y showed that the parametric and nonparametric methods gave comparable results.
As shown in Fig. 4, lower concentrations of EFV clearly stimulate the replication of HIV K101E+G190S and M230L compared to no EFV or at high EFV concentrations. However, this property cannot be recognized by using the traditional sigmoidal model fitting. To test the efficiency of our proposed nonparametric method when the monotonicity property is satisfied, we applied our method to the L74V+K101E+G190S and M41L+K101E+G190S+T215Y datasets. The dose response curves obtained by using the nonparametric model are very similar to those by the sigmoidal model, which confirms the efficiency of the nonparametric method. Table 3 compares the estimated IC values for all datasets obtained by using both methods. When the sigmoidal model works well, as for HIV mutant strains G190S, K101E, L74V+K101E+G190S, M41L+K101E+G190S+T215Y, M41L+G190S+T215Y, M41L+K101E+G190S.
+T215Y(D10), and M41L+G190S+T215Y(D10) (these datasets also satisfy the monotonicity property, as shown in Table 2), the two estimated IC values for the same dataset are close (Table 3). In contrast, because of the lack of monotonicity, the sigmoidal model fails to fit the curves for HIV strains of K101E+G190S, M230L and K101E+G190S(D10) (Table 2). For these datasets, the nonparametric model becomes a better alternative for IC estimations (Table 3).
### Discussion
When the dose response relationship and associated parameters such as IC are studied, data observations, which make the pattern nonmonotonic, are generally deleted in order to use the model (1) or similar monotonic functions. However, by deleting these “unusual” observations, some important information may be lost. For example, in Fig. 2, the observation that a lower EFV concentration stimulates HIV K101E+G190S replication can be neglected if these data points are deleted. Removing the unusual data points leaves only 2 observations in this dataset, making the fitting procedure impossible.
In this paper, we proposed a nonparametric approach as an alternative to the parametric sigmoidal model to fit the dose response curve and estimate IC, and suggested a monotonic check of the dose response relationship at the first stage. If the monotonicity is satisfied, either the traditional sigmoidal model fitting or our local polynomial regression fitting can be applied. If monotonicity is not satisfied, our model is more suited to estimate the IC. Using this new approach, important dose response features will not be omitted. A similar approach has been used to quantify protein lysate assays [16], although no monotonicity needs to be satisfied in that case.
Our proposed method can also be used for other dose response modeling scenarios, such as hormesis dose response curves. In toxicology, hormesis is a special dose response feature characterized by low dose stimulation and high dose inhibition [17][19], giving a J-shape dose response curve. Our nonparametric model is more suited than traditional monotonic models to fit this J-shaped curve. Also, our method can be used to model other nonparametric curves such as U-shaped dose response relationships frequently observed in toxicology and epidemiology studies [20].
The human trefoil peptide (TFF1), a small cysteine-rich secreted protein, stimulates cell migration by chemotaxis. The dose response curve of TFF1 inducing breast cancer cell movement shows a clear bell shape [21]. Similar curves are also seen in may other biomedical studies [22][27]. Our nonparametric model may be more suitable than sigmoidal models to fit such dose response curves and estimate the IC.
Our approach for estimating IC can also be used to estimate the half-maximal effective concentration, which is commonly used when the drug enhances its target's activity, and the lethal dose 50%, or the lethal concentration and time of a toxic substance or radiation represents the dose needed to kill half the tested population. Since the results we obtained are based on large sample theory, a potential limitation of our proposed method is that a moderate sample size may be needed, although a minimum sample size is not determined.
### Acknowledgments
The authors are grateful to Dr. Heckman for sharing her codes, and the Editor and two referees for their valuable comments and suggestions.
### Author Contributions
Conceived and designed the experiments: HZ HL. Performed the experiments: HZ JW HL. Analyzed the data: HZ HL. Contributed reagents/materials/analysis tools: HZ HL. Wrote the paper: HZ JHW HL.
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Ambra 2.9.16 Managed Colocation provided
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https://blender.stackexchange.com/questions/72007/smoke-simulation-not-showing-in-camera | # Smoke Simulation Not Showing in Camera
So I was trying to create a smoke simulation and it shows up fine in the rendered view but as soon as I go to the camera, It disappears.
In the picture, the two sides are the same view and angle but the right is in the camera and the left is not.
Here's the blend file:
Sorry for the late reply for I have been working all day on this. I haven't used the Smoke/Flame physics at all so an extensive amount of experimenting showed that all you have to do is select the Smoke Domain cube and, under Smoke cache, click Bake:
After this your viewport should look something like this:
Then you should be able to see the smoke through the camera and in the final render.
Hope this helps.
• sorry for the super late reply to your late reply but I finally got around to trying this and it works so thanks! – MattTheWaffleCat Jun 14 '17 at 17:00
• Don't mention it – Grimlock Jun 15 '17 at 6:22
This is because left side is rendered mode and right is solid mode. Blender doesn't show smoke and/or fire at full quality in solid mode in the viewport instead it shows only a little bit. If you see closely, in the file you provided, you can see a small white cloud of smoke, but in rendered mode it turns into a dense black mass of smoke.
• if you go into the camera in rendered mode though, does the dense black mass disappear because it does for me – MattTheWaffleCat Jan 24 '17 at 1:32
• I see what you mean. It doesn't show up in the final render either – Grimlock Jan 24 '17 at 23:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4758132994174957, "perplexity": 854.1649279122372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00217.warc.gz"} |
https://www.sdss.org/dr17/algorithms/ancillary/boss/massiveclusters/ | Spectroscopy of Massive Galaxy Cluster Members
Contact
Alexis Finoguenov
University of Helsinki, Finland
Summary
Spectra obtained to find spectroscopic redshifts of massive galaxy clusters selected in x-ray imaging, with the goal of finding the three-dimensional structure of those clusters
Finding Targets
An object whose ANCILLARY_TARGET2 value includes one or more of the bitmasks in the following table was targeted for spectroscopy as part of this ancillary target program. See SDSS bitmasks to learn how to use these values to identify objects in this ancillary target program.
Program (bit name) Bit number Target Description Number of Fibers Number of Unique Primary Objects
CLUSTER_MEMBER 16 Member of a galaxy cluster selected from ROSAT All-Sky Survey data 2,828 2,762
Description
Obtaining spectroscopic redshifts for galaxy clusters serves a broad range of cosmology applications. Spectroscopic confirmation removes the uncertainties and biases associated with photometric redshifts, and is the only method that measures the three-dimensional structure of the cluster and its surroundings. Confirmation of membership removes projection effects and improves the richness estimates of cluster finders. Velocity dispersion estimates for cluster galaxies provide a measurement of dynamical cluster mass that is independent of masses derived from gravitational lensing.
This cluster follow-up program is a prototype for spectroscopic programs that will target cluster candidates in the eROSITA survey. Sources selected for this ancillary target program were candidate member galaxies of clusters, selected as the optical counterparts to faint sources in the ROSAT All-Sky Survey (Voges et al. 1999, Voges et al. 2000).
This ancillary target program increases the number of spectroscopic cluster members from 5 to 20 for rich clusters and from none to a few for low-mass clusters.
Target Selection
Selected sources were optical counterparts to X-ray clusters selected as faint sources in the ROSAT All-Sky Survey (Voges et al. 1999, Voges et al. 2000), identified by applying the redMaPPer (Rykoff et al. 2014) cluster finding algorithm to the position of an X-ray source. The X-ray magnitude limit corresponds roughly to the brightest 30% of clusters that the X-ray satellite eROSITA will find within the BOSS survey area.
Objects denoted by the target flag CLUSTER_MEMBER are selected from the redMaPPer catalog with icmodel < 19.9 and ifib2 < 21.5 to ensure high redshift success rates.
REFERENCES
Rykoff, E. S., et al. 2014,, ApJ, 785, 104
Voges, W., et al. 1999, A&A, 349, 389
Voges, W., et al. 2000, IAU Circ., 7432, 1 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6628455519676208, "perplexity": 4906.911244620526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104277498.71/warc/CC-MAIN-20220703225409-20220704015409-00009.warc.gz"} |
http://math.stackexchange.com/questions/314042/convex-relaxation-for-the-complement-of-lorentz-cone | # Convex relaxation for the complement of Lorentz cone
Is it possible to obtain a convex relaxation for $$\{ (x,t): t \le \|x\|_2\} \in \mathbb{R}^{d+1}$$ where $x \in \mathbb{R}^d$ and $\|x\|_2$ is the usual Euclidean norm, by moving to higher dimensions and projecting back (the so-called lifting)?
