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How Teachers Can Incorporate Math Modeling into Their Regular Curriculum
How Teachers Can Incorporate Math Modeling into Their Regular Curriculum
Math modeling isn’t just for competitions; it can become an integral part of daily classroom lessons. By embedding math modeling into daily curriculum, teachers can provide students with practical
experience that strengthens their math skills and prepares them for success in contests like COMAP’s HiMCM® and MidMCM.
Why Incorporate Math Modeling into Your Curriculum?
Math modeling allows students to see how math is applied in real-world situations. When students work on math modeling problems, they’re not just solving equations—they’re learning how to think
critically, analyze data, and make decisions. Incorporating math modeling into your everyday curriculum encourages these skills and builds confidence in students’ abilities to tackle complex
Practical Ways to Integrate Math Modeling into Daily Lessons
Here are a few strategies for incorporating math modeling into your curriculum.
1. Turn Standard Problems into Real-World Scenarios
Instead of solving standard word problems, transform them into real-life challenges. For example, instead of asking students to calculate the volume of a cube, frame it as a packaging problem for a
new product. This real-world context helps students see the relevance of their math skills and prepares them for future math modeling contests.
2. Introduce Collaborative Problem-Solving
Math modeling contests rely on teamwork, and students can begin practicing this collaboration in the classroom. Encourage group problem-solving sessions where students work together to develop
solutions for open-ended problems. This approach mirrors the experience of math modeling contests, making students more comfortable with the format.
3. Incorporate Current Events as a Source of Problems
Tie lessons to what’s happening in the world. From climate change to economic issues, current events provide rich opportunities for math modeling. For instance, students could calculate the carbon
impact of a city’s transportation system or model the effects of interest rate changes on homeownership. These real-world problems show students the power of math in addressing societal challenges.
4. Create a Weekly Math Modeling Challenge
Set up a "Math Modeling Challenge of the Week" to get students into the habit of thinking critically about real-world issues. These challenges can be small, open-ended problems that give students
practice applying their math knowledge to unfamiliar situations. By doing this weekly, students build their confidence and problem-solving abilities over time.
Preparing Students for Math Modeling Success
Incorporating math modeling into your curriculum doesn’t just make math more engaging—it equips students with the tools they need to succeed in competitions like HiMCM and MidMCM. By making math
modeling a part of daily lessons, you help students bridge the gap between classroom learning and real-world applications, setting them up for success in both math and life.
Encourage your students to participate in COMAP’s math modeling contests this year and watch their problem-solving skills soar! And for more math modeling and contest resources, check out these
Written by
The Consortium for Mathematics and Its Applications is an award-winning non-profit organization whose mission is to improve mathematics education for students of all ages. Since 1980, COMAP has
worked with teachers, students, and business people to create learning environments where mathematics is used to investigate and model real issues in our world.
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). (Conference Paper) | NSF PAGES
Saraf, Shubhangi (Ed.)
We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex
model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability
of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We introduce a
quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on
4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms
via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold.
more » « less
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An Etymological Dictionary of Astronomy and Astrophysics
synodic period
دورهی ِهماگمی
dowre-ye hamâgami
Fr.: période synodique
For planets, the mean interval of time between two successive → conjunctions with or → oppositions to the Sun. For example, → Mars has a synodic period of 779.9 days from Earth; thus Mars'
oppositions occur once roughly 2.135 years. In comparison, the synodic period of → Venus is 583.9 days. If the sideral periods of the two bodies around the third are denoted T[1] and T[2], their
synodic period is given by: 1/T[syn] = |1/T[1] - 1/T[2]|.
→ synodic; → period.
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Ordinal Numbers
KeY Version with Ordinal Number Support (For Tableaux 2017 Reviewers Only)
Download or Start
KeY requires Java version 8 or newer and is tested on Linux, OS X and Microsoft Windows.
• Instant Start: Web Start
• Binary Version: KeY
• Source Code: KeY-src.zip
• GoodsteinBig.java This program computes Goodstein sequences using Java’s BigInteger class. This program is not mentioned in the paper.
• Goodstein.java The file “Goodstein.java” contains the Java program that is analysed in the paper. It does not correctly compute Goodstein sequences once they grow beyond maxInt. But, since in the
default setting, KeY treats Java integers as mathematical integer, the performed analysis is correct. In the paper, the JML annotations of “Goodstein.java” have been reduced to the essential
core. Here the full annotation is shown. Also, the auxiliary method for computing exponentials “intPow” is included
• Program Correctness Proofs This tarfile contains the proofs of the contracts of the three methods in “Goodstein.java”. Proof files can be loaded in the KeY system the same way as annotated Java
programs: in the pull-down menue “file” choose “load” and select the wanted proof file. After loading, you may inspect the finished proof. To keep things simple, the unpacked proof files should
be placed in the same directory as Goodstein.java.
• The technical report contains more material than could be covered in the page resticted paper.
• The program correctness proofs use a number of lemmas on ordinals, the embedding of positive integers into ordinals, and termination functions introduced for Goodstein sequences. Proofs of these
lemmas are contained in the zip-files OrdProofs1 OrdProofs2 OrdProofs3 respectively. The files in these zip-archives use KeY internal names. The correspondence between the lemma whose proof you
want to inspect and the name of the proof file can be retrieved from Fml2Taclet. To inspect the proofs, start the KeY system and select in the pull-down menue “file” subitem “load” the wanted
proof file. Proof files come with extensions .key or .proof. The user will not note any difference after loading them. The .proof files contain saved finished proofs. The .key files contain
commands of a simple, experimental scripting language that construct the proof on the fly. In some rare cases, the proof in a .key file does not complete. It has to be completed manually, which
is in all cases trivial if you know how to handle the prover. This phenomenon is due to deficiencies of the scripting language.
[button type=”primary” size=”lg” link=”https://formal.kastel.kit.edu/key/tableaux17/webstart/KeY.jnlp”][icon type=”play”] Run KeY via Webstart [/button]
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On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives
On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives ()
1. Introduction and Results
Let f and g be two nonconstant meromorphic functions. Let a be a small function of f and
and g share ¥ CM, if
In addition, we need the following definitions:
Definition 1.1. Let f be a non-constant meromorphic function, and let p be a positive integer and
Definition 1.2. Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer. We define
Remark 1.1. From the above inequalities, we have
Definition 1.3. Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer. We define
Remark 1.2. From the above inequality, we have
Definition 1.4. (see [4] ). Let k be a nonnegative integer or infinity. For
We write f, g share
R. Bruck [5] first considered the uniqueness problems of an entire function sharing one value with its derivative and proved the following result.
