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# 4.10: Triangle Similarity using AA and SSS

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Understand that SSA does not necessarily prove triangles are congruent.
• Understand that AA does not necessarily prove triangles are congruent.
• Determine whether triangles are similar.
• Identify corresponding angles and sides of similar polygons from a statement of similarity.
• Calculate and apply scale factors.
• Understand and apply the AA Similarity Postulate.
• Understand and apply the SSS Similarity Postulate.
• Solve problems about similar triangles.

## Proving Triangles are Congruent – Why SSA does NOT always work.

SSA relationships do not necessarily prove congruence. In other words, if you have two sides and a non-included angle (an angle that is not between them), then you cannot prove congruence.

Observe the triangles below. Two sides and a non-included angle from triangle $ABC$ are congruent to the corresponding sides and non-included angle of triangle $PQR$. In fact, the triangles appear to be congruent.

However, below is a diagram that serves as a counterexample to the notion that SSA might work. Again, we see two sides and a non-included angle of triangle $ABC$ which are congruent to their corresponding parts in triangle $PQR$, yet clearly the two triangles are not congruent to one another.

SSA does not prove that two triangles are ________________________________.

A counterexample proves that a statement is not ________________________.

Example 1

Can you prove that the two triangles below are congruent?

Note: Figure is not to scale.

The two triangles above look congruent, but are labeled, so you cannot assume that how they look means that they are congruent!

There are two sides labeled congruent, as well as one angle. Since the angle is not between the two sides, however, this is a case of SSA. You cannot prove that these two triangles are congruent. Also, it is important to note that although two of the angles appear to be right angles, they are not marked that way, so you cannot assume that they are right angles.

1. True/False: SSA is another way of proving that two triangles are congruent.

2. True/False: You CANNOT always trust the drawings, so even if triangles look congruent, you CANNOT assume that they are congruent!

## Proving Triangles are Congruent – Why AA does NOT always work.

Based on the marks in the diagram below, you can see that the three angles of triangle $ABC$ are congruent to the three corresponding angles of $PQR$, yet the triangles are not congruent to one another. Triangles like this that are the same shape but different sizes are called similar triangles.

Similar triangles have the ________________ shape but ___________________________ sizes.

## Similar Polygons

Similar figures have the same shape, but they may have different sizes.

Look at the triangles below:

• The triangles on the left are not similar because they are not the same shape.
• The triangles in the middle are similar. They are all the same shape, no matter what their sizes.
• The triangles on the right are similar. They are all the same shape, no matter how they are turned or what their sizes.

1. Fill in the blanks: Similar figures have the same ________________________ but different _____________________.

2. True/False: If two triangles are similar, then their corresponding angles are congruent.

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3. Can you think of some similar shapes in the real world?

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• The quadrilaterals in the upper left are not similar because they are not the same shape.
• The quadrilaterals in the upper right are similar. They are all the same shape, no matter what their sizes.
• The quadrilaterals in the lower left are similar. They are all the same shape, no matter how they are turned or what their sizes.

Two polygons are similar if and only if:

• they have the same number of sides
• for each angle in either polygon there is a corresponding angle in the other polygon that is congruent
• the lengths of all corresponding sides in the polygons are proportional

Reminder: Just as we did with congruent figures, we name similar polygons according to corresponding parts. The symbol $\sim$ is used to represent “is similar to.”

1. Fill in the blanks: Two polygons are similar if and only if:

They have the same number of ________________________.

Corresponding angles are ________________________________.

Corresponding side lengths are ________________________________.

2. What does the symbol $\sim$ mean?

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3. In the space below, draw two similar rectangles.

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Example 2

Suppose $\Delta ABC \sim \Delta JKL$. Based on this statement, which angles are congruent and which sides are proportional? Write true congruence statements and proportions.

In similar triangles, all matching angles are congruent.

First, list the angles that match up in the names of the triangles:

$\angle A \cong \angle J, \angle B \cong \angle K$, and $\angle C \cong \angle L$

Next, match up the side segments by the order of the letters in the similar triangles:

$\frac{AB}{JK} = \frac{BC}{KL} = \frac{AC}{JL}$

Remember that there are many equivalent ways to write a proportion. The answer above is not the only set of true proportions you can create based on the given similarity statement.

Example 3

Given: $MNPQ \sim RSTU$

What are the values of $x, y,$ and $z$ in the diagram above?

In similar figures, corresponding side lengths are proportional.

Side $x$ in $MNPQ$ corresponds to the side with length 25 in $RSTU$.

Set up a proportion to solve for $x$:

$&\frac{x}{25} = \frac{18}{30} \quad \text{you can reduce} \ \frac{18}{30} \ \text{to be} \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\&\frac{x}{25} = \frac{3}{5}\\&\text{Cross multiply to get} \ 5x = 75 \quad \text{so} \quad x = 15$

Side $y$ in $RSTU$ corresponds to the side with length 15 in $MNPQ$.

Set up a proportion to solve for $y$:

$&\frac{y}{15} = \frac{30}{18} \quad \text{you can reduce}\ \frac{30}{18} \ \text{to be} \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\&\frac{y}{15} = \frac{5}{3}\\&\text{Cross multiply to get} \ 3y = 75 \quad \text{so} \quad y = 25$

Finally, since $Z$ is an angle, we are looking for $m \angle R$:

$Z = m \angle R = m \angle M = 115^\circ$

Because corresponding angles are congruent in similar figures.

Example 4

$ABCD$ is a rectangle with length 12 and width 8.

$UVWX$ is a rectangle with length 24 and width 18.

A. Are corresponding angles in the rectangles congruent?

Yes. Since both are rectangles, all four angles in both figures are congruent right angles.

B. Are the lengths of the sides of the rectangles proportional?

No. The ratio of the lengths is $12 : 24 = 1 : 2$.

The ratio of the widths is $8 : 18 = 4 : 9$ which $\neq 1 : 2$.

Therefore, the lengths of the sides are not proportional.

C. Are the rectangles similar?

No. Although the corresponding angles are congruent, the lengths of corresponding sides are not proportional.

## Scale Factors

If two polygons are similar, we know that the lengths of corresponding sides are proportional.

Similar polygons have ___________________________________ corresponding sides.

If $k$ is the length of a side in one polygon, and $m$ is the length of the corresponding side in the other polygon, then the ratio $\frac{k}{m}$ is called the scale factor relating the first polygon to the second.

Another way to say this is:

The length of every side of the first polygon is $\frac{k}{m}$ times the length of the corresponding side of the other polygon.

• $\frac{k}{m}$ represents the _________________ factor of similar shapes.

If two corresponding segments are shown below, where the segment on the left has a length of 3 and segment on the right has a length of 9, the scale factor relating them is $\frac{3}{9}$ which reduces to $\frac{1}{3}$

Example 5

Look at the diagram below, where $ABCD$ and $AMNP$ are similar rectangles.

A. What is the scale factor?

Since $ABCD \sim AMNP$, then $AM$ and $AB$ are corresponding sides. Since $ABCD$ is a rectangle, you know that $AB = DC = 45$.

The scale factor is the ratio of the lengths of any two corresponding sides.

So the scale factor (relating $ABCD$ to $AMNP$) is $\frac{45}{30} = \frac{3}{2}$ or 1.5.

We now know that the length of each side of $ABCD$ is 1.5 times the length of the corresponding side in $AMNP$.

• Comment: We can turn this relationship around “backwards” and talk about the scale factor relating $AMNP$ to $ABCD$. This scale factor is just $\frac{30}{45} = \frac{2}{3}$, which is the reciprocal of the scale factor relating $ABCD$ to $AMNP$.

B. What is the ratio of the perimeters of the rectangles?

Find each perimeter by adding up the lengths of the sides of the rectangle:

$ABCD$ is a 45 by 60 rectangle. Its perimeter is $45 + 60 + 45 + 60 = 210$.

$AMNP$ is a 30 by 40 rectangle. Its perimeter is $30 + 40 + 30 + 40 = 140$.

The ratio of the perimeters of $ABCD$ to $AMNP$ is $\frac{210}{140} = \frac{3}{2}$.

• Comment: You see from this example that the ratio of the perimeters of the rectangles is the same as the scale factor. This relationship for the perimeters holds true in general for any similar polygons.

The scale factor of similar shapes is the ratio of corresponding sides and the ratio of the ______________________________ of the shapes.

## The AA Rule for Similar Triangles

Suppose that the triangles $\Delta ABC$ and $\Delta MNP$ have two pairs of congruent angles, say $\angle A \cong \angle M$ and $\angle B \cong \angle N$.

But we know that if triangles have two pairs of congruent angles, then the third pair of angles are also congruent (by the Triangle Sum Theorem).

The AAA rule for similar triangles reduces to the AA triangle similarity postulate.

The AA Triangle Similarity Postulate

If two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.

• Two triangles are _____________________________ if two pairs of corresponding angles are congruent.

Example 6

Look at the diagram below:

Yes. They both have congruent right angles, and they both have a $35^\circ$ angle.

These are two pairs of congruent corresponding angles so the triangles are similar by AA.

