text:
Roadway air dispersion modeling is the study of air pollutant transport from a roadway or other linear emitter. Computer models are required to conduct this analysis, because of the complex variables involved, including vehicle emissions, vehicle speed, meteorology, and terrain geometry. Line source dispersion has been studied since at least the 1960s, when the regulatory framework in the United States began requiring quantitative analysis of the air pollution consequences of major roadway and airport projects. By the early 1970s this subset of atmospheric dispersion models were being applied to real world cases of highway planning, even including some controversial court cases.

How the model works

The basic concept of the roadway air dispersion model is to calculate air pollutant levels in the vicinity of a highway or arterial roadway by considering them as line sources. The model takes into account source characteristics such as traffic volume, vehicle speeds, truck mix, and fleet emission controls; in addition, the roadway geometry, surrounding terrain and local meteorology are addressed. For example, many air quality standards require that certain near worst case meteorological conditions be applied.

The calculations are sufficiently complex that a computer model is essential to arrive at authoritative results, although workbook type manuals have been developed as screening techniques. In some cases where results must be refereed (such as legal cases), model validation may be needed with field test data in the local setting; this step is not usually warranted, because the best models have been extensively validated over a wide spectrum of input data variables.

The product of the calculations is usually a set of isopleths (air pollution contour maps), either in plan view or cross sectional view. Typically these might be stated as concentrations of carbon monoxide, total reactive hydrocarbons, oxides of nitrogen, particulate or benzene. The air quality scientist can run the model successively to study techniques of reducing adverse air pollutant concentrations (for example, by redesigning roadway geometry, altering speed controls or limiting certain types of trucks). The model is frequently utilized in an Environmental Impact Statement involving a major new roadway or land use change which will induce new vehicular traffic.

History

The logical building block for this theory was the use of the Gaussian air pollutant dispersion equation for point sourcescite book|author=Turner, D.B.|title=Workbook of atmospheric dispersion estimates: an introduction to dispersion modeling|edition=2nd Edition|publisher=CRC Press|year=1994|id=ISBN 1-56670-023-X [http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=L1023&parent_id=&pc= www.crcpress.com] ] [cite book|author=Beychok, M.R.|title=Fundamentals Of Stack Gas Dispersion|edition=4th Edition|publisher=author-published|year=2005|id=ISBN 0-9644588-0-2 [http://www.air-dispersion.com www.air-dispersion.com] ] . One of the early point source air pollutant plume dispersion equations was derived by Bosanquet and Pearson [C.H. Bosanquet and J.L. Pearson, "The spread of smoke and gases from chimneys", Trans. Faraday Soc., 32:1249, 1936] in 1936. Their equation did not include the effect of ground reflection of the pollutant plume. Sir Graham Sutton derived a point source air pollutant plume dispersion equation in 1947 [O.G. Sutton, "The theoretical distribution of airborne pollution from factory chimneys", QJRMS, 73:426, 1947] which included the assumption of Gaussian distribution for the vertical and crosswind dispersion of the plume and also addressed the effect of ground reflection of the plume. Further advances were made by G. A. Briggs [G.A. Briggs, "A plume rise model compared with observations", JAPCA, 15:433-438, 1965] in model refinement and validation and by D.B. Turner for his user-friendly workbook that included screening calculations which do not require a computer.

In seeing the need to develop a line source model to approach the study of roadway air pollution,Michael Hogan and Richard Venti developed a closed form solution to integrating the point source equation in a series of publications [Michael Hogan, "Theoretical basis for atmospheric diffusion from a linear source", ESL Inc., Environmental Systems Laboratory, Publication IR-29, Sunnyvale, Ca., May 4, 1968] [Richard J. Venti, "Atmospheric diffusion models for roadway sources", ESL Inc., Environmental Systems Laboratory, Publication ET-22, Sunnyvale, Ca., October 5, 1970.] .

While the ESL mathematical model was completed for a line source by 1970, model refinement resulted in a “strip source”, emulating the horizontal extent of the roadway surface. This theory would be the precursor of area source dispersion models. But their focus was roadway simulation, so they proceeded with the development of a computer model by adding to the team Leda Patmore, a computer programmer in the field of atmospheric physics and satellite trajectory calculations. A working computer model was produced by late 1970; then the model was calibrated with carbon monoxide field measurements targeting from traffic on U.S. Route 101 in Sunnyvale, California.

The ESL model received endorsement from the U.S. Environmental Protection Agency (EPA) in the form of a major grant to validate the model using actual roadway tests of tracer gas sulfur hexafluoride dispersion. That gas was chosen since it does not occur naturally or in vehicular emissions and provides a unique tracer for such dispersion studies. Part of the Environmental Protection Agency’s motives may have been to bring the model into public domain. After a successful validation through the EPA research, the model was soon put to use in a variety of settings to forecast air pollution levels in the vicinity of roadways. The ESL group applied the model to the U.S. Route 101 bypass project in Cloverdale, California, the extension of Interstate 66 through Arlington, Virginia, the widening of the New Jersey Turnpike through Raritan and East Brunswick, New Jersey, and several transportation projects in Boston for the Boston Transportation Planning Review.

By the early 1970s at least two other research groups were known to be actively developing some type to roadway air dispersion model: the Environmental Research and Technology group of Lexington, Massachusetts and Caltrans headquarters in Sacramento, California. The Caline model of Caltrans borrowed some of the technology from the ESL Inc. group, since Caltrans funded some of the early model application work in Cloverdale and other locations and was given rights to use parts of their model.

The theory

The resulting solution for an infinite line source is:

$chi = int_\left\{0\right\}^\left\{infty\right\}frac\left\{q\right\}\left\{pileft\left(u c d x^2 ight\right)left\left(cos alpha ight\right)\right\}left\left(expfrac\left\{y^2\right\}\left\{2c^2x^2\right\} ight\right) dx$ where "x" is the distance from the observer to the roadway, "y" is the height of the observer, "u" is the mean wind speed, "α" is the angle of tilt of the line source relative to the reference frame, and "c" and "d" are the standard deviation of horizontal and vertical wind directions (measured in radians) respectively. This equation was integrated into a closed form solution using the error function (erf), and variations in geometry can be performed to include the full infinite line, line segment, elevated line, or arc made from segments. In any case one can calculate three dimensional contours of resulting air pollutant concentrations and use the mathematical model to study alternative roadway designs, various assumptions of worst case meteorology or varying traffic conditions (for example, variations in truck mix, fleet emission controls, or vehicle speed.

The ESL research group also extended their model by introducing the area source concept of a vertical strip to simulate the mixing zone on the highway produced by vehicle turbulence. This model too was validated in 1971 and showed good correlation with field test data.

Example applications of the model

There were several early applications of the model in somewhat dramatic cases. In 1971 the Arlington Coalition on Transportation (ACT) was the plaintiff in an action against the Virginia Highway Commission over the extension of Interstate 66 through Arlington, Virginia , having filed a suit in the U.S. District Court. The ESL model was used to produce calculations of air quality in the vicinity of the proposed highway. ACT won this case after a decision by the U.S. Fourth Circuit Court of Appeals. The court paid special attention to the plaintiff's expert calculations and testimony projecting that air quality levels would violate Federal ambient air quality standards as set forth in the Clean Air Act.

A second contentious case took place in East Brunswick, New Jersey where the New Jersey Turnpike Authority planned a major widening of the Turnpike. Again the roadway air dispersion model was employed to predict levels of air pollution for residences, schools and parks near the Turnpike. After an initial hearing in Superior Court where the ESL model results were set forth, the judge ordered the Turnpike Authority to negotiate with the plaintiff, Concerned Citizens of East Brunswick and develop air quality mitigation for the adverse effects. The Turnpike Authority hired ERT as its expert, and the two research teams negotiated a settlement to this case using the newly created roadway air dispersion models.

More recent model refinements

The CALINE3 model is a steady-state Gaussian dispersion model designed to determine air pollution concentrations at receptor locations downwind of highways located in relatively uncomplicated terrain. CALINE3 is incorporated into the more elaborate CAL3QHC and CAL3QHCR models. CALINE3 is in widespread use due to its user friendly nature and promotion in governmental circles, but it falls short of analyzing the complexity of cases addressed by the original Hogan-Venti model. CAL3QHC and CAL3QHCR models are available in fortran programming language and are limited to carbon monoxide: they do, however, include an embedded traffic model to better assess dynamic features such as stalling traffic [http://www.epa.gov/scram001/dispersion_prefrec.htm] .

In addition, several more recent models have been developed that employ non-steady state Lagrangian puff algorithms. The [http://www.epa.gov/scram001/dispersion_alt.htm#hyroad HYROAD] dispersion model has been developed through the National Cooperative Highway Research Program's [http://www4.trb.org/trb/crp.nsf/All+Projects/NCHRP+25-06 Project 25-06] , incorporating ROADWAY-2 model puff and steady-state plume algorithms [http://www.springerlink.com/(0dvrvwmrr4530lrppvosjvns)/app/home/contribution.asp?referrer=parent&backto=issue,19,54;journal,13,20;linkingpublicationresults,1:106613,1 (Rao et al., 2002)] .

The [http://www.volpe.dot.gov/air/docs/6757fbea9c9c.pdf TRAQSIM] model, developed as part of a Ph.D dissertation with support by the U.S. Department of Transportation's [http://www.volpe.dot.gov/ Volpe National Transportation Systems Center] 's [http://www.volpe.dot.gov/air/index.html Air Quality Facility] is currently under the care of [http://www.wylelabs.com Wyle Laboratories] . The model incorporates dynamic vehicle behavior with a non-steady state Gaussian puff algorithm. Unlike HYROAD, TRAQSIM combines traffic simulation, second-by-second modal emissions, and Gaussian puff dispersion into a fully integrated system (a true simulation) that models individual vehicles as discrete moving sources. TRAQSIM was developed as a next generation model to be the successor to the current CALINE3 and CAL3QHC regulatory models. The next step in the development of TRAQSIM is to incorporate methods to model the dispersion of particulate matter (PM) and hazardous air pollutants (HAPs).