-
The image of a convex set under projection (and any linear map) is convex, so the answer appears to be negative. I may have misunderstood the question. – user53153 Feb 27 '13 at 6:33
Thanks. You are right. What I mean is something which approximates this set in some sense. – passerby51 Feb 27 '13 at 16:21
... I am even open to nonlinear maps as projections if the result is useful. (The question is more or less open-ended.) In deriving convex relaxations/approximations to many problems of interest, this set or its sister $\{ (X,t) : t I \preceq X\}$ comes up. – passerby51 Feb 27 '13 at 16:30
One can map the half-plane $\{t\le 0\}$ onto $\{t\le \|x\|\}$ by a reasonably explicit nonlinear map. Would this help? – user53153 Feb 27 '13 at 16:48
It might. Not quite sure without seeing it. Could you give more details if it is not too much effort? – passerby51 Feb 28 '13 at 18:21
The cone $\{ (x,t): t \le \|x\|_2\}$ cannot be the image of a convex set under a linear map. But it can be realized as the image of the halfspace $\{ (x,t): t \le 0 \}$ set under the following nonlinear map: $F(t,x) = (t+\|x\|_2,x)$. The map is a bijection, the inverse being $F^{-1}(t,x) = (t-\|x\|_2,x)$. It is a kind of shear deformation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9067918658256531, "perplexity": 318.50113374777396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274756.45/warc/CC-MAIN-20160524002114-00178-ip-10-185-217-139.ec2.internal.warc.gz"} |
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# Synthesis of Poly(propyl ether imine) Dendrimers and Evaluation of Their Cytotoxic Properties
Krishna, Thatavarathy Rama and Jayaraman, Narayanaswamy (2003) Synthesis of Poly(propyl ether imine) Dendrimers and Evaluation of Their Cytotoxic Properties. In: Journal of Organic Chemistry, 68 (25). pp. 9694-9704.
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## Abstract
In this paper, we report the synthesis of several poly(propyl ether imine) dendrons and dendrimers. These dendrons and dendrimers were constructed by involving an ether as the linker component and an imine as the branching component. The divergent syntheses of dendrons and dendrimers were established with the aid of two alternate Michael addition reactions and two alternate reduction reactions in a four-step iterative synthetic sequence. Dendrons up to three generations were synthesized and some of the dendrons were attached to a benzenoid core so as to obtain dendrimers up to two generations containing 12 carboxylic acids at the periphery. Divergent synthesis involving ether as the core was found to be more facile, and dendrimers up to three generations having 16 carboxylic acids at the periphery were achieved in good to excellent yields in each individual step. The adopted synthetic sequence allows us to install either alcohol, an amine, or a carboxylic acid at their peripheries. The carboxylic acid-terminated dendrons and dendrimers were evaluated as to their cytotoxic properties, and while most dendrons and dendrimers did not exhibit any measurable cytotoxicity, even up to $100 \mu g/mL$, the second-generation dendrimer with the benzenoid core exhibited a mild toxicity at concentrations above $30 \mu g/mL$.
Item Type: Journal Article The Copyright belongs to American Chemical Society. Division of Chemical Sciences > Organic Chemistry 25 Apr 2006 19 Sep 2010 04:25 http://eprints.iisc.ernet.in/id/eprint/6460
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https://www.dbbrunson.com/tag/2d-models/ | # 2D Models
## Computational Modeling
Demonstrating the use of computers to simulate and study complex systems using mathematics, physics and computer science. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9864914417266846, "perplexity": 1633.012254611709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178364932.30/warc/CC-MAIN-20210302221633-20210303011633-00087.warc.gz"} |
http://blog.tfinley.net/ | ## Polaris and Epsilon Gruis
One of my job responsibilities is interviewing candidates. Finding talent is a high priority so I take it seriously, but I also enjoy it since it also gives me a chance to think about fun puzzles and problems (algorithmic, or otherwise). However, interesting problems are by themselves not good interview questions. Good interview questions must achieve a very delicate and complex balance. They have to be simple enough for a person to have thoughts within a few seconds, but multifarious enough that you get a good “image” of a candidate’s capabilities, and thought process. A question which is good or clever might ultimately be a bad question to ask during an interview. I’m going to write about one such problem which I rejected as it was a poor interview question, but which I still find very interesting.
The question is this: Consider the star Polaris, the North Star. Its prominent cultural significance notwithstanding, Polaris ranks only 48 in apparent brightness to us, compared with other stars. Of course, in a universe with billions upon billions of stars, many more must seem to us less bright. Let’s consider one, incidentally named Epsilon Gruis, which just so happens to appear only one-fourth as luminous as Polaris. Can you estimate for me, please, where Epsilon Gruis ranks on that list of brightest stars?
Some may note I am using “apparent brightness” rather than “apparent magnitude.” Though the latter term is more common, my choice is deliberate since I could hardly expect most people to be conversant on astronomical terminology, and apparent magnitude seems so perverse to the uninitiated: A higher magnitude indicates lesser brightness! Further, it is logarithmically scaled so that if one object has magnitude one higher than a second object, the second object appears 5√100 times more luminous than the first. This definition evolved for historical and not practical reasons. When I “tested” this on my workmates, I found we spent more time clarifying this metric, which they as a rule quite understandably found absurd. By constructing this story about Polaris and Epsilon Gruis (which do in fact have the properties I describe), I remove that difficulty while retaining the essential feature of the problem. Apparent brightness is very easy to explain, as being the amount of energy reaching a certain area on account of that object.
So, let’s see about solving it. The “one-fourth brightness” is itself a hint. Imagine two identical bright objects. If one were twice as far away from us, we would consider that it has brightness roughly one fourth as bright. This can be readily apprehended by imagining an enclosing sphere. The amount of energy (which I express as “brightness”) passing out of that sphere cannot change depending on radius, via the laws of thermodynamics. However, if we double the radius of the enclosing sphere, we quadruple its surface area, and correspondingly cut in one fourth the expected number of photons which would reach a certain area of fixed size on either sphere. (In this case the “fixed size” is a pupil, and the amount of energy reaching that pupil can be thought of as brightness.) You can imagine degenerate cases, e.g., light is emitted with zero beam divergence along some dimension, but it’s impossible to imagine a star would obey that.
Not all stars are identical, and have wildly varying intrinsic brightness. The preceding “sphere” argument however posited two identical objects. The applicability of the preceding to this situation seems questionable, but we can abstract this away by imagining the stated problem as being composed of an infinite number of sub-problems, where each sub-problem is identical to the full problem, with the exception that it does not consider all stars, but only the stars with equal intrinsic brightnesses. Obviously, with identical intrinsic brightness for the stars in each one of these sub-problems, “distance” and “brightness” become easily relatable once again. The original problem is just an integration over those sub-problems. So, this detail can be safely ignored.
We only must believe that for each intrinsic brightness in these sub-problems the local region of space is not special. This is of course taking isotropy a bit too far – or I guess I should say a bit too close? – but this is probably true enough for a toy problem. If we lived in the middle of a cluster or something, more care may be required.
So, we merely have to consider the number of stars in one sphere, versus a sphere with a radius twice as large. A radius twice as large, implies a volume eight times as large. If we suppose stars are generally distributed in three-dimensional space, then we’d expect there to be eight times as many stars. Therefore we’d expect the rank of Epsilon Gruis to be 48×8=384.
However, while I said I resigned myself so isotropy, I lied. As anyone who has finished grade school knows, our sun is a star in a spiral galaxy named the Milky Way, which is easily visible in the night sky, at least, outside of cities. So, there is a distribution in a plane rather than uniformly filling a three dimensional space. If all observable stars (including our own) were co-planar, the stars would not be distributed through volume, but instead across area, as we increase radius. So, we would expect four time as many stars when we double the radius, so the rank of Epislon Gruis would be 48×4=192.