You need _________ pairs of corresponding congruent angles to prove similarity by AA.

B. Write a similarity statement for the triangles.

$\Delta ABC : \Delta TRS$ or equivalent

C. Name all pairs of congruent angles.

Remember, congruent angles match up within the names of the triangles (first letter first letter, and so on...)

$\angle A \cong \angle T, \angle B \cong \angle R,$ and $\angle C \cong \angle S$

D. Write equations stating the proportional side lengths in the triangles.

$\frac{AB}{TR} = \frac{BC}{RS} = \frac{AC}{TS}$ or equivalent

1. True/False: Scale factor is a relationship that is expressed as a ratio or fraction.

2. True/False: The ratio of the perimeters of two polygons is the same as the scale factor of the polygons.

Example 7

Flo wants to measure the height of a windmill. She held a 6 foot vertical pipe with its base touching the level ground, and the pipe’s shadow was 10 feet long. At the same time, the shadow of the windmill tower was 85 feet long. How tall is the windmill?

Draw a diagram:

Note: It is safe to assume that the sun’s rays hit the ground at the same angle. It is also proper to assume that the windmill tower is vertical (perpendicular to the ground).

The diagram shows two similar right triangles. They are similar because each has a right angle, and the angle where the sun’s rays hit the ground is the same for both objects. Because they are similar triangles, their corresponding side lengths are proportional.

We can write a proportion with only one unknown, $x$, the height of the windmill tower:

$\frac{x}{85} &= \frac{6}{10}\\\text{Cross multiply to get}: 10x &= 85 \cdot 6\\10x &= 510\\x &= 51$

Thus, the tower is 51 feet tall.

Note: This method is considered indirect measurement because it would be difficult to directly measure the height of a tall windmill tower. Imagine how difficult it would be to hold a tape measure up to something that is 51 feet tall!

## SSS for Similar Triangles

The SSS Triangle Similarity Postulate

If the lengths of the sides of two triangles are proportional, then the triangles are similar.

• Two triangles with ___________________________________ sides are similar.

Example 8

Imagine a diagram with two triangles. (You may want to draw these in the margin). One triangle has sides 6-8-10. The other has sides 9-12-15.

Are the two triangles similar?

What do you notice? All three side lengths in the two triangles are proportional:

$\frac{6}{9} = \frac{8}{12} = \frac{10}{15} \left ( = \frac{2}{3} \right )$

Yes, the two triangles are similar!

1. The following statement is TRUE. Why? Explain:

All equilateral triangles are similar.

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2. The following statement is FALSE. Why? Explain in words (but you may also draw a picture to show an example):

All isosceles triangles are similar.

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8 , 9 , 10

## Date Created:

Feb 23, 2012

May 12, 2014
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### Home > CALC > Chapter 10 > Lesson 10.1.5 > Problem10-47

10-47.

A solid has a circular base of radius $r = 2$ units. If the cross-sections perpendicular to the base are square, find the volume of the solid.

Review the eTool below. Write an expression for the length of the base of each cross-section.

format: plain


text:
Last edited by Faer
Saturday, May 16, 2020 | History

2 edition of Electron density measurements in an argon inductivelycoupledplasma found in the catalog.

Electron density measurements in an argon inductivelycoupledplasma

Electron density measurements in an argon inductivelycoupledplasma

by

Written in English

Edition Notes

 ID Numbers Statement C. Brown ; supervised by R. Snook. Contributions Snook, R., DIAS. Open Library OL21428052M

Electron temperature and ion density measurements in a glow discharge of an Ar–N2 mixture P.G. Reyes et al Radiation Effects and Defects in Solids 1. Crossref. Comparison of the characteristics of atmospheric pressure plasma jets using different working gases and applications to plasma-cancer cell interactions. In this work, the plasma electron density in argon discharge with inductively coupled plasma (ICP) source was in-situ measured by THz-TDS approach and combined with LTS method, which was acknowledged as a one of the most accurate method to measure the electron density.

The electron density barely increases when a sudden change in the gas composition occurs. The E to H mode transition occurs at a stationary gas composition as the absorbed power increases. The EEDFs of CO 2 plasma, CO plasma, and O 2 plasma were calculated using BOLSIG+, which is a two term Boltzmann solver [G. Hagelaar and L. Author: Kyung-Hyun Kim, Kwan-Yong Kim, Young-Hun Hong, Ho-Jun Moon, Chin-Wook Chung.   Lee and C. W. Chung, “ Experimental measurements of spatial plasma potentials and electron energy distributions in inductively coupled plasma under weakly collisional and nonlocal electron kinetic regimes,” Phys. Plas ().Cited by: 2.

High electron density inductively coupled plasma (ICP) systems are finding increasing applications in the areas of microelectronic processing, cleaning and illumination. In order to understand the fundamental plasma physics mechanisms and to provide data to benchmark codes, many experiments have focused on inductive discharges in argon Cited by: Spearochunica Ada Vol 44B, No 4, pp , /89 S+ 00 Printed 10 Great Bntain Pcrgampn Press pic On the influence of water on the electron density in an argon inductively coupled plasma S. NOWAK*, J. A. M. VAN DER MULLEN, A. C. A. P. VAN LAMMERE^ and D. C. SCHRAM Physics Department, Eindhoven University of Technology, Cited by:

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Electron density measurements in an argon inductivelycoupledplasma by Download PDF EPUB FB2

Electron temperature, electron density, and neutral atom density were measured in a radio‐frequency (rf) inductively coupled plasma using Thomson and Rayleigh scattering of laser radiation. Measurements were made in an argon discharge for pressures from 1 to 20 mTorr and input rf powers from to W.

Spatial distribution profiles were measured for discharges Cited by: Electron temperature, density and energy distribution functions (EEDF) in an argon, planar, radiofrequency inductively-coupled plasma (ICP) were measured using laser Thomson scattering.

In dc plasmas, the EEDF in higher pressure discharges was Maxwellian but in low pressure discharges, a non-Maxwellian distribution was observed. Request PDF | Electron density characterization of inductively-coupled argon plasmas by the terahertz time-domain spectroscopy | Inductively-coupled plasmas (ICP) in the high electron density.

The capability of the method is demonstrated on the basis of exemplary measurements in an inductively coupled plasma (ICP, MHz) suitable for PECVD. Line-integrated electron densities. In the last two decades, the fundamentals of electron- impact excitation processes influencing the populations of the lowest (3p 5 4s, 3p 5 4p, 3p 5 3d, 3p 5 5s, and 3p 5 5p) excited states in argon have been intensively studied, from the threshold to electron energies of hundreds of electronvolts.

Inductivelycoupled plasmas (ICP) in the high electron density regime of the order of 10 13 cm −3 are generated and their electron density characteristics are investigated by the terahertz timedomain spectroscopy (THzTDS) method. In this experiment, the plasma was producedCited by: 6.

Results of measurements of temperature and electron density distribution within an electrodeless discharge in argon at atmospheric pressure are briefly presented. The temperature is found to exhibit an off-axis peak of 9 ° by: Spatial distributions of the effective electron temperature (T eff) and plasma potential were studied from the measurement of an electron energy probability function in a side type ferrite-core inductively coupled plasma with an argon–helium the helium gas was diluted at the fixed total gas pressure of 5 mTorr in an argon discharge, the distribution of the plasma density Author: Duksun Han, Hyo-Chang Lee, Jin-Young Bang, Chin-Wook Chung, Se Youn Moon.

A tuned, cylindrical Langmuir probe has been used to measure the electron energy distribution function (EEDF) in atomic and molecular gases in an inductively coupled plasma. We have discussed the precautions necessary for making Langmuir probe measurements in fluorocarbon plasmas.

The ionic and neutral composition of the plasma is Cited by: The gas pressure at which the transition occurs increases with the discharge power. In contrast to the case of argon plasma, the gas temperature in the nitrogen discharge increases almost linearly with the discharge power.

The electron density in nitrogen plasma is about 10% of that in argon plasma. Finally, the electron density of low-pressure ICP was diagnosed both with this method and the Langmuir probe. The results revealed that the electron density increased with power and the peak value reached $\times 10^{17}\text{m}^{-3}$ at the radio frequency power of : Zhijie Song, Haojun Xu, Xiaolong Wei, Zenghui Chen.

Electron‐density and electron energy distribution functions (EEDFs) are measured in a 20‐cm‐diam by 14‐cm‐long cylindrical, inductively coupled plasma source driven by fields from a planar, spiral coil at MHz.

Radio‐frequency (rf) ‐filtered Langmuir probes are used to obtain spatial profiles of electron population characteristics in argon at powers and pressures of.

electron density has been measured successfully in the region of ×10 19 m −3 to ×10 20 m −3 and electron temperature in the region of eV to by: 6. Laser Thomson scattering in a novel, backscattered configuration was employed to measure the electron temperature (Te) and electron density (ne) in argon dc microdischarges, with an interelectrode.

Experimental measurements of the total energy lost per electron–ion pair lost, ε T, were performed in a low-pressure inductive atomic gases (Ar, He) and molecular gases (O 2, N 2) value of ε T was determined from a power balance based on the electropositive global (volume-averaged) model.