Several models have been developed that handle complex urban meteorology resulting from urban canyons. Examples include the [http://www.tfhrc.gov/ Turner-Fairbank Highway Research Center] 's [http://www.tfhrc.gov/structur/pubs/02036/intro.htm Canyon Plume Box model] , now in version 3 (CPB-3), the [http://www.dmu.dk/International/ National Environmental Research Institute] of Denmark's [http://www2.dmu.dk/1_viden/2_Miljoe-tilstand/3_luft/4_spredningsmodeller/5_ospm/default_en.asp Operational Street Pollution Model (OSPM)] , and the [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VH3-43HK0FJ-K&_user=14684&_coverDate=12%2F31%2F2001&_alid=392204409&_rdoc=1&_fmt=summary&_orig=search&_cdi=6055&_sort=d&_st=4&_docanchor=&_acct=C000001678&_version=1&_urlVersion=0&_userid=14684&md5=f7b2b5a19a190b51ac9930efe57e5442 MICRO-CALGRID] model, which includes photochemistry, allowing for both primary and secondary species to be modeled.

Recent applications in legal cases

Recent health literature indicating that residents near major roads face elevated rates of several adverse health outcomes has prompted legal dispute over the responsibility of transportation agencies to use roadway air dispersion models to characterize the impacts of new and expanded roadways, bus terminals, truck stops, and other sources.

Recently, the Sierra Club of Nevada sued the Nevada Department of Transportation and the Federal Highway Administration over its their failure to assess the impact of the expansion of US-95 in Las Vegas on neighborhood air quality. [http://www.usatoday.com/news/nation/2003-03-06-vegas-highway-usat_x.htm] The Sierra Club asserted that a supplemental Environmental Impact Statement should be issued to address emissions of hazardous air pollutants and particulate matter from new motor vehicle traffic. The plaintiffs asserted that modeling tools were available, including the Environmental Protection Agency's MOBILE6.2 model, the CALINE3 dispersion model, and other relevant models. The defendants won in the U.S. District Court under Judge Philip Pro, who ruled that the transportation agencies had acted in a manner that was not "arbitrary and capricious," despite the agencies' technical arguments regarding the lack of available modeling tools being contradicted by a number of peer-reviewed studies published in scientific journals (e.g. Korenstein and Piazza, Journal of Environmental Health, 2002). On appeal to the 9th Federal Circuit Court, the court stayed new construction on the highway pending the court's final decision. The Sierra Club and the defendants settled out of court, setting up a research program on the air quality impacts of US-95 on nearby schools.

A number of other high-profile cases have prompted environmental groups to call for dispersion modeling to be used to assess the air quality impacts of new transportation projects on nearby communities, but to date state transportation agencies and the Federal Highways Administration has claimed that no tools are available, despite models and guidance available through EPA's Support Center for Regulatory Air Models (SCRAM). [http://www.epa.gov/scram001]

Among the more contentious of cases the Detroit Intermodal Freight Terminal and Detroit River International Crossing (Michigan, USA), and the expansion of Interstate 70 East in Denver (Colorado , USA).

In all of these cases, community-based organizations have asserted that modeling tools are available, but transportation planning agencies have asserted that too much uncertainty exits in all of the steps. A major concern for community-based organizations has been transportation agencies' unwillingness to define the level of uncertainty that they are willing to tolerate in air quality analyses, how that compares to the Environmental Protection Agency's guideline on air quality models, which addresses uncertainty and accuracy in model use. [http://www.epa.gov/scram001/guidance/guide/appw_05.pdf]

References

ee also

*Atmospheric dispersion modeling
*Bibliography of atmospheric dispersion modeling
*Compilation of atmospheric dispersion models
*Global warming
*Peak oil

* [http://www.epa.gov/scram001/ EPA Support Center for Regulatory Atmospheric Modeling]
* [http://www.epa.gov/scram001/dispersion_prefrec.htm EPA Preferred/Recommended Models]
* [http://www.epa.gov/oar/oaqps/organization/emad/aqmg.html EPA's Air Quality Modeling Group (AQMG)]
*
* [http://www.epa.gov/ttn/fera/risk_atra_main.html EPA's Air Toxics Risk Assessment (ATRA) Reference Library]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

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• Air Quality Modeling Group — The Air Quality Modeling Group (AQMG) is in the U.S. EPA s Office of Air and Radiation (OAR) and provides leadership and direction on the full range of air quality models, air pollution dispersion models and other mathematical simulation… …   Wikipedia

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# What is the product of all the roots of the following equation?

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What is the product of all the roots of the following equation?  [#permalink]

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What is the product of all the roots of the following equation?

$$x^5 - 3x^4 = 3x^2 - 3x^3 + 2x$$

A. 28
B. 19
C. 14
D. 1
E. 0

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Originally posted by ChiranjeevSingh on 22 Aug 2017, 19:49.
Last edited by ChiranjeevSingh on 22 Aug 2017, 21:00, edited 5 times in total.
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What is the product of all the roots of the following equation?  [#permalink]

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22 Aug 2017, 20:14
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ChiranjeevSingh wrote:
What is the product of all the roots of the following equation?

$$x^5 - 3x^4 = 3x^2 - 3x^3 + 2x$$

A. 28
B. 14
C. 19
D. 0
E. 1

I'll post the OA in a while!

Hi..

X is common to all terms so one value of x is 0..
No need to go any further..
Product of all terms with one of them as 0 will be 0
E
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Re: What is the product of all the roots of the following equation?  [#permalink]

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22 Aug 2017, 20:55
1
ChiranjeevSingh wrote:
What is the product of all the roots of the following equation?

$$x^5 - 3x^4 = 3x^2 - 3x^3 + 2x$$

A. 28
B. 14
C. 19
D. 0
E. 1

I'll post the OA in a while!

You have to post the OA from the beginning. Also, on the GMAT, answer choices are in ascending or descending order.
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Re: What is the product of all the roots of the following equation?  [#permalink]

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22 Aug 2017, 23:31
ChiranjeevSingh wrote:
What is the product of all the roots of the following equation?

$$x^5 - 3x^4 = 3x^2 - 3x^3 + 2x$$

A. 28
B. 19
C. 14
D. 1
E. 0

since equation becomes
$$x(x^4-3x^3+3x^2-3x-2)=0$$
one of the roots will be x=0

Hence product of all roots will be 0
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Re: What is the product of all the roots of the following equation?  [#permalink]

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23 Aug 2017, 03:07
Taking x common and all the terms on left side ,
x(x4−3x3+3x2−3x−2)=0
Which equuals ,

x=0 or x4-3x3=3x2-2=0

One of the root is 0 . That means product of roots = 0
Re: What is the product of all the roots of the following equation?   [#permalink] 23 Aug 2017, 03:07
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# 6.1: The Leibniz rule

Leibniz’s rule1 allows us to take the time derivative of an integral over a domain that is itself changing in time. Suppose that $$f\left( \vec{x},t \right)$$ is the volumetric concentration of some unspecified property we will call “stuff”. The Leibniz rule is mathematically valid for any function $$f\left(\vec{x},t\right)$$, but it is easiest to interpret physically if we imagine that f is something per unit volume. For a concrete example, imagine that the “stuff” is air, and $$f$$ is then the mass of air molecules per unit volume, i.e., the density. Now consider a closed surface that can change arbitrarily in time (not a material volume, in general). Its area is $$A(t)$$ and the volume it encloses is $$V(t)$$ (Figure $$\PageIndex{1}$$a). The quantity of “stuff” contained in the volume at any given time is $$\int_V f\left( \vec{x},t \right) dV=\int_V \frac{\partial f}{\partial t}dV$$. That quantity can change in time in two ways.

First, the concentration $$f$$ may change in time, e.g., the density of the air may change due to heating or cooling. If this were the only source of change, we could write:

$\frac{d}{d t} \int_{V} f(\vec{x}, t) d V=\int_{V} \frac{\partial f}{\partial t} d V.\label{eqn:1}$

Second, the volume itself can change, for example, the volume could grow, thereby engulfing more “stuff”. Quantifying this second contribution requires a bit more thought. At any point on the boundary we define $$\hat{n}$$ to be the outward-pointing normal vector (Figure $$\PageIndex{1}$$a). The points that make up the boundary have velocity $$\vec{u}_A$$, which varies over the boundary and also in time as the surface evolves. The expansion velocity, $$\vec{u}_A\cdot\hat{n}$$, is the component of $$\vec{u}_A$$ that is perpendicular to the boundary and directed outward. Now consider the motion of a small surface element over a brief time $$dt$$ (Figure $$\PageIndex{1}$$b). The surface element moves by a distance $$\vec{u}_A\cdot\hat{n}dt$$, thereby enclosing a small volume $$dV=\vec{u}_A\cdot\hat{n}dtdA$$. The amount of “stuff” contained in this small volume is $$f dV$$, or $$f\vec{u}_A\cdot\hat{n}dtdA$$. If we now integrate this quantity over the whole surface, we get the amount of “stuff” engulfed (or ejected, if $$\vec{u}_A\cdot\hat{n}<0$$) in time $$dt$$: $$\int_A f\vec{u}_A\cdot\hat{n}dt dA$$. Dividing by $$dt$$ and taking the limit $$dt \rightarrow 0$$, we have the second term that controls the change in the amount of “stuff” enclosed by our surface $$A$$:

$\frac{d}{d t} \int_{V(t)} f(\vec{x}, t) d V=\int_{V(t)} \frac{\partial f}{\partial t} d V+\int_{A(t)} f \vec{u}_{A} \cdot \hat{n} d A.\label{eqn:2}$

This is the most general form of Leibniz’s rule.

## Three special cases

1. If the surface is unchanging in time, then $$\vec{u}_A=0$$ and Equation $$\ref{eqn:2}$$ is the same as $$\ref{eqn:1}$$.

2. Suppose that $$f$$ is a function of only one spatial coordinate and time: $$f = f(x,t)$$. The integral is then an ordinary integral from, say, $$x = a$$ to $$x = b$$, but the boundaries $$a$$ and $$b$$ can vary in time (Figure $$\PageIndex{2}$$). In that case Leibniz’ rule becomes

$\frac{d}{d t} \int_{a(t)}^{b(t)} f(x, t) d x=\int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} d x+f(b, t) \frac{d b}{d t}-f(a, t) \frac{d a}{d t}.\label{eqn:3}$

The second and third terms on the right-hand side are the contributions due to the motion of the boundaries. This is the version of Leibniz’ rule commonly found in calculus textbooks.