Of course neither of these extremes is true; the true fractal geometry is somewhere between two and three. If it were two we would see stars as one brilliant line across the sky and perfect blackness elsewhere, and if it were three there would be no obvious clumping. So what is a happy middle ground? Because I am human, in the face of uncertainty I have a strong, nearly overwhelming bias in favor of uniform priors, and because I am foolish I think the average is the only summary statistic. So, my guess is the average of 192 and 384 is 288, or 48×6=288. The real answer is 285. I am only three off.
That’s it. I never posed this question to anyone aside from (1) coworkers, (2) a couple interviewees that had passed the vanilla questions “early” and were keen to hear what I described as as interesting problem, and (3) my father-in-law. All were flummoxed even to the point of being unable to begin. The first is especially concerning, since obviously any interview question that would lead you to reject colleagues you value is obviously unacceptable.
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## The Implications of the Maximize Button
The maximize button in the Microsoft Windows UI is interesting. While I had obviously been aware that Windows had such an interface idiom, it was not until I’d been required to use Windows in my daily work, used Windows programs day in and day out, and seen the developer culture at Microsoft, that I began to appreciate its profound impact on application design and developer psychology. The fact that there is a button devoted to every window to allow it to take up the entire screen very far from being the ancillary detail. In many respects its presence (or absence) encapsulates the entire philosophy underlining application design on the platform, as well as shape the behavior of users.
In the interest of presenting a counterpoint, I will contrast Windows against the Mac OS interface. I choose Mac OS because it differs on this important aspect. None of the Linux UIs are considered because this is a discussion of UI philosophy; the design of Gnome and other desktop environments or window managers was not really driven by any sort of cohesive unifying UI philosophy at all, beyond “imitate Windows.”
So, let’s begin. Humans perform tasks best when free of distractions. A computer interface should let a user focus solely on his workspace. This is an uncontroversial statement everyone can agree with. The next step is the branching step where there is disagreement: If we suppose “workspace” and “window” are identical, then a maximization button makes sense; all other content is extraneous ex hypothesi, and unused space is wasted space. If we suppose a workspace can consist of multiple windows probably from multiple applications, then a maximization button does not make sense; it is harmful because it hides portions of the workspace.
The presence of such a button in turn informs much of application design. There are practical and psychological consequences for maximizing, or not maximizing.
The consequence for maximizing: The mental cost of switching to another application context is high, especially if that application is yet another maximized application. There is nothing visually connecting one world to the other. There is therefore pressure to engineer applications so that all tasks related to a user’s workflow can be accomplished within the context of a single application. The workflow is integrated.
For not maximizing: When workflow is spread out among multiple applications, there is not a pressure to make any individual program do everything, but there is more pressure to make applications play well with each other. Completing complex tasks requires not one application, but an ensemble of applications. (This bears a strong resemblance to the Unix philosophy of small tools with simple interfaces.) The workflow places value in interoperable applications.
Now, on Windows there are definitely small tools, and on Mac there are definitely all in one applications (increasingly and especially those tools made by Apple itself, the recent phenomenon of the so called Apple ecosystem) but there is a definite idiomatic preference for one type of tool or the other.
I’m going to use making a presentation as an example. On Windows the favored tool for making presentations is PowerPoint, and on the Mac it is Keynote. Keynote embodies this “Mac” philosophy of interoperable applications very clearly. Keynote’s native diagraming tools are, to put it kindly, minimal; but no matter, because one can simply insert diagrams made with OmniGraffle. Neither of these two tools tries to do what the other does – at least not in any meaningful way – nor are they especially aware of each other, but Keynote plays well with graphics produced by other programs. The result is that you can easily produce excellent diagrams in an excellent presentation, despite the fact that both of the involved tools are only excellent at one of those tasks, and minimal or nonfunctional with respect to the other task.
In contrast, PowerPoint, despite having superior native diagramming utilities, plays very poorly with external content. For example, it insists on rasterizing PDFs and other types of vector graphics. If I do the same exercise – copy and paste an OmniGraffle vector graphic into Mac PowerPoint 2008 – it rasterizes the diagram into a badly pixelated graphic with glaring compression artifacts. Why, PowerPoint, why? Windows PowerPoint plays well with Visio, but only on account of special effort between the two programs specifically, as they are part of the same “Office” platform and significant special effort was spent towards making those two programs interoperable between themselves, but without increasing their general interoperability.
On the other hand, often the integrated way has some really strong benefits. As someone whose work involves a fair amount of mathematics, my presentations are rather heavy on equations. Keynote has no equation editing capabilities. The preferred solution is to use a third party utility like LaTeXiT. However, as good as that program is (and it is quite excellent considering the constraints in which it is forced to operate), the overall experience is painfully awkward. Equations don’t even reflow in Keynote, because they’re considered floating graphics; if you have some equations “inline” with your text, you have to put spaces in the text and put the equation in the gap. If you add some text anywhere that moves your existing text, you have to readjust the equations. Further, because they’re simply graphics, they don’t change with, say, themes. (Not that I like themes, for those that do, but it would be nice if resetting the theme was at least possible without having to reset each and every equation, like these are the old days of the letterpress where we are dealing with movable type or something.) The whole experience is pretty awful. The method can’t really compare in ease of use to the general Office (including PowerPoint’s) method of hitting Win-Equals, typing your equation, and being done with it.
That said, the Keynote-LaTeXiT approach does have one very important advantage. Long hard-won experience has made my ability to type LaTeX equations reflexive. For example, in my numerical analysis course in grad school, I typed notes involving some fairly aggressive and visually complex equations as quickly as the professor was able to write them on the board. A LaTeX based solution allows me to draw upon substantial experience to perform a novel task. Moreover, I could use the same macros and special style I had developed for my academic papers in my presentations. The integrated equation editor in PowerPoint, while it does happen to respect a subset of LaTeX syntax for simple symbols, is otherwise a totally unfamiliar environment.
There is a larger point: the integrated solution has the flaw that identical subtasks within different containing tasks are often accomplished in completely different and unfamiliar ways. So, if your workflow involves various combinations of n subtasks, you might not have to learn n different ways of doing things as in the interoperable world, but rather as many as 2n or n!, depending on how tolerant the applications are of reordering of subtasks.
The dual of this from the developer’s perspective is that large chunks of development effort are bent towards simply duplicating functionality that exists elsewhere. This is inefficient; that effort could be spent making the core functionality of the program’s special functionality really efficient and excellently engineered. This pressure on the developer is increased by users of the platform, who are used to integrated solutions and demand them.
To phrase the argument slightly differently, the integrated approach, with its expectation that an application handle not only one task, but all tasks ancillary to that task, leads to lousy software. A single developers will do different things with varying competence and enthusiasm. For example, there are few languages (programming or formatting) that are content in their Windows distributions to be mere languages; they typically feel the need to bundle a specialized editor with them. I am thinking particularly of LaTeX distributions (which nearly always bundle an editor and viewer), and some programming languages (Python’s bundling of IDLE, the so-called Python IDE, far more prominent on Windows than on any other platform). Of course, the effort spent towards making an entirely brand new editor was effort that could have been spent towards improving the support of existing familiar general-purpose editors for that particular language.
Naturally, the counter argument is there is some functionality would have been difficult or impossible to achieve in existing editors, and can be accomplished by starting fresh with an integrated special purpose editor. However, for the user of these tools, they use these specialized advances at the cost of using an editor which is lousy at its primary function. It is rare that it is worth giving up the ability to actually edit stuff properly only on the strength of one or two clever gimmicks.
In the integrated world, because tasks are expected to take place within a single program or application, there is less pressure to make applications interoperable. This bias towards not allowing interoperability occurs even in places where it would be a really fantastic idea, and not hard at all to integrate.
For example, in OS X when you drag a file into a save dialog in any other application (e.g., I drag a file from the Finder, or from the icon in the title bar of a Window), the save dialog moves to the corresponding folder. Very simple, and very useful; often the place you want to save a file is exactly the folder where you are working in some other application. On a related note, if you drag a file to the Terminal, then you get the path of that file; this is fantastically useful as well. Also, there is the near non-existence of modal dialog boxes on the Mac, compared to Windows where modal dialog boxes are the norm – preventing a user from moving on to other tasks is far more poisonous and annoying in an interoperable environment than an integrated environment. These are really tiny petty things, but a truckload of tiny petty things adds up.