Theorem A. Let f be a non-constant entire function satisfying
Bruck [5] further posed the following conjecture.
Conjecture 1.1. Let f be a non-constant entire function
Yang [6] proved that the conjecture is true if f is an entire function of finite order. Yu [7] considered the problem of an entire or meromorphic function sharing one small function with its
derivative and proved the following two theorems.
Theorem B. Let f be a non-constant entire function and
Theorem C. Let f be a non-constant non-entire meromorphic function and
1) f and a have no common poles.
In the same paper, Yu [7] posed the following open questions.
1) Can a CM shared be replaced by an IM share value?
2) Can the condition
3) Can the condition 3) in theorem C be further relaxed?
4) Can in general the condition 1) of theorem C be dropped?
In 2004, Liu and Gu [8] improved theorem B and obtained the following results.
Theorem D. Let f be a non-constant entire function
Lahiri and Sarkar [9] gave some affirmative answers to the first three questions improving some restrictions on the zeros and poles of a. They obtained the following results.
Theorem E. Let f be a non-constant meromorphic function, k be a positive integer, and
1) a has no zero (pole) which is also a zero (pole) of f or
In 2005, Zhang [10] improved the above results and proved the following theorems.
Theorem F. Let f be a non-constant meromorphic function,
In 2015, Jin-Dong Li and Guang-Xiu Huang proved the following Theorem.
Theorem G. Let f be a non-constant meromorphic function,
In this paper, we pay our attention to the uniqueness of more generalized form of a function namely
Theorem 1.1. Let f be a non-constant meromorphic function,
Corollary 1.2. Let f be a non-constant meromorphic function,
2. Lemmas
Lemma 2.1 (see [1] ). Let f be a non-constant meromorphic function,
Lemma 2.2 (see [1] ). Let
where F and G are two non constant meromorphic functions. If F and G share 1 IM and
Lemma 2.3 (see [11] ). Let f be a non-constant meromorphic function and let
be an irreducible rational function in f with constant coefficients
3. Proof of the Theorem
Proof of Theorem 1.1. Let
Case 1. Let
By our assumptions, H have poles only at zeros of
Because F and G share 1 IM, it is easy to see that
By the second fundamental theorem, we see that
Using Lemma 2.2 and (11), (12) and (13), we get
We discuss the following three sub cases.
Sub case 1.1.
Combining (14) and (15), we get
that is
By Lemma 2.1 for
which contradicts with (7).
Sub case 1.2.
Combining (14) and (17) and (18), we get
that is
By Lemma 2.1 for
which contradicts with (8).
Sub case 1.3.
Similarly we have
Combining (14) and (20)-(22), we get
that is
By Lemma 2.1 for
which contradicts with (9).
Case 2. Let
on integration we get from (10)
where C, D are constants and
Sub case 2.1. Suppose
Using the second fundamental theorem for F we get
So, we have
and from which we know
We know from (28) that
So from Lemma 2.1 and the second fundamental theorem we get
which is absurd. So
In view of the first fundamental theorem, we get from above
which is impossible.
Sub case 2.2.
By the second fundamental theorem and Lemma 2.1 for
So, it follows that
This contradicts (7)-(9). Hence
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\times {10^{ - 12}}\
Hint The capacitance of a capacitor is directly proportional to the dielectric constant and inversely proportional to the distance between the plates. Compare the two capacitances to each other by
Formula used: In this solution we will be using the following formulae;
\[C = \dfrac{{K\varepsilon A}}{d}\] where \[C\] is the capacitance of a capacitor, \[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity
of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.
Complete Step-by-Step solution:
According to the question, the first air is between the plates, hence, the dielectric constant is 1. Generally, the capacitance of a capacitor is given by
\[C = \dfrac{{K\varepsilon A}}{d}\] where\[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the
capacitor plates, and \[d\] is the distance between the plates.
Hence, initially,
\[C = \dfrac{{\varepsilon A}}{d} = 8 \times {10^{ - 12}}\]
Then, capacitance after a dielectric material is added, and the distance between the plate is halved will be given as
\[{C_2} = \dfrac{{2K\varepsilon A}}{d}\]
Then, we compare the two capacitances by dividing, we have
\[\dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \div \dfrac{{\varepsilon A}}{d}\]
\[ \Rightarrow \dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \times \dfrac{d}{{\varepsilon A}}\]
Which by cancellation will give the expression
\[\dfrac{{{C_2}}}{C} = 2K\]
\[ \Rightarrow {C_2} = 2KC\]
By inserting all known values, we have
\[{C_2} = 2 \times 6 \times 8 \times {10^{ - 12}} = 9.6 \times {10^{ - 11}}F\]
Note: Noting that in the final expression the area, and permittivity of free space where absent, this implies that we can find the expression without their knowledge. Hence, alternatively, from the
knowledge that the capacitance is proportional to dielectric constant but inversely to distance, we can just write that generally,
\[C = k\dfrac{K}{d}\] where \[k\] is an arbitrary constant. Hence,
\[\dfrac{{{C_2}}}{C} = k\dfrac{K}{{\dfrac{d}{2}}} \div k\dfrac{1}{d} = k\dfrac{{2K}}{d} \times \dfrac{d}{k}\]
\[ \Rightarrow {C_2} = 2KC\]
Watch videos on
A parallel plate capacitor with air between the plates has a capacitance of \[8\mu F\] (\[1\mu F \times {10^{ - 12}}\]). What will be the capacitance if the distance between the plates is reduced by
half, and the space between them is filled with a substance of dielectric constant 6?
Electrostatic Potential and Capacitance Class 12 Physics - NCERT EXERCISE 2.5 | Vishal Kumar Sir
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Laser Beam Expanders
This is Sections 6.1, 6.2, 6.3, 6.4, and 6.5 of the Laser Optics Resource Guide.
Laser beam expanders increase the diameter of a collimated input beam to a larger collimated output beam for applications such as laser scanning, interferometry, and remote sensing. Contemporary
laser beam expanders are afocal systems developed from well-established optical telescope fundamentals. In such systems, the object rays enter parallel to the optical axis of the internal optics and
exit parallel to them. This means that the entire system does not have a focal length.