3. The most important case of Equation $$\PageIndex{2}$$ for fluid mechanics is that in which $$A(t)$$ is a material surface $$A_m(t)$$, always composed of the same fluid particles, and $$V = V_m(t)$$ is therefore a material volume (or fluid parcel). In this case $$\vec{u}_A$$ is just $$\vec{u}\left(\vec{x},t\right)$$, the velocity of the motion, and the time derivative is $$D/Dt$$:

$\frac{D}{D t} \int_{V_{m}(t)} f(\vec{x}, t) d V=\int_{V_{m}(t)} \frac{\partial f}{\partial t} d V+\int_{A_{m}(t)} f \vec{u} \cdot \hat{n} d A.$

Note that the time derivative is defined as $$D/Dt$$ because it is evaluated in a reference frame following the motion. It does not, however, have the form (5.1.3), as it does when applied to a continuous field. Now notice that, in the final term, the integrand is the dot product of the vector $$f\vec{u}$$ and the outward unit normal $$\hat{n}$$. According to the divergence theorem2, we can convert this term to a volume integral:

$\int_{A}(f \vec{u}) \cdot \hat{n} d A=\int_{V} \vec{\nabla} \cdot(f \vec{u}) d V.$

We now have Leibniz’ rule for a material volume:3

$\frac{D}{D t} \int_{V_{m}(t)} f(\vec{x}, t) d V=\int_{V_{m}(t)}\left(\frac{\partial f}{\partial t}+\vec{\nabla} \cdot f \vec{u}\right) d V.\label{eqn:4}$

1Gottfried Wilhelm Leibniz (1646-1716) was a German philosopher and mathematician who invented calculus independently of Isaac Newton. The notation “$$d/dx$$” that we use today comes from Leibniz’s version. As a philosopher, Leibniz espoused the theory of “optimism”, which holds that the universe we inhabit is the best one that God could have created. This cheery attitude is especially admirable given that Newton got all the credit for inventing calculus.

2The divergence theorem reads $$\oint_A\vec{v}\cdot\hat{n}dA=\int_A\vec{\nabla}\cdot\vec{v}dV$$, where $$\vec{v}$$ is a general vector and $$\hat{n}$$ is the unit normal to the closed surface $$A$$. See section 4.2.3 for details.

3Note that the integral on the left-hand side of Equation $$\ref{eqn:4}$$ depends only on time. We write its time derivative as $$D/Dt = \partial/\partial t +\vec{u} \cdot \vec{\nabla}$$, even though only the term $$\partial/\partial t$$ is nonzero, and the derivative could be written just as accurately as $$d/dt$$. We choose the symbol $$D/Dt$$ to remind ourselves that this time derivative is measured by an observer moving with the flow.


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A monomial is an algebraic expression with only one term. Monomial expressions involve the multiplication or division of numbers and variables in some valid combination. $\begin{array}{c|c} z^5 & {\color{#009600}{\Large{\checkmark}}} \\ \hline\\[-1em] 2x & {\color{#009600}{\Large{\checkmark}}} \\ \hline\\[-1em] \dfrac{\text{-} 7yz^4}{2} & {\color{#009600}{\Large{\checkmark}}} \end{array}$ Note, the exponents of the variables in a monomial are whole numbers and a monomial cannot have variables in its denominator. $\begin{array}{c|c} m^{\frac{4}{7}} & {\color{#FF0000}{\Large{\mathbf{\times}}}} \\ \hline\\[-1em] g^{\text{-} 3} & {\color{#FF0000}{\Large{\mathbf{\times}}}} \\ \hline\\[-1em] \dfrac{1}{x} & {\color{#FF0000}{\Large{\mathbf{\times}}}} \end{array}$ When multiple monomials are put together, separated by $+$ or $-$ signs, they are the terms that form polynomials.


text:
TITLE

Spectroscopic measurements of temperature and plasma impurity concentration during magnetic reconnection at the Swarthmore Spheromak Experiment

AUTHOR(S)
Chaplin, V. H.; Brown, M. R.; Cohen, D. H.; Gray, T.; Cothran, C. D.
PUB. DATE
April 2009
SOURCE
Physics of Plasmas;Apr2009, Vol. 16 Issue 4, p042505
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
Electron temperature measurements during counterhelicity spheromak merging studies at the Swarthmore Spheromak Experiment (SSX) [M. R. Brown, Phys. Plasmas 6, 1717 (1999)] are presented. VUV monochromator measurements of impurity emission lines are compared with model spectra produced by the non-LTE excitation kinematics code PRISMSPECT [J. J. MacFarlane et al., in Proceedings of the Third Conference on Inertial Fusion Science and Applications (2004)] to yield the electron temperature in the plasma with 1 Î¼s time resolution. Average Te is seen to increase from 12 to 19 eV during spheromak merging. Average C III ion temperature, measured with a new ion Doppler spectrometer (IDS) [C. D. Cothran et al., Rev. Sci. Instrum. 77, 063504 (2006)], likewise rises during spheromak merging, peaking at âˆ¼22 eV, but a similar increase in Ti is seen during single spheromak discharges with no merging. The VUV emission line measurements are also used to constrain the concentrations of various impurities in the SSX plasma, which are dominated by carbon, but include some oxygen and nitrogen. A burst of soft x-ray emission is seen during reconnection with a new four-channel detector (SXR). There is evidence for spectral changes in the soft x-ray emission as reconnection progresses, although our single-temperature equilibrium spectral models are not able to provide adequate fits to all the SXR data.
ACCESSION #
38611560

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Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Melt-blown PP membrane from face mask Select A Forum Fundamentals   » Chemistry in General   » Organic Chemistry   » Reagents and Apparatus Acquisition   » Beginnings   » Responsible Practices   » Miscellaneous   » The Wiki Special topics   » Technochemistry   » Energetic Materials   » Biochemistry   » Radiochemistry   » Computational Models and Techniques   » Prepublication Non-chemistry   » Forum Matters   » Legal and Societal Issues   » Test Forum

Author: Subject: Melt-blown PP membrane from face mask
Heptylene
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Melt-blown PP membrane from face mask

Face masks, which are suddenly available due to (thanks to?) the coronavirus pandemic contain a polypropylene (PP) filter which is very porous and has a high surface area. Does anyone have an idea what these could be used for in the lab? (apart from wearing them)

PP is very chemically resistant, so I'm thinking they could work well to separate electrode compartments in an electrolytic cell, or as "anode bags" in electroplating.

Perhaps as filtration medium too?

Looks like a nice material and I'm sure there is a way to use them for something.
macckone
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Heptylene,
I have made reference to using vacuum cleaner bags as filters and electrolytic diaphragms.
HEPA vacuum bags have a 5um pore size which is perfect for filtering.
The smaller pore size on n95 masks (good luck actually getting them in the US) would work better for a diaphragm.
The masks that are commonly available are not all PP, some are other materials (including some kind of cellulose or viscose/rayon).
And the pore size is not very consistent on the cheap face masks.

[Edited on 30-9-2020 by macckone]
Herr Haber
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Sounds worth investigating

Sreetips uses vacuum filters for his anode bags. It seems to work perfectly fine for him.
I have purchased but never actually used very dense cotton fabric to use as anode bag. The idea was to eventually add them to the cellulose filters from other steps to incinerate them knowing the only entrant would be carbon.
(Though I guess hitting PP in a crucible with a blowtorch gets the same result)

The spirit of adventure was upon me. Having nitric acid and copper, I had only to learn what the words 'act upon' meant. - Ira Remsen
draculic acid69
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Posts: 1190
Registered: 2-8-2018
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 Quote: Originally posted by Heptylene Face masks, which are suddenly available due to (thanks to?) the coronavirus pandemic contain a polypropylene (PP) filter which is very porous and has a high surface area. Does anyone have an idea what these could be used for in the lab? (apart from wearing them) PP is very chemically resistant, so I'm thinking they could work well to separate electrode compartments in an electrolytic cell, or as "anode bags" in electroplating. Perhaps as filtration medium too? Looks like a nice material and I'm sure there is a way to use them for something.

I was wondering the same thing but if 5um is the pore size I'm not sure if it's helpful for what I have to filter.celite is the only thing that cleans the superfine sediment out of my solution but it takes around 6hrs a litre to filter. Its too long.maybe I'll run it through 3 vac filter papers before I do the celite not sure if the mask pp filters are better than the filter paper

[Edited on 8-10-2020 by draculic acid69]
macckone
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draculic acid69,
The cheap face masks are have pore sizes up to 500um according to this study:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6599448/
But this study says 0.5 to 10um for quality surgical masks:
https://www.ohsu.edu/sites/default/files/2020-03/Proposed%20...

The good n95 masks are unobtainium at the moment.
They should have a 0.5 um pore size to block 95% of 0.3um particles.
zed
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Umm. We were discussing this dillemma about 6 months ago. Some folks had concluded that Harbor Freight Blue PPE shop towels were pretty good mask material. Naturally, Harbor Freight was permanently "Out" of stock. Last I checked, they did have masks, but not necessarily masks I would trust.

I was able to obtain those fancy blue PPE towels from another source. Blain's in the Mid-West beat Harbor Freight's retail price, and shipped them too. Now, I don't know, if the manufacturing method is the same as it was. But, if they prove less than great when tested, they do make good durable shop-towels. I bought two containers, had 'em shipped. Totalled about 35 bucks. Only slightly more than Harbor Freight's retail in store price.

https://www.farmandfleet.com/products/756345-sellars-200-cou...

Now, for my personal breathing, I prefer real protection.
An old school Willson respirator is the minimum. Gas/Vapor cartridge, with Heavy Fiberglass/Fiber prefilter.

Better is a full-face MSA respirator, with a 40mm NATO cartridge. The Optimair Ultravue is OK. The MSA Ultra-Elite, is probably better and certainly more comfortable.

These are great masks, and not terribly expensive, on Ebay. If you attach filtered positive pressure air, you become pretty much bullet-proof as far as Covid is concerned.

