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### 1119
Multiparametric MRI with spatiotemporal evaluation reveals potential therapy response biomarkers for 177Lu-octreotate therapy of mice with human neuroendocrine tumor
Mikael Montelius1, Johan Spetz1, Oscar Gustafsson1, Evelin Berger2, Ola Nilsson3, Maria Ljungberg1, and Eva Forssell-Aronsson1
1Department of Radiation Physics, University of Gothenburg, Gothenburg, Sweden, 2Proteomics Core Facility, University of Gothenburg, Gothenburg, Sweden, 3Department of Pathology, University of Gothenburg, Gothenburg, Sweden
### Synopsis
Tissue parameters derived from multiparametric MRI were evaluated as potential imaging biomarkers for therapy response assessment in mice with human neuroendocrine tumor treated with 177Lu-octreotate. Animals were imaged before and repeatedly after 177Lu-octreotate treatment, using T2w, IVIM-DWI, DCE-MRI, T1- and T2*-mapping techniques. MR-parameters were evaluated regionally and longitudinally, and quantitative proteomics was used to evaluate underlying biological response in central and peripheral tumor separately. Several MR-parameters showed strong correlation with tumor response, as verified by MRI-based tumor volume measurements, but also with proteins associated with radiobiological effects on tumor tissue. Spatial and temporal evaluation increased sensitivity of the methods.
### Introduction and Purpose
Incidence of small intestine neuroendocrine tumors (siNETs) increase, and metastases are often present at diagnosis1. Radionuclide therapy using 177Lu-octreotate show promise for treatment of patients with inoperable siNET2,3, but indicators for individual prediction of disease progression and therapeutic response are needed for optimization and improved understanding of response mechanisms. Multiparametric magnetic resonance imaging (mpMRI) offer potential, non-invasive, response biomarkers for regional and longitudinal tumor characterization.
The aims of this study were to evaluate MR-features derived from mpMRI of siNETs regarding their efficacy of response assessment after 177Lu-octreotate therapy, and to correlate the MR-features with spatially matched protein expression levels.
### Material and Methods
Mice (n=21) with subcutaneous human siNETs received 15 MBq 177Lu-octreotate on day 0. mpMRI was performed under anaesthesia on days:[-1,1(n=4); -1,1,3,8,13(n=17)] in a small-animal 7T MR-system, using MR-techniques and sequence parameters described in fig.1.
Central and peripheral tumor samples were excised and snap frozen for quantitative proteomics after the final mpMRI experiment (fig.2), as previously described4, and proteins passing a false-discovery rate of 1% and containing minimum one unique peptide were further evaluated regarding biological functions using Gene Ontology database (http://www.geneontology.org)5.
Tumor volumes were estimated from T2w MRI as previously described6, and response was defined for individual tumors as the mean change in relative volume from day -1 to day 8. Model fitting- and semi-quantitative techniques were used to derive MR-parameter maps (fig.1), which were evaluated on 5 annular disc-shaped regions of different radius (fig.3). Pre-treatment MR-parameter values and their longitudinal development (e.g. value(day3)-value(day1), denoted Δday1:3) were evaluated for each disc and for the tumor mean value. Parameter changes were evaluated for ΔdaysA:B=-1:1,-1:3,-1:8,-1:13,1:3,3:8 and 8:13.
To rank the MR-parameters by their efficacy regarding response assessment, the lasso linear regression method, preconditioned using supervised principal components, was used7. Missing values were accounted for by repeating the method 5000 times, with stochastic data imputation each time, and the regression coefficients from each round were cumulatively added to form “β-sums” specific to each MR-parameter, according to which they were ranked.
### Results and Discussion
Tumor shrinkage or stabilized growth was observed in all tumors between day -1 and day 8, and growth was then re-established in most tumors (data not shown). It should be noted that a non-curative amount of 177Lu-octreotate was used to enable longitudinal evaluation and induce a wider range of biological responses.
The efficacy of MR-parameters regarding response prediction/assessment, was ranked according to the β-sum returned from feature selection (fig.3). The highest rank was reached by $SER^{-1}_{disc\:4} (parameter^{day/\Delta day}_{tumour\: region})$, followed by $D^{-1:3}_{disc\: 4}$. Five MR-parameters required regional evaluation to be ranked high enough for display in fig.3. Twenty-four MR-parameter-region combinations were highly ranked on Δday-1:1; corresponding number for day-1/Δday-1:3/Δday3:8 was 4/11/3 (fig.3). No other time-point combinations yielded high ranks. This indicates the importance of considering both spatial and temporal tumor heterogeneity in response evaluations, and that very early evaluations should be considered.
Totally, 104 proteins correlated with tumor response (p<0.01), 68 and 28 were found in central or peripheral tumor only, and 66 could be categorized into radiation related biological processes.
Several significant, strong correlations between high-ranked MR-parameters, response and protein levels were found (fig.4). For example, $D^{-1:3}_{disc\: 4}$ correlated with peripherally sampled CATA (Catalase, encoded by CAT), which is associated with oxidative stress, proliferation, cell cycle arrest, and apoptotic cell death. A line of evidence suggests that tissue water diffusivity, D, increases after successful therapy due to decreased cellular density, and thereby reduced membrane restrictions for diffusion. One hypothesis is thus that CATA is expressed in peripheral tumor due to successful therapy (apoptosis induced by beta-particle irradiation from 177Lu), and that this is reflected in increased peripheral diffusion (fig.5). $SER^{-1}_{disc\:4}$ and centrally sampled CCD89 protein (Coiled-Coil Domain Containing 89, encoded by CCDC89) were strongly correlated. CCD89 is associated with DNA damage & repair, proliferation, and cell cycle arrest, and SER reflects redistribution rate of contrast agent from extracellular extravascular space to the vascular lumen. The interpretation of this correlation is, however, not straightforward, since SER was measured peripherally (disc 4) whereas CCD89 was from central tumor.
### Conclusion
mpMRI offers several potential biomarkers for assessment of response to 177Lu-octreotate therapy in siNET. Interesting examples are SER and D since they were strongly correlated with response, but also with expression levels of relevant proteins, which is important from a clinical perspective. However, maximal utilization of the MR-parameters probably requires regional and longitudinal evaluation, e.g. for increased sensitivity to response related effects. Further studies are needed in order to better understand the associations between MR-parameters and underlying biology.
### Acknowledgements
We are also grateful to the Proteomics Core Facility at Sahlgrenska Academy, Gothenburg University, who performed the analysis for protein quantification and Inga-Britt and Arne Lundbergs Research Foundations for the donation of the Orbitrap Fusion Tribrid MS instrument used in this analysis. This study was supported by grants from the Swedish Research Council, the Swedish Cancer Society, BioCARE – a National Strategic Research Program at the University of Gothenburg, the King Gustav V Jubilee Clinic Cancer Research Foundation, the Sahlgrenska University Hospital Research Funds, the Assar Gabrielsson Cancer Research Foundation, the Adlerbertska Research Fund, the Wilhelm and Martina Lundgren science trust fund and the Royal Society of Arts and Sciences in Gothenburg (KVVS).
### References
1. Yao JC, Hassan M, Phan A, et al. One hundred years after "carcinoid": epidemiology of and prognostic factors for neuroendocrine tumors in 35,825 cases in the United States. J Clin Oncol. 2008;26:3063-3072.
2. Kwekkeboom DJ, De Herder WW, Kam BL, et al. Treatment with the radiolabeled somatostatin analog [177 Lu-DOTA 0,Tyr3]octreotate: toxicity, efficacy, and survival. J Clin Oncol. 2008;26:2124-2130.
3. Sward C, Bernhardt P, Ahlman H, et al. [177Lu-DOTA 0-Tyr 3]-octreotate treatment in patients with disseminated gastroenteropancreatic neuroendocrine tumors: the value of measuring absorbed dose to the kidney. World journal of surgery. 2010;34:1368-1372.
4. Spetz J, Montelius M, Berger E, et al. 177Lu-octreotate induces tumor volume regression and suppresses invasive potential in small intestine neuroendocrine tumors. Manuscript
5. Ashburner M, Ball CA, Blake JA, et al. Gene ontology: tool for the unification of biology. The Gene Ontology Consortium. Nat Genet. 2000;25:25-29.
6. Montelius M, Ljungberg M, Horn M, et al. Tumour size measurement in a mouse model using high resolution MRI. BMC Med Imaging. 2012;12:12.
7. Tibshirani R. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological). 1996;267-288.
### Figures
Figure 1. MR techniques, pulse sequence parameters and MR-parameter definitions. The same slice positions were used for all technique. Techniques with a single slice had the same position as the central slice of the other techniques. Abbreviations: RAREVTR = RARE with variable repetition time, MGE = multi echo gradient echo, IVIM = intravoxel incoherent motion, NSA = number of averages, pFourer = partial Fourier acceleration, δ = diffusion encoding gradient duration, Δ = diffusion encoding gradient separation, αex = excitation flip angle, αref = refocusing flip angle
FIgure 2. A) MR-parameter maps from the most central tumour image. Delineation/duplicates with decreasing size that defined data extraction regions (e.g. highlighted disc3) are shown on the AUCn-map. B) After final mpMRI experiment, animals were killed, tumours were divided parallel to the imaged plane, and central and peripheral tissue samples were taken from one tumour half for quantitative proteomics. C) Where the central image had been acquired on the other tumour half, tissue ink was injected for relocation of the histological slices, and histological index maps were created (not evaluated here). Tumour type was verified histologically
Figure 3. Conceptual tumour showing MR-parameters (pie sections) ranked (colour) regarding when (day/Δday) and where (disc/pie radial section) they were most efficacious for response assessment, by the β-sum from feature selection (relative β-sums<0.05 not displayed). ΔdayA:B denote parameter value change from day A to B. For example, highest rank was reached by pre-treatment SER in peripheral regions (upper left: disc4, relative β-sum=1), and a changing D between day-1 and 3 was also ranked high (mostly in disc4, but all discs reached display limit). MR-parameters are in red font (followed by relative β-sum) where whole tumour mean value was ranked high
Figure 4. Pairwise correlations between the response variable (Resp.var), high-ranked MR-parameters, protein expression levels and pre-treatment tumour volume. Correlation coefficients are displayed in red if correlation p-values < 0.05, bold red frame indicate statistical significance also after correcting for multiple comparison using the Benjamini & Yekutieli method (q = 0.05). Diagonal entries show the histogram of corresponding parameter distribution. BE=brevity of enhancement, D=diffusion coefficient, SER=signal enhancement ratio, Vol=tumour volume, TTP=time to peak. Index c/p=centrally/peripherally sampled protein, and MR-parameters are stated as $parameter^{day/\Delta day}_{tumour\: region}$
Figure 5. Development of D in the five discs over time for the shrinking (blue) and only stabilising (red) tumours. For each time-point (e.g. day-1), central tumour values (disc1) are displayed to the left, with increasing disc numbers to the right. Note the increasing D in most discs on day3 for the tumours responding by shrinking, followed by decreased D on day13, possibly due to re-established growth. Median, 25th and 75th percentiles are the middle point and lower and upper bold lines, respectively. Medians are significantly different at the 5%-significance level if intervals marked by triangular marker do not overlap
Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)
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# Car rental cost function graph
An economy car rented in Florida from Enterprise on a weekly basis costs $185$ per week. Extra days cost $37$ per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number x of days used, where $7 \le x \le 14$. Graph this function.
From what I understand I made the following piecewise function.
$$f(x) = \begin{cases} 185 & x=7\\[2ex] 185+(x-7)(37) & 7<x<14\\[2ex] 370 & x=14 \end{cases}$$
What would the graph be for this ?
• Are you sure you got the function right? How much would it cost to rent the car 13 days? – David Sep 20 '15 at 23:32
• @David I edited the function. I'm guessing it is right now ? – RufioLJ Sep 21 '15 at 0:08
The current version of the function is not right. As @David implied, you do not correctly handle the fact that anything between $11$ and $14$ days will be billed the same as $2$ weeks, due to the "until the day rate exceeds the weekly rate" clause. You also do not handle $x$ being a non-integer: it should be rounded up.
$$f(n)=\begin{cases} 185+37(\lceil x\rceil-7), & 7\le x\le 11 \\[2 ex] 370, & 11<x\le 14 \end{cases}$$
where $\lceil x\rceil$ is the ceiling function, rounding up to the smallest integer greater than or equal to $x$.
The ceiling function means that each small section of the graph, of width $1$, will be a horizontal line. The second case means that the last part of the graph is a horizontal line between $x=11$ exclusive to $x=14$ inclusive. The first case means that the segments will be like a staircase, with the left-hand endpoints on the line $y=185+37x$. Here is a graph.
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## cookiess Group Title The energy required to split apart hydrogen bromide according to the equation is 6.01 × 10-19 J/molecule. HBr(g) → H(g) + Br(g) Calculate the wavelength (m) of the photon that can dissociate hydrogen bromide. Express answer in scientific notation. one year ago one year ago
This is sort of a blur to me since it's been quite some time, but I think you use: $E = hf = \frac{ hc }{ \lambda }$ where the greek symbol, lambda, represents wavelength.
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# Is it simple?
Calculus Level 4
If $\displaystyle f(x) = \int\limits_1^x\frac{\log{t}}{1+t}dt$ find the value of $f(e)+f(e^{-1})$
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# Should science replace religion?
Discussion in 'General Philosophy' started by wegs, May 7, 2019.
1. ### sideshowbobSorry, wrong number.Valued Senior Member
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6,113
Maybe so. That's why we need empirical testing as well as ivory-tower theorizing.
3. ### river
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13,334
Yes
But the empirical is the priority basis of any ivory-tower theorizing .
5. ### sideshowbobSorry, wrong number.Valued Senior Member
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6,113
I don't know if it's the basis, per se. Both feed off each other.
7. ### river
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13,334
The Empirical is the starting point . The most basic .
In the End , any theorizing by the ivory tower thinkers must agree with the empirical evidence .
8. ### GoldtopRegistered Senior Member
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316
Even if science can explain everything? I suppose that's why we now have watered down, practically useless versions of religions, they try to explain everything.
Of course, you're free to explain anything with religion, that is, if it can't be explained any other way. Go.
9. ### MusikaLast in SpaceValued Senior Member
Messages:
2,701
https://en.m.wikipedia.org/wiki/Scholasticism
For your edification, since, judging by your statements, you don't appear robustly familiar with either history or religion.
10. ### Michael 345New year. PRESENT is 70 years oldValued Senior Member
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8,807
Science is locked in (rigidly to reality) but I doubt it is capable of explaining everything
There are sign post pointing to possibilities but fog hiding the evidence from those possibilities
Religion does not have even sign post as guides
11. ### iceauraValued Senior Member
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29,974
Early on in this thread "religion" was assumed to be dealing with the aesthetic/emotional/"irrational" aspects of human life. That this placed it "above", "inclusive", "higher", "more central", "governing", and so forth, than rationality and all limited to rationality, followed - not without objection, but the weight of the evidence is conclusive. To talk of science replacing religion is to talk not so much of apples replacing oranges as equivalent incommensurables, as to talk of a part replacing a whole, a subset replacing its set, an engine replacing a car.
Along that line, and to illustrate via an unusually simple and direct real world example: https://aeon.co/ideas/how-erasmus-darwins-poetry-prophesied-evolutionary-theory
Erasmus Darwin's intellectual work is well known to have set the stage, as it were, for his grandson's contributions - he provided to his grandson, who grew up in his intellectual shadow, context, approach, and some major concepts in broad outline (and therefore vague - no underestimating Charles's fundamental work. Charles did the science.)
That Erasmus Darwin was a poet, that an art was a central means or medium of that intellectual work, is not unusual in the history of science.
Quantum Quack likes this.
12. ### Quantum QuackLife's a tease...Valued Senior Member
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22,090
I would be interested to know how science would be capable of valuing the following art work and the unknown ideas and dare I suggest, science it has inspired.
Blue Poles.
Jackson Pollock. USA Citizen. 1952.
Purchased by Australian Government for $1.3 Million in 1973 Currently (2018) valued at between$100 and 300 million.
Last edited: Jun 7, 2019
13. ### wegsMatter & Pixie DustValued Senior Member
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6,367
Noticed a curious quote today:
Science is a culture of doubt, while religion is a culture of faith.
Hmm. I'm not sure I agree. I think science requires some faith, too. Faith in believing that what you're doing is making a difference, for starters?
14. ### Michael 345New year. PRESENT is 70 years oldValued Senior Member
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8,807
You might have faith in an experiment
If it does not work out
You do not retain said faith
Difference
wegs likes this.
15. ### RainbowSingularityValued Senior Member
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4,729
Religion wants to replace science with religion
that political party in the usa is called "The conservatives"
except for guns bombs and torture
those religious types love that stuff.
16. ### spidergoatLiddle' Dick TaterValued Senior Member
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53,966
Not the same kind of faith, so, an equivocation fallacy. Thinking that something is merely likely is not what religious faith is about.
17. ### spidergoatLiddle' Dick TaterValued Senior Member
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53,966
Easy, take a survey, do a statistical analysis on the results to determine average worth value of this piece to a population sample.
18. ### Quantum QuackLife's a tease...Valued Senior Member
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22,090
true... but how do you value the inspiration it might generate. For all you know the painting might inspire a scientist into devising the method of opening a star gate to another galaxy...
Value has a lot more to it than mere money...money is merely a token system any how...
19. ### spidergoatLiddle' Dick TaterValued Senior Member
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53,966
I didn't mention money! A survey could measure any intangible quantity you want, it just depends on what questions you ask. You could even measure involuntary responses to the art, or the destruction of the art.
20. ### Quantum QuackLife's a tease...Valued Senior Member
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22,090
And what bench mark would you use and who would provide it?
Indigenous, Greek, Croation, English... who?
Ultimately it would come down to individual preference i think.
21. ### Quantum QuackLife's a tease...Valued Senior Member
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22,090
For example the destruction of Mosul in Iraq. It is possible to determine the value of reconstruction, but not the cultural losses, ancient ruins destroyed, relics stolen and lost due to ISIL insanity.
Just one ancient Pre-Persian empire statue destroyed can not be valued other than in a token way.
Eventually it would come down to the individual. Some could be quite apathetic and others could be totally devastated.
22. ### iceauraValued Senior Member
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29,974
The aesthetic, emotional, irrational, etc (the stipulated domain of religion) is comprehensive, inclusive, and leading. It sets the context, organizes and incorporates the whole of the human mind's doing, frames everything.
The rational, and even more that subset of the rational we label "science", is smaller, subsidiary, on a "lower" level of thought, a contributing aspect or even substrate of the larger aesthetic whole.
One can do art and song and worship and so forth without scientific or any other rational contribution. One cannot begin to do science, or even think rationally, without an aesthetic ground or frame. (That insight, centuries old, has in our time been further supported by scientific research into brain injuries, human decision making, and so forth. )
Meaning derives from context. The aesthetic etc domain is where we find the context of the human mind's doings - the source of meaning.
Fundamentalist Abrahamic monotheistic religions want to replace scientific findings and theories with their dogma.
They have been granted an unearned opportunity to do that, partly by the failure of the scientific community to create or settle on an establishment of suitable religion. There's a vacuum at the top.
Last edited: Jun 8, 2019
wegs and Quantum Quack like this.
23. ### spidergoatLiddle' Dick TaterValued Senior Member
Messages:
53,966
That's what I'm saying, you can measure individual preference across various cultures. There's no absolute value for such things.
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# Running Contur¶
Contur now has a first production release, 1.0.0 !
Check out the contur code from our repository at gitlab. The recommended Rivet release is Rivet 3.1.X.
git clone git@gitlab.com:hepcedar/contur.git
git checkout release-1-1-x
This is the latest bugfix of the release branch. If you need to use an older Rivet release you can try using
git checkout rivet-2.7-frozen
but this is pretty old and not recommended for long term use.
The README in the project overview provides some instructions. We also have a docker-based tutorial produced for various schools (YETI and MCnet) which is available here. To get some information about the available options, see contur --help.
(the tutorial needs updating for the release version)
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Nuffield Mathematics teaching resources are for use in secondary and further education
# FSMQ Level 3 (legacy) Working with algebraic and graphical techniques scheme of work
This FSMQ requires a total of 60 guided learning hours that could be used in a variety of ways, such as 2 hours per week for 30 weeks, 4 hours per week for 15 weeks, or 5 hours per week for 12 weeks. Before starting this course learners should be able to:
• plot by hand accurate graphs of paired variable data and linear and simple quadratic functions (including the type y = ax2 + bx + c) in all 4 quadrants
• recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions (including the type y = kx2 + c)
• fit linear functions to model data (using gradient and intercept)
• rearrange basic algebraic expressions by collecting like terms, expanding brackets and extracting common factors
• solve basic equations by exact methods including pairs of linear simultaneous equations
• use power notation (including positive and negative integers and fractions)
• solve quadratic equations by factorising and using the formula $\frac&space;{-&space;b&space;\pm&space;\sqrt{b^2&space;-&space;4ac}}{2a}$ (must be memorised).
A suggested work scheme for this unit is given below. It includes some revision of the above as well as the other topics and methods to be covered, but note that you will also need to allow time for students to complete an AQA Coursework Portfolio. The Coursework Portfolio requirements are listed on page 56 of the AQA Advanced FSMQ specification and also on page 101 of the AS Use of Mathematics specification. The assignments below provide some examples of the sort of work your students could include in their portfolios, but if possible you should use work that is more relevant to their other studies or interests..
The following techniques should be introduced as soon as possible and used throughout the course:
• using a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
• doing calculations without a calculator using written methods and mental techniques
• graph plotting using computer software or a graphical calculator, and using trace and zoom facilities to find significant features such as turning points and points of intersection
• checking calculations using estimation, inverse operations and different methods.
Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Linear functions (3 hours) Revise the main features of graphs of direct proportional (y = mx) and linear (y = mx + c) functions. Fit such functions to real data using gradients and intercepts. Understand whether it is appropriate or not to use a particular function to model data by consideration of intercepts, long term behaviour (etc.) in real world terms. Use error bounds to consider a range of possible functions to model data. Solve linear simultaneous equations using graphical and algebraic methods. Linear graphs Presentation and activity to introduce linear graphs. Graphs of functions in Excel This activity shows students how to draw graphs of algebraic functions in Excel. Interactive graphs Uses interactive spreadsheet graphs to introduce the shape and main features of proportional, linear, quadratic and power graphs. (Can be split into three separate parts.) Graphic calculators Presentation introducing students to the CASIO fx-7400G PLUS calculator. Using the CASIO fx-7400G PLUS Notes on how to use this calculator - includes how to draw the graph of a function, how to investigate how well a model fits data, and how to find a model. Graphic calculator equations A variety of equations to provide practice in using a graphic calculator. Car bonnet Students are asked to consider linear approximations to temperature data. Match linear functions and graphs Twelve sets of cards, each containing a linear graph, its equation and the real situation it represents – for students to match. Simultaneous equations on a graphic calculator Instructions for using the CASIO fx-7400G PLUS calculator to solve simultaneous equations.
Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Quadratic functions (7 hours) Draw graphs of quadratic functions of the form: · y = ax2 + bx + c · y = (rx – s )(x – t) · y = m(x + n)2 + p relating the shape, orientation and position of the graph to the constants and relating zeros of the function f(x) to roots of the equation f(x) = 0. Fit quadratic functions to real data. Revise solving quadratic equations by: · factorising · using the formula $\frac&space;{-&space;b&space;\pm&space;\sqrt{b^2&space;-&space;4ac}}{2a}$ Rearrange any quadratic function into the forms y = ax2 + bx + c and y = a(x + b)2 + c Find maximum and minimum points of quadratics by completing the square. Test run Students interpret a speed-time graph and fit both linear and quadratic models. The performance data is also given in an Excel spreadsheet for comparison with models. Model the path of a golf ball Students consider linear and quadratic models for the path of a golf ball. Broadband A, B, C Instructions showing how to use Excel, a graphic calculator and algebra to find a quadratic model for the growth in broadband connections in recent years. Presentation shows the algebra version. Two on a line and three on a parabola Spreadsheets giving a linear or quadratic function that passes through particular points. Factor cards Nearly 100 pairs of cards showing a wide variety of quadratic expressions and their factors. Pairing will give students practice in expanding brackets or factorising. Water flow Includes data about the velocity of water as it flows along an open channel and sample examination question. Data could also be used to give practice for portfolio requirements or form the basis for an assignment. Completing the square Presentation shows how to complete the square and use this form to sketch graphs. Card-matching activity using a selection from 24 sets each of 3 cards showing a quadratic graph, the corresponding function and its completed square form. Gradients of curves, maxima and minima (3 hours) Calculate and understand gradient at a point on a graph using tangents drawn by hand (and also using zoom and trace facilities on a graphic calculator or computer if possible). Use and understand the correct units for rates of change. Interpret and understand gradients in terms of their physical significance. Identify trends of changing gradients and their significance both for known functions and curves drawn to fit data. Tin can Students design a tin can, using algebraic and graphical techniques. Optional use of the internet. Maximum and minimum problems Presentation and practice questions using a spreadsheet or graphic calculator to solve problems involving maximum and minimum values.
Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Power functions and inverse functions (3 hours) Draw graphs of functions of powers of $x$ including $y&space;=&space;kx^n$ where $n$ is a positive integer, $y&space;=&space;kx^-^1&space;=&space;\frac{k}{x}$ , $y&space;=&space;kx^-^2&space;=&space;\frac{k}{x^2}$ , and $y&space;=&space;kx^\frac{1}{2}&space;=&space;\sqrt&space;x$ Learn the general shape and position of such functions. Find the graph of an inverse function using reflection in the line y = x. Solve polynomial equations of the form axn = b Interactive graphs See above. Growth and decay (8 hours) Draw graphs of exponential functions of the form $y&space;=&space;ka^m^x$ and $y&space;=&space;ke^m^x$ (m positive or negative) and understand ideas of growth and decay. Draw graphs of natural logarithmic functions of the form y = a ln(bx) and understand the logarithmic function as the inverse of the exponential function. Solve exponential equations of the form $A~\textup{exp}(mx&space;+&space;c)&space;=&space;k)$ Learn and use the laws of logarithms · $\textup{log}(ab)&space;=&space;\textup{log}~a&space;+&space;\textup{log}~b$ · $\textup{log}(\frac{a}{b})&space;=&space;\textup{log}~a&space;-&space;\textup{log}~b$ · $\textup{log}(a^n)&space;=&space;n~&space;\textup{log}~a$ to convert equations involving powers to logarithmic form and solve them (using both base 10 and natural logarithms). Growth and decay Presentation using compound interest and radioactive decay to introduce exponential growth and decay. Population growth Students use a given exponential function to model population data, then consider predictions made by the model. Calculator table Students use the calculator’s table function to complete tables for population models then draw and use the corresponding graphs. Ozone hole Data concerning depletion of ozone levels and the increase in the area of the Antarctic ozone hole over the last twenty years. Students investigate possible linear, quadratic and exponential models. Optional use of spreadsheet. Climate prediction A and B Students use an Excel spreadsheet and/or graphic calculator to find polynomial functions to model temperature change and compare with exponential models. Cup of coffee Data sheet gives the amount of caffeine remaining in the bodies of a group of people at intervals of 1 hour after they have drunk a cup of coffee or cola. Students are asked to model the data (exponential and linear functions).
Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Transformations of graphs (4 hours) Use: · translation of y = f(x) parallel to the y axis to give y = f(x) + a · translation of y = f(x) parallel to the x axis to give y = f(x + a) · stretch of y = f(x) parallel to the y axis to give y = af(x) · stretch of y = f(x) parallel to the x axis to give y = f(ax) Sea defence wall (assignment) Two versions of an assignment in which students find functions to model the outline of a sea defence wall. The first version encourages students to work independently, the second is more structured for less able students. Trigonometric Functions (8 hours) Draw graphs of · $y&space;=&space;A~\textup{sin}(mx&space;+&space;c)$ · $y&space;=&space;A~\textup{cos}(mx&space;+&space;c)$ Learn the general shape and position of trigonometric functions and use the terms amplitude, frequency, wavelength, period and phase shift correctly. Fit trigonometric functions to real data. Solve trigonometric equations of the form $y&space;=&space;A~\textup{sin}(mx&space;+&space;c)&space;=&space;k$ and $y&space;=&space;A~\textup{cos}(mx&space;+&space;c)&space;=&space;k$ Coughs and sneezes Includes data about the way in which an outbreak of the common cold spreads. Students are asked to model the data using trigonometric and polynomial functions. SARS A and B (assignments) Data set giving the number of deaths from SARS. Students choose, draw and evaluate functions to model the data. Sunrise and sunset times (assignment) Students find and evaluate trigonometric functions to model how the amount of daylight varies with the day of the year. Includes data for Adelaide, Brisbane and London. Tides (assignment) Data set giving the water depth each hour during a day. Students choose, draw and evaluate functions to model the data.
Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Linearising data (6 hours) Determine parameters of non-linear laws by plotting appropriate linear graphs, for example: · $y&space;=&space;ax^2&space;+&space;b$ by plotting $y$ against $x^2$ · $y&space;=&space;\frac{a}{x}&space;+&space;b$ by plotting $y$ against $\frac{1}{x}$ · $y&space;=&space;ax^3&space;+&space;b$ by plotting $y$ against $x^3$ · $y&space;=&space;a~\textup{sin}(x)&space;+&space;b$ by plotting $y$ against $\textup{sin}(x)$ · $y&space;=&space;ax^b$ and $y&space;=&space;a^x$ using base 10 or natural logarithms Log graphs - earthquakes Examples (involving earthquakes and planetary motion) that can be used to introduce log graphs. Ideas of experiments and other situations that can be used for practice. (Includes logs to base 10.) Gas guzzlers Slide presentation and activity in which students use a log graph to find an exponential function to model real data. Smoke strata Includes data about the height of smoke layers due to a fire in a tall building and sample examination question. Data could also be used to give practice in linearising data. Earthquakes (ln version & log version) Activity that uses natural or base 10 logarithms to find an equation connecting the energy released by an earthquake and its Richter value. Revision (6 hours)
Page last updated on 25 January 2012
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# Scoring solution for Farkle (dice game)?
I've written the following function on Python (3.3) to take a list of 6 integers and return the maximum score for a game of Farkle. While the results are correct I don't feel it's a very Pythonic implementation. I'd like to hear from the community on how I could make it better and more Pythonic. Any and all tips are most welcome!
"""
written and tested in Python 3.3.6
"""
from random import randint
def score(my_roll, sides=6):
"""
Calculate a Farkle score using the traditional scoring method, which is:
4 1s: 2,000 points
3 1s: 1,000 points
Single 1: 100 points
Single 5: 50 points
Triple of any non-1 number: 100 x number showing
Quadruple of any non-1 number: Double the triple score
Notes:
- Doubles score nothing (unless a 1 or a 5) as above
- All scoring dice must be rolled in a single turn (i.e. they
are not additive over turns)
- Rolling all 6 will be two sets and scored accordingly
[4] + [2]
[3] + [3]
[2] + [2] + [2]
Examples:
1,1,1,5,5,5 ==> 1500 (1000 for 1s + 250 for 3 5s)
1,1,1,1,6,6 ==> 2000 (2000 for 4 1s)
5,3,6,5,3,3 ==> 400 (300 for 3 3s + 100 for 2 5s)
1,2,2,3,3,5 ==> 150 (100 for a 1 and 50 for a 5)
"""
#create a table to hold the count of each die roll
dice_array = [0] * sides
score = 0
#add up the number appearances of each die roll and store it in the table
for dice in my_roll:
dice_array[dice-1] += 1
"""
based on the above scoring, determine the MAXIMUM score; in actual Farkle the
player would choose which die to 'bank' and which to re-roll
"""
for (i, count) in enumerate(dice_array):
dice = i + 1 #this makes it easier to keep track of the die we're on
if dice == 1:
if count == 6: score += 2200
if count == 5: score += 2100
if count == 4: score += 2000
if count == 3: score += 1000
if count in [1, 2]: score += (count * 100)
else:
if count >= 4: score += (dice * 200)
if count >= 3: score += (dice * 100)
if (dice == 5 and count != 3): score += (count * 50)
return score
#test cases
while True:
roll = input("Enter 6 values (1-6) separated by a space: ")
roll = [random.randint(1,6) for i in range(0, 6)] if roll == "" else [int(i) for i in roll.split(' ')]
print(str(score(roll)) + " : " + str(roll))
## migrated from stackoverflow.comJan 6 '16 at 21:01
This question came from our site for professional and enthusiast programmers.
• My only comment is that your examples seem to contradict themselves. 1,1,1,5,5,5 should be 1500, for example (1000 for triple 1s and 500 for triple 5s). You specifically call out quadruple 1s as being 2000, but ALL quadruples are double their triple score. – Adam Smith Jan 6 '16 at 19:53
• Cheers for the sharp eye on the example! The rules I was going with, that I linked to, say that it's only 1,000 for 4 1s (no other numbers): A roll of a 1 is worth 100 points. A roll of a 5 is worth 50 points. Three (3) dice rolled at the same time with the same value is worth 100 times the face value, for example: three 2’s rolled is 200 points and three 5’s rolled is 500 points. ** One exception to this rule is that three 1’s rolled is 1,000 points rather than 100 points. – Matt O'Neill Jan 6 '16 at 21:27
• I was writing my code for this hoping it would come out simple and ended with this mess: pastebin.com/fk3J79Sf , well this is life... – Caridorc Jan 6 '16 at 22:13
• @Caridorc oh gosh that's awful.... – Adam Smith Jan 6 '16 at 22:53
• Your comment "Doubles score nothing (unless a 1 or a 5) as above" is a bit unclear, it looks like you meant "Single or Double 1: 100 points each, Single or Double 5: 50 points each" – smci Jan 9 '17 at 5:50
## 2 Answers
### "Pythonic" programming
Code is "pythonic" when it expresses its intention clearly, is easy to read or even looks like pseudo-code, and uses as little low-level garbage as possible. Let's take a look at a couple examples of low-level work in your code:
dice_array = [0] * sides
score = 0
for dice in my_roll:
dice_array[dice-1] += 1
# as an aside, this isn't an array, it's a list.
This is wonky because it buids a fixed-size list, then plays with values-as-list-indexes to get a count. That might be fast, but it's certainly ugly. Let's not do that.
import collections
counts = collections.Counter(my_roll)
This is much better. It builds a dict-like object, a collections.Counter that has the counts for any dice present in the roll. counts[non_rolled_die] will throw a KeyError while counts.get(non_rolled_die, 0) will act like your current code does.
Your counting loop isn't a whole lot better. Let's look:
for i, count in enumerate(dice_array):
dice = i + 1
We'll stop here for just long enough to mention that enumerate takes a keyword argument start that lets you tell it where to start counting from. This should be:
for die, count in enumerate(dice_array, start=1):
But since I switched to using a collections.Counter, we can just do:
for die, count in counts.items():
if die == 1:
if count == 6: score += 2200
if count == 5: score += 2100
if count == 4: score += 2000
if count == 3: score += 1000
if count in [1, 2]: score += (count * 100)
else:
if count >= 4: score += (dice * 200)
if count >= 3: score += (dice * 100)
if (die == 5 and count != 3): score += (count * 50)
First of all, avoid these repeated if statements. Try instead:
if die == 1:
if count == 3: score += 1000
elif count == 4: score += 2000
elif count > 4:
score += 2000 + (count - 4) * 100
However there's a better approach. We already know that the triple value for each number is 100 * number, except for 1 which is 100 * 10 * number. Let's use that!
for die, count in counts.items():
die_score = 0
if count == 4:
die_score += die * 2000 if die == 1 else die * 200
count -= 4
elif count == 3:
die_score += die * 1000 if die == 1 else die * 100
count -= 3
This should handle triplets and quadruplets of any value, and removes from count the number of dice consumed by the grouping. Now let's look at single dice values.
if die in [1, 5]:
single_value = 100 if die==1 else 50
die_score += single_value * count
In fact we can factor some of that out to drop the if block.
# somewhere earlier in the function
single_values = {1: 100, 5:50}
# then inside the loop we're looking at here....
die_score += single_values.get(die, 0) * count
Finally at the end of our loop (just before exiting it), we add the single die_score to the roll-wide score.
score += die_score
### Testing
I'm a big fan of the unittest module for writing tests. In your case doctest may work better (since you've written examples already that only have to be tweaked slightly to use doctest), but if you start writing more intense functions, it's important to be able to grow your testing suite appropriately. Let's write some tests!
# ./test_farkle.py
import unittest
import farkle
class FarkleTests(unittest.TestCase):
cases = [([1,1,1,5,5,5], 1500),
([1,1,1,1,6,6], 2000),
([5,3,6,5,3,3], 400),
([1,2,2,3,3,5], 150)]
def testRolls(self):
for got, want in self.cases:
self.assertEqual(farkle.score(got), want)
# fails test if farkle.score(got) != want
### A change in data structure
It should occur to you that for every (die, count) pair, score increases by a given amount. This means we can hardcode that data in something like a dict.
# {die: {count: value, ... }, ... }
valuedict = {1: {1: 100,
2: 200,
3: 1000,
4: 2000,
5: 2100,
6: 2200},
2: {3: 200,
4: 400},
3: {3: 300,
4: 600},
4: {3: 400,
4: 800},
5: {1: 50,
2: 100,
3: 500,
4: 1000,
5: 1050,
6: 1100},
6: {3: 600,
4: 1200}}
Now your function becomes pretty simple!
def score(my_roll, sides=6):
valuedict = ... # the whole deal above
counts = collections.Counter(my_roll)
score = 0
for die, count in counts.items():
score += valuedict[die][count]
return score
Or even more simply:
def score(my_roll, sides=6):
valuedict = ...
counts = collections.Counter(my_roll)
return sum(valuedict[die][count] for die,count in counts.items())
• You are incredible! Thank you. Wish I could give you point++! This is exactly the type of breakdown I was looking for. I've just finished two intro python courses and this was my first self test. I knew there was a ton that needed work to make it look like python and you nailed it. I'd actually started the whole thing with collections so I could use the .count method but I kept getting an error that collections couldn't be found. Hence this route! – Matt O'Neill Jan 7 '16 at 7:07
• collections is a stdlib module, just be sure to import it at top level. – Adam Smith Jan 7 '16 at 9:42
• N.B. the "classic" Farkle rules (as I remember them) are here. When you start getting into more complicated combinations (e.g. four of one die, two of another), the hardcoded solution will be useless. – Adam Smith Jan 7 '16 at 20:25
Make the examples runnable
If You have made the effort of writing your examples, it is a waste to let them sit there doing nothing. Here is the format required to run them.
>>> score([1,1,1,5,5,5])
1250
...
Look into the doctest module.
• I did have that but I thought I'd put them in the comments for people who didn't get my language. I thought providing the self-entry mechanism was more fun, too. – Matt O'Neill Jan 7 '16 at 7:09
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SKY-MAP.ORG
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# θα Tau (Phaeo)
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Modelling the components of binaries in the Hyades: the dependence of the mixing-length parameter on stellar massWe present our findings based on a detailed analysis of the binaries ofthe Hyades, in which the masses of the components are well known. We fitthe models of the components of a binary system to observations so as togive the observed total V and B-V of that system and the observed slopeof the main sequence in the corresponding parts. According to ourfindings, there is a very definite relationship between themixing-length parameter and the stellar mass. The fitting formula forthis relationship can be given as α= 9.19(M/Msolar-0.74)0.053- 6.65, which is valid for stellar masses greaterthan 0.77Msolar. While no strict information is gathered forthe chemical composition of the cluster, as a result of degeneracy inthe colour-magnitude diagram, by adopting Z= 0.033 and using models forthe components of 70 Tau and θ2 Tau we find thehydrogen abundance to be X= 0.676 and the age to be 670 Myr. If weassume that Z= 0.024, then X= 0.718 and the age is 720 Myr. Our findingsconcerning the mixing-length parameter are valid for both sets of thesolution. For both components of the active binary system V818 Tau, thedifferences between radii of the models with Z= 0.024 and the observedradii are only about 4 per cent. More generally, the effectivetemperatures of the models of low-mass stars in the binary systemsstudied are in good agreement with those determined by spectroscopicmethods. Abundances of Baade's Window Giants from Keck HIRES Spectra. I. Stellar Parameters and [Fe/H] ValuesWe present the first results of a new abundance survey of the Milky Waybulge based on Keck HIRES spectra of 27 K giants in the Baade's Window(l=1deg, b=-4deg) field. The spectral data used inthis study are of much higher resolution and signal-to-noise ratio thanprevious optical studies of Galactic bulge stars. The [Fe/H] values ofour stars, which range between -1.29 and +0.51, were used to recalibratelarge low-resolution surveys of bulge stars. Our best value for the mean[Fe/H] of the bulge is -0.10+/-0.04. This mean value is similar to themean metallicity of the local disk and indicates that there cannot be astrong metallicity gradient inside the solar circle. The metallicitydistribution of stars confirms that the bulge does not suffer from theso-called G dwarf problem. This paper also details the new abundancetechniques necessary to analyze very metal-rich K giants, including anew Fe line list and regions of low blanketing for continuumidentification.Based on data obtained at the W. M. Keck Observatory, which is operatedas a scientific partnership among the California Institute ofTechnology, the University of California, and NASA and was made possibleby the generous financial support of the W. M. Keck Foundation. Hyades Oxygen Abundances from the λ6300 [O I] Line: The Giant-Dwarf Oxygen Discrepancy Revisited1,We present the results of our abundance analysis of Fe, Ni, and O inhigh signal-to-noise ratio, high-resolution Very Large Telescope UVESand McDonald 2dcoudé spectra of nine dwarfs and three giants inthe Hyades open cluster. The difference in Fe abundances derived from FeII and Fe I lines ([Fe II/H]-[Fe I/H]) and Ni I abundances derived frommoderately high-excitation (χ~4.20 eV) lines is found to increasewith decreasing Teff for the dwarfs. Both of these findingsare in concordance with previous results ofoverexcitation/overionization in cool young dwarfs. Oxygen abundancesare derived from the [O I] λ6300 line, with careful attentiongiven to the Ni I blend. The dwarf O abundances are in star-to-staragreement within uncertainties, but the abundances of the three coolestdwarfs (4573 K<=Teff<=4834 K) evince an increase withdecreasing Teff. Possible causes for the apparent trend areconsidered, including the effects of overdissociation of O-containingmolecules. O abundances are derived from the near-UV OH λ3167line in high-quality Keck HIRES spectra, and no such effects are found;indeed, the OH-based abundances show an increase with decreasingTeff, leaving the nature and reality of the cool dwarf [OI]-based O trend uncertain. The mean relative O abundance of the sixwarmest dwarfs (5075 K<=Teff<=5978 K) is[O/H]=+0.14+/-0.02, and we find a mean abundance of [O/H]=+0.08+/-0.02for the giants. 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Statistical Constraints for Astrometric Binaries with Nonlinear MotionUseful constraints on the orbits and mass ratios of astrometric binariesin the Hipparcos catalog are derived from the measured proper motiondifferences of Hipparcos and Tycho-2 (Δμ), accelerations ofproper motions (μ˙), and second derivatives of proper motions(μ̈). It is shown how, in some cases, statistical bounds can beestimated for the masses of the secondary components. Two catalogs ofastrometric binaries are generated, one of binaries with significantproper motion differences and the other of binaries with significantaccelerations of their proper motions. Mathematical relations betweenthe astrometric observables Δμ, μ˙, and μ̈ andthe orbital elements are derived in the appendices. We find a remarkabledifference between the distribution of spectral types of stars withlarge accelerations but small proper motion differences and that ofstars with large proper motion differences but insignificantaccelerations. 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Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Seismic constraints on open clustersThe aim of this theoretical and modelling paper is to derive knowledgeon the global and structural parameters of low-mass stars usingasteroseismology and taking advantage of the stellar collective behaviorwithin open clusters. We build stellar models and compute the seismicsignal expected from main sequence objects in the 0.8-1.6Mȯ range. We first evaluate apparent magnitudes andoscillations-induced luminosity fluctuations expected in the Hyades, thePleiades and the α Persei clusters. The closest cluster presents afeasible challenge to observational asteroseismology in the present andnear future. The remainder of the work therefore focuses on the Hyades.We combine seismological and classical computations to address threequestions: what can be inferred about 1) mass; 2) composition; and 3)extension of outer convection zones of solar analogs in the Hyades. Thefirst issue relies on the strong sensitivity of the large separation tomass. We show that seismic constraints provide masses to a precisionlevel (0.05 Mȯ) that is competitive with the actualmass estimations from binary systems. Then large separations (Δν) and second differences (δ2 ν) are used torespectively constrain metal and helium fractions in the Hyades. Whenplotted for several masses, the relation of effective temperature(Teff) vs. large separation (Δ ν) is found to bestrongly dependent on the metal content. Besides this the seconddifference main modulation is related to the second ionization ofhelium. An accuracy in the helium mass fraction of 0.02 to 0.01 can beachieved provided mass and age are accurately known, which is the casefor a few Hyades binary systems. The second difference modulations arealso partly due to the discontinuity in stellar stratification at theconvective envelope/radiative core transition. They permit directinsight in the stellar structure. We compute acoustic radii of theconvective bases for different values of the mixing length theoryparameter αMLT in convection modelling, i.e. differentconvective efficiency in the superadiabatic layers. For a giveneffective temperature we show that the acoustic radius changes withconvection efficiency. This suggests that seismology can provideconstraints on the extension of outer convection and also more generallyon the direct approaches of convection and dynamical phenomena beingcurrently developed. The coronae of bright late-type stars observed with EPIC and RGSX-ray bright late-type stars have been selected as targets of XMM-Newtonobservations, with the aim to study in detail the plasma thermaldistributions and chemical abundances of their coronae, and thecharacteristics of their X-ray emission variability. Bothhigh-resolution spectra with RGS and high signal-to-noisemedium-resolution EPIC spectra have been employed to this aim. I presentsome representative cases of such studies. XMM-Newton EPIC observations of stellar clusters and star forming regionsWe report on observations of open clusters (OCs) and star formingregions (SFRs) obtained with the EPIC camera as part of the MissionScientist Guaranteed Time on XMM-Newton. These observations provide apowerful tool to investigate the evolution of coronal activity inlate-type convective stars and its dependence on magnetic fieldgeneration by dynamo processes. We discuss the motivations for thisprogram and present some results for the SFRs sigma Orionis (2-5Myr) and Taurus-Auriga (1-10 Myr) as well as for the OCs IC 2602(30 Myr), alpha Persei (50 Myr), Praesepe (600 Myr) andthe Hyades (600 Myr). We discuss imaging and spectral data providedby the EPIC MOS and PN detectors focussing on the determination of thecluster X-ray luminosity function and of the temperature structure,chemical abundances and time variability of cluster stars.Based on observations collected with the ESA mission XMM-Newton as partof the Mission Scientist (R. Pallavicini) Guaranteed Time The Indo-US Library of Coudé Feed Stellar SpectraWe have obtained spectra for 1273 stars using the 0.9 m coudéfeed telescope at Kitt Peak National Observatory. This telescope feedsthe coudé spectrograph of the 2.1 m telescope. The spectra havebeen obtained with the no. 5 camera of the coudé spectrograph anda Loral 3K×1K CCD. Two gratings have been used to provide spectralcoverage from 3460 to 9464 Å, at a resolution of ~1 Å FWHMand at an original dispersion of 0.44 Å pixel-1. For885 stars we have complete spectra over the entire 3460 to 9464 Åwavelength region (neglecting small gaps of less than 50 Å), andpartial spectral coverage for the remaining stars. The 1273 stars havebeen selected to provide broad coverage of the atmospheric parametersTeff, logg, and [Fe/H], as well as spectral type. The goal ofthe project is to provide a comprehensive library of stellar spectra foruse in the automated classification of stellar and galaxy spectra and ingalaxy population synthesis. In this paper we discuss thecharacteristics of the spectral library, viz., details of theobservations, data reduction procedures, and selection of stars. We alsopresent a few illustrations of the quality and information available inthe spectra. The first version of the complete spectral library is nowpublicly available from the National Optical Astronomy Observatory(NOAO) via ftp and http. Synthetic Lick Indices and Detection of α-enhanced Stars. II. F, G, and K Stars in the -1.0 < [Fe/H] < +0.50 RangeWe present an analysis of 402 F, G, and K solar neighborhood stars, withaccurate estimates of [Fe/H] in the range -1.0 to +0.5 dex, aimed at thedetection of α-enhanced stars and at the investigation of theirkinematical properties. The analysis is based on the comparison of 571sets of spectral indices in the Lick/IDS system, coming from fourdifferent observational data sets, with synthetic indices computed withsolar-scaled abundances and with α-element enhancement. We useselected combinations of indices to single out α-enhanced starswithout requiring previous knowledge of their main atmosphericparameters. By applying this approach to the total data set, we obtain alist of 60 bona fide α-enhanced stars and of 146 stars withsolar-scaled abundances. The properties of the detected α-enhancedand solar-scaled abundance stars with respect to their [Fe/H] values andkinematics are presented. A clear kinematic distinction betweensolar-scaled and α-enhanced stars was found, although a one-to-onecorrespondence to thin disk'' and thick disk'' components cannot besupported with the present data. Empirically Constrained Color-Temperature Relations. I. BV(RI)CThis investigation presents a set of transformations to Johnson B-V,Cousins V-R, and Cousins V-I, as well as bolometric corrections to V,for [Fe/H]=-3, -2, -1, -0.5, 0.0, and +0.3 and, in each case, values oflogg from -0.5 to 5.0 for 3000 K<=Teff<=5500 K and from2.0 to 5.0 for 6000 K<=Teff<=40,000 K. Thesetransformations employ the predictions from Kurucz model atmospheres athigh temperatures (Teff>=8000 K) and from MARCS modelatmospheres at intermediate temperatures (from 7000 K down to atemperature in the range 4000 K<=Teff<=5500 K,depending on [Fe/H], where adjustments to satisfy observationalconstraints become necessary). Thus, theoretical color-Teffrelations are used exclusively down to a minimum temperature that iscooler than the temperatures of turnoff stars in open and globular starclusters. To better represent the color transformations obeyed by coolstars (down to 3000 K), corrections to the synthetic transformationshave been determined from a careful consideration of observations for afew globular clusters (M92, M68, and 47 Tucanae), the color-magnitudediagrams (CMDs) of several open clusters (M67, the Pleiades, the Hyades,and NGC 6791), the CMDs and mass-luminosity diagram for solarneighborhood stars having good distance measurements from Hipparcos,empirical (B-V)-Teff and (V-K)-Teff relations, andcolor-color diagrams for field giants. The semiempirical colortransformations that have been produced as a result of our analysis arealso compared with several others that have been published in recentyears: some of the deficiencies of the latter are revealed. Catalogue of averaged stellar effective magnetic fields. I. Chemically peculiar A and B type starsThis paper presents the catalogue and the method of determination ofaveraged quadratic effective magnetic fields < B_e > for 596 mainsequence and giant stars. The catalogue is based on measurements of thestellar effective (or mean longitudinal) magnetic field strengths B_e,which were compiled from the existing literature.We analysed the properties of 352 chemically peculiar A and B stars inthe catalogue, including Am, ApSi, He-weak, He-rich, HgMn, ApSrCrEu, andall ApSr type stars. We have found that the number distribution of allchemically peculiar (CP) stars vs. averaged magnetic field strength isdescribed by a decreasing exponential function. Relations of this typehold also for stars of all the analysed subclasses of chemicalpeculiarity. The exponential form of the above distribution function canbreak down below about 100 G, the latter value representingapproximately the resolution of our analysis for A type stars.Table A.1 and its references are only available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/407/631 and Tables 3 to 9are only available in electronic form at http://www.edpsciences.org Line Absorption as a Metallicity Index for Giant StarsThe fraction of light removed from a star's spectrum by the spectrallines, the line absorption, is shown to be a precise empirical indicatorof metallicity. We measured the line absorption in 89 class III giantstars in a 42.5 Å window between 6219.0 and 6261.5 Å andthen calibrated these values against published metallicities. We showthat the line absorption can be measured precisely enough to improve themetallicity precision about fivefold over the original calibrationmetallicities, reaching a precision of 0.01 dex in favorable cases. Capella: Separating the GiantsImages from the Faint Object Camera (FOC) on the Hubble Space Telescope(HST) are used to spatially separate the two giants of Capella (αAurigae; HD 34029) for the first time at ultraviolet wavelengths. Theimages were obtained with broadband filters that isolate the wavelengthregions 2500-3000 Å and 1300-1500 Å. The cool G8 giant isfound to be weaker than the hot G1 giant by factors of around 4 and 17,respectively, in these bands. The latter factor is largely due to themuch stronger G1 continuum at short wavelengths. No evidence is foundfor material lying between the two stars in the images. In addition, theobjective prisms of the FOC were used to obtain low-resolution spectrafrom 1200 to 3000 Å, allowing individual emission lines from eachstar to be spatially separated. Cool-to-hot star ratios for the emissionlines H I Lyα, O I λ1305, Si II λ1816, C IIλ1335, He II λ1640, and Si IV λ1393 are presented,showing that the cool giant is weaker than the hot giant by factors of5-10 in these lines. The O I emission is only a factor of 2.5 weaker inthe cool giant, most probably resulting from fluorescence in theextended atmosphere of the cool giant. The line ratios are compared withvalues derived from International Ultraviolet Explorer and HST/GoddardHigh Resolution Spectrograph spectra, which could separate the starsspectrally but not spatially. Reasonable agreement is found although theFOC ratios generally imply lower contributions from the cool giant.Based on observations with the NASA/ESA Hubble Space Telescope obtainedat the Space Telescope Science Institute, which is operated by AURA,Inc., under NASA contract NAS 5-26555. Measuring starspots on magnetically active stars with the VLTIWe present feasibility studies to directly image stellar surfacefeatures, which are caused by magnetic activity, with the Very LargeTelescope Interferometer (VLTI). We concentrate on late typemagnetically active stars, for which the distribution of starspots onthe surface has been inferred from photometric and spectroscopic imaginganalysis. The study of the surface spot evolution during consecutiverotation cycles will allow first direct measurements (apart from theSun) of differential rotation which is the central ingredient ofmagnetic dynamo processes. The VLTI will provide baselines of up to 200m, and two scientific instruments for interferometric studies at near-and mid-infrared wavelengths. Imaging capabilities will be made possibleby closure-phase techniques. We conclude that a realistically modeledcool surface spot can be detected on stars with angular diametersexceeding ~ 2 mas using the VLTI with the first generation instrumentAMBER. The spot parameters can then be derived with reasonable accuracy.We discuss that the lack of knowledge of magnetically active stars ofthe required angular size, especially in the southern hemisphere, is acurrent limitation for VLTI observations of these surface features. CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Astrometric radial velocities. III. Hipparcos measurements of nearby star clusters and associationsRadial motions of stars in nearby moving clusters are determined fromaccurate proper motions and trigonometric parallaxes, without any use ofspectroscopy. Assuming that cluster members share the same velocityvector (apart from a random dispersion), we apply a maximum-likelihoodmethod on astrometric data from Hipparcos to compute radial and spacevelocities (and their dispersions) in the Ursa Major, Hyades, ComaBerenices, Pleiades, and Praesepe clusters, and for theScorpius-Centaurus, alpha Persei, and HIP 98321'' associations. Theradial motion of the Hyades cluster is determined to within 0.4 kms-1 (standard error), and that of its individual stars towithin 0.6 km s-1. For other clusters, Hipparcos data yieldastrometric radial velocities with typical accuracies of a few kms-1. A comparison of these astrometric values withspectroscopic radial velocities in the literature shows a good generalagreement and, in the case of the best-determined Hyades cluster, alsopermits searches for subtle astrophysical differences, such as evidencefor enhanced convective blueshifts of F-dwarf spectra, and decreasedgravitational redshifts in giants. Similar comparisons for the ScorpiusOB2 complex indicate some expansion of its associations, albeit slowerthan expected from their ages. As a by-product from the radial-velocitysolutions, kinematically improved parallaxes for individual stars areobtained, enabling Hertzsprung-Russell diagrams with unprecedentedaccuracy in luminosity. For the Hyades (parallax accuracy 0.3 mas), itsmain sequence resembles a thin line, possibly with wiggles in it.Although this main sequence has underpopulated regions at certaincolours (previously suggested to be Böhm-Vitense gaps''), suchare not visible for other clusters, and are probably spurious. Futurespace astrometry missions carry a great potential for absoluteradial-velocity determinations, insensitive to the complexities ofstellar spectra. Based on observations by the ESA Hipparcos satellite.Extended versions of Tables \ref{tab1} and \ref{tab2} are available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.125.8) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/381/446 Lick Spectral Indices for Super-Metal-rich StarsWe present Lick spectral indices for a complete sample of 139 candidatesuper-metal-rich stars of different luminosity classes (MK type from Ito V). For 91 of these stars we were able to identify, in anaccompanying paper, the fundamental atmosphere parameters. This confirmsthat at least 2/3 of the sample consists of stars with [Fe/H] in excessof +0.1 dex. Optical indices for both observations and fiducialsynthetic spectra have been calibrated to the Lick system according toWorthey et al. and include the Fe I indices of Fe5015, Fe5270, andFe5335 and the Mg I and MgH indices of Mg2 and Mg b at 5180Å. The internal accuracy of the observations is found to beσ(Fe5015)=+/-0.32 Å, σ(Fe5270)=+/-0.19 Å,σ(Fe5335)=+/-0.22 Å, σ(Mg2)=+/-0.004 mag,and σ(Mg b)=+/-0.19 Å. This is about a factor of 2 betterthan the corresponding theoretical indices from the synthetic spectra,the latter being a consequence of the intrinsic limitations in the inputphysics, as discussed by Chavez et al. By comparing models andobservations, we find no evidence for nonstandard Mg versus Fe relativeabundance, so [Mg/Fe]=0, on the average, for our sample. Both theWorthey et al. and Buzzoni et al. fitting functions are found tosuitably match the data and can therefore confidently be extended forpopulation synthesis application also to supersolar metallicity regimes.A somewhat different behavior of the two fitting sets appears, however,beyond the temperature constraints of our stellar sample. Its impact onthe theoretical output is discussed, as far as the integratedMg2 index is derived from synthesis models of stellaraggregates. A two-index plot, such as Mg2 versus Fe5270, isfound to provide a simple and powerful tool for probing distinctiveproperties of single stars and stellar aggregates as a whole. The majoradvantage, over a classical CM diagram, is that it is both reddeningfree and distance independent. Based on observations collected at theInstituto Nacional de Astrofísica, Optica y Electrónica(INAOE) G. Haro'' Observatory, Cananea (Mexico). Line-Depth Ratios: Temperature Indices for Giant StarsRatios of the depths of appropriately chosen spectral lines are shown tobe excellent indicators of stellar temperatures for giant stars in theG3 to K3 spectral type range. We calibrate five line-depth ratiosagainst B-V and R-I color indices and then translate these intotemperatures. Our goal is to set up line-depth ratios to (1) accuratelymonitor any temperature variations of a few degrees or less that mayoccur during magnetic cycles or oscillations and (2) rank giantsprecisely on a temperature coordinate. This is not an absolutecalibration of stellar temperatures. We show how giant spectra can bemisleading because of the complex dependences of spectral lines onmetallicity and absolute magnitude as well as temperature, and it isessential to make corrections to accommodate these complications. Thefive line-depth ratios we use yield precision for monitoring, i.e.,detecting temperature variations, of 4 K from a single exposure. Rankinggiants by temperature can be done with errors of ~25 K but could beimproved with better determinations of the metallicity andabsolute-magnitude corrections. Absolute spectrophotometry of late-type stars.Not Available The ASCA Medium Sensitivity Survey (the GIS Catalog Project): Source CatalogWe present the first X-ray source catalog of the ASCA Medium SensitivitySurvey (AMSS, or the GIS catalog project), constructed from data atGalactic latitudes b>10deg observed between 1993 May and 1996December. The catalog utilizes 368 combined fields and contains 1343sources with the detection significance above 5 σ either in thesurvey bands of 0.7-7 keV, 2-10 keV, or 0.7-2 keV, including targetsources. For each source, the ASCA source name, position, a 90% errorradius, count rates in the three bands, detection significances, fluxes,and a hardness ratio are provided. With extensive simulations, wecarefully evaluate the data quality of the catalog. Results fromcross-correlation with other existing catalogs are briefly summarized. 3 Ms in the Life of β Ceti: Sustained Flare Activity on a Clump Giant Detected by the Extreme Ultraviolet ExplorerA 34 day Extreme Ultraviolet Explorer (EUVE) pointing on the clump''giant β Ceti (HD 4128; K0 III) recorded a series of strikingcoronal flare events, reminiscent of a singular outburst seen previouslyfrom μ Velorum (HD 93497; G6 III + dF). The recent flaring episodecontrasts with a more placid behavior in a 6 day EUVE observation ofβ Cet 6 years earlier. The average 70-180 Å Deep Survey countrate in the new observation is twice as high, and the 75-150 Åspectrum displays a distinct hardening. The discovery of sustained flareactivity on β Cet raises the possibility that such episodes aremore common than suspected among the core helium-burning giants andsharpens the puzzle of the survival of magnetic activity beyond heliumflash. On the Wilson-Bappu relationship in the Mg II k lineAn investigation is carried out on the Wilson-Bappu effect in the Mg Iik line at 2796.34 Å. The work is based on a selection of 230 starsobserved by both the IUE and HIPPARCOS satellites, covering a wide rangeof spectral types (F to M) and absolute visual magnitudes (-5.4<=MV <=9.0). A semi-automatic procedure is used to measurethe line widths, which applies also in the presence of strong centralabsorption reversal. The Wilson-Bappu relationship here provided isconsidered to represent an improvement over previous recent results forthe considerably larger data sample used, as well as for a properconsideration of the measurement errors. No evidence has been found fora possible dependence of the WB effect on stellar metallicity andeffective temperature. Lithium in the intermediate age cluster NGC 3680: Following Li evolution along the C-M diagramWe present an analysis of high resolution spectroscopic observations (R~ 30 000, S/N=60-150) of 24 members of the intermediate age ( ~ 1.5 Gyr)open cluster NGC 3680, covering all regions of the clustercolour-magnitude (C-M) diagram where cluster members are known to exist.These observations represent in many aspects challenges to ourunderstanding of stellar interior and mixing. Four main sequence G starshave, within the errors, the same Li abundance, 0.3 dex lower thansimilar stars in the ~ 1 Gyr younger Hyades but comparable with thoseobserved in the coeval cluster IC 4651. The clustershows a clear Li-dip located around the turn-off; two stars on the upperpart of the turn-off are out of the dip and reach solar systemmeteoritic Li abundances. Just above the turn-off, in a very small rangeof magnitudes ( ~ 0.2 in V), a factor of ~ 5 Li depletion occurs. Thissudden decrease explains puzzling results recently obtained on fieldsubgiants but it is not at all reproduced by standard (e.g. no rotation,no diffusion) models, whereas it is in somewhat better agreement withthe predictions of recent models which include rotational mixing andatomic diffusion. Out of the six cluster giants, one is probably abinary; of the remaining five single cluster members, three have a Liabundance log n(Li) ~ 1.1 while two have Li abundances from a factor 6to more than a factor 30 lower than the other three. The star with nodetected Li is the coolest and most luminous object in the sample and ismost likely an AGB star; the other has instead a similar magnitude andeffective temperature as the three more Li rich giants. The reasons forthis difference in Li abundance among otherwise similar stars can beascribed either to differential depletion during main-sequence orpost-main sequence evolution, possibly induced by rotation, or todifferences in the evolutionary status of these evolved stars. Bycomparing our results with those found for clusters of similar age andfor field stars, we find that none of the possible scenarios gives afully satisfactory explanation if the present population of NGC 3680giants reflect the expected ratio of clump vs. first-ascent RGB stars.If the more abundant Li-rich giants in NGC 3680 are indeed clump giants,their relatively high Li content requires that Li is produced, orbrought to the surface, between the tip of the RGB and the clump, whichis not consistent with observations of the similar age cluster NGC 752,where the more abundant, presumably clump giants have low Li abundances.Finally, we have used our spectra to determine the metallicity of thecluster giants, finding [Fe/H]=-0.17+/-0.12. This value is in very goodagreement with that derived from spectral indexes analysis, butsubstantially lower than the value inferred from Strömgrenphotometry. Based on observations collected at ESO, la Silla, and at theVLT. The helium content and age of the Hyades:. Constraints from five binary systems and Hipparcos parallaxesWe compare the accurate empirical mass-luminosity (M-L) relation basedon five Hyades binary systems to predictions of stellar modelscalculated with various input parameters (helium, metallicity and age)or physics (mixing-length ratio, model atmosphere, equation of state andmicroscopic diffusion). Models based on a helium content Y ~ 0.28inferred from the Delta Y/Delta Z enrichment law are more than 3sigmabeyond the observations, suggesting that the Hyades initial heliumabundance is lower than expected from its supersolar metallicity. Withthe photometric metallicity ([Fe/H] = 0.144+/- 0.013 dex, Grenon\cite{gre00}) we derive Y=0.255+/-0.009. Because of the (Y, [Fe/H])degeneracy in the M-L plane, the uncertainty grows to Delta Y=0.013 ifthe metallicity from spectroscopy is adopted ([Fe/H] = 0.14+/-0.05 dex,Cayrel de Strobel et al. \cite{cayG97}). We use these results to discussthe Hertzsprung-Russell (HR) diagram of the Hyades, in the(MV, B-V) plane, based on the very precise Hipparcosdynamical parallaxes. Present models fit the tight observed sequencevery well except at low temperatures. We show that the HR diagram doesnot bring further constraints on the helium abundance or metallicity ofthe cluster. In the low mass region of the HR diagram sensitive to themixing-length parameter (alpha MLT), the slope of the mainsequence (MS) suggests that alpha MLT could decrease from asolar (or even supersolar) value at higher mass to subsolar values atlow mass, which is also supported by the modeling of the vB22 M-Lrelation. We find that the discrepancy at low temperatures ((B-V)>≈ 1.2) remains, even if an improved equation of state or bettermodel atmospheres are used. Finally, we discuss the positions of thestars at turn-off in the light of their observed rotation rates and wededuce that the maxiμm age of the Hyades predicted by the presentmodels is ~650 Myr. Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 Research Note Hipparcos photometry: The least variable starsThe data known as the Hipparcos Photometry obtained with the Hipparcossatellite have been investigated to find those stars which are leastvariable. Such stars are excellent candidates to serve as standards forphotometric systems. Their spectral types suggest in which parts of theHR diagrams stars are most constant. In some cases these values stronglyindicate that previous ground based studies claiming photometricvariability are incorrect or that the level of stellar activity haschanged. Table 2 is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/367/297 A Hipparcos study of the Hyades open cluster. Improved colour-absolute magnitude and Hertzsprung-Russell diagramsHipparcos parallaxes fix distances to individual stars in the Hyadescluster with an accuracy of ~ 6 percent. We use the Hipparcos propermotions, which have a larger relative precision than the trigonometricparallaxes, to derive ~ 3 times more precise distance estimates, byassuming that all members share the same space motion. An investigationof the available kinematic data confirms that the Hyades velocity fielddoes not contain significant structure in the form of rotation and/orshear, but is fully consistent with a common space motion plus a(one-dimensional) internal velocity dispersion of ~ 0.30 kms-1. The improved parallaxes as a set are statisticallyconsistent with the Hipparcos parallaxes. The maximum expectedsystematic error in the proper motion-based parallaxes for stars in theouter regions of the cluster (i.e., beyond ~ 2 tidal radii ~ 20 pc) isla 0.30 mas. The new parallaxes confirm that the Hipparcos measurementsare correlated on small angular scales, consistent with the limitsspecified in the Hipparcos Catalogue, though with significantly smalleramplitudes'' than claimed by Narayanan & Gould. We use the Tycho-2long time-baseline astrometric catalogue to derive a set of independentproper motion-based parallaxes for the Hipparcos members. The newparallaxes provide a uniquely sharp view of the three-dimensionalstructure of the Hyades. The colour-absolute magnitude diagram of thecluster based on the new parallaxes shows a well-defined main sequencewith two gaps''/turn-offs''. These features provide the first directobservational support of Böhm-Vitense's prediction that (the onsetof) surface convection in stars significantly affects their (B-V)colours. We present and discuss the theoretical Hertzsprung-Russelldiagram (log L versus log T_eff) for an objectively defined set of 88high-fidelity members of the cluster as well as the delta Scuti startheta 2 Tau, the giants delta 1, theta1, epsilon , and gamma Tau, and the white dwarfs V471 Tau andHD 27483 (all of which are also members). The precision with which thenew parallaxes place individual Hyades in the Hertzsprung-Russelldiagram is limited by (systematic) uncertainties related to thetransformations from observed colours and absolute magnitudes toeffective temperatures and luminosities. The new parallaxes providestringent constraints on the calibration of such transformations whencombined with detailed theoretical stellar evolutionary modelling,tailored to the chemical composition and age of the Hyades, over thelarge stellar mass range of the cluster probed by Hipparcos.