Telescope Theory
Optical telescopes, traditionally used to view distant objects such as celestial bodies in outer space, are divided into two types: refracting and reflecting. Refracting telescopes utilize lenses to
refract, or bend, light, while reflecting telescopes utilize mirrors to reflect light.
There are two categories of refracting telescopes: Keplerian and Galilean. A Keplerian telescope consists of lenses with positive focal lengths separated by the sum of their focal lengths (Figure 1).
The lens closest to the object being viewed, or source image, is called the objective lens, while the lens closest to the eye, or image created, is called the image lens.
Figure 1: Keplerian telescope
A Galilean telescope consists of a positive lens and a negative lens that are also separated by the sum of their focal lengths (Figure 2). However, since one of the lenses is negative, the separation
distance between the two lenses is much shorter than in the Keplerian design. While using the effective focal length of the two lenses will provide a good approximation of the total length, using the
back focal length will provide the most accurate length.
Figure 2: Galilean telescope
(1)$$ \text{Magnifying Power} \left( \text{MP} \right) = \frac{1}{\text{Magnification} \left[ \text{m} \right]} $$
(2)$$ \text{MP} = - \frac{\text{Focal Length}_{\text{Objective Lens}}}{\text{Focal Length}_\text{Image Lens}} $$
If the magnifying power is greater than 1, the telescope magnifies. When the magnifying power is less than 1, the telescope minifies.
Beam Expander Theory
In a laser beam expander, the placement of the objective and image lenses is reversed. Keplerian beam expanders consist of two lenses with positive focal lengths separated by the sum of their focal
lengths. They offer high expansion rations and allow for spatial filtering because the collimated input beam focuses to a spot between the objective and image lenses, producing a point within the
system where the laser's energy is concentrated (Figure 3). However, this heats the air between the lenses, deflecting light rays from their optical path and potentially leading to wavefront errors
especially in high-power laser applications.
Figure 3: Keplerian beam expanders have an internal focus which is detrimental to high power applications, but useful for spatial filtering in lower power applications
Galilean beam expanders, in which an objective lens with a negative focal length and an image lens with a positive focal length are separated by the sum of their focal lengths, are simple, lower-cost
designs that also avoid the internal focus of Keplerian beam expanders (
Figure 4
). The lack of an internal focus makes Galilean beam expanders better suited for high-power laser applications than Keplerian designs.
Figure 4: Galilean beam expanders have no internal foci and are ideally suited for high power lasers applications
When using the Keplerian or Galilean designs in laser beam expander applications, it is important to be able to calculate the output beam divergence. This determines the deviation from a perfectly
collimated source. The beam divergence is dependent on the diameters of the input and output laser beams.
(3)$$ \frac{\text{Input Beam Divergence} \left( \theta_I \right)}{\text{Output Beam Divergence} \left( \theta_O \right)} = \frac{\text{Output Beam Diameter} \left( D_O \right)}{\text{Input Beam
Diameter} \left( D_I \right)} $$
The magnifying power (MP) can now be expressed in terms of the beam divergences or beam diameters.
(4)$$ \text{MP} = \frac{\theta _I}{\theta _O}$$
(5)$$ \text{MP} = \frac{D_O}{D_I} $$
When interpreting Equation 4 and Equation 5, one can see that while the output beam diameter (D[0]) increases, the output beam divergence (θ[O]) decreases and vice versa. Therefore, when using a beam
expander to minimize the beam, its diameter will decrease but the divergence of the laser will increase. The price to pay for a small beam is a large divergence angle.
In addition, it is important to be able to calculate the output beam diameter at a specific working distance (L). The output beam diameter is a function of the input beam diameter and the beam
divergence after a specific working distance (L) (Figure 5).
Figure 5: A laser's input beam diameter and divergence can be used to calculate the output beam diameter at a specific working distance
(6)$$ D_L = D_O + L \cdot \tan{\left( 2 \theta_O \right)} $$
Laser beam divergence is specified in terms of a half angle, which is why a factor of 2 is required in the second term in Equation 6.
A beam expander will increase the input beam and decrease the input divergence by the Magnifying Power. Substituting Equations 4 and 5 with Equation 6 results in the following:
(7)$$ D_L = \left( \text{MP} \times D_I \right) + L \cdot \tan{\left( \frac{2 \theta_I}{\text{MP}} \right)} $$
(8)$$ D_L = \left( \text{MP} \times D_I \right) + L \cdot \tan{\left( 2 \theta_O \right)} $$
Application 1: Reducing Power Density
Beam expanders increase the beam area quadratically with respect to their magnification without significantly affecting the total energy contained within the beam. This results in a reduction of the
beam’s power density and irradiance, which increases the lifetime of laser components, reduces the chances of laser induced damage, and enables the use of more economical coatings and optics.
Application 2: Minimizing Beam Diameter at a Distance
Although it may seem unintuitive, increasing the diameter of a laser using a beam expander may result in a smaller beam diameter far from the laser aperture. A beam expander will increase the input
laser beam by a specific expansion power while decreasing the divergence by the same expansion power, resulting in a smaller collimated beam at a large distance. Laser beam expanders can also be used
in reverse to reduce beam diameter rather than expanding it. This will invert the magnifying power, but divergence will be increased.
A numerical example to explore the previously mentioned beam expander equations:
Initial Parameters
Beam expander magnifying power = MP = 10X
Input beam diameter = 1mm
Input beam divergence = 0.5mrad
Working distance = L = 100m
Calculated Parameter
Beam diameter at distance L:
(9) D_L & = \left( \text{MP} \times D_I \right) + L \cdot \tan{ \left( \frac{2 \theta_I}{\text{MP}} \right)} \\ D_L & = \left( 10 \text{X} \times 1 \text{mm} \right) + 100,000 \text{mm} \cdot \tan{\
left( \frac{2 \cdot 0.5 \text{mrad}}{10 \text{X}} \right)} = 20 \text{mm}
Compare this to the beam diameter without using a beam expander by using Equation 6.
(10) D_L & = D_I + L \cdot \tan{\left( 2 \theta_I \right)} \\ D_L & = 1 \text{mm} + 100,000 \text{mm} \cdot \tan{\left(2 \cdot 0.5 \text{mrad} \right)} = 101 \text{mm}
Using a 10X beam expander reduced the output beam diameter 100m away by over a factor of 5 when compared to the same laser without a beam expander.