Ultra-Elite SCBA Masks, available as fire department surplus for~$50, can be adapted to utilize standard Gas Mask cartridges. Good stuff. https://us.msasafety.com/Air-Purifying-Respirators-%28APR%29... https://www.ebay.com/itm/MSA-Ultra-Elite-Full-Face-Mask-Resp... The Ultra Elite. This particular model, is set up for SCBA, but if you unscrew a few screws and remove the bottom plate, you can unscrew and remove the SCBA Adapter, exposing the native 40mm NATO connection within. Replace the bottom plate, and you have a standard NATO gas mask. It will fit a PAPR unit As-is. If you wish to attach a 40mm NATO cartridge directly, just trim off the obstructing plastic appendages, on the mask. Just an example. Sometimes, they appear to be new. SCBA itself is great, but not for the PPE most of us need. SCBA utilizes a cylinder of compressed air, for about a 30 minute air supply. It's utilized by firefighters, and the mask and tank are accompanied by thousand of dollars worth of sophisticated high tech equipment. Requires a lot of training, just to use the gear. Oddly enough, Gas Mask prices are rather high right now, and stock is low. A year ago, new surplus, was cheaper than what a beater costs now. Supply and demand! [Edited on 17-10-2020 by zed] Mateo_swe Hazard to Others Posts: 228 Registered: 24-8-2019 Location: Within EU Member Is Offline If looking for membranes for electrolytic cells one can buy anion or cation membranes from Fumasep or Fulecellstore online. Not all membranes are very expensive, Fumasep has some membranes that is about 25 Euros for a 30x20cm sheet. https://www.fumatech.com/EN/Onlineshop/index.html Its not very cheap but it will not ruin you. Finding alternative cheap membrane materials are quite interesting though. DavidJR International Hazard Posts: 908 Registered: 1-1-2018 Location: Scotland Member Is Offline Mood: Tired  Quote: Originally posted by zed Oddly enough, Gas Mask prices are rather high right now, and stock is low. A year ago, new surplus, was cheaper than what a beater costs now. Supply and demand! Yeah, I had to wait ~5 months for some backordered 3M filter cartridges. I couldn't get them at my usual supplier and everywhere else was out of stock, which never happened before covid-19. I wonder if it is caused directly by demand for the cartridges, or possibly by 3M's production capacity being redirected towards covid-19 directed products. zed International Hazard Posts: 2202 Registered: 6-9-2008 Location: Great State of Jefferson, City of Portland Member Is Offline Mood: Semi-repentant Sith Lord David, the shortage of 3M stuff, is pretty much world wide. It's decent stuff, but a consumer brand. Well. it has all been consumed. Full face respirators, were for a long while in very short supply, and very expensive. Full list price, plus! And, 3M's full list price, is pretty exhorbitant. No cartridges. None! American retailers were completely out, with no prospects of being restocked. Retailers based in China were offering 3M, but I suspected it was counterfeit. I surely wouldn't trust it. Thus, the reliance upon Nato type gas mask stuff. To start out, it is better stuff. Military spec.. And, the cartridges are universal 40 mm. The cartridges from different manufacturers are interchangable. But, don't believe folks who say their Russians stuff is compatible, it isn't. As for effectiveness, Nato stuff is better. For Covid protection, even long expired cartridges are fine. The Nato Cartridge, generally includes a Hepa filter, plus a back-up of Carbon particles. The Carbon elements may become less effective after decades long storage. Hepa filters do not deteriorate or expire. Where you are, there is an abundance of NATO type stuff available. Moreover, you are close to Israel where every man woman and child, needs to be prepared for gas attacks, at all times. They have lots of inexpensive surplus available, and their Gas Mask Cartridges, are considered some of the finest in the world. Me? I'm on the other side of the planet. But, for you, Israel is just a hop, skip, and a jump, away. The Israelis can supply possibly the least expensive PAPR units on the planet. New, surplus units, have been selling for about$50 Dollars in Israel. A much higher degree of Covid protection than any ordinary respirator. Positive pressure!

https://www.ebay.com/itm/ISRAEL-NEW-PROTECTIVE-HOOD-KIT-WITH...

Like all good things, prices are going up. A less expensive supplier. He's new. Doesn't have an established track record yet. Less, of a problem for you. Like I said, I'm on the far side of the planet. Months away, by mail.


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# RD Sharma Solutions Class 7 Properties Of Triangles Exercise 15.1

## RD Sharma Solutions Class 7 Chapter 15 Exercise 15.1

### RD Sharma Class 7 Solutions Chapter 15 Ex 15.1 PDF Free Download

#### Exercise 15.1

1. Take three non-collinear points A. B and C on a page of your notebook. Join AB, BC and CA. What figure do you get? Name the triangle. Also, name

(i) The side opposite to $\angle B$

(ii) The angle opposite to side AB

(iii) The vertex opposite to side BC

(iv) The side opposite to vertex B.

(i) AC

(ii) $\angle B$

(iii) A

(iv) AC

2. Take three collinear points A, B and Con a page of your note book. Join AB. BC and CA. Is the figure a triangle? If not, why?

No, the figure is not a triangle. By definition a triangle is a plane figure formed by three non-parallel line segments

3. Distinguish between a triangle and its triangular region.

A triangle is a plane figure formed by three non-parallel line segments, whereas, its triangular region includes the interior of the triangle along with the triangle itself.

4. D is a point on side BC of a $\triangle$. AD is joined. Name all the triangles that you can observe in the figure. How many are they?

We can observe the following three triangles in the given figure

(i) $\triangle ABC$

(ii) $\triangle ACD$

(iii) $\triangle ADB$

5. A, B. C and Dare four points, and no three points are collinear. AC and BD intersed at 0. There are eight triangles that you can observe. Name all the triangles

(i) $\triangle ABC$

(ii) $\triangle ABD$

(iii) $\triangle ABO$

(iv) $\triangle BCD$

(v) $\triangle DCO$

(vi) $\triangle AOD$

(vii) $\triangle ACD$

(viii) $\triangle BCD$

6. What is the difference between a triangle and triangular region?

A plane of figure formed by three non-parallel line segments is called a triangle whereastriangular region is the interior of $/triangle ABC$ together with the $/triangle ABC$ itself is called the triangular region ABC

7. Explain the following terms:

(i) Triangle

(a) Parts or elements of a triangle

(iii) Scalene triangle

(iv) Isosceles triangle

(v) Equilateral triangle

(vi) Acute triangle

(vii) Right triangle

(viii) Obtuse triangle

(ix) Interior of a triangle

(x) Exterior of a triangle.

(i) A triangle is a plane figure formed by three non-parallel line segments.

(ii) The three sides and the three angles of a triangle are together known as the parts or elements of that triangle.

(iii) A scalene triangle is a triangle in which no two sides are equal.

(iv) An isosceles triangle is a triangle in which two sides are equal. Isosceles triangle

(v) An equilateral triangle is a triangle in which all three sides are equal. Equilateral triangle

(vi) An acute triangle is a triangle in which all the angles are acute (less than 90°).

(vii) A right angled triangle is a triangle in which one angle is right angled, i.e. 90°.

(viii) An obtuse triangle is a triangle in which one angle is obtuse (more than 90°).

(ix) The interior of a triangle is made up of all such points that are enclosed within the triangle.

(x) The exterior of a triangle is made up of all such points that are not enclosed within the triangle.

8. In Figure, the length (in cm) of each side has been indicated along the side. State for each triangle angle whether it is scalene, isosceles or equilateral:

(i) This triangle is a scalene triangle because no two sides are equal.

(ii) This triangle is an isosceles triangle because two of its sides, viz. PQ and PR, are equal.

(iii) This triangle is an equilateral triangle because all its three sides are equal.

(iv) This triangle is a scalene triangle because no two sides are equal.

(v) This triangle is an isosceles triangle because two of its sides are equal.

9. There are five triangles. The measures of some of their angles have been indicated. State for each triangle whether it is acute, right or obtuse.

(i) This is a right triangle because one of its angles is 90°.

(ii) This is an obtuse triangle because one of its angles is 120°, which is greater than 90°.

(iii) This is an acute triangle because all its angles are acute angles (less than 90° ).

(iv) This is a right triangle because one of its angles is 90°.

(v) This is an obtuse triangle because one of its angles is 110°, which is greater than 90°.

10. Fill in the blanks with the correct word/symbol to make it a true statement:

(i) A triangle has _____ sides.

(ii) A triangle has ______ vertices.

(iii) A triangle has _____ angles.

(iv) A triangle has ________ parts.

(v) A triangle whose no two sides are equal is known as ________

(v0 A triangle whose two sides are equal is known as ________

(vii) A triangle whose all the sides are equal is known as ________

(viii) A triangle whose one angle is a right angle is known as _________

(ix) A triangle whose all the angles are of measure less than 90′ is known as _________

(x) A triangle whose one angle is more than 90′ is known as _________

(i) three

(ii) three

(iii) three

(iv) six (three sides + three angles)

(v) a scalene triangle

(vi) an isosceles triangle

(vii) an equilateral triangle

(viii) a right triangle

(ix) an acute triangle

(x) an obtuse triangle

11. In each of the following, state if the statement is true (T) or false (F):

(i) A triangle has three sides.

(ii) A triangle may have four vertices.

(iii) Any three line-segments make up a triangle.

(iv) The interior of a triangle includes its vertices.

(v) The triangular region includes the vertices of the corresponding triangle.

(vi) The vertices of a triangle are three collinear points.

(vii) An equilateral triangle is isosceles also.

(viii) Every right triangle is scalene.

(ix) Each acute triangle is equilateral.

(x) No isosceles triangle is obtuse.

(i) True.

(ii) False. A triangle has three vertices.

(iii) False. Any three non-parallel line segments can make up a triangle.

(iv) False. The interior of a triangle is the region enclosed by the triangle and the vertices are not enclosed by the triangle.

(v) True. The triangular region includes the interior region and the triangle itself.

(vi) False. The vertices of a triangle are three non-collinear points.

(vii) True. In an equilateral triangle, any two sides are equal.

(viii) False. A right triangle can also be an isosceles triangle.

(ix) False. Each acute triangle is not an equilateral triangle, but each equilateral triangle is an acute triangle.