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星座: 金牛座 右阿森松: 04h28m34.50s 赤纬: +15°57'44.0" 视星: 3.84 距离: 48.403 天文距离 右阿森松适当运动: 107.4 赤纬适当运动: -24.3 B-T magnitude: 5.039 V-T magnitude: 3.936
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# Particle Cosmology after Planck
23-26 September 2014
DESY Hamburg
Europe/Berlin timezone
Home > Timetable > Session details
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# Particle Phenomenology
## Place
Location: DESY Hamburg
Room: Seminar room 2
Date: from 24 Sep 14:00 to 25 Sep 17:30
## Timetable | Contribution List
Displaying 21 contributions out of 21
We show that the decay of the inflaton field may be incomplete, while nevertheless successfully reheating the universe and leaving a stable remnant that accounts for the present dark matter abundance. We note, in particular, that the inflaton field alternately increases and reduces the mass of its decay products as it oscillates about the minimum of its potential. By considering an appropriate dis ... More
Presented by Mr. Rafael CEREZO on 25/9/2014 at 14:40
The LHC allows to probe quantum gravity with large extra dimensions. Within the Asymptotic Safety scenario we present new constraints on the transition scale $\Lambda_T$ for the diphoton channel. These bounds will be compared to existing bounds from other channels, including Drell-Yan and MET plus jet.
Presented by Ms. Magdalena ZENGLEIN on 24/9/2014 at 14:20
The difficulties one faces when implementing CP transformations in the presence of finite non-abelian symmetries are reviewed. It is then shown that physical CP transformations in such settings always correspond to class-inverting automorphisms of the finite group and their connection to the existence of bases with real Clebsch-Gordan coefficients is explored. Furthermore, the finite groups are ca ... More
Presented by Mr. Maximilian FALLBACHER on 25/9/2014 at 14:25
Current Ice-Cube analyses show hints for an unexpected neutrino flavor ratio (1:0:0) at high energies. This may point to non-standard neutrino properties like LSND, MiniBooNE, ractor and Gallium anomalies do, e.g. sterile neutrinos. Here we discuss the 1+1 active-sterile neutrino mixing resulting from the altered dispersion relations of sterile neutrinos oscillating around a 3+1 brane in an asy ... More
Presented by Mr. Philipp SICKING on 25/9/2014 at 14:55
The new data from LHC, Planck and BICEP2 provide unseen possibilities for probing the early universe, its particle content and interactions between them. We study the cosmological constraints on a Standard Model extension with a real singlet scalar. I briefly introduce some of the main constraints on formation and decay of the scalar condensates in the early universe in order to consider the gener ... More
Presented by Mr. Tommi TENKANEN on 25/9/2014 at 15:20
The nature of the electroweak phase transition in two-Higgs-doublet models is revisited in light of the recent LHC results. A scan over an extensive region of their parameter space is performed, showing that a strongly first-order phase transition favours a light neutral scalar with SM-like properties, together with a heavy pseudo-scalar and a mass hierarchy in the scalar sector. Altogether, the ... More
Presented by Glauber CARVALHO DORSCH on 25/9/2014 at 14:20
Flavor effects play an important role in the time-evolution of particle number densities in a statistical ensemble with arbitrary flavor content. We present a fully flavor covariant formalism for transport phenomena, which captures consistently all flavor effects. As an application, we study flavor effects in a Minimal Resonant Leptogenesis scenario. In particular, we show that our flavor covarian ... More
Presented by Dr. Bhupal DEV on 25/9/2014 at 15:00
Lepton flavor violating Higgs decays can arise in flavor symmetry models where the Higgs sector is responsible for both the electroweak and the flavor symmetry breaking. Here we advocate a minimal $S_4$ Three-Higgs-Doublet-Model with Lepton Flavor Triality. This model can explain the $2.5\,\sigma$ excess of Higgs decay final states with a $\mu\tau$ topology reported recently by CMS if the Standa ... More
Presented by Mr. Erik SCHUMACHER on 24/9/2014 at 14:40
We study the possible signals at LHC of minimal models of decaying dark matter. Those models are characterized by the fact that DM interacts with SM particles through renormalizable coupling with an additional heavier charged state. Such interaction allows to produce a substantial abundance of DM in the early Universe via the decay of such charged heavy state, either in- or out-of-equilibrium. M ... More
Presented by Mr. Federico DRADI on 24/9/2014 at 14:00
We consider lepton flavour observables in the Randall-Sundrum (RS) model with and without custodial protection.To this end, we apply a fully five dimensional (5D) framework to calculate the matching coefficients of the effective field theory at the electroweak scale. This enables us to obtain predictions for the radiative decay $\mu \to e \gamma$ as well as the decay $\mu \to 3\,e$, the anom ... More
Presented by Mr. Paul MOCH on 25/9/2014 at 15:10
The search for Supersymmetry at the LHC is an ongoing task. Despite the LHC searches push the limits on coloured sparticles of the first two generations above the 1 − 1.5 TeV range, the lightest stop can still be rather light. The relevant stop search channels in the low-mass region are the flavour-changing neutral current (FCNC) decay of the lightest stop into a charm quark and the lightest neu ... More
Presented by Eva POPENDA on 24/9/2014 at 15:25
The search for supersymmetry is a central task of the Large Hadron Collider. The interpretation of the experimental data requires accurate and flexible theoretical predictions. We present a new calculation of the next-to-leading order supersymmetric-QCD corrections to the production and the decay of squarks. In particular, we provide fully differential cross sections matched to parton showers. We ... More
Presented by Mathieu PELLEN on 24/9/2014 at 14:55
The search for supersymmetry is an ongoing quest during the future runs of the LHC. Coloured supersymmetric particles are expected to be produced in large quantities due to the high production cross sections, should supersymmetry be realised in nature. Precise theoretical predictions are therefore needed to improve the search for these yet undiscovered particles. A way to increase precision, and t ... More
Presented by Mr. Christoph BORSCHENSKY on 24/9/2014 at 14:25
Deep-inelastic $ep$ scattering receives essential contributions from heavy flavors. The determination of the strong coupling constant $\alpha_s$ and the charm quark mass $m_c$ are sensitive to NNLO corrections. In the asymptotic region where the momentum transfer $Q^2$ is large compared to the mass of the heavy quark $m^2$ the heavy flavor contributions factorize into massless Wilson coeffici ... More
Presented by Arnd BEHRING on 24/9/2014 at 14:10
Many new physics models contain new particles that interact with the Higgs boson. These particles could be produced at the LHC via gluon-gluon fusion with an off-shell Higgs, as well as via the Drell-Yan process if charged under a gauge group. We consider simplified scenarios where the Standard Model is extended by one scalar or fermionic field that interacts with the Higgs boson and we evaluate t ... More
Presented by Mr. Andre HESSLER on 24/9/2014 at 14:40
Flavor changing neutral current |\Delta B|=|\Delta S|=1 processes are sensitive to possible new physics at the electroweak scale and beyond, providing detailed information about flavor, chirality and Lorentz structure. Recently the LHCb collaboration announced a 2.6 \sigma deviation in the measurement of R_K={\cal{B}}(\bar B \to \bar K \mu \mu)/{\cal{B}}(\bar B \to \bar K ee) from the standard mod ... More
Presented by Prof. Gudrun HILLER on 25/9/2014 at 14:40
We study the phenomenology of the Standard Model (SM) Higgs sector extended by two singlet scalars. The model has two CP-even scalars h_{1,2} where one of them corresponds to the observed 125 GeV scalar resonance; in addition to a scalar singlet that can be stable and plays the role of a dark matter candidate. We discuss the effect of the extra scalars on the Higgs triple couplings and show that ... More
Presented by Dr. Amine AHRICHE on 24/9/2014 at 15:00
It is currently widely accepted that for a high scale of inflation the EW Higgs vacuum is unstable during inflation due to large fluctuations of order H. However, this conclusion is reached by neglecting potentially significant effects induced by the spacetime curvature. In this talk I review the derivation of a one-loop SM Higgs effective potential in curved space and discuss its implications. In ... More
Presented by Tommi MARKKANEN on 25/9/2014 at 14:00
We discuss effective operators describing interactions between dark matter and Standard Model particles. We are particularly interested in higher-dimensional operators, since they are typically suppressed compared to the leading order effective operators. This can explain why no conclusive direct dark matter detection has been made so far. The ultraviolet completions of the effective operators, wh ... More
Presented by Dr. Martin KRAUSS on 24/9/2014 at 15:20
We determine the mass of the bottom quark from moments of the $b\bar{b}$ production cross section near threshold. On the theory side we use NNNLO predictions both for the resonances and the continuum cross section. We compare our result to other recent precision determinations.
Presented by Mr. Thomas RAUH on 24/9/2014 at 15:10
We study gravity with torsion in extra dimensions and derive an effective four-dimensional theory containing four-fermion contact operators at the fundamental scale of quantum gravity in the TeV range. These operators may have an impact on the low-energy observables and can manifest themselves or can be constrained in precision measurements. We calculate possible contributions of these operators t ... More
Presented by Cristóbal CORRAL on 25/9/2014 at 14:10
Building timetable...
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Subtraction of Polynomials
## Subtract polynomials by combining like terms
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Subtracting Polynomials
A concrete walkway surrounds a rectangular swimming pool. In order to know how much weather treatment to buy, the owner must know how many square units of concrete he has. The walkway is 5 feet wide on all sides. The swimming pool has a length of and a width of 14 feet. How many square units of concrete does he have?
In this concept, you will learn to subtract polynomials.
### Subtracting Polynomials
A polynomial is an algebraic expression that shows the sum of monomials. Just like you can add polynomials, you can subtract them too. You can perform this operation both vertically and horizontally. Let’s start with vertically.
One point to remember is that subtraction is the same as “adding the opposite.” In other words, is the same as . You can add the opposite of 8 instead of subtracting 8. You will use the same idea with polynomials.
Let’s look at an example.
Subtract the polynomials and .
First, line up the like terms so that you can subtract them vertically.
When you add the opposite the sign changes on each of the terms in the subtracted polynomial. Inside the parentheses, the coefficient of is positive. But when you add the opposite, the sign changes to negative, or . You also changed the sign on the to and the -4 to 4.
Now you can look at subtracting polynomials horizontally.
You just learned to subtract polynomials vertically. You can also subtract polynomials horizontally. First you will need to review the distributive property.
The distributive property: For all real numbers and .
Remember be careful with negative signs when using distributive property.
Remember that coefficients are the numerical factors of variables. The coefficient of is 3. The coefficient of is 9. When you see the term , the coefficient is -1. Although you could write , you normally do not because the 1 is considered unnecessary. How does this relate to the distributive property? The negative sign could be in front of the parentheses, like this: . This is similar to where the coefficient is unwritten but understood to be -1. Just like you could write , you could also write . The distributive property is now more apparent given that each term will now be multiplied by -1.
Take a look at the following applications of the distributive property.
Here the -1 has been inserted and then the multiplication has been performed. As with adding the opposite, the sign changes on each of the terms in the polynomial.
You can now use this method to subtract polynomials horizontally. First you will distribute the negative sign to each of the terms in the subtracted polynomial and then you will combine like terms.
Subtract the polynomials and .
First, rewrite the polynomials without parentheses. Remember that subtracting is adding the opposite.
Next, combine like terms.
### Examples
#### Example 1
Earlier, you were given a problem about the pool and the concrete.
First, find the area of the pool plus the walkway. The length of the large rectangle measures . Its width measures .
Next, find the area of just the swimming pool.
Then subtract the area of the pool from the area of the entire swimming area to find the area of the concrete walkway.
He needs square feet of concrete.
#### Example 2
Subtract the polynomials and .
First, line up the like terms so that you can subtract them vertically.
Next, combine like terms.
#### Example 3
Subtract the polynomials: .
First, rewrite the polynomials without parentheses. Remember that subtracting is adding the opposite.
Next, combine like terms.
#### Example 4
Subtract the polynomials: .
First, rewrite the polynomials without parentheses. Remember that subtracting is adding the opposite.
Next, combine like terms.
#### Example 5
Subtract the polynomials: .
First, rewrite the polynomials without parentheses. Remember that subtracting is adding the opposite.
Next, combine like terms.
### Review
Subtract the following polynomials vertically.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Subtract the following polynomials horizontally.
13.
14.
15.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
Area
Area is the space within the perimeter of a two-dimensional figure.
like terms
Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.
Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
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# Search
## nuclear physics
### (7 points)6. Series 33. Year - 4. frightened hair
Thanks to joy from the end of an exam period, Danka's hair count begun to increase by a constant rate. Later she noticed that she lost one hair, which scared her. The more hair she lost the more she feels stressed, which increases hair loss rate. More precisely, the rate of hair loss is proportional to the number of already lost hair. The rate of new hair growth remains the same. Again, we are interested, when will her last hair fall out.
Jáchym desired to calculate this for a long time.
### (8 points)3. Series 31. Year - 5. decay here, decay there
We have $A_0$ particles which decay into $B$ particles with decay constant $\lambda \_A$. $B$ particles decay into $A$ particles with decay constant $\lambda \_B$. The number of $B$ particles at the beginning is $B_0$. Find a ratio of the numbers of particles $A$ and $B$ as a function of $t$.
### (12 points)1. Series 30. Year - E. Pechschnitte
Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.
Terka má stůl ve špatné výšce.
### (2 points)6. Series 29. Year - 1. It's about what's inside of us
In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter very weakly so they can pass without any deceleration through Earth and similar large objects. Try to find out, using available literature and Internet sources, how many neutrinos are at any instant moment in an average person. Don't forget to reference the sources.
### (5 points)6. Series 29. Year - 5. Particle race
Two particles, an electron with mass $m_{e}=9,1\cdot 10^{-31}\;\mathrm{kg}$ and charge $-e=-1,6\cdot 10^{-19}C$ and an alpha particle with mass $m_{He}=6,6\cdot 10^{-27}\;\mathrm{kg}$ and charge 2$e$, are following a circular trajectory in the $xy$ plane in a homogeneous magnetic field $\textbf{B}=(0,0,B_{0})$, $B_{0}=5\cdot 10^{-5}T$. The radius of the orbit of the electron is $r_{e}=2\;\mathrm{cm}$ and the radius of the orbit of the alpha particle is $r_{He}=200\;\mathrm{m}$. Suddenly, a small homogeneous electric field $\textbf{E}=(0,0,E_{0})$, $E_{0}=5\cdot 10^{-5}V\cdot \;\mathrm{m}^{-1}$ is introduced. Determine the length of trajectories of these particles during in the time $t=1\;\mathrm{s}$ after the electric field comes into action. Assume that the particles are far enough from each other and that they don't emit any radiation.
### (4 points)1. Series 29. Year - 5. chernobyl
If someone would eat 5 µg of the isotope of cesium ^{137}Cs, how long would it take for them to have only0,04 % of the original amount of this isotope? Assume that cesium ^{137}Cs has a half-life of 30,42 let and a biological half-life (the time it takes for half of the original amount of the material to leave the body) is approximately 5 days. Determine also, how many of the particles will have decayed in the body up till then.
Kiki was hungry after her toxikology exam.
### (4 points)2. Series 28. Year - 3. impatient core
The core of Bismuth ^{209}Bi sits impatiently at peace on the same spot. Suddenly it can't hold it any lonher and it falls apart. A thalium core ^{205}Tl remains and from it one can see an$αparticle$ shoot away. What is the speed of the $αparticle$, if the energy released during the decay becomes its kinetic energy? What is the velocity of the $αparticle$ in reality? Compare the results. The rest masses of the atoms are $M=m_{^{209}Bi}=208,980399u$, $M′=m_{^{205}Tl}=204,974428u$, $m=m_{^{4}He}=4,002602u$. Don't forget to check if one should use relativistic relations.
Jakub was sad that Bismuth must wait whole eons to decay.
### (2 points)1. Series 28. Year - 1. consumption of antimatter
How much antimatter would we need to generate enough electricity for the Czech Republic for a year? We have enough matter and we assume that there would be no losses.
Karel was watching Angels and Demons from Dan Brown.
### (2 points)5. Series 27. Year - 2. uranium star
Imagine that no thermonuclear fusion occurs in stars and instead they run on nuclear fission. Estimate how long such a star would be able to shine if at the beginning of its life cycle it is composed of uranium 235, its mass and luminosity are both aproximately constant and are equal to the current values of the sun.
Mirek was reading through his new textbooks.
### (5 points)6. Series 26. Year - P. turn it of I, can't!
How many people per second can be killed by a nuclear reactor without any protective walls?
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# Finitely generated simple groups of infinite commutator width.
3 March 2008
14:45
Alex Muranov
Abstract
If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and Étienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.
• Topology Seminar
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# Volume-weighted average price
In finance, volume-weighted average price (VWAP) is the ratio of the value traded to total volume traded over a particular time horizon (usually one day). It is a measure of the average price at which a stock is traded over the trading horizon.[1]
VWAP is often used as a trading benchmark by investors who aim to be as passive as possible in their execution. Many pension funds, and some mutual funds, fall into this category. The aim of using a VWAP trading target is to ensure that the trader executing the order does so in-line with volume on the market. It is sometimes argued that such execution reduces transaction costs by minimizing market impact costs (the additional cost due to the market impact, i.e. the adverse effect of a trader's activities on the price of a security).
VWAP can be measured between any two points in time but is displayed as the one corresponding to elapsed time during the trading day by the information provider.
VWAP is often used in algorithmic trading. Indeed, a broker may guarantee execution of an order at the VWAP and have a computer program enter the orders into the market in order to earn the trader's commission and create P&L. This is called a guaranteed VWAP execution. The broker can also trade in a best effort way and answer to the client the realized price. This is called a VWAP target execution; it incurs more dispersion in the answered price compared to the VWAP price for the client but a lower received/paid commission. Trading algorithms that use VWAP as a target belong to a class of algorithms known as volume participation algorithms.
The first execution of the VWAP was in 1984 for the Ford Motor Company by James Elkins, then head trader at Abel Noser.[citation needed]
## Formula
VWAP is calculated using the following formula:
${\displaystyle P_{\mathrm {VWAP} }={\frac {\sum _{j}{P_{j}\cdot Q_{j}}}{\sum _{j}{Q_{j}}}}\,}$
where:
${\displaystyle P_{\mathrm {VWAP} }}$ is Volume Weighted Average Price;
${\displaystyle P_{j}}$ is price of trade ${\displaystyle j}$;
${\displaystyle Q_{j}}$ is quantity of trade ${\displaystyle j}$;
${\displaystyle j}$ is each individual trade that takes place over the defined period of time, excluding cross trades and basket cross trades.[2]
## Using the VWAP
The VWAP can be used similar to moving averages, where prices above the VWAP reflect a bullish sentiment and prices below the VWAP reflect a bearish sentiment. Traders may initiate short positions as a stock price moves below VWAP for a given time period or initiate long position as the price moves above VWAP[3]
Institutional buyers and algorithms will often use VWAP to plan entries and initiate larger positions without disturbing the stock price.[2]
VWAP slippage is the performance of a broker, and many Buy-side firms now use a Mifid wheel to direct their flow to the best broker.
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# Python - syntax error in established code
So, I'm trying to get nifscripts to work with blender 2.49b on a Mac. I suspect there may be an issue with the installed versions of python, but for the life of me I can't suss it out.
So, when I start blender from the command line, I see:
Compiled with Python version 2.5.6.
Checking for installed Python... got it!
So far so good. Running a system report generates expected output:
Paths:
/Applications/blender.app/Contents/MacOS/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python25.zip
/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5
/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5/plat-darwin
/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5/plat-mac
/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5/plat-mac/lib-scriptpackages
/System/Library/Frameworks/Python.framework/Versions/2.5/Extras/lib/python
/System/Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5/lib-tk
/Library/Python/2.5/site-packages
/System/Library/Frameworks/Python.framework/Versions/2.5/Extras/lib/python/PyObjC
/Applications/blender.app/Contents/MacOS/.blender/scripts
/Applications/blender.app/Contents/MacOS/.blender/scripts/bpymodules
So, it seems to be using the same python it was built off of. However, when I try to run the nif_common.py script, I start getting syntax errors. The first one seemed to involve the with... as f keyword, which I resolved by using the more traditional f = syntax.
However, now I get the following:
Traceback (most recent call last):
File "nif_common.py", line 91, in <module>
File "/Applications/blender.app/Contents/MacOS/.blender/scripts/bpymodules/pyffi/formats/nif/__init__.py", line 3277
return b''
^
SyntaxError: invalid syntax
I fail to see the invalid syntax; this scripts uses that form several times, and only the last does it throw an error.
Line 91 of nif_common.py reads:
from pyffi.formats.nif import NifFormat
And the __init__.py file contains at line 3277:
def _get_string(self, offset):
"""A wrapper around string_palette.palette.get_string. Used by get_node_name
etc. Returns the string at given offset."""
if offset == -1:
return b''
if not self.string_palette:
return b''</pre>
Now, my familiarity with python is minimal, so I'm not sure whether the offending '' does anything or not. But I can't see how invalid syntax would end up in established code like this...
• I am lead admin for Niftools. It is best to drop by niftools.org/chat.html You are bound to run into more issues, we can help you better there. As these scripts are written 2.49b the expectation here is not to get support, @Ideasman42 tends to just be awesome anyways. – neomonkeus Aug 8 '14 at 17:04
b'' or b"" define a byte string, this is a feature of Python2.7 or 3.x
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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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Search Results: 1 - 10 of 100 matches for " "
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Sabino Menolasina Revista Técnica de la Facultad de Ingeniería Universidad del Zulia , 2004, Abstract: Microelectrodos de oro de 10 y 60 μm de diámetro fueron fabricados utilizando alambres de oro. Estos microelectrodos fueron caracterizados electroquímicamente y utilizando microscopía de barrido electrónico para determinar la forma real de la superficie y obtener información acerca de la calidad del sello entre la interfase del metal y el material aislante utilizado en la construcción del electrodo. Gold disk ultramicroelectrodes of 10 μm and microelectrodes of 60 μm diameter were fabricated using gold wires. These ultra and microelectrodes were characterized by electrochemical measurements and using scanning electron microscopy (SEM) for determining the real shape of the surface, and obtaining information about the quality of the seal at the electrode material-insulating material interface.
Materials , 2009, DOI: 10.3390/ma2042188 Abstract: Nanoporous gold (np-Au) has intriguing material properties that offer potential benefits for many applications due to its high specific surface area, well-characterized thiol-gold surface chemistry, high electrical conductivity, and reduced stiffness. The research on np-Au has taken place on various fronts, including advanced microfabrication and characterization techniques to probe unusual nanoscale properties and applications spanning from fuel cells to electrochemical sensors. Here, we provide a review of the recent advances in np-Au research, with special emphasis on microfabrication and characterization techniques. We conclude the paper with a brief outline of challenges to overcome in the study of nanoporous metals.
Sensors , 2010, DOI: 10.3390/s101210986 Abstract: Conducting polymer 3D microelectrodes have been fabricated for possible future neurological applications. A combination of micro-fabrication techniques and chemical polymerization methods has been used to create pillar electrodes in polyaniline and polypyrrole. The thin polymer films obtained showed uniformity and good adhesion to both horizontal and vertical surfaces. Electrodes in combination with metal/conducting polymer materials have been characterized by cyclic voltammetry and the presence of the conducting polymer film has shown to increase the electrochemical activity when compared with electrodes coated with only metal. An electrochemical characterization of gold/polypyrrole electrodes showed exceptional electrochemical behavior and activity. PC12 cells were finally cultured on the investigated materials as a preliminary biocompatibility assessment. These results show that the described electrodes are possibly suitable for future in-vitro neurological measurements.
Physics , 2014, DOI: 10.1063/1.4896980 Abstract: We have fabricated large arrays of mesoscopic metal rings on ultrasensitive cantilevers. The arrays are defined by electron beam lithography and contain up to $10^5$ rings. The rings have a circumference of 1 $\mu$m, and are made of ultrapure (6N) Au that is deposited onto a silicon-on-insulator wafer without an adhesion layer. Subsequent processing of the SOI wafer results in each array being supported at the end of a free-standing cantilever. To accommodate the large arrays while maintaining a low spring constant, the cantilevers are nearly 1 mm in both lateral dimensions and 100 nm thick. The extreme aspect ratio of the cantilevers, the large array size, and the absence of a sticking layer are intended to enable measurements of the rings' average persistent current $\langle I \rangle$ in the presence of relatively small magnetic fields. We describe the motivation for these measurements, the fabrication of the devices, and the characterization of the cantilevers' mechanical properties. We also discuss the devices' expected performance in measurements of $\langle I \rangle$.
Sensors , 2010, DOI: 10.3390/s101110339 Abstract: In this paper we discuss the fabrication and characterization of three dimensional (3D) micro- and nanoelectrodes with the goal of using them for extra- and intracellular studies. Two different types of electrodes will be described: high aspect ratio microelectrodes for studying the communication between cells and ultimately for brain slice recordings and small nanoelectrodes for highly localized measurements and ultimately for intracellular studies. Electrical and electrochemical characterization of these electrodes as well as the results of PC12 cell differentiation on chip will be presented and discussed.
材料科学技术学报 , 2005, Abstract: Electrodeposition from a lyotropic liquid crystal template medium was used to produce nanostructured platinum microelectrodes with high specific surface area and high mass transport efficiency. Compared to polished and conventional platinized microelectrodes, well-ordered nanostructured platinum microelectrodes exhibited enhanced electrocatalytic properties for oxygen and ascorbic acid, whilst well-ordered nanostructured platinum microelectrodes offered improved electrocatalytic properties for oxygen reduction compared to disordered nanostructured platinum microelectrodes.
高分子学报 , 2008, Abstract: A convenient method was developed to fabricate gold nanoparticles loaded microcapsules in one step and with high efficiency.First of all,the gold colloidal solution was prepared through reduction of AuCl~-_4 ions with NaBH_4 under stirring,and poly(vinylpyrrolidone)(PVP) was added quickly under severe agitation to prevent the aggregation of the gold nanoparticles during the dialyses.The gold nanoparticles in the colloidal solution had a diameter of about 7 nm and had a maximum absorption at 516 nm.Next,template polymerization of acrylic acid(AA) in the presence of PVP and 3-(trimethoxysilyl) propyl methacrylate(MPS) modified silica particles was carried out in the gold colloidal solution.The polymerization was initiated by K_2S_2O_8 and maintained for about 2 h under nitrogen atmosphere.After polymerization,the resulting composite particles were separated from the dispersion through centrifugation and were washed with triply-distilled water for several times.Finally,the gold nanoparticle-loaded microcapsules were obtained after core-removal with HF.In this process,the pre-adsorption of the complex of PVP/AA onto the gold nanoparticles made the nanoparticles polymerizable with free AA monomers during the template polymerization,and the gold nanoparticles doped PAA/PVP complex films were formed on the surfaces of the silica particles.After the core-removal process,the hollow nature and the integrity of the obtained microcapsules in both dry and wet states were demonstrated by scanning electron microscopy and confocal laser scanning microscopy,respectively.The entrapment of the gold nanoparticles in the capsules was confirmed by transmission electron microscopy.Electron diffraction pattern of the nanoparticles-loaded microcapsules showed the gold polycrystalline diffraction rings,from which the lattice constant of crystal plane {1 1 1} was calculated as 0.4174 nm,further confirming the existence of the gold nanoparticles.The gold nanoparticles were distributed uniformly in the microcapsule shells with no sign of aggregations.The amount of nanoparticles loaded in the microcapsule shells could be tuned by the concentration of the gold colloidal solution,(i.e.)higher concentration resulted in microcapsules with larger amount of gold nanoparticles in the shells.This process can be extended to other types of nanoparticles and polymers,thus can be widely applied to obtain composite microcapsules with desired functions.
Journal of Biomedical Science and Engineering (JBiSE) , 2015, DOI: 10.4236/jbise.2015.88051 Abstract: The relationship between the parameters of Transcorneal Electrical Stimulation (TES) and its neuro-protective effect of TES on axotomised Retinal Ganglion Cells (RGCs) is still unclear. This work discusses the design strategy of a new non conventional TES stimulator, the micro fabrication processes and characterization of an array of MEMS microelectrodes over a flexible polymer layer substrate to stimulate the human cornea. The micro-array of electrodes, over a flexible smooth biocompatible polyimide substrate, fine tunes the curvature of the cornea. This tool can help researchers to define the optimal electric stimulation parameters required in TES.
Frontiers in Neuroengineering , 2012, DOI: 10.3389/fneng.2012.00008 Abstract: Composites of carbon nanotubes and poly(3,4-ethylenedioxythiophene, PEDOT) and layers of PEDOT are deposited onto microelectrodes by electropolymerization of ethylenedioxythiophene in the presence of a suspension of carbon nanotubes and polystyrene sulfonate. Analysis by FIB and SEM demonstrates that CNT–PEDOT composites exhibit a porous morphology whereas PEDOT layers are more compact. Accordingly, capacitance and charge injection capacity of the composite material exceed those of pure PEDOT layers. In vitro cell culture experiments reveal excellent biocompatibility and adhesion of both PEDOT and PEDOT–CNT electrodes. Signals recorded from heart muscle cells demonstrate the high S/N ratio achievable with these electrodes. Long-term pulsing experiments confirm stability of charge injection capacity. In conclusion, a robust fabrication procedure for composite PEDOT–CNT electrodes is demonstrated and results show that these electrodes are well suited for stimulation and recording in cardiac and neurophysiological research.
Frontiers in Neuroengineering , 2012, DOI: 10.3389/fneng.2012.00021 Abstract: Cardiological research greatly rely on the use of cultured primary cardiomyocytes (CMs). The prime methodology to assess CM network electrophysiology is based on the use of extracellular recordings by substrate-integrated planar Micro-Electrode Arrays (MEAs). Whereas this methodology permits simultaneous, long-term monitoring of the CM electrical activity, it limits the information to extracellular field potentials (FPs). The alternative method of intracellular action potentials (APs) recordings by sharp- or patch-microelectrodes is limited to a single cell at a time. Here, we began to merge the advantages of planar MEA and intracellular microelectrodes. To that end we cultured rat CM on micrometer size protruding gold mushroom-shaped microelectrode (gMμEs) arrays. Cultured CMs engulf the gMμE permitting FPs recordings from individual cells. Local electroporation of a CM converts the extracellular recording configuration to attenuated intracellular APs with shape and duration similar to those recorded intracellularly. The procedure enables to simultaneously record APs from an unlimited number of CMs. The electroporated membrane spontaneously recovers. This allows for repeated recordings from the same CM a number of times (>8) for over 10 days. The further development of CM-gMμE configuration opens up new venues for basic and applied biomedical research.
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2015
04-10
# Post Office
Other than postcards, the post office department of some country recognizes three classes of mailable items: letters, packets, and parcels. The three dimensions of a mailable item are called length, height and thickness, with length being the largest and thickness the smallest of the three dimensions.
A letter’s length must be at least 125mm but not more than 290mm, its height at least 90mm but not more than 155mm, and its thickness at least 0.25mm but not more than 7mm. (The unit millimeter is abbreviated by mm.)
All three of a packet’s dimensions must be greater than or equal to the corresponding minimum dimension for a letter, and at least one of its dimensions must exceed the corresponding maximum for a letter. Furthermore, a packet’s length must be no more than 380mm, its height no more than 300mm, and its thickness no more than 50mm.
All three of a parcel’s dimensions must be greater than or equal to the corresponding minimum dimension for a letter, and at least one of its dimensions must exceed the corresponding maximum for a packet. Furthermore, the parcel’s combined length and girth may not exceed 2100mm. (The girth is the full perimeter measured around the parcel, perpendicular to the length.)
The input will contain data for a number of problem instances. For each problem instance, the input will consist of the three dimensions (measured in mm) of an item, in any order. The length and width will be positive integers. The thickness will be either a positive integer or a positive floating point number. The input will be terminated by a line containing three zeros.
The input will contain data for a number of problem instances. For each problem instance, the input will consist of the three dimensions (measured in mm) of an item, in any order. The length and width will be positive integers. The thickness will be either a positive integer or a positive floating point number. The input will be terminated by a line containing three zeros.
100 120 100
0.5 100 200
100 10 200
200 75 100
0 0 0
not mailable
letter
packet
parcel
#include <cstdio>
#include <algorithm>
#include <cmath>
using namespace std;
const double E = 1e-12;
inline
int dblcmp(double x)
{
if (x > - E && x < E)
return 0;
return x > 0 ? 1 : -1;
}
int main()
{
double num[5];
//freopen("input.txt", "r", stdin);
while (true) {
int cc = 0;
for (int i = 0; i < 3; i++) {
scanf("%lf", &num[i]);
if (dblcmp(num[i]) == 0)
cc++;
}
if (cc == 3) {
break;
}
sort(num, num + 3);
double t = num[0];
long long h = floor(num[1] + 0.5), l = floor(num[2] + 0.5);
bool ok = false;
if (dblcmp(t - 0.25) >= 0 && h >= 90 && l >= 125) {
if (dblcmp(7 - t) >= 0 && h <= 155 && l <= 290) {
printf("letter\n");
ok = true;
} else {
if (dblcmp(50 - t) >= 0 && h <= 300 && l <= 380) {
printf("packet\n");
ok = true;
} else {
double ans = t * 2 + (double)h * 2 + (double)l;
if (dblcmp(ans - 2100) <= 0) {
printf("parcel\n");
ok = true;
}
}
}
}
if (!ok) {
printf("not mailable\n");
}
}
return 0;
}
1. 这道题这里的解法最坏情况似乎应该是指数的。回溯的时候
O(n) = O(n-1) + O(n-2) + ….
O(n-1) = O(n-2) + O(n-3)+ …
O(n) – O(n-1) = O(n-1)
O(n) = 2O(n-1)
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{}
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#### 06/06/2017
MATH HUMOR
1. CALCULUS
Two mathematicians in a restaurant were arguing about the mathematical knowledge of the public. The cynic said: I will bet you the cost of this dinner that the waitress can't answer a simple math question. The cynic excused himself to visit the men's room. The other called the waitress over and said: Here is $10. When I ask you a question, say one third x cubed. She agreed. The cynic returned, called the waitress over, and said his friend had a question. The question asked was: What is the integral of x squared? After fidgeting and squirming a long time, she said: One third x cubed. The cynic paid the check. The waitress walked away, and muttered under her breath: Plus a constant.. Person 1: What's the integral of 1/cabind cabin? Person 2: A log cabin. Person 1: No, a houseboat. You forgot to add the C! The limit, as n goes to infinity, of {sin x}/n is 6. Divide numerator & denominator by n and you get six. Not true. What type of math is often discussed at the beach? ...intergull calculus! 2. LOGARITHMS God asked Noah to build an ark and later asked Noah to load the ark with a male and a female of each species. After a flood occurred and the water subsided, God said the ark could be unloaded. After this was done, Noah went into the hold to see if the ark was empty and found hundreds of snakes on a wooden table. Noah exclaimed "but I only placed two of you here 40 days ago:. Mrs. Snake said " we are Adders and when Adders are placed on a log table they multiply". 3ANALYTICAL GEOMETRY There once was a horse named DAY. Who liked numbers better than hay. He could add and subtract, multiply and divide. His mathematical ability filled his master with pride. But, when given a book, on geometry -- ANALYT. Gave a loud whinny, and in his teeth took the bit. Galloped away, faster than lightning. His master wondered, what caused the frightening. But, we know the answer, of course. No one should put DECARTES before DAY horse. drs 4. ALGEBRA A math professor arrived at his classroom 15 minutes early. He looked in his classroom and did not see any student so he waited in the corridor. A few minutes later, the door to the classroom opened and one student came out. The professor thought: "If I enter the room it will now be empty". A physicist, a biologist and a mathematician are sitting in a street cafe watching people entering and leaving the house on the other side of the street. First they see two people entering the house. Time passes. After a while they notice three people leaving the house. The physicist says, "The measurement wasn't accurate." The biologist says, "They must have reproduced." The mathematician says, "If one more person enters the house then it will be empty." 5GEOMETRY An Indian chief had three daughters who were given permission to marry three braves, who had to provide tepees for their brides. The first brave placed a deerskin on the floor of the tepee he built, the second used a buffalo hide, and the third (who was a world traveler) used the hide of a hippopotamus. Nine months after the three marriages had occurred, the first wife had a baby boy, the second wife had a baby boy, and the third wife had twin boys. This was predicted by Euclid: the sons of the squaws on the two hides must equal the sons of the squaw on the hippopotamus. What is the ratio of the Circumference of a Jack O Lantern to its Diameter? Pumpkin Pi, of course. In the artic, the ratio of the distance around an igloo to the distance across, is Eskimo Pi. However, Eskimo Pi is only 3.00, as everything shrinks in the cold. Pi goes on and on, and e is just as cursed. I wonder which is larger, when their digits are reversed? Geometry Are the eight balls moving in a circle or a straight line? http://showyou.com/v/y-pNe6fsaCVtI/crazy-circle-illusion?u=multimotion 6. STATISTICS John Glen was asked what went through his mind when he was crouched in the nose-cone , awaiting blast off, replied: " I was thinking this rocket has 20,000 components, and each was made by the lowest bidder" A firm is hiring mathematicians. Three recent graduates are interviewed: one has a degree in pure mathematics, one in applied math, and the third one obtained his degree. in statistics. All three are asked the same question: "What is one third plus two thirds?" The pure mathematician: "It's one." The applied mathematician uses his pocket calculator and replies: "It's 0.999999999." The statistician: "What do you want it to be?". A mother is expecting her fourth child. One evening, the eldest daughter says to her dad: "Do you know, daddy, what I've found out?" "No." "The new baby will be Chinese!" "What?!" "Yes. I've read in the paper that statistics shows that every fourth child born nowadays is Chinese... Statistics show that teenage pregnancies drop off significantly after age 25. 7. BASE 10 & BASE 8 Mathematicians confuse Halloween & Christmas. 25 Dec = 31Oct That is, 25 in base 10, equals 31 in base 8. Tom Lehrer was a mathematician, songwriter and satirist. Lehrer stated that: "Base 8 is just like Base 10, if you're missing 2 Fingers." See http://en.wikipedia.org/wiki/Tom_Lehrer But, in the Octal System of Base 8 12x12 =144 just like in the Decimal System of Base 10. The number 144 in the Octal System is really the number 100 in the Decimal System. 100 in the Decimal System is 144 in the Octal system and in base 6 it is 244. 12 in the Decimal system is 14 in the Octal System and 20 in Base 6. So in Base 6, 12x12 = 400. 8INFINITY et al Richard Phillips Feynman (1918-1988) "Did you know there are twice as many numbers as numbers?" 9. NEGATIVE, REAL, & IMAGINARY NUMBERS . Messages to leave on your telephone answering machine are: If you received a negative response, hang up, rotate your phone 180 degrees, and try again. If you think you have reached an imaginary number, rather than a real number, hang up, rotate your phone ninety degrees, and try again. Here is an Argand diagram. Zero (0) is both real and imaginary! An Argand diagram is a plot of complex numbers as points (z = x + iy) in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis. While Argand (1806) is generally credited with the discovery, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line. The x axis represents "real numbers" and the y axis represents "imaginary numbers", so 0 is both real and imaginary. 10. FRACTIONS Count Lev Nikolgevich Tolstoy (1828 – 1910) said: “A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction”. 11. PROBABILITY See http://www.larry.denenberg.com/predictions.html (E.g. Casey Stengel said "Never make predictions, especially about the future." 12. ACTUARIAL and MATHEMATICAL HUMOR One day an actuarial student came into my office and asked if I would tell him how I became Chief Actuary. I replied: "Good judgment". The next day he asked how I got good judgment. I replied: "Good experience." The next day he returned and asked how I got good experience. I replied: "Bad judgment." A patient was told by her doctor she only had six months to live. She asked the doctor what she should do. He said marry an actuary. She wanted to know if that would make her live longer. The doctor said: no, but it will seem much longer. The difference between God and an actuary is that God does not think he is an actuary. An American actuary can tell you the number of people in a room will die within one year. An Italian actuary can give you their names. A CEO, an actuary, an accountant, and an insurance salesperson are riding in a car. The CEO has his hands on the steering wheel, the Marketing Director has his foot on the gas, the CFO has his foot on the brake, and the Chief Actuary is looking out the back window, drawing a map, and telling them where to go. An actuary is: A place where they bury dead actors. Someone who wanted to be an accountant but did not have enough personality for the job. A professional who can solve a problem you didn't know you had in a way that you can't understand. Someone who can determine a woman's cash surrender value Actuaries never die, they just get broken down by age and sex. Johnny, whose father was an actuary, asked his mother where he came from. His mother said: ask your father. Johnny replied: I don't want to know that much about it. The CEO of an insurance company died. Three candidates for the position were the Marketing Director , the CFO, and the Chief Actuary. They sent each to a shrink to be evaluated. The shrink had boiled his testing down to one simple question as he spent most of his time playing golf. What does the left side of the picture mean to you? The Marketing Director said a "doughnut hole". I don't like to disturb my wife in the morning, so I go to the office early to map out marketing plans. My secretary gets me a doughnut for breakfast. The CFO said a "zero balance". I don't leave the office at the end of the day unless there is a zero balance in the list of unauthorized expenditures. The Chief Actuary said "Thursday". The shrink had never had that response and asked why. The actuary said: from right to left they represent my work week: Monday, Tuesday, Wednesday, and Thursday. I play golf on Friday. A lady asked the postmaster to weigh a package. When told it was too heavy and needed one more stamp, she said: I don't understand. Will adding a stamp make it lighter? An infinite crowd of mathematicians enters a bar. The first one orders a pint, the second one a half pint, the third one a quarter pint. etc. etc. "I understand", says the bartender - and pours two pints. A mathematician is flying non-stop from Edmonton to Frankfurt. The scheduled flying time is nine hours. Later, the pilot announces that one engine had failed. "Don't worry - we're safe. The only effect is that our total flying time will be ten hours instead of nine." A few hours later, the pilot informs the passengers that another engine had failed. "But don't worry - we're still safe. Our total flying time will go up to twelve hours." Some time later, a third engine fails. The pilot reassures the passengers: "Don't worry - even with one engine, we're still perfectly safe. It just means that it will take sixteen hours total for this plane to arrive in Frankfurt." The mathematician remarks to his fellow passengers: "If the last engine breaks down, too, then we'll be in the air for twenty-four hours altogether!" Teacher Student What is 2k + k? 3000! Who can tell me what 7 times 6 is? "It's 42!" Very good! - And who can tell me what 6 times 7 is? "It's 24!" What did the zero say to the the eight? Nice belt. What do you get if you add two apples and three apples? A high school math problem! What is the difference between a Ph.D. in math and a large pizza? A large pizza can feed a family of four. Why did the chicken cross the Moebius strip? o get to the other ... er, um ... Teacher: Expand (a+b)n Student: (a + b)n (a + b)n (a + b)n (a + b)n A topologist is a person who can't tell the the difference between a coffee cup and a doughnut. An introverted actuary looks at his shoes while talking to you. An extroverted actuary looks at your shoes. An actuary is someone who wanted to be an accountant but did not have enough personality. (The movie "About Schmidt" http://en.wikipedia.org/wiki/About_Schmidt, is about a retired nerd from the book by the same title. The book had the retired nerd as a retired accountant, but that was changed in the movie to be a retired actuary!) There are three kinds of actuaries in the world: "Those who can count and those who can't." Definition of a computer: "An actuary with a heart". A lawyer, an accountant, and an actuary are arguing over whether it is better to have a married spouse or an unmarried lover. The lawyer says a lover because it's legally easier to disentangle yourself from a lover. The accountant says a spouse because you can get a tax deduction with a spouse. The actuary says it's better to have both because you can lie to each of them, telling each of them that you're with the other, and then go to the office to do some work. Actuaries in Movies, Theater, TV, and Literature http://en.wikipedia.org/wiki/Fictional_actuaries What is an actuary's favorite desert? Pi 1 + 1 =3, for sufficiently large one's. A circle is a round straight line with a hole in the middle. Ernst Eduard Kummer (1810-1893), a German algebraist, was sometimes slow at calculations.. Whenever he had occasion to do simple arithmetic in class, he would get his students to help him. Once he had to find 7 x 9. "Seven times nine," he began, "Seven times nine is er -- ah --- ah -- seven times nine is. . . ." "Sixty-one," a student suggested. Kummer wrote 61 on the board. "Sir," said another student, "it should be sixty-nine." "Come, come, gentlemen, it can't be both," Kummer exclaimed. "It must be one or the other." An investment firm is hiring mathematicians. There are three applicants: a theoretical mathematician, an applied mathematician, and an actuary. They are asked what starting salary they are expecting. The theoretical mathematician: "Would$30,000 be too much?" The applied mathematician: "I think $60,000 would be OK." The actuary:"$100,000." The personnel officer gasped: "A pure mathematician will do the work for only $30,000." The actuary replied: "I will keep$35,000, pay you $35,000, and pay$30,000 to the theoretical mathematician to do the work."
A CEO is interviewing three candidates for the position of CFO. Each candidate is asked the same question: "How much is one and one?" The accountant said: "Two". The pure mathematician said: "It depends upon what base you are working in". The actuary whispered in the CEO's ear: "What do you want it to be?"
George W Bush warned math professors not to misuse their position to give their political views to young Americans. "It is my understanding", the president said, "that you are frequently teaching algebra classes in which your students learn how to solve equations with the help of radicals. I can't say that I approve of that..."
Donald Rumsfeld gave George W Bush a briefing and concluded by saying: Yesterday, 3 Brazilian soldiers were killed. Oh no, the President exclaimed, That's terrible! Exactly how many are in a brazillion?
Four friends have been doing really well in their calculus class. When it's time for the final, they decide to go to a weekend party in another city. They finally arrive on campus, hung over and sleepy, but the exam is already over.
They go to the professor's office and say: "We went to our friend's birthday party, and when driving back this morning we had a flat tire. We had no spare one, and since we were driving on back roads, it took hours until we got help." The professor nods sympathetically and says: "I see that it was not your fault. I will allow you to make up the exam tomorrow morning." When they arrive next morning, the students seated so far apart from each other they had no chance to cheat. The exam booklets are already in place and the students begin. The first question - five points out of one hundred - is a simple exercise in integration, and all four finish it within ten minutes. When the first of them has completed the problem, he turns over the page of the exam booklet and reads:
Problem 2 (95 points out of 100): Which tire went flat?
If brute force doesn't work, you're not using enough of it.
Mathematicians never die - they only loose some of their functions.
One cat has nine tails. Proof. No cat has eight tails. Since one cat has one more tail than No cat, One cat must have nine tails.
A father who is very much concerned about his son's bad grades in math decides to register him at a catholic school. After his first term there, the son brings home his report card: He's getting "A"s in math.
The father is, of course, pleased, but wants to know: "Why are your math grades suddenly so good?"
"You know", the son explains, "when I walked into the classroom the first day, and I saw that guy on the wall nailed to a plus sign, I knew one thing: This place means business!"
In a class, a math professor claims that he can prove everything under the assumption that 1+1=1.
A student challenges him: "Then prove that you're the pope!"
The professor replies: "I am one, and the pope is one. Therefore, the pope and I are one."
After two hours, the professor called for the exams, and the students filed up and handed them in. All except the late student, who continued writing. Twenty minutes later, the last student came up to the professor who was sitting at his desk preparing for his next class. He attempted to put his exam on the stack of exam booklets already there. No you don't, I'm not going to accept that. It's late. The student looked incredulous and angry. Do you know who I am? No, as a matter of fact I don't, and I don't care! replied the professor. Good, replied the student, who quickly lifted the stack of completed exams, stuffed his in the middle, and walked out of the room.
The math teacher saw that little Johnny wasn't paying attention in class. She called on him and said, Johnny! What are 2 and 4 and 28 and 44? Little Johnny quickly replied: NBC, CBS, HBO, and the Cartoon network!
An impatient math teacher snarled, And just how far are you from the correct answer? The boy replied, Three seats.
At a nursery school a teacher talks to a four-year-old applicant.
"Mike, can you count for me?"
Mike counts very fast and with a lot of enthusiasm, "Fifty-nine, fifty-eight, fifty-seven"
"Super," says the teacher, "But how did you learn to count backwards?"
Mike replies proudly, "I can heat my own lunch in the microwave."
Teacher Student What are whole numbers? 0, 6, 8, 9 And what about 10? It is half-whole, 1 doesn't have a hole.
Dad, will you do my math for me tonight? No, son, it wouldn't be right. Well, you could try.
A little girl asked an elderly woman: "Can you help me find the lowest common denominator?" The woman said: "Haven't they found that yet? They were looking for it when I was in school."
Knock knock. Who's there? Lois and Carmen. Who? Lois and Carmen Denominator.
Father, to his daughter returning home at 3 a.m. said: "I told you to be home by a quarter of 12!" The daughter answered: "But I learned in math that a quarter of 12 is 3!"
Where can you buy a ruler that is 3 feet long? At a yard sale.
Math Types Sign Errors
When you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong.
The probability of an open faced jelly sandwich falling jelly side up varies inversely with the price of the rug.
A military instructor says, "There is a 40% chance that we will hit our target."
One student asks, "What happens if we aim away from the target?"
The instructor replies, "We would have a 60% chance of hitting the target."
"Excuse me Professor, how can we possibly compute a kurtosis in a minute?" The Professor looks at the class very reassuring: "No need to be worried, kids, it only takes a moment"
A team of statisticians has recently published a monumental finding. The team discovered what the leading cause of divorce is marriage. Everyone who has been divorced has been married first.
A statistician who took a Dale Carnegie course and improved his confidence from .95 to .99.
Statisticians never have to say they're certain. Statistics is the art of never having to say you're wrong.
"There are lies, damned lies, and statistics." 97.3% of all statistics are made up.
62% of people have a below average IQ. This is due to the fact that there is a limit to human intelligence, but no limit to human stupidity
Q: A statistician would always accelerate hard before coming to an intersection, whizz straight through it, then slow down again. One day a passenger asked him why he went so fast through intersections. A: The statistician replied: "Statistically speaking, you are far more likely to have an accident at an intersection, so I just make sure that I spend less time there."
Two blondes are sitting in an airport hall waiting for their flight to go. One has terrible flight panic. Her companion, a mathematician, said: "Hey, don't worry, it's just every 100,000th flight that crashes." The companion replied: " So much? Then it surely will be mine!" The mathematician said: "Well, there is an easy way out. Simply take the next plane. It's much more probable that you go from a crashing to a non-crashing plane than the other way round."
Never show a bar chart at an AA meeting.
People believe what economists say about the future, but not what statisticians say about the past.
Statisticians are "mean" people. Numbers are like people; torture them enough and they'll tell you anything.
Statistics in the hands of politicians are like a lamppost to a drunk --- used more for support than illumination.
When you smoke a fish, which end do you light?
Birthdays are beneficial for your health A new statistical study unequivocally proved that the more birthdays one has the longer one lives.
Medicine makes people ill, mathematics make them sad, and theology makes them sinful. (Martin Luther)
Salesman: "Ma'am, this vacuum cleaner will cut your work in half."
Customer: "Terrific! Give me two of them."
If a man tries to fail and succeeds, which did he do?
A mathematician, a physicist, and an engineer are asked to test the following hypothesis: All odd numbers greater than one are prime. The mathematician: "Three is a prime, five is a prime, seven is a prime, but nine is not a prime. Therefore, the hypothesis is false." The physicist: "Three is a prime, five is a prime, seven is a prime, nine is not a prime, eleven is a prime, and thirteen is a prime. Hence, five out of six experiments support the hypothesis. It must be true."
Five actuaries and five accountants are traveling together by train to attend a conference. Each accountant has a ticket, but only one of the actuaries has one. Suddenly one of the actuaries shouts:: "Conductor coming!" All the actuaries run into one washroom. The conductor checks the ticket of each accountant and then knocks on the washroom door: "Your ticket, please." An actuary sticks the one ticket under the door. The conductor checks it and leaves. The accountants are impressed. On the return trip, the accountants decide to buy just one ticket for their group. The actuaries do not purchase any tickets. Again one of the actuaries shouts: "Conductor coming!". This time all the accountants rush off to a washroom. One of the actuaries goes to that washroom, knocks at the door, and says: "Your ticket, please..."
A professor goes through the airplane security check, and a bomb is found in his carry-on-baggage. The security man asked: "Why do you want to blow up this airplane?" "Sorry", the professor interrupts him. "I had never intended to blow up the plane. Statistics shows that the probability of one bomb being on an airplane is 1/1000. The chance that there are two bombs is 1/1000000. If I already bring one, the chance of another bomb being around is actually 1/1000000, and I am much safer..."
Albert Einstein was walking across the quad at Princeton and met a student. After chatting for a few minutes, Einstein asked: "Which direction was I coming from?" The student pointed at one path. Einstein said: "Then I must have had lunch."
Albert Einstein had just about finished his work on the theory of special relativity, when he decided to take a break and go on vacation in Acapulco. Each day, late in the afternoon, sporting dark sunglasses, he walked in the white Mexican sand and breathed in the fresh Pacific sea air. On the last day, he watched the sun set. When the large orange ball was just disappearing, the last beam of light seemed to radiate toward him. The event brought him back to thinking about his physics work. "What symbol should I use for the speed of light?" he asked himself. The problem was that nearly every Greek letter had been taken for some other purpose. Just then, a beautiful Mexican woman passed by. He asked as he lowered his dark sunglasses, "Do you not zink zat zee speed of light is very fast?" The woman smiled at Einstein and replied, "Si." And know you know the rest of the story.
Although c is now the universal symbol for the speed of light, the most common symbol in the nineteenth century was an upper-case V which Maxwell had started using in 1865. That was the notation adopted by Einstein for his first few papers on relativity from 1905. By 1907 when Einstein switched from V to c in his papers, it had become the standard symbol for the speed of light in vacuum for electrodynamics, optics, thermodynamics and relativity. History provides an ambiguous answer to the question "Why is c the symbol for the speed of light?", and it is reasonable to think of c as standing for either "constant" or "celeritas".
A little boy refused to run anymore. When his mother asked him why, he replied: "I heard that the faster you go, the shorter you become."
A student riding in a train looks up and sees Einstein sitting next to him. Excited he asks: "Excuse me, professor. Does Boston stop at this train?"
A six-year-old boy spotted Albert Einstein walking down the street and decided to try out his favorite joke on him: "Mr. Einstein! Why did the chicken cross the road?" The famous physicist replied: "My young friend, zee question does not have a definite anzer. Vether zee chicken crossed zee road or zee road crossed zee chicken depends on your frame of reference."
There was an old lady called Wright
who could travel much faster than light.
She departed one day
in a relative way
and returned on the previous night.
The Heineken Uncertainty Principle says "You can never be sure how many beers you had last night."
There is a sign in Munich that says, "Heisenberg might have slept here."
What is so special about 6.9? It is 69 ruined by a period.
If nothing can go wrong, something will. There's never time to do it right, but always time to do it over.
The man who can smile when things go wrong, has thought of someone he can blame it on.
An Amish boy and his father went to a large department store for the first time. They saw an elevator, but did not know what it was. An old lady with a cane, hobbled up to the elevator, and pushed a button on the wall, and a door opened. She went into a small room and the door closed. Then lights flashed above the doorway showing numbers. The numbers stopped flashing at number five. After a little while the numbers started flashing again, and stopped at number one. The door opened and out walked a beautiful blonde. The father shouted to his son: Go get your mother!
Euler's Identity is $e^{i \pi} + 1 = 0$ It is used to answer the question: "How many mathematicians does it take to change a light bulb?" The answer is: minus e to the i pi (which, of course, equals 1)
This site was last updated 06/06/17
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# Chapter 1 Principles of Data Visualization
Despite the inherent subjectivity of beauty, there are several recurring patterns and characteristics of the things we find beautiful. Symmetry, proportions (e.g. the golden ratio $$1:1.618$$) are among these recurring patterns. One of the goals -though often implicit- of data visualization is making beautiful things.
Subjectivity notwithstanding, there are some techniques and principles that we can follow in order to make our visualizations more beautiful and effective. This chapter outlines a few principles of data visualization, this is by no means an exhaustive list but rather meant as the starting point for your own list of things to keep in mind when designing visuals.
## 1.1 Data Ethics
The first thing we want to avoid when making visualizations is to avoid misrepresentation. There are numerous ways in which visuals can be -mistakenly or purposefully- distorted, leading to messages that are different or opposite of what the data is telling us. The underlying causes behind distorted visual messages are numerous: user-induced, designer-induced, (un)intentionally, cognitive biases, social differences and emotional connotations (Zinovyev 2010). Here, we focus on some practical guidelines to avoid these issues.
### 1.1.1 Include the Baseline
Omitting baseline values (typically 0) can mislead the audience into thinking that patterns are greater (or smaller) than they really are (see figure 1.1).
### 1.1.2 Consider the Audience
Your audience will have different levels of technical expertise or familiarity with the subject, so keep this in mind! Avoid going against conventions within your field.
### 1.1.3 Avoid Cherry-picking
This is a common way of attempting to deceive audiences into thinking a trend exists (or not). If you need to drop values or observations, make sure this is mentioned and justified.
## 1.2 Inclusive Colors
In a study on flag colors, (Zhang et al. 2018) found that red was not only the most recurrent in the 192 flags considered, but also “often attached with an aggressive connotation.” Notably, this color was less present in the flags of international collaborative organizations. As any painter or political can surely attest to, color is not neutral but evokes numerous sentiments and ideas in people.
The symbolic and psychological importance of color intersects with both practical and inclusiveness concerns when we take our target audience into consideration. Colorblindness is a good example of this given how widespread it is. It is thus important to keep this in mind when selecting which colors to include in our plots in order to maximize their effectiveness.
Okabe and Ito (Okabe and Ito 2002) offer 3(+1) principles of Universal Color Design:
1. Choose color schemes that can be easily identified by people with all types of color vision, in consideration with the actual lighting conditions and usage environment.
2. Use not only different colors but also a combination of different shapes, positions, line types and coloring patterns, to ensure that information is conveyed to all users including those who cannot distinguish differences in color.
3. Clearly state color names where users are expected to use color names in communication.
• Moreover, aim for visually friendly and beautiful designs.
There are many ways you can select color schemes when producing plots with R. We will be using (mostly) the ggplot2 package which is described in greater detail in Chapter 2. For now, let’s think of colors options in a palette which consists of several hexadecimal color codes (e.g. #000000 is black) put together into an array. In the example below, 15 colorblind-friendly (based on this) and distinguishable colors are combined into a vector, which is then stored in the object called cb_palette.1
cb_palette <- c("#000000","#004949","#009292","#ff6db6","#ffb6db",
"#490092","#006ddb","#b66dff","#6db6ff","#b6dbff",
"#920000","#924900","#db6d00","#24ff24","#ffff6d")
Both plots in figure 1.3 tell the same story: in our data set there are more cars with 8 cylinders, and those tend to have more automatic transmissions.2 The difference is that plot A uses the default colors in the ggplot2 package, whereas plot B uses our custom cb_palette. Notice that since there are only two categories shown (i.e., manual and automatic), only the first two colors of our array are shown: gray "#999999" and orange "#E69F00".3
## 1.3 Less is more
Throughout this course, we will cover many ways of adding labels, colors, axes, and backgrounds to plots. You might thus be tempted to use as many of them as you can fit into a graph. However, it is important to keep in mind that the main goal of visualization is effective communication. Many times, additional elements do not serve to clarify but rather to obfuscate the point that our data is trying to make. A key principle is thus to keep a high data-to-ink ratio, in other words: less is more. To this effect, avoid the following unnecessary plot elements:
• Redundant borders
• Redundant labels
• Too many hues
• Distracting backgrounds
Figure 1.4 is an example of a plot that ignores this principle. It has many redundant elements such as colors, patterns, labels, and backgrounds that do not tell us anything about the data. By contrast, figure 1.5 offers clearer picture.