Application 3: Minimizing Focused Spot Size
Spot size is typically defined as the radial distance from the center point of maximum irradiance to the point where the intensity drops to 1/e^2 of the initial value (Figure 6). The focused spot
size of an ideal lens can be calculated by using wavelength (λ), the focal length of the lens (f), the input beam diameter (D[I]), the refractive index of the lens (n), and the beam’s M^2 factor,
which represents the degree of variation from an ideal Gaussian beam.
(11)$$ \definecolor{Diffraction}{RGB}{0, 0, 255} \definecolor{Aberration}{RGB}{255, 0, 0} \phi_{\text{Spot Size}} = \color{Diffraction} \phi_{\text{Diffraction}} \color{black} + \color{Aberration} \
phi_{\text{Aberration}} \color{black} = \color{Diffraction} \frac{4 \lambda M^2 f}{\pi D_I} \color{black} + \color{Aberration} \frac{n D_I ^3}{f^2} $$
Figure 6: Spot size is usually measured at the point where the intensity I(r) drops to 1/e^2 of the initial value I[0]
Spot size is fundamentally determined by the combination of diffraction and aberrations illustrated by red and blue, respectively, in Figure 7. Generally, when focusing laser beams, spherical
aberration is assumed to be the only and dominant type of aberration, which is why Equation 11 only takes spherical aberration into account. In regards to diffraction, the shorter the focal length,
the smaller the spot size. More importantly, the larger the input beam diameter the smaller the spot size.
By expanding the beam within the system, the input diameter is increased by a factor of MP, reducing the divergence by a factor of MP. When the beam is focused down to a small spot, the spot is a
factor of MP smaller than that of an unexpanded beam for an ideal, diffraction-limited spot. However, there is a tradeoff with spherical aberration because it increases along with the input beam
Figure 7: At small input beam diameters, the focused spot size is diffraction limited. As the input beam diameter increases, spherical aberration starts to dominate the spot size
Application 4: Compensating for Input Laser Beam Variability
Most commercial lasers specify an output beam diameter of the laser at the aperture with a tolerance that is often on the order of 10% or more. For many laser applications, a specific beam diameter
is required at the end of the system. A variable beam expander can be inserted into the system to compensate for variability between individual laser units, ensuring that the final beam diameter is
consistent for all systems.
Beam Expander Selection Criteria
When choosing a beam expander for an application, certain criteria must be determined in order to achieve the correct performance.
Sliding vs. Rotating Divergence Adjustment:
The mechanics used to focus a beam expander or change the magnification of a variable beam expander are typically classified into two different types: sliding and rotating. Rotating divergence
adjustment, such as threaded focusing tubes, rotate the optical elements during translation. They have a lower cost than sliding divergence adjustment due to their simplified mechanics, but they
create the potential for beam wander due to the element rotation (Figure 8).
Figure 8: Exaggerated illustration of the beam wander that may be caused by rotating focus mechanisms
Sliding divergence adjustment, such as helicoid barrels, translate the internal optics without rotating them, thus minimizing beam wander. However, this requires more complex mechanics than those of
rotating focus mechanisms, increasing system cost. Poorly designed sliding optics may also have too much freedom of movement in the mechanics. While the pointing error in these poorly designed
designs will not rotate when adjusted, it will be larger than for rotating optics or correctly designed sliding optics.
Internal Focus:
Keplerian beam expanders contain an internal focus that may be problematic in high power systems. The intense focused spot can ionize the air or lead to wavefront errors as a result of the heat
deflecting light rays. Because of this, most beam expanders are Galilean to avoid complications caused by internal focusing. However, certain applications require spatial filtering which is only
possible in Keplerian designs because of the internal focus capability.
Reflective vs. Transmissive:
Reflective beam expanders utilize curved mirrors instead of transmissive lenses to expand a beam (Figure 9). Reflective beam expanders are much less common than transmissive beam expanders, but have
several advantages that make them the right choice for certain applications. Reflective beam expanders do not suffer from chromatic aberration, whereas the magnification and output beam collimation
of transmissive beam expanders is wavelength dependent. While this is not relevant for many laser applications because lasers tend to lase at a single wavelength, it may be critical in broadband
applications. The achromatic performance of reflective beam expanders is required for multi-laser systems, some tunable lasers, and ultrafast lasers. Ultrafast lasers inherently span a broader
wavelength range than other lasers due to their extremely short pulse duration. Quantum cascade lasers also benefit from reflective beam expanders as transmissive options may not exist at their
operating wavelengths.
Figure 9: Unlike transmissive beam expanders, the curved mirrors of this Canopus Reflective Beam Expander expand the incident laser beam. The holes on the side of the beam expander are integrated
mounting features
Edmund Optics Products
The TECHSPEC^® Scorpii Nd:YAG Beam Expanders are available for applications where cost is the driving factor. Featuring 2-element Galilean designs with diffraction limited performance at YAG
wavelengths, the Scorpii Nd:YAG Beam Expanders offer a variety of magnification ranges from 2X to 10X ideal for prototyping and OEM integration.
The TECHSPEC^® Vega Laser Line Beam Expanders provide excellent value with λ/10 performance at design wavelength for apertures up to 4 mm. Featuring laser line V-coats for Nd:YAG harmonics down to
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1. Greivenkamp, John E. Field Guide to Geometrical Optics. Vol. FG01. Bellingham, WA: SPIE—The International Society for Optical Engineers, 2004.
2. Smith, Warren J. Modern Optical Engineering. 3rd ed. New York, NY: McGraw-Hill Education, 2000.
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A triangle is a shape with three straight sides. There are several types of triangle. Here is a video that describes them:
The main types of triangle are:
• Equilateral triangles, that have three equal sides and three equal angles.
• Isosceles triangles, that have two equal sides and two equal angles.
• Scalene triangles, where none of the sides are equal and none of the angles are equal.
Isosceles and scalene triangles can be divided into three types:
• Acute, where all three angles are less than 90 degrees.
• Right-angled, where one angle is exactly 90 degrees.
• Obtuse, where one angle is greater than 90 degrees.
Angles of a triangle
The angles inside any triangle add up to 180 degrees.
Equilateral triangles
An equilateral triangle is a triangle where all three sides are the same length, and all three angles are equal:
The tick marks on the three sides indicate that the sides are the same length.