(x) False. An isosceles triangle can be an obtuse triangle, a right triangle or an acute triangle

#### Practise This Question

What is the sum of x3+4xy+3y2 and 3x2+5xy ?


text:
# Cheminformatics on a Mac

I gave a talk at the Cambridge Cheminformatics meeting last week, I’ve put the slides here. It was more of a demonstration than a talk but the slides give an overview and links to the various tools.


text:
# ellipse

0th

Percentile

##### Elliptical Window.

Create an elliptical window.

Keywords
spatial, datagen
##### Usage
ellipse(a, b, centre=c(0,0), phi=0, …, mask=FALSE, npoly = 128)
##### Arguments
a,b

The half-lengths of the axes of the ellipse.

centre

The centre of the ellipse.

phi

The (anti-clockwise) angle through which the ellipse should be rotated (about its centre) starting from an orientation in which the axis of half-length a is horizontal.

Logical value controlling the type of approximation to a perfect ellipse. See Details.

Arguments passed to as.mask to determine the pixel resolution, if mask is TRUE.

npoly

The number of edges in the polygonal approximation to the ellipse.

##### Details

This command creates a window object representing an ellipse with the given centre and axes.

By default, the ellipse is approximated by a polygon with npoly edges.

If mask=TRUE, then the ellipse is approximated by a binary pixel mask. The resolution of the mask is controlled by the arguments … which are passed to as.mask.

The arguments a and b must be single positive numbers. The argument centre specifies the ellipse centre: it can be either a numeric vector of length 2 giving the coordinates, or a list(x,y) giving the coordinates of exactly one point, or a point pattern (object of class "ppp") containing exactly one point.

##### Value

An object of class owin (either of type “polygonal” or of type “mask”) specifying an elliptical window.

disc, owin.object, owin, as.mask

• ellipse
##### Examples
# NOT RUN {
W <- ellipse(a=5,b=2,centre=c(5,1),phi=pi/6)
plot(W,lwd=2,border="red")


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18.3: ABL Structure and Evolution

The fair-weather ABL consists of the components sketched in Fig. 18.8. During daytime there is a statically unstable mixed layer (ML). At night, a statically stable boundary layer (SBL) forms under a statically neutral residual layer (RL). The RL contains the pollutants and moisture from the previous day’s mixed layer, but is not very turbulent.

The bottom 20 to 200 m of the ABL is called the surface layer (SL, Fig. 18.9). Here frictional drag, heat conduction, and evaporation from the surface cause substantial variations of wind speed, temperature, and humidity with height. However, turbulent fluxes are relatively uniform with height; hence, the surface layer is known as the constant flux layer.

Separating the free atmosphere (FA) from the mixed layer is a strongly stable entrainment zone (EZ) of intermittent turbulence. Mixed-layer depth zi is the distance between the ground and the middle of the EZ. At night, turbulence in the EZ ceases, leaving a non-turbulent layer called the capping inversion (CI) that is still strongly statically stable.

Typical vertical profiles of temperature, potential temperature, humidity (mixing ratio), and wind speed are sketched in Fig. 18.9. The “day” portion (Fig. 18.9a) corresponds to the 3 PM time indicated in Fig. 18.8, while “night” (Fig. 18.9b) is for 3 AM.

Next, we will look at ABL temperature, winds, and turbulence in more detail.

This page titled 18.3: ABL Structure and Evolution is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Roland Stull via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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AIMS Mathematics, 2018, 3(1): 96-130. doi: 10.3934/Math.2018.1.96

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Minimization solutions to conservation laws with non-smooth and non-strictly convex flux

University of Pittsburgh, Mathematics Department, 301 Thackeray Hall, Pittsburgh PA 15260

## Abstract    Full Text(HTML)    Figure/Table    Related pages

Conservation laws are usually studied in the context of suffcient regularity conditionsimposed on the flux function, usually C2 and uniform convexity. Some results are proven with theaid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show thatmany of these classical results can be extended to a flux function that is not necessarily smooth oruniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longerhold, by considering the greatest (or supremum, where applicable) of all possible minimizers, wecan successfully extend the results. One specific nonlinear case is that of a piecewise linear fluxfunction, for which we prove existence and uniqueness results. We also approximate it by a smoothed,superlinearized version parameterized by ε and consider the characterization of the minimizers for thesmooth version and limiting behavior as ε ↓ 0 to that of the sharp, polygonal problem. In proving akey result for the solution in terms of the value of the initial condition, we provide a stepping stone toanalyzing the system under stochastic processes, which will be explored further in a future paper.
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# References

1. D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 2 Eds, Cambridge: Cambridge University Press, 2009.

2. J. Bertoin, Levy Processes, Cambridge: Cambridge University Press, 1996.

3. Y. Brienier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317–2328.

4. M. Chabanol and J. Duchon, Markovian solutions of inviscid Burgers equation, J. Stat. Phys., 114 (2004), 525–534.

5. M. Crandall, L. Evans and P. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502.

6. C. Dafermos, Polygonal Approximations of Solutions of the Initial Value Problem for a Conservation Law, J. Math. Anal. Appl., 38 (1972), 33–41.

7. C. Dafermos, Hyberbolic Conservation Laws in Continuum Physics, 3 Eds, New York: Springer, 2010.

8. E.Weinan, G. Rykov and G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Commun. Math. Phys., 177 (1996), 349–380.

9. C. Evans, Partial Differential Equations, 2 Eds, New York: Springer, 2010.

10. L. Frachebourg and P. Martin, Exact statistical properties of the Burgers equation, J. Fluid. Mech., 417 (2000), 323–349.

11. D. G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer, 1977.

12. P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Rel., 81 (1989), 79–109.

13. H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, New York: Springer, 2015.

14. E. Hopf, The partial differential equation ut + uux = μuxx , Comm. Pure Appl. Math., 3 (1950), 201–230.

15. D. Kaspar and F. Rezakhanlou, Scalar conservation laws with monotone pure-jump Markov initial conditions, Probab. Theory Rel., 165 (2016), 867–899.

16. K. Karlsen and N. Risebro, A note on front tracking and equivalence between viscosity solutions of Hamilton-Jacobi equations and entropy solutions of scalar conservation law, J. Nonlin. Anal., 50 (2002), 455–469.

17. P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537–566.

18. P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, CBMS-NSF Regional Conference Series in Applied Mathematics, 1973.

19. G. Menon, Complete integrability of shock clustering and Burgers turbulence, Arch. Ration. Mech. An., 203 (2012), 853–882.

20. G. Menon and R. Pego, Universality classes in Burgers turbulence, Commun. Math. Phys., 273 (2007), 177–202.

21. G. Menon and R. Srinivasan, Kinetic theory and Lax equations for shock clustering and Burgers turbulence, J. Stat. Phys., 140 (2010), 1195–1223.

22. H. L. Royden and P. Fitzpatrick, Real Analysis, 4 Eds, Boston: Prentice Hall, 2010.

23. W. Rudin, Real and Complex Analysis, 3 Eds, Boston: McGraw-Hill, 1987.

24. Z. Schuss, Theory and Applications of Stochastic Processes, New York: Springer, 2010.

25. A. Vol'pert, Spaces BV and quasilinear equations, Mat. Sb. (N.S.), 73 (1967), 255–302.


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# A simple lambda interpreter

Posted on 2018 03 27 by phimuemue

I was never an expert of lambda-calculus, but I always wanted to learn (at least the basics of) it. So, I decided to write a very simplistic interpreter for the untyped lambda calculus.

## Goals for the project

The goals for this small tool are relatively simply to describe:

• Simple user interface: command-line based, repl-loop-like. In particular, I wanted to help the user with manual lambda calculations. For this reason, I wanted support for some things:
• Abbreviations/macros, so that the user does not need to input too many things at once.
• Step-by-step evaluation of lambda expressions.
• Simple codebase.
• Language of choice: Haskell (using Parsec to parse user input).

Here's a screenshot of an early version:

As you can see, the program supports colors for specific parts of a lambda expression. A beta-reduction that's going to take place in the next step is displayed such that the function that is applied is displayed in red, while the argument is printed blue.

## Lambda expressions

A very straightforward approach to representing expressions in Haskell could look as follows:

data Expression =
Variable String
| Abstraction String Expression
| Application Expression Expression

However, I decided to equip abstractions and applications with an additional flag indicating whether the respective abstraction can be eta-reduced or the respective application can be beta-reduced. This results in the following declaration:

data Expression =
Variable String
-- abstractions may be eta-convertible
| Abstraction Bool String Expression
-- applications may be beta-convertible
| Application Bool Expression Expression

While this makes pattern matching more cumbersome in many places, it gives more freedom in other contexts: Different reduction strategies can be implemented in a more general manner and we do not have to update the color-printing routine for each reduction strategy: We simply set the corresponding flags (the ones that shall be reduced in the next step), pretty-print the expression and apply the reductions.

I also installed some smart constructors that automatically create abstractions or applications with the flag set to False. From now on, we will also call these flags tags.

## Parsing lambda expressions

I never used a Parser combinator before and was pleasantly surprised how comfortable it is to use Parsec. We can parse lambda expressions in under 40 lines of code (and I suppose this code can be improved):

variableParser :: Parser Expression
variableParser = do r <- many1 (noneOf " .\\()")
return (Variable r)

applicationParser :: Parser Expression
applicationParser = do content <- endBy1 vl spaces
return $foldl1 (\a b -> application a b) content where vl = (try variableParser <|> try parensLambdaParser) abstractionParser :: Parser Expression abstractionParser = do char '\\' spaces varlist <- (endBy1 variableParser spaces) char '.' spaces body <- lambdaParser spaces return$ foldr (\(Variable v) a -> abstraction v a)
body varlist

parensLambdaParser :: Parser Expression
parensLambdaParser = do char '('
spaces
content <- lambdaParser
spaces
char ')'
_ <- spaces
return content

lambdaParser :: Parser Expression
lambdaParser = (try applicationParser)
<|> (try variableParser)
<|> (try abstractionParser)
<|> (try parensLambdaParser)

parseLambda s = case parse lambdaParser "(unknown)" s of
Right a -> Just a
Left _ -> Nothing

## Rewriting lambda expressions

### Normal-order

For now, let us simply consider how we can implement the normal order reduction. As said before, we first of all set only those tags to True whose beta-reductions shall be done. The following code does exactly this:

betaReducible term = case term of
Variable _ -> False
e@(Application _ (Abstraction _ _ _) _) -> True
Application _ f x -> (betaReducible f) || (betaReducible x)
Abstraction _ x f -> betaReducible f

normalOrderTags term = case term of
Application _ e@(Abstraction _ x f) y -> Application True e y
Application _ f x -> if betaReducible f
then application (normalOrderTags f) x
else application f (normalOrderTags x)
Abstraction _ x f -> abstraction x (normalOrderTags f)
e -> e

The simple case distinction of normalOrderTags works as follows:

• If we have an application whose first part is an abstraction, we will be able to perform a beta reduction.
• If we have an application whose first part is not an abstraction, we check whether its first component is beta-reducible. If so, we install the proper tags in the first subexpression and leave the second as it is. If not, we install possible tags in the second subexpression.
• If we have an abstraction, we have to compute the tags within the body of the abstraction.
• If we have something else (i.e. a variable), we simply return it as it is.