What is somewhat interesting is that the Mac platform is changing. Apple has in the last decade or so taken to making “integrated” applications. Consider iMovie and iPhoto, which support the entire workflow of viewing, categorizing, editing, exporting, and publishing either movies or photos, respectively. Perhaps this is a symptom of their growing success; as a developer grows more popular, the temptation to control every aspect of a task must become overwhelming. Naturally, in these applications, the “zoom” button (the green plus) acts like a maximize button.
## The Awful Tychus Findlay
I’ve been enjoying the single player campaign of Starcraft II. It’s great fun. The tremendous variety of goals and mechanics of each level makes each level a very unique fresh experience. They are not so much “levels” per se, but more mini-games with rules all their own which just happen to use Starcraft units as gamepieces. The Mass-Effect-like mechanics of collecting resources which you spend on troop upgrades in between missions is likewise really engaging.
The story of that campaign is another matter entirely. Though the technical mechanics of the storytelling have advanced considerably beyond the disjoint unsynced-talking-heads-in-squares of yore, the actual writing is horrifyingly bad, a Transformers-2-esque grab-bag of cliches. It’s not that I expect high literature, but some small measure of coherence and sense would be appreciated. What is more, the shoddy writing is quite incongruous with the obvious care and love of craftsmanship that went into gameplay.
Things were rough from the start. When we first reacquaint ourselves with Jim Raynor, the game’s hero, he’s drinking hard liquor and watching TV. As it happens, one of the game’s villains is on TV, talking about him! This angers Jim, so we’re treated to a hearty “it ain’t over till it’s over you son of a bitch!” And he shoots the TV with a revolver. Wow.
That was actually one of the more tolerable sequences, if for no better reason than for the absence of our second hero, Jim’s old buddy Tychus Findlay. Like all the characters, Findlay’s conversational repertoire is mostly limited to trite one liners, but he is rendered all the more intolerable by his characterization; he’s a bulky muscle-headed half-wit who talks in a languid southern drawl. It’s somehow much more annoying than I just made it sound.
I list here a few gems of Tychus, prefixed by their context:
The zerg have attacked the human worlds. The death toll is in the billions (somehow billions of humans arose from forty thousand original settlers in the span of two hundred years; whatever). Tychus has just finished watching a video where Kerrigan herself rips apart a squad of armed soldiers with her bare hands. His appraisal of this threat? “Seems this queen of blades got everybody runnin’ scared! She don’t look so tough!” Right. You can take her Tychus! Please, go for it.
Raynor rescues survivors of a human colony. Their leader, a doctor who becomes your resident scientist, has seen the deaths of her fellow colonists, the destruction of her home, and other miscellaneous horror, doom, destruction, etc. How does Tychus comfort her? “I asked that sweet thang if she’d like to give me a physical.” Yeah… I think most people outgrow that chestnut by the time they reach their teens.
These selfsame colonists then settle on a new world, whereupon they are infested with the “zerg virus” which turns them into shambling zombies. His expert analysis? “Those colonists sure do have some zerg troubles!”
Oh, Tychus… Tychus Tychus. You insufferable jackass.
He’s utterly redundant, and from his introduction he’s clearly there so that he can betray Raynor at some critical point. I’m a little unclear on whether we’re supposed to know this or not; it seems really clear, but they keep foreshadowing in a way which suggests it’s supposed to be a mystery to us? I do wish they’d get it over with and be done with the beef slab. However, owing to the “multipath” mission structure, it would be technically awkward for it to happen anytime soon.
OK, I have now finished Starcraft II. The campaign missions remained delightfully fun and varied. They even managed to make the RPG squad based missions tolerable. Unfortunately, the writing remained execrable and senseless. I’m rather amused by how inept it is, so I think I’ll continue my screed.
Regarding the betrayal, Tychus waited to the absolute end to betray Raynor, so we had to suffer through his mouth breathing charm the entire game. Throughout the game it seemed clear the writer didn’t expect us to know Tychus was going to betray Raynor, which I found really confusing, given Findlay’s introduction with Mengsk talking about the price he’s going to pay for his freedom. The betrayal was: he had been blackmailed by Mengsk into killing Kerrigan. Yeah. So, I’m trying to comprehend Mengsk’s plan here. Mengsk wanted to kill Kerrigan; fair enough. He somehow felt that the best way to accomplish this was to let this random criminal Tychus go free, and blackmail Tychus so that, on the off chance he encountered Kerrigan, and Kerrigan was vulnerable, Tychus would shoot her. As far as plans go, that’s impossibly stupid; its success relies on a confluence of highly implausible events and happenstances that would have been impossible to anticipate at the time when Tychus was released. What is more, at the time Tychus was released, absolutely no one would have needed to be blackmailed into killing Kerrigan – anyone would have been only too eager to do so. The plan, and hence the plot, doesn’t make one iota of sense.
On the subject of Kerrigan, she was one of the more regrettable casualties of the terrible writing. Starcraft I’s writing and story were uneven, perhaps, but it often succeeded in making Kerrigan delightfully evil. Under the tender mercies of Starcraft II’s writers, though, she has been diminished to petulant incoherent outbursts. Here are two examples. She gives her first line in the campaign after you’ve stolen an alien artifact from under her nose: “I forgot how resourceful you were, Jim. I won’t make that mistake twice!” Aside from being trite, such a statement renders her impotent. Obviously you’re going to win, and continue being resourceful, so this declaration becomes nothing more than empty words. She clearly makes the mistake not just twice, but repeatedly again and again through the course of the campaign. So, we lead off by making our lead villain appear ineffective and weak. Nice. The last thing she says in the final mission is “you will pay for this treachery” after you beat back the last of her attacks. I found the comment bewildering. So… um, defense is treachery? This rather makes me suspect the writer doesn’t know the definition of treachery. This seems likely given the volume of other malaprops.
Then we have our hero’s characterization. Jim Raynor is supposed to be a pathetic drunk. We are treated to lots of scenes of him drinking, and in any one of these scenes he drinks a lot, gulping down in a single instant twice what I’d be hard pressed to imbibe over the course of a lengthy evening. Curiously, in no way is he ever affected at all by his drinking. Part of having a character be a pathetic drunk is that he is drunk. That’s kind of an important component of that characterization that they somehow missed. I wouldn’t have guessed it was even possible to make that oversight.
Aside from content, we have some weaknesses introduced by the technical aspects. The non-linear structure has some disconcerting effects. You go from discrediting Mengsk and fighting his primary military commander (the imaginatively named “General Warfield”), to fighting shoulder to shoulder alongside this same commander. He accepts your help without so much as a sidelong glance, and with Mengsk seemingly as powerful as ever. More broadly, the nonlinear progress often gets in the way of telling a cohesive story; the pieces of the story are necessarily modular, but a story itself is almost by definition non-modular; a modular part of a story is a part you can do without. Further, that you can talk to people in any order means that all conversations occur seemingly in a vacuum. The overall effect is a heightened sense of emotional flatness and meaninglessness, almost like everyone has been lobotomized, or that I was observing it through a dream. I think this might be a general problem with non-linear structure; I observed a similar uncanny effect with Mass Effect 2 – though strangely, Dragon Age had similar non-linear mechanics and avoided the uncanny effect somehow.
All this said, I’m perfectly delighted with Starcraft II. In the end I bought a game, not a novel, and the game itself is stellar. It’s not like story matters all that much in an RTS compared to, say, an RPG. I’m just rather amused the story is as bad as it is.
## New Blog: Defaults and Sidebars
This is my new blog site.
When my homepage was booted from the Cornell CS department webservers and forced to make its own way in the world, I quickly gave it a home with a web hosting company. Their most basic plan offers far more capacity and features than I would probably ever use. My website is not terribly spectacular or sophisticated: it’s a motley collection of PHP and HTML I had thrown together to document the artifacts of my graduate career, an intentionally minimalist work mostly describing myself and my formal projects as briefly as I thought would do.