A floating harmonic method was employed to measure ion fluxes and electron Cited by: 7. on the electron number density and indicates that the electron density decreases with increase of gas flow rate and filling pressure, which may be explained as follows.

The mechanism of excitation and ionization of atomic and ionic species in argon plasma is supposed to occur mainly by electron Size: KB. Electron energy distribution functions (EEDFs) are measured in an argon inductively coupled plasma. It is observed that the measured EEDFs are nearly Maxwellian distributions, and the electron temperature derived from the EEDFs decreases with increasing input radio-frequency (rf) power.

It appears that additional ionization processes by: The characteristics of pulse modulated inductively coupled plasmas in argon and chlorine have been experimentally investigated.

Measurements were performed for peak rf powers between and W, duty cycles between 10 and 90%, pulse repetition frequencies between 3 and 20 kHz, and argon or chlorine gas. surface wavelength. (~ The electron density of an atmospheric-pressure discharge has been measured using Stark broadening.

(z) The objective of this paper is to summarize and compare measurements of spatially averaged electron densities (Ne) made in argon, microwave-generated ( GHz) surface wave discharges. In this regard, we have performed fully three dimensional (3D), time dependent measurements of the magnetic field, electron density, and electron temperature for an ICP sustained in argon in an industrial reactor designed for plasma etching in microelectronics : Jia Han, Patrick Pribyl, Walter Gekelman, Alex Paterson, Steven J.

Lanham, Chenhui Qu, Mark J. Kushn. A study of the dissociation of a small concentration of cyclic-C4F8(c-C4F8) in a predominately argon, low pressure inductively coupled plasma is reported. Measurements of electron density, plasma potential, and electron energy distribution function.The net electron density in the final system will be the sum of all the singly ionized species (ignoring double ionization for the analyte), thus we can write; - n^ + "^ + "^ (11) ^) (^) (^') where nur, "AT, and n^ are the total concentrations of argon (atom and ion), metal (atom and ion), and excited state argon by: Neutral Argon Plasma: Measurements of Electron Temperature and Density, and of Ion Acoustic Speeds Dean Henze, Janet Chao, Kent Lee, Patrick Smith University of San diego Department of Physics (Dated: Ap ) I.

INTRODUCTION Plasma is the most common phase of matter over the universe since stars are plasmas (they are scarce to usFile Size: 2MB.


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# statsmodels.discrete.count_model.ZeroInflatedGeneralizedPoissonResults.resid¶

method

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# Subrings of Integers are Sets of Integer Multiples/Examples

Let $2 \Z$ be the set of even integers.
Then $\struct {2 \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$.


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SIAM News Blog
SIAM News

University of Minnesota Student Chapter’s Math Modeling Contest

Each fall semester since 2015, the SIAM Student Chapter at the University of Minnesota has held a local mathematical modeling contest. This modeling contest is designed to encourage undergraduate students to get more involved in the math community at the University of Minnesota, raise awareness for the international Mathematical Contest in Modeling, and recruit teams to participate in the international competition.

The Mathematical Contest in Modeling (MCM) is a five-day international competition held in late January or early February, in which teams of undergraduates and high school students use mathematical modeling to solve open-ended, real world problems. Students participate in teams of up to three members, and the competition is conducted remotely. On the first day of the competition, teams select a problem to work on from the MCM website and, over the remainder of the competition, design a model addressing the issues raised in the problem statement. The team’s submission at the end of the competition consists of a solution paper which restates the problem in the students’ own words, explains any assumptions that are made in the solution, describes the model and results, and discusses the strengths and weaknesses of the model in light of any model testing, sensitivity analysis, or error analysis that was conducted. These solution papers are usually upwards of twenty pages in length. The team must also submit a summary of their solution, usually addressed to a technical lay person.

The MCM gives undergraduates and high school students the opportunity to practice and enhance their skills in modeling real life problems using mathematics they have learned in their classes. They get experience solving problems where there is no one “correct” solution with the added bonus that it looks great on a resume. Additionally, the competition shows students that you must be able to work collaboratively and as a team to be successful in mathematics. The team format is one factor that is attributed to the high proportion of women participating in the MCM (42% in 2018) relative to other national and international math competitions.

Compared to the international MCM, the local contest run by the UMN SIAM Student Chapter is a low key affair. Students still compete in teams of up to three members, but the format of the competition is adjusted to better suit the UMN student body and encourage first-time participants. Usually, teams are presented with only one problem on which they can work (the problem given in Fall 2017 appears at the end of this article), and are given about a week to prepare their submission. Because the competition is held at the end of fall semester when UMN students are busy taking midterms and preparing for finals, teams are asked to submit a proposal up to three pages in length about how they would address the problem if it were given in the real MCM competition. They must also submit a one page personal statement about why the team members want to participate in the international contest. Team proposals are reviewed by a panel of student chapter officers and graded on both the mathematical content and writing quality.

Using chapter funds, the top two to three teams from the local competition are sponsored to take part in the international competition. In the roughly two months between the two competitions, chapter officers provide the teams with feedback on how they could improve their work and help the top teams prepare for the international competition. These officers also register the students for the international competition and serve as their advisors.

Chapter officers organize all parts of the local competition, from writing the problem to advertising in addition to the judging and mentoring components outlined above. Officers actively recruit participants by giving short presentations in math, statistics, and science classes where students learn techniques that may make them successful in the international competition. Furthermore, if a student is interested in participating in the local competition but does not have a full group, chapter officers will connect interested parties so that a full group can be formed.

The local competition is a great opportunity for graduate students to gain experience mentoring and for undergraduates to learn about using mathematics outside of a classroom setting. We have found this to be a useful and rewarding event at the University of Minnesota. If you have any further questions about our local competition, we would love to hear from you. Send us an email at [email protected]. Also, be sure to check our website. For more information on the international MCM, please visit their website.

Local MCM Competition Problem Fall 2017: Say It Ain’t Snow

Your local Minneapolis government has caught the Math Bug and realized that their snow plowing strategy is inefficient. They have hired a team of quantitatively inclined people (that’s you!) to come up with a better method. But hurry! Winter is coming! You can operate up to X snow plows at any given time and need to plow part of a city street system for as little money as possible. In the figure are four neighborhoods of Minneapolis, with major roads in bold. It is in the interest of the neighborhood to plow the bold streets first. Plows cost Y dollars per mile plus Z dollars per hour to operate. It takes, on average, W hours to clear a single lane of a cubic kilometer of snow.

Your task, should you choose to accept it, is to come up with and analyze a plowing model for a small region of the city. Your model should answer some, but not all, of the following questions:

• What other variables do you need to take into account to have the optimal strategy?
• When should the plowing start? In particular, what does the start time depend on?
• How do you determine whether or not you need to plow?
• Consider the costs and benefits of salting vs. plowing. Note: it costs $$S$$ dollars per mile ($$S>Y$$) plus $$T$$ dollars per hour ($$T<Z$$) to operate a salt truck, and salting is only effective for small quantities of snow.
• If you make a mistake in the determination on whether or not to plow, what are the costs? What do these costs depend on? Are you more worried about sending out plows too often or not often enough?
• How do your strategies change as you get close to going over budget?
• How does your model scale to the entire city?
• How should you balance the safety of plow and salt truck operators vs. the need to plow the city, in particular for emergency vehicles?
• Explicitly address and justify any assumptions, approximations, or simplifications you make in your model and analysis.

Know of any high school students interested a math modeling competition? Check out the SIAM-organized MathWorks Math Modeling Challenge.

Alice Nadeau, Harini Chandramouli, and Sarah Milstein are math graduate students at the University of Minnesota. Sarah is the current president of the SIAM student chapter, and Alice and Harini have both been presidents in the past. They study a wide variety of mathematics from dynamical systems to algebraic topology.


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# How to convert specific humidity ERA5 in kg/kg to g/kg or mm/day?

The specific humidity of ERA5 data is kg/kg, but some articles work with this variable in g/kg. How can I convert this?

Or to compare with precipitation, evaporation, and other variables, how to convert to mm/day?

It is possible?

Thank you!

• Yes, a dimensional analysis shows that relative humidity is dimensionless, while the dimension of precipitation is LT$^{-1}$. Sep 13, 2022 at 15:51


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# pointy problem

Geometry Level 4

There are 20 points inside a square. They are connected by non intersecting segments with each other and with the vertices of the square ,in such a way that the square is dissected into triangles. How many triangles do we have?

×


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On the Free Energy of a Directed Polymer in a Brownian Environment

#### J. Moriarty, N. O'Connell

2007, v.13, Issue 2, 251-266

ABSTRACT

We prove a formula conjectured in [N. O'Connell and M. Yor, Brownian analogues of Burke's theorem, Stoch. Process. Appl., 2001, v. 96, N 2, 285-304] for the free energy density of a directed polymer in a Brownian environment in $1+1$ dimensions.