The case distinction in betaReducible is structured similar.

If we now want to apply all beta-reductions that we have found by using normalOrderTags, we can use the following function:

-- applyTags takes a (possibly) tagged expression and returns an untagged one
applyTags :: Expression -> Expression
applyTags term = case term of
Variable x -> Variable x
Application False f x -> application (applyTags f) (applyTags x)
Application True (Abstraction _ x f) y -> replaceVariable f x y
Application True _ y -> error "Disallowed tagged application."
Abstraction False x f -> abstraction x (applyTags f)
Abstraction True x (Application _ f y) -> applyTags f
Abstraction True x _ -> error "Disallowed tagged abstraction."

Again, we have simple case distinctions, but now incorporating the value of the tag so that we can determine whether the current subterm shall be reduced or not.

### All outermost reductions

Another, more aggressive reduction strategy is to take all outermost possible reductions and perform them. The relevant code could look as follows:

allOutermostTags :: Expression -> Expression
allOutermostTags term = case term of
-- eta-reducible abstraction
Abstraction _ x e@(Application _ f (Variable y)) ->
if x==y
then Abstraction True x e
else Abstraction False x $allOutermostTags e -- beta-reducible application Application _ e@(Abstraction _ x f) y -> Application True e y -- "normal stuff" Variable x -> Variable x Abstraction _ x f -> Abstraction False x$ allOutermostTags f
Application _ f x -> Application False tf tx
where tf = allOutermostTags f
tx = allOutermostTags x

Note that this does not include "nested" reductions. I.e. if we set the tag for a beta reduction, we won't set any tags for its two subexpressions.

## Code

The full code is hosted over there at github.


text:
## Monday, March 31, 2014

### 78: That Set

An updated version:

And a video zooming in ... 2 * 10275

## Sunday, March 30, 2014

### 77: Some Probability Calculations are All Wet

Because it's Sunday, here's another:

So ... what do you think? How likely is it that you'll get wet if you walk outside today?

### 76: Fractions and decimals

Which do you think works better?
"Decimals are a subset of fractions." OR "Fractions are a subset of decimals."

Say What?

## Friday, March 28, 2014

### 74: The One where we discuss Size.

After we laugh at the joke,

Consider:

Here is the NPR discussion on this.

## Thursday, March 27, 2014

### 73: Four-Squares

2² ends in a 4
12² ends with 44
Find a square that ends with 444.
Find another that ends with 4444.

## Wednesday, March 26, 2014

### 72: Trends in the Data

FiveThirtyEight had this graph and the comment. "It looks to me as if Mr. Cain had been on a positive trajectory before, perhaps having moved up to about 28 percent of the Republican vote."

Is that blue trendline reasonable?

Some background information. From Wikipedia, "Cain ran as a Washington outsider and became a front-runner in the race in the fall of 2011. However, Cain's support plummeted after several women alleged that he had engaged in sexual harassment or, in one case, a 13-year extramarital affair. Cain and his wife unequivocally said the accusations were false, but Cain, citing the toll the allegations had taken on his family and his political support, suspended his campaign on December 3, 2011."

## Tuesday, March 25, 2014

### 71: Is it easier to be wrong?

Which is easier ... a completely correct bracket or a completely wrong one?

## Monday, March 24, 2014

### 70: Logarithmic Fun

What is $log_2 3 * log_3 4 * log_4 5 *log_5 6 * log_6 7 * log_7 8$ ?

## Sunday, March 23, 2014

### 69: Triangular Fibonacci Numbers

1, 3, 21, 55 only triangular Fibonacci numbers.

Any tetrahedral beyond 1?

## Saturday, March 22, 2014

### 68: The cool pythagorean triangle.

I lied. It's not a Pythagorean triangle. It's TWO Pythagorean triangles that make up a Heronian triangle.
What other Pythagorean Triangle combinations make a triangle like this?

## Friday, March 21, 2014

### 67: Two Questions of Average Difficulty

From Gabriel Rosenberg via email

True or False?
If my stock increases in value by 28% one year, and then decreased in value by 2% the following year, then on average it increased by 13% per year.

If I powerwalk to school at a constant speed of 4 miles per hour, and then saunter home at a constant speed of 2 miles per hour, then I averaged 3 miles per hour for my trip.

## Thursday, March 20, 2014

### 66: Go Fish with Primes

3 William Carey, on a Dan Meyer post:
We play Prime fish with the sixth and seventh graders. You need a special deck of cards, but it’s an easy deck to make (edited, - ed.):
Ten cards with a "2" on them (or two fish)
Ten cards with a "3" on them (or three squid)
Eight cards with a "5" on them (or five eels)
Five cards with a "7" on them (or seven sea-slugs)
Two cards with a "11" on them (or eleven shrimp)
Each player draws four cards. They multiply their hand together, and announce only the product (!) to the group. They then play go-fish.

It’s fun to watch the kids debate whether there’s a strategy to the game. It’s more fun to watch them work out the strategy once they decide that there is a strategy.

What's the strategy?

## Wednesday, March 19, 2014

### 65: Square Fibonacci Numbers

What are the only square Fibonacci numbers?
How can we show that these are the only ones?

## Tuesday, March 18, 2014

### 64: Who's Better at Packing

Which student would you choose to pack your parachute?

## Monday, March 17, 2014

### 63: Who's Better at Doodle Jump?

Dan might not even remember this ...

How many different ways can you compare these two to decide that Dan is better?

How many different ways can you compare these two to decide that Mike is actually better?

Posted years ago by @ddmeyer

I think my pre-algebra came up with seven different ways to decide.

## Sunday, March 16, 2014

There are 68 teams in the U.S. College Basketball Tournament.  The bottom eight play an extra round; the winners get the bottom seeds in the main tournament.  The main bracket has 64 teams in a single elimination tournament.

In the main bracket, how many games will be played, total?

If you just flipped a coin, what is the probability that you'd win them all?

Warren Buffet has offered $1 Billion Dollars to anyone who does pick all the games. What are the chances he will lose 1/60th of his money? If you'd like to join the fun. with the rest of the Mathtwitterblogosphere. ## Saturday, March 15, 2014 ### 61: Textbook Error You can tell by the font this is a textbook. You can tell by my tone that this is an error. Fix that Graph ... Fix That Question. ## Friday, March 14, 2014 ### 60: We watch someone else talk about pi Well, recite it, actually ... 3.14159265 3589793238 4626433832 7950288419 7169399375 1058209749 4459230781 6406286208 9986280348 2534211706 7982148086 5132823066 4709384460 9550582231 7253594081 2848111745 0284102701 9385211055 5964462294 8954930381 9644288109 7566593344 6128475648 2337867831 6527120190 9145648566 9234603486 1045432664 8213393607 2602491412 7372458700 6606315588 1748815209 2096282925 4091715364 3678925903 6001133053 0548820466 5213841469 5194151160 9433057270 3657595919 5309218611 7381932611 7931051185 4807446237 9962749567 3518857527 2489122793 8183011949 1298336733 6244065664 3086021394 9463952247 3719070217 9860943702 7705392171 7629317675 2384674818 4676694051 3200056812 7145263560 8277857713 4275778960 9173637178 7214684409 0122495343 0146549585 3710507922 7968925892 3542019956 1121290219 6086403441 8159813629 7747713099 6051870721 1349999998 3729780499 5105973173 2816096318 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4532207770 9212019051 6609628049 0926360197 5988281613 3231666365 2861932668 6336062735 6763035447 7628035045 0777235547 1058595487 0279081435 6240145171 8062464362 6794561275 3181340783 3033625423 2783944975 3824372058 3531147711 9926063813 3467768796 9597030983 3913077109 8704085913 3746414428 2277263465 9470474587 ### 59: In which we talk about pi on pi-day From Gabriel Rosenberg via email: If π is a non-repeating and non-terminating value, is it a number? ### 58: It's not Hump Day.$3+\dfrac{4}{2*3*4}-\dfrac{4}{4*5*6}+\dfrac{4}{6*7*8}-\dfrac{4}{8*9*10}+ ....$### 57: 10,000 Random Dots. To repeat the experiment, refresh the page or ... Do it Again. Your browser does not support the HTML5 canvas tag. ### What are we looking at, here? Feel free to steal the code. I converted to javascript from the Original python code by @rjallain. ## Thursday, March 13, 2014 ### 56: Squaring the Circle I don't care if it's a RealWorld question. It's still fascinating. Exactly when are the two areas equal? From @ddmeyer ## Wednesday, March 12, 2014 ### 55: The Fastest Way Down. Where is the steepest part of this bowl? What is that slope? What is the angle in degrees? (Click to embiggen) ## Tuesday, March 11, 2014 ### 54: Cheating Is this evidence of cheating? ## Monday, March 10, 2014 ### 53: Right six, CARE, tightens, gnarl. Jiggy Jiggy, Fifty, crest. Right three minus Go full size: The 7.6-mile-long Mt. Washington Road is lined with trees, drop-offs, and winds on the way to the 6,288-foot summit. Travis Pastrana in a 2011 Subaru Impreza WRX STI rally car, with an officially timed run of 6 minutes 20.47 seconds. There are questions all over the place. 1. Average speed 2. Average slope of the road. 3. What do the calls mean? 4. Add more in the comments. This one includes rally notes system details. "Increases the radius, decreases the radius" "Can you take it wide open?" "If the notes are wrong, we go off a cliff." ## Sunday, March 9, 2014 ### 52: Pythagorean Triples If you've worked with the Pythagorean Theorem, you've come across some integer solutions. 3-4-5 and 5-12-13 come to mind, especially if you've studied for the SAT; 8-15-17 is another good one. How many can you find? Which one is your favorite? Is there a way to find as many as you want? (For those who haven't seen this trivia snippet) pick distinct positive integer values for u and v: perhaps 3 and 4. The three sides are 2*u*v, u² + v², and |u² - v²|, thus 24, 25, 7 The hypotenuse is simply the longest of the three sides, but won't be generated by the same expression all the time. ## Saturday, March 8, 2014 ### 51: Dividing by a fraction Why is dividing by a fraction equivalent to multiplying by the reciprocal ... Here's one explanation, from my other blog: We "invert and multiply", "multiply by the reciprocal" or insist on using the fraction key because we can't remember or were never really taught the reasons or the algorithm. Is there a simple explanation for the method we old farts memorized years ago in third or fourth grade? Why does it work? Let's start with a problem:$\frac{3}{4} \div \frac{5}{6}$and change to a compound fraction:$\dfrac{\frac{3}{4}}{\frac{5}{6}}$Now what? Dividing by a fraction is confusing, but dividing by 1 is obvious. So we turn$\frac{5}{6}$into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by$\frac{6}{5}$without changing its value, so we multiply by 1:$\dfrac{\frac{6}{5}}{\frac{6}{5}}$All in one image:$\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}\$