The angle arcs in the corners indicate that the angles are all equal.
Since the three angles of any triangle always add up to 180 degrees, it follows that each angle in an equilateral triangle must be equal to 60 degrees.
And equilateral triangle is a regular polygon. All equilateral triangles are the same shape, but can be different sizes (we say they are similar).
Isosceles triangles
An isosceles triangle has two sides that are equal in length. The third side is not equal to the other two (it can be longer or shorter).
We sometimes call the two equal sides the legs of the triangle, and the other side the base.
Each leg makes an equal angle with the base.
Scalene triangles
A scalene triangle has three unequal sides.
All three angles in a scalene triangle are also unequal.
Acute triangles
A triangle is acute if all three of its angles are less than 90 degrees.
All the triangles shown above are acute.
Obtuse triangles
A triangle is obtuse if one of its angles is greater than 90 degrees.
Here is an obtuse isosceles triangle:
An obtuse isosceles triangle has a wide base, which means that the top angle (the angle where the legs meet) is larger than 90 degrees. The two legs are the same length, and the two angles at the
base are equal.
Here is an obtuse scalene triangle:
In this case, all the sides have different lengths and all the angles are different, but the shape has one angle that is greater than 90 degrees.
It is impossible for a triangle to have more than one angle that is greater than 90 degrees because the sum of all three angles is 180 degrees.
An equilateral triangle can't be obtuse, because all the angles of an equilateral triangle are 60 degrees.
Right-angled triangles
A right-angled triangle has one angle of exactly 90 degrees.
Here is a right-angled isosceles triangle:
As before the two legs of an isosceles triangle are the same length and the two angles at the base are equal.
Since we know that the other angle is 90 degrees, this means that the two angles at the base must both be equal to 45 degrees (because 90 + 45 + 45 is 180).
This means that all right-angled isosceles triangles are the same shape. They can be different sizes (they are similar).
A right-angled scalene triangle is similar, but the legs are different sizes:
The sum of the two angles at the base is 90 degrees.
An equilateral triangle can't be right-angled, because all the angles of an equilateral triangle are 60 degrees.
There are seven types of triangle:
• Equilateral triangles (that are always acute).
• Isosceles triangles (that can be acute, obtuse, or right-angled).
• Scalene triangles (that can be acute, obtuse, or right-angled).
See also
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To Log or Not to Log, Part IV
Do stock price indices follow a random walk? No, but log stock price indices do…kind of
Here’s a graph of the Standard and Poors 500 price index, level (blue) and the log (red).
Figure 1: S&P 500 stock price index (blue, left scale) and log (red, right scale). Source: FRED, and author’s calculations.
Both processes look nonstationary, but the levels process more so (technically, “explosive”). Which one is the more appropriate to examine? A little theory might be helpful. Consider an asset pricing
model for stocks. Ignoring dividends,
P[t] = (E[t]P[t+1])/(1+i)
Where P is the stock price, i is an interest or equity discount rate, and E[t](.) is an expectations operator, conditional on time t information. Notice if errors are normally distributed, then the
price level at t+1 is a multiple of that at time t (imposing an auxiliary assumption that the interest rate is constant and nonstochastic). Assume instead expectations errors are log-normally
distributed (where p denotes log(P)). Then:
p[t] = E[t]p[t+1] – ln(1+i)
p[t+1] ≈ p[t] + i + e[t+1]
Where e[t+1] is a true innovation. If one runs a regression in levels, the coefficient on the lagged variable is 1.04, and the process appears explosive. In logs, one obtains:
p[t] = 0.0052 + 1.0000p[t-1] + u[t]
Adj.-R^2 = 0.999, SER = 0.035, Nobs = 702, DW = 1.47.
The coefficient on lagged (log) price is not statistically significantly different from one.
Another hint about what specification is more appropriate comes from inspection of the first differences. Figure 2 depicts both.
Figure 2: First difference of S&P 500 stock price index (blue, left scale) and first difference of log (red, right scale). Source: FRED, and author’s calculations.
The log first differences exhibit much less heteroskedasticity, and the distribution is less fat-tailed.
Note that in neither levels nor logs are these classic random walks (where the residuals are true innovations). The errors reject the no serial correlation null and no ARCH effects. But less so for
the logs specification.
Digression: Great post from David Giles, on retrieving elasticities from different specifications (h/t Mark Thoma).
11 thoughts on “To Log or Not to Log, Part IV”
1. jonathan
The annual NYSE volume in the mid-60’s was about 1 billion. I didn’t want to get into this deeply but I know the annual volume in the early 2000’s was about 350 billion. Comparing those eras
requires a scaling method … like logs.
2. Rick Stryker
I’m not sure I understand your point in this post. Some questions:
1) How did you conclude that the coefficient on the lagged logged variable is not statistically significant from one? If you used that same argument on the coefficient of the level variable that
estimated explosive, could you also conclude that the coefficient is not statistically different from one?
2) Why do we care that the log difference specification shows less heteroskedasticity and less fat tails? Why does that induce you to prefer the log specification?
1. Menzie Chinn Post author
Rick Stryker:
1) Standard errors for levels specification indicate I could reject a coefficient of unity on the lagged dependent variable. On the other hand, in the log-level specification, I cannot reject
the unit coefficient.
2) Example: Consider ADF test regression (a zero lag ADF is isomorphic to this regression). Proper inference requires normally distributed residuals…
1. Rick Stryker
I guess I’m not seeing it unless I misunderstand what you are saying.
1) If you can reject rho = 1 on the levels specification, are you then saying it’s a stationary series, i.e., rho < 1? That's not plausible although the alternative that the series is
explosive is even less plausible. How did you do this test? Is it the standard regression test, i.e., (rho – 1)/s.e.(rho) distributed asymptotically normal or something else?
2) Why do you need normally distributed residuals for an ADF test regression? Asymptotically, you don't need that and you do have a pretty large sample here.
1. Menzie Chinn Post author
Rick Stryker:
1) In level Y[t] = alpha + 1.04Y[t-1]. Using the standard errors churned out by OLS, I can reject a unit coefficient. Now, it’s likely that in finite samples the estimated coefficient
is far from the true, so the standard errors are likely distorted. But this was a quick pass. More formal testing discussed in this paper. To be clear, I haven’t done this.
2) I dunno, is 700 observations big or small? Depends on what you’re looking at. Testing hypotheses serial correlation properties, I like to remember Summers’ paper at such times.