Keywords: directed polymers,exponential functionals of Brownianmotion,quasi-reversibility,Brownian queues,large deviations


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Main Content

# canoncorr

Canonical correlation

## Description

[A,B] = canoncorr(X,Y) computes the sample canonical coefficients for the data matrices X and Y.

[A,B,r] = canoncorr(X,Y) also returns r, a vector of the sample canonical correlations.

example

[A,B,r,U,V] = canoncorr(X,Y) also returns U and V, matrices of the canonical scores for X and Y, respectively.

[A,B,r,U,V,stats] = canoncorr(X,Y) also returns stats, a structure containing information related to testing the sequence of hypotheses that the remaining correlations are all zero.

## Examples

collapse all

Perform canonical correlation analysis for a sample data set.

The data set carbig contains measurements for 406 cars from the years 1970 to 1982.

Load the sample data.

load carbig;
data = [Displacement Horsepower Weight Acceleration MPG];

Define X as the matrix of displacement, horsepower, and weight observations, and Y as the matrix of acceleration and MPG observations. Omit rows with insufficient data.

nans = sum(isnan(data),2) > 0;
X = data(~nans,1:3);
Y = data(~nans,4:5);

Compute the sample canonical correlation.

[A,B,r,U,V] = canoncorr(X,Y);

View the output of A to determine the linear combinations of displacement, horsepower, and weight that make up the canonical variables of X.

A
A = 3×2

0.0025    0.0048
0.0202    0.0409
-0.0000   -0.0027

A(3,1) is displayed as —0.000 because it is very small. Display A(3,1) separately.

A(3,1)
ans = -2.4737e-05

The first canonical variable of X is u1 = 0.0025*Disp + 0.0202*HP — 0.000025*Wgt.

The second canonical variable of X is u2 = 0.0048*Disp + 0.0409*HP — 0.0027*Wgt.

View the output of B to determine the linear combinations of acceleration and MPG that make up the canonical variables of Y.

B
B = 2×2

-0.1666   -0.3637
-0.0916    0.1078

The first canonical variable of Y is v1 = 0.1666*Accel — 0.0916*MPG.

The second canonical variable of Y is v2 = —0.3637*Accel + 0.1078*MPG.

Plot the scores of the canonical variables of X and Y against each other.

t = tiledlayout(2,2);
title(t,'Canonical Scores of X vs Canonical Scores of Y')
xlabel(t,'Canonical Variables of X')
ylabel(t,'Canonical Variables of Y')
t.TileSpacing = 'compact';

nexttile
plot(U(:,1),V(:,1),'.')
xlabel('u1')
ylabel('v1')

nexttile
plot(U(:,2),V(:,1),'.')
xlabel('u2')
ylabel('v1')

nexttile
plot(U(:,1),V(:,2),'.')
xlabel('u1')
ylabel('v2')

nexttile
plot(U(:,2),V(:,2),'.')
xlabel('u2')
ylabel('v2')

The pairs of canonical variables $\left\{{u}_{i},{v}_{i}\right\}$ are ordered from the strongest to weakest correlation, with all other pairs independent.

Return the correlation coefficient of the variables u1 and v1.

r(1)
ans = 0.8782

## Input Arguments

collapse all

Input matrix, specified as an n-by-d1 matrix. The rows of X correspond to observations, and the columns correspond to variables.

Data Types: single | double

Input matrix, specified as an n-by-d2 matrix where X is an n-by-d1 matrix. The rows of Y correspond to observations, and the columns correspond to variables.

Data Types: single | double

## Output Arguments

collapse all

Sample canonical coefficients for the variables in X, returned as a d1-by-d matrix, where d = min(rank(X),rank(Y)).

The jth column of A contains the linear combination of variables that makes up the jth canonical variable for X.

If X is less than full rank, canoncorr gives a warning and returns zeros in the rows of A corresponding to dependent columns of X.

Sample canonical coefficients for the variables in Y, returned as a d2-by-d matrix, where d = min(rank(X),rank(Y)).

The jth column of B contains the linear combination of variables that makes up the jth canonical variable for Y.

If Y is less than full rank, canoncorr gives a warning and returns zeros in the rows of B corresponding to dependent columns of Y.

Sample canonical correlations, returned as a 1-by-d vector, where d = min(rank(X),rank(Y)).

The jth element of r is the correlation between the jth columns of U and V.

Canonical scores for the variables in X, returned as an n-by-d matrix, where X is an n-by-d1 matrix and d = min(rank(X),rank(Y)).

Canonical scores for the variables in Y, returned as an n-by-d matrix, where Y is an n-by-d2 matrix and d = min(rank(X),rank(Y)).

Hypothesis test information, returned as a structure. This information relates to the sequence of d null hypotheses ${H}_{0}^{\left(k\right)}$ that the (k+1)st through dth correlations are all zero for k=1,…,d-1, and d = min(rank(X),rank(Y)).

The fields of stats are 1-by-d vectors with elements corresponding to the values of k.

FieldDescription
Wilks

Wilks' lambda (likelihood ratio) statistic

df1

Degrees of freedom for the chi-squared statistic, and the numerator degrees of freedom for the F statistic

df2

Denominator degrees of freedom for the F statistic

F

Rao's approximate F statistic for ${H}_{0}^{\left(k\right)}$

pF

Right-tail significance level for F

chisq

Bartlett's approximate chi-squared statistic for ${H}_{0}^{\left(k\right)}$ with Lawley's modification

pChisq

Right-tail significance level for chisq

stats has two other fields (dfe and p), which are equal to df1 and pChisq, respectively, and exist for historical reasons.

Data Types: struct

## More About

collapse all

### Canonical Correlation Analysis

The canonical scores of the data matrices X and Y are defined as

$\begin{array}{c}{U}_{i}=X{a}_{i}\\ {V}_{i}=Y{b}_{i}\end{array}$

where ai and bi maximize the Pearson correlation coefficient ρ(Ui,Vi) subject to being uncorrelated to all previous canonical scores and scaled so that Ui and Vi have zero mean and unit variance.

The canonical coefficients of X and Y are the matrices A and B with columns ai and bi, respectively.

The canonical variables of X and Y are the linear combinations of the columns of X and Y given by the canonical coefficients in A and B respectively.

The canonical correlations are the values ρ(Ui,Vi) measuring the correlation of each pair of canonical variables of X and Y.

## Algorithms

canoncorr computes A, B, and r using qr and svd. canoncorr computes U and V as U = (X—mean(X))*A and V = (Y—mean(Y))*B.

## References

[1] Krzanowski, W. J. Principles of Multivariate Analysis: A User's Perspective. New York: Oxford University Press, 1988.

[2] Seber, G. A. F. Multivariate Observations. Hoboken, NJ: John Wiley & Sons, Inc., 1984.

## See Also

Introduced before R2006a

Download ebook


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A rectangular box is 10 inches wide, 10 inches long, and 5 inches high

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A rectangular box is 10 inches wide, 10 inches long, and 5 inches high [#permalink]

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06 Dec 2017, 03:21
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A rectangular box is 10 inches wide, 10 inches long, and 5 inches high. What is the greatest possible (straight-line) distance, in inches, between any two points on the box?

(A) 15
(B) 20
(C) 25
(D) 10√2
(E) 10√3
[Reveal] Spoiler: OA

_________________

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Re: A rectangular box is 10 inches wide, 10 inches long, and 5 inches high [#permalink]

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06 Dec 2017, 03:37
We can use super Pythagoras theorem here:

l^2+b^2+w^2
100+100+25
225
√225

A. 15

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Re: A rectangular box is 10 inches wide, 10 inches long, and 5 inches high [#permalink]

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06 Dec 2017, 03:47
Bunuel wrote:
A rectangular box is 10 inches wide, 10 inches long, and 5 inches high. What is the greatest possible (straight-line) distance, in inches, between any two points on the box?

(A) 15
(B) 20
(C) 25
(D) 10√2
(E) 10√3

$$Diagnoal of Cube = \sqrt{l^2 + b^2 + h^2}$$
$$\sqrt{100 + 100 + 25}$$
15
A
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Re: A rectangular box is 10 inches wide, 10 inches long, and 5 inches high [#permalink]

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12 Dec 2017, 14:36
What level question would this be?

Kudos [?]: 8 [0], given: 830

Re: A rectangular box is 10 inches wide, 10 inches long, and 5 inches high   [#permalink] 12 Dec 2017, 14:36
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text:
## Wednesday, January 28, 2015

### Grading Stories: "Cheese Weight" and Thusforthwith

One thing I love about the internet is being able to share stories and moments from everyday life. Here are a couple about something I'm sure other academics will be able to relate to: grading stories.

Cheese Weight

My alma mater enforced mathematical writing guidelines and the use of $\LaTeX$ very strongly. Yet some people, notably non-majors, chose to ignore those guidelines completely and complain when points were taken off for writing. Some people handed in scratch work done in pen on graph paper in consistently gigantic writing. Some people *coughEigenpetercough* printed out the questions in $\LaTeX$... two problems to one page, in landscape form... then did them out by hand in tiny writing. Some people *coughalsoEigenpetercough* did the homework in $\LaTeX$ but omitted large amounts of information to fit every proof-based problem on one side of one page.