Nonetheless, with shiny new features and capabilities comes the temptation to use them. I may as well; I’m paying for it. So, I set up a blog using one of my ten allowed databases and one of my ten allowed subdomains. I already had a blog I started in May, but I wanted to maintain it all myself. However, my reasons for wanting to do it myself were rooted in technical interest, totally divorced from any practical considerations. Among these practical considerations:
• I don’t have time. I work and then come home to help my wife take care of a baby. On a good day I have one hour of strained leisure time.
• I am not really the blogging type. Many years ago while in undergrad when “blog” was still a novel term, I set one up only to discover I didn’t have anything I particularly wanted to say to the world. This hasn’t changed much.
• Blog software is largely database driven; this makes backups irritating to the point where, frankly, I probably won’t bother. I’m going to regret this later when inevitably my webhost loses all my data. Also, given that I do not give much care for the “dynamic” features of a blog – comments, tracebacks, etc. – why not just have a series of static HTML files?
On the other hand, if one wishes to have the fun of toying with a lightweight database driven web application, a personal blog is a fairly ideal choice: there’s minimal cost to get to something working, and it requires neither a cohesive master plan, nor participating third parties.
Like everyone else and their dog I used WordPress, fortuitous timing as WordPress 3.0 just came out. I have no idea what that means and the release notes referenced features that meant nothing to me, but heck, 3.0 just looks prettier than 2.9. I wanted to use WordPress primarily because I read it was the software that integrated most easily with $\LaTeX$, good for mathematical content. Setup is easy; their claims of a “famous five-minute installation” are pretty much spot on. I spent more time randomly generating passwords than I did actually installing the thing. I had only four posts in my other blog, so transfer wasn’t difficult, images being the main annoyance.
The real problem though, and where I spent nearly all my effort, is personalization of appearance. Sticking with the default theme seemed kind of lame and lazy, but the free themes available either: violated my minimal aesthetic (though some were quite beautiful), were poorly implemented, or shockingly wasteful of screen real estate. The sad thing is, I really like the TwentyTen default theme. It’s the best theme by far. So, I decided to not replace it, but adapt it. (I discovered after I had finished my adaptation that best practice requires the use of child themes; I may refactor my changes later in my copious spare time.)
Adaptation involved a pleasing amount of futzing about in PHP files of the theme – I say pleasing because my primary motivation was technical interest and a desire to learn, and as I futzed about I found myself repeatedly pleasantly surprised at WordPress’s elegance and customizability. I didn’t have to touch anything outside of the theme directory.
I started with a number of modest changes: I changed the date/byline information under each title (“Posted on July 23, 2010 by John Doe”) to just contain the date and time. Clarifying that I’m the author on every post is clearly unnecessary. The “leave a comment” was changed to “no comments.” I did not intend to ever allow comments, so inviting them seemed odd. “Proudly powered by WordPress” in the footer seemed a bit much, so I took out the “Proudly,” though I do think it only fair to acknowledge WordPress.
Next I made the title and description inside the image, rather than simply above it – basically the title and description are now in a DIV with my header image as the background, rather than having the header image be in an `img` tag. I reproduced that leading black divider at the top of the image since I think it looks pretty sharp.
Getting rid of the right sidebar was a priority for me. The right sidebar is a ubiquitous feature in blogs, but I have always had a strong dislike of them. I always found them distracting. I especially never liked how space continued to be reserved for them below the navigational content, so space was wasted and your content was forever oddly off center, even when the reason for that wastage had long since been scrolled past. Also, simply wrapping around the associated `div` produced lines which were uncomfortably long.
This took the most work, not because removing it was the problem – though frankly it was far more trouble than it rightly should have been – but duplicating what functionality and content I wished to present to end users posted some small design challenges. For this I abused the header’s navigational bar. While this element is intended for the sole use of a main menu, with some light modifications, one can continue to use the left of the bar for the menu, and the right for some special elements, like the search box and the RSS feed. This also gave me a chance to customize the search box, which I did by putting the search button inside the form element a la Wikipedia or Bing. I also omitted the login and comment RSS feeds; these were totally extraneous for my purpose.
Archives and recent posts are typically in the right sidebar, and these are things I would like. I don’t need or want a monthly archive though. I don’t actually intend to write anything most months, and I need a monthly archive, somehow? My current solution is to just put a link to my posts under “Archive.” I’m ambivalent on this. “Recent posts” and “archive” both provide immediate means for people to easily see what is present beyond that first post. At best, I’ve moved that one click away. I’m unsure how I’ll solve this, but I emphatically do not want a right sidebar.
Photographs are rectangular. Historically, this makes sense; when photography was first invented, it was the film (or rather, photographic plates) that cost the most money versus, say, the lens, and obviously one makes the best use of the raw material of plates by cutting it into rectangles, and later the lengthly rolls of plastic film were put to best use by partitioning it into rectangles. (Hexagons could potentially work for no wastage, but this would be far more awkward, especially when it came time to put them in a roll.)
Digital cameras don’t use plates or film any more, but reusable sensors. A single sensor is reused for many thousands of photos, and so there is no reason to efficiently partition a physical medium into many subparts as there is with film.
Lenses are circular, and correspondingly image degradation (whether vignetting or other distortion) increases roughly as the distance from some center on the photographic plane. If we suppose there’s some maximum distance beyond which one cannot get a good image, the “useful” area of a photographic plane is circular. To accomodate rectangular film without wasting any of this film, whatever portion of the circle we use is a rectangle inscribed in this useful circle, meaning we are wasting at least (1-2/π) ≈ 36% of the useful circle, and that’s only if the rectangle is a square. With my own camera’s 2:3 aspect ratio, that’s about 41% wastage. (As a practical matter, it is more like 63% or so for an EF lens, I think, since it’s a crop sensor; that’s astonishing waste.) As megapixel sensor density continues to increase, we shall feel the cost of the corresponding lens wastage keenly, since the amount of meaningful detail sensors can gather begins to exceed the capabilities of all but very expensive lenses.
So, now we come to my point: digital cameras have obviated film wastage, but the rectangular shape inherited from film cameras continues to waste lens resolution. Clearly this is wrong. It is time to revisit the concept of the “rectangular” photo, to literally think outside the box. Rather than having rectangular digital sensors, let us have circular ones, and spread out that sensor resolution to cover all of the useful circle of the photographic plane.
Obviously for many applications one will still want rectangular photos, but if you wish to subset and crop your photos to be rectangular, that will be your choice. In the present rectangular paradigm, one has no choice at all.
The primary advantage is that when you buy a lens, you get to use all of it, not just about half of it. We pay good money for the glass only to ignore a sizable chunk of it.
There are some practical cost reasons why one could not make a photographic plane cover the entire useful field. In such a case, even if your image sensor remains the same size but has been reformed into a circle, you have given up the lower quality corners for real estate much closer to the lens center, which presumably suffers from less distortion.
Second, a photograph is more or less identical no matter the orientation of the camera. There’s no need to think about whether to photograph a subject landscape or portrait, nor even to be sure you’re keeping the horizon level. These would be details left to post processing, and totally irrelevant considerations at the time the photo is actually taken. This frees up one’s attention for other matters.
Third, lots of subjects benefit stylistically from being photographed in a circle, right up to the edge of vignetting and distortion – typical shots of faces, flowers, birds not in flight, and lengthy roads winding into the distance come to mind right off the bat – which is currently an effect one has to do with filters or post processing.
## Zoom vs. Prime: Focal Length Usage
I recently observed an online discussion of camera lenses. People were debating the merits of zoom vs. prime lenses, falling into the predictable points of flexibility vs. performance. One of those defending the merits of prime lenses posited that if one were to plot the usage of focal length with respect to number of photographs taken, the tendency would be for photographs to cluster around the extremes of the focal lengths.
Since I was curious about how I would fall into this range, and decided to run the experiment for myself. As a bit of background, I have a Digital Rebel XT with which I use four lenses:
• EF 28-135mm IS USM is my midrange general purpose lens. I have this lens mostly because when I got it in 2005, the passable EF-S midrange lenses did not exist yet. I believe it was the right choice for the time, but I can’t pretend that I’d even consider it today. The 28mm is still way too long in a crop sensor.
• EF 70-300mm DO US USM is my telephoto zoom. For roughly the same price I could have bought the EF 100-400mm L IS USM, but that thing is huge and conspicuous, and I wanted something I could carry everywhere and use without feeling self-conscious, as I’m kind of a shy guy.