Divide by one. Seems simple to me.

## Friday, March 7, 2014

### 50: Rectangle made up of Squares

"A perfect fit with no overlapping"

Take the squares as defined below and fit them into one big rectangle with no gaps or spaces between the squares or in the corners.

The squares have sides of the following lengths 2, 5, 7, 9, 16, 25, 28, 33, & 36.

## Thursday, March 6, 2014

### 49: Pizza, pizza

Pizza ... makes you hungry doesn't it?
What if you had ten toppings available. How many different two-topping pizza variations can you make ...

What definition did you have to "adjust" in order to get David's answer of 55?

Do you agree that this is a fair interpretation of the problem?

source:

## Wednesday, March 5, 2014

### 48: The One Where We Label Axes.

Why are coordinate axes perpendicular?
Do they have to be?
What if there are more than two dimensions?

## Tuesday, March 4, 2014

Submitted without comment:

Oh, the Humanity ...

## Monday, March 3, 2014

### 46: The Dinosaur Bone

What questions does this image bring up?

What part of the dinosaur is that?

How big was it?

What measurement system are we using?

## Sunday, March 2, 2014

### 45: Complicated Crossings.

Explain WHY this works for adding fractions.

Three very similar-looking, but very different, ideas:


text:
# IMO 1987 - function

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.

-

Here is an alternative approach. It is obvious that such an $f$ must be an injection. Now, look at sets $\mathbb{N}$, $A = f(\mathbb{N})$ and $B = f(f(\mathbb{N})) = \{n + 1987 \ | \ n \in \mathbb{N}\}$.

It is easy to see that $B \subset A \subset \mathbb{N}$, and also from the injectivity of $f$ that $f$ induces a bijection between the disjoint sets $\mathbb{N} \setminus A$ and $A \setminus B$. Therefore, $\mathbb{N} \setminus B = (\mathbb{N} \setminus A) \cup (A \setminus B)$ must contain an even number of elements. But $|\mathbb{N} \setminus B| = 1987$, which is a contradiction, and we are done.

-
Very interesting solution. should say $f(f(n)) = n + 1987$ at the second last line. –  user58512 Mar 9 '13 at 15:33
@user58512 Thanks for the correction! –  Dan Shved Mar 9 '13 at 15:35
+1 Great answer, mush easier to follow than the other ones. Only I would have said $B=f(f(\Bbb N))=\{\,i+1987\mid i\in\Bbb N\,\}$ right away, which avoids making a case distinction (and also like your answer avoids any commitment to where the natural numbers start exactly ;-). –  Marc van Leeuwen Mar 9 '13 at 15:40
@MarcvanLeeuwen I've updated the answer. Is this what you had in mind? If not, feel free to improve it! –  Dan Shved Mar 9 '13 at 15:53

Here is a generalisation.

Proposition. For positive integers $k,l$, the functional equation $$f^k(n)=n+l\qquad\text{for all n}$$ has no solutions $f$ in the functions $\Bbb Z\to\Bbb Z$, or in the functions $\Bbb N\to\Bbb N$, unless $k$ divides $l$.

In the case $k\mid l$, one does of course have solutions, for instance $n\mapsto n+l/k$.

Proof. Assume that $f$ satisfies the functional equation.

• For all $n$ one has $f(n+l)=f^{k+1}(n)=f(n)+l$.
• Therefore $f(n+l)-(n+l)=f(n)-n$, and $f(n)-n$ is constant when $n$ varies over any congruence class modulo $l$.
• The image under $f$ of such a congruence class is therefore another congruence class, and $f$ gives rise to a function $\bar f:\Bbb Z/l\Bbb Z\to\Bbb Z/l\Bbb Z$.
• Since $(\bar f)^k(\bar n)=\bar n$ for all $\bar n\in\Bbb Z/l\Bbb Z$, this function $\bar f$ is a bijection.
• Put $S=\sum_{i=0}^{l-1}(f(i)-i)$; then $l$ divides $S$ since $\sum_{i=0}^{l-1}f(i)\equiv\binom l2=\sum_{i=0}^{l-1}i\pmod l$, so put $s=S/l$.
• From the functional equation $\sum_{i=0}^{l-1}\bigl(f^k(i)-i\bigr)=\sum_{i=0}^{l-1}l=l^2$.
• By a telescopic sum, $f^k(i)-i=\sum_{j=0}^{k-1}\bigl(f^{j+1}(i)-f^j(i)\bigr)$.
• Since each $(\bar f)^j$ is a bijection, this implies $$\sum_{i=0}^{l-1}\bigl(f^k(i)-i\bigr) =\sum_{j=0}^{k-1}\sum_{i=0}^{l-1}(f^{j+1}(i)-f^j(i)\bigr) =\sum_{j=0}^{k-1}S=kS=kls.$$
• Therefore $l^2=kls$, so $l=ks$ and $k$ divides $l$.
-

Just a minor variation of the answer by user58512, for fun; like that answer it works even if we replace $\Bbb N$ by $\Bbb Z$. The equation $f(n+1987)=f(n)+1987$ in that answer means that $$f(n+1987)-(n+1987)=f(n)-n,$$ so it shows that $f(n)-n$ only depends on the class of $n$ modulo $1987$. Also the map $\bar f$ that $f$ induces on $\Bbb Z/1987\Bbb Z$ is surjective (the class of $n$ is in the image of $\bar f^2$) hence bijective: $f(n)$ runs over all congruence classes as $n$ runs over all congruence classes modulo $1987$.

Now let $S$ be the sum of $f(n)-n$ over all those congruence classes, obviously an integer. But computing the sum of $1987=f(f(n))-n=\bigl(f(f(n))-f(n)\bigl)+\bigr(f(n)-n\bigr)$ over all those congruence classes, one gets on one hand the odd number $1987^2$, and on the other hand the even number $2S$; contradiction.

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Hint $\$mod $\rm\,p\!:\ f(f(n)) = n+p\equiv n\:$ is an involution, hence $\rm\:p\:$ odd $\Rightarrow$ that it has a fixed point $\rm\:f(n)\equiv n,\:$ so $\rm\:f(n) = n\!+\!k p,\ k\in\Bbb Z.\:$ Now the orbit of $\rm\:f\:$ on $\rm\,n\,$ yields a contradiction

$$\rm\begin{array}{rlcl} &\rm n &\rm \to &\rm \color{#C00}{n+kp} \\ \to &\rm \color{#0A0}{n\!+\!p} &\rm \to &\rm n\!+\!(k\!+\!1)p \\ \to &\rm n\!+\!2p &\rm \to &\rm n\!+\!(k\!+\!2)p\\ \to &\rm n\!+\!3p &\rm \to &\rm n\!+\!(k\!+\!3)p\\ &\rm\ \cdots &\rm &\rm \quad\cdots \\ \to &\rm \color{#C00}{n\!+\!kp} &\rm \to &\rm n\!+\!(k\!+\!k)p = \color{#0A0}{n\!+\!p}\ \Rightarrow\ 2k=1\ \Rightarrow\Leftarrow\: k\in\Bbb Z\\ \end{array}$$

Remark $\$ This was a hint I posted many years ago in another math forum. You can find many further posts here exploiting parity and involutions by searching on involution and Wilson (group theoretical form of Wilson's theorem).

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We prove that if $f(f(n)) = n + k$ for all $n$, where $k$ is a fixed positive integer, then $k$ must be even. If $k = 2h$, then we may take $f(n) = n + h$.

Suppose,

$f(m) = n$ with $m= n (mod$ k $)$.

Then by an easy induction on $r$ we find $f(m + kr) = n + kr, f(n + kr) = m + k(r+1)$.

Suppose $m < n$,

$n = m + ks$ for some $s > 0$ $\implies f(n) = f(m + ks) = n + ks$.

But $f(n) = m + k, so m = n + k(s - 1) >= n$. Contradiction.

So we must have $m\ge n$, so $m = n + ks$ for some $s \ge 0$.

But now,

$f(m + k) = f(n + k(s+1)) = m + k(s + 2)$.

But $f(m + k) = n + k$, so $n = m + k(s + 1) > n$. Contradiction again.

So if $f(m) = n$, then $m$ and $n$ have different residues $(mod$ k $)$.

Suppose they have $r_1$ and $r_2$ respectively.

Then the same induction shows that:

All sufficiently large $s= r_1 (mod k)$ have $f(s)= r_2(mod k)$,

and that all sufficiently large $s = r_2 (mod k)$ have $f(s) = r_1 (mod k)$.

Hence if $m$ has a different residue $r(mod k)$, then $f(m) cannot have residue$r_1$or$r_2$. For if f(m) had residue$r_1$, then the same argument would show that all sufficiently large numbers with residue$r_1$had$f(m) = r (mod k)$. Thus the residues form pairs, so that if a number is congruent to a particular residue, then f of the number is congruent to the pair of the residue. But this is impossible for$k\$ odd.