Then there are the people with just plain bad handwriting. While grading with a friend, I encountered a homework that exemplified this while grading a core class; apparently, one of the people in the class was secretly a chicken tied to a Ouija board. Here's how it went down.

Me: Hey, do you have any idea what these two words are?

Friend: ..........

Me: It looks like it says "cheese weight".

Friend: It does, but that doesn't have anything to do with the problem.

Me: Can you tell from context?

Friend: .... no.... (to another person) Hey, do you know what this says?

Someone else: ..... looks like "cheese weight"?

Yet another person: I have no idea.

Me: Well, "cheese weight" it is then.

And that's how someone got their work back with "what's a cheese weight?" written as a comment.

Runner-up for best handwriting-related mishap goes to the person who tried to write "I used Professor X's code," but botched the last two letters in "code" in a way that evoked, erm, Little Professor X.

---------------------------------------

Thusforthwith!

As a fan of both analysis and silly things, I can't help but enjoy when they're combined. This story is about a friend who perfected this combination.

My friend, at the time, was taking the same real analysis course I was grading, so I mentioned to him how funny it was when people used archaic connecting words: "thusly", "wither" and the like. From there we started trying to come up with the most ridiculous word. Thenceforth! Thuswith! Whencehence!

So of course every homework I got from this friend had at least one made-up connecting word (despite being typed up quite nicely). This continued without incident, until one day:

Me: This is hilarious! I'm worried about you slipping up and doing it on the test, though.

Him: Why not?

Me: Well... the professor might notice, and you might get docked some points...

Him: Hmm...

Which obviously culminated in him PUTTING FAKE WORDS ON THE ANALYSIS TEST.

And guess what?

THE PROFESSOR DIDN'T NOTICE.

Jesus.

------------------------------------------

Readers, do you have any grading stories? Let me know if anyone tries to pull off using fake connecting words---not everyone may be as lucky!

## Monday, January 26, 2015

### Adventures in Linear Algebra with the Prismatoy

As the nature of the first few posts here should somewhat suggest, my fiancé and I spend a whole lot of time talking to each other about math. He needs a nom. Let's call him Eigenpeter.

The latest installment of "Peter finds an interesting idea, spends 1 hour worth of whiteboard lecture on representation theory to his algebra-phobic lover and makes a Mathematica toy in 15 minutes" is brought to you by Prismatoy, a cube that can be collapsed into a parallelpiped:

Basically, we wandered into a puzzle store where he picked one of these up and didn't put it down. (We did pay before leaving!)

I like this because, when restricted to any one of the 6 faces, it gives a visualization of the linear transformation
$$\left[\begin{array}{cc}1 & \cos\theta\\ 0 & \sin\theta\end{array}\right]$$
(up to transformations, scaling and unitary operations) with $0<\theta\leq \frac{\tau}{4}$* being the acute angle in the final configuration. You could derive this quickly at home by imagining one face as a unit square, then exploiting some basic trig to find that the transformation maps (0,1) to ($\cos\theta$,$\sin\theta$), (1,1) to ($1+\cos\theta$, $\sin\theta$), and leaves the bottom side of the square unchanged. The above then follows from knowing how the transformation acts on the standard basis vectors for $\mathbb{R}^2$.

Peter noted that the volume of this structure is given by the area of the base times the height, which, in this case, is the determinant of the linear transformation that takes it from cube form to its parallelpiped shape. To demonstrate this, we name the three vectors along the given three sides $\vec{a}$, $\vec{b}$ and $\vec{c}$:

The area of the base is given by $\lvert\vec{a} \times \vec{b}\rvert$---one can see this because
$$\lvert\vec{a}\times\vec{b}\rvert=\vec{a}\vec{b}\sin\theta,$$
which corresponds exactly to the area in the first picture, except the vectors are no longer of unit length. Now recall that $\vec{a}\times\vec{b}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$. Taking the dot product $(\vec{a} \times \vec{b})\cdot \vec{c}$ only takes into account the component of $\vec{c}$ that is parallel to $\vec{a}\times\vec{b}$---in other words, perpendicular to both $\vec{a}$ and $\vec{b}$---in other words, the height of the parallelpiped! Hence taking the magnitude of this quantity gives us base times height, which is volume.

But wait, there's more! The quantity $\left\lvert(\vec{a} \times \vec{b})\cdot \vec{c}\right\rvert$ can be written as
$$(\vec{a} \times \vec{b})\cdot \vec{c}=\sum_{i=1}^3\left(\sum_{j=1}^3\sum_{k=1}^3 \epsilon_{ijk}a_jb_k\right)c_i$$
where (in case the reader hasn't seen it before) the Levi-Civita symbol $\epsilon_{ijk}$ essentially acts as the 'opposite' of the Kroenecker $\delta$ function, i.e.
$$\epsilon_{ijk}=\left\{\begin{array}{ll}1 & i=1, j=2, k=3;\ i=3, j=1, k=2;\ i=2, j=3, k=1\\ 0 & i=k=j\\ -1 & \textrm{else}\end{array}\right..$$
Now imagine taking the determinant of the matrix
$$\left[\begin{array}{ccc} \lvert & \lvert & \lvert\\ \vec{c} & \vec{a} & \vec{b}\\ \lvert & \lvert & \lvert\end{array}\right].$$
I won't put the algebra all out here, but calculating the determinant according to the definition and rearranging it will give the previous nested sum. This technique can also be used to prove that
$$(\vec{a} \times \vec{b})\cdot \vec{c}=(\vec{b} \times \vec{c})\cdot \vec{a}=(\vec{c} \times \vec{a})\cdot \vec{b}.$$
Yay! It is now evident that
$$V_{ppiped} = \left\lvert(\vec{a} \times \vec{b})\cdot \vec{c}\right\rvert = \textrm{det}[\vec{c}\ \vec{a}\ \vec{b}].$$
More generally, the determinant of a matrix is a factor indicating what change in volume (or area, or the appropriate dimensional quality) it produces.

EPILOGUE: He spent most of Saturday trying to sketch the manifold of possible shape configurations of this object, then trying to determine whether the set of operations on the toy in SL(3) was a group, then made several demonstrations of these and similar phenomena in Mathematica. However, he did not do the dishes. I cut the lecture for brevity.

*YES, TAU.

## Wednesday, January 21, 2015

### 5 New High-Level Math Jokes

We here at the Analysisters enjoy puns, and are happy to contribute to the already-gigantic list of math jokes every once in a while. Here are some we've come up with over the years!

1. A topologist walks into $\bar A$. It's closed.

2. (3 variants) Q: If chocolate is a Hilbert space and peanut butter is its dual, why can every element in peanut butter be written as an inner product $\langle y,x \rangle$, where $y$, $x$ are chocolates and $x$ is uniquely fixed?
A: Reese's Representation Theorem.

Q: Why is peanut butter the adjoint of chocolate? Why is chocolate the adjoint of peanut butter?
A: Reese's Representation Theorem.
(Thanks, Paul!)

Q: What's a mathematician's favorite candy?
A: Riesz's Pieces.

3. What did the analyst have for dinner?
Limsup.

4. Your mama isn't Lebesgue integrable because she doesn't vanish at infinity!

5. Eight mathematicians walk into a diner. The first one says they're not hungry, and orders nothing. The next three order a beer, a hamburger, and french fries, respectively. The next three order a burger and fries, fries and a beer, and a burger and a beer, respectively. The last one orders a burger, fries, and a beer.

"I'm sorry, I can't fill your order," she says.

"Why is that?"

"This is only a partial order."

BONUS: Not necessarily in joke format, but I refuse to refer to
$$F_n = \frac{1}{\sqrt{5}}(\phi^n - \bar{\phi}^{n})$$
as anything except "that formula that shoots water up your butt".

## Monday, January 19, 2015

### MIT Mystery Hunt 2015: Let's Hear it for Random Hall!

(Note: Links may temporarily not work without a login. I will update them as the situation changes.)

To puzzles.

He wakes up in the morning and does five crosswords, usually in five minutes each; is never caught without a copy of The Enigma; plays puzzle games when he comes home; and, over dinner, tells me about the latest puzzles he solved. He also participates in a minimum of 10 puzzlehunts a year with his MIT puzzlehunting buddies. I generally stay out of his way for the online ones, but there's no physically avoiding the MIT Mystery Hunt. So off I was dragged.

Let me admit: I was expecting to play a supporting role, making sure my dude gets enough food and sleep. I was not expecting to personally crack several puzzles, including 2 metas. Guess what happened.

As everyone on my team agreed, this year's puzzlehunt was particularly well-constructed! This year's theme was 20,000 Leagues Under the Sea and was presented by One Fish, Two Fish, Random Fish, Blue Fish. The short format of many of the puzzles made it possible for individuals to solve puzzles alone, which is a great confidence boost for newer puzzlers such as myself; and the way in which the puzzles were released (solving puzzles gave more Deep, which revealed puzzles hiding in the ocean) made it possible to work around roadblocks, eliminating the frustration of being stuck on everything. Feeling Bluefin was probably the team favorite, and a lot of people liked Nautilus's Duplicated Quest as well. I could not believe there was a Dresden Codak puzzle* released so early---we love Aaron Diaz here! Mad props!