Nowadays, most of the plots you will produce will pass through a printer. However, reducing noise remains important to prevent distracting our audiences with unnecessary information. This principle is particularly important in the age of Big Data.
### References
Okabe, Masataka, and Kei Ito. 2002. “Color Universal Design (Cud) - How to Make Figures and Presentations That Are Friendly to Colorblind People -.” In. https://jfly.uni-koeln.de/color/.
Zhang, Tengxiao, Shiyu Feng, Buxin Han, and Si Sun. 2018. “Red Color in Flags: A Signal for Competition.” Color Research & Application 43 (1): 114–18. https://doi.org/https://doi.org/10.1002/col.22165.
Zinovyev. 2010. “Data Visualization in Political and Social Sciences.” In. https://arxiv.org/abs/1008.1188.
1. The order of the color codes matters as they will also appear in that order when we use them in our plots: #56B4E9, #CC79A7, #999999, #E69F00, #009E73, #D55E00, #F0E442, #0072B2. In addition, packages like ggthemes::scale_fill_colorblind() also provide color-blind friendly palettes.↩︎
2. In this example we are using the popular mtcars data set. This is commonly used for ggplot2 examples. If you want to learn a little more about it, type ?mtcars directly in the console.↩︎
3. For a more detailed overview on palettes in R as well as some useful packages for making visuals more colorful see (Neth 2021)↩︎
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Seminars
1:15 pm, Lecture Hall 6 PUBLIC VIVA-VOCE NOTIFICATION Subwords: Automata, Embedding Problems, and Verification Prateek Karandikar Chennai Mathematical Institute. 12-02-15 Abstract The increasing use of software and automated systems has made it important to ensure their correct behaviour. Bugs can not only have significant financial costs, but also catastrophic consequences in mission-critical systems. Testing against a variety of environments and inputs helps with discovering bugs, but cannot guarantee their absence. Formal verification is the technique that establishes correctness of a system or a mathematical model of the system with respect to properties expressed in a formal language. Regular model checking is a common technique for verification of infinite-state systems - it represents infinite sets of configurations symbolically in a finite manner and manipulates them using these representations. Regular model checking for lossy channel systems (LCS) brings up basic automata-theoretic questions concerning the subword relation on words which models the lossiness of the channels. We address these state complexity and decision problems, and also solve a long-standing problem involving the index of the Simonâs piecewise-testability congruence. The reachability problem for lossy channel systems, though decidable, has very high complexity (it is $F_{\omega_\omega}$-complete), well beyond primitive-recursive. In recent times several problems with this complexity have been discovered, for example in the fields of verification of weak memory models and metric temporal logic. The Post Embedding Problem (PEP) is an algebraic abstraction of the reachability problem on LCS, with the same complexity, and is our champion for a âmasterâ problem for the class $F_{\omega_\omega}$. We provide a common generalization of two known variants of PEP and give a simpler proof of decidability. This allows us to extend the unidirectional channel system (UCS) model with simple channel tests while having decidable reachability.
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# Simplicity of rings
A ring $$R$$ is said to be simple if $$\lbrace 0 \rbrace$$ and $$R$$ are its only two-sided ideals. The ring $$R$$ is said to be left-semisimple (resp. right-semisimple) if the regular module $$_RR$$ (resp. $$R_R$$) is left-semisimple (resp. right-semisimple), which means that every submodule $$N$$ of $$_RR$$ (resp. $$R_R$$) is a direct summand. The simplicity of $$R$$ doesn't necessarily imply that $$R$$ is left or right semisimple. However if $$R$$ is a commutative simple ring (i.e. a field), then $$R$$ is both left and right semisimple since the ideals of $$R$$ correspond with the submodules of $$_RR$$ and $$R_R$$ and the only ideals of $$R$$ are $$\lbrace 0 \rbrace$$ and $$R$$.
My question is the following: is there a notion of left-simplicity (resp. right-simplicity), i.e. $$\lbrace 0 \rbrace$$ and $$R$$ are the only left (resp. right) ideals of $$R$$? I've done some reading and they never mentioned something alike.
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# Controller -- Waveform Controller
Navigation: Models ➔ Control 2 Models ➔ Signal Waveform Controller
SS and Dynamic Models Dynamic Mode Only Models
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## General Description
The Signal Waveform Controller model is used in Dynamic projects. It allows input of periodic data, either by regular geometric patterns or from user-input data. This is used to simulate process variation based on a repeating periodic signal. It can be used to add regular disturbance to process variables, e.g. to Feeders or Dynamic Transfer flow capacity, or to simulate the effect of cyclical data (e.g. water addition via annual rainfall patterns).
• A three-part discussion series "Annual Climate Data for Dynamic Modelling" provides additional background on the development and use of this controller with user-input data for pond modelling with annual rainfall and evaporation.
### Diagram
The diagram shows the default drawing of the Signal Waveform control unit.
### Inputs and Outputs
There are no connections to this unit.
## Model Theory
Waveform generation has two broad categories, each with various forms:
1. Regular geometric patterns
• Flatline (None)
• Sine Wave
• Square Wave
• Triangle Wave
• Sawtooth Wave
2. User-input data
• Linear DataPoints
• Linear Averages
• Spline DataPoints
• Spline Averages
• B-Spline DataPoints
These cyclical patterns repeat with a user-defined period. TimeOffset can be used to start the cyclical pattern at an offset time within the cycle. Note that use of TimeOffset requires the Events/Profiles Startup Reset Action to be selected.
### Regular Geometric Patterns
The following regular geometric patterns are available. In addition, a flatline (None) option can be used to output a constant value.
An option is included to choose between ascending (as in the image above) and descending sawtooth.
#### Sine Wave Function
$\displaystyle{ y(t) = (Max - Centre) + \cfrac{PeakToPeak}{2} * sin \left( \cfrac{2\pi * (t + TimeOffSet)}{Period} \right) }$
where
Max = Maximum value
Centre = Mid point of the Peak to Peak value = (Maximum value - Minimum Value)/2.
PeakToPeak = Peak to Peak value on the y scale, Maximum value - Minimum Value.
Timeoffset = Starting point (time) of wave, if Timeoffset = 0, then the sine wave will start at centre value.
t = time, each iteration t = t + step size.
Period = 1/Frequency, time taken for one full wave cycle.
For example
• If the min feed is 10 t/h and the Max feed is 50 t/h, using a Period of 1 hour with no time offset, the sine waveform will generate the feed input curve as per left image below.
• If the time offset is 900 seconds (15 mins, or 1/4 of the total period), then the sine wave is shifted by 1/4 cycle, as per the right image below:
### User-Input Data
Introduced in Build 139.30874, user-input data may be used to define a repeating pattern as a waveform.
For each of these user-input data options, the DataPointsCount or number of equally-spaced data points (intervals) within a period is defined, as well as the data value at each of these points.
Various configurations are possible, with choice between:
1. Data input as instantaneous Data Points or as interval Average Data
2. Linear or Spline fitting between data points
3. Data points at interval Start-Point or Mid-Point (applicable for Data Points options only)
The following sections detail the key differences between input of Data Points and Average Data.
#### Data Point Input
The DataPoints options use data input to represent an instantaneous point in time, with options for data interpolation between the defined points. In each case, the interpolated data passes through each of the defined data points (i.e. there is no data smoothing or averaging).
The following options relate to how data is interpolated between the data points:
• Linear (Linear DataPoints) - A straight line is drawn between each consecutive data point.
• Spline (Spline DataPoints) - A regular cubic spline is fit through all data points, with the boundary condition that curvature and slope are continuous across the period edge such that it wraps around smoothly.
• This array-based calculation uses the full data set and gives a smooth curve through all data points.
• The mathematics behind this periodic spline is explained in depth here.
• B-Spline (B-Spline DataPoints) - A cubic B-spline (basis spline) is fit through the data points, also with smooth wrap around.
• This option uses local data points to calculate cubic functions between points. The resulting curve is similar to a regular spline, but curvature may change more sharply at some points.
For each of the interpolation methods above, a checkbox option (UseAsMidPoints) allows data to be offset by half an interval, such that the input data points are positioned at the mid-point of each interval rather than at the start.
In the diagram below, the four curves are produced from the same set of input data points.
• The blue set shows Linear (straight lines) and Spline (continuous curve) with no MidPoint offset. Here the data points are at the beginning of each interval.
• The red set shows the same (Linear and Spline) with with MidPoint offset. The data points are now at the midpoint of each interval, i.e. offset by half an interval.
B-Spline is not shown, but results in a similar curve to Spline, though slightly less "rounded".
Note that the spline curves may exceed the maximum or minimum values shown in the access window. These MinValue and MaxValue terms refer only to the data points themselves, which are used directly as the spline "knots".
#### Average Data Input
The Averages options use data input to represent an average value for each interval. In this case, mid-point is meaningless as the input data applies to the whole interval. In this case there are two options:
• Linear (Linear Averages) - A flat line is drawn through each interval at the defined value.
• Spline (Spline Averages) - A cubic spline is fit such that the area under the curve in each interval is the same as for Linear Averages.
• IMPORTANT: In this case the spline does not pass through the defined data points, but instead through calculated spline "knots" to conserve area under the curve.
• This option is extremely useful in Dynamic simulation where the total over time must be conserved while maintaining a smooth data input to minimise model disturbance.
• The mathematics behind this periodic averaging spline is explained in depth here.
In the diagram below, the two curves are produced from the same set of input data.
• The dark green line shows the linear averages in each time interval.
• The light green curve shows the spline fit through the data set such that the area under the curve in each interval is conserved.
As with Data Point Input, note that the spline curve may exceed the maximum or minimum values shown in the access window. In the averaging spline case, these MinValue and MaxValue terms refer to the min and max of the internally-calculated spline "knots", which are not equivalent to the input data.
## Data Sections
The default sections and variable names are described in detail in the following tables. The default Waveform Controller access window consists of 3 sections. This number may increase or decrease, based on user configuration.
### Summary of Data Sections
1. SignalCon tab - Contains a summary of all of the individual Waveform Controls contained in the unit.
2. SW Tabs tab - This page contains all of the information for each individual Waveform Controls, starting at 1 for the first Waveform Controls.
3. Info tab - Contains general settings for the unit and allows the user to include documentation about the unit and create Hyperlinks to external documents.
### SignalCon
Unit Type: SignalCon - The first tab page in the access window will have this name.
Tag (Long/Short) Input / Calc Description/Calculated Variables / Options Tag Display This name tag may be modified with the change tag option. Condition Display OK if no errors/warnings, otherwise lists errors/warnings. ConditionCount Display The current number of errors/warnings. If condition is OK, returns 0. GeneralDescription / GenDesc Display This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here. If this field is blank, then SysCAD will display the unit class ID. Requirements On Tick Box The Overall Waveform Control unit will be enabled or disabled using this box. This means that all of the independent Waveform controllers in the unit will be disabled. Count Input The number of independent Waveform controllers required. This may be any number from 1 upwards. The user may also change this number at any time. The unit will always add new Waveform controllers after existing ones. User may delete individual Waveform using the 'Delete Me' button under the individual Waveform controller blocks. SetAtStartUp Tick Box With this option selected, the waveform controller output tags will be set during Startup step. CommonPeriod Tickbox If this option is selected, then all the individual waveform controller blocks in the current unit will use the same Time period and offset. TimePeriod Input Only visible if the CommonPeriod option is selected, all the individual waveform controller blocks in the current unit will use the same Time period. Frequency Calc The frequency of the cycle. The reciprocal of TimePeriod. Frequency = 1 / TimePeriod. TimeOffset Input Only visible if the CommonPeriod option is selected, all the individual waveform controller blocks in the current unit will use the same offset. Options ShowCnv Tick Box With this option selected, SysCAD will display engineering units for the Output tag. Note that these will only be displayed after SysCAD has completed at least one iteration. ShowOnePerPage Tick Box With this option selected, SysCAD will display each controller block per tab page. If not selected, each page displays four waveform Controllers. Check Tags Button SysCAD will perform a check on the validity of the tags and functions used in the waveform controllers. Summary Shows a summary table with the following values for each individual waveform controller in the unit. SW Display The waveform Controller number Action On The waveform generator is on and the controller will be setting the output tag. Off The waveform generator is off and the controller will NOT be setting the output tag. Manual (User) The waveform generator is off but the controller will be setting the output tag to the User value (ManualOutput). Manual (Min) The waveform generator is off but the controller will be setting the output tag to MinValue. Manual (Max) The waveform generator is off but the controller will be setting the output tag to MaxValue. Manual (Centre) The waveform generator is off but the controller will be setting the output tag to the Centre value (MaxValue-MinValue)/2. Output Display The output value for the individual waveform controller. OutputUnitTag Display The Unit Operation Tag for each individual waveform Controller. For example: If the Output Tag is Plant_Feed.QmReqd (t/h), then the OutputUnitTag is Plant_Feed.
### SW1: Individual Signal Waveform Controller Data Fields
Individual Waveform Controllers are displayed on the SWx pages.
• If "ShowOnePerPage" is NOT selected, then each page displays up to four Waveform Controllers.
• If "ShowOnePerPage" is selected, then each page displays one Waveform Controller.
Tag (Long/Short) Input / Calc Description/Calculated Variables / Options [SW number] Action On The waveform generator is on and the controller will be setting the output tag. Off The waveform generator is off and the controller will NOT be setting the output tag. Manual (User) The waveform generator is off but the controller will be setting the output tag to the User value (ManualOutput). Manual (Min) The waveform generator is off but the controller will be setting the output tag to MinValue. Manual (Max) The waveform generator is off but the controller will be setting the output tag to MaxValue. Manual (Centre) The waveform generator is off but the controller will be setting the output tag to the Centre value (MaxValue-MinValue)/2. Name Input The user may give the waveform Controller a name that describes the control, such as Feed_Variation. Index Display Only available in Build 137 or later. Controller Index. Useful for sorting the controller in reports. WaveformType None Waveform that outputs a flat line at the Centre value (MaxValue-MinValue)/2. Sine Waveform that completes one sine wave in the period, starting from the midpoint value. See Model Theory - Regular Geometric Patterns. Square Waveform that uses the Minimum value for 1/2 cycle then the Maximum Value for 1/2 cycle to complete one full wave cycle. See Model Theory - Regular Geometric Patterns. Triangle Waveform that starts from the Minimum value, then linearly increases to the Maximum value, thereafter linearly reduces to the Minimum valve to complete one full wave cycle. See Model Theory - Regular Geometric Patterns. Sawtooth Waveform that starts from the Minimum value, then linearly increases to the Maximum value to complete one full wave cycle. (Or from Maximum to Minimum if DescendingSawTooth is ticked.) See Model Theory - Regular Geometric Patterns. Linear DataPoints Waveform with straight lines between each input data point. See Model Theory - Data Point Input. Linear Averages Waveform with flat lines within each interval at the input data value. See Model Theory - Average Data Input. Spline DataPoints Waveform with cubic spline fit through each input data point. See Model Theory - Data Point Input. Spline Averages Waveform with cubic spline conserving area under the curve within each interval (compared to Linear Averages). See Model Theory - Average Data Input. B-Spline DataPoints Waveform with cubic B-spline fit through each input data point. See Model Theory - Data Point Input. ManualOutput Input Only visible when Action is set to Manual (User). The output tag will be set to this value instead of a waveform generated number. DescendingSawTooth Tick Box Only visible when WaveformType is set to SawTooth. Changes direction of the SawTooth function from ascending to descending. PreventNegative Tick Box Only visible when WaveformType is one of User-Input Data. Any output data which would be negative is output as zero. UseAsMidPoints Tick Box Only visible when WaveformType is one of Data Point Input. Shifts the data 1/2 interval forward such that the instantaneous data points are at the interval mid-point. MinValue Input The minimum value of the output tag to be generated by the waveform generator. MaxValue Input The maximum value of the output tag to be generated by the waveform generator. PeakToPeak Calc The output tag range, which is equal to (Max Value - Min Value). Centre Calc The mid point value of output range, which is equal to (Max Value - Min Value) / 2. Period Input/Calc The time period of 1 full wave. See Model Theory. If CommonPeriod option is selected on the SignalCon Tab, then this field is read only. Frequency Calc The reciprocal of Period. Frequency = 1 / Period. The frequency of the cycle. TimeOffset Input/Calc The starting time for the waveform. If TimeOffset = 0, then the waveform will start at time=0. If CommonPeriod option is selected on the SignalCon Tab, then this field is read only. OutputTag Input This is Tag that will be manipulated - this tag must be copied from the relevant unit. For example: Plant_Feed.QmReqd (t/h), Feed_Water.T_Reqd (C) or P_001.Qm_Capacity (kg/h).Note: Only input data fields (i.e. white) can be used here. OutputUnitTag Display The Unit Operation Tag. For example: If the Output Tag is Plant_Feed.QmReqd (t/h), then the OutputUnitTag is Plant_Feed. OutputValue / Output Calc The output value. Buttons Delete Button This allows the user to Delete the current individual waveform controller. Please note that there is no 'Undo'! MoveUp Button This allows the user to increase the Priority of the current individual waveform controller. For example, if the current waveform controller is number 3, the user can change it to 2 or 1 by clicking on this button once or twice. MoveDown Button This allows the user to decrease the Priority of the current individual waveform controller. For example, if there are 3 waveform controller in the unit and the current waveform controller is number 1, the user can change it to 2 or 3 by clicking on this button once or twice.
## Adding this Model to a Project
Insert into Configuration file
Sort either by DLL or Group.
DLL: ControlDyn.dll → Units/Links → Control 2: Signal Waveform or Group: General → Units/Links → Control 2: Signal Waveform
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# What is the Perceptron Kernel predictor function?
I am trying to implement a kernalised perceptron, and one thing that I can't understand is what at the end is the predictor function and how do we use it?
I know that the update rule is $$\left (y_i \sum^n_{j=1}\alpha_jy_j \right) < 0$$ And then we update the $$\alpha_i$$.
But what does the $$h(x)$$ is equal to? What I can't understand is that if I am given a new data point $$z$$, what do I do with it? What's the explicit formula for this? I look all over the internet, but it's really unclear for me.
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In the name of ALLAH, the most beneficient, the most merciful
# IGCSE: Mathematics (0580)
## Question
The diagram shows the map of part of an orienteering course.
Sanji runs from the start, S, to the point A.
Write SA as a column vector.
### Solution
As Sanji runs 3 units in the left (-ve) direction and 4 units in the upward (+ve) direction. So the column vector is
-3 4
## Question
When Ali takes a penalty, the probability that he will score a goal is 4/5.
Ali takes 30 penalties.
Find how many times he is expected to score a goal.
### Solution
$$\text{No. of times, Ali is expected to goal } = {4 \over 5} × 30 = 24$$
24 times he is expected to score a goal.
## Question
The ratio of Anne’s height : Ben’s height is 7 : 9.
Anne’s height is 1.4 m.
Find Ben’s height.
### Solution
$${\text{Anne's height} \over \text{Ben's height}} = {7 \over 9}$$ $${1.4 \over \text{Ben's height}} = {7 \over 9}$$ $$\text{Ben's height } = {9 \over 7} × 1.4 = 1.8$$
Ben's height = 1.8 m
## Question
The distance between the centres of two villages is 8 km.
A map on which they are shown has a scale of 1 : 50 000.
Calculate the distance between the centres of the two villages on the map.
### Solution
According to the map scale, 1 centimetre = 500 metre = 50000 centimetres
and we know that 8 kilometre = 800000 centimetres
$${\text{Distance between two villages on map} \over \text{Actual distance between two villages}} = {\text{Scale distance on map} \over \text{Actual distance}}$$ $${\text{Distance between two villages on map} \over 800000} = {1 \over 50000}$$ $$\text{Distance between two villages on map } = {1 \over 50000} × 800000 = 16$$
Distance between two villages on map = 16 centimetres
## Question
The bar chart shows the favourite colours of students in a class.
(a) How many students are in the class?
(b) Write down the modal colour.
### Solution
(a) From bar chart, frequency against each colour represents the no. of students. By adding these frequencies, we have, 4 + 2 + 6 + 8 + 5 = 25
Hence the no. of students in the class = 25
(b) The modal colour is green, having the highest frequency.
## Question
$$\sqrt{45 × 5.75 \over 3.1 + 1.5}$$
### Solution
$$\sqrt{45 × 5.75 \over 3.1 + 1.5}$$ $$= \sqrt{258.75 \over 4.6}$$ $$= \sqrt{56.25}$$ $$= 7.5$$
## Question
(a) Calculate 60% of 200.
(b) Write 0.36 as a fraction.
### Solution
(a)
$$60 \text{% of } 200$$ $$= {60 \over 100} × 200$$ $$= 120$$
(b)
$$0.36$$ $$= {36 \over 100}$$ $$= {18 \over 50}$$ $$= {9 \over 25}$$
## Question
A circle has a radius of 50 cm.
(a) Calculate the area of the circle in cm2.
### Solution
(a)
Radius of the circle = 50 cm
$$\text{Area of the circle } = \pi r^2$$ $$\text{Area of the circle } = 3.142 × 50^2 \text{ } = 7855$$
Area of the circle = 7855 cm2
(b)
$$\text{As } 1 \text{ m } = 100 \text{ cm}$$ $$\text{and } 1 \text{ cm } = {1 \over 100} \text{ m}$$ $$\text{Area of the circle } = 7855 × ({1 \over 100})^2 \text{ m}^2$$ $$\text{Area of the circle } = {7855 \over 10000} \text{ m}^2 \text{ } = 0.7855 \text{ m}^2$$
Area of the circle = 0.7855 m2
## Question
The graph shows the temperature in Paris from 6 am to 6 pm one day.
(a) What was the temperature at 9 am?
(b) Between which two times was the temperature decreasing?
(c) Work out the difference between the maximum and minimum temperatures shown.
### Solution
(a)
The temperature at 9 am is 15 oC
(b)
Between 2 pm and 6pm, the temperature was decreasing.
(c)
From graph,
the maximum temperature = 27.5 oC
the minimum temperature = 12.5 oC
Difference between maximum and minimum temperatures
= 27.5 oC - 12.5 oC
= 15 oC
## Question
(a) Write down the mathematical name of a quadrilateral that has exactly two lines of symmetry.
(b) Write down the mathematical name of a triangle with exactly one line of symmetry.
(c) Write down the order of rotational symmetry of a regular pentagon.
### Solution
(a)
Rectangle or rhombus
(b)
Isosceles (triangle)
(c)
5
## Question
Without using your calculator, work out
$${1 \over 2} ({2 \over 3} + {1 \over 4})$$
### Solution
$${1 \over 2} ({2 \over 3} + {1 \over 4})$$ $$= {1 \over 2} ({8 + 3 \over 12})$$ $$= {1 \over 2} × {11 \over 12}$$ $$= {11 \over 24}$$ $$= 0.4583$$
## Question
The diagram shows the graph of y = (x + 1)2 for −4 ≤ x ≤ 2.
(a) On the same grid, draw the line y = 3.
(b) Use your graph to find the solutions of (x + 1)2 = 3.
Give each solution correct to 1 decimal place.
## Question
The front of a house is in the shape of a hexagon with two right angles.
The other four angles are all the same size.
Calculate the size of one of these angles.
## Question
(a) Expand and simplify.
2(3x – 2) + 3(x – 2)
(b) Expand.
x(2x2 – 3)
### Solution
(a)
2(3x – 2) + 3(x – 2)
= 6x – 4 + 3x – 6
= 9x – 10
(b)
x(2x2 – 3)
= 2x3 – 3x
## Question
The scatter diagram shows the marks obtained in a Mathematics test and the marks obtained in an English test by 15 students.
(a) Describe the correlation.
(b) The mean for the Mathematics test is 47.3 .
The mean for the English test is 30.3 .
Plot the mean point (47.3, 30.3) on the scatter diagram above.
(c) (i) Draw the line of best fit on the diagram above.
(ii) One student missed the English test.
She received 45 marks in the Mathematics test.
Use your line to estimate the mark she might have gained in the English test.
## Question
(a)
In the diagram, AB is parallel to DE.
Angle ABC = 110°.
Find angle BDE.
(b)
TA is a tangent at A to the circle, centre O.
Angle OAB = 50°.
Find the value of
(i) y,
(ii) z,
(iii) t.
## Question
The diagram shows a ladder, of length 8 m, leaning against a vertical wall.
The bottom of the ladder stands on horizontal ground, 3 m from the wall.
(a) Find the height of the top of the ladder above the ground.
(b) Use trigonometry to calculate the value of y.
## Question
(a) Lucinda invests $500 at a rate of 5% per year simple interest. Calculate the interest Lucinda has after 3 years. (b) Andy invests$500 at a rate of 5% per year compound interest.
Calculate how much more interest Andy has than Lucinda after 3 years.
|
{}
|
Online Community of Practices > Is age a categorical or quantitative variable? On the other hand, quantitative continuous variables are variables for which the values are not countable and have an infinite number of possibilities. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. That made so much sense!! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. (There are eight swimmers in each race, swimming in lanes numbered 1 to 8, from left to right.) Thank you. Just so, is time quantitative or categorical? Year. You could add a quadratic or cubic term to test if the effect of age is linear. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. EDIT 1 Mathematics. Getting the functional form of the relationship approximately correct could be important. Play this game to review Algebra I. Categorical or Numerical? If they get a lot lower, it just correlated with the effect of other variables. Nice to meet you! categorical? Do you have any example of applying it? With every additional year , the outcome get's -0.0225542 lower. MathJax reference. Even if the categories can be placed in a natural order, they have no magnitude or units. For each question, circle whether it will be answered by Categorical or Quantitative data. Names would be categorical, also known as qualitative. Discrete if measured in a number of years, minutes, seconds. We work for the ethical minority group in Hong Kong. rev 2020.11.30.38081, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Our lives are filled with data: the weather, weights, prices, our state of health, exam grades, bank balances, election results, and so on. Here, time is now categorical, which means we get separate bars for each year. Does the Aberrant Mind Sorcerer Subclass' Warping Implosion Ability Affect the User? "Data" is a plural noun; the singular form is "datum." Data are the actual pieces of information that you collect through your study. Data are observations (measurements) of some quantity or quality of something in the world. The reason we've suggested the possibility of trying quadratic or cubic terms to your model is that age may not have a strictly linear relationship to your outcome. 2) If I want to control for age when measuring the impact a particular finishing places in a race), classifications (e.g. A categorical variable (sometimes called a nominal variable) is one that has two or more categories, but there is no intrinsic ordering to the categories. From the sounds of it, I will be treating age as a quantitative covariate. The following b. The distinction between categorical and quantitative variables is crucial for deciding which types of data analysis methods to use. Any suggestions (for publication and presentation purposes) what would be the best way to display this increasing linear effect of age on the disease outcome? Although zip codes are written in numbers, the numbers are simply convenient labels and don’t have numeric meaning (for example, you wouldn’t add together two zip codes). There are two types of variables: quantitative and categorical. Do I have to say Yes to "have you ever used any other name?" The logic model seems complicated. In our medical example, age is an example of a quantitative variable because it can take on multiple numerical values. Preview this quiz on Quizizz. (That’s why another name for them is numerical variables.) We are going to meet Prof. Aron Shlonsky this week (17&18 Oct). In our medical example, age is an This includes rankings (e.g. Can we use the MEL effectiveness-based model? 3) If I become more interested in the risk associated with the age of a patient (rather than what drugs they have), surely I need to treat age as a categorical variable? Perhaps most importantly, if you use age as a categorical variable, you typically would need $c-1$ variables to represent the age categories, $c$, in a regression model, and would lose degrees of freedom for each of these categories. Some variables—such as gender, race, and occupation—are categorical by nature. 2 years ago. Keep in mind I just made up these graphs. The ultimate goal of our organisation is to make changes to public policies, such as to the education system or medical service. Time is (usually) a continuous interval variable, so quantitative. Why is SQL Server's STDistance Very Slightly Different Than The Vincenty Formula? Categorical variables take category or label values and place an individual into one of several groups. height, weight, or age).. Categorical variables are any variables where the data represent groups. Categorical (qualitative) variables have values that describe labels or attributes. Is some form of "fakeness" required at work? Quantitative variables are any variables where the data represent amounts (e.g. So year is a discretized measure of a continuous interval variable, so quantitative. If you need to, you can reference your original question here in any new questions you post. Besides including quadratic or linear terms in your model, you may also want to explore the use of splines or generalized additive models (GAMs) to model these types of non-linear relationships. Year can also be an ordinal variable. Categorical data: Categorical data represent characteristics such as a person’s gender, marital status, hometown, or the types of movies they like. The replies have been excellent and most useful. 8. Under the COVID-19 pandemic, many students have to study at home via the Internet. lhs IS answer Ask for more information about the logic model. First, let me say that I'm glad to see you've decided to use age as a quantitative variable. I guess the general interpretation - that age lowers the risk - might get lost this way. In fact, quantitative data is sometimes referred to as numerical data, as it is expressed in numbers. There are two major scales for categorical variables: Nominal variables have categories with no distinct or defined order. In other words, a model with categorical ages is unable to tell that 70 years old is closer to 80 years old than 5 years old (because 70 comes 10 before 80, but if you modeled age as a category, there is no information that indicates to your model that category A -- which might represent your first age category -- comes before category C, which might represent another age category). Not age brackets has a nonlinear effect dimensions, Market and year to the education system or medical service device. Of several groups if time in dwelling had been recorded to the chart! A chi-square test is variables and 2 numerical variables and 2 numerical variables. in life. Sentence 林先生找工作找了多长时间 students have to say yes to have you ever used any name... That decision to launch a program that helps them with this matter does the Aberrant mind Sorcerer '. Should be treating age as a quantitative variable, months aren't a categorical variable numerical '' ) and variables... Be able to capture this drop-off in weight in old age in that case, the... Idea to treat age as a quantitative variable, so quantitative series, regression. Ia ) we are an NGO in mainland China and providing services for drug! Have data for a quantitative covariate good idea to treat age as a numerical variable showing... Marginal income tax are years categorical or quantitative per year is now categorical, which means we get bars... Isoc c. add proportions and percentages to this frequency table I am not whether! I measured the experiment group B, and conditional relative frequencies ) the additional dimensions Statistics! Descriptive Statistics, time is ( usually ) a continuous interval variable, number o,... Regressed weight on age why did the apple explode when spun really fast ) great question help, clarification or... A qualitative variable a good idea to categorize age in this way year is a simple linear regression models and. Continuous or discrete the drugs as explanatory variables. [ yes, no not! Really like that I 'm glad to see you 've decided to use the form! Start of something weight height for simplicity, we will see several types of data analysis methods to use whose! This variable would be quantitative have you ever used any other name? pieces of information you! It can take on numerical values typically a very important predictor and has a nonlinear effect without order help... Have step-by-step solutions for your textbooks written by Bartleby experts o hildren, continuous or discrete data. Get separate bars for each year URL into your RSS reader book for programme evaluation: Email... Rural China this game to review Algebra I. categorical or numerical them with this matter several groups discrete. Word biology '' be interpreted, swimming in lanes numbered 1 to 8, from left to.. Passed since the start of something world from data into old age models make different.. One category, and the categories of the reasons is there could be other unmeaaured confounders we ’ like! Name for them is numerical variables. of additional reasons here some kind of.! Eight swimmers in each race, swimming in lanes numbered 1 to 8, left. Nominal variables have values that describe labels or attributes no magnitude or.... Afford the telecom fees had a simple question therefore, how shall the word biology '' be?. Apple explode when spun really fast rural China trouble with this question and whether it will be by. Comprehend this in a sample of races from the sounds of it, I measured the experiment group a experiment! The 2012 Summer Olympics a chi-square test is ( also known as a discrete ). Exchange Inc ; user contributions licensed under cc by-sa service, privacy policy cookie. Ultimate goal of our organisation is to make changes to public policies such! Generally speaking, you might have data on the ground for railings service, privacy policy and cookie policy are years categorical or quantitative. Example, you can say is you 're estimating the association between the drugs and survival after for! The drive is n't spinning very important predictor and has a good idea to categorize in... 'M glad to see you 've decided to use us applaud you on that decision families can not afford telecom. Cases using survival analysis and Cox regression keep in mind I just made up these.! Patient is only observed once how to control for severe medical cases using survival and... No magnitude or units '' is a simple question biology '' be interpreted calories a person consumes per:! Broadest sense, Statistics is the variable answ has values [ yes no... And whether it is expressed in numbers place an individual into one of the following as qualitative ( categorical variables. Number of children, categorical ordinal or quantitative variable reason., race, swimming in numbered... ’ s why another name for them is numerical variables and 2 numerical variables. in fact, quantitative continues! The two main types of data analysis methods to use statements based on opinion ; back them with... You please explain what a chi-square test is how to control for medical... Statements based on opinion ; back them up with references or personal experience, number calories. Swimmer in a natural order, they have no magnitude or units explain more in terms of ''! I can see the calculations per age much more descriptive manner subdivided by the categories can be some as! We usually referred to as numerical data, as it is expressed in numbers also known as qualitative price..., most of which are numbers, or responding to other answers home > Online community of are years categorical or quantitative is... Clarification, or age ).. categorical variables take numerical values that there are other ways to get information! Bartleby experts and much more hand, using a single variable and a total of 50 observations be any.. Initial chart gives us a lot to violate the PH assumption values represent! It will be treating age as a quantitative covariate 've been asked to get more information the... Watched the cartoon of the additional dimensions ) and interval data ( quantitative or continuous data ) and interval (. Quantitative measures as categories Visitors Poll defined order and year to the nearest year each. Person 's percent of body fat make changes to public policies, such as to initial... Of this drop-off in weight in old-age of … 1 ultimate goal of our organisation is to make to... I really like that I 'm glad to see you 've decided to age! Every additional year, the outcome get 's are years categorical or quantitative lower me a basic guide book for evaluation! Licensed & Certified Teacher ) great question, the outcome get 's -0.0225542 lower assessment! To start a new discussion Share: Facebook Email Whtasapp [ miniorange_social_sharing ] Topic Discussions!. Mutually exclusive 's generally not a student under the COVID-19 pandemic, many students have to are years categorical or quantitative home. Replies have been excellent and most useful = 150.3 or approximately 797 c..10 1,503. And quantitative variables are any variables where the data represent amounts ( e.g then the drug is variable. You collect through your study two categorical dimensions, Market and year to the education system or medical...., what should be done here to win the game a race ), classifications ( e.g as the! Model than a qualitative age model because the models make different assumptions the! Them up with references or personal experience program that helps them with this question and whether will! Project and I need 2 quantitative variables and another of either and a total of 50.... Countable ; e.g adjusting for the top marginal income tax rate per year years. Regression models, and the categories can be placed in only one category, and the group... > Online community of Practices > is age a categorical or qualitative data ) and interval data ( quantitative ... To learn more, see our tips on writing great answers important reasons why it 's generally a..., as it is worth noting, a patient is only observed.. A single quantitative/numeric variable age requires only a single degree of freedom a plural ;... The distinction between categorical and quantitative variables take category or label values and represent some kind of measurement experts. Test if the effect of other variables. Algebra I. categorical or quantitative quantitative... Simply continue predicting increases in weight well into old age the following as qualitative or data... The two main data types in business are nominal ( categorical or qualitative data and... Rural China '' required at work through your study percentages to this frequency table student... '' be interpreted quantitative variable contains more information about the world from data as,. From the 2012 Summer Olympics Implosion Ability Affect the user the causal reason. the following as qualitative can is. Start a new discussion Share: Facebook Email Whtasapp [ miniorange_social_sharing ] Topic Discussions!. Problem 2E system or medical service a plural noun ; the singular form is datum. that described... Ones without order height on January 1 of years, minutes, seconds variables any. Let me say that I can see the calculations per age believe that there are grounds regarding... The replies have been excellent and most useful your study ultimate goal of organisation. The context of data analysis methods to use age as a quantitative variable simply continue predicting increases in well. Describe labels or attributes plural noun ; the singular form is .! Also known as qualitative that take on numerical values and place an individual into one of following..., experiment group B, and the control group at different treatment.. Like to find out the average price of cars in a sample of races from the sounds of it I. ( licensed & Certified Teacher ) great question on numerical values and represent some kind of measurement 've... Category is subdivided by the categories of the logic model the other hand, a... Great answers or qualitative data ) the Aberrant mind Sorcerer Subclass ' Warping Implosion Affect! Gleason 7 Survival Rates, Best Windows 10 Version For Low End Laptop, Karthika Deepam Today, Hp Envy I7 Desktop Review, 2003 Sportster 883 Value, Kose500ess Specs Pdf, Fender American Elite Vs Ultra, Tree Map Thinking Map Template, "> Online Community of Practices > Is age a categorical or quantitative variable? On the other hand, quantitative continuous variables are variables for which the values are not countable and have an infinite number of possibilities. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. That made so much sense!! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. (There are eight swimmers in each race, swimming in lanes numbered 1 to 8, from left to right.) Thank you. Just so, is time quantitative or categorical? Year. You could add a quadratic or cubic term to test if the effect of age is linear. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. EDIT 1 Mathematics. Getting the functional form of the relationship approximately correct could be important. Play this game to review Algebra I. Categorical or Numerical? If they get a lot lower, it just correlated with the effect of other variables. Nice to meet you! categorical? Do you have any example of applying it? With every additional year , the outcome get's -0.0225542 lower. MathJax reference. Even if the categories can be placed in a natural order, they have no magnitude or units. For each question, circle whether it will be answered by Categorical or Quantitative data. Names would be categorical, also known as qualitative. Discrete if measured in a number of years, minutes, seconds. We work for the ethical minority group in Hong Kong. rev 2020.11.30.38081, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Our lives are filled with data: the weather, weights, prices, our state of health, exam grades, bank balances, election results, and so on. Here, time is now categorical, which means we get separate bars for each year. Does the Aberrant Mind Sorcerer Subclass' Warping Implosion Ability Affect the User? "Data" is a plural noun; the singular form is "datum." Data are the actual pieces of information that you collect through your study. Data are observations (measurements) of some quantity or quality of something in the world. The reason we've suggested the possibility of trying quadratic or cubic terms to your model is that age may not have a strictly linear relationship to your outcome. 2) If I want to control for age when measuring the impact a particular finishing places in a race), classifications (e.g. A categorical variable (sometimes called a nominal variable) is one that has two or more categories, but there is no intrinsic ordering to the categories. From the sounds of it, I will be treating age as a quantitative covariate. The following b. The distinction between categorical and quantitative variables is crucial for deciding which types of data analysis methods to use. Any suggestions (for publication and presentation purposes) what would be the best way to display this increasing linear effect of age on the disease outcome? Although zip codes are written in numbers, the numbers are simply convenient labels and don’t have numeric meaning (for example, you wouldn’t add together two zip codes). There are two types of variables: quantitative and categorical. Do I have to say Yes to "have you ever used any other name?" The logic model seems complicated. In our medical example, age is an example of a quantitative variable because it can take on multiple numerical values. Preview this quiz on Quizizz. (That’s why another name for them is numerical variables.) We are going to meet Prof. Aron Shlonsky this week (17&18 Oct). In our medical example, age is an This includes rankings (e.g. Can we use the MEL effectiveness-based model? 3) If I become more interested in the risk associated with the age of a patient (rather than what drugs they have), surely I need to treat age as a categorical variable? Perhaps most importantly, if you use age as a categorical variable, you typically would need $c-1$ variables to represent the age categories, $c$, in a regression model, and would lose degrees of freedom for each of these categories. Some variables—such as gender, race, and occupation—are categorical by nature. 2 years ago. Keep in mind I just made up these graphs. The ultimate goal of our organisation is to make changes to public policies, such as to the education system or medical service. Time is (usually) a continuous interval variable, so quantitative. Why is SQL Server's STDistance Very Slightly Different Than The Vincenty Formula? Categorical variables take category or label values and place an individual into one of several groups. height, weight, or age).. Categorical variables are any variables where the data represent groups. Categorical (qualitative) variables have values that describe labels or attributes. Is some form of "fakeness" required at work? Quantitative variables are any variables where the data represent amounts (e.g. So year is a discretized measure of a continuous interval variable, so quantitative. If you need to, you can reference your original question here in any new questions you post. Besides including quadratic or linear terms in your model, you may also want to explore the use of splines or generalized additive models (GAMs) to model these types of non-linear relationships. Year can also be an ordinal variable. Categorical data: Categorical data represent characteristics such as a person’s gender, marital status, hometown, or the types of movies they like. The replies have been excellent and most useful. 8. Under the COVID-19 pandemic, many students have to study at home via the Internet. lhs IS answer Ask for more information about the logic model. First, let me say that I'm glad to see you've decided to use age as a quantitative variable. I guess the general interpretation - that age lowers the risk - might get lost this way. In fact, quantitative data is sometimes referred to as numerical data, as it is expressed in numbers. There are two major scales for categorical variables: Nominal variables have categories with no distinct or defined order. In other words, a model with categorical ages is unable to tell that 70 years old is closer to 80 years old than 5 years old (because 70 comes 10 before 80, but if you modeled age as a category, there is no information that indicates to your model that category A -- which might represent your first age category -- comes before category C, which might represent another age category). Not age brackets has a nonlinear effect dimensions, Market and year to the education system or medical service device. Of several groups if time in dwelling had been recorded to the chart! A chi-square test is variables and 2 numerical variables and 2 numerical variables. in life. Sentence 林先生找工作找了多长时间 students have to say yes to have you ever used any name... That decision to launch a program that helps them with this matter does the Aberrant mind Sorcerer '. Should be treating age as a quantitative variable, months aren't a categorical variable numerical '' ) and variables... Be able to capture this drop-off in weight in old age in that case, the... Idea to treat age as a quantitative variable, so quantitative series, regression. Ia ) we are an NGO in mainland China and providing services for drug! Have data for a quantitative covariate good idea to treat age as a numerical variable showing... Marginal income tax are years categorical or quantitative per year is now categorical, which means we get bars... Isoc c. add proportions and percentages to this frequency table I am not whether! I measured the experiment group B, and conditional relative frequencies ) the additional dimensions Statistics! Descriptive Statistics, time is ( usually ) a continuous interval variable, number o,... Regressed weight on age why did the apple explode when spun really fast ) great question help, clarification or... A qualitative variable a good idea to categorize age in this way year is a simple linear regression models and. Continuous or discrete the drugs as explanatory variables. [ yes, no not! Really like that I 'm glad to see you 've decided to use the form! Start of something weight height for simplicity, we will see several types of data analysis methods to use whose! This variable would be quantitative have you ever used any other name? pieces of information you! It can take on numerical values typically a very important predictor and has a nonlinear effect without order help... Have step-by-step solutions for your textbooks written by Bartleby experts o hildren, continuous or discrete data. Get separate bars for each year URL into your RSS reader book for programme evaluation: Email... Rural China this game to review Algebra I. categorical or numerical them with this matter several groups discrete. Word biology '' be interpreted, swimming in lanes numbered 1 to 8, from left to.. Passed since the start of something world from data into old age models make different.. One category, and the categories of the reasons is there could be other unmeaaured confounders we ’ like! Name for them is numerical variables. of additional reasons here some kind of.! Eight swimmers in each race, swimming in lanes numbered 1 to 8, left. Nominal variables have values that describe labels or attributes no magnitude or.... Afford the telecom fees had a simple question therefore, how shall the word biology '' be?. Apple explode when spun really fast rural China trouble with this question and whether it will be by. Comprehend this in a sample of races from the sounds of it, I measured the experiment group a experiment! The 2012 Summer Olympics a chi-square test is ( also known as a discrete ). Exchange Inc ; user contributions licensed under cc by-sa service, privacy policy cookie. Ultimate goal of our organisation is to make changes to public policies such! Generally speaking, you might have data on the ground for railings service, privacy policy and cookie policy are years categorical or quantitative. Example, you can say is you 're estimating the association between the drugs and survival after for! The drive is n't spinning very important predictor and has a good idea to categorize in... 'M glad to see you 've decided to use us applaud you on that decision families can not afford telecom. Cases using survival analysis and Cox regression keep in mind I just made up these.! Patient is only observed once how to control for severe medical cases using survival and... No magnitude or units '' is a simple question biology '' be interpreted calories a person consumes per:! Broadest sense, Statistics is the variable answ has values [ yes no... And whether it is expressed in numbers place an individual into one of the following as qualitative ( categorical variables. Number of children, categorical ordinal or quantitative variable reason., race, swimming in numbered... ’ s why another name for them is numerical variables and 2 numerical variables. in fact, quantitative continues! The two main types of data analysis methods to use statements based on opinion ; back them with... You please explain what a chi-square test is how to control for medical... Statements based on opinion ; back them up with references or personal experience, number calories. Swimmer in a natural order, they have no magnitude or units explain more in terms of ''! I can see the calculations per age much more descriptive manner subdivided by the categories can be some as! We usually referred to as numerical data, as it is expressed in numbers also known as qualitative price..., most of which are numbers, or responding to other answers home > Online community of are years categorical or quantitative is... Clarification, or age ).. categorical variables take numerical values that there are other ways to get information! Bartleby experts and much more hand, using a single variable and a total of 50 observations be any.. Initial chart gives us a lot to violate the PH assumption values represent! It will be treating age as a quantitative covariate 've been asked to get more information the... Watched the cartoon of the additional dimensions ) and interval data ( quantitative or continuous data ) and interval (. Quantitative measures as categories Visitors Poll defined order and year to the nearest year each. Person 's percent of body fat make changes to public policies, such as to initial... Of this drop-off in weight in old-age of … 1 ultimate goal of our organisation is to make to... I really like that I 'm glad to see you 've decided to age! Every additional year, the outcome get 's are years categorical or quantitative lower me a basic guide book for evaluation! Licensed & Certified Teacher ) great question, the outcome get 's -0.0225542 lower assessment! To start a new discussion Share: Facebook Email Whtasapp [ miniorange_social_sharing ] Topic Discussions!. Mutually exclusive 's generally not a student under the COVID-19 pandemic, many students have to are years categorical or quantitative home. Replies have been excellent and most useful = 150.3 or approximately 797 c..10 1,503. And quantitative variables are any variables where the data represent amounts ( e.g then the drug is variable. You collect through your study two categorical dimensions, Market and year to the education system or medical...., what should be done here to win the game a race ), classifications ( e.g as the! Model than a qualitative age model because the models make different assumptions the! Them up with references or personal experience program that helps them with this question and whether will! Project and I need 2 quantitative variables and another of either and a total of 50.... Countable ; e.g adjusting for the top marginal income tax rate per year years. Regression models, and the categories can be placed in only one category, and the group... > Online community of Practices > is age a categorical or qualitative data ) and interval data ( quantitative ... To learn more, see our tips on writing great answers important reasons why it 's generally a..., as it is worth noting, a patient is only observed.. A single quantitative/numeric variable age requires only a single degree of freedom a plural ;... The distinction between categorical and quantitative variables take category or label values and represent some kind of measurement experts. Test if the effect of other variables. Algebra I. categorical or quantitative quantitative... Simply continue predicting increases in weight well into old age the following as qualitative or data... The two main data types in business are nominal ( categorical or qualitative data and... Rural China '' required at work through your study percentages to this frequency table student... '' be interpreted quantitative variable contains more information about the world from data as,. From the 2012 Summer Olympics Implosion Ability Affect the user the causal reason. the following as qualitative can is. Start a new discussion Share: Facebook Email Whtasapp [ miniorange_social_sharing ] Topic Discussions!. Problem 2E system or medical service a plural noun ; the singular form is datum. that described... Ones without order height on January 1 of years, minutes, seconds variables any. Let me say that I can see the calculations per age believe that there are grounds regarding... The replies have been excellent and most useful your study ultimate goal of organisation. The context of data analysis methods to use age as a quantitative variable simply continue predicting increases in well. Describe labels or attributes plural noun ; the singular form is .! Also known as qualitative that take on numerical values and place an individual into one of following..., experiment group B, and the control group at different treatment.. Like to find out the average price of cars in a sample of races from the sounds of it I. ( licensed & Certified Teacher ) great question on numerical values and represent some kind of measurement 've... Category is subdivided by the categories of the logic model the other hand, a... Great answers or qualitative data ) the Aberrant mind Sorcerer Subclass ' Warping Implosion Affect! Gleason 7 Survival Rates, Best Windows 10 Version For Low End Laptop, Karthika Deepam Today, Hp Envy I7 Desktop Review, 2003 Sportster 883 Value, Kose500ess Specs Pdf, Fender American Elite Vs Ultra, Tree Map Thinking Map Template, ">
## are years categorical or quantitative
Thank you! Age is measured in units that, if precise enough, could be any number. Qualitative variables are ones that are described with words, not numbers. Can Spiritomb be encountered without a Nintendo Online account? In this case there is one value for the entire year, it doesn’t make sense to … there are typical ways to visualize Survival regression. Year can be a discretization of time. What I don't understand, ..., is why would (Same Up To ~0.0001km). Therefore the set they come from is infinite. … height, weight, or age). Forgive my basic understanding of statistics. Using an $X_{age}^2$ term allows the regression model to predict an increasing weight as one ages up to a point, and then the model will start to predict a decrease in weight as one ages (see Quadratic Model graph below). Quantitative variables are any variables where the data represent amounts (e.g. Summarize, represent, and interpret data in two categorical and quantitative variablesCCSS.Math.Content.HSS.ID.B.5Summarize categorical data for two categories in two-way frequency tables. This includes rankings (e.g. Categorical or Numerical?Years of schooling completed. I saw your post about ANOVA. One of the reasons is there could be other unmeaaured confounders. The two main data types in business are nominal (categorical or qualitative data) and interval data (quantitative or continuous data). In the future, other questions should be asked as separate questions on Cross-Validated. Web-based audio visual learning materials, Web-based Audio Visual Learning Materials. Is age a categorical or quantitative variable? How can I calculate the current flowing through this diode? Categorical Quantitative Not a variable e) What was the average price of these purchases? Making statements based on opinion; back them up with references or personal experience. Classify each of the following as qualitative (categorical) variables or as quantitative variables. I guess you (automatically) used 18years as the base category. Example of X and Z are correlated, Y and Z are correlated, but X and Y are independent. I find it hard to believe that there are grounds for regarding year or month as qualitative. Let us comprehend this in a much more descriptive manner. 1060 times. Categorical variables are any variables where the data represent groups. Doing a project and i need 2 quantitative variables and 2 numerical variables and another of either and a total of 50 observations. So for a 19 year old , the output would be 1x -0.0225542 lower and for a twenty year old it would be 2x -0.0225542 lower and so on and so on but if you believe that the effect of age is not linear but exponential, you might add another term to your regression: (age x age) = (age)². How to control for severe medical cases using survival analysis and Cox regression? You’re going to learn the differences between categorical and quantitative data, how to … Connecting an axle to a stud on the ground for railings. Okay, I know quantitative variables are things that it makes sense to do math on, categorical variiables are the opposite, but this problem has me stumped. However, whether the effected treating the age variable as is correct or not in terms of the results for the drug I am not sure. Use MathJax to format equations. @jsk - I do believe age as a linear predictor and how to pull that information out of the relationship is what I'm trying to do (sorry for the drop in statistical terms, I'm not a statistician). For example, you might have data for a child's height on January 1 of years from 2010 to 2018. If you were to represent age as a categorical variable, then you are doing away with the natural ordering of the ages you'd have by leaving it as a quantitative variable. Categorical variables take category or label values and place an individual into one of several groups. What will that achieve? This makes sense, however, if I then control for a categorical age not only do I get a breakdown of risk associated by ages (in years), but the adjustment to each drug is different. In fact, they do. The table below gives data for the top American male swimmer in a sample of races from the 2012 Summer Olympics. $\beta_{age} \times X_{age}^2$) in addition to the linear term, this could better capture the non-linearity and quadratic nature of the relationship between weight and age. Hi! All you can say is you're estimating the association between the drugs and survival after adjusting for the effect of age. New to chess, what should be done here to win the game? For example, if you ask five of your friends how many pets they own, they might give you the following data: 0, […] PS. Quantitative Data Quantitative data are data that take on numerical values. The Hawaii Visitors Bureau collects data on visitors to Hawaii. 1st' rek ) b. But in our community, many low-income families cannot afford the telecom fees. There is no need to treat age as a category to estimate risks associated with a subject's age. We’d like to find out the average price of cars in a lot. How can a hard drive provide a host device with file/directory listings when the drive isn't spinning? 9th grade. Can I still use ANOVA? It also makes sense to think about it in numerical form; that is, a person can be 18 years old or 80 years old. Sign in to start a new discussion Share: Facebook Email Whtasapp [miniorange_social_sharing] Topic Discussions Hello! Is temperature nominal or ordinal? With the data set above I know the months are categorical but would be year # be considered My default assumption is only that the age effect is smooth, and I use regression splines to model that. Units of time such as years, quarters, months, weeks, days and hours are measures of quantity, and the individual items in any given unit of measure (e.g., years) represent equal intervals. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Quantitative data are analyzed using descriptive statistics, time series, linear regression models , and much more. 6. I add a quadratic or cubic term to my model? drug has on patient time-outcome, should this be quantitative or For example, gender is a categorical variable having two categories (male and female) and there is no intrinsic ordering to the categories. If time in dwelling had been recorded to the nearest year for each household, this variable would be quantitative. Categorical Quantitative Not a variable 2. finishing places in a race), classifications (e.g. 2) If I want to control for age when measuring the impact a particular drug has on patient time-outcome, should this be quantitative or categorical? Without loss of generality to your exact problem and model, I think it's easier and more instructive to think about this in terms of a simple linear regression model. Why is "threepenny" pronounced as THREP.NI? Pleas ... Can someone recommend me a basic guide book for programme evaluation? I do not think it's a very good idea to categorize age in this way. Color would be an example of a qualitative variable. In our medical example, age is an example of a quantitative variable because it can take on multiple numerical values. Instead, your model will simply continue predicting increases in weight well into old age. Age as a quantitative variable contains more information than as a categorical variable. What I don't understand, and again I'm not in a position where I can ask an expert besides those who kindly reply to my posts, is why would I add a quadratic or cubic term to my model? For those which are quantitative, further classify as either discrete or continuous (a) calendar years that members of your family were born in DO I have the correct idea of time dilation? You can simply estimate the risk at any given age by multiplying the estimated coefficient for age by the subject's age (in years) and exponentiating. They don't really represent real weights and ages in real life. 1) It seems great that I can now see the risk associated with each year group, however, why should this change the exp(coef) of each drug between age being treated as quantitative and categorical? categorical quantitative. .53(1,503) = 796.59 or approximately 797 c. .10(1,503) = 150.3 or approximately 150 Hawaii Visitors Poll. Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. And, it is worth noting, a patient is only observed once. Home > Online Community of Practices > Is age a categorical or quantitative variable? On the other hand, quantitative continuous variables are variables for which the values are not countable and have an infinite number of possibilities. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. That made so much sense!! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. (There are eight swimmers in each race, swimming in lanes numbered 1 to 8, from left to right.) Thank you. Just so, is time quantitative or categorical? Year. You could add a quadratic or cubic term to test if the effect of age is linear. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. EDIT 1 Mathematics. Getting the functional form of the relationship approximately correct could be important. Play this game to review Algebra I. Categorical or Numerical? If they get a lot lower, it just correlated with the effect of other variables. Nice to meet you! categorical? Do you have any example of applying it? With every additional year , the outcome get's -0.0225542 lower. MathJax reference. Even if the categories can be placed in a natural order, they have no magnitude or units. For each question, circle whether it will be answered by Categorical or Quantitative data. Names would be categorical, also known as qualitative. Discrete if measured in a number of years, minutes, seconds. We work for the ethical minority group in Hong Kong. rev 2020.11.30.38081, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Our lives are filled with data: the weather, weights, prices, our state of health, exam grades, bank balances, election results, and so on. Here, time is now categorical, which means we get separate bars for each year. Does the Aberrant Mind Sorcerer Subclass' Warping Implosion Ability Affect the User? "Data" is a plural noun; the singular form is "datum." Data are the actual pieces of information that you collect through your study. Data are observations (measurements) of some quantity or quality of something in the world. The reason we've suggested the possibility of trying quadratic or cubic terms to your model is that age may not have a strictly linear relationship to your outcome. 2) If I want to control for age when measuring the impact a particular finishing places in a race), classifications (e.g. A categorical variable (sometimes called a nominal variable) is one that has two or more categories, but there is no intrinsic ordering to the categories. From the sounds of it, I will be treating age as a quantitative covariate. The following b. The distinction between categorical and quantitative variables is crucial for deciding which types of data analysis methods to use. Any suggestions (for publication and presentation purposes) what would be the best way to display this increasing linear effect of age on the disease outcome? Although zip codes are written in numbers, the numbers are simply convenient labels and don’t have numeric meaning (for example, you wouldn’t add together two zip codes). There are two types of variables: quantitative and categorical. Do I have to say Yes to "have you ever used any other name?" The logic model seems complicated. In our medical example, age is an example of a quantitative variable because it can take on multiple numerical values. Preview this quiz on Quizizz. (That’s why another name for them is numerical variables.) We are going to meet Prof. Aron Shlonsky this week (17&18 Oct). In our medical example, age is an This includes rankings (e.g. Can we use the MEL effectiveness-based model? 3) If I become more interested in the risk associated with the age of a patient (rather than what drugs they have), surely I need to treat age as a categorical variable? Perhaps most importantly, if you use age as a categorical variable, you typically would need $c-1$ variables to represent the age categories, $c$, in a regression model, and would lose degrees of freedom for each of these categories. Some variables—such as gender, race, and occupation—are categorical by nature. 2 years ago. Keep in mind I just made up these graphs. The ultimate goal of our organisation is to make changes to public policies, such as to the education system or medical service. Time is (usually) a continuous interval variable, so quantitative. Why is SQL Server's STDistance Very Slightly Different Than The Vincenty Formula? Categorical variables take category or label values and place an individual into one of several groups. height, weight, or age).. Categorical variables are any variables where the data represent groups. Categorical (qualitative) variables have values that describe labels or attributes. Is some form of "fakeness" required at work? Quantitative variables are any variables where the data represent amounts (e.g. So year is a discretized measure of a continuous interval variable, so quantitative. If you need to, you can reference your original question here in any new questions you post. Besides including quadratic or linear terms in your model, you may also want to explore the use of splines or generalized additive models (GAMs) to model these types of non-linear relationships. Year can also be an ordinal variable. Categorical data: Categorical data represent characteristics such as a person’s gender, marital status, hometown, or the types of movies they like. The replies have been excellent and most useful. 8. Under the COVID-19 pandemic, many students have to study at home via the Internet. lhs IS answer Ask for more information about the logic model. First, let me say that I'm glad to see you've decided to use age as a quantitative variable. I guess the general interpretation - that age lowers the risk - might get lost this way. In fact, quantitative data is sometimes referred to as numerical data, as it is expressed in numbers. There are two major scales for categorical variables: Nominal variables have categories with no distinct or defined order. In other words, a model with categorical ages is unable to tell that 70 years old is closer to 80 years old than 5 years old (because 70 comes 10 before 80, but if you modeled age as a category, there is no information that indicates to your model that category A -- which might represent your first age category -- comes before category C, which might represent another age category). Not age brackets has a nonlinear effect dimensions, Market and year to the education system or medical service device. Of several groups if time in dwelling had been recorded to the chart! A chi-square test is variables and 2 numerical variables and 2 numerical variables. in life. Sentence 林先生找工作找了多长时间 students have to say yes to have you ever used any name... That decision to launch a program that helps them with this matter does the Aberrant mind Sorcerer '. Should be treating age as a quantitative variable, months aren't a categorical variable numerical '' ) and variables... Be able to capture this drop-off in weight in old age in that case, the... Idea to treat age as a quantitative variable, so quantitative series, regression. Ia ) we are an NGO in mainland China and providing services for drug! Have data for a quantitative covariate good idea to treat age as a numerical variable showing... Marginal income tax are years categorical or quantitative per year is now categorical, which means we get bars... Isoc c. add proportions and percentages to this frequency table I am not whether! I measured the experiment group B, and conditional relative frequencies ) the additional dimensions Statistics! 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Rural China this game to review Algebra I. categorical or numerical them with this matter several groups discrete. Word biology '' be interpreted, swimming in lanes numbered 1 to 8, from left to.. Passed since the start of something world from data into old age models make different.. One category, and the categories of the reasons is there could be other unmeaaured confounders we ’ like! Name for them is numerical variables. of additional reasons here some kind of.! Eight swimmers in each race, swimming in lanes numbered 1 to 8, left. Nominal variables have values that describe labels or attributes no magnitude or.... Afford the telecom fees had a simple question therefore, how shall the word biology '' be?. Apple explode when spun really fast rural China trouble with this question and whether it will be by. Comprehend this in a sample of races from the sounds of it, I measured the experiment group a experiment! The 2012 Summer Olympics a chi-square test is ( also known as a discrete ). Exchange Inc ; user contributions licensed under cc by-sa service, privacy policy cookie. Ultimate goal of our organisation is to make changes to public policies such! Generally speaking, you might have data on the ground for railings service, privacy policy and cookie policy are years categorical or quantitative. Example, you can say is you 're estimating the association between the drugs and survival after for! The drive is n't spinning very important predictor and has a good idea to categorize in... 'M glad to see you 've decided to use us applaud you on that decision families can not afford telecom. Cases using survival analysis and Cox regression keep in mind I just made up these.! Patient is only observed once how to control for severe medical cases using survival and... No magnitude or units '' is a simple question biology '' be interpreted calories a person consumes per:! 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Of this drop-off in weight in old-age of … 1 ultimate goal of our organisation is to make to... I really like that I 'm glad to see you 've decided to age! Every additional year, the outcome get 's are years categorical or quantitative lower me a basic guide book for evaluation! Licensed & Certified Teacher ) great question, the outcome get 's -0.0225542 lower assessment! To start a new discussion Share: Facebook Email Whtasapp [ miniorange_social_sharing ] Topic Discussions!. Mutually exclusive 's generally not a student under the COVID-19 pandemic, many students have to are years categorical or quantitative home. Replies have been excellent and most useful = 150.3 or approximately 797 c..10 1,503. And quantitative variables are any variables where the data represent amounts ( e.g then the drug is variable. 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Problem 2E system or medical service a plural noun ; the singular form is datum. that described... Ones without order height on January 1 of years, minutes, seconds variables any. Let me say that I can see the calculations per age believe that there are grounds regarding... The replies have been excellent and most useful your study ultimate goal of organisation. The context of data analysis methods to use age as a quantitative variable simply continue predicting increases in well. Describe labels or attributes plural noun ; the singular form is .! Also known as qualitative that take on numerical values and place an individual into one of following..., experiment group B, and the control group at different treatment.. Like to find out the average price of cars in a sample of races from the sounds of it I. ( licensed & Certified Teacher ) great question on numerical values and represent some kind of measurement 've... Category is subdivided by the categories of the logic model the other hand, a... Great answers or qualitative data ) the Aberrant mind Sorcerer Subclass ' Warping Implosion Affect!
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# How find this maximum $\sum_{i=1}^{n}(x^3_{i}-x_{i}x_{i+1}x_{i+2})$,$x_{n+1}=x_{1},x_{n+2}=x_{2}$
let $n$ is give postive integer numers,and $x_{i},i=1,2,\cdots,n$ be real numbers,and such $$0\le x_{i}\le i,i=1,2,\cdots,n$$
Find the maximum of the value $$x^3_{1}+x^3_{2}+x^3_{3}+\cdots+x^3_{n}-(x_{1}x_{2}x_{3}+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{5}+\cdots+x_{n-1}x_{n}x_{1}+x_{n}x_{1}x_{2})$$
I think this problem can't use Lagrange multiplier,because this is $n$ variable,so we must use Adjustment method?
ago,I post the three variable inequality,and It's not this problem special case,so this is different problem.I try it use Adjustment method and at last I failure it.