1. Rick Stryker
But what you did do is invalid. The whole point of the unit root literature is that you can’t do standard OLS hypothesis tests under the null hypothesis that rho = 1, since the
distributions of the test statistics are non-standard, being functionals of Brownian motion.
Since these series are trending, you would need to have estimated the regressions with a time trend included and then tested the hypothesis that rho = 1. Given that the residuals
show evidence of heteroskedasticity, you could account for that using something like the Phillips-Perron adjustment to the test statistics. I’m not disputing that you would still
find that rho = 1 for the log of the S&P 500, which I think is a standard result. But I would think it a pretty weird situation if you didn’t conclude the same for the level of
the S&P. That the level of the series is truly explosive just isn’t plausible. But I’ve never done the tests so I don’t know what you’d find.
I don’t get your second point either. These tests are generally justified asymptotically and in this case the number of data points you have should make you feel more comfortable
about this assumption, given the typical size of data sets. If you had to establish that the residuals are normal before doing ADF tests, which they never are, no one would ever
do these tests at all.
Moreover, I don’t understand why you are worried about heteroskedasticity and fat tails. The change in the logs of the S&P 500 on higher frequency data, such as daily data, can be
well-described by an EGARCH process in which the conditional (as well as the unconditional) residual distribution has fat tails. Heteroskedasticity and fat tails are a feature of
the data.
2. Menzie Chinn Post author
Rick Stryker:
1) Concur I didn’t the “right” thing. But what you propose is not “the right thing” either. As far as I can ell, the Phillips article I linked to is more “the right thing”. By the
way, if I do a Phillips Perron test w/constant, I get a positive t-stat consistent with explosive series on levels, not so in logs.
2) See page 120 here. I think in an ADF test you do want at least homoskedasticity.
3. 2slugbaits
I think acres burned data would make some of these points more forcefully. For example, in levels the data are clearly heteroskedastic. In logs, not so much.
It would probably be a good idea to explain the difference between an explosive coefficient (e.g., 1.04) and a unit root coefficient that is not significantly different from 1.00.
4. Rick Stryker
No, I think the procedure I proposed is the right thing. You’d want to start the analysis with the standard tests for both series, which is what I proposed. If you did reject the null that the
coefficient rho is 1 on the levels series using Phillips-Perron, then you’d want to explain that. As I mentioned, given your simple autoregressive specification, neither alternative possibility
is plausible: rho 1 implies the levels series is explosive. Phillips, Shi, and Yu agree with my earlier point that rho > 1 (1.04) in your regression is implausible, since they say about equation
(5) in their paper, which corresponds to your levels regression, that:
“Model (5) is formulated with a non-zero intercept and produces a dominating deterministic component that has an empirically unrealistic explosive form (P. C. B. Phillips and J. Yu, Unpublished
Manuscript 2009, PY hereafter). Similar characteristics apply a fortiori in the case of the inclusion of a deterministic trend term in model (6). These forms are unreasonable for most economic
and financial time series and an empirically more realistic description of explosive behaviour is given by models (4) and (7), which are both formulated without an intercept or a deterministic
Put another way, if I believed that your levels regression of the S&P 500 were correct and the series really is explosive, I think I’d want to re-allocate my 401K!
Phillips, Yi, and Shu form a more realistic alternative hypothesis of a time series that exhibits bubble behavior that periodically collapses, so that it is not permanently explosive. I’d agree
with you that this is a sensible test to run to explain your finding on the level of prices.
1. Rick Stryker
Sorry, that should read above:
neither alternative possibility is plausible: rho 1 implies the levels series is explosive.
1. Rick Stryker
That’s interesting. The blog software is truncating the words so that I got the same error the second time. Anyway, I’m trying to say that the alternative to a unit coefficient in the
regression is that the levels series is either stationary or explosive, neither of which is plausible.
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Model Machine Learning Algorithms for Better Predictions - reason.townModel Machine Learning Algorithms for Better Predictions
Model Machine Learning Algorithms for Better Predictions
Predictive modeling is a powerful way to apply machine learning to real-world problems. In this blog post, we’ll show you how to use Python to build predictive models and make better predictions.
Checkout this video:
In this guide, we are going to look at different ways to model machine learning algorithms. In particular, we are going to look at how to use different machine learning models to make predictions. We
will also look at how to evaluate these models and choose the best one for a given problem.
Data pre-processing
In order to make better predictions with machine learning models, it is important to understand the data pre-processing stage. This is the stage where the data is prepared for modeling. Data
pre-processing includes tasks such as cleaning the data, imputing missing values, scaling numerical columns, and encoding categorical columns.
Cleaning the data involves removing invalid or incorrect observations from the dataset. Invalid observations can occur due to errors in data collection or recording. Incorrect observations can occur
when there is a misunderstanding of the variables in the dataset. For example, if a column is supposed to contain only numerical values but some non-numerical values are found, those values would be
considered incorrect and should be removed.
Imputing missing values is the process of replacing missing values with estimated values. This is done because machine learning models cannot operate on datasets with missing values. There are
several methods for imputing missing values, such as using the mean or median value of the column, using a value from another similar observation, or using a prediction from a machine learning model.
Scaling numerical columns is important because machine learning models often perform better when all features are on a similar scale. There are two common methods for scaling numerical columns:
normalization and standardization. Normalization scales all feature values to be between 0 and 1. Standardization scales all feature values so that they have a mean of 0 and a standard deviation of
Encoding categorical columns is also important for building machine learning models. Categorical variables are variables that can take on one of a limited number of values, such as “male” or
“female”. These variables need to be encoded so that they can be represented numerically by the machine learning model. One common method for encoding categorical columns is one-hot encoding. This
creates a new column for each possible value of the categorical column and assigns a 1 to indicate that the observation belongs to that category and a 0 otherwise.
Data partitioning
Different types of data partitioning are used in machine learning to support the selection of training and test sets, as well as to validate models. The most common data partitions are based on
random sampling, such as:
-Simple Random Sampling: A simple random sample is a subset of a data set in which each element has an equal chance of being selected.
-Stratified Sampling: Stratified sampling is a method of sampling that involves dividing a population into strata and selecting a representative sample from each stratum.
-Cluster Sampling: Cluster sampling is a method of sampling that involves dividing a population into clusters and selecting a random sample of clusters.