There are always a couple puzzles that require special knowledge, which is great if you spent the last 5 years listening to showtunes instead of doing puzzles. I definitely enjoyed the auditory (directly or otherwise) puzzles such as the theater one*, Nina and Topsy-Turvy. Someone in Random Hall has pipes! The best 'esoteric knowledge' moment came when one of the puzzles required reading a diving chart*---apparently one of the members of our team was an experienced diver all along.

My fiancé, being a long-time language puzzle master, enjoyed everything from Flat Containers to The Curse of The Atlantean's Tomb, both of which he made me help with. Representative Characters (math!) was also a nice surprise (math!) because it required understanding his field (math!) in order to solve. Although our team concentrated on solving earlier metas rather than the Atlantean puzzles towards the end, he also glanced at Practice in Theory (physics!) and enthused at me about it for an hour.

Other than Feeling Bluefin, the puzzles with the cutest premises were Follow the Bees!, Montages and MIT Mystery Hunt. So adorable.

Our team enjoyed all of the physical puzzles (even the meta!). We had people solving cubes, picking locks, cracking the gelt puzzle so we could eat it; I decoded the knitted square, and one girl with INFINITE PATIENCE sat on the floor for hours putting together the paper jigsaw that had scared off everyone else. Seriously, I was amazed by her persistence.

This was a very different experience from last year's hunt: for me, personally, most of the change in experience quality was due to being on a smaller team. Fewer people means having more opportunity to get an 'aha!' moment and a higher fun-to-automation ratio. The large number of easy puzzles (School of Fish round) also made the hunt continuously accessible to new puzzlers, but I found myself avoiding them in preference of harder puzzles as time went on.

It didn't hurt that everyone on our team was awesome and cracked jokes the entire time! Even my fiancé, who is normally reserved and academic, was dropping the sass left and right. Highlights include giving Ariel a bottle of hair dye when she asked for us to give her a soul (she's a ginger) and the phrase 'Chocolate Rain' being used to describe makin' it rain with gelt. Nevertheless, Puzzfeed has outdone us all.

If you're new to the puzzle scene, get on a smaller team (10-30 people) with some experienced puzzlers and some MIT/Boston residents. You can also look at any of the puzzles listed above for at least an entire year (and Random Fish plans to publish a fancy book of the School of Fish puzzles.) I'm not even going to preface that with 'if you want to'. GET ON A TEAM. IT IS FUN.

Now for some flats before bed...

## Wednesday, January 14, 2015

### Why Learning Math is Important (General)

A common question I get from tutees is, why learn math at all? Many others on the internet have provided satisfactory answers to this question, so I'll try my best to come up with a couple new points.

(A side note to those who are highly educated in humanities, social sciences, etc.: I am not claiming that every one of these benefits is unique to mathematical learning. Those disciplines are useful for critical thinking as well!)

Problem-solving practice

If you don't go into a technical career, the odds that you'll have to use algebra or calculus every day are slim. However, no matter what you do in life, you will have to know how to effectively solve problems. It's unfortunately difficult to practice and develop good problem-solving skills on their own---that often comes with experience---but doing relatively simple math problems is a good substitute.

How would that work? When you start a new videogame, the game doesn't immediately drop you into the final boss fight; you start by doing the tutorial instead. This is what schools are hoping to accomplish by giving you simple problems to solve: they may be in a weird format, and you may not be sure how they connect with day-to-day life, but you're being given them because these number problems are some of the simplest problems that are possible to solve. Furthermore, it's not a bad thing that algebra problems are disconnected from your real life---if they were, then there would be much greater punishments for failure to solve them. I'm certain you'll agree a broken friendship or broken arm is much worse than losing a couple of test points!

To summarize: think of grade school math problems as the tutorial level to real-life problems, and their disconnect from regular life as a protective safety net.

Increased ability to communicate abstract ideas

Math can be seen as not only a tool, or a scientific discipline, but also a language. Understanding and communicating mathematical ideas requires a set of symbols and vocabulary that people wouldn't learn just by going through life. Ideas related to math do pop up from time to time, and it feels great when you know what the answer is, and how to explain it!

Here's an example: let's say you and some friends are trying to get to a frozen yogurt place a block away. We call your current location A and the location of the frozen yogurt place B. Also, let's call the frozen yogurt place Froyomorphism for the sake of puns. Your friends, whose favorite colors are purple, green and blue, suggest the following three paths on the map:

Now you are asked to choose the best path. Because you are good at judging distances, you know that the purple and green paths are the same length, but the blue path is much longer. Could you communicate this concept to your friends without using math? Without using the word 'sum' or 'length' or evoking a visual proof? Probably, but it would be much harder. The point is: even if you're right, you may end up taking the blue path if no one can explain to Blue why the other two paths are shorter.

If two people have a shared vocabulary that can be used to talk about abstract objects, they can exchange information about what essentially amounts to different lines of thought. This is how people get smarter and better at problem solving.

Protection against being exploited

Most people think they are smart. However, as I'm sure you've figured out by now, not everyone is. Several people/institutions/etc. have realized this and use people's lack of mathematical awareness to make a living. Gambling is a classic example: also see the Monty Hall Problem or Bertrand's Box Paradox for situations where common sense can be deceiving.

However, not only can ill-meaning people use your unwillingness to think about mathematics (and academic prospects in general) to separate you from your money, they can twist information to separate you from your ideals and beliefs as well. Most people like the idea of experiments being able to prove, disprove, support, or refute ideas, but don't want to dig through heavily written academic papers to find the point. This is where exploitative people come in. If they can bank on the audience being too busy or unable to read the source material, they can make their audience believe whatever they want---even if it comes at the expense of the audience! See Flaws and Fallacies in Statistical Thinking for tons of real-world examples; Stephen Campbell explains this better than a blog post ever could.

The only way to protect yourself against this is to be able to read and analyze scientific papers without needing someone to tell you what you mean. In many cases, this requires some knowledge of statistics (math), experimentalism versus mathematical modeling and the implications, or what conclusions can be drawn from the data presented (logic, which is part of math).

Not looking like a tool on the internet

If you've spent any amount of time on the internet, at all, you may have come across someone who is angry at their opponents for not understanding "logic" and "reason." You may have seen someone make a statement along the lines of "that doesn't make any logical sense" without noting what the error is (or invoking a fallacy incorrectly). You may have seen someone who is incapable of understanding that a smart person may disagree with them, and who concludes that if someone disagrees with them, that person is stupid.

Judging by the relatively low proportion of people with bachelor's degrees in mathematics or philosophy, it stands to reason that very few of these people have had real training in formal logic. Someone with completely illogical arguments would have no way of knowing so (i.e., a special case of the Dunning-Kruger Effect.) On the other hand, someone who has studied higher-level mathematics can recognize what is and is not logically consistent, which affects how they act in everyday life as well. This makes everyday life a lot---a lot!---easier. (I could go on for days about this; but that's a story for another blog post... or twelve.)

Yet embarrassing oneself is often caused by a lack of empathy---what about that? Math has no relation to that, unfortunately. [Sad face.]

Concluding remarks

Oh, and talking about math is awesome, and we have the best jokes.

(Check back in the future for specific examples of how some topics you may have seen are actually used by mathematicians and scientists!)

## Monday, January 12, 2015

### Vector Calculus... with Poles?!

A couple days ago, my partner and I were about to go to sleep, when I wondered out loud whether Stokes' Theorem and the Divergence Theorem would hold for functions that were analytic except at finitely many poles (it's what engaged couples do in bed!).

Since my fiancé is a physicist, he already knew the answer for the Divergence Theorem, and was happy to clue me in: for $f(r, \theta,\phi)$ in polar coordinates,
$$\iiint_\Omega \nabla \cdot f(r,\theta,\phi) \; d\Omega=\oint_{\partial \Omega} f(r,\theta,\phi)\;d\partial\Omega.$$
appears to break down when a pole occurs in the interior of $\Omega$. In order to demonstrate this, we consider the function $f(r,\theta,\phi)=\frac{1}{r^2}$, which has the classical divergence (in polar coordinates)
$$\nabla \cdot \frac{1}{r^2} = \frac{\partial }{\partial r}\frac{r^2}{r^2}=0,$$
forcing the left-hand side to be zero. Yet, because $\frac{1}{r^2}$ is constant on the surface of the sphere, the right-hand side evaluates to
$$\oint_{\partial \Omega}\frac{1}{r^2}\hat{r}\cdot\hat{r}\;dA=\frac{1}{r^2}\oint_{\partial \Omega}dA=4\pi,$$
which is nonzero! Hence the Divergence Theorem does not hold in this case... for classical forms of divergence.