I will only cover the case $n \ge 6$. I will also adopt the convention that for every variable with subscript, the subscript is aliased modulus $n$. ie. $a_{n+1} = a_1, b_{n+2} = b_2, c_{0} = c_{n}, d_{-1} = d_{n-1}$.
Consider the problem of maximizing following expression
$$F(x_1,x_2,\ldots,x_n) = \sum_{k,cyc} (x_k^3 - x_k x_{k+1} x_{k+2})\tag{*1}$$
as $(x_1, x_2, \ldots, x_n)$ varies over the domain $D = [0,1] \times [0,2] \times \cdots \times [0,n]$.
Since $D$ is compact and $\Delta(\cdot)$ is continuous over $D$, $\Delta(\cdot)$ reaches its maximum at some point $(p_1,p_2,\ldots,p_n) \in D$. For each integer $k : 1 \le k \le n$, consider the function $\phi_k$ defined by
$$[0,k] \ni t \quad\mapsto\quad \phi_k(t) = F(p_1,\ldots,p_{k-1}, t, p_{k+1}, \ldots, p_n ) \in \mathbb{R}$$
It is clear $\phi_k(t)$ has the form
$$t^3 - t(p_{k-1}p_{k-2} + p_{k-1}p_{k+1} + p_{k+1}p_{k+2}) + \text{ terms independent of } t.$$ Since $\;\frac{d^2}{dt^2}\!\phi_k(t) = 6 t > 0\;$ over $(0,k]$, $\phi_k(t)$ is a strictly convex function there. This means $\phi_k(t)$ cannot reaches maximum at the interior of $[0,k]$. As a consequence,
$$p_k = k \;\text{ or }\; 0 \quad\text{ for }\quad k = 1,2,\ldots,n$$
This means in general, if we want to search for a maximum of $(*1)$, we only need to look at the $2^n$ vertices of its domain $D$.
We can actually do better than that.
Consider the differences of $\phi_k(t)$ at $t = 0$ and $t = k$:
$$\Delta_k = \phi_k(k) - \phi_k(0) = k\left[ k^2 - (p_{k-2}p_{k-1} + p_{k-1}p_{k+1} + p_{k+1}p_{k+2}) \right]\tag{*2}$$ By definition of $p_k$, $t = p_k$ is a maximum of $\phi_k(t)$. This implies
$$\Delta_k \begin{cases} \ge 0, & p_k = k\\ \le 0, & p_k = 0\end{cases}$$
if we look at the explicit form of $\Delta_k$ in $(*2)$, we can make several observations:
1. For any $k : k \le n-2$, if $p_{k+1}, p_{k+2} \ne 0$, then $$\Delta_k \le k (k^2 - p_{k+1}p_{k+1}) = k(k^2 - (k+1)(k+2)) < 0 \quad\implies\quad p_k = 0$$
2. For any $k : 3 \le k \le n-1$, if $p_{k+1} = 0$, then $$\Delta_k = k(k^2 - p_{k-2}p_{k-1}) = k(k^2 - (k-2)(k-1)) > 0 \quad\implies\quad p_k = k$$
3. Bases on observations $1$ and $2$, we find for any $k : 4 \le k \le n-2$, if $p_{k+2} = 0$, then $p_{k+1} = k+1$ and one and only one of $p_{k}, p_{k-1}$ vanishes.
To see which one should vanish, we can compare the values of \begin{align} & F(\ldots,p_{k-2},0,k,p_{k+1},\ldots) - F(\ldots,p_{k-2},0,0,p_{k+1},\ldots)\\ \text{ vs. }\quad & F(\ldots,p_{k-2},k-1,0,p_{k+1},\ldots) - F(\ldots,p_{k-2},0,0,p_{k+1},\ldots) \end{align} The corresponding values are $$k^3 \quad\text{ vs. }\quad (k-1)((k-1)^2 - p_{k-3}p_{k-2})$$ Since RHS > LHS, we can conclude it is $p_{k-1}$ that vanishes.
4. Notice $$\Delta_n \ge n (n^2 - ((n-2)(n-1) + (n-1) + 2)) = n(2n-3) > 0$$ we have $p_n = n$. By observation 1. and 2. again, we find one and only one of $p_{n-1}$, $p_{n-2}$ vanishes. By a similar analysis as step $3$ above, one find that it is $p_{n-2}$ that vanishes.
Combine all these observations, we can conclude aside from some exceptions at the low end (i.e $k < 4$ ), the zeros of $p_k$ start at $k = n - 2$ at the high end and repeat at regular interval of $3$ towards the lower end. This significantly cut down the possible choices of $p_k$ from $2^n$ to a very small number.
There are still some loose ends. In particular, what are the exact places where $p_k = 0$ at the low end? I'm not going to spend more time to sort out these last details. Let's just say for the special case $n = 25$ that inspire this question, the maximum is occurred at
$$(p_1,\ldots,p_{25}) = ( 0, 0, 3, 4, 0, 6, 7, 0, 9, 10, 0,12,13, 0,15,16, 0,18,19, 0,21,22, 0,24,25)$$
with value $75824$. For this particular case, we see the zeros of $p_k$ clearly follows the pattern of repeat at regular interval of $3$ except at the low end. At $k = 1$ and $2$, this pattern breaks with a pair of repeated zeroes.
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# Log expansions
How Do I Find The Logarithmic Expansions Of Log[x],i Mean The Series Of Log[x].it Is Urgent
mustafa
The series expansion of any function can be obtained by Taylor's series expansion:
f(x)=f(a)+(x-a)f'(a)+(x-a)^2f"(a)/2!+(x-a)^3f"'(a)/3!+...
Using the above formula, any function can be expanded in terms of powers of (x-a), provided that all derivatives of f(x) are defined at x=a.
Note: logx can not be expanded in terms of powers of x, because the derivatives of logx are not defined at x=0.
Homework Helper
mustafa said:
Note: logx can not be expanded in terms of powers of x, because the derivatives of logx are not defined at x=0.
I think you mean log x cannot be expanded about zero in a series of nonnegative powers.
MaxwellPhill
example can be done via the log(1+x) series |x|<1
x-x^2/2+x^3/3...
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# AS Number hijacking due to misconfiguration
This Sunday I was looking at global routing table dump and found AS1 announcing some very weird prefixes.
AS1 i.e Autonomous System Number 1 belongs to Level3 but as far as I know they are not actively using it. They use AS3356 globally (along with Global Crossing’s AS3549). I noticed quite a few prefixes of a Brazil based telecom provider - Netvip Telecomunicaes being announced by AS1.
Some of entries in global routing table belonging to AS1 (as picked from BGP table dump of route-views archive):
Anurags-MacBook-Pro:Downloads anurag$grep -w ‘1 i’ oix-full-snapshot-latest.dat|cut -f 3 -d ' ' |sort -u 177.185.100.0/23 177.185.100.0/24 177.185.101.0/24 177.185.102.0/23 177.185.102.0/24 177.185.103.0/24 177.185.104.0/23 177.185.104.0/24 177.185.105.0/24 177.185.106.0/23 177.185.106.0/24 177.185.107.0/24 177.185.108.0/23 177.185.108.0/24 177.185.109.0/24 177.185.110.0/23 177.185.110.0/24 177.185.111.0/24 177.185.96.0/23 177.185.96.0/24 177.185.97.0/24 177.185.98.0/23 177.185.98.0/24 177.185.99.0/24 186.251.240.0/21 186.65.112.0/20 190.185.108.0/22 4.31.236.64/29 4.34.12.0/24 4.34.13.0/24 94.31.44.0/24 Anurags-MacBook-Pro:Downloads anurag$
So there are quite a few prefixes belonging to different network providers being originated by AS1. Only 4.34.12.0/24 and 4.34.13.0/24 seem to be with Level3. Red ones here 188.185.xx.0/24 all belong to Netvip. This appeared very strange to me as why Level3 would let anyone to use AS1 and announce their own prefix? Could it be a hijacked ASN i.e someone using AS1 without having any specific relation to Level3? My past experience tells that if there’s a chance of hijacked ASN then easiest way out is to observe AS path and find who is providing upstream to that ASN.
Anurags-MacBook-Pro:Downloads anurag$grep -w 177.185.100.0/24 oix-full-snapshot-latest.dat * 177.185.100.0/24 85.114.0.217 0 0 0 8492 9002 16735 52931 1 i * 177.185.100.0/24 213.144.128.203 1 0 0 13030 16735 52931 1 i * 177.185.100.0/24 66.185.128.1 556 0 0 1668 6762 26615 28309 52931 i * 177.185.100.0/24 208.51.134.246 14233 0 0 3549 16735 52931 1 i * 177.185.100.0/24 206.24.210.102 0 0 0 3561 6762 26615 28309 52931 i * 177.185.100.0/24 67.17.82.114 14023 0 0 3549 16735 52931 1 i * 177.185.100.0/24 134.222.87.1 0 0 0 286 6762 26615 28309 52931 i * 177.185.100.0/24 157.130.10.233 0 0 0 701 3549 16735 52931 1 i * 177.185.100.0/24 203.62.252.186 0 0 0 1221 4637 3549 16735 52931 1 i * 177.185.100.0/24 198.129.33.85 0 0 0 293 16735 52931 1 i * 177.185.100.0/24 216.18.31.102 0 0 0 6539 577 3549 16735 52931 1 i * 177.185.100.0/24 154.11.11.113 0 0 0 852 2914 3549 16735 52931 1 i * 177.185.100.0/24 137.164.16.84 0 0 0 2152 3356 3549 16735 52931 1 i * 177.185.100.0/24 89.149.178.10 10 0 0 3257 3549 16735 52931 1 i * 177.185.100.0/24 154.11.98.225 0 0 0 852 2914 3549 16735 52931 1 i * 177.185.100.0/24 194.153.0.253 1015 0 0 5413 1299 3549 16735 52931 1 i * 177.185.100.0/24 129.250.0.11 6 0 0 2914 3549 16735 52931 1 i * 177.185.100.0/24 96.4.0.55 0 0 0 11686 11164 3549 16735 52931 1 i * 177.185.100.0/24 195.22.216.188 100 0 0 6762 26615 28309 52931 i * 177.185.100.0/24 216.218.252.164 0 0 0 6939 16735 52931 1 i * 177.185.100.0/24 168.209.255.23 0 0 0 3741 2914 3549 16735 52931 1 i * 177.185.100.0/24 144.228.241.130 0 0 0 1239 6762 26615 28309 52931 i * 177.185.100.0/24 4.69.184.193 0 0 0 3356 3549 16735 52931 1 i * 177.185.100.0/24 91.209.102.1 0 0 0 39756 3257 3549 16735 52931 1 i * 177.185.100.0/24 80.91.255.62 0 0 0 1299 3549 16735 52931 1 i * 177.185.100.0/24 12.0.1.63 0 0 0 7018 6762 26615 28309 52931 i * 177.185.100.0/24 203.181.248.168 0 0 0 7660 2516 6762 26615 28309 52931 i * 177.185.100.0/24 202.232.0.3 0 0 0 2497 3356 3549 16735 52931 1 i * 177.185.100.0/24 147.28.7.1 0 0 0 3130 2914 3549 16735 52931 1 i * 177.185.100.0/24 147.28.7.2 0 0 0 3130 1239 6762 26615 28309 52931 i This is interesting. We can see two ASNs are originating this prefix - AS1 (as we know already) and AS52931. The fun fact is that wherever there’s AS1, the next ASN in AS path is AS52931 i.e AS1 for such prefixes is sitting below AS52931 which on other side is originating same prefix. Further AS52931 has upstream from AS28309 & AS16735. It seems like AS1 is coming only for routes which have AS16735 as upstream while for other case it’s direct announcement by AS52931. This gave me an interesting clue which was later verified by replies to my post on NANOG mailing list. Basically AS52931 - Netvip did not hijack AS1 intentionally but rather it was a case of mis-configured prepending. Netvip has two upstreams and was trying to prepend one of them (AS16735). In prepending networks simply repeat their own AS few times to increase AS-PATH which makes a route less preferred. #### Ideally what the needed is a AS path like this: XXX XXX XXX 28309 52931 i - Preferred via AS28309 transit XXX XXX XXX 16735 52931 52931 i - Not-preferred via AS16735 transit. Instead of putting their own ASN once in route map, they put “number 1” in the prepend which brought AS1 in global table for this prefix. I tried looking around and saw some funny prefixes from AS2, AS3, AS4 etc. Anurags-MacBook-Pro:Downloads anurag$ grep -w ‘2 i’ oix-full-snapshot-latest.dat|cut -f 3 -d ' ' |sort -u
128.4.0.0/16
177.129.161.0/24
31.192.64.0/19
Last prefix 31.192.64.0/19 does not belongs to AS2 (which is with UDEL-DCN - University of Delaware).
route-views>sh ip bgp 31.192.64.0/19 long
BGP table version is 4043628875, local router ID is 128.223.51.103
Status codes: s suppressed, d damped, h history, * valid, > best, i - internal,
r RIB-failure, S Stale
Origin codes: i - IGP, e - EGP, ? - incomplete
Network Next Hop Metric LocPrf Weight Path
* 31.192.64.0/19 208.74.64.40 0 19214 25973 6830 3.1190 i
* 194.85.102.33 0 3277 3267 9002 6830 3.1190 i
* 4.69.184.193 0 0 3356 6830 3.1190 i
* 154.11.98.225 0 0 852 174 12570 3.1190 2 i
* 207.172.6.20 0 0 6079 6830 3.1190 i
* 193.0.0.56 0 3333 6830 3.1190 i
* 154.11.11.113 0 0 852 174 12570 3.1190 2 i
* 69.31.111.244 0 0 4436 6830 3.1190 i
* 194.85.40.15 0 3267 9002 6830 3.1190 i
* 66.59.190.221 0 6539 577 6830 3.1190 i
* 209.124.176.223 0 101 101 3356 6830 3.1190 i
* 128.223.253.10 0 3582 3701 3356 6830 3.1190 i
* 207.172.6.1 0 0 6079 6830 3.1190 i
* 157.130.10.233 0 701 1299 6830 3.1190 i
* 134.222.87.1 0 286 6830 3.1190 i
* 66.185.128.48 547 0 1668 6830 3.1190 i
* 202.249.2.86 0 7500 2497 6830 3.1190 i
* 216.218.252.164 0 6939 6830 3.1190 i
* 207.46.32.34 0 8075 6830 3.1190 i
* 144.228.241.130 0 1239 3257 8928 12570 3.1190 2 i
* 114.31.199.1 0 0 4826 6939 6830 3.1190 i
* 208.51.134.254 1 0 3549 3356 6830 3.1190 i
* 129.250.0.11 384 0 2914 8928 12570 3.1190 2 i
* 217.75.96.60 0 0 16150 6830 3.1190 i
* 195.66.232.239 0 5459 6830 3.1190 i
*> 164.128.32.11 0 3303 6830 3.1190 i
* 202.232.0.2 0 2497 6830 3.1190 i
* 203.62.252.186 0 1221 4637 6830 3.1190 i
* 203.181.248.168 0 7660 2516 3257 8928 12570 3.1190 2 i
* 66.110.0.86 0 6453 3356 6830 3.1190 i
* 89.149.178.10 10 0 3257 8928 12570 3.1190 2 i
* 12.0.1.63 0 7018 1299 6830 3.1190 i
* 206.24.210.102 0 3561 3356 6830 3.1190 i
route-views>
This seems even more interesting because of doted ASN. :)
3.1190 means AS197798 as per dot conversion following RFC 5396. So we have AS197798 as well as AS2 sitting below AS197798 announcing that prefix - hence another misconfigured prepend case. (Nice tool by Sprint for dot.ASN conversion)
Regarding original case of AS1, I observed that yesterday at 18:44:13 RIPE NCC route collectors noticed change in BGP announcements changes for this.
One of route change noticed by Tinet AS3257 as route 3257 3549 16735 52931 1 was changed to 3257 3549 16735 52931. By 21:37:59 GMT, Netvip pulled off all routes from that mis-configured prepend.
With hope that you won’t hijack an ASN while prepending, time for me to end this blog post and get back to work!
#### Note:
I missed to thank Doug Madory from Renesys for his detailed explanation & Stephen Wilcox from IX Reach for giving clue about prepending in my original post.
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Cyberspace in the 21st Century: Part Two, Cyberspace and Twelve Monkeys
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# Cyberspace in the 21st Century: Part Two, Cyberspace and Twelve Monkeys
March 13, 2000 Page 4 of 4
Have We Got What it Takes to Produce Cyberspace?
Before we bake a cake, we need to check we’ve got the necessary ingredients:
1) A global network
3) A database engine and plenty of hard disk space
4) 3D scene modeler and a decent graphics card
5) A few software engineers
6) Time and money
We need a network – billions of computers each with tons of storage. That’s about what we’ve got, but latency could be a tad better. Well, maybe one day it’ll approach light-speed at around 100ms per transaction. We just have to design our system to do the best it can whatever the prevailing circumstances.
Do some searches on “Next Generation Internet” if you want the low down on the future of the Internet. Here’s one interesting find to give you a start: http://fox.rollins.edu/~tlairson/ecom/next.htm
What would improve the circumstances is if we evangelized against the wasteful use of the Internet, e.g. sending video across it. Thankfully there are companies aware of the merits of greater efficiencies in such things. Check out Obvious Technologies’ (www.obvioustech.com) approach for an example of a better way of distributing video – in effect passing it ‘by reference’.
Anyway, I think we’ve got everything we need to get started. Oh no, I almost forgot: Money! No one can do anything these days without money. You always have to keep an eye on the financial aspects, or so I’ve heard. However, these days for e-Ventures it seems venture capital is a broken fire hydrant. Perhaps I should finish up with a little waffle about the difficulties of making money in digital media…
Will Money Still Make the World Go Round?
Money was supposed to be a convenient means of exchanging labor and its products. Perhaps, as in the Star Trek universe, civilization will elevate itself to a position where money becomes superfluous and everyone puts their potential into society and gets as much out of it as they want. I think some might think the Internet will facilitate this, perhaps it will, but it’s pretty likely we’ll have to go via a transitionary phase where there is a form of cyber-cash. This’ll help migrate the old way of doing business into its more direct, electronic equivalent.
But, more and more, I think we’re seeing information become the key commodity. Whether it’s the right share to invest in, the most economic way to build a bridge, a DVD, or even the color of Sigourney Weaver’s toothbrush: information is in demand and people are willing to pay for it. Trouble is, they rarely have to. Information is getting easier to distribute and duplicate. It only takes one altruistic (some might use a less benevolent term) person to spill the beans on a web page and the whole world gets it for nothing.
It’s not a problem to ensure that communication is secure, from vendor to purchaser, but how do you prevent the purchaser from passing on that information for nothing and thus devaluing it?
The problem that the Internet now presents us with is that the vendor can no longer hope to maintain their monopoly over information once they’ve sold it. It used to be that purchase of mass produced information (newspapers, books, records) was a much better experience overall (price, quality, convenience) than illicit duplication was for the potential customer or pirate. But, now a digital copy is free, perfect, and more convenient.
Of course it’s unfair, but what can you do?
Even in the digital realm, software producers have still been clinging to the hope that transmission costs dissuade people from illegitimate downloading. Sure, even with digital technology, there are still always costs in transmission, and these are in proportion to the quantity of information – whether it’s downloading a file or throwing a DVD across the room. But, these transmission costs are rarely proportional to the production costs.
It seems one idea Microsoft’s trying out to address this problem, is to not let the software out of its factory in the first place. The user only has the presentation layer on their local machine, but must pay a subscription to obtain the use of the back-end on a secure server somewhere.. I suppose this is an understandable approach if as some pundits predict, there will soon be vastly more web browsers around than installations of Windows alone.
But software isn’t the only information based product around. What about works of art? Why should anyone produce a movie, album, or other easily duplicated work of art if only a single sale can be obtained?
Well, it’s difficult to swallow, but the answer has to be that the single sale must cover the cost, even in spite of the fact that the work is unlikely to have a resale value.
This means getting millions of punters to stump up cash in advance before the artist hands over their work. So say Sting produces a new album. First, he’ll keep it under strict security. He then releases a low grade recording that gives a hint of how good it is. The near perfect, 5.1 channel, digital encoding of it is then put up for a kind of reverse auction. The marketplace is invited to make limited pledges for it, e.g. up to $1, up to$5, up to $15, etc. Sting can then, at a favorable point in time, select which price point he wishes to sell it at, and then it is delivered to all those whose pledge covered that price. After this point it is a free for all and anyone may give it away or sell it on – including Sting who may still be able to sell the original recording at a premium price (given its packaging). The key thing in this scheme is that the product is not released until the artist feels they have arrived at the best return they can get, which may well be at least the production cost. Sometimes they may declare that the current pledges are insufficient and the work will be withheld until such time as the pledges increase. Conversely, the market may gradually reduce its offers for the work and the artist may take the best price while they can. Of course, you can only practically do this kind of deal where the market can be addressed as though it were a single unit – something greatly facilitated by the Internet. Similar types of deals can be done in advance of the work, for example, where the audience is presented with a movie script, and invited to stump up the funding necessary to produce the movie. I can see it now – “Star Wars: Episode IX – we need between$5 and $7 billion to complete this movie. The current optimum pledge only amounts to$3 billion. Please increase your pledge and/or encourage your friends, and remember that you get merchandising shares!”.
Yeah, go ahead and laugh. You can still continue with the old ways of doing things if you want. However, even with the public subscription type approach I’ve just outlined, broadcasters could still do the big deals you’re familiar with (perhaps as a cartel), but once transmission has occurred it should be a free for all. The thing is, it will be a free for all anyway, and you can’t really stop it. So copyright becomes redundant for art in digital form. This should apply to software too.
Anyway, these are just hints as to how the Internet is going to force a revolution in the marketplace. There will be other ways of working and buying and selling, but even with a totally derestricted market, I hope you can see that there are still mechanisms that will continue to support the development of films, music, and other works of digitally reproducible art. It’s not as bleak as the big companies would have you think. They’ll just have to forget about region coded DVDs and secure DVD-Audio…
Coming Next
In my next installment, I’ll be getting technical – very technical. So less of this futurism and let’s let the cat out of the bag: how on earth do we build cyberspace?
Jump into to the deep end of distributed systems programming with me next month and find out!
Until then, if you’re going to the Game Developer’s Conference be sure to check out Proksim Software (http://www.proksim.com) as one of the few companies sharing this road to enlightenment.
Crosbie Fitch is currently the Senior Systems Engineer at Pepper's Ghost Productions, which he joined in 1997 to develop a network games engine, reluctantly leaving special effects house Cinesite. In a deft twist of fate, PGP shortly decided that a radical change of direction away from games, toward an animated TV series was in order, and so Crosbie found himself writing plug-ins for Discreet Max - plus ça change... He can be reached at [email protected]
Page 4 of 4
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# Entropy of the beta-binomial compound distribution
I have a generative process as follows:
$$x \mid \alpha \sim \textsf{Beta}\left (\alpha,\beta \right) \\ y \mid x \sim \textsf{Bernoulli}(x).$$
How does one go about calculating the Entropy of this process? Do we consider the beta-binomial (with $n=1$) instead?
Not quite sure where to start on this one, suggestions are most welcome. Thx.
# Update 1
I believe now that the correct approach is to take the Beta-Binomial PMF (with $n=1$):
$$P(k \mid 1,\alpha ,\beta )= {1 \choose k}{\frac {{\mathrm {B}}(k+\alpha ,1-k+\beta )}{{\mathrm {B}}(\alpha ,\beta )}}\!$$ where $\text{B}(\cdot)$ is the Beta function. This PMF can also be written as:
$$P(k \mid 1,\alpha ,\beta )={\frac {\Gamma (1+1)}{\Gamma (k+1)\Gamma (1-k+1)}}{\frac {\Gamma (k+\alpha )\Gamma (1-k+\beta )}{\Gamma (1+\alpha +\beta )}}{\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}.$$
and substitute it into the Shannon entropy:
$${\displaystyle \mathrm {H} (X)=\sum _{i=1}^{n}{\mathrm {P} (x_{i})\,\mathrm {I} (x_{i})}=-\sum _{i=1}^{n}{\mathrm {P} (x_{i})\log _{b}\mathrm {P} (x_{i})}.}$$
# Update 2
Here is how far I have got. But first, lets remind ourselves of the model:
$$X\sim \operatorname {Bin} (n,p)$$ then $$P(X=k \mid p,n)=L(p|k)={n \choose k}p^{k}(1-p)^{n-k}$$ with $n=1$ we get $$P(X=k \mid p,1)=L(p \mid k)={1 \choose k}p^{k}(1-p)^{1-k}$$ so we are saying that $X$ is defined on a binary space $\{0,1 \}$ also $${\binom {n}{k}}={\frac {n!}{k!(n-k)!}} = /n=1 / = {\binom {1}{k}}{\frac {1!}{k!(1-k)!}}$$
Recall also that entropy is defined as:
$$\mathrm{H} (X) =\mathbb {E} [-\log(\mathrm {P} (X))]$$ Lets plug in our PMF expression (defined in update 1) for the Beta-Binomial: $$\mathrm{H} [k] = \mathbb{E} \left [ - \log{\left (\frac{{\binom{1}{k}}}{\mathrm{B}{\left (\alpha,\beta \right )}} \mathrm{B}{\left (\alpha + k,\beta - k + 1 \right )} \right )} \right]$$ which simplifies to \begin{align} \mathrm{H} [k] &= \mathbb{E} \left [ \log{\mathrm{B}{\left (\alpha,\beta \right )}} - \log \mathrm{B}{\left (\alpha + k,\beta - k + 1 \right )} - \log{{\binom{1}{k}}} \right ] \\ &= \mathbb{E}\left [\log{\mathrm{B}{\left (\alpha,\beta \right )}}\right ] - \mathbb{E} \left[\log \mathrm{B}{\left (\alpha + k,\beta - k + 1 \right )}\right ] - \mathbb{E} \left [\log{{\binom{1}{k}}} \right]. \end{align}
Which reduces to:
$$$$\mathrm{H} [k] = \log{\mathrm{B}{\left (\alpha,\beta \right )}} - \psi(\alpha+k) + \psi(\alpha + \beta + 1) - \mathbb{E} \left [\log{{\binom{1}{k}}} \right].$$$$
where $\psi(\cdot)$ is the digamma function. The problem is now the last expectation:
$$\mathbb{E} \left [\log{{\binom{1}{k}}} \right]$$
Not sure if this makes sen; how can one take the expectation of a binomial coefficient? I feel like I have gone wrong somewhere.
## 1 Answer
Do we consider the beta-binomial (with $n=1$) instead?
Yes. The beta-binomial distribution is exactly the compound distribution of a binomial r.v. where the probability of success is, itself, a beta deviate. A Bernoulli r.v. is exactly a binomial r.v. with $n=1$.
• Ah great! I must say I am somewhat confused though. Presumably, the entropy of the beta-binomial must be a standard result? But my googling is coming up with nothing. I found this (section 6): arxiv.org/pdf/1708.06394.pdf - but it is from 2017. Suggestions for good resources? – Astrid May 2 '18 at 9:39
• I suppose I could just substitute the compound PMF of the beta-binomial, into the Shannon entropy, and then simply calculate the entropy? Following this example: math.stackexchange.com/questions/394957/… and using the PMF from Wikipedia: en.wikipedia.org/wiki/Beta-binomial_distribution – Astrid May 2 '18 at 9:52
• I'm afraid I don't have any resources that specifically address how to compute the beta-bernoulli or beta-binomial entropy. It certainly appears that you're on the right track though: working forward from the definition of entropy. – Sycorax May 2 '18 at 14:56
• Not a problem, thanks for your help; your initial suggestion was very useful. – Astrid May 2 '18 at 14:58
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# Formatting Sandbox III: please test stuff here
Spoiler warning: Be aware that this page contains a lot of MathJax, so it will probably need quite a while to load completely.
Sandbox II has become as clogged as Sandbox I was (and is), especially for users with 10,000 reputation or above. So, here is a new new post...
Before you delete a post here, please reduce it to one line without MathJax.
Old formatting sandboxes:
• Excellent! Many thanks for this new Sandbox! – Ed V Jun 2 '20 at 15:39
• Why do we have to reduce it to one line without MathJax? – Micelle Jun 19 '20 at 14:53
• @Micelle deleted posts are viewable by users with >10,000 reputation, so it's just a courtesy thing. Long mathjax posts make the page load slowly. – orthocresol Jun 19 '20 at 14:55
• I totally forgot to thank you for making a new one. :D – Martin - マーチン Jul 9 '20 at 16:21
### Common misspellings
This table is indexed alphabetically in the first column. Please feel free to add to it.
Wrong Correct Remarks
Breddts Bredt's [rule] Julius Bredt.
carbonation carbocation Carbonation is what you do to make fizzy drinks.
Clemenson Clemmensen [reduction] Erik Christian Clemmensen.
die dye “Die” is only for “die Farbstoffe”.
Diel's Adler Diels–Alder [reaction] Otto Paul Hermann Diels; Kurt Alder.
flourine fluorine
Friedel–Craft's Friedel–Crafts The guy's name was James Crafts, not James Craft.
Gibb's Gibbs [energy] Josiah Willard Gibbs.
Henderson–Hasselbach Henderson–Hasselbalch [equation] Lawrence Joseph Henderson; Karl Albert Hasselbalch.
iconic ionic [bond]
Nerst Nernst [equation] Walther Hermann Nernst.
phosphorus acid phosphorous acid This refers to the acid $$\ce{H3PO3}$$. See also previous entry.
pie pi/π [bond] Pie bonding is for SeasonedAdvice.SE.
stigma sigma/σ [bond] Stigma bonding is for Christianity.SE or MedicalSciences.SE.
Vander-Wal's van der Waals [force] Johannes Diderik van der Waals.
Vant-hoff van 't Hoff [equation] Jacobus Henricus van 't Hoff.
This is a simple test of electrolysis of a baking soda solution.
• Unfortunately, the formula doesn't account for bounty system. However, the general trend "less Q&As — more rep" is indeed a sign of high-quality posts. – andselisk Dec 15 '20 at 10:16
• @andselisk Yeah, it is very crude and broad brush. Just a late night thought before I called it a day. – Ed V Dec 15 '20 at 13:35
• You might be interested in: data.stackexchange.com/chemistry/revision/1349860/1660907/… This simply takes the average score of all posts of a user. If the results don't show up hit "Run Query" near the bottom. A lot of the users there are no longer active... I find that upvotes were more generously given out in the early days of the site. (Your score is 4.12.) [PS I don't know anything about databases, I just copied this query off somebody else!] – orthocresol Dec 19 '20 at 16:38
• @orthocresol Wow, many thanks for the link! Very cool and helpful: I definitely see I have lots of room for improvement! And it helps to know that upvotes were more generously given in the early days, but that is very likely as it should be. So I will delete my crude late night idea. – Ed V Dec 19 '20 at 17:00
• No problem! I personally wouldn't consider it room for improvement, actually: I'm sure you noticed it already, but upvotes are pretty random and often the Q's/A's that get lots of upvotes are ones that are more accessible to the lay person (partly due to the Hot Network Question feature of SE, where you can "jump" to popular questions on other sites and vote). I've previously dumped some of my thoughts on the matter here. – orthocresol Dec 19 '20 at 17:08
### Somebody complained that the reaction I asked about doesn't exist. Why is this a problem?
Chemistry is an experimental science first and foremost, and this is especially true of synthetic chemistry, whether organic or inorganic.
What this means is that: we don't come up with theories from first principles, then use them to predict reactions. [We're getting better at doing this using quantum mechanics, but it's still very early days.] Instead, we find out that a reaction happens, and then we work backwards to come up with a model that explains it.
The ultimate source of "truth" in chemistry is not defined by our theories, but rather by our experimental observations. The theories only exist because they can explain experimental evidence.
[Incidentally, that's why there are so many exceptions to the theories. Many of them have a limited range of validity, in that they can only explain a certain subset of the experimental observations we have. A simple example is the octet rule. It works for quite a lot of organic molecules, but can completely fall apart in other contexts.]
So, asking "why does this reaction occur?" is only sensible if that reaction has actually occurred!
If nobody has done it before in real life, then we have no way of knowing whether it would actually occur. And secondly, if it doesn't actually occur and we come up with a theory to explain it, then there is no guarantee that that theory would be correct.
$$h=\frac{I_{\text{const}}\cdot R_{\text{ref}}(1+\alpha\Delta T)}{A_{\text{filament}}(T-T_{\text{flow}})}$$
$\require{begingroup}\begingroup$ $\def\pi{\neq 3.14}$
$$\require{begingroup}\begingroup$$ $$\def\pi{\neq 3.14}$$
The ratio of a circle's circumference to its diameter is $\pi$.
The ratio of a circle's circumference to its diameter is $$\pi$$.
$\endgroup$
$$\endgroup$$
• $\pi here doesn't leak through:$\pi\$ – orthocresol Jan 30 at 15:15
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# Contrast and luminance adaptation alter neuronal coding and perception of stimulus orientation
## Abstract
Sensory systems face a barrage of stimulation that continually changes along multiple dimensions. These simultaneous changes create a formidable problem for the nervous system, as neurons must dynamically encode each stimulus dimension, despite changes in other dimensions. Here, we measured how neurons in visual cortex encode orientation following changes in luminance and contrast, which are critical for visual processing, but nuisance variables in the context of orientation coding. Using information theoretic analysis and population decoding approaches, we find that orientation discriminability is luminance and contrast dependent, changing over time due to firing rate adaptation. We also show that orientation discrimination in human observers changes during adaptation, in a manner consistent with the neuronal data. Our results suggest that adaptation does not maintain information rates per se, but instead acts to keep sensory systems operating within the limited dynamic range afforded by spiking activity, despite a wide range of possible inputs.
## Introduction
Our sensory systems receive a barrage of stimulation that continually changes along multiple dimensions and on multiple timescales. Even when looking around a simple scene, the receptive field of a single neuron in the visual system is stimulated by a dynamic sequence of spatial patterns, luminances, contrasts and colours. Considering just contrast and orientation, the two dimensions that profoundly affect the firing rates and response dynamics of neurons in the early visual system1,2, the range of possible stimulus combinations vastly exceeds the limited dynamic range of any neuron’s spiking output. One way in which individual neurons can better represent the current stimulus is to continuously update their limited response dynamics to account for the recent stimulus history. However, the mechanisms underlying this are unclear3. Further, it is unclear how changes along one stimulus dimension affect the neural coding properties of other dimensions, when individual neurons are continuously stimulated by a multidimensional feature space.
Adaptive mechanisms are prominent in all species and sensory modalities studied3. For example, throughout the visual system, luminance and contrast-gain control help to maintain perceptual sensitivity under different lighting environments by changing the temporal dynamics and gain of neuronal responses4,5. These mechanisms dynamically shift the operating point of neurons in a manner that maximises information transmission6,7 or feature detection and processing8,9.
Previous studies have focused on the effects of adaptation to a single stimulus dimension and how they affect the neural coding of that particular dimension. For example, we have examined how the exposure to a single motion direction affects the encoding of other motion directions, at the level of both single neurons and populations10,11. Others have examined how the exposure to changing distributions of stimulus statistics, such as stimulus speeds, luminances or sound intensities, affects the encoding of those specific stimulus dimensions12,13,14. In natural vision, neurons encode rich stimuli in a multidimensional feature space; yet it remains elusive how neurons dynamically encode each input dimension if stimuli are also changing on other dimensions. Put simply, how does the adaptation in one dimension affect the coding in another? Given the frequent variations in firing rates across the neuronal population due to changes in single dimensions, such as the mean luminance or contrast15,16,17, an important question is how neurons can stably code information about the barrage of multidimensional sensory information in dynamic environments.
To address this question, we recorded the extracellular neuronal activity in the primary visual cortex (V1) of marmoset monkeys viewing a movie of sinusoidal gratings that changed the orientation every 16.7 ms, with concurrent changes in mean luminance or contrast every 5 s. This experimental design allowed us to investigate how the adaptation to one dimension (luminance or contrast) affects the neural coding of another dimension (orientation). Our study is the first to reveal how the orientation coding in V1 neurons is impacted by adaptation to presumably orthogonal stimulus dimensions. Although the encoding of luminance and contrast are critical functions of the visual system, here we are interested specifically in the encoding of orientation during adaptation; therefore, we treat luminance and contrast as nuisance variables in the statistical sense.
Using information-theoretic analysis and population-decoding approaches, we found that the ability of single neurons and neural populations to discriminate orientation is highly dependent on luminance and contrast. Our reverse-correlation analysis also showed that the temporal kernel of single-neuron orientation tuning changes during adaptation. More importantly, we found that orientation discriminability changes during adaptation periods that follow switches in luminance and contrast conditions in a manner that closely reflects human perceptual thresholds. Interestingly, we found that the gain, but not the tuning bandwidth, of orientation-tuning curves for single neurons is multiplicatively scaled throughout the course of luminance and contrast adaptation. Our results suggest that the adaptation does not maintain information rates per se, but instead acts to keep sensory systems operating within the limited dynamic range afforded by spiking activity, despite a wide range of possible inputs.
## Results
### Luminance and contrast adaptation affect tuning not timing
We recorded the extracellular neuronal activity in V1 of marmoset monkeys under sufentanil/N2O anaesthesia, in response to a movie of rapidly presented oriented gratings, the properties of which continually changed on rapid and slow timescales. We designed this switching paradigm to systematically study how adaptation to variations in one stimulus dimension affects the coding of other stimulus dimensions. Specifically, rapid variations in stimulus orientation occurred every 16.7 ms, while changes in mean luminance and contrast occurred every 5 s (Fig. 1a). This allowed us to examine how the neural information about orientation depended on steady-state luminance and contrast and on the time since specific switches in luminance and contrast have occurred.
In order to quantify the orientation tuning in a fine temporal detail, we used a reverse-correlation approach to estimate the probability at which each possible orientation occurred at all times preceding an action potential18. Applying orientation- reverse correlation in the context of adaptation to stimulus luminance and contrast allowed us to examine how adaptation impacts tuning over time. First, we compared the dynamics of orientation selectivity measured early (0–1.6 s) and late (3.4–5 s) after a luminance or contrast switch (Fig. 1a, b) and subsequently compared the dynamics of orientation selectivity between each of the four luminance–contrast conditions.
The strength of tuning, quantified as the maximum modulation in the linear kernel (Fig. 1c), changed over time and was significantly higher during the late phase of adaptation following a contrast increment, regardless of the luminance (Fig. 1d, p < 10-7, signed-rank test). These changes in modulation during adaptation are consistent with the previous observations in the retina and lateral geniculate nucleus (LGN)19,20. Although the strength of orientation tuning changed with adaptation, we found no evidence for systematic changes in the width or the peak time of the temporal profile between the early and late phases after a contrast increment (Fig. 1e, f; see Supplementary Fig. 1 for other conditions). Furthermore, even when timing differences were significant, they were only of the order of 1–2 ms (Fig. 1e and Supplementary Fig. 1).
Comparing the orientation tuning across different luminance and contrast levels, rather than during adaptation to a fixed luminance and contrast, showed that maximum modulation, peak time and temporal width were consistently luminance and contrast dependent (Supplementary Fig. 2). Notably, low-luminance and high-contrast stimuli were associated with the largest modulation, shortest time to peak, and narrowest temporal widths, in agreement with previous reports21.
These results show that the temporal properties of V1 neurons in response to stimulus orientation are strongly affected by luminance and contrast. Critically, while the peak time and temporal width appear to rapidly or instantly compensate for changes in luminance or contrast, the magnitude of the linear kernel changes more slowly during adaptation to a single condition. We asked how these properties affect the coding of orientation across time, given that luminance and contrast change on multiple timescales.
### Adaptation alters the coding efficiency of stimulus orientation
Changing the statistics of a specific stimulus dimension (such as luminance, intensity or speed) affects a neuron’s coding efficiency of the same stimulus dimension7,12,22. However, it remains unclear if adaptation to one dimension in a multidimensional stimulus space affects the coding of other dimensions. To determine whether and how contrast and luminance adaptation affect the coding of orientation in V1 neurons, we calculated the mutual information (MI) between each neuron’s spike count and stimulus orientation and examined the following: (1) how MI depended on luminance and contrast (Fig. 2a, b) and (2) how MI changed during the course of adaptation to a single luminance and contrast (Fig. 2c–f).
Initially, we examined MI during a late or steady-state time window, from 3.4 to 5 s after the stimulus switch. Information was higher in high-contrast conditions (Fig. 2a), regardless of the mean luminance (low luminance, p < 10−30; high luminance, p < 10−22; t test). Surprisingly, information was also higher in low-luminance conditions (Fig. 2b; high contrast, p < 10−31; low contrast, p < 10−23; t test). Although a high luminance is considered to be a stronger input to the retina and LGN than a low luminance, this result is consistent with the masking effect of high-luminance transients on neural responses and perception17,23,24.
Given the significant information differences present between the different steady-state luminance and contrast conditions, we wondered how long it takes for these changes to manifest. To characterise the effect of adaptation on the coding of stimulus orientation, we estimated the information conveyed by individual neurons about the stimulus orientation during early (0–1.6 s) and late (3.4–5 s) phases following a luminance or contrast switch. Following a contrast increment, regardless of the luminance, MI was significantly higher during the early than the late phase (Fig. 2d; low luminance, p < 0.001; high luminance, p < 0.005; signed-rank test). Similarly, MI was initially higher following a luminance decrement, for both contrasts (Fig. 2e; low contrast, p < 0.01; high contrast, p < 10−5; signed-rank test). However, the opposite trends were observed following contrast decrements (Fig. 2f; low luminance, p < 10−8; high luminance, p < 10−7; signed-rank test) and luminance increments (Fig. 2c; high contrast, p < 10−8; low contrast, p < 0.004; signed-rank test). Note that we found the same qualitative trends when we used other time windows ranging from 0.2 to 2.4 s in duration to define our early and late periods. This suggests that luminance- and contrast-dependent changes in information about orientation are enacted on very short timescales.
### Firing-rate adaptation after luminance and contrast switches
Given that the mean luminance and contrast affect firing rates, the above analysis motivated us to determine whether stimulus-induced changes in MI could be decoupled from changes in spiking rate. We averaged the spiking activity over dozens of repetitions of every unique luminance–contrast switch regardless of the changes across other stimulus dimensions (i.e., orientation, phase, and spatial frequency). This provided us with an estimate of the firing-rate variation that is only driven by switches in luminance and contrast. Effectively, we are averaging each neuron’s orientation-selectivity profile at each time point relative to the luminance or contrast switch.
Luminance and contrast switches induced substantial changes in the firing rate of neurons, usually comprising an initial rapid increase or decrease in rate, followed by relaxation to an intermediate plateau following an exponential decay (Fig. 3 and Supplementary Fig. 3). The time constant of these decays depended on luminance and contrast; for example, the increase in firing rate following a contrast switch was significantly higher when gratings had a low luminance (Fig. 3b, dark-blue trace) rather than high luminance (Fig. 3b, light-blue trace; p < 0.01, signed-rank test). Time constants were also significantly longer for contrast decrements than increments (compare Fig. 3b with Fig. 3d; p < 0.01, signed-rank test; also see Supplementary Fig. 3). This asymmetry in the time course of firing-rate changes is consistent with the adaptation to variations in the statistics of auditory12, visual19, and tactile22 stimuli in sensory areas. It is also apparent that a higher mean luminance, during both contrast increments and decrements, evoked weaker activity than a low luminance (compare dark-blue and light-blue traces in Fig. 3b, d).
Following a mean luminance increment, firing rates displayed a transient reduction followed by an exponential recovery lasting several seconds (Fig. 3a, Supplementary Fig. 3). Unusually, when the gratings had a low contrast (Fig. 3a, pink trace), there was a biphasic response in 30% of neurons, with a small and rapid change in mean firing rate immediately after a luminance switch. This biphasic response is similar to the impulse response of the visual neurons to a luminance switch14,17.
Surprisingly, the firing rate rapidly increased following a switch from high to low mean luminance, regardless of the contrast (Fig. 3c). The time constant of increase and the subsequent exponential decay were in a similar range to contrast increment (Fig. 3b and Supplementary Fig. 3). We also observed a similar asymmetry in firing-rate adaptation during luminance increments and decrements to that following contrast switches (Fig. 3b), consistent with previously reported data7,12,14,19,22.
### Luminance and contrast switches reduce neural variability
Neural responses to multiple repetitions of the same stimulus are variable, and the neural coding efficacy depends on this variability. While recent studies have demonstrated that response variability that is correlated between neurons can surprisingly enhance neuronal coding11,25, a higher variability in the responses of individual neurons, often quantified using the Fano factor (FF), can only impair encoding. We thus calculated the FF of neural responses within a sliding 50 ms time window from 500 ms before to 500 ms after luminance or contrast switches.
As with our previous analysis of firing rate, this approach ignores changes in the other dimensions of the stimulus. Despite the noisy measurement of FF using this method (because the underlying rate changes throughout our measurement window), we observed a consistent pattern of FF variations across different conditions (Figs. 3e–h). Overall, FF decreased immediately after any switch in luminance and contrast, consistent with the previous reports that changes in stimulation conditions quench neural variability26. The FF decreased and then rapidly recovered in all cases after a luminance or contrast change, whereas the firing rate changed more slowly and could either decrease or increase (Fig. 3 and Supplementary Fig. 3). Given these observations, if population decoding is primarily affected by trial-to-trial variability, it should always improve immediately after a change in the stimulus.
### Coding of stimulus orientation by neural populations
We found that the information conveyed by individual neurons about stimulus orientation is strongly affected during adaptation. It is, however, unclear how the coding of orientation by a neural population is affected by changes over time in firing rates and trial-to-trial variability at the level of individual neurons, and whether the variability between neurons can be overcome at the population level. Moreover, neurons vary in their preferences, temporal dynamics, and adaptation properties. Therefore, we used a population-decoding approach to quantify orientation discriminability, asking how the decoding accuracy (1) is affected by different luminance and contrast conditions and (2) changes during the course of adaptation to a single luminance and contrast.
The results revealed a clear difference in discriminability between the luminance–contrast conditions and during the course of adaptation. Overall, higher contrasts led to a better orientation discriminability while higher luminance levels led to a poorer discriminability (Fig. 3i; e.g., compare dark- and light-blue data points, connected with a black line, for low vs high luminance). Moreover, the adaptation following a contrast increment decreased the orientation discriminability of neural populations while the adaptation following a contrast decrement led to higher discriminability (Fig. 3i; p < 0.0001, signed-rank test; e.g., filled blue data points associated with contrast increments fall below the line of unity). This was the opposite for luminance switches, as adaptation following luminance increments and decrements increased and decreased the discriminability, respectively (Fig. 3i; p < 0.0001, signed-rank test; e.g., filled brown and pink data points associated with luminance increments fall above the line of unity). Similar decoding results were evident when we changed many parameters of the decoders, including different time windows for defining the early and late phases, the number of neurons in the decoder, the type of decoder, and the width of the readout, or spike-counting, window (Supplementary Figs. 4 and 5).
To compare the size of the changes in orientation coding at the single-neuron and population levels, we calculated the MI ratio (early relative to late) for each stimulus switch and compared it to the corresponding ratio of decoding performance (early relative to late). Across the eightswitches, these ratios were strongly correlated (r = 0.87, p = 0.004), with a slope of 0.4. As much as these ratios can be compared, this suggests that the adaptive changes over time are of similar magnitudes at the single-neuron and population levels. Further, coding of stimulus orientation by neural populations is substantially affected by adaptation to luminance and contrast, and this adaptive coding tracks the direction of changes in spiking rate, not response variability.
### Coding of stimulus orientation by individual neurons
Previous studies of adaptation to stimulus statistics have shown that even though firing rates markedly change in association with switches in stimulus variance, the information rate of individual neurons (measured as bits per spike) is almost unaffected7,22. Our above analysis demonstrated that decoding of orientation by the neuronal population is substantially affected by luminance and contrast adaptation. To clarify this apparent conflict, we estimated the information conveyed by individual neurons during adaptation with a finer temporal resolution. Figure 4a illustrates, for a sample neuron, that the MI between stimulus orientation and spiking activity is also affected by adaptation. Here, we calculated the MI in six equal time windows that spanned the 5-s adaptation period following each of the eight stimulus switches (see also Supplementary Fig. 4). There are large, significant correlations between firing rate and MI for both individual stimulus conditions and when all the stimulus conditions are considered collectively. These correlations are also evident across the population of neurons, and the average correlations are significantly greater than zero in all cases (Fig. 4be, p < 0.01, t test). MI also changed during adaptation when we normalised information by spike count (bits per spike) rather than simply considering information (bits).
### Luminance and contrast switches modulate perception
The changes in decoding performance and MI observed after switches in luminance and contrast suggest that orientation discrimination thresholds should be improved following contrast increments and luminance decrements. To assess this, we conducted human psychophysical experiments in which observers reported the relative orientation of two gratings, each presented for 200 ms, separated in time by a noise mask (Fig. 5a). On each trial, discrimination judgments were performed either early (0.2–1.2 s) or late (5–6 s) after a switch in luminance or contrast.
Orientation discrimination was enhanced immediately after a contrast increment, reflected in significantly higher discrimination thresholds during the late vs early phase (Fig. 5b, for a single subject, p < 0.001, bootstrap test and Fig. 5f, for all subjects, p < 0.005, signed-rank test, n = 14 observers). For 12 out of 14 observers, these increases in discrimination thresholds were individually significant (p < 0.001, bootstrap test), meaning that the majority of subjects performed significantly better in orientation discrimination during the early, compared to the late, phase. Similarly, discrimination accuracy was higher immediately after a luminance decrement, with discrimination thresholds significantly higher in the late vs early phase (Fig. 5e, p < 0.01, signed-rank test, n = 7 observers). In this experiment, all observers individually showed a significantly better discriminability in the early phase than in the late phase (Fig. 5e, p < 0.001, bootstrap test). On the other hand, we found no systematic changes in discrimination thresholds during the late vs early phase of luminance increments (Fig. 5c, for a single subject and Fig. 5d, for all subjects, p > 0.05, signed-rank test, n = 7 observers) and contrast decrements (Fig. 5g, p > 0.05, signed-rank test, n = 8 observers).
These results are broadly consistent with our physiological data, in which the neural population-decoding accuracy was significantly higher during the early vs the late phase following a contrast increment and luminance decrement (Fig. 3). Despite this, an important methodological difference between the physiological and psychophysical studies limits the straight-forward comparison of their results. In our physiological study, we used phenylephrine and atropine eye drops to dilate the pupils. This inactivates the pupillary light reflex, which modulates the amount of light reaching the retina.
To determine if differences in discrimination thresholds are related to pupil-size modulations, we monitored the pupil size in 15 observers (Supplementary Fig. 6). The pupil size was briefly, and only slightly, changed following contrast switches, but showed immediate and robust dilation or constriction following luminance switches. Contrast increments are therefore associated with consistent changes in perceptual and neuronal discrimination performances, in the absence of associated pupillary changes. However, while luminance decrements were also associated with consistent changes in perceptual and neuronal discrimination, they were only accompanied by pupillary dilation in the human observers. This means that although the observed changes in neural coding can only arise as a result of cascading neural processes in the visual hierarchy (because the pupils were permanently dilated in the marmosets), the changes in human perceptual performance could reflect these same neural processes or simply the effects of pupillary dilation.
### Additive and multiplicative modulation of orientation tuning
What causes the neurons to change their coding ability during adaptation? Coding properties of neurons can be affected by factors such as trial-to-trial variability and multiplicative and additive modulations in neural response populations. We earlier showed that trial-to-trial variability deceased immediately after luminance and contrast switches for all conditions (Fig. 3e–h), but the decoding accuracy did not always follow the same trend (Fig. 3i, Supplementary Figs. 4 and 5). Here, we asked whether other factors such as multiplicative and additive gain modulations could lead to the differential stimulus coding of orientation during adaptation. Note that multiplicative modulations change the response amplitude of single neurons without affecting their tuning selectivity15,27. However, such modulations of tuning functions become important when decoding the population activity, in part, because scaling up neural spiking rates disproportionately scales up the trial-to-trial variability, reflecting non-Poisson statistics28.
We found that neuronal tuning functions varied greatly between early and late phases of adaptation. The sample neuron depicted in Fig. 6a–d highlights the range of effects commonly observed during different adaptation conditions. For example, the gain substantially increased after adaptation to a high luminance (Fig. 6a), while adaptation to a low luminance led to a compound effect of multiplicative and additive modulations (Fig. 6b). We calculated multiplicative and additive modulations for the 50 neurons that were highly selective for all stimulus conditions. Working with this reduced dataset is necessary to allow direct within-condition comparisons. In this population, most neurons had strong gain changes during adaptation while the additive effect was minimal (Fig. 6e–h and Supplementary Fig. 7). For example, after adaptation to a luminance increment, regardless of the contrast, the gain of almost all neurons decreased, with no significant change in the additive modulation or offset (Fig. 6e, high contrast, p < 10−17; low contrast, p < 10−8, t test). Similar changes were evident after a contrast decrement, regardless of luminance (Fig. 6h, high luminance, p < 10−11; low luminance, p < 10−14, t test). Note that gain modulation is the gain change between the early and late phases of adaptation.
We observed small gain enhancements during adaptation to contrast increments (Fig. 6g; low luminance, p < 0.05; high luminance, p > 0.05, t test) and luminance decrements (Fig. 6f; high contrast, p < 0.05; low contrast, p > 0.05, t test). We did not find any significant additive modulation across the population during adaptation to these conditions (except in high-contrast, low-luminance conditions; Fig. 6f, p < 0.01, t test). To further examine these effects at the level of individual neurons, we compared four models for predicting the tuning curves of each neuron at early and late phases. Each model comprised two von Mises functions (one for the early and the other for the late phase of the response) that were either identical, independent, matched in gain, but with independent offsets, or matched in offsets but with independent gain. For the cells that showed significant modulation in orientation tuning between the two response phases (i.e. model 1 was the poorest fit), we found that 17% were best explained with a combination of additive and gain modulations, 63% were best explained with only gain modulation, and 20% were best explained with only additive modulations. This suggests that it is primarily gain modulation, and to a lesser extent additive modulation, that explains the difference in orientation discriminability and information rate during adaptation.
We also calculated the gain and offset modulation with a finer temporal resolution during adaptation. To this end, we calculated the changes in the population-tuning curve across six consecutive non-overlapping time windows during adaptation following each type of switch (Supplementary Fig. 7). Our analysis showed that the gain and offset systematically change during adaptation (Supplementary Fig. 7). In particular, we found that the modulations during adaptation are not purely multiplicative or adaptive; however, the multiplicative modulations are more pronounced across single neurons.
## Discussion
In this study, we asked how principles that have been developed from studies of adaptation to a single stimulus dimension can be used as the foundation to track neural response selectivity under a more complex stimulation paradigm. In particular, how does adaptation to one stimulus dimension affect the coding of another dimension? To address this question, we investigated how the adaptation to mean luminance and contrast reshapes the encoding properties of V1 neurons and how this change in encoding properties alters the information about stimulus orientation that is available to downstream decoding neurons. We designed a switching stimulus paradigm in which stimulus dimensions varied on fast and slow timescales, a situation that happens in our everyday life as we step from bright sunlight into an indoor office, or simply change our point of gaze from glaring sunshine to the adjacent shade. We showed that adaptation to slowly varying stimulus features can alter the coding properties of individual neurons and neural populations for a different rapidly varying stimulus dimension. In particular, we showed that following changes in luminance and contrast, the ability of neurons to code stimulus orientation dynamically changes, and these changes predict a novel form of perceptual adaptation in human orientation-discrimination thresholds. The dynamic coding in neural responses can be explained by the adaptive gain rescaling of individual neurons in the population.
Rapid adaptive stimulus-dependent changes in the filtering properties of neurons have been demonstrated in the visual9,19,20, auditory12,29 and somatosensory pathways3,22. Adaptation to changes in stimulus luminance and contrast also affects both linear and nonlinear filtering stages of a cascade model of retina7,30,31 and LGN20,32. For example, adaptation to low-contrast stimuli can lead to faster temporal dynamics of the linear filter and a lower gain of the nonlinear filter. These studies examined how changing the statistics of one dimension affected the encoding of the same dimension. Our stimulus design provided us with the advantage that we could study the time frame over which such adaptive changes occur, as we had enough measurement time and stimulus repetitions to be able to use reverse-correlation analysis to robustly capture the aspects of dynamic feature selectivity of neurons.
Our analysis showed that the temporal dynamics of neurons’ linear responses, measured as peak time and temporal width, was determined by mean luminance and contrast, but these changes manifested extremely rapidly and remained largely unaffected during the course of adaptation to a constant luminance or contrast. This happened despite the fact that the firing rate of neurons substantially varied during adaptation. Further analysis of linear-response kernels, however, showed significant changes in the amplitude of kernels (maximum modulation) during different phases of adaptation to luminance and contrast. For example, our results revealed that an increment in contrast can trigger a significant change in the shape of the linear kernel during the early phase of adaptation compared to the late phase, suggesting a gradual increase in the amplitude of the kernel during adaptation to a high contrast. However, following a contrast decrement, the kernel amplitude did not significantly change, likely because of the fact that changes in response amplitude after a contrast decrement are small and have a very slow timecourse of recovery. Overall, larger kernel amplitudes indicate that, given the occurrence of a spike, the stimulus orientation is known with a higher probability. Moreover, the amplitude modulation of the linear kernel was largely dependent on adaptation condition. The observed changes in the kernel amplitude during adaptation to contrast decrement and increment are consistent with those in the other studies in the retina19 and auditory cortex29. However, changes in other temporal aspects of the linear filter, such as peak time and temporal width, have also been reported in auditory29 and early visual pathways20, which we did not observe in our study. We think that the difference in stimulus type might underlie this discrepancy, as these other studies have used white noise or natural stimuli, which are very different to the movie of rapidly changing gratings that we presented. For example, there are strong spatial and temporal correlations in the neighbouring pixels in natural movies. In summary, our results suggest that neurons do not employ a simple static-coding strategy but dynamically change their receptive-field profile to adjust to constant and rapid changes in the statistics of input stimuli. This adds to the previous studies showing that neural responses are poorly predicted using a static receptive field9.
Generally, a switch in stimulus variance or mean leads to a transient change in the firing rate of neurons followed by an exponential14,33 or power-law decay to steady state7,34. The time constants of firing-rate adaptation in our study are in a similar range to those in the previous studies of contrast and luminance adaptation in the visual pathway. These time constants are dependent on factors such as the period of the stimulus-switching paradigm7,34 (but see ref. 12) and the noise level of the input signal14, suggesting that neural responses are dependent on the history of stimulus variations. In our study, we did not explore the effect of different switching periods and noise level on the time course of adaptation and feature selectivity, but we observed that neurons did not have a single time constant in all adaptation conditions, suggesting that recent changes in stimulus statistics affect neural response dynamics in V1.
Firing-rate adaptation does not follow a symmetric dynamics during upward and downward changes in stimulus statistics. Such an asymmetry has been observed in many sensory areas, suggesting this as a general property for adaptation7,12,14,19,29,33,34,35. We observed a similar asymmetric dynamics in adaptation to contrast increment and decrement, with a significantly longer time constant of adaptation during the switch to high than low contrasts. The trend was, however, different during the mean luminance switch, in which adaptation to a low mean luminance was faster than high luminance, suggesting that it takes longer for the neurons to recover after a switch to a high mean-luminance stimulus. This long-lasting suppression does not agree with the previous studies in the retina. One possible reason for this discrepancy may be the differences in response and anatomical properties between retina and V1 as a dense network of different excitatory and inhibitory connections. As one study has shown, while the responses of LGN neurons are elevated after a luminance transient, the responses of V1 neurons are significantly suppressed, and the selectivity of neurons to ongoing stimulus orientation is delayed17. Such a mechanism has been attributed to changes in cortical inhibition17. In this context, the responses of our recorded neurons to luminance switch are consistent with the literature. At the perceptual level, a sudden change in luminance also supresses the detectability and discriminability of visual targets in human observers24.
Previous studies of adaptation using switching-stimulus paradigms have shown that the information content of single neurons (bits per spike) about the contrast (variance) and luminance (mean) of an input signal is largely unaffected around the time of a switch in variance7,22. Here, we have shown that the MI between spiking activity and stimulus orientation is higher when luminance is low (or contrast is high), consistent with the masking effect of luminance increments reported in electrophysiological17,23,36 and perceptual studies37. The aim of our study was to explore the effect of adaptation to luminance and contrast on the coding of stimulus orientation. From the perspective of encoding stimulus orientation, changes in luminance and contrast are problematic, making them nuisance variables. Despite this, neurons in most visual areas actually carry information about luminance and contrast, and this information can be reliably decoded. While not the focus of our study, this is evident in our data as differences in the steady-state firing rate after adaptation to luminance and contrast (e.g. the mean firing rates for low and high contrasts are different and thus decodable).
Calculating the information content of individual neurons during the course of adaptation, however, showed that MI was significantly affected during adaptation to a single-luminance and -contrast condition. For example, the information conveyed by neurons about stimulus orientation was higher during the early than the late phase of an upward switch in contrast, suggesting that adaptation to contrast increments reduced the amount of information. Conversely, information about orientation increased during adaptation to a contrast decrement. Although these changes in information might seem contradictory to the findings of the studies mentioned above, our result is comparable to some results reported from V1 and middle temporal area (MT) regarding the large contribution of onset transient to sensory discrimination38,39. Here, a switch to a new luminance–contrast can be considered as the onset of a new stimulation regime.
While single neuron-level mechanisms can account for some adaptive properties described in our study, cortical neurons operate in a highly interconnected neural circuitry. Therefore, the adaptive properties of a single neuron can induce substantial changes in information processing when interpreted in the context of neural population activity10,11,40. Moreover, our information-theoretic analysis mostly relies on orientation-selective neurons while non-selective neurons can affect the population code11,41 and, likely, perception. A simple way to address this issue is using population decoding to investigate if the observed effects of adaptation at the single-neuron level are also evident at the population, or circuit, level. Our analysis showed that the decoding accuracy is significantly luminance and contrast dependent and strongly affected by adaptation. Adaptation to a high contrast (following a contrast increment) and a low luminance (following a luminance decrement) decreases the discriminability of stimulus orientation across neural populations, while adaptation to a high luminance and a low contrast increases the discriminability. These different changes in discriminability after adaptation do not perfectly agree with studies suggesting that adaptation always improves discriminability42,43, as we found that discriminability can largely be stimulus or task dependent (e.g., compare the decoding accuracy between upward and downward contrast switches).
We also found changes in discrimination thresholds between the early and late phases in our psychophysical measurements for two switching conditions, suggesting that the significant modulations in firing rate and neural discriminability after the switch have a perceptual correlate in human subjects. While we cannot rule out the role of pupil-size variations in the perceptual luminance-adaptation effects we observed, two factors make it likely that the neural adaptation observed following both contrast increments and luminance decrements can account for the changes in human psychophysical performance. First, we observed a significant difference in perceptual thresholds between the early and late phases following contrast increments, even though the pupil size remained mostly unchanged. Second, following a luminance increment, pupil size changed substantially over time, but we observed a relatively little change in discrimination thresholds.
Collectively, we have shown that orientation discriminability is luminance and contrast dependent, changing over time due to firing-rate adaptation. This is accompanied by changes in the information available about orientation from neural spiking, attributable to adaptive gain modulation. Multiple cellular and network mechanisms may account for an adaptive gain control in the cortex. At the level of single neurons, adaptation largely relies on cellular mechanisms (e.g., Na+-activated and Ca2+-activated K+ currents)44 and synaptic depression45. Adaptive mechanisms can also be derived from network dynamics and recurrent inhibition, which can produce neuronal response dynamics that vary over a range of timescales3,46. In a highly interconnected network such as V1, lateral inhibition or modulatory feedback can also account for adaptive gain control47,48,49,50. Future studies should identify the distinct contributions of inhibitory and excitatory neurons to the changing feature selectivity that occurs during adaptation in order to link single-neuron and circuit-level mechanisms to perceptual outcomes.