There are also methods for partitioning data that are not based on random sampling, such as:
-Holdout Method: The holdout method is a type of data partitioning in which a certain portion of the data set is withheld from the training process.
-Cross-Validation: Cross-validation is a type of data partitioning in which the data set is divided into folds, and each fold is used in turn to train and test the model.
Training the model
Before a model can make predictions, it must be trained. Training is the process of exposing the model to a set of training data, so that the model can learn to map the input data to the output
labels. The quality of the predictions made by the model is directly proportional to the amount and quality of training data that the model has been exposed to.
There are two general types of training data: labeled and unlabeled. Labeled data is a set of training examples where each example has a known output label. For example, in a classification task, the
labels could be positive or negative sentiment. In a regression task, the labels could be continuous values such as dollars or temperatures. Unlabeled data is a set of training examples where the
output labels are not known.
In order to train a model, we need both labeled and unlabeled data. However, we usually have more unlabeled data than labeled data. This is because it is usually easier and cheaper to collect
unlabeled data than it is to label it. For example, we can easily collect millions of tweets from Twitter without knowing their sentiment ahead of time. However, labeling those tweets would require
manually reading and assessing each one, which would be prohibitively expensive.
Fortunately, there are ways to train models even when we only have access to unlabeled data. These methods are called unsupervised learning algorithms. With unsupervised learning algorithms, we can
still train models even though we don’t have any labeled data!
Evaluating the model
Evaluating the model by its prediction error is a commonly used approach to model selection and algorithm evaluation. The objective is to find the model that minimizes the sum of the prediction error
across all examples in the dataset. This approach can be used for both classification and regression problems.
There are a number of ways to measure the prediction error, including the mean squared error (MSE), mean absolute error (MAE), and root mean squared error (RMSE). The MSE is the most commonly used
metric for regression problems, while the MAE is more commonly used for classification problems.
Fine-tuning the model
In order to make the most accurate predictions possible, it is important to fine-tune the machine learning algorithms that you are using. This process involves adjusting the parameters of the
algorithm in order to improve its performance.
There are a few different methods that can be used to fine-tune machine learning algorithms, and the best approach will vary depending on the specific algorithm and on the data set that you are
using. One common method is called cross-validation, which involves training the algorithm on a portion of the data set and then testing it on another portion.
Another method is called grid search, which involves training the algorithm on a range of different parameter values and then finding the combination that gives the best results.
No matter which method you use, it is important to tune your machine learning algorithms carefully in order to get the most accurate predictions possible.
Saving and loading the model
Storing the model is important so that you can load and use the model at a later time. You may also want to send the model to someone else so that they can use it without having to retrain the model
themselves. There are two ways to save and load models: 1) as a .pmml file or 2) as a .pkl file.
The .pmml file is the industry standard for storing predictive models. The file format is XML-based and is portable across languages and platforms. To save a model as a .pmml file, you will need to
install the pymc library. Once pymc is installed, you can use the following code to save your model:
import pymc
model = pymc.MCMC(model) # loads your PyMC3 model
model_file = ‘my_model.pmml’
pymc.createXML(model, model_file) # saves your model as my_model.pmml in the current directory
Alternatively, you can save your model as a .pkl file, which is a binary pickle format. The advantage of this format is that it can be used to store any type of object, not just models. To save a
model as a pickle, use the following code:
import pickle
with open(‘my_model.pkl’, ‘wb’) as f: # wb means “write bytes”
pickle.dump(model, f) # saves your PyMC3 model under my_model.pkl in the current directory
Making predictions
Whether we’re trying to predict the weather, the stock market, or consumer behavior, machine learning is a powerful tool that can help us make better predictions. But what are the different types of
machine learning algorithms, and how do they work?
In this article, we’ll take a look at some of the most popular machine learning algorithms and how they can be used to make predictions.
Linear regression is one of the most basic and popular machine learning algorithms. It’s used to find relationships between variables in data sets. For example, you might use linear regression to
predict how much money a person will spend on a given day, based on their income.
Logistic regression is another popular machine learning algorithm. It’s used to predict the probability that an event will occur, based on past data. For example, you might use logistic regression to
predict whether or not a person will vote in an upcoming election, based on their age and voting history.
Decision trees are a type of machine learning algorithm that are used to create models that can be used to make predictions. Decision trees are created by splitting data sets into smaller groups,
based on certain characteristics. For example, you might use a decision tree to predict whether or not a person will buy a product, based on their age, gender, and whether or not they’ve bought
similar products in the past.
Random forest is a type of machine learning algorithm that builds multiple decision trees and then combines them to make predictions. Random forest is often used for classification tasks (predicting
whether an instance belongs to one class or another), but can also be used for regression (predicting a numeric value).
Gradient Boosted Machines is another type of machine learning algorithm that builds multiple models and then combines them to make predictions. Gradient Boosted Machines are often used for
classification tasks, but can also be used for regression.
In conclusion, it is important to remember that no single model is right for every problem. Each model has its own strengths and weaknesses, and what works well for one problem might not work as well
for another. The best way to find the right model is to try out a few different ones and see which one gives the best results.
Further reading
If you want to learn more about model machine learning algorithms, here are some resources that can help:
-“A Few Useful Things to Know about Machine Learning”, Pedro Domingos, Communications of the ACM, Vol. 56, No. 10 (2013), pp. 78-87.
-“Pattern Recognition and Machine Learning”, Christopher Bishop, Springer 2006.
-“Machine Learning: A Probabilistic Perspective”, Kevin Murphy, MIT Press 2012.
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APHEO - 10 Standardization of Rates
Core Indicators Work Group Recommendations: Standardization of Rates
Based on Standardization of Rates paper by Nam Bains
Introductory points:
Both direct and indirect methods of standardization have their own merits and are appropriate to use depending on the situation.
• Direct standardization produces age adjusted rates or SRATES. Indirect standardization results in Standardized Incidence Ratios (SIRs) or Standardized Morbidity/Mortality Ratios (SMRs).
• Indirect standardization is more stable when studying rates based on small numbers. The ratio of observed to expected cases is also intuitively easy to understand. Because the age-distribution of
the study population is used to calculate SIRs or SMRs, it is incorrect to compare indirectly adjusted rates for different areas. The same age-specific rates (from a standard) applied to study
populations with different age distributions will result in very different SMRs.