Knowing my interest in analysis (see blog title, above), my fiancé clarified that when a function has poles, we redefine divergence in the sense of distributions so that Green's Theorem does hold. Apparently then
$$\nabla \cdot \frac{1}{r^2} = 4\pi\delta(r)$$
which I plan to check rigorously in the future (no money for books). It hadn't occurred to me yet that that distributions could also be used to generalize multivariate forms of the derivative, so this was an interesting way for the conversation to go.

We quickly noted that for Stoke's Theorem in 2D, with $f(z)=p(z)+iq(z)$ and $z=x+iy$,
$$\int\int_\Omega \frac{\partial p}{\partial x} - \frac{\partial q}{\partial y} \;d\Omega=\int_{\partial \Omega} p\; dx - q\;dy$$
for the real part and
$$\int\int_\Omega \frac{\partial p}{\partial y} + \frac{\partial q}{\partial x} \;d\Omega=\int_{\partial \Omega} p\; dy + q\;dx$$
for the imaginary part. Assuming that $f(z)$ is analytic leads to one of the proofs of Cauchy's Integral Theorem, but, as differentiability is a condition for Stokes' Theorem, it is expected to break down under classical conditions for a complex function with finite poles. However, given my partner's earlier insight with the Divergence Theorem, I wouldn't be surprised if a distributional equivalent existed for Stokes' Theorem as well.

I'm certain there are a few texts that could clear up exactly how this happens rigorously... sounds like something fun to do in the future!

(Please excuse my slightly incorrect use of notation. Some symbols are not supported in MathJax.)

## Wednesday, January 7, 2015

### Funny Papers: Overly Expressive Lab Mouse

Scientists are people too, and they have a sense of humor. We don't claim to be a research humor blog (that's what Annals of Improbable Research is for.) However, sometimes we come across something that is so hilarious it has to be shared.

In this case, it's a silly figure in a medical research paper. I doubt the legality of posting an image from someone else's protected academic work, so here's the source (Figure 2) and a short description of what's happening in the figure:

• The first image in the chain is a newly infected lab mouse with a darling smile on its little face.
• Endotoxins produce IL-12, TNF and IFN-gamma in the mouse, contributing to shock. The mouse's smile has been reverted; it is now unhappy at its predicament.
• The toxic shock leads to weight loss. Here, weight loss is manifested as someone copy-pasting an image of the sad mouse and shrinking it a bit.
• The poor mouse dies. The authors have helpfully put 'x's in the eyes and put the mouse's little feet up to communicate to the reader that the mouse is, indeed, dead.
A shout-out to C.A. Biron and R.T. Gazzinelli for demonstrating that, while mathematicians have the best jokes, experimental biologists have us completely beat on black humor!

## Monday, January 5, 2015

### Math Test-taking Strategy

Math testing strategy varies a bit from the strategies that would normally be useful for classes requiring a lot of rote memorization. This is mostly because, in addition requiring a student to interpret and regurgitate information beforehand, math tests also involve a performance element that tests short-term problem solving ability. You may recognize this as an important skill to have! Before we list some specific ways to prepare, keep in mind that:
• Mathematical ability can be improved. Some people have a tendency to give up on math if they aren't good at it immediately. However, if you spend more time on math than your classmates; whether it is going to math camp, working with a private tutor, thinking about outside problems, or figuring out how things work; you will get better faster. Nobody comes out of the womb being able to solve every type of math problem. It's like playing a musical instrument or learning to dance: the more practice, the better.
• Getting an A in most classes requires a level of understanding that is not taught in the class. In American schools, an A grade is meant to signify that a student is going above and beyond what is required, even if it looks like all of the testing material is being taught in the class. There is often far more to the material! For example, a lot of students understand the general material being taught, but get slammed on small mistakes such as minus sign errors. Catching and being aware of these errors is something that the student must develop on their own, and it is hard to explicitly teach. Small things like this are often the difference between an A and a C.
Got that? Good! If you're studying for a test and aren't quite sure what to start with, there are several 'levels' of understanding the material, which, for your purposes, we'll express in three separate categories:
• Basic understanding. You're at this level if you can read and understand everything in the textbook chapter, and know how to do the problems that aren't word problems.
• Familiarity. You're at this level if you understand what the answers should 'look' like, why the answers 'look' that way, and how to fix something if you've made a mistake. To get to this level, you have to be observant and look for patterns in the work you're doing. Knowing where mistakes can arise in a certain type of problem is very powerful!
• Creative application. If basic understanding is like knowing how to get home, and familiarity is knowing how to get home even if you've taken a wrong turn, creative application is like getting home by parkour. You're at this level if you understand the technique so well that even when it's not mentioned, you know when it has to be used. This type of understanding is the most important for word problems.
Let's use (scalar) multiplication as an example. Someone would have basic understanding if they could multiply 45x25 on paper (or in their head! Can you?). They're familiar with how multiplication works if they can explain why the answer cannot be 725, or 329670, or 2621. Lastly, they will have mastered multiplication as a concept if they can use it to solve problems such as 'how much do 45 vending-machine gumballs cost'? or use multiplicative identities to prove the exponent rules.

So, in a nutshell, if you understand the book and can do all the problems, you're still not guaranteed to do well on tests. Nooo! How can this be?

The primary problem is that very few books go into detail on how things work and how to recognize mistakes. Yet recognizing where mistakes happen and how to fix them is vital---not to mention that knowing how methods work is the only way to understand word problems! Here are a few helpful things you can do to improve your math testing ability:
• Know exactly where your abilities are for each type of problem. Can you do mental multiplication? Can you do basic algebra problems? Can you prove that L^2 is complete? Being honest about where your weaknesses are makes it much easier to conquer them. Needing more practice on a specific type of problem isn't a bad thing, and you'll save time if you focus on only the hardest problems.
• Develop 'sanity checks'. 'Sanity checks' are what I call pieces of information you remember to check whether you've made a mistake in the problem. Using the multiplication example above, a good example of a sanity check would be 'an odd number times an odd number cannot be an even number.' This helps build your familiarity with the material and could save you from losing tons of points.
• Build the test. Even if you don't know which specific problems will be on the test, you can generally work out how many of each type of problem will be on the test. This will help you direct your attention towards whatever will win you the most points. For example, you might be okay with everything on the test except one very, very hard type of problem: would it be better to focus on figuring out the hard problems, or making sure you don't mess up on the moderate problems? This depends on how many of the hard problems will be on the test.
• Talk with friends. We don't mean about videogames. There aren't a lot of ways to develop creative thinking for mathematics other than 'think about math a lot and try to come up with and solve math problems outside of school', but this is one of the more fun ones. Your friends may have some insight about math that hasn't occurred to you yet, or you may be able to solve a hard problem by working together. In any case, talking directly to people who know more than you will teach you a lot.
• Look for patterns. If you don't want to talk to your friends about math, or have some pride about developing things on your own, remember that math is all about finding and exploiting patterns. Once you've found a pattern, try to figure out where it comes from. This line of thinking often leads to developing newer, faster ways of solving problems. See our mental math post for some simple examples of pattern exploitation.
• Hire a private tutor. We said earlier that talking to people who know more than you will improve your skills very quickly: well, private tutors know a lot of math and they can teach you a lot about math. There's no shame in needing one! If someone is very good at the guitar and wants to become even better, they hire a private guitar teacher. The same is true for math. This is how you should see a private math tutor: someone who can help you improve very, very quickly and understand mathematics far beyond what you are taught in class.
These are all things you would do before the test happens (we've left out the obvious "read the textbook and do the problems" advice.) While the test is happening, try to:
• Read the test beforehand and do the easiest problems first.
• Temporarily improve performance by stretching, chewing gum, drinking caffeine, or listening to energetic music.
Now you should be focused and in the moment!

Lastly: even if progress seems slow sometimes, or you're having trouble catching up, don't feel hopeless. If you spend time thinking about how the methods work, you will improve.

Good luck, and happy testing!

### Introduction

Good day, internet!

We are (currently) two flat broke prospective scientists: one future physicist with no money and one mathematician with no money, who thought it might be a good idea to start a blog in order to spread the love. Here you'll find insight gained from teaching, stories about research, discussion of odd problems, jokes about LaTeX, and possibly an informal paper review once in a while.

We had intended on starting a free tutoring video series, but had to quit due to severe technical difficulties (see 'broke,' above). Boo! Perhaps that's what the future holds.


text:
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_robust_corr_estim (g02hkc)

## 1  Purpose

nag_robust_corr_estim (g02hkc) computes a robust estimate of the covariance matrix for an expected fraction of gross errors.