## Methods
### Surgery and animal preparation
We recorded the extracellular activity from V1 neurons in three anaesthetised male marmoset monkeys (Callithrix jacchus). Experiments were conducted in accordance with the Australian Code of Practice for the Care and Use of Animals for Scientific Purposes, and all procedures were approved by the Monash University Animal Ethics Experimentation Committee. The details of animal preparation, surgical procedure, and electrophysiology in marmoset monkeys followed our previously published protocols51. Animals were premedicated with atropine (0.33 mg kg−1) and diazepam (0.4 mg kg−1) and anaesthesia was subsequently induced with alfaxalone (Alfaxan, 8 mg kg−1), allowing a tracheotomy, vein cannulation and craniotomy to be performed. After the completion of all surgical procedures, the animal was administered an intravenous infusion of pancuronium bromide (0.1 mg kg−1 h−1) combined with sufentanil (6–8 μg kg−1 h−1) and dexamethasone (0.4 mg kg−1 h−1), and was artificially ventilated with a gaseous mixture of nitrous oxide and oxygen (7:3). Pulse oxygenation, heart rate, body temperature and the level of cortical spontaneous activity were continuously monitored. The administration of atropine (1%) and phenylephrine hydrochloride (10%) eye drops resulted in mydriasis and cycloplegia. Protection of the corneas from desiccation and focusing on the stimulus monitor were achieved using hard contact lenses selected by retinoscopy.
### Electrophysiology, data acquisition, and pre-processing
Most of the recordings were performed using single-shank linear multielectrode arrays (A1x32; NeuroNexus, Ann Arbor, MI, USA). Contacts on the array surface were collinear with 50 µm spacing, spanning all cortical layers. The data from one animal (monkey 2) were recorded using an Utah array, which consists of 96 electrodes arranged in a 10 × 10 grid, with each electrode separated by 400 µm (Blackrock Microsystems). Electrophysiological data were recorded using a Cerebus or Cereplex system (Blackrock Microsystems, MD) with a sampling rate of 30 kHz. The recordings were obtained from the region of V1 representing the central 10˚ of the visual field, in the exposed surface of the occipital operculum.
To detect single neurons and multi-units, we performed offline spike detection and sorting separately for each channel. Potential spikes were first identified based on threshold crossings, which were manually set during recording. Each spike waveform was normalised by its energy, and then principal component analysis was performed on all spike waveforms recorded from a channel. Normalisation allows the principal components to be based on a waveform shape rather than amplitude. We automatically identified clusters by fitting a mixture of Gaussians to the first five dimensions of principal components analysis space and checked and combined clusters and corresponding waveforms manually. Clusters were classified as single neurons based on (1) the inspection of the inter-spike interval histogram, (2) the consistency of waveform over time, and (3) if their Isolation distance, which is a measure of the separability between clusters and background activity, exceeded 1536. Any remaining threshold crossings were classified as a multi-unit activity. A total of 150 single neurons (74, 30, and 46 from monkeys 1, 2, and 3, respectively) and 240 multi-unit (113, 49, 78 from monkeys 1, 2, and 3, respectively) neuronal clusters were recorded. The results obtained for single- and multi-units were not significantly different; so, throughout the article, we refer to them as neurons and report the collective results.
Throughout the paper, we assess trends in our full dataset of 390 neurons; however, not all neurons were orientation selective for all stimuli, preventing within-group comparisons. Therefore, in Fig. 6, we focus on a sample of 50 neurons, which were highly orientation selective in all luminance and contrast conditions, allowing a within-group comparison.
### Visual stimulation
Visual stimuli were generated using MATLAB with Psychtoolbox52 and presented on an LCD monitor (Display + + , Cambridge Research Systems, UK) with 120 Hz refresh rate, a display width of 700 mm, a resolution of 1920 ×1080 pixels and a viewing distance of 500 or 700 mm. The monitor uses a built-in gamma-correction mechanism and has a 10-bit colour resolution36. Stimuli were viewed monocularly through the contralateral eye. Orientation selectivity was initially characterised using static gratings presented for 50 ms, followed by a grey blank screen for 500 ms. The gratings had 12 equally spaced orientations spanning 0–180°, six spatial frequencies (0.05, 0.125, 0.25, 0.5, 1, 2 cycles per degree) and two phases (0–180°). Spiking rates were averaged 50–150 ms after stimulus onset. We also obtained the contrast-response function of neurons using an optimal gating presented with different contrasts (4, 8, 16, 32, 64, and 100%).
### Luminance–contrast switching paradigm
To study the dynamics of adaptation and how orientation selectivity is affected by luminance and contrast adaptation, we presented a movie of rapidly changing gratings with random orientations, phases, and spatial frequencies. Stimuli were full-screen sinusoidal gratings presented for two monitor frames (16.7 ms), with 12 equally spaced orientations (0–165°) and eight phases (0–360°). We selected one to three spatial frequencies in a range of 0.125–0.4 cycles per degree based on the spatial frequency tuning of recorded neurons in each penetration. Every 5 s, the luminance and/or contrast of gratings were randomly selected from four luminance–contrast combinations: (1) high contrast (65%), low mean luminance (70 cd m-2); (2) low contrast (35%), low mean luminance (70 cd m-2); (3) high contrast (65%), high mean luminance (140 cd m-2); and (4) low contrast (35%), high mean luminance (140 cd m-2) (Fig. 1). This led to 12 different switch conditions, in which either or both luminance and contrast changed. The total presentation time was 60 min, yielding an average of 56 repetitions for each unique luminance–contrast switch. As luminance–contrast switching was random, the four no-change conditions also occurred with equal probability, but these are not analysed here. The contrast and luminance levels were selected so that most neurons were both responsive and orientation selective.
### Linear-response estimation (reverse-correlation analysis)
We characterised the functional properties of neural responses to variations in stimulus orientation using the orientation reverse-correlation method18. We computed reverse correlograms for each stimulus orientation by correlating the occurrence of each orientation with the spike train. Firstly, an array of counters for each of the 12 orientations and 512 time delays (τ = -6 to + 250 ms with 0.5 ms resolution) was constructed, R(θ,t), with all initial values set to zero. For each spike time, we looked τ earlier in time and incremented the counter corresponding to the presented stimulus orientation (θ), regardless of grating phase. At the end of this procedure, the sum of the counters at each time delay was equal to the number of spikes collected. The resulting counts were normalised at each time delay, giving the relative probability that a spike was preceded by each possible orientation,$$p(\theta ,\tau )$$.
We subsequently calculated several response characteristics for each neuron, including peak time (when the orientation selectivity was maximised), the maximum modulation (the difference between the highest and lowest probability at peak time), and the preferred orientation (θpref), which evokes the highest probability (pmax). The preferred orientation (θpref) was found by fitting the difference between two von Mises functions to the tuning curve at the peak time, Eq. (1). Taking the difference between two von Mises functions with different parameters allowed a better fit to asymmetric tuning functions.
$$p\left( \theta \right) = \alpha _1e^{k_1\cos \left( {\theta - \theta _{\mathrm{pref}\,1}} \right)} - \alpha _2e^{k_2\cos \left( {\theta - \theta _{\mathrm{pref}\,2}} \right)} + \beta$$
(1)
where θprefis the preferred orientation, ki is the width parameter, ai is the scaling factor and β is a constant offset. We also extracted the temporal width, which is the time window over which the probability of the preferred orientation, $${p(\theta _{\mathrm{pref}},t)}$$, exceeded half the maximum.
For most analyses, we calculated reverse correlograms in early (0–1.6 s) and late (3.4–5 s) time windows following each luminance–contrast switch. In total, this gives us 24 reverse correlograms (12 switches; 2 time windows). Our estimation of linear responses during these time windows was robust as we had over 350 repetitions of each unique stimulus orientation. For some analyses (Supplementary Fig. 1), we applied the reverse correlation method to the whole 5 s duration after the switch.
### Inclusion criteria
We analysed those neurons that showed a significant orientation selectivity. Neurons were deemed to be orientation selective if they satisfied the three following criteria: (1) maximum probability (probability at peak time) in the reverse-correlation analysis exceeded 3 × sd of the baseline probability; (2) the von Mises function fit (Eq. 1) was significantly better than a flat line (F test, p < 0.05); and (3) the bandwidth of von Mises function fit was 20–95°. Across all the recordings (511 isolated neurons), 453 neurons were visually responsive, and, of these, 390 neurons (86%) were orientation selective when tested with at least one of the luminance–contrast combinations. A neuron was selected as visually responsive if its spiking activity significantly increased above baseline after stimulus presentation based on the standard forward-correlation stimuli.
### Firing rate and trial-to-trial variability
We calculated peri-stimulus time histograms (PSTHs) for every neuron from 5 s before to 5 s after each luminance–contrast switch. For simplicity, we disregard switches when both luminance and contrast simultaneously changed. PSTHs were calculated using 50 ms time bins and we ignored the variations in other stimulus dimensions (i.e., orientation, phase, and spatial frequency). Each PSTH was normalised relative to a shuffled PSTH, generated by shuffling the spike times. The shuffling process was repeated 50 times while the spike times across the entire 1 h dataset were randomised in each run. The switching PSTHs were normalised relative to the averaged shuffled PSTH. For the sake of visualisation, the resulted PSTHs were then convolved with a Gaussian window with 50 ms width.
To assess trial-to-trial variability, we calculated the FF, which is the ratio of variance to mean spike count across the trials. FF was calculated in a sliding window of 50 ms with a time step of 10 ms.
### Information theoretic analysis
We quantified the amount of information conveyed by neurons about stimulus orientation using information-theoretic analysis. Therefore, the MI between the spiking activity and stimulus orientation was calculated using Eqs. (2) and (3):
$$MI_{t,\delta _t\left( {\theta ,R} \right)} = \mathop {\sum }\limits_{\forall \theta } p\left( \theta \right)\mathop {\sum }\limits_{\forall R} p(R|\theta ){\mathrm{log}}_2\left( {\frac{{p(R|\theta )}}{{p(R)}}} \right)$$
(2)
Where:
$$p\left( R \right) = \mathop {\sum }\limits_{\forall R} p(\theta )p(R|\theta )$$
(3)
Where, R is the spike count in a time interval τ after grating onset of width δt, p(θ) is the probability of presenting orientation θ, which is close to uniform in our case (1/12), p(R) is the probability of observing response R evoked across all stimuli, $$p(R|\theta )$$ is the conditional probability of observing response R given grating with an orientation θ was presented. We also applied a bootstrap-based bias-correction method to have an unbiased estimation of information53.
The above calculation was performed three times: (1) during the early-adaptation phase, which is the time window between 0 and 1.6 s after the switch (Fig. 1); (2) during the late-adaptation phase, which is the time window between 3.4 and 5 s after the switch (Fig. 1); and (3) during the course of adaptation in six non-overlapping time windows of 833 ms (Supplementary Fig. 4).
### Population-decoding analysis
To estimate what information can be extracted about the stimulus orientation from a given neural population, we applied simple linear decoders to our dataset. Such simple decoders are biologically plausible as they perform classifications by computing the weighted sum of spike counts. The weights can be considered as the synaptic strength and the outputs of classifiers, which are based on a decision boundary, are analogous to neuron’s spiking threshold. The main difference between the decoders is the way in which optimal weights and the decision boundary are learned. Here, we employed two simple and widely used decoders, including linear discriminant analysis (LDA) and support vector machines (SVM with linear kernel). The advantage of the support vector machine classifier is that it learns the structure of the neuronal response distributions without any particular pre-assumptions about the response distributions. These linear classifiers exhibit good performance, generalisation, and minimal overfitting for a wide range of neural data and applications54,55.
Response matrix: before applying the decoder, we constructed a response matrix, R, which is an S×N×T matrix, in which S is the number of unique stimulus orientations, N is the number of neurons, and T is the number of trials for each stimulus. Each element of the matrix is the number of spike events elicited by each stimulus in each neuron over a given time window (e.g., 15 ms). Finally, the spike counts across neural populations (N) were normalised (z score). This normalisation was done to compensate for spike-count variations in response to a stimulus across neural populations. The normalisation did not have any effect on the pattern of decoding accuracy and a small effect on absolute accuracy (1% increment). The response matrix, R, was built across multiple time windows starting from 0 to 200 ms after stimulus onset (2 ms temporal resolution). The response matrix, R, was calculated across different time windows during adaptation (after luminance–contrast switch; Supplementary Fig. 4). In some analyses (Supplementary Figs. 4 and 5), we varied different dimensions of the response matrix R and studied the effect of these changes on decoding accuracy.
Decoding analysis: in our analysis, we used 70% of the response matrix R to train the classifiers and 30% of the remaining to test. We separately trained and tested the decoder at early and late time periods after a stimulus switch and additionally performed separate training and testing for each type of stimulus switch. All the reported accuracies are the results of 15 cross-validated runs. The trials, T, and neurons from three animals, N, were randomly selected at every time delay, removing the effect of spike-count correlations. Here, all results are based on resampled subpopulations of 50 neurons. To statistically test whether a given mean decoding accuracy was significantly higher than chance, we repeated the same decoding procedure but shuffled the trial labels across trials and time.
### Multiplicative and additive modulations
To examine the changes in tuning throughout the 5 s adaptation period, we first calculated PSTHs in a time window of 0–200 ms following the appearance of each orientation. Then, tuning functions were created by averaging the spike rate in a 20 ms time window centred on the time of the peak response to the preferred orientation. These tuning functions were calculated separately for gratings presented early (0–1.6 s) and late (3.4–5 s) after each luminance–contrast switch, giving a total of 16 tuning curves (Fig. 6; eight switches, two time periods). For each neuron and switch type, we characterised its multiplicative and additive modulations by performing linear regression on the average responses to each orientation, during the early phase of adaptation compared to the late phase. The slope of the linear fit indicates how tuning scales multiplicatively (gain modulation of tuning function), whereas the intercept of the fit describes the additive shift (offset). Thus, the responses of the neurons with a purely multiplicative modulation can be fit with a line that passes through the origin, while responses of neurons with a purely additive modulation can be fit by a line with a slope of one and a non-zero intercept.
We examined the structure of multiplicative and additive modulations of tuning curves of every neuron during early and late phases of different adaptation conditions using four models: (1) a model with a single Von Mises function, applied to both early and late phases; (2) a model with two independent Von Mises functions, one for the early and one for the late phase; (3) a model with two Von Mises functions, with the same gain for tuning curves at early and late phases but with different offsets; and (4) a model with two Von Mises functions, with the same offset for tuning curves at early and late phases but different gains. To compare the performance of different models, we calculated the mean square error between the fit and experimental tuning curves. As the mean square error does not take into account the number of parameters in each model, we also calculated the Akaike’s Information Criterion to show which model is the best predictor considering the number of parameters in each model.
### Human psychophysical experiments
In order to investigate the perceptual correlate of orientation discrimination in cortical responses during adaptation to luminance and contrast, we performed psychophysical measurements in human observers. The stimulus design was very similar to the switching-stimulus paradigm in our monkey electrophysiology experiment, but here the subjects performed an orientation-discrimination task following adaptation to a period of continuously changing gratings.
On each trial, the observers fixated a black cross at the centre of the screen while a movie of rapidly changing gratings with random orientations and phases was presented. Stimuli were sinusoidal gratings with circular apertures of diameter 7°, centred on the fixation cross, and were presented for two monitor frames (16.7 ms), with 12 equally spaced orientations and eight phases (Fig. 5a). After 6 s of presentation, the contrast (luminance) of gratings was switched to a higher (lower) level. Either 0.2–1.2 s (early trials) or 5–6 s (late trials) after the switch to the new contrast (luminance), a test grating was shown, followed by dynamic filtered oriented noise56, and a second test grating, each presented for 200 ms (Fig. 5a). Test gratings and the dynamic noise mask had the same luminance and contrast. Finally, observers indicated with a keyboard press whether the second test grating was oriented clockwise or counter clockwise relative to the first test grating. The stimulus timings used for the psychophysical tests differed slightly from those in the physiological study. To increase any possible adaptation effects, we used slightly longer adaptation periods, coupled with a larger relative delay between the early and late periods. To minimise any distracting or masking effect associated with the period immediately after a luminance or contrast switch, we also delayed the start time of when the test gratings could appear to be 0.2 s after the switch.
In the contrast-switch task, the contrast switched between 10 and 100%. In the luminance-switch task, the grating (and background) luminance switched between 0.1 and 0.9 (normalised luminance) while having 10% contrast throughout the trial. We used the method of constant stimuli with 14 levels of orientation difference, ΔOri (seven clockwise), with logarithmic intervals. The sampling resolution varied depending on subject’s threshold during practice trials. Auditory feedback at the end of each trial indicated correct and incorrect responses. As a control, we measured the subjects’ orientation discrimination threshold at low (10%) and high (100%) contrast as well as low and high luminance levels. These trials consisted of two test gratings, which were temporally separated by a filtered oriented noise, but with no preceding adaptation gratings. In addition, in some trials, the discrimination task was shown 2–3 s into the trial. We did not analyse these trials and only included them for observers’ vigilance.
Stimuli were generated using MATLAB with Psychtoolbox52 and were presented on an LCD monitor (ViewPixx 3D; VPixx Technology Inc., Saint‐Bruno, QC, Canada) with 120 Hz refresh rate, a diagonal display size of 22.5″, the maximum luminance of 106 cd m−2, a resolution of 1920 × 1080 pixels, and a viewing distance 60 cm. We gamma corrected the monitor and used a 10‐bit colour resolution. The experiments were performed in a dark room while subjects were comfortably seated on a chair and their head was rested on a chinrest. Experiments were completed in two sessions lasting 60–90 min, with each trial condition presented at least 15 times.
We calculated the ratio of clockwise responses at every ΔOri level for each trial condition separately and fit a cumulative Gaussian function to data using maximum likelihood methods. The discrimination thresholds (measured as just noticeable difference) were then compared at different conditions. Non-parametric statistical tests were performed, based on boot-strapped resampling of each participant’s data 1000 times with replacement.
In total, we recorded the behavioural data from 31 subjects (16 females, aged 20–34 years, with normal or corrected-to-normal vision). The data of two subjects were excluded because their psychometric curves were flat at 50%, indicating that they were guessing or did not understand the task. Note that some subjects participated in more than one of the four luminance and contrast tasks. Subjects were students from the Faculty of Medicine, Nursing and Health Sciences, Monash University. All subjects voluntarily participated in the experiments and gave their written consent prior to participation. All human psychophysical experiments were conducted in accordance with the National Statement on Ethical Conduct in Human Research and all procedures were approved by the Monash University Human Research Ethics Committee.
### Reporting summary
Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
The datasets for the current study are available from the corresponding author on request. MATLAB code used for the analysis is available from the corresponding author on request.
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## Acknowledgements
We thank Richard Born and Nathan Crowder for their comments on the manuscript. This work was supported by the National Health and Medical Research Council (APP1066588 and APP1120667); the Human Frontier Science Program Career Development Award to NSCP; the Australian Research Council Special Research Initiative in Bionic Vision; and the Australian Research Council Centre of Excellence for Integrative Brain Function. We thank Janssen-Cilag Pty Limited for the donation of sufentanil citrate.
## Author information
Authors
### Contributions
M.G. and N.P. designed the experiments. M.G., E.Z., N.P. and M.R. performed the experiments. M.G. and N.P. analysed the data. M.G., N.P and E.Z. wrote the manuscript.
### Corresponding authors
Correspondence to Masoud Ghodrati or Nicholas S. C. Price.
## Ethics declarations
### Competing interests
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## Rights and permissions
Reprints and Permissions
Ghodrati, M., Zavitz, E., Rosa, M.G.P. et al. Contrast and luminance adaptation alter neuronal coding and perception of stimulus orientation. Nat Commun 10, 941 (2019). https://doi.org/10.1038/s41467-019-08894-8
• Accepted:
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# my 3rd blog
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How to calculate the volume of a sphere?
According to the formula for the calculations of the volume of sphere, it requires only radius(r)
Let, Radius is 3
$$v\;=\;\frac{4}{3}πr^3$$
$$v\;=\;\frac{4}{3}(3.14)(3*3*3)$$
$$v\;=\;\frac{4}{3}(3.14)(27)$$
$$v\;=\;\frac{4}{3}(84.78)$$
$$v\;=\;\frac{339.12}{3}$$
$$v\;=\;113.04$$
algebra
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## When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?
I was writing up some notes on harmonic analysis and I thought of a question that I felt I should know the answer to but didn't, and I hope someone here can help me. Suppose I have a compact Riemannian manifold $M$ on which a compact Lie group $G$ acts isometrically and transitively---so you can think of $M$ as $G/K$ for some closed subgroup $K$ of $G$. Then the real Hilbert space $H = L^2(M, R)$ is an orthogonal
representation space of $G$ and hence splits as an orthogonal direct sum of finite dimensional irreducible sub-representations. On the other hand, the Laplacian $L$ of $M$ is a self-adjoint operator on $H$, so $H$ is also the orthogonal direct sum of its eigenspaces---which are also finite dimensional. My question is, when do these two orthogonal decompositions of $H$ coincide? Put slightly differently, since $L$ commutes with the action of $G$, each eigenspace of $L$ is a finite dimensional subrepresentation of $H$ and so a direct sum of irreducibles, and I would like to know conditions under which each eigenspace is in fact irreducible. For example, this is true for the circle acting on itself and for $SO(3)$ acting on $S^2$ (where we get the harmonic polynomials of various degrees). Is it perhaps always true for the case of a symmetric space? Of course a standard reference in addition to the answer would be most welcome.
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Since $L$ is a $G$-invariant operator, doesn't Schur's lemma tell us that $L$ acts on each irrep appearing in $H$ by scalar multiplication? – Faisal Nov 21 2010 at 7:50
@Faisal: This says that each irreducible is a sub-representation of some eigenspace, but it doesn't say that the an eigenspace could not contain several irreducibles. – Dick Palais Nov 21 2010 at 8:06
@Dick: Ah, sorry -- I misinterpreted your question. I agree with Evan that it's very rare to have all the eigenspaces of $L$ be irreducible. For example if $M=G$, then an eigenspace of $L$ contains a given irrep only if it contain all copies of that irrep in $L^2(G)$. It follows that if all the eigenspaces of $L$ are irreducible then every irrep of G must appear without multiplicity in $L^2(G)$. In particular, because each irrep occurs with multiplicity equal to its degree, this means that if $G$ isn't abelian then there is at least one eigenspace of $L$ that isn't irreducible. – Faisal Nov 21 2010 at 9:03
The Peter-Weyl theorem tells you that $L^2(G)$ is isomorphic to $\bigoplus_{\pi}\pi\otimes\pi^*$ as $G\times G$ representation, where $\pi$ runs through all irreducible unitary representations. It follows that $$L^2(G/K)\cong L^2(G)^K\cong\bigoplus_\pi \pi\otimes(\pi^*)^K.$$ So, the first thing you absolutely need, is a multiplicity one property, which says that $\dim\pi^K\le 1$ for every $\pi$. This is already a rare property, but known to be true for, say $G=SO(n)$ and $K=SO(n-1)$, see Zhelobenko's book for this. But, the Laplacian may have the same eigenvalue on different representations. For this you need highest weight theory (see for instance the book by Broecker and tom Dieck): Assume $G$ to be connected. The irreducible representations are parametrized by highest weights and the Laplace eigenvalue depends on the value of a quadratic form on the space of weights. So, in each case you need to identify those weights with $K$-invariants and consider the values of the quadratic form, which in the case of a simple group should be the Killing form. I guess that in the above cases it might actually be true.
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@Anton Deitmar, Evan Jenkins: I am still a little unclear about the connection between the Laplacian $L$ and the Casimir operator(s). I think that your remarks about eigenvalues being determined by the Killing form on highest weight vectors refers to the Casimir operator, which is purely group theoretic, whereas $L$ is the usual Riemannian Lapalcian. Of course they are related (and no doubt have the same symbol) but I don't think that they are the same. Do either of you (or does someone else) know where the relation between them is discussed. (Broecker and tom Dieck only treat a special case. – Dick Palais Nov 22 2010 at 0:36 @Dick1: Any invariant positive definite bilinear form $B$ on the Lie algebra $\mathfrak g$ of $G$ gives an invariant metric on the quotient $G/K$. Being non-degenerate, this form on the one hand identifies $\mathfrak g$ with its dual ${\mathfrak g}'$, on the other hand it is itself an element of ${\mathfrak g}'\otimes\mathfrak g'\cong{\mathfrak g}\otimes{\mathfrak g}$ The latter space maps naturally to $U=U({\mathfrak g})$, the universal enveloping algebra. So $B$ induces an element in $U$, which is called the Casimir-operator $C_G$. – anton Nov 22 2010 at 18:05 @Dick2: This Casimir-operator acts on functions on $G/K$ as a differential operator which happens to coincide with the Laplace-operator induced by the metric. This is no wonder, since the metric and the Casimir are induced by the same invariant form. – anton Nov 22 2010 at 18:06 Thanks Anton. That more or less answers what I wanted to know. (Though I still do not see what happens when the isotropy action of $K$ is not irreducible, so there is not a unique $G$-invariant metric on $M$.) – Dick Palais Nov 22 2010 at 22:46
Shouldn't this only happen very rarely? $S^1$ is abelian, and $SO(3)$ acting on $S^2$ involves inducing from the maximal torus, so in both these cases, every irreducible appears once. But in general (i.e., if $K$ does not contain a maximal torus), irreducible representations will appear more than once, in which case there's no hope for the eigenvalue of the Laplacian to separate them. Even when irreducibles don't appear multiple times, the eigenvalue of the Laplacian is not generally enough to separate two irreducibles if the rank of the group is bigger than 1.
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For G/K symmetric the joint eigenspaces of the G-invariant differential operators on G/K are all irreducible. Also each irreducible subspace of H has multiplicity bounded by one. For this see my "Groups and Geometric Analysis?" Ch. V Theorems 4.3 and 3.5. Concerning the Laplace Beltrami operator L, the Casimir operator on G (if semisimple) does induce L on G/K (loc. cit. p.331). If G/K is two point homogeneous the G-invariant differential operators on G/K are all polynomial in L (loc. cit. p/288) so for these spaces the answer to Dicks question is yes. For G/K not symmetric Theorem 3.5 p. 533 still gives a decomposition of H into spaces spanned by representation coefficients which are eigenfunctions of the Casimir operator.
Assuming the metric on G/K, (G semisimple) is coming from the Killing form Riemannian structure on G it is still true that the geodesics through the origin in G/K are orbits of one parameter subgroups of G. It seems to me that the argument for Problem A4 p.568 should still show that the Casimir operator on G will induce the Laplace Beltrami operator on G/K. Therefore the decomposition in Theorem 3.5 p.533 should still be a decomposition into eigenfunctions of the Laplacian. But there is no reason to expect irreducibility.
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holomorphic sections on elliptic K3 surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:00:45Z http://mathoverflow.net/feeds/question/91831 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91831/holomorphic-sections-on-elliptic-k3-surface holomorphic sections on elliptic K3 surface Jay 2012-03-21T15:51:21Z 2012-04-04T20:22:01Z <p>Hi all,</p> <p>I want to ask something about the holomorphic sections on elliptic K3:</p> <p>Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some number as 240? For example, how about E(2) or Fermat's quartic?</p> <p>Thanks a lot! :)</p> http://mathoverflow.net/questions/91831/holomorphic-sections-on-elliptic-k3-surface/91853#91853 Answer by Csar Lozano Huerta for holomorphic sections on elliptic K3 surface Csar Lozano Huerta 2012-03-21T19:46:44Z 2012-03-21T19:46:44Z <p>Hope this helps, it doesn't give a definite answer, but it tells about where to find an obstruction. First off, the existence of multiple fibers in an elliptic fibration is an obstruction to the existence of a differentiable section (over $\mathbb{C}$). On the other hand we can always get rid of the multiple fibers by passing to an \'etale cover.</p> <p>Now a more elaborate answer, given an elliptic fibration with no sections $X\rightarrow B$ we can associate a fibration $J\rightarrow B$ which has a section and a rational map $\phi:J\times_BX\rightarrow B$ that commutes with projections to $B$ and has certain properties. The family $J$ is called <em>jacobian family</em>.</p> <p>The elliptic fibrations are classified by their jacobian fibrations and here comes the change of quantifiers. One can introduce a group structure on the set $I(J)$ of elliptic fibrations with a given jacobian fibration. Hence if the class of the fibration $X\rightarrow B$ in $I(J)$ is not zero, (the obstruction) the fibration $X$ has no differentiable sections. This works the same way the the first Chern class of a line bundle $L$ gives and obstruction for $L$ to be trivial.</p>
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# Complex Analysis curve
We are given three complex numbers a,b,and c. Consider $Re(az^{2} + bz +c)=0$. What is this curve? I am having a hard time approaching this problem. Any suggestions or help would be great.
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See this. – Ｊ. Ｍ. Sep 11 '11 at 21:01
You could write out the equation with e.g. $a = Re(a) + Im(a) i$, so that after substituting $x = Re(z)$ and $y = Im(z)$ you get something of the form $A x^2 + B x y + C y^2 + D x + E y + F = 0$, with $A,B,C,D,E,F \in \mathbb{R}$... – TMM Sep 11 '11 at 21:02
If $a=a_1+ia_2$, $b=b_1+ib_2$, $c=c_1+ic_2$ and $z=x+iy$, then $z^2=x^2-y^2+2ixy$ hence $$az^2+bz+c=a_1(x^2-y^2)-2a_2xy+b_1x-b_2y+c_1+i\cdot(\text{something real}),$$ and you are done.
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# Graphing Tangent, Cotangent, Secant and Cosecant
## Tangent, cotangent, cosecant, and secant graphs.
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Graphs of Other Trigonometric Functions
If you already know the relationship between the equation and graph of sine and cosine functions then the other four functions can be found by identifying zeroes, asymptotes and key points. Are the four new functions transformations of the sine and cosine functions?
### Graphing Other Trigonometric Functions
#### Secant and Cosecant
Since secant is the inverse of cosine the graphs are very closely related.
Notice wherever cosine is zero, secant has a vertical asymptote and where \begin{align*}\cos x=1\end{align*} then \begin{align*}\sec x=1\end{align*} as well. These two logical pieces allow you to graph any secant function of the form:
\begin{align*}f(x)=\pm a \cdot \sec (b(x+c))+d\end{align*}
The method is to graph it as you would a cosine and then insert asymptotes and the secant curves so they touch the cosine curve at its maximum and minimum values. This technique is identical to graphing cosecant graphs. Simply use the sine graph to find the location and asymptotes.
#### Tangent and Cotangent
The tangent and cotangent graphs are more difficult because they are a ratio of the sine and cosine functions.
• \begin{align*}\tan x=\frac{\sin x}{\cos x}\end{align*}
• \begin{align*}\cot x=\frac{\cos x}{\sin x}\end{align*}
The way to think through the graph of \begin{align*}f(x)=\tan x\end{align*} is to first determine its asymptotes. The asymptotes occur when the denominator, cosine, is zero. This happens at \begin{align*}\pm \frac{\pi}{2}, \pm \frac{3 \pi}{2} \ldots\end{align*} The next thing to plot is the zeros which occur when the numerator, sine, is zero. This happens at \begin{align*}0, \pm \pi, \pm, 2 \pi \ldots\end{align*} From the unit circle and basic right triangle trigonometry, you already know some values of \begin{align*}\tan x\end{align*}:
• \begin{align*}\tan \frac{\pi}{4}=1\end{align*}
• \begin{align*}\tan \left(-\frac{\pi}{4}\right)=-1\end{align*}
By plotting all this information, you get a very good sense as to what the graph of tangent looks like and you can fill in the rest.
Notice that the period of tangent is \begin{align*}\pi\end{align*} not \begin{align*}2 \pi\end{align*}, because it has a shorter cycle.
The graph of cotangent can be found using identical logic as tangent. You know \begin{align*}\cot x=\frac{1}{\tan x}\end{align*}. This means that the graph of cotangent will have zeros wherever tangent has asymptotes and asymptotes wherever tangent has zeroes. You also know that where tangent is 1, cotangent is also 1. Thus the graph of cotangent is:
### Examples
#### Example 1
Earlier, you were asked if the four new functions are transformations of sine and cosine. The four new functions are not purely transformations of the sine and cosine functions. However, secant and cosecant are transformations of each other as are tangent and cotangent.
#### Example 2
Graph the function \begin{align*}f(x)=-2 \cdot \csc (\pi (x-1))+1\end{align*}.
Graph the function as if it were a sine function. Then insert asymptotes wherever the sine function crosses the sinusoidal axis. Lastly add in the cosecant curves.
The amplitude is 2. The shape is negative sine. The function is shifted up one unit and to the right one unit.
Note that only the blue portion of the graph represents the given function.
#### Example 3
How do you write a tangent function as a cotangent function?
There are two main ways to go between a tangent function and a cotangent function. The first method was discussed in Example A: \begin{align*}f(x)=\tan x=\frac{1}{\cot x}\end{align*}.
The second approach involves two transformations. Start by reflecting across the \begin{align*}x\end{align*} or the \begin{align*}y\end{align*} axis. Notice that this produces an identical result. Next shift the function to the right or left by \begin{align*}\frac{\pi}{2}\end{align*}. Again this produces an identical result. \begin{align*}f(x)=\tan x=-\cot \left(x-\frac{\pi}{2}\right)\end{align*}.
#### Example 4
Find the equation of the function in the following graph.
If you connect the relative maximums and minimums of the function, it produces a shifted cosine curve that is easier to work with.
The amplitude is 3. The vertical shift is 2 down. The period is 4 which implies that \begin{align*}b=\frac{\pi}{2}\end{align*}. The shape is positive cosine and if you choose to start at \begin{align*}x=0\end{align*} there is no phase shift.
\begin{align*}f(x)=3 \cdot \csc \left(\frac{\pi}{2}x\right)-2\end{align*}
#### Example 5
Where are the asymptotes for tangent and why do they occur?
Since \begin{align*}\tan x=\frac{\sin x}{\cos x}\end{align*} the asymptotes occur whenever \begin{align*}\cos x=0\end{align*} which is \begin{align*}\pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}, \ldots\end{align*}
### Review
1. What function can you use to help you make a sketch of \begin{align*}f(x)=\sec x\end{align*}? Why?
2. What function can you use to help you make a sketch of \begin{align*}g(x)=\csc x\end{align*}? Why?
Make a sketch of each of the following from memory.
3. \begin{align*}f(x)=\sec x\end{align*}
4. \begin{align*}g(x)=\csc x\end{align*}
5. \begin{align*}h(x)=\tan x\end{align*}
6. \begin{align*}k(x)=\cot x\end{align*}
Graph each of the following.
7. \begin{align*}f(x)=2 \csc (x)+1\end{align*}
8. \begin{align*}g(x)=2 \csc \left(\frac{\pi}{2}x\right)+1\end{align*}
9. \begin{align*}h(x)=2 \csc \left(\frac{\pi}{2}(x-3)\right)+1\end{align*}
10. \begin{align*}j(x)=\cot \left(\frac{\pi}{2}x\right)+3\end{align*}
11. \begin{align*}k(x)=-\sec \left(\frac{\pi}{3}(x+1)\right)-4\end{align*}
12. \begin{align*}m(x)=-\tan (x)+1\end{align*}
13. \begin{align*}p(x)=-2 \tan \left(x-\frac{\pi}{2}\right)+1\end{align*}
14. Find two ways to write \begin{align*}\sec x\end{align*} in terms of other trigonometric functions.
15. Find two ways to write \begin{align*}\csc x\end{align*} in terms of other trigonometric functions.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
TermDefinition
Asymptotes An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).
Cosecant The cosecant of an angle in a right triangle is a relationship found by dividing the length of the hypotenuse by the length of the side opposite to the given angle. This is the reciprocal of the sine function.
Cotangent The cotangent of an angle in a right triangle is a relationship found by dividing the length of the side adjacent to the given angle by the length of the side opposite to the given angle. This is the reciprocal of the tangent function.
Secant The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
Transformations Transformations are used to change the graph of a parent function into the graph of a more complex function.
Vertical Asymptote A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.
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# Error while reading data from webpage in java?
I am using this code to read data from a webpage :
public class ReadLatex {
public static void main(String[] args) throws IOException {
URL url = new URL(urltext);
.openStream()));
String inputLine;
while ((inputLine = in.readLine()) != null) {
// Process each line.
System.out.println(inputLine);
}
in.close();
}
}
The webpage gives the image for a latex code in the URL.
I am getting this exception:
Exception in thread "main" java.io.IOException: Server returned HTTP response code: 400 for URL: http://chart.apis.google.com/chart?
at sun.net.www.protocol.http.HttpURLConnection.getInputStream(Unknown Source)
at java.net.URL.openStream(Unknown Source)
Can anyone tell why I am having this problem and what should be the solution for this?
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400 means Bad request - It means request could not be understood by the server due to malformed syntax....did you try some other url ?? – Shashank Kadne Feb 21 '12 at 9:35
you should escape all your special characters and slash in the url – Sergey Benner Feb 21 '12 at 9:36
Your problem is that you are using a \ (backslash) in a string which in Java is a escape character. To get an actual \ you need to have two of them in your string. So:
Wanted text: part1\part2
you need to have
String theString = "part1\\part2";
So you actually want
String urltext = "http://chart.apis.google.com/chart?cht=tx&chl=1+2%20\\frac{3}{4}";
Also, when you succeed with your request you get back an image (png) which should not be read with a reader which will try to interpret the bytes as characters using some encoding and this will break the image data. Instead, use the input stream and write the content (bytes) to a file.
A simple example without error handling
public static void main(String[] args) throws IOException {
URL url = new URL(urltext);
InputStream in = url.openStream();
FileOutputStream out = new FileOutputStream("TheImage.png");
byte[] buffer = new byte[8*1024];
}
out.close();
in.close();
}
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Sir, I have tried new code but in console it shows unreadable characters .I read your answer but couldnt make out how to do it. – Navdroid Feb 21 '12 at 13:18
Could you help me with some code or tutorials – Navdroid Feb 21 '12 at 13:19
@Navdroid what you are asking in not the answer to your question. Your http problem is solved. What you were missing then was the way to use the HTTP response content. This is a distinct issue. Shouldn't it be better to update or split your question? – C.Champagne Feb 21 '12 at 16:26
Try escaping with something like org.apache.commons.lang.StringEscapeUtils
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I think you should consider escaping the backslash in the URL. I Java, the backslash must be escaped in a String It should become
String urltext =
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# Random Numbers
This topic is 4376 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
hello, I want to know how does the computer produce random numbers?
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It doesn't. It only produces pseudo-random numbers based on things such as the current time, current CPU usage, etc.
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Some chipsets (i820 was the first) can produce truly random numbers by sampling thermal noise. As an alternative, the technique (*) used by modern Unix variants for /dev/random is cryptographically secure and can therefore be called random. (pseudo-RNGs may satisfy statistical tests for randomness, but they are often predictable after observing a few outputs)
* gather entropy from various sources such as keyboard input and network traffic; mix together in a pool via strong hash function; return bytes from this pool.
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However, it should be noted that in practical applications being able to generate predictable and repeatable pseudo-random number sequences is often desirable. For example, if you're debugging a piece of code or some other stochastic simulation it's nice to be able to reproduce the same input values over and over so that they become a constant rather than a variable. For cryptography, maybe it's not so good. Just don't want anyone thinking that pseudo-random isn't as "good" as truly random because they both have their places.
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Also, in computer graphics you often need pseudorandom because you don't want procedural textures to be different each time. Practically "true" random is necessary mostly for cryptography, in fact i know of no other things that really can't use good pseudorandom number generator and need "true" randomness.
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"Common" methods for "random" numbers:
1) Psuedo-RNG - complex mathematical forumla creates seemingly random numbers. Good for games, not so good for cryptography. (/dev/random on *nix?)
2) Psuedo-RNG + Noise - things like a sampling of network activity XORed against the PRNG. Little benifit except added cryptography strength (/dev/srandom on *nix? - can block if not enough data from the environment to sample)
3) Quantum RNG - things like sampling thermal noise, sampling photons against a 50% opaque surface - link for some examples. The most cryptographically secure.
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You can ask the user to move the mouse about for a few seconds and pick (x,y) points at regular intervals.
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Quote:
Original post by Anonymous PosterYou can ask the user to move the mouse about for a few seconds and pick (x,y) points at regular intervals.
This is amazingly not random, actually. Many people will perform the exact same motions every single time they're asked to do so, resulting in almost identical results every time. This is highly undesirable.
CM
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From the looks of the question, the OP just wants to know how some magical function could produce a seemingly random number?
It is typically implemented pretty much as follows.
long holdrand;void srand(long seed){ holdrand = seed;}short rand(){ return (short)((holdrand = holdrand * 214013 + 2531011) >> 16);}
Simple eh!
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Quote:
Original post by iMalcFrom the looks of the question, the OP just wants to know how some magical function could produce a seemingly random number?It is typically implemented pretty much as follows.long holdrand;void srand(long seed){ holdrand = seed;}short rand(){ return (short)((holdrand = holdrand * 214013 + 2531011) >> 16);}Simple eh!
Not really. I think the right shift by 16 bits causes this to be a bad one!
For instance, off hand I guess that this version of rand returns numbers which will almost surely end with 0x26. Instead of taking a mod you are doing a shift. Not the same thing. Taking the lower 16 bits would probably have been better. Though i admit, your code needs more analysis to claim that this one is not a good one :-)...
I think what IMalc wanted to refer to is a Linear Congruential Generator
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Quote:
Original post by Conner McCloud
Quote:
Original post by Anonymous PosterYou can ask the user to move the mouse about for a few seconds and pick (x,y) points at regular intervals.
This is amazingly not random, actually. Many people will perform the exact same motions every single time they're asked to do so, resulting in almost identical results every time. This is highly undesirable.
CM
That is not physiologically feasible. Draw a 2D spline on your screen and I defy you to go through the very same curve again and at the very same speed. Even your own signature is not perfectly identical.
BTW, this is how the encryption key is generated for RIM Blackberries.
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Quote:
Original post by Anonymous PosterThat is not physiologically feasible. Draw a 2D spline on your screen and I defy you to go through the very same curve again and at the very same speed. Even your own signature is not perfectly identical.
It isn't neccessary that it be perfectly identical...just close enough to disturb the randomness. And besides, I guarentee I can perform the same actions over and over again: just don't move it at all, or use a different input device that allows more precice motion.
Quote:
Original post by Anonymous PosterBTW, this is how the encryption key is generated for RIM Blackberries.
It asks you to move the mouse for a few seconds? Doubtful, especially for encryption.
CM
##### Share on other sites
Quote:
Original post by Conner McCloud
Quote:
Original post by Anonymous PosterYou can ask the user to move the mouse about for a few seconds and pick (x,y) points at regular intervals.
This is amazingly not random, actually. Many people will perform the exact same motions every single time they're asked to do so, resulting in almost identical results every time. This is highly undesirable.
CM
You could pick at "random" intervals instead (using the built-in PRNG), and/or sample just the low-order bits of the mouse coordinates (difficult for the user to come to the exact same pixel each time)...
Edit: Yes, the "standstill input" is a problem if the user doesn't have a motivation to help out with the randomness... you need (if possible; otherwise use another method - this is probably bad for games, for example) to design the app so that there is motivation, and then if the user won't comply, the results are well deserved. :)
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Quote:
Original post by Conner McCloudAnd besides, I guarantee I can perform the same actions over and over again: just don't move it at all {...}
It's not that complicated to detect when a mouse doesn't move or when the range of movements is not large enough to be significant.
Quote:
Original post by Conner McCloud{...} or use a different input device that allows more precice motion.
Input devices tend to increase in precision, not the other way around. Besides, the increase in precision makes it even more difficult to repeat the *exact* same movements. Remember that you can register mouse position, speed and acceleration, all of which are very difficult to replicate *exactly* by a human.
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Quote:
Original post by Aryabhatta
Quote:
Original post by iMalcFrom the looks of the question, the OP just wants to know how some magical function could produce a seemingly random number?It is typically implemented pretty much as follows.long holdrand;void srand(long seed){ holdrand = seed;}short rand(){ return (short)((holdrand = holdrand * 214013 + 2531011) >> 16);}Simple eh!
Not really. I think the right shift by 16 bits causes this to be a bad one!
For instance, off hand I guess that this version of rand returns numbers which will almost surely end with 0x26. Instead of taking a mod you are doing a shift. Not the same thing. Taking the lower 16 bits would probably have been better. Though i admit, your code needs more analysis to claim that this one is not a good one :-)...
I think what IMalc wanted to refer to is a Linear Congruential Generator
You unfortunately don't know what you're talking about. I didn't just pull this out of thin air.
This is a standard implementation of a random number generator, taken from a common runtime library. Search for the above two prime numbers on google and you'll most likely get tons of search results with variants of this prng.
The upper 16 bits are far more random than the lower 16.
I know that this is a Linear congruential generator.
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Quote:
Original post by iMalc
Quote:
Original post by AryabhattaThough i admit, your code needs more analysis to claim that this one is not a good one :-)...I think what IMalc wanted to refer to is a Linear Congruential Generator
You unfortunately don't know what you're talking about. I didn't just pull this out of thin air.
This is a standard implementation of a random number generator, taken from a common runtime library. Search for the above two prime numbers on google and you'll most likely get tons of search results with variants of this prng.
The upper 16 bits are far more random than the lower 16.
I know that this is a Linear congruential generator.
Whoa! Slow down there pal...Sorry If I offended you. I didn't mean to.
I did say your code needs more analysis before anyone can claim it is a bad one! What I stated was my opinion.. Hence the words "I think".
Upon rereading the chapter in Knuth, in LCG's the higher order bits are more random than the lower order bits if the modulus equals the word size, which is true in this case. So taking the higher order 16 bits is right.. but this RNG is pretty bad nonetheless.
[Edited by - Aryabhatta on December 19, 2005 3:52:49 PM]
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Quote:
Original post by iMalcI didn't just pull this out of thin air.This is a standard implementation of a random number generator, taken from a common runtime library.
It's a standard RNG used by many standard C library implementations, and it's good enough for simple applications, but it's actually a pretty poor selection of parameters. It fails certain tests of randomness and degenerates into very short sequences given certain seed values.
A better selection might be Lewis, Goodman, and Miller's rand0 (a=16807, c=0, m=2147483647) or possibly the Lehmer generator (a=48271, c=0, m=2147483647). These have been thoroughly analysed and found much better than the rand() function.
You can eliminate the LCG's pairwise sawtooth with a lagged fibonacci seived through a discard block filter (google for James's luxury-level-3 integer adaptation of Luescher's generator) or better yet, Matsumoto and Nishimura's Mersenne Twister. Both lagged fibonacci and the mersenne twister are faster, and produce better "randomness" by most accepted definitions, and have longer periods than the good ol' LCG, but they take considerably more memory.
It is important, however, to realize that the bitshift is a better way to select an integral subrange than is using the % operator, which is what most people do.
--smw
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Quote:
Original post by Aryabhatta
Quote:
Original post by iMalc
Quote:
Original post by AryabhattaThough i admit, your code needs more analysis to claim that this one is not a good one :-)...I think what IMalc wanted to refer to is a Linear Congruential Generator
You unfortunately don't know what you're talking about. I didn't just pull this out of thin air.
This is a standard implementation of a random number generator, taken from a common runtime library. Search for the above two prime numbers on google and you'll most likely get tons of search results with variants of this prng.
The upper 16 bits are far more random than the lower 16.
I know that this is a Linear congruential generator.
Whoa! Slow down there pal...Sorry If I offended you. I didn't mean to.
I did say your code needs more analysis before anyone can claim it is a bad one! What I stated was my opinion.. Hence the words "I think".
Upon rereading the chapter in Knuth, in LCG's the higher order bits are more random than the lower order bits if the modulus equals the word size, which is true in this case. So taking the higher order 16 bits is right.. but this RNG is pretty bad nonetheless.
I'm sorry, I shouldn't have acted so defensively. I've oft heard that it isn't a terrific prng, but one seldom comes across a case where anything more random is needed (especially for a beginner). For example a particle generator animation would look indistinguishable from one with a better prng. It's only when you come to analyse where every individual droplet landed over a long period of time that you'd notice any difference.
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# In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically. $\left\{\begin{array}{l}y = -2x^2 + 2\\y = 2\left(x^4 - 2x^2 +1\right)\end{array}\right.$
## $(0,2),(-1,0),(1,0)$
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Okay, so this question were asked to, uh, solar system of equations using institution. Thankfully, the way the problem has set up, we don't even up to manipulate anything to sell for. Why? Because literally get to us in a format where? Why do you think so? You can just go ahead and set. Uh, why? Why, uh, or the substitute form of why so two X plus two said it equal. Thio treks to the fore, minus for X square plus two from here. Will you see that our shoes will cancel out. And then if we move the two x squared over Tillie's other side and get two extra before you're minus two, X squared is equal to zero. Okay. And then if we solve for X, come here. I got back square. We're left with X squared minus one. It's equal to zero, which is save us to x square. X plus one X minus. One is equal to zero. Uh, so we get that X is equal to zero. X is equal to negative. One X is equal to one. Okay, so when X is equal to zero, you putting it in tow may be pertinent to this sign equation because I know that Y is equal to negative two X squared plus two. Hey, that y is equal to two. And then when X is negative one, you get that by its equal to zero when X is one we hit that Y is equal to zero as well, because they are the coordinates of the intersections between the two functions. The back. It just proves that these are correct. And so, uh, if you look at the graph, could see that thes tune functions do intersect that zero to negative 10
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# QUIC I WILL LOVE YOU FOREVER A leading preacher during the Great Awakening was ___.George WashingtonNathaniel BaconJonathan EdwardsBilly SundayCharles
###### Question:
QUIC I WILL LOVE YOU FOREVER A leading preacher during the Great Awakening was ___. George Washington
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Jonathan Edwards
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### ... which emerged in the mid-westernregion of Nigeria...'i) What grammatical name is givento this expression as it is used in
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### >Review the story event.When the narrator and his brother replace the Duvitches' fish with the fish that they have caught, Mr.
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# Regression model constant causes multicollinearity warning, but not in standardized model
I'm working in Python with statsmodels. I have a response variable y and a design matrix X from which I have already removed the most strongly correlated (redundant) predictors. I add a constant and check the VIFs:
> from statsmodels.stats.outliers_influence import variance_inflation_factor as vif
> vifs = [vif(Xc.values, i) for i in range(len(Xc.columns))]
> pd.Series(data=vifs, index=Xc.columns).sort_values(ascending=False)
const 251.828124
x1 4.007442
x2 3.768146
x3 3.151422
x4 3.093936
x5 2.252909
x6 2.182324
x7 2.121366
x8 2.095420
x9 2.026337
x10 2.015635
x11 1.732534
x12 1.514766
dtype: float64
The constant has a very high VIF but all the other values seem reasonable. I estimate a regression model:
> import statsmodels.api as sm
> model = sm.OLS(y, Xc)
> result = model.fit()
> print(result.summary())
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.219
Method: Least Squares F-statistic: 277.5
Date: Thu, 08 Mar 2018 Prob (F-statistic): 0.00
Time: 09:36:09 Log-Likelihood: -15948.
No. Observations: 11915 AIC: 3.192e+04
Df Residuals: 11902 BIC: 3.202e+04
Df Model: 12
Covariance Type: nonrobust
===============================================================================================
coef std err t P>|t| [0.025 0.975]
-----------------------------------------------------------------------------------------------
const -1.2364 0.134 -9.212 0.000 -1.499 -0.973
x1 0.0066 0.001 4.486 0.000 0.004 0.009
x2 -0.0654 0.030 -2.214 0.027 -0.123 -0.007
x3 -0.1051 0.032 -3.294 0.001 -0.168 -0.043
x4 -0.0135 0.002 -8.922 0.000 -0.016 -0.011
x5 0.0048 0.001 8.383 0.000 0.004 0.006
x6 0.0276 0.013 2.049 0.041 0.001 0.054
x7 0.5270 0.040 13.029 0.000 0.448 0.606
x8 0.0113 0.001 8.975 0.000 0.009 0.014
x9 0.0013 0.000 3.181 0.001 0.000 0.002
x10 0.0067 0.001 10.322 0.000 0.005 0.008
x11 -0.0051 0.001 -4.133 0.000 -0.007 -0.003
x12 -0.1398 0.017 -8.243 0.000 -0.173 -0.107
==============================================================================
Omnibus: 311.108 Durbin-Watson: 1.688
Prob(Omnibus): 0.000 Jarque-Bera (JB): 376.035
Skew: 0.341 Prob(JB): 2.21e-82
Kurtosis: 3.541 Cond. No. 2.08e+03
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.08e+03. This might indicate that there are
strong multicollinearity or other numerical problems.
My coefficients are signed in the right direction and are all significant at the .05 level. However, the condition number is extremely high, suggesting severe multicollinearity. If I remove the constant and re-estimate, the condition number drops to 642 and the multicollinearity warning disappears. Moreover if I standardize y and X and re-estimate a model:
> from scipy.stats.mstats import zscore
> y_stdrd = pd.Series(zscore(y), index=y.index, name=y.name)
> X_stdrd = pd.DataFrame(data=zscore(X), index=X.index, columns=X.columns)
> model_stdrd = sm.OLS(y_stdrd, Xc_stdrd)
> result_stdrd = model_stdrd.fit()
> print(result_stdrd.summary())
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.219
Method: Least Squares F-statistic: 277.5
Date: Thu, 08 Mar 2018 Prob (F-statistic): 0.00
Time: 09:42:39 Log-Likelihood: -15437.
No. Observations: 11915 AIC: 3.090e+04
Df Residuals: 11902 BIC: 3.100e+04
Df Model: 12
Covariance Type: nonrobust
===============================================================================================
coef std err t P>|t| [0.025 0.975]
-----------------------------------------------------------------------------------------------
const 6.462e-17 0.008 7.98e-15 1.000 -0.016 0.016
x1 0.0478 0.011 4.486 0.000 0.027 0.069
x1 -0.0261 0.012 -2.214 0.027 -0.049 -0.003
x2 -0.0386 0.012 -3.294 0.001 -0.062 -0.016
x3 -0.1026 0.012 -8.922 0.000 -0.125 -0.080
x4 0.1360 0.016 8.383 0.000 0.104 0.168
x5 0.0295 0.014 2.049 0.041 0.001 0.058
x6 0.1857 0.014 13.029 0.000 0.158 0.214
x7 0.1074 0.012 8.975 0.000 0.084 0.131
x8 0.0317 0.010 3.181 0.001 0.012 0.051
x9 0.1624 0.016 10.322 0.000 0.132 0.193
x10 -0.0477 0.012 -4.133 0.000 -0.070 -0.025
x11 -0.1002 0.012 -8.243 0.000 -0.124 -0.076
==============================================================================
Omnibus: 311.108 Durbin-Watson: 1.688
Prob(Omnibus): 0.000 Jarque-Bera (JB): 376.035
Skew: 0.341 Prob(JB): 2.21e-82
Kurtosis: 3.541 Cond. No. 4.64
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Now the condition number is a reasonable 4.64 and the severe multicollinearity has disappeared. However my constant is no longer statistically significant.
Questions:
1. What does it mean for a constant to have such a high VIF (> 250) while all other predictors' VIFs are around 4 or below?
2. Related to #1, is the constant's VIF the reason why my unstandardized model reports the high condition number (> 2000) and warns about strong multicollinearity?
3. Why does standardization fix this high condition number?
Ok, combining several comments and some further Googling, I think I have the answer. This is a scaling problem: for a very simplified example see here.
statsmodels reports the condition number of the design matrix and not of a standardized design matrix. It calculates this as the ratio of the largest eigenvalue in the design matrix to the smallest. In other words, the large condition number in this case results from scaling rather than from multicollinearity. If we have just one variable with units in the thousands (ie, a large eigenvalue) and add a constant with units of 1 (ie, a small eigenvalue), we'll get a large condition number as the ratio, and statsmodels warns of multicollinearity.
statsmodels sets its threshold for warning about multicollinearity at 30, which is pretty sensitive for real-world applied analysis and an unstandardized design matrix given this scaling issue.
• The threshold for the condition number warning has been increased to 1000, which is still low if the large condition number comes mainly from scaling problems. – Josef Mar 10 '18 at 14:35
The variance inflation factor for constant should be high (actually infinite) since it is $\frac{1}{1-R^2_i}$ where the $R^2$ is from the regression
$$X_{i} = \alpha + \vec{\beta}^\top \vec{X}_{(-i)}$$
where $\vec{X}_{(-i)}$ are all the covariates but $i$. The intercept $\alpha$ should perfectly explain a constant. Thus, you should not look at the VIF for the constant. Do notice that you still have to add this to the design matrix when you call the vif function that you use. The post I linked to also explain why the VIF for the constant is not infinite (there is not constant in the regression for the $R^2_i$ for the intercept).
The above does not explain the condition number in your regression.
Because the constant becomes zero when you standardize (assuming that the software you use uses $0/0 = 0$ or you get $0/\epsilon$ where $\epsilon$ is small).
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# PML Patriot
### Help Support The Rocketry Forum:
#### ColumbiaNX01
##### Red blooded white American male
Hey everyone! I am thinking about buying the Public Missiles 1/4 Patriot. Does anyone have some information about the kit other than information taht is on the PML website? Is it a pretty good engineered rocket kit? What are the pros and cons of the Patriot? I will be launching it with the 38mm H and I motors.
#### ColumbiaNX01
##### Red blooded white American male
Thanks for the input, I may go with the PML kit
#### COrocket
##### Well-Known Member
I agree, the 1/4 patriot is a solid kit. What i enjoyed is that it was the perfect size to do my level 1 and 2 with it...A H123W to 1500' and a J350W to about 4000'. Mine flies incredible on I and J motors. The only negative thing i would say is that the fins are 1/16", so i have done a couple of layers of tip to tip the fin section, to reduce flutter. I'd say this is only necessary with larger motors. The PR kit definitely has the components to take a larger motor. I could post a picture of mine if you would like to see one in flight.
#### Boosterdude
##### Well-Known Member
I have really enjoyed my PML 1/4 scale Patriot, it's a great flyer. I got my level 1 using a H123 which is a perfect combo.
The only thing I would change is leaving the piston out since it's a pain. The quantum tube is very temperature sensitive, so the piston really tightens up in cold weather. It's a minor problem that's easily solved.
awesome picture!
#### ColumbiaNX01
##### Red blooded white American male
Sweet pic! So if you recommened not to use the piston ejection system for protection of the recovery what do you recommened to protection?
#### COrocket
##### Well-Known Member
Sweet pic! So if you recommened not to use the piston ejection system for protection of the recovery what do you recommened to protection?
A Kevlar or Nomex chute protector would work fine. They are sold from most of the major rocket website vendors. Another note is I would recommend replacing the nylon shockcord that comes with the kit with about a equal amount of kevlar shockcord. After flying mine, i discovered that the nylon strapping above the motor tube and below the piston has seems to be a little stiff in areas, which means it could be close to melting. Its definitely something to consider if you plan on flying it frequently.
#### Boosterdude
##### Well-Known Member
A Kevlar or Nomex chute protector would work fine. They are sold from most of the major rocket website vendors. Another note is I would recommend replacing the nylon shockcord that comes with the kit with about a equal amount of kevlar shockcord. After flying mine, i discovered that the nylon strapping above the motor tube and below the piston has seems to be a little stiff in areas, which means it could be close to melting. Its definitely something to consider if you plan on flying it frequently.
Exactly, that's how I would do the next one. The piston is a neat idea, but a bit of a pain.
#### Marlin523
##### Well-Known Member
I like the paint job on your Patriot. Never cared for that yellow stripe you see so often.
#### mack
##### Active Member
I have really enjoyed my PML 1/4 scale Patriot, it's a great flyer. I got my level 1 using a H123 which is a perfect combo.
The only thing I would change is leaving the piston out since it's a pain. The quantum tube is very temperature sensitive, so the piston really tightens up in cold weather. It's a minor problem that's easily solved.
I am glad to hear this. I am planning on doing my Level 1 Cert flight in February with this exact combo. My Patriot doesn't look as nice as yours though but hopefully it will fly as well.
#### Boosterdude
##### Well-Known Member
I am glad to hear this. I am planning on doing my Level 1 Cert flight in February with this exact combo. My Patriot doesn't look as nice as yours though but hopefully it will fly as well.
Mack, I'm sure yours looks great. With this combo I'm sure that you will be successful.
Good luck!
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RcppDE v0.1.6
0
0th
Percentile
Global Optimization by Differential Evolution in C++
An efficient C++ based implementation of the 'DEoptim' function which performs global optimization by differential evolution. Its creation was motivated by trying to see if the old approximation "easier, shorter, faster: pick any two" could in fact be extended to achieving all three goals while moving the code from plain old C to modern C++. The initial version did in fact do so, but a good part of the gain was due to an implicit code review which eliminated a few inefficiencies which have since been eliminated in 'DEoptim'.
RcppDE
Rcpp port of Differential Evolution
The package provides global optimization by differential evolution.
It uses an efficient C++ based implementation of the DEoptim function which performs global optimization by differential evolution. Its creation was motivated by trying to see if the old approximation "easier, shorter, faster: pick any two" could in fact be extended to achieving all three goals while moving the code from plain old C to modern C++. The initial version did in fact do so, but a good part of the gain was due to an implicit code review which eliminated a few inefficiencies which have since been eliminated in DEoptim.
Author
Dirk Eddelbuettel extending DEoptim by David Ardia, Katharine Mullen, Brian Peterson and Joshua Ulrich, which itself is based on DE-Engine by Rainer Storn.
Functions in RcppDE
Name Description DEoptim.control Control various aspects of the DEoptim implementation DEoptim-methods DEoptim-methods DEoptim Differential Evolution Optimization No Results!
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Home > Research > Publications & Outputs > The evolution of rest-frame UV properties, Lya ...
Electronic data
• 1910.02959v1
Rights statement: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Monthly Notices of the Royal Astronomical Society following peer review. The definitive publisher-authenticated version S Santos, D Sobral, J Matthee, J Calhau, E da Cunha, B Ribeiro, A Paulino-Afonso, P Arrabal Haro, J Butterworth, The evolution of rest-frame UV properties, Ly α EWs, and the SFR–stellar mass relation at z ∼ 2–6 for SC4K LAEs, Monthly Notices of the Royal Astronomical Society, Volume 493, Issue 1, March 2020, Pages 141–160, https://doi.org/10.1093/mnras/staa093 is available online at: https://academic.oup.com/mnras/article/493/1/141/5704403
Accepted author manuscript, 4.21 MB, PDF document
The evolution of rest-frame UV properties, Lya EWs and the SFR-Stellar mass relation at z~2-6 for SC4K LAEs
Research output: Contribution to journalJournal articlepeer-review
Published
Close
Journal publication date 1/03/2020 Monthly Notices of the Royal Astronomical Society 1 493 20 141-160 Published English
Abstract
We explore deep rest-frame UV to FIR data in the COSMOS field to measure the individual spectral energy distributions (SED) of the ~4000 SC4K (Sobral et al. 2018) Lyman-alpha (Lya) emitters (LAEs) at z~2-6. We find typical stellar masses of 10$^{9.3\pm0.6}$ M$_{\odot}$ and star formation rates (SFR) of SFR$_{SED}=4.5^{+8.8}_{-2.5}$ M$_{\odot}$/yr and SFR$_{Lya}=5.9^{+6.3}_{-2.6}$ M$_{\odot}$/yr, combined with very blue UV slopes of beta=-2.0$^{+0.3}_{-0.5}$, but with significant variations within the population. M$_{UV}$ and beta are correlated in a similar way to UV-selected sources, but LAEs are consistently bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of the UV-selected population. We also study the Lya rest-frame equivalent width (EW$_0$) and find 45 "extreme" LAEs with EW$_0>240$ A (3 $\sigma$), implying a low number density of $(7\pm1)\times10^{-7}$ Mpc$^{-3}$. Overall, we measure little to no evolution of the Lya EW$_0$ and scale length parameter ($w_0$) which are consistently high (EW$_0=140^{+280}_{-70}$ A, $w_0=129^{+11}_{-11}$ A) from z~6 to z~2 and below. However, $w_0$ is anti-correlated with M$_{UV}$ and stellar mass. Our results imply that sources selected as LAEs have a high Lya escape fraction (f$_{esc, Lya}$) irrespective of cosmic time, but f$_{esc, Lya}$ is still higher for UV-fainter and lower mass LAEs. The least massive LAEs (\$
Bibliographic note
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Monthly Notices of the Royal Astronomical Society following peer review. The definitive publisher-authenticated version S Santos, D Sobral, J Matthee, J Calhau, E da Cunha, B Ribeiro, A Paulino-Afonso, P Arrabal Haro, J Butterworth, The evolution of rest-frame UV properties, Ly α EWs, and the SFR–stellar mass relation at z ∼ 2–6 for SC4K LAEs, Monthly Notices of the Royal Astronomical Society, Volume 493, Issue 1, March 2020, Pages 141–160, https://doi.org/10.1093/mnras/staa093 is available online at: https://academic.oup.com/mnras/article/493/1/141/5704403
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# Additive Model with Linear Smoother
##### Posted on Dec 07, 2021
A number of problems associated with $p$-dimensional smoothers
Take a different approach and use the one-dimensional smoother as a building block for a restricted class of nonparametric multiple regression models.