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# Tag Info

86

Haskell, 26 bytes (++"[]").((++":").show=<<) Try it online! Explanation: In non-pointfree notation and using concatMap instead of =<<, this becomes f s = concatMap(\c-> show c ++ ":")s ++ "[]" Given a string s, we map each char c to a string "'c':" using the show function which returns a string representation of most Haskell types. ...

45

Define infix operators instead of binary functions This saves usually one or two spaces per definition or call. 0!(y:_)=y x!(y:z)=(x-1)!z vs. f 0(y:_)=y f x(y:z)=f(x-1)z The available symbols for 1-byte operators are !, #, %, &, and ?. All other ASCII punctuation is either already defined as an operator by the Prelude (such as $) or has a special ... 44 Haskell, 33 28 26 bytes foldr((.(':':)).shows)"[]" Try it online! fold the given pointfree function from the right into the input string starting with []. The function is: show char as a Haskell char, i.e. surrounded with ' and concatenate with the result so far after putting a : in front of it. Edit: @Ørjan Johansen saved two bytes. Thanks! 29 Use guards not conditionals: f a=if a>0 then 3 else 7 g a|a>0=3|True=7 Use semicolons not indents f a=do this that g a=do this;that Use boolean expressions for boolean functions f a=if zzz then True else f yyy g a=zzz||f yyy (SO is being a pain about letting me post these separately) 29 Use pointless (or -free) notation where appropriate Often a function with one or two parameters can be written point free. So a lookup for a list of tuples whose elements are swapped is naïvely written as: revlookup :: Eq b => b -> [(a, b)] -> Maybe a revlookup e l=lookup e(map swap l) (the type is there just to help you understand what it's ... 28 Use the list monad A quick review: xs >> ys = concat$ replicate (length xs) ys xs >>= f = concatMap f xs mapM id[a,b,c] = cartesian product of lists: a × b × c mapM f[a,b,c] = cartesian product of lists: f a × f b × f c Examples: Repeating a list twice Prelude> "aa">>[1..5] [1,2,3,4,5,1,2,3,4,5] Shorter ...

24

Use GHC 7.10 The first version of GHC that contained this stuff was released on March 27, 2015. It's the latest version, and Prelude got some new additions that are useful for golfing: The (<$>) and (<*>) operators These useful operators from Data.Applicative made it in! <$> is just fmap, so you can replace map f x and fmap f x with f&...

24

interact :: (String → String) → IO () People often forget that this function exists - it grabs all of stdin and applies it to a (pure) function. I often see main-code along the lines of main=getContents>>=print.foo while main=interact$show.foo is quite a bit shorter. It is in the Prelude so no need for imports! 24 MagicHash, 30 bytes x=1 y#a=2 x#a=1 main=print$x#x -XMagicHash outputs 1, -XNoMagicHash outputs 2 MagicHash allows variable names to terminate in a #. Therefore with the extension, this defines two functions y# and x# which each take a value and return a constant 2, or 1. x#x will return 1 (because it is x# applied to 1) Without the extension, this ...

23

Use 1<2 instead of True and 1>2 instead of False. g x|x<10=10|True=x f x|x<10=10|1<2=x

22

Use list comprehensions (in clever ways) Everyone knows they're useful syntax, often shorter than map + a lambda: Prelude> [[1..x]>>show x|x<-[1..9]] ["1","22","333","4444","55555","666666","7777777","88888888","999999999"] Or filter (and optionally a map at the same time): Prelude> [show x|x<-[1..60],mod 60x<1] ["1","2","3","4","5","...

19

Python 3, 32 bytes lambda s:"%r:"*len(s)%(*s,)+"[]" Try it online!

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CPP, 33 20 bytes main=print$0-- \ +1 Prints 0 with -XCPP and 1 with -XNoCPP. With -XCPP, a slash \ before a newline removes the newline, thus the code becomes main=print$0-- +1 and only 0 is printed as the +1 is now part of the comment. Without the flag the comment is ignored and the second line is parsed as a part of the previous line because it is ...

18

Know your Prelude Fire up GHCi and scroll through the Prelude documentation. Whenever you cross a function that has a short name, it can pay off to look for some cases where it might be useful. For example, suppose you wish to transform a string s = "abc\ndef\nghi" into one that's space-separated, "abc def ghi". The obvious way is: unwords$lines s But ... 17 JavaScript ES6, 42 40 31 bytes s=>s.replace(/./g,"'$&':")+"[]" Replaces each char with '<char>':, then adds [] to the end Try it online!

17

Haskell, 169 148 bytes init.tail.fst.([]%) p:k="(," l%('(':r)|(y,x:s)<-[]%r,m<-y:l=last$m%(p:s):[(p:p:(l>>k)++x:foldl(\r x->x++[' '|x>k,r>k]++r)[x]m,s)|x<','] l%r=lex r!!0 Try it online! Takes the tuple as a string. init.tail.fst.([]%) is the anonymous main function. Bind it to e.g. f and use like f "(3,(14,1),4,7)", which yields "(,... 16 Match a constant value A list comprehension can pattern match on a constant. [0|0<-l] This extracts the 0's of a list l, i.e. makes a list of as many 0's as are in l. [1|[]<-f<$>l] This makes a list of as many 1's as there are elements of l that f takes to the empty list (using <$> as infix map). Apply sum to count these elements. ... 16 Common Lisp, 50 42 bytes (format t"~{'~a':~}[]"(coerce(read)'list)) Try it online! Reduced thanks to the comment of @coredump, by using read instead of defining a function. 15 Know your monadic functions 1)simulate monadic functions using mapM. a lot of times code will have sequence(map f xs), but it can be replaced with mapM f xs. even when just using sequence alone it is longer then mapM id. 2)combine functions using (>>=) (or (=<<)) the function monad version of (>>=) is defined as so: (f >>= g) x = g (f ... 15 Arguments can be shorter than definitions I just got outgolfed in a very curious way by henkma. If an auxiliary function f in your answer uses an operator that isn’t used elsewhere in your answer, and f is called once, make the operator an argument of f. This: (!)=take f a=5!a++3!a reverse.f Is two bytes longer than this: f(!)a=5!a++3!a reverse.f take 14 Use words instead of a long list of strings. This isn't really specific to Haskell, other languages have similar tricks too. ["foo","bar"] words"foo bar" -- 1 byte longer ["foo","bar","baz"] words"foo bar baz" -- 1 byte shorter ["foo","bar","baz","qux"] words"foo bar baz qux" -- 3 bytes shorter 14 BinaryLiterals, 57 bytes b1=1 instance Show(a->b)where;show _="" main=print$(+)0b1 -XBinaryLiterals prints a single newline. -XNoBinaryLiterals prints a 1. I am sure there is a better way to do this. If you find one, please post it.

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NumDecimals, 14 bytes main=print 1e1 -XNumDecimals prints 10. -XNoNumDecimals prints 10.0.

12

Working with the minus sign The minus sign - is an annoying exception to many syntax rules. This tip lists some short ways of expressing negation and subtraction in Haskell. Please let me know if I've missed something. Negation To negate an expression e, just do -e. For example, -length[1,2] gives -2. If e is even moderately complex, you will need ...

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Shorter conditional last$x:[y|b] is equivalent to if b then y else x Here's how it works: [y|b] x:[y|b] last$x:[y|b] if... +-------------------------------- b == False | [] [x] x b == True | [y] [x,y] y

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Use the cons operator (:) when concatenating lists, if the first is of length 1 then use : instead. a++" "++b a++' ':b -- one character shorter [3]++l 3:l -- three characters shorter

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Use pattern guards They're shorter than a let or a lambda that deconstructs the arguments of a function you're defining. This helps when you need something like fromJust from Data.Maybe: f x=let Just c=… in c is longer than f x=(\(Just c)->c)$… is longer than m(Just c)=c;f x=m$… is longer than f x|Just c<-…=c In fact, they’re shorter even when ...

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MonomorphismRestriction + 7 others, 107 bytes This uses TH which requires the flag -XTemplateHaskell at all times. File T.hs, 81 + 4 bytes module T where import Language.Haskell.TH p=(+) t=reify(mkName"p")>>=stringE.show Main, 22 bytes import T main=print $t Compiling with the flag MonomorphismRestriction forces the type of p to Integer -> ... 11 Try rearranging function definitions and/or arguments You can sometimes save a couple of bytes by changing the order of pattern-matching cases in a function definition. These savings are cheap, but easy to overlook. As an example, consider the following earlier version of (a part of) this answer: (g?x)[]=x (g?x)(a:b)=g(g?x$b)a This is a recursive ...

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Don't use backticks too often. Backticks are a cool tool for making sections of prefix functions, but can sometimes be misused. Once I saw someone write this subexpression: (xv) Although it is the same as just v x. Another example is writing (x+1)divy as opposed to div(x+1)y. I see it happen around div and elem more often because these functions are ...

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text:
# Scaling¶

Methods for manipulating irradiance for temporal or spatial considerations

 scaling.wvm(clearsky_index, positions, …) Compute spatial aggregation time series smoothing on clear sky index based on the Wavelet Variability model of Lave et al.


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# How do you solve  4x^2 + 2x - 1 = 0 by completing the square?

$x = \frac{\sqrt{5} - 1}{4}$ or $x = - \frac{\sqrt{5} + 1}{4}$
Dividing by $4$ we get
${x}^{2} + \frac{1}{2} x - \frac{1}{4} = 0$
${\left(x + \frac{1}{4}\right)}^{2} = \frac{5}{16}$


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# If the diameter of a cylinder is 28 cm and its height is 20 cm

Question:

If the diameter of a cylinder is 28 cm and its height is 20 cm, then its curved surface area is
(a) 880 cm2
(b) 1760 cm2
(c) 3520 cm2
(d) 2640 cm2

Solution:

(b) $1760 \mathrm{~cm}^{2}$

Curved surface area of the cylinder $=2 \pi r h$

$=2 \times \frac{22}{7} \times 14 \times 20$

$=44 \times 40$

$=1760 \mathrm{~cm}^{2}$