• EF-S 10-22mm USM is my wide angle. If you have a crop sensor and want a wide angle, it’s this or nothing.
• EF 50mm f/1.4 USM is my only prime. I got this a relatively short time ago when my baby girl was born, and I found my 28-135 utterly inadequate in low light. However, because it is so new, and because the purpose of this analysis is to see which focal lengths I used (in a prime, there’s no choice!), it is not covered in this analysis.
With the help of a small Python script to and EXIF.py, I was able to read and aggregate all of the focal lengths and lens types for a period a couple weeks after I got the 10-22 wide lens, up until the day my daughter was born. Naturally my photographic habits changed with both events, but between these two events my behavior may have been stable. This covered a period of roughly two years, with 1366 photographs. Naturally, these are just the photographs that I did not delete.
In the following plots I give focal length on the horizontal axis, and the number of photographs in the Y-axis in a cumulative histogram. So, pay less attention to the value of the vertical axis, and more to the size of the jumps.
We lead off with my general purpose lens. Unsurprisingly, I do have a tendency to use the entire range of this lens. I have a slight tendency to “top off” at around 80-100mm or so, before having a strong tendency to shoot at the maximum 135mm. The most notable and obvious trend, though, is my heavy usage of the 28mm focal length, perhaps reflecting my frustration with the lens being too long for its own good for my camera body.
Next we have the telephoto and wide angle zooms, in that order. My behavior with these is quite different from my general purpose lens. Indeed, as posited, there is a strong tendency to use the extreme endpoints. I like my longs long and my wides wide, clearly. Nonetheless, I do shoot a substantial amount along the range of both, and I do exhibit a slight tendency to shoot with my telephoto wide (probably because I’m too lazy to swap in the general purpose lens in many circumstances), so clearly I get substantial benefit from the zoom.
I find this quite interesting, not only for the sake of this isolated analysis, but because it seems like there’s a tremendous amount of interesting data we could mine regarding people’s photographic habits if we could just collect everyone’s EXIF tags. What do good photographers do that bad photographers don’t, or vice versa? If we multiplied aperture times ISO times shutter speed, would the resultant value tend to decrease in summer, and increase in winter? What if we could somehow limit the analysis to “good” photographs? It’s unclear whether any of this could be put to any practical worth, but it’s still interesting.
## Webpage Exile
Making a new website was something I should have done earlier. I had procrastinated partly because of the effort and cost, but actually more because of I could not choose a hosting company. They are a vast legion of largely indistinguishable anonymous entities. The number of services with their slight variations in their services is bewildering, and there are so many suspicious looking review sites to the point where objective information is hard to call out, and nearly everyone has something bad to say about everyone. How could one tell the good hosters from the bad? Making an informed choice seemed impossible to the point where I did not want to decide at all.
My needs are few and my traffic quite light, but I did have some criteria: I wanted databases, Python, Ruby, Ruby on Rails, PHP, and a hoster that did not offer “unlimited” plans. (Seriously, who’s kidding whom with that “unlimited” nonsense?) In the end, I went with a host a friend and colleague of mine used for his own site and recommended as reliable.
Hence, tfinley.net. Owing to the aforementioned birth of my child, my ambitions were limited: it’s basically my Cornell site with the verbs attached to my graduate student career put in the past tense, and the URLs updated. Of course, between the parade of feedings and diaper changes, even these modest changes took me a couple days to finalize.
### Subdomain on Localhost
When developing the site, I found it helpful to host a site on my local web server, to make writing and debugging my site more efficient. However, I could not work in the root directory of localhost, as I was already using this to host some resources for my home LAN. I initially worked in a subdirectory, but this becomes really irritating as absolute paths do not work. An `<a href="/page.php">` would mean entirely separate things locally and on the actual website.
My solution was to add a subdomain to localhost. The address `http://localhost` would point to my existing web server, but I added a subdomain to localhost so that `http://tfinleynet.localhost` pointed to the directory where I was developing the website. This allowed absolute paths to work properly. The requisite steps are not difficult, but they are sufficiently non-obvious to the point where others may benefit from my research. The steps are for a Mac OS X machine.
1. Add a line resembling the following to `/etc/hosts`, of course replacing `foo` with your desired subdomain:
`127.0.0.1 foo.localhost`
2. In `/private/etc/apache2/httpd.conf`, uncomment (remove the preceding `#` from) the line:
`Include /private/etc/apache2/extra/httpd-vhosts.conf`
3. Edit the file `/private/etc/apache2/extra/httpd-vhosts.conf` whose inclusion you just uncommented, so that the uncommented lines read as follows, again with `foo` replaced with your desired subdomain, and the `DocumentRoot` set to the desired directory.
```NameVirtualHost *:80
<VirtualHost *:80>
DocumentRoot "/Library/WebServer/Documents"
ServerName localhost
</VirtualHost>
<VirtualHost *:80>
DocumentRoot "/path/to/local/development/directory"
ServerName foo.localhost
ErrorLog "/private/var/log/apache2/foo-error_log"
CustomLog "/private/var/log/apache2/foo-access_log" com\$
</VirtualHost>
```
4. Restart Apache, most easily accomplished by unchecking and checking “web sharing” in the Sharing pane of system preferences.
5. Point your browser to `http://foo.localhost`, and observe the mighty workings.
### Favicons and Robots
Technically, `favicon.ico` and `robots.txt` are optional for a website. However, a few days with an error log containing a megabyte or so of lines upon lines of `File does not exist: /foo/bar/biz/httpdocs/favicon.ico` and the like are strong motivators to create them, optional or not. I am on a shared server so I cannot reconfigure the website to just return 404 errors for these files, so I just bit the bullet and created them.
The favicon made me nervous. Sure, I was creating it just to get my error log to shut up, but any visitor would assume I did so because I thought what I had done was better than nothing. It’s not like I’m an artist that can create something beautiful in a sparse 256 pixels, nor yet a corporation or university with some universally recognizable brand or logo. Nonetheless, I think I acquitted myself adequately: I fired up the GIMP, put a gradient in a circle, some alpha-heavy black in an ellipse under that, and my initials in white. Boom. Instant favicon.
For robots, I’m not too picky, so I allow everything with the following `robots.txt` file.
```User-agent: *
Disallow:```
## Orion: The Space Flight Simulator
When I was much younger, still on my first computer, my parents gave me a voluminous 240 MB hard drive for my Mac IIcx. Upon connecting the drive, I found among other contents a shareware spaceflight simulator, Orion, the work of one Robert Munafo. This serendipitous acquisition quickly became one of my favorite programs. With this program, one controls a spacecraft by applying thrust, roll, pitch, and yaw, and has a capability to travel to any star system within a 32 light-year radius (all of these helpfully had 9 planets, Pluto being considered a planet at the time). By flying at a planet in just right way and executing some slightly deft maneuvering, one could enter orbit. By executing the maneuver incorrectly, you’d fall into the planet or, more likely, get flung out past escape velocity back into the star system.
Pictured is a screenshot of Orion’s mighty workings. All controls are shown in the lower ribbon. By modern standards the control system is obtuse, but for 1988 at a time when people were still figuring out how computer interfaces should work, it was quite excellent. One turned the ship by holding the mouse down in the lower left box, with the direction and magnitude of the turn indicated by the offset of the click from the center dot. The F/S/R buttons controlled the motion of the ship, being forward/stop/reverse. The next box allowed one to perform lateral thrust. The top bar allowed one to roll. Then there was a control for easy mode, with non-easy mode resulting in Newtonian motion (once you applied thrust in a direction, turning did not redirect your velocity) and the application of gravity. Remaining controls toggled display elements.
This program was enormously fun and instructive.