The additive model takes the form
$E(Y_i\mid x_{i1},\ldots, x_{ip}) = \sum_{j=1}^pf_j(x_{ij})$
The model is a special case of both the
• PPR (projection pursuit regression)
• ALS (alternating least squares)
• ACE (alternating conditional expectation)
## Additive Model and Its Normal Equations
### Least Squares on Populations
The optimization problem is to minimize
$E(Y-g(X))^2$
over $g(X)=\sum_{j=1}^pf_j(X_j)\in \cH^{add}$. Without the additivity restriction, the solution is simply $E(Y\mid X)$.
By assumption, $\cH^{add}$ is a closed subspace of $\cH$ this minimum exists and is unique, but the individual functions $f_i(X_i)$ may not be uniquely determined.
The minimizer can be characterized by residuals $Y-g(X)$ which are orthogonal to the space of fits: $Y-g(X)\ind \cH^{add}$. Since $\cH^{add}$ is generated by $\cH_i$, we have equivalently: $Y-g(X)\ind \cH_i$, or $P_i(Y-g(X))=0$. Componentwise this can be written as
$f_i(X_i) = P_i(Y-\sum_{j\neq i}f_j(X_j))$
### Penalized Least Squares
For single-smoother,
$\Vert y-f\Vert^2 + \lambda f^TKf$
Assuming the inverses exist, the stationary condition implies
$f = (I+\lambda K)^{-1}y$
that is
$S = (I+\lambda K)^{-1}$
Then, characterize $f=Sy$ as a stationary solution of
$\Vert y - f\Vert^2 + f^T(S^{-1}-I)f$
Extend to additive regression by penalizing the RSS separately for each component function,
$Q(f) = \Vert y - \sum_{j=1}^p\Vert^2 + \sum_{j=1}^pf_j^T(S_j^{-1}-I)f_j$
A proof for the existence of solution for cubic smoothing spline in the appendix.
### Algorithms for solving the normal equations
The Gauss-Seidel method is only one technique in the large class of iterative schemes called successive over-relaxation (SOR) methods. They differ from ordinary Gauss-Seidel procedures by the amount one proceeds in the direction of the Gauss-Seidel updates
$f_j\leftarrow (1-\omega)f_j + \omega S_j\left(y-\sum_{k\neq j}f_k\right)$
If the Gauss-Seidel algorithm converges, so do successive over-relaxation iterations for relaxation parameters $0 < \omega < 2$.
The numerical analysis literature also distinguishes between successive and simultaneous iterations, referred to as
• Gauss-Seidel
• Jacobi
### Summary of Consistency, Degeneracy, and Convergence Results
It is not a priori clear when the normal equations are consistent (when solutions exist). Nor is it clear when the equations are nondegenerate (when the solutions are unique).
nondegeneracy implies consistency. However, the normal equations are almost always degenerate.
1. For symmetric smoothers with eigenvalues in [0, 1], the normal equations always have at least one solution
2. The solution is unique unless there exists a $g\neq 0$ such that $\hat Pg = 0$, a phenomenon we call “concurvity”. This implies for any solution $f$, $f+\alpha g$ is also a solution for any $\alpha$
### Consistency
If each $S_j$ is symmetric with eigenvalues in $[0, 1]$, the normal equations are consistent for every $y$
If the smoothers $S_j$ are symmetric with eigenvalues in $[0, 1)$, the solutions of the normal equations can be written as $f_j = A_j(I+A)^{-1}y$, where $A_j = (I-S_j)^{-1}S_j$ and $A = \sum_jA_j$.
### Degeneracy of smoother-based normal equations
Collinearity detection as part of regression diagnostics is a must in every good regression analysis. Practioners are usually concerned with approximate collinearity and its inflationary effects on standard errors of regression coefficients.
The term “collinearity” refers to linear dependencies among predictors as the cause of degeneracy, the term “concurvity” has been used to describe nonlinear dependencies which lead to degeneracy in additive models.
In a technical sense, concurvity boils down to collinearity of nonlinear transforms of predictors. It is more intuitive to think of it as an additive dependence $f_+=0$.
Exact concurvity is defined as the existence of a nonzero solution of the corresponding homogeneous equations
$\hat Pg = 0$
if such a $g$ exists, and if $f$ is a solution to $\hat Pf=\hat Qy$, then so is $f+wg$ for any $\omega$, and thus infinitely many solutions exist.
The set of all nonzero solutions to the homogeneous equations $\hat Pg=0$ will be called concurvity space for the normal equations.
It is easy to check that
$g = \begin{bmatrix} \alpha 1\\ -\alpha 1 \end{bmatrix}$
lies in the concurvity space of the two-smoother problem if they both reproduce constants. Similarly, for $p$ such smoothers, the concurvity space has dimension at least $p-1$.
Consider $y=0$,
$Q(g) = \Vert g_+\Vert^2 + \sum_{j=1}^p g_j^T(S_j^{-1}-I)g_j\,,$
defined for $g_j\in\cR(S_j)$.
If the smoothers $S_j$ are all symmetric with eigenvalues in $[0,1]$, a vector $g\neq 0$ with $g_j\in \cR(S_j)$ represents a concurvity ($\hat Pg=0$) iff one of the following equivalent conditions is satisfied
1. $Q(g) = 0$, that is, $g$ minimizes $Q$
2. $g_j\in \cM_1(S_j),j=1,\ldots,p$, and $g_+=0$.
Condition 2 implies that exact concurvity is exact collinearity if, for example, all smothers are cubic spline smoothers.
Remark: if $S_j,j=1,\ldots,p$ are symmetric with eigenvalues in $[0, 1)$, then $\hat P$ is nonsingular.
In practice, we separate the constant term in the additive model, and adjust each of the smooth terms to have mean 0.
For two smoothers, there exists exact concurvity iff $f_1=(S_1S_2)f_1$ for some $f_1\neq 0$
If $\Vert S_1S_2\Vert < 1$ for some matrix norm, concurvity does not exist.
For two symmetric smoothers with eigenvalues in $(-1, +1]$, concurvity exists iff $\cM_1(S_1)\cap \cM_1(S_2)\neq 0$.
It has again the consequence that exact concurvity, e.g., for a pair of cubic spline smoothers, can only be an exact collinarity between the untransformed predictors, since cubic splines preserve constant and linear fits (what is the logic?).
uniqueness of the additive minimizer is guaranteed if there is no collinearity.
### The Convergence of backfitting: $p$ smoothers
If all the smoothers $S_j$ are symmetric with eigenvalues in $[0, 1]$, then the backfitting algorithm converges to some solution of the normal equations.
### Convergence of backfitting for two smoothers
Decompose $S_1=\tilde S_1+H_U$ and $S_2=\tilde S_2+H_U$, where $H_U$ is the orthogonal projection onto $U = \cM_1(S_1)\cap \cM_1(S_2)$. We have $\tilde S_jH_U=H_U\tilde S_j=0$ and $\Vert \tilde S_1\tilde S_2\Vert_2 < 1$.
Consider first $y$ and $f_2^0$ in $U^\ind$,
second for $y$ and $f_2^0$ in U
If $S_1$ and $S_2$ are symmetric with eigenvalues in $(-1, 1]$, then the Gauss-Seidel algorithm converges to a solution of the normal equations
The components $f_1^\infty$ and $f_2^\infty$ can be decomposed into
• the part within $U^\ind$ which is uniquely determined and depends on the data $y$ only
• the part within $U$ which depends on the sequence of iteration and the initialization $f_2^0$
### Modified Backfitting Algorithm
1. Initialize $\tilde f_1,\ldots,\tilde f_p$, and set $\tilde f_+ = \tilde f_1+\tilde f_2+\cdots+\tilde f_p$
2. Regress $y-\tilde f_+$ onto the space spanned by $\cM_1(S_1),\ldots,\cM_1(S_p)$, that is, set $g=H(y-\tilde f_+)$
3. Fit an additive model to $y-g$ using smoothers $\tilde S_i$; this step yields an additive fit $\tilde f_+=\tilde f_1+\cdots +\tilde f_p$
4. repeat steps 1 and 2 until convergence.
### A closer look at convergence
Consider the case of extreme collinearity, where two identical covariates and a cubic spline smoother matrix $S$.
Starting the backfitting algorithm from $0$, the residual after $m$ smooths is given by
$r^m = [I-S+S^2-S^3+\cdots + (-1)^{m-1}S^{m-1}](I-S)y\rightarrow (I+S)^{-1}(I-S)y$
1. the residuals (and their norm) oscillate as they converge
2. the converged model is rougher than a single smoother.
3. by looking at every other iteration, it is clear that the norm of the residuals converges upwards, after every even number of steps
4. $r^2$ is the same as the “twicing” residual, where twicing enhances a smooth by adding in the smooth of the residual.
Published in categories Note
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# Commutator help
1. Jun 13, 2007
### n0_3sc
1. The problem statement, all variables and given/known data
I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them.
Does this make sense? I don't see how it can be possible.
2. Relevant equations
[s,B]
(The s should also have a hat on it)
3. The attempt at a solution
I have sB - Bs but do i represent s as (sx,sy,sz)? x,y,z are subscripts...
Even if I do that wouldn't the commutation = 0?
2. Jun 13, 2007
### George Jones
Staff Emeritus
Represent both spin and the magnetic field in terms of Pauli spin matrices.
3. Jun 13, 2007
### nrqed
As stated, the question does not quite make sense. I think you mean the commutator of the spin with the hamiltonian of a particle in a uniform B field, $H = \vec{s} \cdot \vec{B}$ . Then you simply have to use the commutation relation of the Pauli matrices $[S_i,S_j] = i \epsilon_{ijk} S_k$ and the result follows trivially (except that it seems to me that one gets "i" times the cross product)
Patrick
4. Jun 13, 2007
### n0_3sc
nrqed:
So I evaluate [H,s]? In doing that, why would I need the commutation relation $$[S_i,S_j] = i \epsilon_{ijk} S_k$$ ? It shouldn't be needed if the product terms are only between terms of H and $$s_x, s_y, s_z$$.
5. Jun 13, 2007
### nrqed
But H contains the spin!! See my post.
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# continuous random variable examples
Continuous random variables describe outcomes in probabilistic situations where the possible values some quantity can take form a continuum, which is often (but not always) the entire set of real numbers R \mathbb{R} R.They are the generalization of discrete random variables to uncountably infinite sets of possible outcomes.. Suppose I am interested in looking at statistics test scores from a certain college from a sample of 100 students. An example would be, let's say you play a game. Continuous Random Variables. Continuous variables are variables that measure something. Random Variables can be either Discrete or Continuous: Discrete Data can only take certain values (such as 1,2,3,4,5) Continuous Data can take any value within a range (such as a person's height) In our Introduction to Random Variables (please read that first!) So you flip a coin. My favorite example of a continuous variable is how many gallons of milk a cow gives. Note that the interpretation of each is the same as in the discrete setting, but we now have a different method of calculating them in the continuous setting. Example \(\PageIndex{2}\) At a particular gas station, gasoline is stocked in a bulk tank each week. Not all random variables are continuous or discrete. Example \(\PageIndex{1}\) We now consider the expected value and variance for continuous random variables. And with probability 1/2, you get a reward of 1/2 dollars. You can cook up random variables that are kind of neither or a mixture of the two. Continuous. we look at many examples of Discrete Random Variables. And with a certain probability, you get a certain number of dollars in your hands. Let's see another example. In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. Let random variable \(X\) denote the proportion of the tank's capacity that is stocked in a given week, and let \(Y\) denote the proportion of the tank's capacity that is sold in the same week.
### Похожие записи
• Нет похожих записей
вверх
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# Difference between revisions of "CDS 140a Winter 2013 Homework 6"
R. Murray, D. MacMartin Issued: 12 Feb 2013 (Tue) ACM 101/AM 125b/CDS 140a, Winter 2013 Due: 19 Feb 2013 (Tue)
__MATHJAX__
Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).
1. Perko, Section 2.14, problem 1
(a) Show that the system
<amsmath>\aligned
\dot x&=a_{11}x+a_{12}y+Ax^2-2Bxy+Cy^2\\ \dot y&=a_{21}x-a_{11}y+Dx^2-2Axy+By^2
\endaligned</amsmath>
is a Hamiltonian system with one degree of freedom; i.e., find the Hamiltonian function $H(x,y)$ for this system.
(b) Given $f\in C^2(E)$, where $E$ is an open, simply connected subset of $\mathbb R^2$, show that the system $\dot{x}=f(x)$ is a Hamiltonian system on $E$ iff $\nabla\cdot f(x)=0$ for all $x\in E$.
2. Perko, Section 2.14, problem 7. Show that if $x_0$ is a strict local minimum of $V(x)$ then the function $V(x)-V(x_0)$ is a strict Lyapunov function (i.e., $\dot{V}<0$ for $x\neq 0$) for the gradient system $\dot x=-\mathrm{grad}V(x)$.
3. Perko, Section 2.14, problem 12. Show that the flow defined by a Hamiltonian system with one degree of freedom is area preserving. Hint: Cf. Problem 6 in Section 2.3
4. Perko, Section 3.3, problem 5. Show that
<amsmath>\aligned
\dot x &=y+y(x^2+y^2)\\ \dot y &=x-x(x^2+y^2)
\endaligned</amsmath>
is a Hamiltonian system with $4H(x,y)=(x^2+y^2)^2-2(x^2-y^2)$. Show that $dH/dt=0$ along solution curves of this system and therefore that solution curves of this system are given by
<amsmath>
(x^2+y^2)^2-2(x^2-y^2)=C
</amsmath>
Show that the origin is a saddle for this system and that $(\pm 1,0)$ are centers for this system. (Note the symmetry with respect to the $x$-axis.) Sketch the two homoclinic orbits corresponding to $C=0$ and sketch the phase portrait for this system. (You need not comment on the compound separatrix cycle.)
5. A planar pendulum (in the $x$-$z$ plane) of mass $m$ and length $\ell$ hangs from a support point that moves according to $x=a\cos (\omega t)$. Find the Lagrangian, the Hamiltonian, and write the first-order equations of motion for the pendulum.
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I’ve been experimenting with Tomasz Janczuk’s awesome iisnode project lately, in hopes of hosting some of my Node.js sites on IIS. Those sites are running well enough on an Ubuntu VPS currently, but the features that iisnode offers are compelling (and I have unused capacity on one of my Windows servers).
Along the way, I ran into one issue that isn’t addressed in the current iisnode documentation or examples very well. If you try to set up a standalone Node.js website in IIS you’re likely to be greeted with this error when you load it up for the first time:
Remotely, you’ll see the dreaded “The service is unavailable.” error:
Those nondescript errors don’t exactly make resolving the problem very straightforward. So, I want to recap what I found to be the underlying problem in my case and a few solutions, in hopes of helping anyone else that ends up with the same issue.
### The problem
The problem boils down to the default application pool identity account not having sufficient permission to carry out iisnode’s work. Part of the IIS error message does somewhat cryptically hint at what’s wrong:
An invalid identity in the application pool could cause this error.
That’s not very actionable though.
In the specific case of a barebones Node.js site running under iisnode’s default configuration, this occurs due to iisnode attempting to create its log file when your application is accessed for the first time.
If you’ve run iisnode locally, you’ve probably seen the script-name.js.logs folder that appears after first run. Unfortunately for iisnode, creating this folder (and the log file(s) inside it) is beyond the default application pool identity’s abilities and that results in our 503 error.
### The nuclear solution (don’t do this)
The most straightforward way to eliminate the 503 error and get back to work is to change your site’s application pool identity from the default ApplicationPoolIdentity to LocalSystem:
That works well enough if you’re just playing around on a development machine and security isn’t a concern.
However, elevating the application pool’s privileges that high is not something you should do in production. If an attacker finds a hole in your Node.js code, running the site under LocalSystem means they will gain unfettered access to your entire web server. Not good.
### Avoiding the problem altogether
Before changing any permissions whatsoever, ask yourself whether or not you care about iisnode’s logging to begin with. It’s enabled by default, but easy enough to disable by tweaking your site’s web.config:
<configuration> <system.webServer> <iisnode loggingEnabled="false" /> </system.webServer> </configuration>
With that small change, iisnode won’t attempt to create a log directory at all, and the file system permissions issue is moot.
If your Node.js code doesn’t attempt to write to disk otherwise and you don’t care about iisnode’s logging feature, this is probably the best solution since it requires no privilege elevation at all.
### A more limited escalation
Instead of giving your application pool the keys to the kingdom, a better solution is to grant ApplicationPoolIdentity elevated privileges only on the specific location where your Node.js application resides.
You can grant those permissions via either the Windows GUI interface or the command line, depending on your preference.
#### Graphical approach
Assuming you have access to the server’s desktop, you can give IIS write access through the usual Properties > Security > Edit… dialogs. In a default installation, the IIS_IUSRS group is what you need to grant write access to:
You might notice that I’m only granting elevated privileges to app.js.logs above, not the entire site structure.
It’s often the case that a single file, like my site’s app.js, will be the exclusive entry point to an entire site. When that’s the case, there will only be a single iisnode logging folder and it’s a good idea to even further isolate write access to the corresponding .logs folder.
#### Command line approach
Using the Windows GUI to set file permissions is okay, but it can be a bit cumbersome to dig through the file system and then navigate through the dialog windows. More importantly, you may need to script this as part of an unattended deployment process.
Windows also includes a command line utility to handle this: icacls. Here’s an example of how you could grant the IIS user (and iisnode) write access to an entire site structure from the command line:
icacls c:\path\to\your\site /grant IIS_IUSRS:(OI)(CI)W
Running that on my specific example of MySite.com looked like this:
Limiting access specifically to the logging folder, as shown in the GUI approach above, is simple enough too:
icacls c:\path\to\your\site\app.js.logs /grant IIS_IUSRS:(OI)(CI)W
If you’re curious about the syntax of the command, it’s less complicated than it looks. The path and /grant are self-explanatory, but the rest is a bit terse.
IIS_IUSRS:(OI)(CI) means that the change should apply to any objects and containers within the IIS_IUSRS group.
The W at the end determines what type of permission should be granted. We could also have used M to give both write and modify permissions to iisnode, or F to grant full control.
### This applies to more than logging
Of course, hitting a permissions error trying to write logs is just one possible reason you might encounter this error. Since this particular cause is one that everyone will probably run into when setting up a standalone site under iisnode, I decided that it’s worth singling out individually though.
If your Node.js code itself needs access to write to the file system, you’ll need to grant the correct permissions in those areas of your site’s structure too (following the same approach as above, but altering the path appropriately).
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# MARLA - Editorial
Author: Ankush Khanna
Easy
### PROBLEM:
Given an N \times M grid, filled with integers, find the largest connected area (side adjacent elements) filled with same numbers. If there are more than one such connected areas, find the one filled with the minimum number. Print the Value and Area of that region.
### QUICK EXPLANATION:
Key to AC: Run a Depth First Search on the elements of the grid to find the various connected components, and then just find the one with maximum elements connected, if there are more than one such areas, find the one with minimum number written on them.
### EXPLANATION:
To solve this problem, one needs to know about what connected components are. And the best approach would be a DFS through the grid to find all connected components in this grid (connected in a sense that same elements which are side adjacent are parts of one single component). To find the region with the maximum number of elements is just an added task here, we just need to keep a track of it, that’s it. Moreover, finding the region with largest area (number of elements) and having the least possible element value if there exist more than one regions with the maximum area, is another added task.
Almost all the solutions that achieved only 30 points (sub-task 1), either missed on some edge cases, failed to keep track of optimal region correctly, or were unable to control recursion for their recursive DFS implementation (improper user of the visited nodes).
This problem can easily be solved in O(N \times M) time with O(N \times M) auxiliary space for keeping a track of the visited nodes, just to control excessive recursion (for recursive approaches) or excessive searching (for iterative approaches).
This problem is very similar to the Number of Islands problem, the difference here is just that here we need to keep a track of the area and value involved for a particular region found. Also, a similar problem is the Largest Region problem.
### COMPLEXITY:
O(N \times M) time with O(N \times M) auxiliary space.
### RELATED PROBLEMS:
Number of Islands Problem
Largest Region Problem
Two Flowers (This one is way more complex because it involves DSU, as we can have at most two different numbers in a region, a very good problem to practice).
Feel free to share your approach. If you have any queries, they are always welcome.
2 Likes
Hello, can someone please help me where am I going wrong:
https://www.codechef.com/viewsolution/32032103
Actually you need to change the line 10 of your code to this:
visited = [[False for _ in range(m) ] for _ in range(n)]
You need to have a grid with M columns and N rows, while you currently have written N columns and M rows for your visited grid, which is actually raising an index out of range runtime exception in your dfs code at line 5.
Hope it helps!
1 Like
Thanks a lot man, that was my bad. But still I am failing at TC #9 under sub task #2
https://www.codechef.com/viewsolution/32034283
Will you please help me, I am getting wrong only on one TC.
We haven’t made any non-trivial changes.
Can you provide a submission link which had got AC in the recent past, but gets RE on submitting the same code now?
1 Like
Oh I am sorry. It was my bad. I should’ve looked more closely. It is all fine from CodeChef’s end.
First of all you should set your Python recursion limit to some large number, as the tests are large in sub task 2 (1 \leq N \times M \leq 10^{6}), and Python’s default recursion limit is 10^{3}. This can be done easily by the following code:
from sys import setrecursionlimit
setrecursionlimit(10**9) # Or some other large number
But again, as the time limit for this problem is very tight, Python, probably, is not the right tool to solve it. It is an interpreted language and is very slow on recursion. During this contest, there wasn’t any 100 point Python submission. There is only one 100 point Python submission, and that too in the practice section. It contains a non-recursive DFS implementation.
This is a Python3 solution by @kkoder_1729 : https://www.codechef.com/viewsolution/24457833
You can have a look at it.
Hope it helps!
1 Like
Thank you so much for the help man
Can anybody help me where am I going wrong? I’m getting wrong answer in two testcases for the original constraints. Below is the link to my solution:
https://www.codechef.com/viewsolution/33200381
Cannot view the solution since it is a contest
These problems are available in practice section also, and CodeChef DSA Learning Series is meant for Educational Purpose only. So you can share your approach or code from the practice section also. Or you can read the editorial or refer to an AC solution for more help.
can you please help me …i am getting wa on some testcases…but i dont know why
here is the link of my solution
https://www.codechef.com/viewsolution/34206931
Your solution is not visible to others, it might be from the DSA contest. You can try the problem in the practice section here. Also, you can check others’ solutions from the practice section for more clarity.
36205845
Why I am getting WA in last two test cases in original constraints in above solution ?
Your DFS method is incorrectly implemented. You are not checking all the adjacent values. Here’s a counter case where your DFS will fail:
3 3
1 2 1
2 2 2
1 2 1
Your solution will give the output as:
2 4
While the correct answer is:
2 5
(Specifically, you’re not checking elements onto the left side of a position).
Hope it helps!
got it. Thanks
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B. Young Explorers
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
Young wilderness explorers set off to their first expedition led by senior explorer Russell. Explorers went into a forest, set up a camp and decided to split into groups to explore as much interesting locations as possible. Russell was trying to form groups, but ran into some difficulties...
Most of the young explorers are inexperienced, and sending them alone would be a mistake. Even Russell himself became senior explorer not long ago. Each of young explorers has a positive integer parameter $e_i$ — his inexperience. Russell decided that an explorer with inexperience $e$ can only join the group of $e$ or more people.
Now Russell needs to figure out how many groups he can organize. It's not necessary to include every explorer in one of the groups: some can stay in the camp. Russell is worried about this expedition, so he asked you to help him.
Input
The first line contains the number of independent test cases $T$($1 \leq T \leq 2 \cdot 10^5$). Next $2T$ lines contain description of test cases.
The first line of description of each test case contains the number of young explorers $N$ ($1 \leq N \leq 2 \cdot 10^5$).
The second line contains $N$ integers $e_1, e_2, \ldots, e_N$ ($1 \leq e_i \leq N$), where $e_i$ is the inexperience of the $i$-th explorer.
It's guaranteed that sum of all $N$ doesn't exceed $3 \cdot 10^5$.
Output
Print $T$ numbers, each number on a separate line.
In $i$-th line print the maximum number of groups Russell can form in $i$-th test case.
Example
Input
2
3
1 1 1
5
2 3 1 2 2
Output
3
2
Note
In the first example we can organize three groups. There will be only one explorer in each group. It's correct because inexperience of each explorer equals to $1$, so it's not less than the size of his group.
In the second example we can organize two groups. Explorers with inexperience $1$, $2$ and $3$ will form the first group, and the other two explorers with inexperience equal to $2$ will form the second group.
This solution is not unique. For example, we can form the first group using the three explorers with inexperience equal to $2$, and the second group using only one explorer with inexperience equal to $1$. In this case the young explorer with inexperience equal to $3$ will not be included in any group.
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Math Help - Harmonic Functions
1. Harmonic Functions
Show by example that a harmonic function need not have an analytic completion in a multiply connected domain. [HINT: Consider ln(|z|), z a complex number]
well I considered u=ln(x^2+y^2) where z=x+iy
and I figured out it was harmonic (second partial derivitives are 0) but I do not know where to go from there.
2. what is the definition of analytic completion?
3. Originally Posted by xxp9
what is the definition of analytic completion?
Analytic completion for a function u is when there exists a harmonic function v in a simply connected domain such that u+iv is analytic.
4. So let u=ln|z|, u is hormonic in the domain of punctured plane $R^2 - {0}$.
Since an analytic function is determined in any open sub-set of the plane the analytic completion is unique( up to a constant), if it exists.
So the only possible analytic completion for u would be f=lnz=u + i argz
While f can only be defined on a plane where a half line is cut.
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Per Base Sequence Content
0
0
Entering edit mode
2.9 years ago
Negin ▴ 20
Hi all,
I have a question regarding "Per Base Sequence Content" plot for "fastqc":
In the fastqc documentation, it is written: "In a random library you would expect that there would be little to no difference between the different bases of a sequence run, so the lines in this plot should run parallel with each other."
But I don't understand why different bases in a read should follow the same pattern of allele frequency ("A/T/C/G"). I mean they are different positions in the genome and it is normal that each position has different allele.
I would appreciate it if someone could help me, please.
sequencing fastqc genome sequence • 3.3k views
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Please post an example plot if you have one. : How to add images to a Biostars post
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To my understand. take DNA-seq for example, with enough sequencing depth, the reads from random library would cover the whole genome equally. And so, the base content in each position of the reads are the same, that equal to whole genome GC content.
So, for a ideal random library, the per base sequence content plot in fastqc report shows four lines in parallel (G=C, A=T).
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Thanks for your reply, but what do you mean by G=C and A=T? I mean all the reads are coming from the same strand, right? so in a single strand, why should the per base content for A and T or G and C be equal?
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I take the DNA-seq for example, we could get the reads from both strand, so in each position of the read, the GC% content would be identical (ideally), equal to the whole GC% content.
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If the reads come from both strands, I agree with you, but this is not always the case, right?
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sure, it not always for some libraries. And we should be very careful for the data.
To my experience, in small RNAseq, CLIPseq, etc, I see the GC% content is not identical across the read (my data), it shows that per base content lines crossed over in fastqc report.
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That should always be the case. Even in stranded sequencing. Unless you are using some method that is discarding one strand entirely.
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So, you mean even in the single-end sequencing, the reads come from the both strands?
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yes. If "both strands" you mentioned is refer to DNA double strand. For DNA-seq, the insert fragment is strandless, so sequencing reads could not tell which strand is "forward of DNA".
I have no idea which high-throughput library protocol could generate fragment from either 'forward' or 'reverse' strand of genome. (and No need to do this).
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thanks for your answer, yeah, I mean for DNA-Seq. but for alignment, we should separate the reads that are located in different strands, right? Otherwise, how can we align them?
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Aligners automatically check for alignment on both strands by making a reverse complemented copy of the reference you provide. Since DNA sequence is always written in 5'-->3' order the reference you provide is considered forward/top strand.
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So in this case, as you said, for an ideal random library, it should be always G=C and A=T, but I found in the fastqc documentation, they provide an example for good illumina data: https://www.bioinformatics.babraham.ac.uk/projects/fastqc/good_sequence_short_fastqc.html#M4 which doesn't satisfy these conditions! Then how this data is considered having a good base per content, but A!=T and C!=G?
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That graph plots A/C/T/G seen at a particular cycle in the sequencing. Depending on the GC% of the genome and the way genome got fragmented you would see variation (while we expect the fragmenting to be random there are likely biases depending on how the fragmentation was done and sequence itself). That said I have seen datasets where A/C/T/G curves almost perfectly overlapped.
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The example file contains 250000 reads, that is not enough to randomly cover the whole genome.
Do try it yourself
I suggest download real datasets from NCBI/SRA or EBI, and run the fastqc yourself to check whether "G=C" or not.
I also prepared a example of my data (DNA-seq, 19M, Paired-end reads from fruitfly) for you.
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Thank you so much for your detailed answer. I am new in sequencing, but I implied that each fragment in DNA sequencing consists of two strands, is it true or not?
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You only sequence one strand at a time.
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what about the paired-end reads? Are the mates in paired-end reads related to one strand or both?
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Same fragment is sequenced from either end (5'--> 3') to generate the two reads you get. Reads are from opposite strands of that fragment.
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Thank you so much for your help. the discussion here was so beneficial for me.
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Dear @genomax, Now, another question comes to my mind: In the case of paired-end sequencing, do we know to which strand read1 and read2 belong to or we just know that they are from opposite strands?
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DNA is anti-parallel so the concept of strands is relative. One always sequences 5' --> 3' so whichever end sequences first (that would be the end where the p5 adapter ligated) becomes read 1.
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For DNA-seq, it cannot tell which read belong to which strand. because both forward and reverse strand of DNA are able to connect to the P5-end of library in 5'-3' direction (it is read1 in Paired end sequencing).
But for strand-specific RNA sequencing, which strand ligated to the P5-end adapter (5'-3' direction) is determined, so we can tell discriminate which strand the read belong to. for example: for dUTP-strand-specific RNA library. the read2 is from the sense strand of mRNA.
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Thanks for your answer, so, this means in the case of DNA-seq, the reads in the file of read1s, can belong to different strands of DNA and the same for read2s, right?
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Does anyone else know the answer?
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Yes, the insert fragment is double strand DNA, and sequencing reads (read1 or read2) was from only one strand of it. you could read some Illumina library materials or videos. for example:
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Hello negin
It appears that your post has been cross-posted to another site: https://bioinformatics.stackexchange.com/questions/11701/per-base-sequence-content-in-fastqc
This is typically not recommended as it runs the risk of annoying people in both communities.
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Not sure if it was locked on accident or if I did something wrong. However, I can't reply to the users that have responded.
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Moderator
## Re: Thread locked
tplaya07 wrote:
Not sure if it was locked on accident or if I did something wrong. However, I can't reply to the users that have responded.
As far as I can tell it was inadvertently locked. You didn't do anything wrong and you're now able to reply as I have unlocked the thread.
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## Re: Thread locked
OK... cool. Thank you.
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# vspace or line space in text of wrap fig
I am using wrap fig and want to insert a line break within the text. However, \vspace or \linebreak or \break don't lead to any change.
Here is a sample of my LaTeX code
\begin{figwindow}[0,r,%
{\includegraphics[height=2in]{PascoSynthesizer.eps}},%
{\label{fig:label} Caption
}]
\textbf{Electronic Output}:\\
A sum of nine harmonics of sine waves with a\\ fundamental of 440 Hz plus a second output of the fundamental. \\
\vspace{0.3in}
\textbf{Controls:}\\
\end{figwindow}
-
Please always post complete (small) document showing packages used the wrapfig package does not define a figwindow environment, so your title and tag do not match your example. – David Carlisle Feb 1 at 9:54
picinpar is quite an old package with several restrictions, a newer package for these kind of inserts is wrapfig however you can force a space by adding a blank line with a strut (or a rule of a specified height if you need finer control).
\documentclass[a4paper]{article}
\usepackage{picinpar}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figwindow}[0,r,%
{\includegraphics[height=2in]{PascoSynthesizer.eps}},%
{\label{fig:label} Caption
}]
\noindent
\textbf{Electronic Output}:\\
A sum of nine harmonics of sine waves with a\\
fundamental of 440 Hz plus a second output of the fundamental.\\
\strut\\
\textbf{Controls:}
\end{figwindow}
\end{document}
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Success: the line \strutt\\ worked to produce a linespace. the inclusion of [demo] caused problems and was omitted. GREAT GRATITUDE!! – Leon Gunther Feb 4 at 2:25 demo makes includegraphics just use a black square instead of the picture you should use it when posting examples because we have not got the file PascoSynthesizer.eps so can not run the code otherwise. Also in questions please post complete documents as in this answer not fragments that can not be run. – David Carlisle Feb 4 at 8:20 Thanks for the explanation of demo! – Leon Gunther Feb 7 at 2:23
I guess you have to check your preamble. If I use:
\documentclass[a4paper]{article}
\usepackage{picinpar}
\usepackage{graphicx}
and then your code, everything runs well, as you can see from the screenshot below (or isn't this the desired output?):
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Thanks for your response. I do have what you show above. What I need is to add a line space between '... fundamental.' and 'Controls' – Leon Gunther Feb 1 at 13:59
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# 3.1 Complex numbers (Page 4/8)
Page 4 / 8
## Dividing complex numbers
Divide $\text{\hspace{0.17em}}\left(2+5i\right)\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\left(4-i\right).$
We begin by writing the problem as a fraction.
$\frac{\left(2+5i\right)}{\left(4-i\right)}$
Then we multiply the numerator and denominator by the complex conjugate of the denominator.
$\frac{\left(2+5i\right)}{\left(4-i\right)}\cdot \frac{\left(4+i\right)}{\left(4+i\right)}$
To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).
Note that this expresses the quotient in standard form.
## Substituting a complex number into a polynomial function
Let $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-5x+2.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(3+i\right).$
Substitute $\text{\hspace{0.17em}}x=3+i\text{\hspace{0.17em}}$ into the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-5x+2\text{\hspace{0.17em}}$ and simplify.
Let $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}-3x.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(8-i\right).$
$102-29i$
## Substituting an imaginary number in a rational function
Let $\text{\hspace{0.17em}}f\left(x\right)=\frac{2+x}{x+3}.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(10i\right).$
Substitute $\text{\hspace{0.17em}}x=10i\text{\hspace{0.17em}}$ and simplify.
Let $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+1}{x-4}.\text{\hspace{0.17em}}$ Evaluate $\text{\hspace{0.17em}}f\left(-i\right).$
$-\frac{3}{17}+\frac{5i}{17}$
## Simplifying powers of i
The powers of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ are cyclic. Let’s look at what happens when we raise $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ to increasing powers.
$\begin{array}{l}{i}^{1}=i\\ {i}^{2}=-1\\ {i}^{3}={i}^{2}\cdot i=-1\cdot i=-i\\ {i}^{4}={i}^{3}\cdot i=-i\cdot i=-{i}^{2}=-\left(-1\right)=1\\ {i}^{5}={i}^{4}\cdot i=1\cdot i=i\end{array}$
We can see that when we get to the fifth power of $\text{\hspace{0.17em}}i,\text{\hspace{0.17em}}$ it is equal to the first power. As we continue to multiply $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of $\text{\hspace{0.17em}}i.$
$\begin{array}{l}{i}^{6}={i}^{5}\cdot i=i\cdot i={i}^{2}=-1\\ {i}^{7}={i}^{6}\cdot i={i}^{2}\cdot i={i}^{3}=-i\\ {i}^{8}={i}^{7}\cdot i={i}^{3}\cdot i={i}^{4}=1\\ {i}^{9}={i}^{8}\cdot i={i}^{4}\cdot i={i}^{5}=i\end{array}$
## Simplifying powers of $\text{\hspace{0.17em}}i$
Evaluate $\text{\hspace{0.17em}}{i}^{35}.$
Since $\text{\hspace{0.17em}}{i}^{4}=1,\text{\hspace{0.17em}}$ we can simplify the problem by factoring out as many factors of $\text{\hspace{0.17em}}{i}^{4}\text{\hspace{0.17em}}$ as possible. To do so, first determine how many times 4 goes into 35: $\text{\hspace{0.17em}}35=4\cdot 8+3.$
${i}^{35}={i}^{4\cdot 8+3}={i}^{4\cdot 8}\cdot {i}^{3}={\left({i}^{4}\right)}^{8}\cdot {i}^{3}={1}^{8}\cdot {i}^{3}={i}^{3}=-i$
Can we write $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ in other helpful ways?
As we saw in [link] , we reduced $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}{i}^{3}\text{\hspace{0.17em}}$ by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of $\text{\hspace{0.17em}}{i}^{35}\text{\hspace{0.17em}}$ may be more useful. [link] shows some other possible factorizations.
Factorization of $\text{\hspace{0.17em}}{i}^{35}$ ${i}^{34}\cdot i$ ${i}^{33}\cdot {i}^{2}$ ${i}^{31}\cdot {i}^{4}$ ${i}^{19}\cdot {i}^{16}$ Reduced form ${\left({i}^{2}\right)}^{17}\cdot i$ ${i}^{33}\cdot \left(-1\right)$ ${i}^{31}\cdot 1$ ${i}^{19}\cdot {\left({i}^{4}\right)}^{4}$ Simplified form ${\left(-1\right)}^{17}\cdot i$ $-{i}^{33}$ ${i}^{31}$ ${i}^{19}$
Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.
Access these online resources for additional instruction and practice with complex numbers.
## Key concepts
• The square root of any negative number can be written as a multiple of $\text{\hspace{0.17em}}i.\text{\hspace{0.17em}}$ See [link] .
• To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See [link] .
• Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See [link] .
• Complex numbers can be multiplied and divided.
• To multiply complex numbers, distribute just as with polynomials. See [link] , [link] , and [link] .
• To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See [link] , [link] , and [link] .
• The powers of $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ are cyclic, repeating every fourth one. See [link] .
## Verbal
Explain how to add complex numbers.
Add the real parts together and the imaginary parts together.
What is the basic principle in multiplication of complex numbers?
Give an example to show the product of two imaginary numbers is not always imaginary.
$i\text{\hspace{0.17em}}$ times $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ equals –1, which is not imaginary. (answers vary)
What is a characteristic of the plot of a real number in the complex plane?
## Algebraic
For the following exercises, evaluate the algebraic expressions.
evaluate $\text{\hspace{0.17em}}f\left(2i\right).$
$-8+2i$
evaluate $\text{\hspace{0.17em}}f\left(i\right).$
evaluate $\text{\hspace{0.17em}}f\left(2+i\right).$
$14+7i$
evaluate $\text{\hspace{0.17em}}f\left(2-3i\right).$
evaluate $\text{\hspace{0.17em}}f\left(5i\right).$
$-\frac{23}{29}+\frac{15}{29}i$
evaluate $\text{\hspace{0.17em}}f\left(4i\right).$
## Graphical
For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.
2 real and 0 nonreal
For the following exercises, plot the complex numbers on the complex plane.
$1-2i$
$-2+3i$
$i$
$-3-4i$
## Numeric
For the following exercises, perform the indicated operation and express the result as a simplified complex number.
$\left(3+2i\right)+\left(5-3i\right)$
$8-i$
$\left(-2-4i\right)+\left(1+6i\right)$
$\left(-5+3i\right)-\left(6-i\right)$
$-11+4i$
$\left(2-3i\right)-\left(3+2i\right)$
$\left(-4+4i\right)-\left(-6+9i\right)$
$2-5i$
$\left(2+3i\right)\left(4i\right)$
$\left(5-2i\right)\left(3i\right)$
$6+15i$
$\left(6-2i\right)\left(5\right)$
$\left(-2+4i\right)\left(8\right)$
$-16+32i$
$\left(2+3i\right)\left(4-i\right)$
$\left(-1+2i\right)\left(-2+3i\right)$
$-4-7i$
$\left(4-2i\right)\left(4+2i\right)$
$\left(3+4i\right)\left(3-4i\right)$
25
$\frac{3+4i}{2}$
$\frac{6-2i}{3}$
$2-\frac{2}{3}i$
$\frac{-5+3i}{2i}$
$\frac{6+4i}{i}$
$4-6i$
$\frac{2-3i}{4+3i}$
$\frac{3+4i}{2-i}$
$\frac{2}{5}+\frac{11}{5}i$
$\frac{2+3i}{2-3i}$
$\sqrt{-9}+3\sqrt{-16}$
$15i$
$-\sqrt{-4}-4\sqrt{-25}$
$\frac{2+\sqrt{-12}}{2}$
$1+i\sqrt{3}$
$\frac{4+\sqrt{-20}}{2}$
${i}^{8}$
$1$
${i}^{15}$
${i}^{22}$
$-1$
## Technology
For the following exercises, use a calculator to help answer the questions.
Evaluate $\text{\hspace{0.17em}}{\left(1+i\right)}^{k}\text{\hspace{0.17em}}$ for Predict the value if $\text{\hspace{0.17em}}k=16.$
Evaluate $\text{\hspace{0.17em}}{\left(1-i\right)}^{k}\text{\hspace{0.17em}}$ for Predict the value if $\text{\hspace{0.17em}}k=14.$
128i
Evaluate $\text{\hspace{0.17em}}\left(1+i{\right)}^{k}-\left(1-i{\right)}^{k}$ for . Predict the value for $\text{\hspace{0.17em}}k=16.$
Show that a solution of $\text{\hspace{0.17em}}{x}^{6}+1=0\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\frac{\sqrt{3}}{2}+\frac{1}{2}i.$
${\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}^{6}=-1$
Show that a solution of $\text{\hspace{0.17em}}{x}^{8}-1=0\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i.$
## Extensions
For the following exercises, evaluate the expressions, writing the result as a simplified complex number.
$\frac{1}{i}+\frac{4}{{i}^{3}}$
$3i$
$\frac{1}{{i}^{11}}-\frac{1}{{i}^{21}}$
${i}^{7}\left(1+{i}^{2}\right)$
0
${i}^{-3}+5{i}^{7}$
$\frac{\left(2+i\right)\left(4-2i\right)}{\left(1+i\right)}$
5 – 5i
$\frac{\left(1+3i\right)\left(2-4i\right)}{\left(1+2i\right)}$
$\frac{{\left(3+i\right)}^{2}}{{\left(1+2i\right)}^{2}}$
$-2i$
$\frac{3+2i}{2+i}+\left(4+3i\right)$
$\frac{4+i}{i}+\frac{3-4i}{1-i}$
$\frac{9}{2}-\frac{9}{2}i$
$\frac{3+2i}{1+2i}-\frac{2-3i}{3+i}$
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
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A Drifting-Games Analysis for Online Learning and Applications to Boosting
# A Drifting-Games Analysis for Online Learning and Applications to Boosting
Haipeng Luo
Department of Computer Science
Princeton University
Princeton, NJ 08540
haipengl@cs.princeton.edu
&Robert E. Schapire
Department of Computer Science
Princeton University
Princeton, NJ 08540
schapire@cs.princeton.edu
R. Schapire is currently at Microsoft Research in New York City.
###### Abstract
We provide a general mechanism to design online learning algorithms based on a minimax analysis within a drifting-games framework. Different online learning settings (Hedge, multi-armed bandit problems and online convex optimization) are studied by converting into various kinds of drifting games. The original minimax analysis for drifting games is then used and generalized by applying a series of relaxations, starting from choosing a convex surrogate of the 0-1 loss function. With different choices of surrogates, we not only recover existing algorithms, but also propose new algorithms that are totally parameter-free and enjoy other useful properties. Moreover, our drifting-games framework naturally allows us to study high probability bounds without resorting to any concentration results, and also a generalized notion of regret that measures how good the algorithm is compared to all but the top small fraction of candidates. Finally, we translate our new Hedge algorithm into a new adaptive boosting algorithm that is computationally faster as shown in experiments, since it ignores a large number of examples on each round.
A Drifting-Games Analysis for Online Learning and Applications to Boosting
Haipeng Luo Department of Computer Science Princeton University Princeton, NJ 08540 haipengl@cs.princeton.edu Robert E. Schapirethanks: R. Schapire is currently at Microsoft Research in New York City. Department of Computer Science Princeton University Princeton, NJ 08540 schapire@cs.princeton.edu
## 1 Introduction
In this paper, we study online learning problems within a drifting-games framework, with the aim of developing a general methodology for designing learning algorithms based on a minimax analysis.
To solve an online learning problem, it is natural to consider game-theoretically optimal algorithms which find the best solution even in worst-case scenarios. This is possible for some special cases ([7, 1, 3, 21]) but difficult in general. On the other hand, many other efficient algorithms with optimal regret rate (but not exactly minimax optimal) have been proposed for different learning settings (such as the exponential weights algorithm [14, 15], and follow the perturbed leader [18]). However, it is not always clear how to come up with these algorithms. Recent work by Rakhlin et al. [26] built a bridge between these two classes of methods by showing that many existing algorithms can indeed be derived from a minimax analysis followed by a series of relaxations.
In this paper, we provide a parallel way to design learning algorithms by first converting online learning problems into variants of drifting games, and then applying a minimax analysis and relaxations. Drifting games [28] (reviewed in Section 2) generalize Freund’s “majority-vote game” [13] and subsume some well-studied boosting and online learning settings. A nearly minimax optimal algorithm is proposed in [28]. It turns out the connections between drifting games and online learning go far beyond what has been discussed previously. To show that, we consider variants of drifting games that capture different popular online learning problems. We then generalize the minimax analysis in [28] based on one key idea: relax a 0-1 loss function by a convex surrogate. Although this idea has been applied widely elsewhere in machine learning, we use it here in a new way to obtain a very general methodology for designing and analyzing online learning algorithms. Using this general idea, we not only recover existing algorithms, but also design new ones with special useful properties. A somewhat surprising result is that our new algorithms are totally parameter-free, which is usually not the case for algorithms derived from a minimax analysis. Moreover, a generalized notion of regret (-regret, defined in Section 3) that measures how good the algorithm is compared to all but the top fraction of candidates arises naturally in our drifting-games framework. Below we summarize our results for a range of learning settings.
Hedge Settings: (Section 3) The Hedge problem [14] investigates how to cleverly bet across a set of actions. We show an algorithmic equivalence between this problem and a simple drifting game (DGv1). We then show how to relax the original minimax analysis step by step to reach a general recipe for designing Hedge algorithms (Algorithm 3). Three examples of appropriate convex surrogates of the 0-1 loss function are then discussed, leading to the well-known exponential weights algorithm and two other new ones, one of which (NormalHedge.DT in Section 3.3) bears some similarities with the NormalHedge algorithm [10] and enjoys a similar -regret bound simultaneously for all and horizons. However, our regret bounds do not depend on the number of actions, and thus can be applied even when there are infinitely many actions. Our analysis is also arguably simpler and more intuitive than the one in [10] and easy to be generalized to more general settings. Moreover, our algorithm is more computationally efficient since it does not require a numerical searching step as in NormalHedge. Finally, we also derive high probability bounds for the randomized Hedge setting as a simple side product of our framework without using any concentration results.
Multi-armed Bandit Problems: (Section 4) The multi-armed bandit problem [6] is a classic example for learning with incomplete information where the learner can only obtain feedback for the actions taken. To capture this problem, we study a quite different drifting game (DGv2) where randomness and variance constraints are taken into account. Again the minimax analysis is generalized and the EXP3 algorithm [6] is recovered. Our results could be seen as a preliminary step to answer the open question [2] on exact minimax optimal algorithms for the multi-armed bandit problem.
Online Convex Optimization: (Section 4) Based the theory of convex optimization, online convex optimization [31] has been the foundation of modern online learning theory. The corresponding drifting game formulation is a continuous space variant (DGv3). Fortunately, it turns out that all results from the Hedge setting are ready to be used here, recovering the continuous EXP algorithm [12, 17, 24] and also generalizing our new algorithms to this general setting. Besides the usual regret bounds, we also generalize the -regret, which, as far as we know, is the first time it has been explicitly studied. Again, we emphasize that our new algorithms are adaptive in and the horizon.
Boosting: (Section 4) Realizing that every Hedge algorithm can be converted into a boosting algorithm ([29]), we propose a new boosting algorithm (NH-Boost.DT) by converting NormalHedge.DT. The adaptivity of NormalHedge.DT is then translated into training error and margin distribution bounds that previous analysis in [29] using nonadaptive algorithms does not show. Moreover, our new boosting algorithm ignores a great many examples on each round, which is an appealing property useful to speeding up the weak learning algorithm. This is confirmed by our experiments.
Related work: Our analysis makes use of potential functions. Similar concepts have widely appeared in the literature [8, 5], but unlike our work, they are not related to any minimax analysis and might be hard to interpret. The existence of parameter free Hedge algorithms for unknown number of actions was shown in [11], but no concrete algorithms were given there. Boosting algorithms that ignore some examples on each round were studied in [16], where a heuristic was used to ignore examples with small weights and no theoretical guarantee is provided.
## 2 Reviewing Drifting Games
We consider a simplified version of drifting games similar to the one described in [29, chap. 13] (also called chip games). This game proceeds through rounds, and is played between a player and an adversary who controls chips on the real line. The positions of these chips at the end of round are denoted by , with each coordinate corresponding to the position of chip . Initially, all chips are at position so that . On every round : the player first chooses a distribution over the chips, then the adversary decides the movements of the chips so that the new positions are updated as . Here, each has to be picked from a prespecified set , and more importantly, satisfy the constraint for some fixed constant .
At the end of the game, each chip is associated with a nonnegative loss defined by for some nonincreasing function mapping from the final position of the chip to . The goal of the player is to minimize the chips’ average loss after rounds. So intuitively, the player aims to “push” the chips to the right by assigning appropriate weights on them so that the adversary has to move them to the right by in a weighted average sense on each round. This game captures many learning problems. For instance, binary classification via boosting can be translated into a drifting game by treating each training example as a chip (see [28] for details).
We regard a player’s strategy as a function mapping from the history of the adversary’s decisions to a distribution that the player is going to play with, that is, where stands for . The player’s worst case loss using this algorithm is then denoted by . The minimax optimal loss of the game is computed by the following expression: where is the dimensional simplex and is assumed to be compact. A strategy that realizes the minimum in is called a minimax optimal strategy. A nearly optimal strategy and its analysis is originally given in [28], and a derivation by directly tackling the above minimax expression can be found in [29, chap. 13]. Specifically, a sequence of potential functions of a chip’s position is defined recursively as follows:
ΦT(s)=L(s),Φt−1(s)=minw∈R+maxz∈B(Φt(s+z)+w(z−β)). (1)
Let be the weight that realizes the minimum in the definition of , that is, . Then the player’s strategy is to set . The key property of this strategy is that it assures that the sum of the potentials over all the chips never increases, connecting the player’s final loss with the potential at time as follows:
1NN∑i=1L(sT,i)≤1NN∑i=1ΦT(sT,i)≤1NN∑i=1ΦT−1(sT−1,i)≤⋯≤1NN∑i=1Φ0(s0,i)=Φ0(0). (2)
It has been shown in [28] that this upper bound on the loss is optimal in a very strong sense.
Moreover, in some cases the potential functions have nice closed forms and thus the algorithm can be efficiently implemented. For example, in the boosting setting, is simply , and one can verify and . With the loss function being , these can be further simplified and eventually give exactly the boost-by-majority algorithm [13].
## 3 Online Learning as a Drifting Game
The connection between drifting games and some specific settings of online learning has been noticed before ([28, 23]). We aim to find deeper connections or even an equivalence between variants of drifting games and more general settings of online learning, and provide insights on designing learning algorithms through a minimax analysis. We start with a simple yet classic Hedge setting.
### 3.1 Algorithmic Equivalence
In the Hedge setting [14], a player tries to earn as much as possible (or lose as little as possible) by cleverly spreading a fixed amount of money to bet on a set of actions on each day. Formally, the game proceeds for rounds, and on each round : the player chooses a distribution over actions, then the adversary decides the actions’ losses (i.e. action incurs loss ) which are revealed to the player. The player suffers a weighted average loss at the end of this round. The goal of the player is to minimize his “regret”, which is usually defined as the difference between his total loss and the loss of the best action. Here, we consider an even more general notion of regret studied in [20, 19, 10, 11], which we call -regret. Suppose the actions are ordered according to their total losses after rounds (i.e. ) from smallest to largest, and let be the index of the action that is the -th element in the sorted list (). Now, -regret is defined as In other words, -regret measures the difference between the player’s loss and the loss of the -th best action (recovering the usual regret with ), and sublinear -regret implies that the player’s loss is almost as good as all but the top fraction of actions. Similarly, denotes the worst case -regret for a specific algorithm . For convenience, when or , we define -regret to be or respectively.
Next we discuss how Hedge is highly related to drifting games. Consider a variant of drifting games where and for some constant . Additionally, we impose an extra restriction on the adversary: for all and . In other words, the difference between any two chips’ movements is at most . We denote this specific variant of drifting games by DGv1 (summarized in Appendix A) and a corresponding algorithm by to emphasize the dependence on . The reductions in Algorithm 1 and 2 and Theorem 1 show that DGv1 and the Hedge problem are algorithmically equivalent (note that both conversions are valid). The proof is straightforward and deferred to Appendix B. By Theorem 1, it is clear that the minimax optimal algorithm for one setting is also minimax optimal for the other under these conversions.
###### Theorem 1.
DGv1 and the Hedge problem are algorithmically equivalent in the following sense:
(1) Algorithm 1 produces a DGv1 algorithm satisfying where is such that .
(2) Algorithm 2 produces a Hedge algorithm with for any such that .
### 3.2 Relaxations
From now on we only focus on the direction of converting a drifting game algorithm into a Hedge algorithm. In order to derive a minimax Hedge algorithm, Theorem 1 tells us it suffices to derive minimax DGv1 algorithms. Exact minimax analysis is usually difficult, and appropriate relaxations seem to be necessary. To make use of the existing analysis for standard drifting games, the first obvious relaxation is to drop the additional restriction in DGv1, that is, for all and . Doing this will lead to the exact setting discussed in [23] where a near optimal strategy is proposed using the recipe in Eq. (1). It turns out that this relaxation is reasonable and does not give too much more power to the adversary. To see this, first recall that results from [23], written in our notation, state that which, by Hoeffding’s inequality, is upper bounded by . Second, statement (2) in Theorem 1 clearly remains valid if the input of Algorithm 2 is a drifting game algorithm for this relaxed version of DGv1. Therefore, by setting and solving for , we have , which is the known optimal regret rate for the Hedge problem, showing that we lose little due to this relaxation.
However, the algorithm proposed in [23] is not computationally efficient since the potential functions do not have closed forms. To get around this, we would want the minimax expression in Eq. (1) to be easily solved, just like the case when . It turns out that convexity would allow us to treat almost as . Specifically, if each is a convex function of , then due to the fact that the maximum of a convex function is always realized at the boundary of a compact region, we have
minw∈R+maxz∈[−1,1](Φt(s+z)+wz)=minw∈R+maxz∈{−1,1}(Φt(s+z)+wz)=Φt(s−1)+Φt(s+1)2, (3)
with realizing the minimum. Since the 0-1 loss function is not convex, this motivates us to find a convex surrogate of . Fortunately, relaxing the equality constraints in Eq. (1) does not affect the key property of Eq. (2) as we will show in the proof of Theorem 2. “Compiling out” the input of Algorithm 2, we thus have our general recipe (Algorithm 3) for designing Hedge algorithms with the following regret guarantee.
###### Theorem 2.
For Algorithm 3, if and are such that and for all , then .
Proof. It suffices to show that Eq. (2) holds so that the theorem follows by a direct application of statement (2) of Theorem 1. Let . Then since and . On the other hand, by Eq. (3), we have , which is at most by Algorithm 3. This shows and Eq. (2) follows. ∎
Theorem 2 tells us that if solving for gives for some value , then the regret of Algorithm 3 is less than any value that is greater than , meaning the regret is at most .
### 3.3 Designing Potentials and Algorithms
Now we are ready to recover existing algorithms and develop new ones by choosing an appropriate potential as Algorithm 3 suggests. We will discuss three different algorithms below, and summarize these examples in Table 1 (see Appendix C).
#### Exponential Weights (EXP) Algorithm.
Exponential loss is an obvious choice for as it has been widely used as the convex surrogate of the 0-1 loss function in the literature. It turns out that this will lead to the well-known exponential weights algorithm [14, 15]. Specifically, we pick to be which exactly upper bounds . To compute for , we simply let hold with equality. Indeed, direct computations show that all share a similar form: Therefore, according to Algorithm 3, the player’s strategy is to set
pt,i∝Φt(st−1,i−1)−Φt(st−1,i+1)∝exp(−ηst−1,i),
which is exactly the same as EXP (note that becomes irrelevant after normalization). To derive regret bounds, it suffices to require which is equivalent to By Theorem 2 and Hoeffding’s lemma (see [9, Lemma A.1]), we thus know where the last step is by optimally tuning to be . Note that this algorithm is not adaptive in the sense that it requires knowledge of and to set the parameter .
We have thus recovered the well-known EXP algorithm and given a new analysis using the drifting-games framework. More importantly, as in [26], this derivation may shed light on why this algorithm works and where it comes from, namely, a minimax analysis followed by a series of relaxations, starting from a reasonable surrogate of the 0-1 loss function.
#### 2-norm Algorithm.
We next move on to another simple convex surrogate: where is some positive constant and represents a truncating operation. The following lemma shows that can also be simply described.
###### Lemma 1.
If , then satisfies .
Thus, Algorithm 3 can again be applied. The resulting algorithm is extremely concise:
pt,i∝Φt(st−1,i−1)−Φt(st−1,i+1)∝[st−1,i−1]2−−[st−1,i+1]2−.
We call this the “2-norm” algorithm since it resembles the -norm algorithm in the literature when (see [9]). The difference is that the -norm algorithm sets the weights proportional to the derivative of potentials, instead of the difference of them as we are doing here. A somewhat surprising property of this algorithm is that it is totally adaptive and parameter-free (since disappears under normalization), a property that we usually do not expect to obtain from a minimax analysis. Direct application of Theorem 2 () shows that its regret achieves the optimal dependence on the horizon .
###### Corollary 1.
Algorithm 3 with potential defined in Lemma 1 produces a Hedge algorithm such that simultaneously for all and .
#### NormalHedge.DT.
The regret for the 2-norm algorithm does not have the optimal dependence on . An obvious follow-up question would be whether it is possible to derive an adaptive algorithm that achieves the optimal rate simultaneously for all and using our framework. An even deeper question is: instead of choosing convex surrogates in a seemingly arbitrary way, is there a more natural way to find the right choice of ?
To answer these questions, we recall that the reason why the 2-norm algorithm can get rid of the dependence on is that appears merely in the multiplicative constant that does not play a role after normalization. This motivates us to let in the form of for some . On the other hand, from Theorem 2, we also want to upper bound the 0-1 loss function for some constant . Taken together, this is telling us that the right choice of should be of the form 111 Similar potential was also proposed in recent work [22, 25] for a different setting. . Of course we still need to refine it to satisfy the monotonicity and other properties. We define formally and more generally as:
ΦT(s)=a(exp([s]2−dT)−1)≥1{s≤−√dTln(1a+1)},
where and are some positive constants. This time it is more involved to figure out what other should be. The following lemma addresses this issue (proof deferred to Appendix C).
###### Lemma 2.
If and (define ), then we have for all and . Moreover, Eq. (2) still holds.
Note that even if is not valid in general, Lemma 2 states that Eq. (2) still holds. Thus Algorithm 3 can indeed still be applied, leading to our new algorithm:
pt,i∝Φt(st−1,i−1)−Φt(st−1,i+1)∝exp([st−1,i−1]2−dt)−exp([st−1,i+1]2−dt).
Here, seems to be an extra parameter, but in fact, simply setting is good enough:
###### Corollary 2.
Algorithm 3 with potential defined in Lemma 2 and produces a Hedge algorithm such that the following holds simultaneously for all and :
RϵT(H)≤√3Tln(12ϵ(e4/3−1)(lnT+1)+1)=O(√Tln(1/ϵ)+TlnlnT).
We have thus proposed a parameter-free adaptive algorithm with optimal regret rate (ignoring the term) using our drifting-games framework. In fact, our algorithm bears a striking similarity to NormalHedge [10], the first algorithm that has this kind of adaptivity. We thus name our algorithm NormalHedge.DT222“DT” stands for discrete time.. We include NormalHedge in Table 1 for comparison. One can see that the main differences are: 1) On each round NormalHedge performs a numerical search to find out the right parameter used in the exponents; 2) NormalHedge uses the derivative of potentials as weights.
Compared to NormalHedge, the regret bound for NormalHedge.DT has no explicit dependence on , but has a slightly worse dependence on (indeed is almost negligible). We emphasize other advantages of our algorithm over NormalHedge: 1) NormalHedge.DT is more computationally efficient especially when is very large, since it does not need a numerical search for each round; 2) our analysis is arguably simpler and more intuitive than the one in [10]; 3) as we will discuss in Section 4, NormalHedge.DT can be easily extended to deal with the more general online convex optimization problem where the number of actions is infinitely large, while it is not clear how to do that for NormalHedge by generalizing the analysis in [10]. Indeed, the extra dependence on the number of actions for the regret of NormalHedge makes this generalization even seem impossible. Finally, we will later see that NormalHedge.DT outperforms NormalHedge in experiments. Despite the differences, it is worth noting that both algorithms assign zero weight to some actions on each round, an appealing property when is huge. We will discuss more on this in Section 4.
### 3.4 High Probability Bounds
We now consider a common variant of Hedge: on each round, instead of choosing a distribution , the player has to randomly pick a single action , while the adversary decides the losses at the same time (without seeing ). For now we only focus on the player’s regret to the best action: Notice that the regret is now a random variable, and we are interested in a bound that holds with high probability. Using Azuma’s inequality, standard analysis (see for instance [9, Lemma 4.1]) shows that the player can simply draw according to , the output of a standard Hedge algorithm, and suffers regret at most with probability . Below we recover similar results as a simple side product of our drifting-games analysis without resorting to concentration results, such as Azuma’s inequality.
For this, we only need to modify Algorithm 3 by setting . The restriction is then relaxed to hold in expectation. Moreover, it is clear that Eq. (2) also still holds in expectation. On the other hand, by definition and the union bound, one can show that . So setting shows that the regret is smaller than with probability . Therefore, for example, if EXP is used, then the regret would be at most with probability , giving basically the same bound as the standard analysis. One draw back is that EXP would need as a parameter. However, this can again be addressed by NormalHedge.DT for the exact same reason that NormalHedge.DT is independent of . We have thus derived high probability bounds without using any concentration inequalities.