## 2  Specification

 #include #include
 void nag_robust_corr_estim (Integer n, Integer m, const double x[], Integer tdx, double eps, double cov[], double theta[], Integer max_iter, Integer print_iter, const char *outfile, double tol, Integer *iter, NagError *fail)

## 3  Description

For a set $n$ observations on $m$ variables in a matrix $X$, a robust estimate of the covariance matrix, $C$, and a robust estimate of location, $\theta$, are given by:
 $C = τ 2 AT A -1$
where ${\tau }^{2}$ is a correction factor and $A$ is a lower triangular matrix found as the solution to the following equations.
 $z i = A x i - θ$
 $1 n ∑ i=1 n w z i 2 z i = 0$
and
 $1 n ∑ i=1 n u z i 2 z i ziT - I = 0 ,$
 where ${x}_{i}$ is a vector of length $m$ containing the elements of the $i$th row of X, ${z}_{i}$ is a vector of length $m$, $I$ is the identity matrix and 0 is the zero matrix, and $w$ and $u$ are suitable functions.
nag_robust_corr_estim (g02hkc) uses weight functions:
 $u t = a u t 2 , if  t < a u 2 u t = 1 , if  a u 2 ≤ t ≤ b u 2 u t = b u t 2 , if  t > b u 2$
and
 $w t = 1 , if  t ≤ c w w t = c w t , if  t > c w$
for constants ${a}_{u}$, ${b}_{u}$ and ${c}_{w}$.
These functions solve a minimax problem considered by Huber (1981).
The values of ${a}_{u}$, ${b}_{u}$ and ${c}_{w}$ are calculated from the expected fraction of gross errors, $\epsilon$ (see Huber (1981) and Marazzi (1987)). The expected fraction of gross errors is the estimated proportion of outliers in the sample.
In order to make the estimate asymptotically unbiased under a Normal model a correction factor, ${\tau }^{2}$, is calculated, (see Huber (1981) and Marazzi (1987)).
Initial estimates of ${\theta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$, are given by the median of the $j$th column of $X$ and the initial value of $A$ is based on the median absolute deviation (see Marazzi (1987)). nag_robust_corr_estim (g02hkc) is based on routines in ROBETH, (see Marazzi (1987)).

## 4  References

Huber P J (1981) Robust Statistics Wiley
Marazzi A (1987) Weights for bounded influence regression in ROBETH Cah. Rech. Doc. IUMSP, No. 3 ROB 3 Institut Universitaire de Médecine Sociale et Préventive, Lausanne

## 5  Arguments

1:     nIntegerInput
On entry: the number of observations, $n$.
Constraint: ${\mathbf{n}}>1$.
2:     mIntegerInput
On entry: the number of columns of the matrix $X$, i.e., number of independent variables, $m$.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
3:     x[${\mathbf{n}}×{\mathbf{tdx}}$]const doubleInput
On entry: ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{tdx}}+\mathit{j}-1\right]$ must contain the $\mathit{i}$th observation for the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
4:     tdxIntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: ${\mathbf{tdx}}\ge {\mathbf{m}}$.
5:     epsdoubleInput
On entry: the expected fraction of gross errors expected in the sample, $\epsilon$.
Constraint: $0.0\le {\mathbf{eps}}<1.0$.
6:     cov[${\mathbf{m}}×\left({\mathbf{m}}+1\right)/2$]doubleOutput
On exit: the ${\mathbf{m}}×\left({\mathbf{m}}+1\right)$/2 elements of cov contain the upper triangular part of the covariance matrix. They are stored packed by column, i.e., ${C}_{\mathit{i}\mathit{j}}$, $\mathit{j}\ge \mathit{i}$, is stored in ${\mathbf{cov}}\left[\mathit{j}\left(\mathit{j}+1\right)/2+\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{m}}-1$ and $\mathit{j}=i,\dots ,{\mathbf{m}}-1$.
7:     theta[m]doubleOutput
On exit: the robust estimate of the location arguments ${\theta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$.
8:     max_iterIntegerInput
On entry: the maximum number of iterations that will be used during the calculation of the covariance matrix.
Suggested value: ${\mathbf{max_iter}}=150$.
Constraint: ${\mathbf{max_iter}}>0$.
On entry: indicates if the printing of information on the iterations is required and the rate at which printing is produced.
${\mathbf{print_iter}}\le 0$
No iteration monitoring is printed.
${\mathbf{print_iter}}>0$
The value of $A$, $\theta$ and $\delta$ (see Section 8) will be printed at the first and every print_iter iterations.
10:   outfileconst char *Input
On entry: a null terminated character string giving the name of the file to which results should be printed. If outfile is NULL or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.
11:   toldoubleInput
On entry: the relative precision for the final estimates of the covariance matrix.
Constraint: ${\mathbf{tol}}>0.0$.
12:   iterInteger *Output
On exit: the number of iterations performed.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_GT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{m}}\le {\mathbf{n}}$.
NE_2_INT_ARG_LT
On entry, ${\mathbf{tdx}}=〈\mathit{\text{value}}〉$ while ${\mathbf{m}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdx}}\ge {\mathbf{m}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_C_ITER_UNSTABLE
The iterative procedure to find $C$ has become unstable. This may happen if the value of eps is too large.
NE_CONST_COL
On entry, column $〈\mathit{\text{value}}〉$ of array x has constant value.
NE_INT_ARG_LE
On entry, max_iter must not be less than or equal to 0: ${\mathbf{max_iter}}=〈\mathit{\text{value}}〉$.
NE_INT_ARG_LT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
NE_NOT_APPEND_FILE
Cannot open file $〈\mathit{string}〉$ for appending.
NE_NOT_CLOSE_FILE
Cannot close file $〈\mathit{string}〉$.
NE_REAL_ARG_GE
On entry, eps must be not be greater than or equal to 1.0: ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LE
On entry, tol must not be less than or equal to 0.0: ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LT
On entry, eps must not be less than 0.0: ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
NE_TOO_MANY
Too many iterations($〈\mathit{\text{value}}〉$ ).
The iterative procedure to find the co-variance matrix $C$, has failed to converge in max_iter iterations.

## 7  Accuracy

On successful exit the accuracy of the results is related to the value of tol, see Section 5. At an iteration let
 (i) $d1=\text{}$ the maximum value of the absolute relative change in $A$ (ii) $d2=\text{}$ the maximum absolute change in $u\left({‖{z}_{i}‖}_{2}\right)$ (iii) $d3=\text{}$ the maximum absolute relative change in ${\theta }_{j}$
and let $\delta =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(d1,d2,d3\right)$. Then the iterative procedure is assumed to have converged when $\delta <{\mathbf{tol}}$.

The existence of $A$, and hence $c$, will depend upon the function $u$, (see Marazzi (1987)), also if $X$ is not of full rank a value of $A$ will not be found. If the columns of $X$ are almost linearly related, then convergence will be slow.

## 9  Example

A sample of 10 observations on three variables is read in and the robust estimate of the covariance matrix is computed assuming 10% gross errors are to be expected. The robust covariance is then printed.

### 9.1  Program Text

Program Text (g02hkce.c)

### 9.2  Program Data

Program Data (g02hkce.d)

### 9.3  Program Results

Program Results (g02hkce.r)


text:
# C6x assembly programming  (Page 2/11)

 Page 2 / 11

Every instruction in the C6x can execute conditionally. There are five registers available for conditions (on the C67x processors): A1, A2, B0, B1, and B2. If blank, the instruction always executes. Conditions can take a form such as [A1] where the instruction will execute if A1 is not zero. This can be handy for making loops were the counter is put in a register like A1 and is counted down to zero. The condition is put on the branch instruction that branches back to the beginning of the loop.

## Register files

Where are the numbers stored in the CPU? In the C6000, the numbers used in operations are stored in register. Because the registers are directly accessible though the data path of the CPU, accessing the registers is much faster than accessing data in the external memory.

The C6000 CPU has two register files (A and B). Each of these files consists of sixteen 32-bit registers (A0-A15 for file A and B0-B15 for file B). The general-purpose registers can be used for data, data address pointers, or condition registers. The following figure shows a block diagram of the C67x processors. This basic structure is similar to other processors in the C6000 family.

The general-purpose register files support data ranging in size from 16-bit data through 40-bit fixed-point. Values larger than 32 bits, such as 40-bit long quantities, are stored in register pairs. In a register pair, the 32 LSB's of data are placed in an even-numbered register and the remaining 8 MSB's in the next upper register (which is always an odd-numbered register). In assembly language syntax, a colon between two register names denotes the register pairs, and the odd-numbered register is specified first. For example, A1:A0 represents the register pair consisting of A0 and A1.

Let's for now focus on file A only. The registers in the register file A are named A0 to A15. Each register can store a 32-bit binary number. Then numbers such as a , x , prod and y above are stored in these registers. For example, register A0 stores a . For now, let's assume we interpret all 32-bit numbers stored in registers as unsigned integer. Therefore the range of values we can represent is 0 to ${2}^{\text{32}}$ −1. Let's assume the numbers a , x , prod and y are in the registers A0, A1, A3, A4, respectively. Then the above assembly instructions can be written specifically:

MPY .M1 A0,A1,A3 ADD .L1 A4,A3,A4

The TI C6000 CPU has a load/store architecture. This means that all the numbers must be stored in the registers before being used as operands of the operations for instructions such as MPY and ADD . The numbers can be read from a memory location to a register (using, for example, LDW , LDB instructions) or a register can be loaded with a constant value. The content of a register can be stored to a memory location (using, for example, STW , STB instructions).

In addition to the general-purpose register files, the CPU has a separate register file for the control registers. The control registers are used to control various CPU functions such as addressing mode, interrupts, etc.

#### Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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