First, it gave me a healthy appreciation of the vast scale of the universe. At first blush it may sound like a 32 light-year long leash is not terribly restrictive, but the paucity of “interesting” stars that lie in that sphere soon becomes clear. That limit quickly seems unbearably oppressive. (For example, none of the interesting stars actually in the constellation Orion can be visited within the program Orion. )
It also gave me an intuitive feel for Kepler’s laws, especially as it applied to elliptical orbits and the inverse relationship between orbital distance and velocity as I flew in orbit around a planet, many many years before I learned that they were called Kepler’s laws. (“Hey, when I’m twice as far away, I’m going half the speed…”)
Quite apart from anything having to do with space and more appealing to my nascent interest in software engineering, seeing my orbital parameters (major/minor axis) slowly change over time when they realistically should not have changed gave me an appreciation for the role approximate arithmetic takes in software simulations of physical systems, and the dangers of accumulated numerical error.
Less intellectually, then, as now, I was a fan of Star Trek TNG and liked to pretend I was flying in the USS Enterprise. The supposedly impossibly fast Warp 9 amounts to a paltry and pitiful 1,516 times the speed of light (roughly 3 AU per second), meaning that a trip from earth to the Alpha Centauri system would take almost exactly 24 hours. My preteen brain found this a sobering and disappointing realization. Nonetheless, I decided to make the trip one day. Pretending my bedroom was a spaceship cabin and my computer the helm (naturally allowances were made to leave the room for the bathroom and food), I set my sights for Rigel Kent and accelerated to Warp 9. During the daylong journey I read, periodically corrected my course, and overnight I slept. Roughly a day later I arrived at Rigel Kent, whereupon I “explored” all the planets. I was oddly proud of myself for completing the journey, but the tedium did give the lie to the notion of Jean Luc Picard and company casually dropping by Earth from the far reaches with the same ease of someone today traveling from New York to Boston.
With the help of Basilisk II, I was able to use Orion once again for the first time in about two decades. Aside from the usual shock I encounter using ancient software — e.g., the old Mac design philosophy of never using keyboard controls, the help box with a snail-mail address and no sign of a @ or .com anywhere, the usual hiccups of software when running on a computer thousands of times the speed of the Soviet era hardware it was designed for — it was quite enjoyable, and wonderfully nostalgic. So, thank you Robert! Sorry I never sent you \$7, but my ten year old brain didn’t quite have a fully developed sense of social responsibility. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48272690176963806, "perplexity": 1739.7682077264194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864958.91/warc/CC-MAIN-20180623113131-20180623133131-00082.warc.gz"} |
http://clay6.com/qa/52209/a-very-small-amount-of-a-non-volatile-solute-that-does-not-dissociate-is-di | Browse Questions
# A very small amount of a non-volatile solute (that does not dissociate) is dissolved in $56.8cm^3$ of benzene (density $=0.889gcm^{-3}$).At room temperature vapour pressure of this solution is 98.88mm Hg while that of benzene is 100 mm Hg.Find the molality of this solution.If the freezing temperature of this solution is 0.73 degree lower than that of benzene .What is the value of $K_f$ for benzene?
$\begin{array}{1 1}0.1452,5.027Kmolality^{-1}.S\\0.1252,3.027Kmolality^{-1}.S\\0.1352,4.027Kmolality^{-1}.S\\1.1452,6.027Kmolality^{-1}.S\end{array}$
We know that
$\large\frac{P^0-P_s}{P_s}=\frac{w\times M}{m\times W}$
Substituting values
$\large\frac{100-98.88}{98.88}=\frac{w}{m}\times \frac{78}{W}\times \frac{1000}{1000}$
$\Rightarrow$molality $\times \large\frac{78}{1000}$
Molecular weight of benzene =78
Or,Molality $\big(\large\frac{w\times 1000}{m\times W}\big)=\frac{1.12\times 1000}{78\times 98.88}$
$\Rightarrow 0.1452$
We know that
$\Delta T=K_f\times$ molality
$\therefore K_f=\large\frac{\Delta T}{\text{molality}}$
$\Rightarrow \large\frac{0.73}{0.1452}$
$\Rightarrow 5.027K molality^{-1}.S$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3423731327056885, "perplexity": 3048.038006666673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172050.87/warc/CC-MAIN-20170219104612-00301-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://brilliant.org/problems/decay-time-of-damped-oscillator/ | # Decay Time of Damped Oscillator
A $$10 \text{ kg}$$ mass is attached to a spring of spring constant $$10 \text{ N}/\text{m}$$. The entire system is submerged in water, which exerts a viscous damping force on the mass $$F_d = -(2 \text{ N}\cdot \text{s}/\text{m}) \:v$$. The mass is pulled so that the spring is displaced from equilibrium by $$.1 \text{ m}$$ and is released. Find the $$1/e$$ decay time of oscillation in seconds.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994506299495697, "perplexity": 180.61520275421523}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255092.55/warc/CC-MAIN-20190519181530-20190519203530-00067.warc.gz"} |
https://calculus.subwiki.org/w/index.php?title=Variation_of_first_derivative_test_for_discontinuous_function_with_one-sided_limits&oldid=1367 | # Variation of first derivative test for discontinuous function with one-sided limits
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## Statement
Suppose $f$ is a function and $c$ is a point in the domain of $f$. This statement is a variation of the first derivative test that helps deal with situations where a function has one-sided limits but is not necessarily continuous at the critical point of interest. It can be viewed as a remedy for the fact that the first derivative test fails for function that is discontinuous at the critical point.
### One-sided version
Limit existence assumption Comparison of one-sided limit and value Conclusion
Left hand limit of $f$ at $c$ exists Left hand limit is less than the value, i.e., $\lim_{x \to c^-} f(x) < f(c)$ $f$ has a strict local maximum from the left at $c$, regardless of the way $f'$ behaves on the left of $c$
Left hand limit of $f$ at $c$ exists Left hand limit is greater than the value, i.e., $\lim_{x \to c^-} f(x) > f(c)$ $f$ has a strict local minimum from the left at $c$, regardless of the way $f'$ behaves on the left of $c$
Left hand limit of $f$ at $c$ exists Left hand limit equals value, i.e., $\lim_{x \to c^-} f(x) = f(c)$, so $f$ is left continuous We can try to use the one-sided version of the first derivative test:
If $f'(x) \ge 0$ on the immediate left, then local maximum from the left
If $f'(x) \le 0$ on the immediate left, then local minimum from the left.
Right hand limit of $f$ at $c$ exists Right hand limit is less than the value, i.e., $\lim_{x \to c^+} f(x) < f(c)$ $f$ has a strict local maximum from the right at $c$, regardless of the way $f'$ behaves on the right of $c$
Right hand limit of $f$ at $c$ exists Right hand limit is greater than the value, i.e., $\lim_{x \to c^+} f(x) > f(c)$ $f$ has a strict local minimum from the right at $c$, regardless of the way $f'$ behaves on the right of $c$
Right hand limit of $f$ at $c$ exists Right hand limit equals value, i.e., $\lim_{x \to c^+} f(x) = f(c)$, so $f$ is right continuous We can try to use the one-sided version of the first derivative test:
If $f'(x) \ge 0$ on the immediate right, then local minimum from the right
If $f'(x) \le 0$ on the immediate right, then local maximum from the right.
### Two-sided version
We list the strict cases:
Case for left side behavior Case for right side behavior Conclusion for behavior of $f$ at $c$
Either $\lim_{x \to c^-} f(x) < f(c)$ or ($f$ is left continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) < f(c)$ or ($f$ is right continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate right of $c$) strict local maximum
Either $\lim_{x \to c^-} f(x) > f(c)$ or ($f$ is left continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) > f(c)$ or ($f$ is right continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate right of $c$) strict local minimum
Either $\lim_{x \to c^-} f(x) < f(c)$ or ($f$ is left continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate left of $c$) ither $\lim_{x \to c^+} f(x) > f(c)$ or ($f$ is right continuous at $c$ and $f'(x) > 0$ for $x$ on the immediate right of $c$) neither local maximum nor local minimum
Either $\lim_{x \to c^-} f(x) > f(c)$ or ($f$ is left continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate left of $c$) Either $\lim_{x \to c^+} f(x) < f(c)$ or ($f$ is right continuous at $c$ and $f'(x) < 0$ for $x$ on the immediate right of $c$) neither local maximum nor local minimum | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 93, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9737976789474487, "perplexity": 177.7488939359911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304309.59/warc/CC-MAIN-20220123202547-20220123232547-00326.warc.gz"} |
http://mathoverflow.net/questions/40176/terminology-metric-space-with-product-and-unit-and-the-opposite-of-a-nonexpans | # Terminology: metric space with product and unit, and the opposite of a nonexpansive map
Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.