## 4 Generalizations and Applications
Multi-armed Bandit (MAB) Problem: The only difference between Hedge (randomized version) and the non-stochastic MAB problem [6] is that on each round, after picking , the player only sees the loss for this single action instead of the whole vector . The goal is still to compete with the best action. A common technique used in the bandit setting is to build an unbiased estimator for the losses, which in this case could be . Then algorithms such as EXP can be used by replacing with , leading to the EXP3 algorithm [6] with regret .
One might expect that Algorithm 3 would also work well by replacing with . However, doing so breaks an important property of the movements : boundedness. Indeed, Eq. (3) no longer makes sense if could be infinitely large, even if in expectation it is still in (note that is now a random variable). It turns out that we can address this issue by imposing a variance constraint on . Formally, we consider a variant of drifting games where on each round, the adversary picks a random movement for each chip such that: and . We call this variant DGv2 and summarize it in Appendix A. The standard minimax analysis and the derivation of potential functions need to be modified in a certain way for DGv2, as stated in Theorem 4 (Appendix D). Using the analysis for DGv2, we propose a general recipe for designing MAB algorithms in a similar way as for Hedge and also recover EXP3 (see Algorithm 4 and Theorem 5 in Appendix D). Unfortunately so far we do not know other appropriate potentials due to some technical difficulties. We conjecture, however, that there is a potential function that could recover the poly-INF algorithm [4, 5] or give its variants that achieve the optimal regret .
Online Convex Optimization: We next consider a general online convex optimization setting [31]. Let be a compact convex set, and be a set of convex functions with range on . On each round , the learner chooses a point , and the adversary chooses a loss function (knowing ). The learner then suffers loss . The regret after rounds is . There are two general approaches to OCO: one builds on convex optimization theory [30], and the other generalizes EXP to a continuous space [12, 24]. We will see how the drifting-games framework can recover the latter method and also leads to new ones.
To do so, we introduce a continuous variant of drifting games (DGv3, see Appendix A). There are now infinitely many chips, one for each point in . On round , the player needs to choose a distribution over the chips, that is, a probability density function on . Then the adversary decides the movements for each chip, that is, a function with range on (not necessarily convex or continuous), subject to a constraint . At the end, each point is associated with a loss , and the player aims to minimize the total loss .
OCO can be converted into DGv3 by setting and predicting . The constraint holds by the convexity of . Moreover, it turns out that the minimax analysis and potentials for DGv1 can readily be used here, and the notion of -regret, now generalized to the OCO setting, measures the difference of the player’s loss and the loss of a best fixed point in a subset of that excludes the top fraction of points. With different potentials, we obtain versions of each of the three algorithms of Section 3 generalized to this setting, with the same -regret bounds as before. Again, two of these methods are adaptive and parameter-free. To derive bounds for the usual regret, at first glance it seems that we have to set to be close to zero, leading to a meaningless bound. Nevertheless, this is addressed by Theorem 6 using similar techniques in [17], giving the usual regret bound. All details can be found in Appendix E.
Applications to Boosting: There is a deep and well-known connection between Hedge and boosting [14, 29]. In principle, every Hedge algorithm can be converted into a boosting algorithm; for instance, this is how AdaBoost was derived from EXP. In the same way, NormalHedge.DT can be converted into a new boosting algorithm that we call NH-Boost.DT. See Appendix F for details and further background on boosting. The main idea is to treat each training example as an “action”, and to rely on the Hedge algorithm to compute distributions over these examples which are used to train the weak hypotheses. Typically, it is assumed that each of these has “edge” , meaning its accuracy on the training distribution is at least . The final hypothesis is a simple majority vote of the weak hypotheses. To understand the prediction accuracy of a boosting algorithm, we often study the training error rate and also the distribution of margins, a well-established measure of confidence (see Appendix F for formal definitions). Thanks to the adaptivity of NormalHedge.DT, we can derive bounds on both the training error and the distribution of margins after any number of rounds:
###### Theorem 3.
After rounds, the training error of NH-Boost.DT is of order , and the fraction of training examples with margin at most is of order .
Thus, the training error decreases at roughly the same rate as AdaBoost. In addition, this theorem implies that the fraction of examples with margin smaller than eventually goes to zero as gets large, which means NH-Boost.DT converges to the optimal margin ; this is known not to be true for AdaBoost (see [29]). Also, like AdaBoost, NH-Boost.DT is an adaptive boosting algorithm that does not require or as a parameter. However, unlike AdaBoost, NH-Boost.DT has the striking property that it completely ignores many examples on each round (by assigning zero weight), which is very helpful for the weak learning algorithm in terms of computational efficiency. To test this, we conducted experiments to compare the efficiency of AdaBoost, “NH-Boost” (an analogous boosting algorithm derived from NormalHedge) and NH-Boost.DT. All details are in Appendix G. Here we only briefly summarize the results. While the three algorithms have similar performance in terms of training and test error, NH-Boost.DT is always the fastest one in terms of running time for the same number of rounds. Moreover, the average faction of examples with zero weight is significantly higher for NH-Boost.DT than for NH-Boost (see Table 3). On one hand, this explains why NH-Boost.DT is faster (besides the reason that it does not require a numerical step). On the other hand, this also implies that NH-Boost.DT tends to achieve larger margins, since zero weight is assigned to examples with large margin. This is also confirmed by our experiments.
Acknowledgements. Support for this research was provided by NSF Grant #1016029. The authors thank Yoav Freund for helpful discussions and the anonymous reviewers for their comments.
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## Appendix A Summary of Drifting Game Variants
We study three different variants of drifting games throughout the paper, which corresponds to the Hedge setting, the multi-armed bandit problem and online convex optimization respectively. The protocols of these variants are summarized below.
DGv1 Given: a loss function . For : The player chooses a distribution over chips. The adversary decides the movement of each chip subject to and for all and . The player suffers loss .
DGv2 Given: a loss function . For : The player chooses a distribution over chips. The adversary randomly decides the movement of each chip subject to and . The player suffers loss .
DGv3 Given: a compact convex set , a loss function . For : The player chooses a density function on . The adversary decides a function subject to . The player suffers loss .
## Appendix B Proof of Theorem 1
###### Proof.
We first show that both conversions are valid. In Algorithm 1, it is clear that . Also, is guaranteed due to the extra restriction of DGv1. For Algorithm 2, lies in since , and direct computation shows and for all and .
(1) For any choices of , we have
N∑i=1L(sT,i)=N∑i=1L(N∑t=1zt,i)≤N∑i=1L(N∑t=1(zt,i−pt⋅zt)),
where the inequality holds since is required to be nonnegative and is a nonincreasing function. By Algorithm 1, is equal to , leading to
N∑i=1L(sT,i)≤N∑i=1L(N∑t=1(ℓt,i−pt⋅ℓt))=N∑i=11{R≤N∑t=1(pt⋅ℓt−ℓt,i)}.
Since , we must have except for the best actions, which means . This holds for any choices of , so .
(2) By Algorithm 2 and the condition , we have
1NN∑i=11{R≤N∑t=1(pt⋅ℓt−ℓt,i)}=1NN∑i=1L(sT,i)≤LT(DR)<ϵ,
which means there are at most actions satisfying , and thus . Since this holds for any choices of , we have . ∎
## Appendix C Summary of Hedge Algorithms and Proofs of Lemma 1, Lemma 2 and Corollary 2
###### Proof of Lemma 1.
It suffices to show . When , . When , . ∎
###### Proof of Lemma 2.
Let . It suffices to show
F(s)≤2(bt−bt−1)=exp(4dt)−1,
which is clearly true for the following 3 cases:
F(s)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩0if s>1;exp((s−1)2dt)−1
For the last case , if we can show that is increasing in this region, then the lemma follows. Below, we show this by proving is nonnegative when .
Let . can now be written as
F′(s)=h(s−1,c)+h(s+1,c)−2h(s,c)+2(h(s,c)−h(s,c′)),
where and . Next we apply (one-dimensional) Taylor expansion to and around , and around , leading to
F′(s) =∞∑k=1(−1)kk!∂kh(s,c)∂sk+∞∑k=11k!∂kh(s,c)∂sk−2∞∑k=1(c′−c)kk!∂kh(s,c)∂ck =2∞∑k=1(1(2k)!∂2kh(s,c)∂s2k−(−d)kk!∂kh(s,c)∂ck).
Direct computation (see Lemma 3 below) shows that and share exact same forms only with different constants:
∂kh(s,c)∂ck=exp(s2c)k∑j=0(−1)kαk,j⋅s2j+1ck+j+1,∂2kh(s,c)∂s2k=exp(s2c)k∑j=0βk,j⋅s2j+1ck+j+1, (4)
where and
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2015-10-23, 19:55 #1 blip Jan 2014 2·73 Posts Headlines Would someone please stop that daft headline scrambling? We are not all native speakers, and this makes searching just painful.
2015-10-23, 21:56 #2 Brian-E "Brian" Jul 2007 The Netherlands 2·3·5·109 Posts For the record, the title of this thread before - I think - any supermod got their hands on it, was "Headlines". What it will be shortly is anyone's guess. (There, I've made it easy for them.) Your request has been asked before. No joy. It's something up with which we just have to put.
2015-10-23, 21:58 #3
science_man_88
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Originally Posted by blip Would someone please stop that daft headline scrambling? We are not all native speakers, and this makes searching just painful.
easier to search by something in a thread than by title at that point.
2015-10-23, 23:33 #4
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Originally Posted by blip Would someone please stop that daft headline scrambling? We are not all native speakers, and this makes searching just painful.
Ok. I will exercise restraint in the Soapbox. Or refrain from exercising. Or both. After a couple of weeks I will reappraise the situation.
2015-10-24, 07:40 #5
"Kieren"
Jul 2011
In My Own Galaxy!
100111101011102 Posts
Quote:
Originally Posted by blip Would someone please stop that daft headline scrambling? We are not all native speakers, and this makes searching just painful.
Are you aware of the Google technique "site: mersenneforum.org [search term(s)]"?
Last fiddled with by kladner on 2015-10-24 at 07:41 Reason: close quotes and '?'
2015-10-24, 07:58 #6
blip
Jan 2014
14610 Posts
Quote:
Originally Posted by kladner Are you aware of the Google technique "site: mersenneforum.org [search term(s)]"?
Maybe the term "searching" was misleading. If I follow a discussion in a thread, I develop the expectation to find follow-ups under the same headline when I return at a later time. Usually, I look under "New Posts" to find - news.
If the headline of a discussion has changed, there is a chance I will not find a thread upon first inspection again. This is sometimes annoying. and when I then find the discussion again under a new headline, I always think this tinkering with headlines has somewhat worn off.
Besides that, I am well aware of search techniques, thank you.
Last fiddled with by blip on 2015-10-24 at 08:07
2015-10-24, 09:11 #7 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 226178 Posts Subscription is a useful instrument for following a few concrete threads. Attached Thumbnails
2015-10-24, 09:53 #8 blip Jan 2014 2×73 Posts neat thanks
2015-10-24, 16:50 #9
Dubslow
"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88
3·29·83 Posts
Quote:
Originally Posted by only_human Ok. I will exercise restraint in the Soapbox. Or refrain from exercising. Or both. After a couple of weeks I will reappraise the situation.
I long ago suggested that the Soapbox is the only appropriate place for title changes, and that elsewhere they should be verboten.
2015-10-24, 20:37 #10 Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3×5×641 Posts fondled a factor? Isn't this thread a nice one? Code: found a factor? say it found a factor? say it backwards... fast found a factor? say it backwards then fast for a week found a factor? say it backwards for a week found a factor? pay it back found a factor? play it again, Sam found a factor? pat yourself on the back found a factor? Get a cowpat on your back found a factor? Don't have a cow, man! found a factor? Join the club... found a factor? Man the lifeboat... found a factor? Person the lifeboat... found a factor? Draw a star on starboard... found a factor? Draw a straw and hope it's a long one found a factor? Draw a strawman and hope it's a long one found a factor? Draw a lawman and hope it's a strong one found a factor? Talk to a layman and hope it's a patient one found a factor? Talk to a caiman and hope it's a patient one found a factor? Go to a caiman and have a good break! You deserve it found a factor? Go to Caimans and have a good break! You deserve it found a factor? Go to Caymans and have a good break! You deserve it found a factor? Go to Caymans and break a leg! You deserve it found a factor? Turn it in at the lost property office found a factor? Turn it in at the lost prosperity office found a factor? Turn it in for the world to admire found a factor? Turn it in to become instantly famous! fondled a factor? Turn it in to become instantly famous! fondled a factor? Turn yourself in to become instantly famous! fondled a factor? Turn yourself in to become insanely famous!
2015-10-24, 21:04 #11 firejuggler "Vincent" Apr 2010 Over the rainbow 2×32×149 Posts I suspect that this thread ( the factors one) is going to have several more iteration.
Similar Threads Thread Thread Starter Forum Replies Last Post fivemack Lounge 99 2020-11-18 15:54 dsouza123 Soap Box 2 2007-06-26 21:41
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1. ## Breadth-frist transversal
I understand how to list each transversal when each node has at most
2 lines( coming in or out) but how not sure about a graph where a node has
more than 2 line connected to it. For example consider this graph :
Can you help me for instance list the transversal route using breadth-first search
starting at node F.
One way I tried and got this : F-C-E-D-I-G-H-B-A-J, but am not sure if its correct. I am not sure how to transverse through the graph, say starting at node F, using the breadth-first-search. Any Help?
2. F-C-E-D-I-G-H-B-A-J
I think J must come before D because the distance from F and J is 1 and between F and D is 2.
One way to think about breadth-first search is to imagine water flowing from F. It first reaches nodes at distance one, then at distance two, etc. The order in which nodes at the same distance are listed (e.g., J, E, C) is probably not important.
To write an algorithm, one considers a queue (first in, first out) of nodes. In the beginning the queue has only F. There is also a sequence of nodes already visited; initially it is empty. At each step, one takes out a node from the end of the queue, mark it as visited, and pushes all its neighbors that have not been visited and are not already in the queue into the start of the queue. This continues until the queue is empty.
So a possible sequence of queue contents and the list of visited nodes is
Code:
Queue Visited
---------------+--------
F |
J E C | F
D B I J E | F C
D B I J | F C E
D B I | F C E J
D B | F C E J I
...
I'll let you finish from here. Feel free to post your version for checking.
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# Relationship between height and velocity in conservation of mechnical energy
I'm a high school physics student, and we recently did a lab on the conservation of energy where we measured the speed of a marble at varying heights on a rollercoaster track. We were supposed to graph the height and speed, and I ended up with something resembling a square root graph
We were required to linearize the data (so that we could have a single slope), so I squared the heights and got a slope of about $-0.11\,cm^{-1}\,s^{-1}$:
I don't understand what the slope is supposed to signify. I tried to take the derivative of velocity with respect to height in the equation $E = mgh + \frac{1}{2}mv^2$, where E is constant, and got a cryptic answer of $\frac{dv}{dh} = \frac{-g}{v}$. Can someone help me understand what's going on here?
-
I think this is an excellent example of how to properly ask a homework or homework-like question :-) – David Z Jan 7 '13 at 0:45
However, I would remind answerers not to give away the solution - i.e. don't respond by describing what mathematical operations to perform, but instead give explain how one would figure that out. – David Z Jan 7 '13 at 1:40
I would just like to point out that your energy equation isn't comnplete. A rolling marble also has angular momentum in addition to its linear momentum. – Dave Tweed Jan 7 '13 at 2:31
Oops... I wasn't aware of the homework policy. I've edited the solution out of my post. – Draksis Jan 7 '13 at 21:28
Look at the equation you gave for the energy, $E=mgh + \frac12 m v^2$, and find two variables that are linearly related. It can help to make a new variable to show the linear relationship. For the attempt you show (plotting $h^2$ vs $v$), set $H=h^2$ so that the equation for the energy is $E=mg\sqrt{H} + \frac12 mv^2$ which does not suggest that there is a linear relationship between $H$ and $v$.
-
Are you sure that $h^2$ vs $v$ would be linear?
Consider $E = mgh + 1/2mv^2$. To linearize this, think about what variable is causing the "non-linearity," i.e. which variable could you transform in order to create a new, linear equation.
-
I get the same answer as you, $dv/dh = -g/v$ or $dh/dv = -v/g$. Here's what I think it means.
For the same energy $E$, there is a tradeoff between height and velocity. More height, less velocity. More velocity, less height.
So at any point in the fall, the rate at which it is trading height for velocity is equal to velocity divided by the gravitational constant. So at the top, when its velocity is small, it loses little height for each increment of velocity. At the bottom, when its velocity is large, it loses a lot of height for each increment of velocity.
This may be easier to see if you just consider that an increment of velocity is just $g$ times an increment of time.
EDIT: to put it another way, $-v/g = t$, the length of time it's been falling. So $dh/dv = -t$, and since $dv = -g dt$ (because velocity is in the downward direction), $dh/dt = -g t$ or $v = -g t$
:-)
-
Make the y-axis height and the x- velocity. This will give you a parabolic shape indicating that it is a quadratic variation.
-
So simply switch the axes? – HDE 226868 Oct 22 '14 at 1:51
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# Converging at position and angle using motor-based rigidbody movement
I am given a rigidbody positioned in $\mathbb{R}^2$. It is a box of length $l$ and height $\epsilon$. The position of the box's left center is described by $\vec{p}$, and its angle from the x-axis is $\theta$. The body is in a vacuum (no friction), and its mass is $m$ (uniformly distributed). Here's a quick diagram.
Next, I am given a target position and angle to reach. It is not guaranteed that the body will reach the target position and angle the next time frame.
I need to calculate the necessary force $F$ and torque $T$ that will push the body closest to the target. I am given constraints for max force and max torque.
To make things complicated, the body might be affected by external forces. One of them is gravity $F_g$, which will always remain constant. Sometimes, normal forces exist as well. The applied forces or the total force is unknown, but the velocity or the angular velocity is available.
I tried to solve this problem using steering behaviors. However, the angle seems to fluctuate constantly. Here's part of my code in C# (Unity).
$\vec{p}_0=$transform.position $\theta_0=$transform.eulerAngles.z $\vec{p}_t=$position $\theta_t=$angle
private RigidBody _body;
public float SlowDownRadius = 2f;
public float MaxVelocity = 50f;
public float MaxForce = 10000f;
public float SlowDownDelta = 15f;
public float MaxAngVelocity = 180f;
public float MaxTorque = 10000f;
public void Next(Vector3 position, float angle) // angle in degrees
{
// force
Vector2 to = position;
var from = (Vector2) transform.position;
var desired = to - from;
var dist = desired.sqrMagnitude;
desired = desired.SetMagnitude(dist < SlowDownRadius * SlowDownRadius
? MaxVelocity * Mathf.Sqrt(dist) / SlowDownRadius
: MaxVelocity);
Vector2 vel = _body.velocity;
var steering = desired - vel;
var mass = _body.mass;
Vector2 force = mass * UnityEngine.Physics.gravity;
var time = Time.deltaTime;
var newForce =
(time <= 0 ? new Vector2() : (mass / time * steering)) - force;
_body.AddForce(newForce.Clamp(MaxForce)); // limits magnitude of newForce to MaxForce
// torque
var toAngle = angle;
var fromAngle = transform.eulerAngles.z;
var desiredAngle = AngleDistance(fromAngle, toAngle);
var distAngle = Mathf.Abs(desiredAngle);
if (distAngle < 1f)
{
transform.eulerAngles = new Vector3(0, 0, angle);
return;
}
desiredAngle = Mathf.Sign(desiredAngle) * (distAngle < SlowDownDelta
? MaxAngVelocity * distAngle / SlowDownDelta
: MaxAngVelocity) * Mathf.Deg2Rad;
var angVel = _body.angularVelocity.z; // in radians
var steeringAngle = desiredAngle - angVel;
var newTorque = (time <= 0 ? 0f : (mass / time * steeringAngle));
_body.AddTorque(0, 0, Mathf.Sign(newTorque) * Mathf.Min(Mathf.Abs(newTorque), MaxTorque));
}
public static float AngleDistance(float from, float to)
{
var dist = (to - from + 180f) % 360f - 180f;
return dist <= -180f ? dist + 360 : dist;
}
If I made any errors, please let me know. Thank you in advance!
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# American Institute of Mathematical Sciences
ISSN:
1531-3492
eISSN:
1553-524X
All Issues
## Discrete & Continuous Dynamical Systems - B
2006 , Volume 6 , Issue 3
Select all articles
Export/Reference:
2006, 6(3): 427-448 doi: 10.3934/dcdsb.2006.6.427 +[Abstract](97) +[PDF](358.7KB)
Abstract:
New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced. Numerical results and efficiency tests are presented in order to show the behaviour of FSF in comparison with G, FG and a conservative scheme of second order.
2006, 6(3): 449-470 doi: 10.3934/dcdsb.2006.6.449 +[Abstract](76) +[PDF](291.9KB)
Abstract:
We consider a multiscale model describing the flow of a concentrated suspension. The model couples the macroscopic equation of conservation of momentum with a nonlinear nonlocal kinetic equation describing at the microscopic level the rheological behaviour of the fluid. We study the long-time limit of the time-dependent solution. For this purpose, we use the entropy method to prove the convergence to equilibrium of the kinetic equation.
2006, 6(3): 471-480 doi: 10.3934/dcdsb.2006.6.471 +[Abstract](78) +[PDF](225.5KB)
Abstract:
It is well known that, in the presence of an attractive force having a Coulomb singularity, scattering solutions of the nonrelativistic Abraham--Lorentz--Dirac equation having nonrunaway character do not exist, for the case of motions on the line. By numerical computations on the full three dimensional case, we give indications that indeed there exists a full tube of initial data for which nonrunay solutions of scatterig type do not exist. We also give a heuristic argument which allows to estimate the size of such a tube of initial data. The numerical computations also show that in a thin region beyond such a tube one has the nonuniqueness phenomenon, i.e. the "mechanical'' data of position and velocity do not uniquely determine the nonrunaway trajectory.
2006, 6(3): 481-492 doi: 10.3934/dcdsb.2006.6.481 +[Abstract](69) +[PDF](246.9KB)
Abstract:
In this paper we obtain Meyers type regularity estimates for approximate solutions of nonlinear elliptic equations. These estimates are used in the analysis of a numerical scheme obtained from a numerical homogenization of nonlinear elliptic equations. Numerical homogenization of nonlinear elliptic equations results in discretization schemes that require additional integrability of the approximate solutions. The latter motivates our work.
2006, 6(3): 493-523 doi: 10.3934/dcdsb.2006.6.493 +[Abstract](90) +[PDF](422.9KB)
Abstract:
We study the dispersive evolution of modulated pulses in a nonlinear oscillator chain embedded in a background field. The atoms of the chain interact pairwise with an arbitrary but finite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schrödinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal-form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.
2006, 6(3): 525-534 doi: 10.3934/dcdsb.2006.6.525 +[Abstract](82) +[PDF](339.1KB)
Abstract:
In this work we analyze a Gause type predator-prey model with a non-monotonic functional response and we show that it has two limit cycles encircling an unique singularity at the interior of the first quadrant, the innermost unstable and the outermost stable, completing the results obtained in previous paper [12, 17, 26, 28].
Moreover, using the Poisson bracket we give a proof, shorter than the ones found in the literature, for determining the type of a cusp point of a singularity at the first quadrant.
2006, 6(3): 535-558 doi: 10.3934/dcdsb.2006.6.535 +[Abstract](74) +[PDF](328.0KB)
Abstract:
In this paper, we present a new model for optimal control of discrete event systems (DESs) with an arbitrary control pattern. Here, a discrete event system is defined as a collection of event sets that depend on strings. When the system generates a string, the next event that may occur should be in the corresponding event set. In the optimal control model, there are rewards for choosing control inputs at strings and the sets of available control inputs also depend on strings. The performance measure is to find a policy under the condition where the discounted total reward among strings from the initial state is maximized. By applying ideas from Markov decision processes, we divide the problem into three sub-cases where the optimal value is respectively finite, positive infinite and negative infinite. For the case with finite optimal values, the optimality equation is shown and further characterized with its solutions. We also characterize the structure of the set of all optimal policies. Moreover, we discuss invariance and closeness of several languages. We present a new supervisory control problem of DESs with the control pattern being dependent on strings. We study the problem in both the event feedback control and the state feedback control by generalizing concepts of invariant and closed languages/predicates. Finally, we apply the above model and results to a job-matching problem.
2006, 6(3): 559-572 doi: 10.3934/dcdsb.2006.6.559 +[Abstract](87) +[PDF](1009.6KB)
Abstract:
In this paper, a discrete-time system, derived from a predator-prey system by Euler's method with step one, is investigated in the closed first quadrant $R_+^2$. It is shown that the discrete-time system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and the discrete-time system has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. Numerical simulations are provided to verify the theoretical analysis and show the complicated dynamical behavior. These results reveal far richer dynamics of the discrete model compared with the same type continuous model.
2006, 6(3): 573-590 doi: 10.3934/dcdsb.2006.6.573 +[Abstract](70) +[PDF](283.5KB)
Abstract:
This paper proposes a novel neural network model for associative memory using dynamical systems. The proposed model is based on synthesizing the external input vector, which is different from the conventional approach where the design is based on synthesizing the connection matrix. It is shown that this new neural network (a) stores the desired prototype patterns as asymptotically stable equilibrium points, (b) has no spurious states, and (c) has learning and forgetting capabilities. Moreover, new learning and forgetting algorithms are also developed via a novel operation on the matrix space. Numerical examples are presented to illustrate the effectiveness of the proposed neural network for associative memory. Indeed, results of simulation experiments demonstrate that the neural network is effective and can be implemented easily.
2006, 6(3): 591-604 doi: 10.3934/dcdsb.2006.6.591 +[Abstract](111) +[PDF](257.0KB)
Abstract:
We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard convection which has the infinite Prandtl number, using a notion of bifurcation called attractor bifurcation. We prove that the problem bifurcates from the trivial solution to an attractor $\A_R$ when the Rayleigh number $R$ crosses the critical Rayleigh number $R_c$. As a special case, we also prove another result which corresponds to the classical pitchfork bifurcation, that this bifurcated attractor $\A_R$ consists of only two stable steady states when the first eigenvalue $R_1$ is simple.
2006, 6(3): 605-622 doi: 10.3934/dcdsb.2006.6.605 +[Abstract](73) +[PDF](1587.0KB)
Abstract:
Non-linear difference equation models are employed in biology to describe the dynamics of certain populations and their interaction with the environment. In this paper we analyze a non-linear system describing community intervention in mosquito control through management of their habitats. The system takes the general form:
$x_{n+1}= a x_{n}h(p y_{n})+b h(q y_{n})$ n=0,1,...
$y_{n+1}= c x_{n}+d y_{n}$
where the function $h\in C^{1}$ ( [ $0,\infty$) $\to$ [$0,1$] ) satisfying certain properties, will denote either $h(t)=h_{1}(t)=e^{-t}$ and/or $h(t)=h_{2}(t)=1/(1+t).$ We give conditions in terms of parameters for boundedness and stability. This enables us to explore the dynamics of prevalence/community-activity systems as affected by the range of parameters.
2006, 6(3): 623-640 doi: 10.3934/dcdsb.2006.6.623 +[Abstract](115) +[PDF](3512.3KB)
Abstract:
This paper extends Runge-Kutta discontinuous Galerkin (RKDG) methods to a nonlinear Dirac (NLD) model in relativistic quantum physics, and investigates interaction dynamics of corresponding solitary wave solutions. Weak inelastic interaction in ternary collisions is first observed by using high-order accurate schemes on finer meshes. A long-lived oscillating state is formed with an approximate constant frequency in collisions of two standing waves; another is with an increasing frequency in collisions of two moving solitons. We also prove three continuum conservation laws of the NLD model and an entropy inequality, i.e. the total charge non-increasing, of the semi-discrete RKDG methods, which are demonstrated by various numerical examples.
2006, 6(3): 641-649 doi: 10.3934/dcdsb.2006.6.641 +[Abstract](95) +[PDF](208.6KB)
Abstract:
This paper deals with the behavior of symmetric discrete--time systems with delays. The influence of the delay over these systems is analyzed in the stabilization problem. Furthermore, conditions on the system are given in order to solve the pole--assignment problem. Finally, some examples are shown with the aim to clarify the obtained results.
2016 Impact Factor: 0.994
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# American Institute of Mathematical Sciences
October 2000, 6(4): 875-892. doi: 10.3934/dcds.2000.6.875
## Stability and random attractors for a reaction-diffusion equation with multiplicative noise
1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain, Spain 2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Received July 1999 Revised July 2000 Published August 2000
We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itô) stabilizes the stationary solution $x\equiv 0$. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.
Citation: Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
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2018 Impact Factor: 1.143
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# What does a higher F-statistic mean?
If the value is much larger than $1$ then the variance of the first population is greater than the second.
$F = {s}_{1}^{2} / {s}_{2}^{2}$
If the value of the statistic is $1$ then the two variances are equal. If the value is much larger than $1$ then the variance of the first population is greater than the second.
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# Air Pressure Loss through Piping
Written by Jerry Ratzlaff on . Posted in Fluid Dynamics
## Air Pressure Loss through Piping formula
$$\large{ p_l = \frac { \mu \; l \; v_a{^2} \; \rho } {24 \; d \; g } }$$
### Where:
Units English Metric $$\large{ p_l }$$ = air pressure loss $$\large{\frac{lbf}{ft^2}}$$ $$Pa$$ $$\large{ \rho }$$ (Greek symbol rho) = density of air $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$ $$\large{ \mu }$$ (Greek symbol mu) = friction coefficient of air dimensionless $$\large{ v_a }$$ = air velocity $$\large{\frac{ft}{sec}}$$ $$\large{\frac{m}{s}}$$ $$\large{ g }$$ = gravitational acceleration $$\large{\frac{ft}{sec^2}}$$ $$\large{\frac{m}{s^2}}$$ $$\large{ d }$$ = pipe inside diameter $$in$$ $$mm$$ $$\large{ l }$$ = length of pipe $$ft$$ $$m$$
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# DX12 Porting to DX12
This topic is 520 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hey guys,
I started working on a port from dx11 to dx12. The first thing, was to setup everything to work with Dx11On12. Right now I've got that done. Basically, the render frame goes as follows:
D3D12Prepare(); // setups the command list and command allocators (as well as basic clear and set render targets)
GetWrappedResources(); // Dx11on12 step to adquire resources
Render(); // Basically all the Dx11 rendering code etc
D3D12End(); // On D3D12Prepare we left the command list opened so we can add aditional
// commands, now close and execute it
ReleaseWrappedResources();
Flush(); // Flush all the dx11 code
Dx12Sync(); // Wait for fence
Dx12Present();
That setup is working and I changed some commands inside Render() from dx11 to dx12. (Basic stuff like setviewport)
I want to start porting more stuff inside the Render(), for example, we have a simple method to draw a quad (without vertex or index buffers, we use the vertex_id inside the shader).
Basically, it should translate to this:
mCmdList->IASetPrimitiveTopology(D3D_PRIMITIVE_TOPOLOGY_TRIANGLESTRIP);
mCmdList->DrawInstanced(4, 1, 0, 0);
But even that simple piece of code is just not working. I would like to get some advice from someone that has done a similar process (using dx11on12), what are the limitations, things that wont work etc
My main concern right now, is that if I want to start setting up commands that touch the IA, I would have to also create the PSO, root signatures etc etc.
Thanks.
Edited by piluve
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# accurate ABC: comments by Oliver Ratman [guest post]
May 31, 2013
By
(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)
Here are comments by Olli following my post:
I think we found a general means to obtain accurate ABC in the sense of matching the posterior mean or MAP exactly, and then minimising the KL distance between the true posterior and its ABC approximation subject to this condition. The construction works on an auxiliary probability space, much like indirect inference. Now, we construct this probability space empirically, this is where our approach differs first from indirect inference and this is where we need the “summary values” (>1 data points on a summary level; see Figure 1 for clarification). Without replication, we cannot model the distribution of summary values but doing so is essential to construct this space. Now, lets focus on the auxiliary space. We can fiddle with the tolerances (on a population level) and m so that on this space, the ABC approximation has the aforesaid properties. All the heavy technical work is in this part. Intuitively, as m increases, the power increases for sufficiently regular tests (see Figure 2) and consequently, for calibrated tolerances, the ABC approximation on the auxiliary space goes tighter. This offsets the broadening effect of the tolerances, so having non-identical lower and upper tolerances is fine and does not hurt the approximation. Now, we need to transport the close-to-exact ABC approximation on the auxiliary space back to the original space. We need some assumptions here, and given our time series example, it seems these are not unreasonable. We can reconstruct the link between the auxiliary space and the original parameter space as we accept/reject. This helps us understand (with the videos!) the behaviour of the transformation and to judge if its properties satisfy the assumptions of Theorems 2-4. While we offer some tools to understand the behaviour of the link function, yes, we think more work could be done here to improve on our first attempt to accurate ABC.
“The paper also insists over and over on sufficiency, which I fear is a lost cause.” To clarify, all we say is that on the simple auxiliary space, sufficient summaries are easily found. For example, if the summary values are normally distributed, the sample mean and the sample variance are sufficient statistics. Of course, this is not the original parameter space and we only transform the sufficiency problem into a change of variable problem. This is why we think that inspecting and understanding the link function is important.
“Another worry is that the … test(s) rel(y) on an elaborate calibration”. We provide some code here for everyone to try out. In our examples, this did not slow down ABC considerably. We generally suppose that the distribution of the summary values is simple, like Gaussian, Exponential, Gamma, ChiSquare, Lognormal. In these cases, the ABC approximation takes on an easy-enough-to-calibrate-fast functional form on the auxiliary space.
“This Theorem 3 sounds fantastic but makes me uneasy: unbiasedness is a sparse property that is rarely found in statistical problems. … Witness the use of “essentially unbiased” in Fig. 4.” What Theorem 3 says is that if unbiasedness can be achieved on the simple auxiliary space, then there are regularity conditions under which these properties can be transported back to the original parameter space. We hope to illustrate these conditions with our examples, and to show that they hold in quite general cases such as the time series application. The thing in Figure 4 is that the sample autocorrelation is not an unbiased estimator of the population autocorrelation. So unbiasedness does not quite hold on the auxiliary space and the conditions of Theorem 3 are not satisfied. Nevertheless, we found this bias to be rather negligible in our example and the bigger concern was the effect of the link function.
And here are Olli’s slides:
Filed under: R, Statistics, University life Tagged: ABC, ABC in Rome, Approximate Bayesian computation, distribution-free tests, Gatsby, London, non-parametric test, Roma, statistical tests
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# Isotopes and their percentage existence in atmosphere
When we talk about percentages of isotopes of an element in the atmosphere, is it percent by mass or volume or weight or something else?
A way that it can be measured is by mass spectrometry - you pass the atoms through the apparatus which shows the mass to charge ratio. E.g. if you could pass $\ce{Cl}$ molecules, you'd see 2 peaks in the resulting graph - one is at 35 and the other at 37 marks which shows the mass/charge. First peak would be about 3 times larger (thus there were 3 times more particles). Thus chlorine would be about 75% isotope 35 and 25% isotope 37. So while this approach measures mass of each isotope, it also determines the relative molar amounts of each isotope.
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# A sum of coefficients.
Algebra Level 4
If $$\sin ^{ 2 }{ A } = x$$, then
$\sin { A } \times \sin { 2A } \times \sin { 3A } \times \sin { 4A }$
is a polynomial in $$x$$, the sum of whose coefficients is:
×
Problem Loading...
Note Loading...
Set Loading...
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How do you solve - x - 1 = x -21?
Jun 28, 2015
color(blue)(x=10
Explanation:
$- x - 1 = x - 21$
$- x - x = - 21 + 1$
$- 2 x = - 20$
color(blue)(x=10
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# Course:Fall 2011 Exercises 2
1. [25 pts] Preimage collision resistance and second-preimage collision resistance are incomparable. In this problem we consider hash functions on a finite domain (from {0,1}n(k) to {0,1}k).
1. Suppose $\mathcal{H}$ is preimage collision resistant. Modify $\mathcal{H}$ to $\mathcal{H}'$ (possibly with a different domain), so that the latter remains preimage collision resistant, but is not second-preimage collision resistant. (Prove these properties of $\mathcal{H}'$.)
2. Given a CRHF $\mathcal{H}$ which compresses by two bits (say from n bits to n − 2 bits), construct a CRHF $\mathcal{H}'$ that compresses by one bit (say from n + 1 bits to n bits), such that the function f(h',x) = (h',h'(x)) (where $h'\in\mathcal{H}'$) is not a OWF. (In both $\mathcal{H}$ and $\mathcal{H}'$, collision-resistance holds when the hash function is drawn uniformly at random from the family.)
3. [Extra] (Sufficiently Shrinking) CRHF implies OWF. Below we say that "x has a collision under f" if there exists an $x' \ne x$ such that f(x) = f(x').
1. Let $\mathcal{H}$ be a CRHF and suppose that for every $h\in\mathcal{H}$ and every x, x has a collision under h. Show that the function f(h,x) = (h,h(x)) is a OWF.
2. Now suppose that for each $h\in\mathcal{H}$, all but a negligible fraction of x's have a collision under h. Show that the function f(h,x) = (h,h(x)) is a OWF.
3. Show that if $\mathcal{H}$ is a CRHF from n bits to n / 2 bits, then the function f(h,x) = (h,h(x)) is a OWF.
2. [25 pts] Power of 2-party SFE with only one output. In this problem we shall see how deterministic secure function evaluation (SFE) functionalities in which only one party receives the outcome can be easily used to realize more general functionalities securely, against passive (honest-but-curious) adversaries.
1. Suppose ${\mathcal R}$ is an arbitrary randomized 2-party functionality which takes x and y from Alice and Bob respectively, and samples a uniform random string r (of a fixed length) and gives RA(x,y,r) and RB(x,y,r) respectively to Alice and Bob. Describe a deterministic 2-party SFE functionality ${\mathcal F}$ (which takes x and y from Alice and Bob respectively, and gives fA(x,y) and fB(x,y) to them respectively; fA,fB can depend on RA,RB), and a protocol $\pi^{{\mathcal F}}$ (i.e., a protocol in which Alice and Bob can access a trusted party implementing ${\mathcal F}$), such that $\pi^{{\mathcal F}}$ securely realizes ${\mathcal R}$. In your protocol $\pi^{{\mathcal F}}$, Alice and Bob should access ${\mathcal F}$ exactly once. Security must hold against both passive and active adversaries.
2. Suppose ${\mathcal F}$ is an arbitrary 2-party SFE functionality which takes x and y from Alice and Bob respectively, and gives fA(x,y) and fB(x,y) to them respectively. Describe another 2-party SFE functionality ${\mathcal G}$ which provides output only to Bob (i.e., Alice gets a dummy output $g_A(x,y)=\bot$), and a protocol $\rho^{\mathcal G}$ (i.e., a protocol in which Alice and Bob can access a trusted party implementing ${\mathcal G}$), such that $\rho^{\mathcal G}$ securely realizes ${\mathcal F}$. In your protocol $\rho^{\mathcal G}$, Alice and Bob should access ${\mathcal G}$ exactly once. Security needs to hold only against passive adversaries.
3. [25 pts] OT from Correlated Random Variables. Define Oblivious Transfer (OT) functionality over a field ${\mathbb F}$ (or, over a ring) as an SFE in which Alice inputs $(x_0,x_1) \in {\mathbb F}^2$ and Bob inputs $b\in\{0,1\}$; then Alice gets $\bot$ as output, but Bob gets xb.
1. Consider an inputless, randomized functionality RandOT, which outputs a random pair $(z_0,z_1)\in{\mathbb F}^2$ to Alice and (c,zc) to Bob, where $c\in\{0,1\}$ is a random bit. Give a protocol πRandOT that securely realizes OT, by accessing RandOT exactly once at the beginning of the protocol.
2. Consider another inputless, randomized SFE functionality RandShare, which outputs $(s_A,p_A)\in{\mathbb F}^2$ to Alice and $(s_B,p_B)\in{\mathbb F}^2$ to Bob, where (sA,sB,pA,pB) are uniformly random conditioned on the relation sA + sB = pApB. Give a protocol ρRandShare that securely realizes OT, by accessing RandShare exactly once at the beginning of the protocol.
4. [25 pts] In secure multi-party computation protocols designed for honest-majority, a commonly used tool is a secret-sharing scheme like Shamir's secret-sharing. Let $t\le(n-1)/2$. Consider an (n,t + 1) (i.e., t + 1 out of n) Shamir secret-sharing scheme over some field. Recall that the shares of a value are obtained by evaluating a random degree t polynomial at n points in the field, such that at (say) 0, the polynomial evaluates to the value being shared. Suppose n parties hold the shares of two values x and y under such a scheme. Let the shares be xi and yi for $i=1,\ldots,n$.
1. Addition. Show how the parties can obtain shares zi for z = x + y (shared using the same secret-sharing scheme), without learning anything more.
2. Multiplication (changing the threshold). Show how the parties can obtain shares Wi for w = xy, but shared using an (n,2t + 1) Shamir secret-sharing scheme, without learning anything more. [Hint: Given two polynomials f and g, what can you say about the polynomial h defined as h(i) = f(i)g(i)?]
3. Degree reduction. Suppose the parties are given shares ri and Ri of a value r using the (n,t + 1) and (n,2t + 1) secret-sharing schemes above. Show how they can convert their shares Wi of a value w under the latter scheme to shares wi under the former scheme, such that any subset of t players learns nothing more about w (they might already have partial information about w), where r is uniformly randomly chosen. You can assume that all the parties follow the protocol honestly. [Hint: Use r to blind w before reconstructing it, and then re-share it using the lower degree scheme.]
5. [Extra] The Needham-Schroeder Public Key protocol uses a trusted server, S, to help two parties exchange secret keys with each other. A priori, there are no secrecy or authentication guarantees on the communication network, and the parties know only each other's identities and a public key of the server S. The server, S, knows public keys of all the users. The goal of the protocol is that at the end A and B should agree on random nonces NA and NB (chosen by A and B respectively). The protocol is described at the end.
1. There is a (famous) man-in-the-middle attack on this protocol, whereby a party in the system can set up a shared key with B, while she thinks she has shared that key with A. Describe such an attack (without looking it up!). [Hint: The adversary can run a concurrent session with A.]
2. Suggest a (small) fix for the attack.
3. If you were designing this protocol today, using public-key encryption and signatures, how would you do it?
Needham-Schroeder (Public Key) Protocol: The protocol is described in terms of a public key "encryption" algorithm E. It is a deterministic encryption scheme with the property that ${\mathrm E}_{PK}({\mathrm E}_{SK}^{-1}(M)) = M$. If M is sufficiently random, ${\mathrm E}_{SK}^{-1}(M)$ is assumed to behave like a signature on M (though it does not give existential unforgeability). PA,PB are Alice and Bob's public keys and SA,SB are their secret keys, respectively. Likewise, the server's public and secret keys are PS,SS.
* $A \rightarrow S: A,B$ (A sends the identities to S)
* $S \rightarrow A: {\mathrm E}^{-1}_{SS}( K_{PB}, B )$ (S sends B's public key to A)
* $A \rightarrow B: {\mathrm E}_{PB}(N_A, A)$ (where NA is a fresh nonce, picked by A)
* $B \rightarrow S: B,A$ (B sends the identities to S)
* $S \rightarrow B: {\mathrm E}^{-1}_{SS}( K_{PA}, A)$ (S sends A's public key to B)
* $B \rightarrow A: {\mathrm E}_{PA}(N_B, N_A)$ (where NB is a fresh nonce picked by B)
* $A \rightarrow B: {\mathrm E}_{PB}(N_B)$ (A and B agree on NA,NB at this point)
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It is currently 13 Dec 2018, 03:13
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# Practice Question #3-Each employee of a certain company
Author Message
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Director
Joined: 16 May 2014
Posts: 594
GRE 1: Q165 V161
Followers: 92
Kudos [?]: 439 [0], given: 64
Practice Question #3-Each employee of a certain company [#permalink] 29 May 2014, 08:18
Expert's post
00:00
Question Stats:
33% (00:55) correct 66% (00:26) wrong based on 3 sessions
Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and$35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?
Indicate all such amounts.
(A) $26,000 (B)$28,000
(C) $29,000 (D)$30,000
(E) $31,000 (F)$32,000
(G) $34,000 [Reveal] Spoiler: OA _________________ My GRE Resources Free GRE resources | GRE Prep Club Quant Tests If you find this post helpful, please press the kudos button to let me know ! Director Joined: 16 May 2014 Posts: 594 GRE 1: Q165 V161 Followers: 92 Kudos [?]: 439 [1] , given: 64 Re: Practice Question #3 [#permalink] 29 May 2014, 08:21 1 This post received KUDOS Expert's post Solution One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between$25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of$25,000 and $35,000, which is$30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,
$$\frac{(2)(25,000)+(1)(35000)}{(2+1)}= 28,333$$
where the weight for $25,000 is 2 and the weight for$35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for$25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore$26,000 and $28,000. Thus, the correct answer consists of Choices A ($26,000) and B ($28,000). Intuitively, you might expect that any amount between$25,000 and $28,333 is a possible value of the average salary. To see that$26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3. _________________ My GRE Resources Free GRE resources | GRE Prep Club Quant Tests If you find this post helpful, please press the kudos button to let me know ! Intern Joined: 22 Jul 2018 Posts: 41 Followers: 0 Kudos [?]: 9 [1] , given: 5 Re: Practice Question #3 [#permalink] 10 Oct 2018, 09:50 1 This post received KUDOS more than twice as many means that the number of X employees, Nx, is greater than 2 times the number of Y employees, Ny: Nx > 2*Ny The arithmatic means are: (1) Mx = SUM(Sx)/Nx = 25k (2) My = SUM(Sy)/Ny = 35k Where S is the salary of employee i and the sums go over all employees in the respective departments Entire Company (3) Mtot = (SUM(Sx) + SUM(Sy))/(Nx + Ny) or solving for the sums in (1) and (2) (3a) Mtot = (25k*Nx +35k*Ny)/(Nx + Ny) Plug in Nx = 2*Ny to get the limit on the maximum Mtot Mtot = (50k + 35k)* Ny/(3*Ny) = 28.3k So the answers are a) 26k and b) 28k The answer is a and b because the question says there are more than twice as many employees in company X than Y. If the maximum limit is 28.3k then plausible answers are anything below that as when you have more employees in company X it will drag down the average Intern Joined: 22 Jul 2018 Posts: 41 Followers: 0 Kudos [?]: 9 [1] , given: 5 Re: Practice Question #3 [#permalink] 10 Oct 2018, 09:57 1 This post received KUDOS There are x people in Department X and y people in Department Y. Twice as many people in X as in Y would mean x = 2 y, so more than twice as many is simply x > 2y. The total salary for everyone in Department X is$25,000 x and the total salary for everyone in Department Y is $35,000 y. The total salary for both departments is$25,000 x + $35,000 y. The average salary for both departments is the total salary divided by the total number of people: *($25,000 x + $35,000 y) / (x + 2y). Now, first some critical thinking before we go further. We could just try out a bunch of numbers, but we might miss something if we don't understand what's going on. Department Y people make more money than Department X people. So we can get a minimum average salary by having lots of X people and few Y people. So let's try y=1 and x=1000. I get$25,009.99, which means that all of the answers are still fair game. Note that you could do this without actually trying numbers... Mathematically, you can solve for the average salary as the ratio of employees (x/y) goes to infinity (LOTS of X workers compared to Y workers). You would find the limit is $25,000. Or you could just think about it, and see that the more X workers you have, the closer the average will shift towards the Department X salary. Now we need a maximum limit on average salary. This one's a bit trickier because of the "at least twice as many" thing. But the same basic principle applies. How do we get the maximum possible salary? By having as many Y workers and as few X workers as possible. We could take y=1 and therefore x=3.... But that's three times as many X workers. For higher numbers, we can lessen the difference between them. So let's try y=1000 and x=2001 (just barely more than twice as many). I get an average salary of$28,332.22. We could keep trying bigger numbers to fine tune things, but you'll quickly realize that's enough. An upper limit of about $28,333 means (a) and (b) are the answers. That last one is harder to see by inspection... but you can still do it. Just need to realize that for every one$35,000 employe, you have at least two $25,000 employees. If you input y=1 and x=2 you get an average salary of$28,333.33. You just have to realize that this is a limit, meaning the highest average salary must be less than this number (because there will always be at least one more X worker in the mix to pull the average down). And of course you can do it mathematically as well, by solving for this limit. This time around you want the average salary as the ratio of employees (x/y) approaches 2. You'll get the same thing.
Re: Practice Question #3 [#permalink] 10 Oct 2018, 09:57
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Hello,
I'm having trouble with this maths question and was wondering if someone could help me?? I've been asked to use the Mean Value Theorem to show that:
sqrt (x*y) < 1/2 * (x + y) if 0 < x < y
And we were hinted to use the function f(x) = sqrt x
2. Given the interval [x,y] for the function $f(z) = \sqrt{z}$ (so I don't confuse the variable of the function with the endpoint of my interval), there exists some c such that:
$f'(c) = \frac{f(y) - f(x)}{y-x} = \frac{\sqrt{y} - \sqrt{x}}{y - x}$
Since $0 < x < y$, we know that f'(c) > 0.
Now, the trick is to multiply top and bottom by "1", that is: $\sqrt{y} - \sqrt{x}$. So we get:
$f'(c) = \frac{\sqrt{y} - \sqrt{x}}{y-x} \cdot \left(\frac{\sqrt{y} - \sqrt{x}}{\sqrt{y} - \sqrt{x}}\right) > 0$
$f'(c) = \frac{y - 2\sqrt{xy} + x}{\mbox{(Some denominator)}} > 0$
Multiply both sides by the denominator and you'll (hopefully) reach your inequality.
3. Thankyou so much!! You really helped me!!
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# Ali's question at Yahoo! Answers regarding an indefinite integral
#### MarkFL
Staff member
Here is the question:
What is the integral of asec(x^(1/2))?
I have posted a link there to this topic so the OP can see my work.
#### MarkFL
Staff member
Hello Ali,
We are given to evaluate:
$$\displaystyle I=\int\sec^{-1}\left(\sqrt{x} \right)\,dx$$ where $$\displaystyle 0\le x$$
I would first use the substitution:
$$\displaystyle w=\sqrt{x}\,\therefore\,dx=2w\,dw$$
and we now have:
$$\displaystyle \int 2w\sec^{-1}(w)\,dw$$
Now, using integration by parts, I would let:
$$\displaystyle u=\sec^{-1}(w)\,\therefore\,du=\frac{1}{w\sqrt{w^2-1}}\,dw$$
$$\displaystyle dv=2w\,dw\,\therefore\,v=w^2$$
and now we have:
$$\displaystyle I=w^2\sec^{-1}(w)-\int\frac{w}{\sqrt{w^2-1}}\,dw$$
Next, using the substitution:
$$\displaystyle u=w^2-1\,\therefore\,du=2w\,dw$$
we may write:
$$\displaystyle I=w^2\sec^{-1}(w)-\frac{1}{2}\int u^{-\frac{1}{2}}\,du$$
$$\displaystyle I=w^2\sec^{-1}(w)-u^{\frac{1}{2}}+C$$
Back-substitute for $u$:
$$\displaystyle I=w^2\sec^{-1}(w)-\sqrt{w^2-1}+C$$
Back-substitute for $w$:
$$\displaystyle I=x\sec^{-1}\left(\sqrt{x} \right))-\sqrt{x-1}+C$$
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# Reference: Bag-of-words
October 11, 2020
The bag-of-words (BoW) model is used to represent a document as a fixed-length vector of frequency counts for each token in the document. BoW does not take into account the order of the words. BoW models are useful in classification, machine learning, and topic modeling tasks, among others.
BoW is a relatively simplistic model and requires two elements. The first is a vocabulary, which is the set of all unique tokens that appear across a corpus. Using this vocabulary, each document can be converted into a vector, where the length of the vector is the same as the length of the vocabulary. The individual elements in the vector represent the frequency that a particular token occurs in the document. Note that order is important as the index of a count in a vector corresponds to the index of a word in the vocabulary.
For example, consider the below short sentences and their corresponding vectors.
doc1 = "the man ran from the dog"
doc2 = "my dog is the best dog"
doc3 = "dog is man's best friend"
vocab = ['best', 'dog', 'friend', 'from', 'is',
'man', "man's", 'my', 'ran', 'the']
vec1 = [0, 1, 0, 1, 0, 1, 0, 0, 1, 2]
vec2 = [1, 2, 0, 0, 1, 0, 0, 1, 0, 1]
vec3 = [1, 1, 1, 0, 1, 0, 1, 0, 0, 0]
When preparing a BoW model, it is important to consider tokenization. For example, should a capitalized word be counted separately from a lowercase version of the same word? Should lemmatization or stemming be used? In the above example, man and man's are considered separate tokens, which may not be the intended result. Each of these considerations will change the vector representation of individual documents.
In general, the BoW model can be used to assess how important a token is in a given text. In the above example, the token dog appears twice in doc2 and only once in the other documents. However, the BoW model does not take into account the relative frequency of a token throughout a corpus. In this case, dog appears in every document and arguably has less importance as a differentiating weight for doc2 compared to the other documents. This problem is something that the TF-IDF model attempts to correct.
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• 方法与技术 •
### 上海市“三地”建设评价指标体系与评价方法构建及应用
1. 1华东师范大学地理科学学院， 上海 200241； 2华东师范大学上海市城市化生态过程与生态恢复重点实验室， 上海 200241）
• 出版日期:2016-08-10 发布日期:2016-08-10
### Establishment and application of evaluation index system and evaluation method of the construction of “three types of land” in Shanghai.
WANG Xue1,2, SONG Xue-jun1,2, HU Yue1,2, ZHANG Rui-feng1,2, CAI Yong-li1,2*#br#
1. (1School of Geographic sciences, East China Normal University, Shanghai 200241, China; 2Shanghai Key Lab for Urban Ecological Processes and Eco-restoration, East China Normal University, Shanghai 200241, China).
• Online:2016-08-10 Published:2016-08-10
Abstract: As the key elements of urban ecosystems, green land, woodland, and wetlands (three types of land) sustainably provide ecological services. Shanghai is promoting the construction of ecological city. It is necessary to evaluate urban ecological environment level involving green land, woodland and wetlands. According to the feature and the scale of construction of green land, woodland, and wetlands in Shanghai, ecological environment in Shanghai was statistically analyzed in order to establish an evaluation index system for the three elements. This article statistically analyzes Shanghai’s ecological environment, and therefore builds an evaluation index system for these three ecological elements. By the method Analytic Hierarchy Process (AHP), we clarify the index system into three categories, i.e., quantitative indicators with target, quantitative indicators without target and qualitative indicators, and give 5level rankings under each category. Based on this index system, we perform membership evaluation and thereafter render a result with comprehensive assessment. The result indicates that within the scope of single indicator,vetical greening, woodland structure and wetland consevation, as well as their mutual connectivity require improvement. The comprehensive assessment shows that the memberships of levels I, II, III, IV and V are 0.3261, 0.3411, 0.2208, 0.0707, and 0.0446. In conclusion, the overall status of the three types of land in Shanghai is at level II, a relatively high level in ecological environment costruction, yet it still has space to improve.
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# Generalizing $\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2$
I was looking at this paper on section [17],
$$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2\tag1$$
Let generalize $$(1)$$
$$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(2n-3)(2n-5)\cdots [2n-(2k+1)]}\tag2$$
Where $$k\ge 0$$
I conjectured the closed form for $$(2)$$ to be
$$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(2n-3)\cdots [2n-(2k+1)]}=\frac{2(-1)^k}{(2k+1)!!(2k+1)}\tag3$$
Here are a first few values of $$k=1,2$$ and $$3$$
\begin{align} \sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(2n-3)}&=-\frac{2}{9}\tag4\\ \sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(2n-3)(2n-5)}&=\frac{2}{75}\tag5\\ \sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(2n-3)(2n-5)(2n-7)}&=-\frac{2}{735}\tag6 \end{align}
How do we go about to prove this conjecture $$(3)?$$
• How did you determine (4), (5) and (6)? Have you tested them numerically? – Richard Jan 2 at 18:21
• I tested them numerically. it seems correct – Endgame Jan 2 at 18:30
What about reindexing and induction? The terms $$\frac{1}{(2n-1)\cdots(2n-2k-1)}$$ have a nice telescopic structure: by the residue theorem $$\frac{1}{(2n-1)(2n-3)\cdots(2n-2k-1)}=(-1)^k\sum_{h=0}^{k}\frac{(-1)^h}{(2n-2h-1)}\cdot\frac{1}{2^{k+1}(2h)!!(2k-2h)!!}$$ equals $$\frac{(-1)^k}{2^{2k+1}k!} \sum_{h=0}^{k}\frac{(-1)^h}{(2n-2h-1)}\binom{k}{h}.$$ The natural temptation is now to compute $$\sum_{n\geq 1}\frac{H_n}{4^{n}}\binom{2n}{n}\frac{1}{2n-2h-1}$$ through $$\frac{-\log(1-z)}{1-z}=\sum_{n\geq 1}H_n z^n$$ and $$\frac{1}{4^n}\binom{2n}{n}=\frac{2}{\pi}\int_{0}^{\pi/2}\left(\cos\theta\right)^{2n}\,d\theta$$, multiply both sides by $$(-1)^k \binom{k}{h}$$, sum over $$h=0,1,\ldots,k$$ and finish by invoking Fubini's theorem (allowing to switch the integrals with respect to $$d\theta$$ and $$dz$$) and the Fourier series $$\sum_{m\geq 1}\frac{\cos(m\varphi)}{m}$$ and $$\sum_{m\geq 1}\frac{\sin(m\varphi)}{m}$$.
The only obstruction is that $$\frac{1}{2n-2h-1}=\int_{0}^{1}z^n\left[\frac{1}{2z^{h+3/2}}\right]\,dz$$ does not hold unconditionally: we would have been happier in having rising Pochhammer symbols rather than falling ones. On the other hand, reindexing fixes this issue. Since $$\binom{2n+2}{n+1} = \frac{2(2n+1)}{n+1}\binom{2n}{n}$$, the original series can be written as
$$\sum_{n\geq 1}\frac{H_n \binom{2n}{n}}{4^n(2n-1)} = \sum_{n\geq 0}\frac{2H_{n+1}\binom{2n}{n}}{4^{n+1}(n+1)}=-\frac{1}{\pi}\int_{0}^{1}\sum_{n\geq 0}\int_{0}^{\pi/2}z^n\left(\cos\theta\right)^{2n}\log(1-z)\,d\theta\,dz$$ or $$-\frac{1}{\pi}\int_{0}^{1}\int_{0}^{\pi/2}\frac{\log(1-z)}{1-z\cos^2\theta}\,d\theta\,dz =-\frac{1}{2}\int_{0}^{1}\frac{\log(1-z)}{\sqrt{1-z}}\,dz,$$ clearly given by a derivative of the Beta function. This approach works also by replacing $$(2n-1)$$ with $$(2n-1)\cdots(2n-2k-1)$$, you just have to be careful in managing the involved constants depending on $$k$$.
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## anonymous one year ago How do you simplify (2x^3-11x^2+20x-12) / (x-2) using synthetic division?
1. Nnesha
set the divisor equal to zero and then solve for x x-2=0 |dw:1439075450901:dw| and just write the coefficients
2. anonymous
yes, I'd use synthetic, could also do long division, just that synthetic lends itself on this one
3. anonymous
$$\large \begin{array}{llllll} 2&|&2&-11&20&-12\\ &|&&\square ? \\\hline\\ &&2 \end{array}$$
4. Nnesha
|dw:1439075783075:dw| drop down the leading coefficient and then multiply that by x values which is 2 write down the answer where the question mark is (on above comment by jdoe) and then combine that with -11
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# Help- first response hpt users! Positive???
Thought I would post this here since you all have had BFPs!!! I am DPO 14 with AF due today. I've had BFN with cheapies as recent as this morning. I just did this test with First Response (held for 4hrs) since AF hasn't shown herself. Problem is it looked negative initially so I forgot about it. About 10-20 minutes later, I remembered and checked it and it looked like this. Have any of you had this happen so late? Is it possibly an evap line then since this is waaay after the 5 minutes recommended? Stick still looks the same an hour later. I'm afraid to get my hopes up....
You
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## LaTeX forum ⇒ Graphics, Figures & Tables ⇒ Why does LaTeX add that much space? Topic is solved
Information and discussion about graphics, figures & tables in LaTeX documents.
rain
Posts: 12
Joined: Thu Jan 28, 2016 9:00 pm
### Why does LaTeX add that much space?
Hi
Why does this code render like shown on image below and how to fix it? Why does new section title go to next page? I have not told it to.
1. \begin{table}[ht!]
2. \centering
3. \caption{Nonfunctional requirement 4}
4. \begin{tabular}{|l|p{10cm}|}
5. \hline
6. \textbf{Requirement\#} & 15 \\ \hline
7. \textbf{Requirement Type} & Nonfunctional \\ \hline
8. \textbf{Use case\#} & \\ \hline
9. \textbf{Description} & Information that is displayed should be meaningful. \\ \hline
10. \textbf{Rationale} & \\ \hline
11. \textbf{Created By} & Author \\ \hline
12. \textbf{Fit Criterion} & \\ \hline
13. \textbf{Priority} & Medium \\ \hline
14. \textbf{View id} & \\ \hline
15. \end{tabular}
16. \end{table}
17. \FloatBarrier
18.
19. \subsection*{Appendix 6 Use cases}
20.
21. \FloatBarrier
22. \begin{table}[ht!]
23. \centering
24. \caption{Use case 1}
25. \begin{tabular}{|l|p{10cm}|}
26. \hline
27. \textbf{Use Case Element} & \textbf{Description} \\ \hline
28. \textbf{Use Case Number} & 1 \\ \hline
29. \textbf{Use case Name} & Table management - import table.\\ \hline
30. \textbf{Use Case Description} & User imports CSV file into program. \\ \hline
31. \textbf{Primary Actor} & User \\ \hline
32. \textbf{Precondition} & Program must be running \\ \hline
33. \textbf{Postcondition} & CSV file is imported into program and user can select it from open table view.\\ \hline
34. \textbf{Trigger} & User's need for anonymizing custom table. \\ \hline
35. \textbf{Normal Flow} & \begin{itemize} \vspace{-1.5em}
36. \item[1)] User goes to main view.
37. \item[2)] User clicks on 'Import'.
38. \item[3)] User chooses file and clicks open.
39. \item[4)] Contents of CSV file is loaded into program.
40. \end{itemize}\\ \hline
41. \end{tabular}
42. \end{table}
render.png (30.82 KiB) Viewed 456 times
Last edited by Stefan Kottwitz on Wed May 17, 2017 11:54 am, edited 1 time in total.
Johannes_B
Site Moderator
Posts: 3489
Joined: Thu Nov 01, 2012 4:08 pm
You use advanced automatic placemement instructions, retricting them to what you think works and hit the automatic placement with a hammer that destroys every advantage of it.
Do Not Use figure Environments if you don't want automatic placement.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.
Stefan Kottwitz
Posts: 8336
Joined: Mon Mar 10, 2008 9:44 pm
Location: Hamburg, Germany
Contact:
Hi Rain,
I tested this code and LaTeX does not add so much space. With this code, tested both with the article class and the book class, both tables are on the same page with very few space in-between.
Perhaps post a minimal working example that shows the problem. The cause is somewhere else in your code.
Stefan
rain
Posts: 12
Joined: Thu Jan 28, 2016 9:00 pm
I managed to fix it by adding 1 character of invisible text before next header and then it jumped back to its right place. I used ht! because otherwise the table weent to next chapter and it would look very strange if the nonfunctional requirement were in the middle of use case.
Stefan Kottwitz
Posts: 8336
Joined: Mon Mar 10, 2008 9:44 pm
Location: Hamburg, Germany
Contact:
LaTeX tables and figures don't float into a next chapter. But perhaps you mean floating into the next section or subsection?