1) Suppose $X$ is a metric space equipped with an associative product and a unit element. Let $m: X \times X \to X$ be the product. Suppose also that $m$ is nonexpansive, i.e. $$d\big( m(x,x'),m(y,y') \big) \le d\big((x,x'),(y,y')\big)$$ say when $X\times X$ is given the Euclidean metric with respect to the given metric of $X$. Is there a standard name for this type of structure?
2) Let $f : X \to Y$ be a map between metric spaces with the property that for all $x,y \in X$ $$d(f(x), f(y)) \geq d(x,y).$$ So in a way this is the opposite of a nonexpansive map. Is there a name for such maps?
Thanks.
-
2) Expansive. – Bill Johnson Sep 27 '10 at 18:01
2) noncontracting map – Anton Petrunin Sep 28 '10 at 0:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9757629036903381, "perplexity": 142.0934128780622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988860.88/warc/CC-MAIN-20150728002308-00065-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://ageconsearch.umn.edu/record/99010 | Formats
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Abstract
In this farewell lecture some concepts important for (agricultural) economics are discussed. Special emphasis is given to the Invisible Hand of Adam Smith, sunk costs, information economics, path dependency, error tolerance and redundancy. It is proposed that agricultural economists follow closely new developments in economics proper. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8921995759010315, "perplexity": 7099.630768409251}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487634616.65/warc/CC-MAIN-20210618013013-20210618043013-00245.warc.gz"} |
http://cpr-astrophhe.blogspot.com/2013/06/13065389-rupak-roy-et-al.html | ## SN 2007uy - metamorphosis of an aspheric Type Ib explosion [PDF]
Rupak Roy, Brijesh Kumar, Justyn R. Maund, Patricia Schady, Felipe Olivares E., Daniele Malesani, Giorgos Leloudas, Sumana Nandi, Nial Tanvir, Dan Milisavljevic, Jens Hjorth, Kuntal Misra, Brajesh Kumar, S. B. Pandey, Ram Sagar, H. C. Chandola
The supernovae of Type Ibc are rare and the detailed characteristics of these explosions have been studied only for a few events. Unlike Type II SNe, the progenitors of Type Ibc have never been detected in pre-explosion images. So, to understand the nature of their progenitors and the characteristics of the explosions, investigation of proximate events are necessary. Here we present the results of multi-wavelength observations of Type Ib SN 2007uy in the nearby ($\sim$ 29.5 Mpc) galaxy NGC 2770. Analysis of the photometric observations revealed this explosion as an energetic event with peak absolute R band magnitude $-18.5\pm0.16$, which is about one mag brighter than the mean value ($-17.6\pm0.6$) derived for well observed Type Ibc events. The SN is highly extinguished, E(B-V) = 0.63$\pm$0.15 mag, mainly due to foreground material present in the host galaxy. From optical light curve modeling we determine that about 0.3 M$_{\odot}$ radioactive $^{56}$Ni is produced and roughly 4.4 M$_{\odot}$ material is ejected during this explosion with liberated energy $\sim 15\times10^{51}$ erg, indicating the event to be an energetic one. Through optical spectroscopy, we have noticed a clear aspheric evolution of several line forming regions, but no dependency of asymmetry is seen on the distribution of $^{56}$Ni inside the ejecta. The SN shock interaction with the circumburst material is clearly noticeable in radio follow-up, presenting a Synchrotron Self Absorption (SSA) dominated light curve with a contribution of Free Free Absorption (FFA) during the early phases. Assuming a WR star, with wind velocity $\ga 10^3 {\rm km s}^{-1}$, as a progenitor, we derive a lower limit to the mass loss rate inferred from the radio data as $\dot{M} \ga 2.4\times10^{-5}$ M$_{\odot}$, yr$^{-1}$, which is consistent with the results obtained for other Type Ibc SNe bright at radio frequencies.
View original: http://arxiv.org/abs/1306.5389 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8226867318153381, "perplexity": 6391.797474646632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187826283.88/warc/CC-MAIN-20171023183146-20171023203146-00498.warc.gz"} |
http://tahirhassan.blogspot.com/2016/03/sql-server-security-model.html | # Tahir Hassan's Blog
My Technical Notes
## Thursday, 10 March 2016
### SQL Server Security Model
#### Heirarchical Organisation
There are different levels of organisation within SQL Server. A Server contains databases, a database contains schemas and schemas contain tables, stored procedures, functions and so on. We can represent this by talking about the $[Things] that something contains.$[ServerThings] include $[Databases],$[DatabaseThings] include $[Schemas],$[SchemaThings] include $[Tables]. All things are a union of these three sets (levels of organisation) $$Things = ServerThings \cup DatabaseThings \cup SchemaThings$$ If we use$[*] to represent things at a level or below that level we can see: $$SchemaThings^* \subset DatabaseThings^* \subset ServerThings^*$$
#### Schemas
Schemas are a $[DatabaseThing]. Given a$[Database\ d], we can take its schemas. Schemas include the familiar $[dbo]: $$schemas(d) = \{ dbo, \ldots \}$$ We let$[Schemas] denote this set for the database under consideration:
$$Schemas \eqdef schemas(d)$$
#### Objects
Objects are contained within a schema, therefore they are $[SchemaThings]. Objects include things like tables, views, stored prodedures and so on. Given a$[Schema\ s] We can take its objects: $$objects(s) = tables(s) \cup views(s) \cup storedprocs(s) \cup \ldots$$
We use $[Objects] when the$[Schema\ s] is understood:
$$Objects \eqdef objects(s)$$
Given an $[o \in Objects], we can take it's schema: $$schema(o) = s$$ Given an$[o \in objects(s)] we can say that $[o] is in$[s]'s scope.
Logins are $[ServerThings]. On installation, an$[sa] login will be created (System Administrator). Apparently, their correct name is Server Principles. Given a $[Server\ S], we can take its logins: $$logins(S) = \{ sa, \ldots \}$$ We let$[Logins] denote this set when the server $[S] is understood: $$Logins \eqdef logins(S)$$ #### Users Users are$[DatabaseThings]. However, a $[User\ u] is only valid if they exist in$[Logins].
$$validUser(u) \iff u \in Logins$$
In order to create a user for a database, we have to first create a login and then create the user.
In the case where we restore a database but a $[User\ u] does not exist in$[Logins] then \$[\neg validUser(u)] holds. Such a user is called an orphaned user.
TODO: Complete this. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.805362343788147, "perplexity": 11285.14940261577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102891.30/warc/CC-MAIN-20170817032523-20170817052523-00069.warc.gz"} |
http://mathhelpforum.com/calculus/130072-sum-sequence.html | # Thread: Sum of a sequence
1. ## Sum of a sequence
$\sum_{x=0}^{\infty}\frac{x}{2^x}$
All I know is that it converges. Really have nothing more to say.
2. Hello, lausing!
$S \;=\;\sum^{\infty}_{x=1} \frac{x}{2^x}$
$\begin{array}{cccccc}\text{We have:} & S &=& \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \hdots \\ \\[-3mm]
\text{Multiply by }\frac{1}{2}\!: & \frac{1}{2}S &=& \quad\;\;\dfrac{1}{2^2} + \dfrac{2}{2^3} + \dfrac{3}{2^4} + \hdots \end{array}$
. . $\text{Subtract: }\;\;\tfrac{1}{2}S \;\;=\;\;\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots$ .[1]
The right side is a geometric series with: . $a = \tfrac{1}{2},\;\;r = \tfrac{1}{2}$
. . Its sum is: . $\frac{\frac{1}{2}}{1-\frac{1}{2}} \:=\:1$
Hence [1] becomes: . $\tfrac{1}{2}S \;=\;1$
Therefore: . $S \;=\;2$
3. Thanks!
Also, the question is erroneously posted in calculus rather than linear algebra or something, since it orginally was about derivatives. Whoops! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9757276177406311, "perplexity": 4328.568111904157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280364.67/warc/CC-MAIN-20170116095120-00004-ip-10-171-10-70.ec2.internal.warc.gz"} |