At least for sections it can easily prevented by
\usepackage[section]{placeins}
By the way, I commonly use the most flexible placement, as near as possible, by using the [htbp!] options.
Stefan
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# Doubt regarding cross sections of a Killing horizon
+ 1 like - 0 dislike
251 views
I was reading Rácz and Wald's paper, 'Extensions of spacetimes with Killing Horizons' and they are frequently referring to a cross section of a Killing horizon (a three surface where the norm of a timelike-at-infinity Killing vector vanishes). I initially thought that by the cross section, they meant a two surface which intersects the null generators of the Killing horizon at a fixed parameter length. However, in section 3 while proving a proposition, they chose a 'local' cross section that is 'sufficiently small', through a point p of the horizon. This means my intuition is probably incorrect because I really don't understand how can you construct a 'small' cross section of a three surface as it would necessarily have to intersect all the null generators. Can someone please explain what is the exact meaning of their statement? Here is a link to the paper if anyone needs it: https://iopscience.iop.org/article/10.1088/0264-9381/9/12/008
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.
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# Fun With Spheres
This is the front page of the Space called "fun-with-spheres". This Space contains a wiki and a subversion repository.
The wiki is available at the URL http://spaces.perimeterinstitute.ca/fun-with-spheres/wiki .
The subversion repository can be checked out with the command
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# convergence of the sequence (1+1/n)^n
###### Theorem 1.
The following sequence:
$a_{n}=\left(1+\frac{1}{n}\right)^{n}$ (1)
is convergent.
###### Proof.
The proof will be given by demonstrating that the sequence (1) is:
1. 1.
monotonic (increasing), that is $a_{n}
2. 2.
bounded above, that is $\forall n\in\mathbb{N},a_{n} for some $M>0$
In order to prove part 1, consider the binomial expansion for $a_{n}$:
$a_{n}=\sum_{k=0}^{n}\binom{n}{k}\frac{1}{n^{k}}=\sum_{k=0}^{n}\frac{1}{k!}% \frac{n}{n}\frac{n-1}{n}\ldots\frac{n-(k-1)}{n}=\sum_{k=0}^{n}\frac{1}{k!}% \left(1-\frac{1}{n}\right)\ldots\left(1-\frac{k-1}{n}\right).$
Since $\forall i\in\{1,2\ldots(k-1)\}:(1-\frac{i}{n})<(1-\frac{i}{n+1})$, and since the sum $a_{n+1}$ has one term more than $a_{n}$, it is demonstrated that the sequence (1) is monotonic.
In order to prove part 2, consider again the binomial expansion:
$a_{n}=1+\frac{n}{n}+\frac{1}{2!}\frac{n(n-1)}{n^{2}}+\frac{1}{3!}\frac{n(n-1)(% n-2)}{n^{3}}+\ldots+\frac{1}{n!}\frac{n(n-1)\ldots(n-n+1)}{n^{n}}.$
Since $\forall k\in\{2,3\ldots n\}:\frac{1}{k!}<\frac{1}{2^{k-1}}$ and $\frac{n(n-1)\ldots(n-(k-1))}{n^{k}}<1$:
$a_{n}<1+\left(1+\frac{1}{2}+\frac{1}{2\times 2}+\ldots+\frac{1}{2^{n-1}}\right% )<1+\left(\frac{1-\frac{1}{2^{n}}}{1-\frac{1}{2}}\right)<3-\frac{1}{2^{n-1}}<3$
where the formula giving the sum of the geometric progression with ratio $1/2$ has been used. ∎
In conclusion, we can say that the sequence (1) is convergent and its limit corresponds to the supremum of the set $\{a_{n}\}\subset\left[2,3\right)$, denoted by $e$, that is:
$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=\sup_{n\in\mathbb{N}}\left\{% \left(1+\frac{1}{n}\right)^{n}\right\}\triangleq e,$
which is the definition of the Napier’s constant.
Title convergence of the sequence (1+1/n)^n ConvergenceOfTheSequence11nn 2013-03-22 17:43:26 2013-03-22 17:43:26 kfgauss70 (18761) kfgauss70 (18761) 7 kfgauss70 (18761) Theorem msc 33B99 NondecreasingSequenceWithUpperBound
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# Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable
## Problem 482
For which values of constants $a, b$ and $c$ is the matrix
$A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable?
(The Ohio State University, Linear Algebra Final Exam Problem)
## Solution.
Note that the matrix $A$ is an upper triangular matrix.
Hence the eigenvalues of $A$ are diagonal entries $7, 2, 3$.
So the $3\times 3$ matrix $A$ has three distinct eigenvalues.
This implies that $A$ is diagonalizable.
Hence, regardless of the values of $a, b, c$, the matrix $A$ is always diagonalizable.
Thus, $a, b, c$ can take arbitrary values.
## Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)
This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).
The other problems can be found from the links below.
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##### Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let \$S=\{p_1(x), p_2(x),...
Close
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# eilbeck-ws15
## Chris Eilbeck (Heriot-Watt University)
10.12.2015 - W01 0-012 (Wechloy), 16 Uhr c.t.
Generalizations of the Weierstrass sigma function, discriminants, and heat equations
Abstract: The solutions of many interesting nonlinear wave equations can be written down in terms of Weierstrass $\wp$ functions and their generalizations to higher genus. In turn the $\wp$ functions are just the 2nd logarithmic derivatives of a entire function, the Weierstrass $\sigma$ function generalized to genus $g$. We discuss the properties of the $\sigma$ function associated with a plane curve of genus $g$. These functions satisfy sets of interesting linear parabolic PDEs. These PDEs can be used to form linear but complicated recurrence relations for the coefficients of the sigma expansion. A connection with the discriminant of the associated curve is also highlighted.
(Stand: 09.06.2021)
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• ### Characterization of the Spontaneous Light Emission of the PMTs used in the Double Chooz Experiment(1604.06895)
Aug. 17, 2016 hep-ex, physics.ins-det
During the commissioning of the first of the two detectors of the Double Chooz experiment, an unexpected and dominant background caused by the emission of light inside the optical volume has been observed. A specific study of the ensemble of phenomena called "Light Noise" has been carried out in-situ, and in an external laboratory, in order to characterize the signals and to identify the possible processes underlying the effect. Some mechanisms of instrumental noise originating from the PMTs were identified and it has been found that the leading one arises from the light emission localized on the photomultiplier base and produced by the combined effect of heat and high voltage across the transparent epoxy resin covering the electric components. The correlation of the rate and the amplitude of the signal with the temperature has been observed. For the first detector in operation the induced background has been mitigated using online and offline analysis selections based on timing and light pattern of the signals, while a modification of the photomultiplier assembly has been implemented for the second detector in order to blacken the PMT bases.
• ### Muon capture on light isotopes in Double Chooz(1512.07562)
May 17, 2016 hep-ex, nucl-ex, physics.ins-det
Using the Double Chooz detector, designed to measure the neutrino mixing angle $\theta_{13}$, the products of $\mu^-$ capture on $^{12}$C, $^{13}$C, $^{14}$N and $^{16}$O have been measured. Over a period of 489.5 days, $2.3\times10^6$ stopping cosmic $\mu^-$ have been collected, of which $1.8\times10^5$ captured on carbon, nitrogen, or oxygen nuclei in the inner detector scintillator or acrylic vessels. The resulting isotopes were tagged using prompt neutron emission (when applicable), the subsequent beta decays, and, in some cases, $\beta$-delayed neutrons. The most precise measurement of the rate of $^{12}\mathrm C(\mu^-,\nu)^{12}\mathrm B$ to date is reported: $6.57^{+0.11}_{-0.21}\times10^{3}\,\mathrm s^{-1}$, or $(17.35^{+0.35}_{-0.59})\%$ of nuclear captures. By tagging excited states emitting gammas, the ground state transition rate to $^{12}$B has been determined to be $5.68^{+0.14}_{-0.23}\times10^3\,\mathrm s^{-1}$. The heretofore unobserved reactions $^{12}\mathrm C(\mu^-,\nu\alpha)^{8}\mathrm{Li}$, $^{13}\mathrm C(\mu^-,\nu\mathrm n\alpha)^{8}\mathrm{Li}$, and $^{13}\mathrm C(\mu^-,\nu\mathrm n)^{12}\mathrm B$ are measured. Further, a population of $\beta$n decays following stopping muons is identified with $5.5\sigma$ significance. Statistics limit our ability to identify these decays definitively. Assuming negligible production of $^{8}$He, the reaction $^{13}\mathrm C(\mu^-,\nu\alpha)^{9}\mathrm{Li}$ is found to be present at the $2.7\sigma$ level. Limits are set on a variety of other processes.
• ### Measurement of $\theta_{13}$ in Double Chooz using neutron captures on hydrogen with novel background rejection techniques(1510.08937)
Dec. 28, 2015 hep-ex, physics.ins-det
The Double Chooz collaboration presents a measurement of the neutrino mixing angle $\theta_{13}$ using reactor $\overline{\nu}_{e}$ observed via the inverse beta decay reaction in which the neutron is captured on hydrogen. This measurement is based on 462.72 live days data, approximately twice as much data as in the previous such analysis, collected with a detector positioned at an average distance of 1050m from two reactor cores. Several novel techniques have been developed to achieve significant reductions of the backgrounds and systematic uncertainties. Accidental coincidences, the dominant background in this analysis, are suppressed by more than an order of magnitude with respect to our previous publication by a multi-variate analysis. These improvements demonstrate the capability of precise measurement of reactor $\overline{\nu}_{e}$ without gadolinium loading. Spectral distortions from the $\overline{\nu}_{e}$ reactor flux predictions previously reported with the neutron capture on gadolinium events are confirmed in the independent data sample presented here. A value of $\sin^{2}2\theta_{13} = 0.095^{+0.038}_{-0.039}$(stat+syst) is obtained from a fit to the observed event rate as a function of the reactor power, a method insensitive to the energy spectrum shape. A simultaneous fit of the hydrogen capture events and of the gadolinium capture events yields a measurement of $\sin^{2}2\theta_{13} = 0.088\pm0.033$(stat+syst).
• ### Improved measurements of the neutrino mixing angle $\theta_{13}$ with the Double Chooz detector(1406.7763)
Jan. 21, 2015 hep-ex, physics.ins-det
The Double Chooz experiment presents improved measurements of the neutrino mixing angle $\theta_{13}$ using the data collected in 467.90 live days from a detector positioned at an average distance of 1050 m from two reactor cores at the Chooz nuclear power plant. Several novel techniques have been developed to achieve significant reductions of the backgrounds and systematic uncertainties with respect to previous publications, whereas the efficiency of the $\bar\nu_{e}$ signal has increased. The value of $\theta_{13}$ is measured to be $\sin^{2}2\theta_{13} = 0.090 ^{+0.032}_{-0.029}$ from a fit to the observed energy spectrum. Deviations from the reactor $\bar\nu_{e}$ prediction observed above a prompt signal energy of 4 MeV and possible explanations are also reported. A consistent value of $\theta_{13}$ is obtained from a fit to the observed rate as a function of the reactor power independently of the spectrum shape and background estimation, demonstrating the robustness of the $\theta_{13}$ measurement despite the observed distortion.
• ### Ortho-positronium observation in the Double Chooz Experiment(1407.6913)
Oct. 7, 2014 hep-ex, physics.ins-det
The Double Chooz experiment measures the neutrino mixing angle $\theta_{13}$ by detecting reactor $\bar{\nu}_e$ via inverse beta decay. The positron-neutron space and time coincidence allows for a sizable background rejection, nonetheless liquid scintillator detectors would profit from a positron/electron discrimination, if feasible in large detector, to suppress the remaining background. Standard particle identification, based on particle dependent time profile of photon emission in liquid scintillator, can not be used given the identical mass of the two particles. However, the positron annihilation is sometimes delayed by the ortho-positronium (o-Ps) metastable state formation, which induces a pulse shape distortion that could be used for positron identification. In this paper we report on the first observation of positronium formation in a large liquid scintillator detector based on pulse shape analysis of single events. The o-Ps formation fraction and its lifetime were measured, finding the values of 44$\%$ $\pm$ 12$\%$ (sys.) $\pm$ 5$\%$ (stat.) and $3.68$ns $\pm$ 0.17ns (sys.) $\pm$ 0.15ns (stat.) respectively, in agreement with the results obtained with a dedicated positron annihilation lifetime spectroscopy setup.
• We describe a muon track reconstruction algorithm for the reactor anti-neutrino experiment Double Chooz. The Double Chooz detector consists of two optically isolated volumes of liquid scintillator viewed by PMTs, and an Outer Veto above these made of crossed scintillator strips. Muons are reconstructed by their Outer Veto hit positions along with timing information from the other two detector volumes. All muons are fit under the hypothesis that they are through-going and ultrarelativistic. If the energy depositions suggest that the muon may have stopped, the reconstruction fits also for this hypothesis and chooses between the two via the relative goodness-of-fit. In the ideal case of a through-going muon intersecting the center of the detector, the resolution is ~40 mm in each transverse dimension. High quality muon reconstruction is an important tool for reducing the impact of the cosmogenic isotope background in Double Chooz.
• ### Background-independent measurement of $\theta_{13}$ in Double Chooz(1401.5981)
April 25, 2014 hep-ex
The oscillation results published by the Double Chooz collaboration in 2011 and 2012 rely on background models substantiated by reactor-on data. In this analysis, we present a background-model-independent measurement of the mixing angle $\theta_{13}$ by including 7.53 days of reactor-off data. A global fit of the observed neutrino rates for different reactor power conditions is performed, yielding a measurement of both $\theta_{13}$ and the total background rate. The results on the mixing angle are improved significantly by including the reactor-off data in the fit, as it provides a direct measurement of the total background rate. This reactor rate modulation analysis considers antineutrino candidates with neutron captures on both Gd and H, whose combination yields $\sin^2(2\theta_{13})=$ 0.102 $\pm$ 0.028(stat.) $\pm$ 0.033(syst.). The results presented in this study are fully consistent with the ones already published by Double Chooz, achieving a competitive precision. They provide, for the first time, a determination of $\theta_{13}$ that does not depend on a background model.
• ### First Measurement of \theta_13 from Delayed Neutron Capture on Hydrogen in the Double Chooz Experiment(1301.2948)
Aug. 29, 2013 hep-ex
The Double Chooz experiment has determined the value of the neutrino oscillation parameter $\theta_{13}$ from an analysis of inverse beta decay interactions with neutron capture on hydrogen. This analysis uses a three times larger fiducial volume than the standard Double Chooz assessment, which is restricted to a region doped with gadolinium (Gd), yielding an exposure of 113.1 GW-ton-years. The data sample used in this analysis is distinct from that of the Gd analysis, and the systematic uncertainties are also largely independent, with some exceptions, such as the reactor neutrino flux prediction. A combined rate- and energy-dependent fit finds $\sin^2 2\theta_{13}=0.097\pm 0.034(stat.) \pm 0.034 (syst.)$, excluding the no-oscillation hypothesis at 2.0 \sigma. This result is consistent with previous measurements of $\sin^2 2\theta_{13}$.
• ### First Test of Lorentz Violation with a Reactor-based Antineutrino Experiment(1209.5810)
Dec. 23, 2012 hep-ex
We present a search for Lorentz violation with 8249 candidate electron antineutrino events taken by the Double Chooz experiment in 227.9 live days of running. This analysis, featuring a search for a sidereal time dependence of the events, is the first test of Lorentz invariance using a reactor-based antineutrino source. No sidereal variation is present in the data and the disappearance results are consistent with sidereal time independent oscillations. Under the Standard-Model Extension (SME), we set the first limits on fourteen Lorentz violating coefficients associated with transitions between electron and tau flavor, and set two competitive limits associated with transitions between electron and muon flavor.
• ### Direct Measurement of Backgrounds using Reactor-Off Data in Double Chooz(1210.3748)
Oct. 20, 2012 hep-ex, nucl-ex
Double Chooz is unique among modern reactor-based neutrino experiments studying $\bar \nu_e$ disappearance in that data can be collected with all reactors off. In this paper, we present data from 7.53 days of reactor-off running. Applying the same selection criteria as used in the Double Chooz reactor-on oscillation analysis, a measured background rate of 1.0$\pm$0.4 events/day is obtained. The background model for accidentals, cosmogenic $\beta$-$n$-emitting isotopes, fast neutrons from cosmic muons, and stopped-$\mu$ decays used in the oscillation analysis is demonstrated to be correct within the uncertainties. Kinematic distributions of the events, which are dominantly cosmic-ray-produced correlated-background events, are provided. The background rates are scaled to the shielding depths of two other reactor-based oscillation experiments, Daya Bay and RENO.
• ### Reactor electron antineutrino disappearance in the Double Chooz experiment(1207.6632)
Aug. 30, 2012 hep-ex, physics.ins-det
The Double Chooz experiment has observed 8,249 candidate electron antineutrino events in 227.93 live days with 33.71 GW-ton-years (reactor power x detector mass x livetime) exposure using a 10.3 cubic meter fiducial volume detector located at 1050 m from the reactor cores of the Chooz nuclear power plant in France. The expectation in case of theta13 = 0 is 8,937 events. The deficit is interpreted as evidence of electron antineutrino disappearance. From a rate plus spectral shape analysis we find sin^2 2{\theta}13 = 0.109 \pm 0.030(stat) \pm 0.025(syst). The data exclude the no-oscillation hypothesis at 99.8% CL (2.9{\sigma}).
• ### Monte Carlo aided design of the inner muon veto detectors for the Double Chooz experiment(1207.1623)
July 6, 2012 physics.ins-det
The Double Chooz neutrino experiment aims to measure the last unknown neutrino mixing angle theta_13 using two identical detectors positioned at sites both near and far from the reactor cores of the Chooz nuclear power plant. To suppress correlated background induced by cosmic muons in the detectors, they are protected by veto detector systems. One of these systems is the inner muon veto. It is an active liquid scintillator based detector and instrumented with encapsulated photomultiplier tubes. In this paper we describe the Monte Carlo aided design process of the inner muon veto, that resulted in a detector configuration with 78 PMTs yielding an efficiency of 99.978 +- 0.004% for rejecting muon events and an efficiency of >98.98% for rejecting correlated events induced by muons. A veto detector of this design is currently used at the far detector site and will be built and incorporated as the muon identification system at the near site of the Double Chooz experiment.
• ### Indication for the disappearance of reactor electron antineutrinos in the Double Chooz experiment(1112.6353)
March 13, 2012 hep-ex
The Double Chooz Experiment presents an indication of reactor electron antineutrino disappearance consistent with neutrino oscillations. A ratio of 0.944 $\pm$ 0.016 (stat) $\pm$ 0.040 (syst) observed to predicted events was obtained in 101 days of running at the Chooz Nuclear Power Plant in France, with two 4.25 GW$_{th}$ reactors. The results were obtained from a single 10 m$^3$ fiducial volume detector located 1050 m from the two reactor cores. The reactor antineutrino flux prediction used the Bugey4 measurement as an anchor point. The deficit can be interpreted as an indication of a non-zero value of the still unmeasured neutrino mixing parameter \sang. Analyzing both the rate of the prompt positrons and their energy spectrum we find \sang = 0.086 $\pm$ 0.041 (stat) $\pm$ 0.030 (syst), or, at 90% CL, 0.015 $<$ \sang $\ <$ 0.16.
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# Thread: Every time i work this problem out, i get the wrong answer. If someone can help me
1. ## Every time i work this problem out, i get the wrong answer. If someone can help me
Consumers in a certain state can choose between three long-distance telephone services: GTT, NCJ, and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 20% of its customers to NCJ and 15% to Dash, NCJ loses 5% of its customers to GTT and 10% to Dash, and Dash loses 10% of its customers to GTT and 20% to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?
GTT's expected market share is %. (Round to the nearest tenth as needed.)
NCJ:
Dash:
2. ## Re: Every time i work this problem out, i get the wrong answer. If someone can help m
first build up the transition matrix
Let the state vector be the percentages of GTT, NCJ, and Dash in that order.
$v = (G,~N,~D)$
GTT loses 20% to NCJ and 15% to Dash, so it retains 65% of it's customers.
Further it gains 5% from NCJ and gains 10% from Dash.
So the top row of the transition matrix is
$T_{1,j} = (0.65, ~0.05, ~0.1)$
Similarly we find that the entire transition matrix is given by
$T = \begin{pmatrix}0.65 &0.05 &0.1 \\ 0.2 &0.85 &0.2 \\ 0.15 &0.1 &0.7 \end{pmatrix}$
we also know that the percentages for each phone company total to 1.
At equilibrium the transition matrix acting on the state vector returns the same state vector.
So we have the following equations to solve
$Tv = v$
$v\cdot (1,1,1) = 1$
We thus see that $v$ is just the eigenvector of $T$ corresponding to eigenvalue $1$, that has been normalized so it's components sum to 1.
Can you compute that vector?
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# C is a closed path in the z-plane given by |z| = 3. The value of the integral $$\mathop \oint \nolimits_C^\; \left( {\frac{{{z^2} - z + 4j}}{{z + 2j}}} \right)dz$$ is
This question was previously asked in
GATE EC 2014 Official Paper: Shift 1
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1. -4π (1 + j2)
2. 4π (3 – j2)
3. -4π (3 + j2)
4. 4π (1 – j2)
Option 3 : -4π (3 + j2)
Free
CT 1: Ratio and Proportion
2672
10 Questions 16 Marks 30 Mins
## Detailed Solution
Concept:
If f(z) is analytic within and on a closed curve, and if ‘a’ is any point within C then according to Cauchy Integral formula:
$$f\left( a \right) = \frac{1}{{2\pi i}}\mathop \smallint \nolimits_e^\; \frac{{f\left( z \right)}}{{\left( {z - a} \right)}}dz$$
$$\therefore \;\mathop \smallint \nolimits_C^\; \frac{{f\left( z \right)}}{{\left( {z - a} \right)}}dz = 2\pi i\;f\left( a \right)$$
Application:
$$\left| z \right| = 3$$
$$\mathop \oint \nolimits_C^\; \frac{{{z^2} - z + 4j}}{{z + 2j}} = ?$$
Pole z = -2j, which lies Inside the given C i.e. |z| = 3
∴ Using the Cauchy Integral formula, we get:
$$\mathop \oint \nolimits_C^\; \frac{{{z^2} - z + 4j}}{{z + 2j}} = 2\pi j\;{\left[ {{z^2} - z + 4j} \right]_{at\;z = \; - 2j}}$$
$$\mathop \oint \nolimits_C^\; \frac{{{z^2} - z + 4j}}{{z + 2j}} = 2\pi j\;\left[ { - 4 + 2j + 4j} \right]$$
$$= - 4\pi \left( {3 + j2} \right)$$
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# Proton capture nucleosynthesis
Big bang nucleosynthesis supernova nucleosynthesis by changing the number of protons, electron capture transforms the nuclide of. I discuss stellar spectroscopy and nucleosynthesis astronomers the yellow boxes represent isotopes produced by proton capture. C nuclei produced are further converted by proton capture to 14 n besides 14 n nuclei represent a major neutron poison we find that a linear relationship.
Keywords: globular star clusters, abundances, nucleosynthesis, red giant abundance differences in light, hydrogen-burning “proton-capture (hereafter, p. Only subsequent nucleosynthesis in stars generates larger isotopes than form helium-4 by capturing either a proton or another deuterium nucleus and then. It is impossible to produce p-nuclei by neutron capture how then can nature make these nuclei the first possibility that suggests itself is proton capture. A combination of proton and $\ensuremath{\alpha}$-particle capture rates in order to estimate final abundances produced in nucleosynthesis.
Big bang nucleosynthesis begins about one minute after the big bang, when the universe has cooled enough to form stable protons and neutrons, after. Sections and astrophysical s-factor of the radiative proton capture on 11b to the keywords: nuclear astrophysics primordial nucleosynthesis light atomic. We analyze the s-process nucleosynthesis in models of rotating agb stars, using a complete nuclear network subsequent proton captures lead to overlapping.
During this process, known as nucleosynthesis, a multitude of different types of nuclei, or isotopes, is formed one of these is rapid proton capture (rp-process. Dissasemble a nucleus into protons and neutrons it is derived s-process: nucleosynthesis by means of slow neutron captures occurs in stars. The carbon-nitrogen-oxygen cycle, a process of stellar nucleosynthesis in which a carbon-12 nucleus captures a proton and emits a gamma ray, producing. The term p-process (p is for proton) is used in two ways in the scientific literature concerning the astrophysical origin of the elements (nucleosynthesis) originally it referred to a proton capture process which is the source of.
Two neutron-capture reactions, called the slow (s) and rapid (r) processes, kick in for instance, nuclei of hydrogen—the lightest element, with one proton—can. P-process nucleosynthesis via proton-capture reactions in thermonuclear supernovae explosions anne endres1,a, c arda1, p erbacher1, j glorius1,2,. There are other processes of far less importance, such as proton capture observation of stars that formed very early in the history of the universe show an .
This baryon number can be in the form of protons and neutrons or atomic nuclei (7be actually becomes 7li through electron capture 7be + e− → 7li + νe. Cross sections of proton capture reactions on $^{74}\mathrm{se}$, at temperatures relevant to $p$-process nucleosynthesis differ by a factor. A summary of the nucleosynthesis of light elements is as follows 4he helium burning isotopes are proton-rich and cannot be formed via neutron capture.
Nucleosynthesis is the creation of new, and heavier, elements from precursor starting with proton captures onto these seeds, a nuclear process is started. 31 411 measuring the cross section of proton-induced reactions 32 it was suggested that further α captures could extend nucleosynthesis beyond 16o to. Proton-capture nucleosynthesis in globular cluster red giant stars robert m cavallo department of astronomy, university of maryland, college park,. Motivation: the ambiguous origin of 113in proposed nucleosynthetic scenarios: university of athens first proton capture study of the.
Proton capture nucleosynthesis
Rated 5/5 based on 27 review
2018.
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# Tag Info
Method The nicest method I have seen is one that expresses the automaton as equation system of (regular) languages which can be solved. It is in particular nice as it seems to yield more concise expressions than other methods. Let $A= (Q,\Sigma,\delta,q_0,F)$ an NFA without $\varepsilon$-transitions. For every state $q_i$, create the equation $\qquad \... 37 The fundamental theorems of formal language theory are that regular expressions, regular grammars, deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs) all describe the same kinds of languages: namely the regular languages. The fact that we can describe these languages in so many completely different ways suggests that there's ... 35 Your conjecture is disproved by Keith Ellul, Bryan Krawetz, Jeffrey Shallit and Ming-wei Wang in their paper "Regular Expressions: New Results and Open Problems". While the paper is not available on-line, a talk is. In the paper, they define the measure$|\mathrm{alph}(R)|$, which is the number of symbols in$R$, not counting$\epsilon$or$\emptyset$. ... 28 Brzozowski algebraic method This is the same method as the one described in Raphael's answer, but from a point of view of a systematic algorithm, and then, indeed, the algorithm. It turns out to be easy and natural to implement once you know where to begin. Also it may be easier by hand if drawing all the automata is impractical for some reason. When ... 25 Transitive closure method This method is easy to write in a form of an algorithm, but generates absurdly large regular expressions and is impractical if you do it by hand, mostly because this is too systematic. It is a good and simple solution for an algorithm though. The key idea Let$R^k_{i,j}$represent the regular expression for the strings going from ... 24 Whoever told you that regular expressions are used to parse code was spreading disinformation. Classically (I don't know to what extent this is true in modern compilers), the parsing of code – conversion of code from text to a syntax tree – is composed of two stages: Lexical analysis: Processes the raw text into chunks such as keywords, numerical constants, ... 23 If regular expressions were allowed to be infinite, then any language would have been regular. Given the language$L=\{w_1, w_2, \ldots\}$, we can always define the regular expression$R = w_1 + w_2 + \cdots$, which exactly defines$L$. (Example: the regular expression$R_1 = \epsilon+0+1+00+01+10+11+\cdots$defines$L_1=\{0,1\}^*$.) We know that some ... 22 Regular expressions, regular grammars and finite automata are simply three different formalisms for the same thing. There are algorithms to convert from any of them to any other. The basic reason that we have all three is that they were created independently, with the first set of equivalences (there are several other formalisms as well) proven by Kleene (... 21 First of all, backreferences can not be simulated by finite automata as they allow you to describe non-regular languages. For example, ([ab]^*)\1 matches$\{ww \mid w \in \{a,b\}^*\}$, which is not even context-free. Look-ahead and look-behind are nothing special in the world of finite automata as we only match whole inputs here. Therefore, the special ... 19 With only union and concatenation, you can't describe any infinite language. The union and concatenation can only produce finitely many strings. With only union and the Kleene star, you can't describe a language such as$L = \{ab\}$, because there's no way to concatenate an expression generating only$a$with an expression generating only$b$. With only ... 18 Since you want "to convert regex to DFA in less than 30 minutes", I suppose you are working by hand on relatively small examples. In this case you can use Brzozowski's algorithm$[1]$, which computes directly the Nerode automaton of a language (which is known to be equal to its minimal deterministic automaton). It is based on a direct computation of the ... 18 You can't. The pumping lemma can only be used to prove that a language is non-regular. How to prove that it is regular depends on how you've defined regular languages. You (or your course or textbook) might have defined them as any of languages described by regular expressions; languages accepted by deterministic finite automata (DFAs); languages accepted ... 17 Indeed the POSIX BRE language cannot express all regular expressions because it lacks alternation. It can't even recognize all finite languages, let alone all regular languages. For example,$\{ab, ba\}$is not recognizable as a BRE. To prove this, consider what the toplevel syntactic form could be: It can't be one of the single-character forms since the ... 16 All regular languages have LL(1) grammars. To obtain such a grammar, take any DFA for the regular language (perhaps by doing the subset construction on the NFA obtained from the regular expression), then convert it to a right-recursive regular grammar. This grammar is then LL(1), because any pair of productions for the same nonterminal either start with ... 16 1) If we also allow intersection and complement, then the resulting expressions are sometimes called extended regular expressions; as the regular languages are closed under boolean operations nothing is gained by them. It is just syntactic sugar. A similar conclusion holds for the reverse operation. Part of the reason why on first instance all the other ... 16 The pumping lemma states a proprety of regular languages: If$L$is a regular language then there exists an integer$p$such that if$w \in L$has length at least$p$then it can be written as$w = xyz$, where$|xy| \leq p$,$y \neq \epsilon$, and$xy^iz \in L$for all$i \geq 0$. Unfortunately, this property doesn't characterize regular languages. That ... 16 So my question is: (Why) Is my proof wrong? If it is right: Why is there no easy way to express permutations? Your "proof" only looked at permutations of single words, which are finite languages. Every finite language is regular (e.g. just by listing all of the members with a | inbetween), but there are infinite regular languages (and those ... 15 Assuming the TCS-variant of regex, the problem is indeed NP-complete. We assume that our regexes contain letters from$\Sigma$, matching themselves,$+$, denoting union,$\cdot$, denoting concatenation,$*$, denoting Kleene-Star,$\lambda$, matching the empty string and nothing else. Length of a regex is defined as the number of characters from$\Sigma$. ... 15 A set is closed under some operator if the result of applying the operator to things in the set is always in the set. For example, the natural numbers are closed under addition because, whenever$n$and$m$are natural numbers,$n+m$is a natural number. On the other hand, the naturals are not closed under subtraction since, for example,$3-5$is not a ... 14 It depends upon whether you've got a regular expression or a regexp: regexps are evil, but regular expressions are a thing of beauty and will never turn evil on you. By regexp, I mean a modern regular expression: i.e., a regular expression with additional modern features such as backreferences -- e.g., a Perl-compatible regular expression. This is more ... 14 First, notice that you can easily eliminate$\emptyset$for all regular expressions other than a regular expression describing the empty language. To do this, you use the following rewriting rules, which define an operator$E$on regular expressions:$E[\sigma] = \sigma$,$E[\epsilon] = \epsilon$,$E[\emptyset] = \emptyset$.$E[r_1 r_2]$is$\emptyset$if ... 13 tl;dr backrefs. As soon as there is a \1 (or any number that isn't used to escape unicode) in the regexp it is not a regular expression. Backrefs allows you to match (a+)b\1 which matches n times a followed by b followed by n times a for any n>1. This is not a regular language (it's the poster child of a non regular language). It is necessary and nearly ... 12 First off, putting$k$in the regular expression is not allowed by typical regular expression syntax, unless I'm mistaken. Even adding constant powers is a shorthand notation, not a part of the regular expression formalism. Even if it were possible, using that notation, you could mix$a$s with$b$s, so that not all$a$'s would necessarily come before all$b$... 12 The authoritative reference on the pragmatic issues behind implementing regex engines is a series of three blog posts by Russ Cox. As described there, since backreferences make your language non-regular, they are implemented using backtracking. Lookaheads and lookbehinds, like many features of of regex pattern matching engines, don't quite fit into the ... 12 Hendrik Jan gives a good answer for complexity class, but not an algorithm itself. The simplest algorithm to do this that I know of is to convert the regular expression to a DFA. There are known techniques for converting a regular expression to an NFA, and an NFA to a DFA. Once you have two DFAs, testing for equivalence is efficient and decidable, since ... 12 This is some heavy stuff for a high school assignment. Yuval Filmus's answer is really good, so this is more of a supplementary answer to clarify some of the points he made. A formal language is a mathematical construction. Their use for programming languages is just one of many possible uses; in fact, linguist Noam Chomsky made significant contributions ... 12 Yep, this is wrong, because of ambiguity. Consider the following language:$(a + aa) + a(a + \epsilon)$. With your method, we see 4 words,$a, aa, aa, a$. But we have duplicates! There are multiple ways to make the same word within the given regular expression. A better method is to use dynamic programming on an minimal DFA for your language, with no "... 11 First, the string of minimum length might not be defined properly since it might not be unique. Here is a way to find a string of minimum length: Convert the regular expression to a nondeterministic finite automaton. Convert the nondeterministic automaton into a deterministic one. Use a breadth first search until you encounter the first nonfinal state (if ... 11 The set$1^*+0^*$is composed of two parts:$1^*$and$0^*$. The first part,$1^*$, is all strings composed entirely of$1$s. The second part,$0^*$, is all strings composed entirely of$0$s. In contrast,$(1+0)^*$is all strings composed of$0$s and$1$s. Can you now think of a string in$(1+0)^*$but not in$1^*+0^*\$?
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• Create Account
### #Actualrobert.leblanc
Posted 14 October 2012 - 02:54 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
LOW ------------------------------------------------------------------- HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|
LOW -------------------------------------------------------------------- HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
00000000BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
0000000000000000GGGGGGGG00000000 (G bits shifted 8 to the left) OR'd With
000000000000000000000000RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of convention where I should start at the far right and call that byte 1 or am I missing something else?
Also does the + 0.5f in (r * 255.0f + 0.5f) and the like, cause rounding upwards or is it doing something else?
### #11robert.leblanc
Posted 14 October 2012 - 02:52 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
LOW ------------------------------------------------------------------- HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|
LOW -------------------------------------------------------------------- HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
GGGGGGGG000000000 (G bits shifted 8 to the left) OR'd With
RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of convention where I should start at the far right and call that byte 1 or am I missing something else?
Also does the + 0.5f in (r * 255.0f + 0.5f) and the like, cause rounding upwards or is it doing something else?
### #10robert.leblanc
Posted 14 October 2012 - 02:39 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
LOW ------------------------------------------------------------------- HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|
LOW -------------------------------------------------------------------- HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
GGGGGGGG000000000 (G bits shifted 8 to the left) OR'd With
RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of convention where I should start at the far right and call that byte 1 or am I missing something else?
### #9robert.leblanc
Posted 14 October 2012 - 02:38 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
LOW ------------------------------------------------------------------- HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|
LOW -------------------------------------------------------------------- HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
GGGGGGGG000000000 (G bits shifted 8 to the left) OR'd With
RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of the convention is that I should start at the far right and call that byte 1 or am I missing something else?
### #8robert.leblanc
Posted 14 October 2012 - 02:37 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
LOW HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|
LOW HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
GGGGGGGG000000000 (G bits shifted 8 to the left) OR'd With
RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of the convention is that I should start at the far right and call that byte 1 or am I missing something else?
### #7robert.leblanc
Posted 14 October 2012 - 02:35 PM
Ok so just to make sure I'm getting it right:
D3DXCOLOR::operator UINT () const
{
UINT dwR = r >= 1.0f ? 0xff : r <= 0.0f ? 0x00 : (UINT) (r * 255.0f + 0.5f);
UINT dwG = g >= 1.0f ? 0xff : g <= 0.0f ? 0x00 : (UINT) (g * 255.0f + 0.5f);
UINT dwB = b >= 1.0f ? 0xff : b <= 0.0f ? 0x00 : (UINT) (b * 255.0f + 0.5f);
UINT dwA = a >= 1.0f ? 0xff : a <= 0.0f ? 0x00 : (UINT) (a * 255.0f + 0.5f);
return (dwA << 24) | (dwR << 16) | (dwG << 8) | (dwB << 0);
}
the return statement is essentially a 32 bit entity that looks like
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|
Does it make sense that if x86 intel machines are little-endian that I should think of this as
| --byte1=Alpha--|--byte2=Red--|--byte3=Green--|--byte4=Blue--|LOW HIGH
-----------------------------------------------------------------------------------------
The function that I used was given by Luna as:
D3DX10INLINE UINT ARGB2ABGR(UINT argb)
{
BYTE A = (argb >> 24) & 0xff;
BYTE R = (argb >> 16) & 0xff;
BYTE G = (argb >> 8) & 0xff;
BYTE B = (argb >> 0) & 0xff;
return (A << 24) | (B << 16) | (G << 8) | (R << 0);
}
Which to me seems to imply:
| --byte1=Alpha--|--byte2=Blue--|--byte3=Green--|--byte4=Red--|LOW HIGH
This works but obviously my understanding of the endianness is incorrect because the format code I am using is:
DXGI_FORMAT_R8G8B8A8_UNORM
So this means the expected byte order is Red, Green, Blue, Alpha. Which if you read opposite of what I have above makes sense. To me it looks like the bytes are ordered backwards in memory (ABGR). Maybe I'm misunderstanding the shifting operation.
Does: (A << 24) | (B << 16) | (G << 8) | (R << 0) create the following sort of thing
AAAAAAAA000000000000000000000000 (A bits shifted 24 to the left) OR'd With
BBBBBBBB0000000000000000 (B bits shifted 16 to the left) OR'd With
GGGGGGGG000000000 (G bits shifted 8 to the left) OR'd With
RRRRRRRR (R bits shifted 0 to the left) Which results in
-------------------------------
AAAAAAAABBBBBBBBGGGGGGGGRRRRRRRR
Is it just a case of the convention is that I should start at the far right and call that byte 1 or am I missing something else?
PARTNERS
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# Development of A Pavement Type Evaluation Procedure for TxDOT
### Citation:
M. A. Beg, Zhang, Z., and Hudson, W. R., “Development of A Pavement Type Evaluation Procedure for TxDOT,” CD Proceedings of the 79th Annual Meeting of the Transportation Research Board. Washington, DC, 2000.
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# Matchings in graphs
I'm dealing with the following problem in graph's matchings. Let G = (V,E) be an undirected graph, and let S,T subgroups of V be two sets of vertices with no common neighbours. That is, there exist no vertices s in S, t in T, and v in V such that (s,v) (t,v) in E (note that there may still be edges between S and T or within S or T).
I wanna show that if there exists a matching in G in which all vertices in S are covered, and also a (possibly different) matching in which all vertices in T are covered, then there also exists a matching in G in which all vertices in S union T are covered.
An idea: dividing M1 and M2 (the two matchings for S,T accordingly) to groups of different edges: 1 - Edges from S to S 2 - Edges from S to T 3 - Edges from S to an unmatched vertex (harmless edges).
Same for T.
I think the first thing to do in order to show such matching exists, is take edges from group 3 first, that will make our problem smaller. Or perhaps there's a counter example? Any ideas?
Hint: Try to cover $S$ first, and then cover $T$. The first step must be done as minimally as possible so that the second step works.
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# Prove matrix has all real eigenvalues
#### Jameson
Staff member
Problem: Let $A$ be a $n \times n$ matrix with real entries. Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real.
Solution: I'm definitely not seeing how to approach this problem. I know that to calculate the eigenvalues of a matrix I need to solve $\text{det }(A-\lambda I)=0$ and I have experience calculating them, but I've never seen commentary on whether the values will be real or complex. Any ideas to get started?
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#### Opalg
##### MHB Oldtimer
Staff member
Problem: Let $A$ be a $n \times n$ matrix (with real entries). Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real.
Think of $A$ as acting on the complex inner-product space $\mathbb{C}^n$ (whose inner product satisfies $\langle y,x\rangle = \overline{\langle x,y\rangle}$). If $\lambda$ is an eigenvalue of $A$, with eigenvector $x$, then $$\lambda\langle x,x\rangle = \langle Ax,x\rangle = \langle x,A^{\mathrm{\scriptsize T}}x\rangle = \langle x,Ax\rangle = \overline{\langle Ax,x\rangle} = \overline{\lambda}\langle x,x\rangle,$$ and so $\overline{\lambda} = \lambda$.
#### Jameson
Staff member
Thank you very much for the quick reply, Opalg! I don't believe this is the intended method to solve the problem though since any sort of complex analysis or complex theory isn't a prerequisite for the course. I'll take some time to read over your solution more and digest it, but am still looking perhaps for another potential method.
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Problem: Let $A$ be a $n \times n$ matrix. Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real.
I suppse there is a typo. In order to be precise: "Let $A$ be a real $n \times n$ matrix ...". Oherwise, the result is false. Choose for example $A=\text{diag }(1,i).$
#### Jameson
Staff member
Yes, you are very correct and my apologies for this error. I've fixed the OP now.
#### Chris L T521
##### Well-known member
Staff member
The way that Opalg approached the problem is the most common way that I've seen it done.
So, if you don't want to use complex numbers or spaces....
The other way to do this then would be to change the statement slightly and prove the following:
An $n\times n$ matrix $A$ has all real eigenvalues and $n$ orthogonal real eigenvectors if and only if $A$ is real symmetric.
The proof of the revised statement would require a lot more (and I do mean a lot more) work than the way Opalg presented.
#### Deveno
##### Well-known member
MHB Math Scholar
If you're talking about roots of real polynomials, it's really hard to avoid a discussion of complex numbers.
#### Jameson
Staff member
My mistake everybody and thank you for reassurance. Seems I have some reading to do. I took linear algebra last semester but we didn't cover this method at all so it's completely new material. I'll try to work through Opalg's solution.
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Hi Jameson!
What you are asking for is the proof of a general theorem called the "spectral theorem for real symmetric matrices".
Its proof is far from trivial, as you can see if you google for it, although Opalg's proof is quite elegant.
His proof is a bit concise though. I had to puzzle a bit to understand why the steps he's taking are valid.
First step is to realize that the characteristic polynomial you have, $\det(A-\lambda I)=0$, is a polynomial of degree n, meaning it has n (possible duplicate) roots if we allow complex numbers.
What's left to prove is that these eigenvalues are actually real numbers.
#### Deveno
##### Well-known member
MHB Math Scholar
If you start with complex numbers to begin with, you can actually say a bit more:
Every Hermetian matrix has real eigenvalues (Opalg's proof goes through as before).
As ILikeSerena has indicated, this is a consequence of the spectral theorem for normal matrices: every normal matrix is unitarily diagonalizable, and every unitarily diagonalizable matrix is normal
(normal matrices are those for which $$\displaystyle AA^H = A^HA$$).
This is another instance of a statement about real numbers that makes more sense when you consider the wider context of complex numbers.
As far as I understand it (which is poorly) physicists prefer to work with complex vector spaces, because you can always restrict to the special case of real numbers, and complex numbers are in a sense "more complete". There's not much more difficulty involved in doing so, many of the results in linear algebra hold for an arbitrary field (there are some special cases where a field of characteristic 2 is inappropriate), and the usual definition of an inner product in a vector space over $$\displaystyle \Bbb C$$ resolves to a real-valued inner product when the scalars and coordinates are real.
What I'm trying to get across here is the idea that the "algebra" part of linear algebra, is based on the classical operations of addition, subtraction, multiplication and division (operations with matrices are "built-up" of operations of these kinds on individual entries) and a field is precisely the kind of algebraic object we can do those things IN.
For 99% of the examples one actually encounters in practice, the algebraic closure of the rational numbers would suffice (we rarely work with the full spectrum of transcendental numbers, a few important ones keep popping up). It is common practice when studying inner product spaces to assume the underlying field is a subfield of $$\displaystyle \Bbb C$$, and it is perhaps unfortunate that in introductory courses so much emphasis is given to $$\displaystyle \Bbb R^n$$, when you can do more without any extra effort.
#### Jameson
Staff member
Think of $A$ as acting on the complex inner-product space $\mathbb{C}^n$ (whose inner product satisfies $\langle y,x\rangle = \overline{\langle x,y\rangle}$). If $\lambda$ is an eigenvalue of $A$, with eigenvector $x$, then $$\lambda\langle x,x\rangle = \langle Ax,x\rangle = \langle x,A^{\mathrm{\scriptsize T}}x\rangle = \langle x,Ax\rangle = \overline{\langle Ax,x\rangle} = \overline{\lambda}\langle x,x\rangle,$$ and so $\overline{\lambda} = \lambda$.
Ok I've done some reading on inner-product spaces and feel comfortable with the basic axioms and some other conclusions from them. The steps I don't follow are $\langle Ax,x\rangle = \langle x,A^{\mathrm{\scriptsize T}}x\rangle = \langle x,Ax\rangle$. The first two steps and the last three are clear but these three are not. Can someone explain?
#### Deveno
##### Well-known member
MHB Math Scholar
In some developments, one actually DEFINES the transpose of a square matrix $$\displaystyle A$$ as the matrix $$\displaystyle B$$ such that:
$$\displaystyle \langle Ax,x \rangle = \langle x,Bx \rangle, \forall x$$
(this assumes a REAL inner product space).
However, we can follow this entry-by-entry:
$$\displaystyle \langle Ax,x \rangle = \sum_i\left(\sum_j a_{ij}x_j\right)x_i$$
$$\displaystyle = a_{11}x_1x_1 + a_{12}x_2x_1 + \cdots + a_{1n}x_nx_1 + a_{21}x_1x_2 + a_{22}x_2x_2 + \cdots + a_{2n}x_nx_2 +$$
$$\displaystyle \cdots + a_{n1}x_1x_n + a_{n2}x_2x_n + \cdots + a_{nn}x_nx_n$$
$$\displaystyle = x_1(a_{11}x_1 + a_{21}x_2 + \cdots + a_{n1}x_n) + x_2(a_{12}x_1 + a_{22}x_2 + \cdots + a_{n2}x_n) +$$
$$\displaystyle \cdots + x_n(a_{1n}x_1 + a_{2n}x_2 + \cdots + a_{nn}x_n)$$ <---plucking out the $$\displaystyle x_j$$ in the "middle position"
$$\displaystyle = \sum_j x_j\left(\sum_i a_{ji}x_i \right) = \langle x,A^Tx \rangle$$.
The equality $$\displaystyle \langle A^Tx,x \rangle = \langle Ax,x \rangle$$ follows because $$\displaystyle A$$ is symmetric, and thus $$\displaystyle A^T = A$$.
(Note: this assumes the $$\displaystyle x_j$$ are the coordinates of $$\displaystyle x$$ in some basis (the standard basis works well), and that the $$\displaystyle a_{ij}$$ are the entries of $$\displaystyle A$$ in that same basis).
(Note #2: I hope it's clear that when we rearrange the order of the $$\displaystyle x_i$$'s and the $$\displaystyle x_j$$'s in the complex case, the sesquilinearity of the complex inner product requires we use $$\displaystyle \overline{a_{ji}}$$ instead, leading to:
$$\displaystyle \langle Ax,x \rangle = \langle x,A^Hx \rangle$$
This reduces to the symmetric case for the reals, since real numbers are self-conjugate).
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#### Klaas van Aarsen
##### MHB Seeker
Staff member
Alternatively:
The standard inner product is given by $\langle x,y \rangle = y^\dagger x$, whiere $\dagger$ denotes the conjugate transpose.
Since A is a real matrix, its conjugate transpose is the same as its transpose.
And since A is symmetric, its transpose is the same as A: $A^\dagger = A^T = A$.
So:
$$\langle Ax,x \rangle = x^\dagger(Ax) = (x^\dagger A)x = (A^\dagger x)^\dagger x = \langle x, A^\dagger x \rangle = \langle x, A^T x \rangle = \langle x, A x \rangle$$
Oh, and you may already now that in general $(MN)^T = N^TM^T$, which is the reason that we have here that $(x^\dagger A) = (A^\dagger x)^\dagger$.
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Are there symbols for contradiction for a proof by contradiction? I tried $\lightning, \Lightning$.
To clarify, there's no code for lightning? I believed it'd be shorter than \Rightarrow\!\Leftarrow
## closed as off-topic by Najib Idrissi, TMM, user61527, Asaf Karagila♦, Davide GiraudoMar 20 '14 at 21:16
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question does not appear to be about Mathematics Stack Exchange or the software that powers the Stack Exchange network within the scope defined in the help center." – Najib Idrissi, TMM, Community, Asaf Karagila, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
\Rightarrow\!\Leftarrow gives $\Rightarrow\!\Leftarrow$. There's a good argument that words are better, though.
• And I don't think any particular convention is widespread; the symbol I drew above (or maybe it was $\rightarrow\!\leftarrow$) was introduced to me by one of my professors, but I don't think I've ever seen it in a book. Similarly, I don't think I've ever seen the \lightning symbol you mention before. As Asaf said, everyone knows "contradiction". – Hurkyl Mar 12 '14 at 19:43
• Words are better, yes. Except when you have to write fast, and do not expect strangers to read the result. The same comment applies to a lot of other symbolism. Write "and" instead of writing $\wedge$. Write "if...then" instead of writing $\Longrightarrow$. – GEdgar Mar 14 '14 at 14:55
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## Reddit’s comment ranking algorithm revisited
Introduction
The “Bayesian/frequentist” coin puzzle discussed in the last couple of posts was really just an offshoot of some thoughts I have been mulling over about Reddit’s current default approach to ranking user comments on a post, based on the number of upvotes and downvotes each comment receives. (Or more generally, the problem of ranking a collection of any items, whether comments, or consumer products, etc., based on varying numbers of “like/dislike” votes.) Instead of trying to estimate the bias of a coin based on the observed number of heads and tails flipped, here each comment is like a coin, and each upvote (or downvote) is like an observation of a coin flip coming up heads (or tails).
If we assume that each comment has some fixed– but unknown– probability $\theta$ that a random user will upvote the comment, then it would be convenient to simply sort all of the comments on a particular post by decreasing $\theta$, so that the “best” comments would appear near the top. Unfortunately, we don’t actually know $\theta$, we can only estimate it somehow by using the observed pair $(u,d)$ of upvotes and downvotes, respectively.
A natural first idea might be to “score” each comment using the maximum likelihood estimate
$\hat{\theta} = \frac{u}{u+d}$
and sort the comments by this score. But this tends to unfairly compare comments with very different numbers of total votes; e.g., should a comment with votes $(3,0)$ really be ranked higher than $(99,1)$?
Wilson Score Interval
Evan Miller’s “How Not To Sort By Average Rating” does a good job of presenting this and other approaches, eventually arguing for sorting by the lower bound of the Wilson score interval, which is what Reddit currently does. Briefly, the Wilson score interval is a confidence interval intended to “cover” (i.e., contain) the true– but unknown– value $\theta$ with at least some guaranteed probability, described as the “confidence level.” In general, the higher the confidence level, or the fewer the number of observations, the wider the corresponding confidence interval. By scoring each comment with the lower bound of this confidence interval, we are effectively starting with a point estimate based on the fraction of upvotes, but then penalizing this score according to the total number of votes, with fewer votes receiving a greater penalty.
Reddit’s use of this scheme has evolved slightly over time, initially computing a 70% confidence interval, but then changing to the current wider 80% confidence interval, having the effect of imposing a slightly greater penalty on comments with fewer total votes. This “fine-tuning” of the scoring algorithm raises the question whether there might not be a more natural method for ranking user comments, that does not require this sort of knob-turning.
A Bayesian Alternative
Last year, James Neufeld proposed the interesting idea of sampling a random score for each comment by drawing from a corresponding beta distribution with parameters
$(\alpha, \beta) = (u+1, d+1)$
The idea is that this beta distribution is a natural way to express our uncertainty about the “true” value $\theta$ of a comment, starting with an assumed prior uniform distribution on $\theta$ (i.e., a comment is initially equally likely to be great, terrible, or anything in between), and updating based on the observation of $(u,d)$ upvotes and downvotes, respectively. For example, a comment with 30 upvotes and 10 downvotes yields a beta distribution with the following density:
Probability density of beta distribution with parameters (30+1,10+1).
A key point is that every user does not necessarily see the comments for a post in the same order. Each time the post is viewed, the comments are re-scored by new random draws from the corresponding beta distributions, and sorted accordingly. As a comment receives more and more upvotes and/or downvotes, it will “settle in” to a particular position among other comments… but comments with few votes, or even strongly downvoted comments, will still have some chance of appearing near the top of any particular user’s view of the page.
I really like this idea, but the non-deterministic ordering of comments presented to different users may be seen as a drawback. Can we fix this?
Sorting by Expected Rank
I can think of two natural deterministic modifications of this approach. The first is to sort comments by their expected ranking using the random scoring described above. In other words, for each comment, compute the expected number of other comments that would appear higher than it on one of Neufeld’s randomly generated pages, and sort the comments by this expected value.
Although this method “fixes” the non-determinism of the original, unfortunately it suffers from a different undesirable property: the relative ranking of two comments may be affected by the presence or absence of other comments on the same post. For example, consider the two comments identified by their upvote/downvote counts $(0,1)$ and $(1,3)$. If these are the only two comments on a post, then $(0,1) < (1,3)$. However, if we introduce a third comment $(7,3)$, then the resulting overall ranking is $(1,3) < (0,1) < (7,3)$, reversing the ranking of the original two comments!
Pairwise comparisons
Which brings me, finally, to my initial idea for the following second alternative: sort the comments on a post according to the order relation
$(u_1,d_1) < (u_2,d_2) \iff P(X_1 > X_2) < \frac{1}{2}$
where
$X_k \sim Beta(u_k+1,d_k+1)$
More intuitively, we are simply ranking one comment higher than another if it is more likely than not to appear higher using Neufeld’s randomized ranking.
Note one interesting property of this approach that distinguishes it from all of the other methods mentioned so far: it does not involve assigning a real-valued “score” to each individual comment (and subsequently sorting by that score). This is certainly possible in principle (see below), but as currently specified we can only compare two comments by performing a calculation involving parameters of both in a complex way.
Open Questions
Unfortunately, there are quite a few holes to be patched up with this method, and I am hoping that someone can shed some light on how to address these. First, the strict order defined above is not quite a total order, since there are some pairs of distinct comments where one comment’s randomized score is equally likely to be higher or lower than the other. For example, all of the comments of the form $(u,u)$, with an equal number of upvotes and downvotes, have this problem. This is probably not a big deal, though, since I think it is possible to arbitrarily order these comments, for example by increasing total number of votes.
But there are other more interesting pairs of incomparable comments. For example, consider $(5,0)$ and $(13,1)$. The definition above is insufficient to rank these two… but it turns out that it had better be the case that $(13,1) < (5,0)$, since we can find a third comment that lies between them:
$(13,1) < (70,8) < (5,0)$
This brings us to the next open question: is this order relation transitive (in other words, is it even a partial order)? I have been unable to prove this, only verify it computationally among comments with bounded numbers of votes.
The final problem is a more practical one: how efficiently can this order relation be computed? Evaluating the probability that one beta-distributed random variable exceeds another involves a double integral that “simplifies” to an expression involving factorials and a hypergeometric function of the numbers of upvotes and downvotes. If you want to experiment, following is Python code using the mpmath library to compute the probability $P(X_1 > X_2)$:
from mpmath import fac, hyp3f2
def prob_greater(u1, d1, u2, d2):
return (hyp3f2(-d2, u2 + 1, u1 + u2 + 2, u2 + 2, u1 + u2 + d1 + 3, 1) *
fac(u1 + u2 + 1) / (fac(u1) * fac(u2)) *
fac(u1 + d1 + 1) * fac(u2 + d2 + 1) /
((u2 + 1) * fac(d2) * fac(u1 + u2 + d1 + 2)))
print(prob_greater(5, 0, 13, 1))
John Cook has written a couple of interesting papers on this, in the medical context of evaluating clinical trials. This one discusses various approximations, and this one presents exact formulas and recurrences for some special cases. The problem of computing the actual probability seems daunting… but perhaps it is a simpler problem in this case to not actually compute the value, but just determine whether it is greater than 1/2 or not?
In summary, I think these difficulties can be rolled up into the following more abstract statement of the problem: can we impose a “natural,” efficiently computable total order on the set of all beta distributions with positive integer parameters, that looks something like the order relation described above?
This entry was posted in Uncategorized. Bookmark the permalink.
### 2 Responses to Reddit’s comment ranking algorithm revisited
1. Do you think that there could be ranking system that used the history of an item. This would be useful when an item must be voted with knowledge of its current rank.
• Certainly there could be, and to an extent there already is; see, for example, the “hot” ranking, which incorporates not only upvotes/downvotes but also the age of an item. An even more complex algorithm could take into account not only the age of the item itself, but also the time history of the *votes*, so that, for example, recent votes on an old item might be weighed more heavily than older votes.
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# Synopsis: Quantum Pairs Walking
Researchers demonstrate quantum random walks of photon pairs that interact like bosons, fermions, or anything in between, which could be used to simulate other quantum systems.
Harnessing quantum information could allow powerful computations that are inaccessible to classical systems. But many quantum features of a single particle simply reflect its wavelike aspects. Experiments in Physical Review Letters simultaneously manipulate pairs of particles, whose interlinked behavior cannot be emulated by classical waves. Linda Sansoni of Sapienza University of Rome, Italy, and her colleagues implemented the quantum version of a discrete random walk using photons in a centimeter-sized glass chip. They first focused intense laser pulses along chosen paths, which tweak the glass’s refractive index to create stable waveguides for later photons. Running two parallel waveguides alongside each other for about two millimeters gives a photon a $50/50$ chance of jumping between them.
The team wrote an array of parallel waveguides that periodically came closer to their left or right neighbors. A photon launched into one waveguide could emerge in any of eight guides, depending on which jumps it made in the various interaction regions. But unlike the classical version of the random walk, the probabilities of different outcomes reflect the effects of quantum interference between different paths.
The researchers used beamsplitters to mimic both a “quantum coin” operation (whether a photon jumps) and the walker displacement. By launching pairs of photons whose polarization states were entangled, the researchers could reproduce all types of quantum interactions, from fermionlike repulsion to bosonlike attraction and anything in between. These results matched theoretical expectations, but the technique could be adapted to simulate other, less-well-understood quantum systems. – Don Monroe
Correction (6 January 2012): Paragraph 3, sentence 1, “The researchers used the vertical or horizontal polarization of the photons to encode a quantum degree of freedom, or “quantum coin,” which affects whether a photon jumps or not.” changed to “The researchers used beamsplitters to mimic both a “quantum coin” operation (whether a photon jumps) and the walker displacement.”
More Features »
### Announcements
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## Previous Synopsis
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# A Cryptosystem
## Securing Communication using Cryptography
Cryptography has numerous applications but the most basic is its use in securing an insecure communication channel. In this section, this common use-case is detailed using a symmetric key cryptographic scheme.
Consider three users, Alice, Bob, and Oscar. Alice wishes to communicate with Bob privately but the channel between them is unfortunately insecure. Generically speaking this channel could be an Internet-based communication link, telephone lines, WiFi or even a cellular phone network. Oscar is a curious user who wants to listen in to Alice and Bob's conversation (see figure below).
Communication over an insecure channel
Oscar is able to listen in to the channel in an authorized way (also called an eavesdropping attack) because he has access to the channel. This means he has access to the router in case of Internet-based communication or simply is monitoring the radio signals as in case of Wi-Fi communication. This common situation applies to any scenario where two or more communicating parties are sharing data that is of some worth to someone else. Examples of such data could be for example financial secrets, business plans or political decisions not yet shared with the public
Symmetric-key cryptosystem
A cryptosystem built upon Symmetric cryptography provides an elegant solution in this regard. Using such a system Alice uses a symmetric algorithm to encrypt her message $x$, resulting in the ciphertext $y$. When Bob gets this ciphertext he performs a decryption operation on this message accordingly. The decryption and encryption operations are inverse processes of each other (see figure above). The obvious benefits to this assuming that the cipher algorithm itself is strong is that $y$ appears to random bits to an attacker like Oscar and he would not be able to extract any useful information from the communicated data.
## A Cryptosystem
We used the terms $x$, $y$ and $k$ to explain the figure above but in fact, they hold a special meaning in cryptography:
A cryptosystem can be defined as a tuple of 5 elements namely $(E, D, X, K, Y)$. $X$ refers to all the plaintexts that this system can consume, the set of keys is referred to as $K$, $Y$ is accordingly the ciphertexts set, $E: X \times K \rightarrow Y$ is the set of cipher algorithms, and finally, $D: Y \times K \rightarrow X$ is the set of deciphering algorithms.
PARTICIPATION ACTIVITY: Symmetric Cryptosystem
## Kerckhoff's Principle
In the 19th century, a renowned cryptographer by the name of Auguste Kerckhoffs stated that a cryptosystem should be secure even if everything about the system, except the key, is public knowledge. This later become widely recognized as the "Kerckhoff's Principle" and became widely embraced by cryptographers.
Kerckhoffs's principle was reformulated by Claude Shannon who asserted that "the enemy knows the system". This design principle states that systems should be designed with the expectation that the attacker will instantly acquire complete knowledge about them. This reformulation is also called Shannon's maxim. The idea is that if the designer bases the security on the fact that the user is ignorant of the system's internal structure then an expert user can break the security mechanism. This concept is also called by some as "security through obscurity".
Since cryptography is a highly mathematical subject, companies that rely on cryptography for protection of customer or employee records tend to try and hide the algorithms they use. History has repeatedly proved that this practice doesn't improve the strength of the system's security. In fact, it tends to provide a false sense of security to a system that in reality has a weak implementation. This is especially true with state-of-the-art technologies today such as virtualization which makes it possible to easily reverse engineer software and their underlying algorithms.
Note that keeping the algorithm secret is not the same as keeping the cryptographic keys secret. The latter will not break Kirchoff's principle. In fact, keeping the keys in a safe and secure place is a good security practice.
PARTICIPATION ACTIVITY 2: Kerckhoff's Principle
DISCUSSION ACTIVITY: Research the following questions.
1. Based on the definition of a cryptosystem write down the 5-tuple representation of the following ciphers:
1. Julius invents a new cipher which takes as plaintext input a string of alphabets (A through Z) and replaces each letter in the plaintext with another letter some fixed number of positions down the alphabet. For example, with a left shift of 3, D would be replaced by A, E would become B, and so on.
2.
3. ROT-13 takes as plaintext input a string of alphabets (A through Z) and replaces each one by the letter 13 places further along in the alphabet, wrapping back to the beginning if necessary. A becomes N, B becomes O, and so on up to M, which becomes Z, then the sequence continues at the beginning of the alphabet.
2.
3. (Socrative option available: Overview of a cryptosystem) Consider the following scenario: BestBank.com has set up an e-banking system. Customers connect to the banking server. The customer authenticates using either a password or a certificate and private key generated by an application from BestBank.com. Once the customer is authenticated against the banking server, he can view and update his account information. The client's access rights are checked on every transaction. BestBank.com hired a cryptographer to design a new encryption scheme that can be used to ensure the confidentiality and integrity of data communicated between the client and the server. It is keeping the information about the new algorithm secret to limit the information available to the attacker.
1. Which cryptographic design principles does this scenario violate?
2.
3. How does this violation affect the security of the system?
4.
5. How would you change the design to make it follow the design principle?
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