• Direct standardization preserves the consistency between different study populations; since many study populations can be adjusted to the same standard, the resulting rates can be compared
against each other. This is important when comparing study populations over place or time.
1. Examine crude rates, age-specific rates and counts before calculating adjusted rates.
2. When there is little to no variation across age-specific rates or where there is no dfference in the age structure of the populations over time and geography, crude rates can be valid for
comparisons over time and geography.
3. Only consider direct standardization if there are 20 or more events.
4. Only consider indirect standardization if there are 10 or more events.
5. Consider suppressing age-adjusted rates if the Relative Standard Error is greater than 23%. RSE is similar to a coefficient of variation (CV); the larger the RSE (or the CV), the less reliable is
the estimate.
6. When using direct standardization, use the 1991 Canadian population structure as the standard population.
7. Although there is not a recommended number of age categories to use when calculating age-adjusted rates, epidemiologists should be aware of the issues around age categories and the factors that
should be considered before determining the number of age groups.
8. When using direct standardization, for age strata with zero events, epidemiologists should consider combining multiple years, collapsing geographies or age strata where feasible. If this is not
feasible, substitute a small number (e.g. 0.1) for zero events or impute from a higher level geography.
9. When using direct standardization, confidence intervals should be calculated using the Poisson approximation.
10. When using indirect standardization, confidence intervals should be calculated using the Armitage and Berry method.
This page last updated: July 2, 2009
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Complexity of push_back()
07-15-2009 08:26 AM
I wasn't able to find this information neither in the Reference Manual, nor by Google - what is the time complexity of the push_back() method of parallel_vector ? According to the Wikipedia article
for std::vector this operation has an amortized complexity O(1). How is it for the parallel relative ?
Thank you in advance and best regards
07-16-2009 01:37 AM
Quoting - bartlomiej
what is the time complexity of the push_back() method of parallel_vector ? According to the Wikipedia article for std::vector this operation has an amortized complexity O(1). How is it for the
parallel relative ?
07-16-2009 01:37 AM
Quoting - bartlomiej
what is the time complexity of the push_back() method of parallel_vector ? According to the Wikipedia article for std::vector this operation has an amortized complexity O(1). How is it for the
parallel relative ?
07-17-2009 09:50 AM
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Regression Analysis
A Regression Analysis is a way of gauging the relationships between different variables by looking at the behavior of the system. There are many analysis techniques used to determine the relationship
between dependent and independent variables. However, a Regression Analysis is one of the best. For example, transformations can be used to reduce the higher-order terms in the model.
Remember the equation for a line that you learned in high school? Y = mx + b where m is the slope of the line, and b is the point on the y-axis where the line intercepts. Given the slope (m) and the
y-intercept (b), you can plug in any value for X and get a result y. Very simple and very useful. That’s what we are trying to do in root cause analysis when we say “solve for y.”
Though statistical linear models are described as a classic straight line, often linear models are shown as curvilinear graphs. While non-linear regression aka Attributes Data Analysis is used to
explain the nonlinear relationship between a response variable and one or more than one predictor variable (mostly curve line).
Unfortunately, real life systems do not always boil down to a simple math problem. Sometimes you just have a collection of points on a graph, and your boss tells you to make sense of them. That’s
where regression analysis comes into play; you are basically trying to derive an equation from the graph of your data.
“In the business world, the rear view mirror is always clearer than the windshield.”
Warren Buffet
Linear Regression Analysis
The easiest kind of regression is linear regression. Imagine that all of your data is lined up in a neat row. If you could draw a straight line through all points that would model a simple equation Y
= mx + b that we talked about earlier. That would give you a model that would predict what your system would do given any input of x.
But what if your data doesn’t look like a line?
In that case, Multiple linear regression is an extension of the methodology of simple linear regression.
Method of Least Squares
The Method of least squares is a method to create the best possible approximation of a line given the data set.
How well the created line fits the data can be determined by the Standard Error of Estimate. The larger the Standard Error of the Estimate, the greater the distance of the charted points from the
The normal rules of Standard Deviation apply here; 68% of the points should be within +/- 1 Standard Error of the line; 95.5% of the points within +/- 2 Standard Error.
For more examples of Least Squares, see Linear Regression.
Coefficient of Determination (R^2 aka R Squared)
The Coefficient of Determination provides the percentage of variation in Y that is explained by the regression line.
Coefficient of Correlation is r.
Just take the square root of the Coefficient of Determination – Sqrt(R Squared)
Go here for more on the Correlation Coefficient.
Measuring the validity of the model
Use the F statistic to find a p-value of the system. The degrees of freedom for the regression are equal to the number of Xs in the equation (in a linear regression analysis, this is 1 because there
is only 1 x in y=mx+b).
Null Hypothesis: Suggests that there is no statistical significance between the two variables in the study.
Alternative Hypothesis: Suggests that there is statistical significance between the two variables in the study.
The smaller the p-value, the better. But really, you judge this by finding the acceptable level of alpha risk and seeing if that % is greater than the p-value. A P-value of 0.05 means that 5% of the
time, we will falsely reject the null hypothesis. It means 5% of the time we might falsely have concluded a relationship.
For example, if your alpha risk level is 5% and the p-value is 0.014, then we can reject the null hypothesis and may conclude that there exists a relationship between the variables. – in this case,
you’d accept the line as there is a significant relationship between the variables.
Additional Helpful Resources
Residual Analysis: “Since a linear regression model is not always appropriate for the data, assess the appropriateness of the model by defining residuals and examining residual plots.”
What is the difference between Residual Analysis and Regression Analysis?
Regression models, a residual measures how far away a point is from the regression line. In a residual analysis, residuals are used to assess the validity of a statistical or ML model. The model
is considered a good fit if the residuals are randomly distributed.
• When should we use regression analysis?
Regression Analysis and Correlation Videos
ASQ Six Sigma Black Belt Exam Regression Analysis Questions
Question: In regression analysis, which of the following techniques can be used to reduce the higher-order terms in the model?
A) Large samples.
B) Dummy variables.
C) Transformations.
D) Blocking.
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Transformations. Once you have identified a working equation for the system, you can often reduce the higher-order terms (the messier and more difficult work) into equations that are easier to work
by transforming them.
Comments (8)
Ted, what exactly does “transformations can be used to reduce the hire-order terms in the model” mean. What are ‘higher-order terms’? Why do they need to be reduced? Reduced from what? To what?
Hi Andrew,
Typo there – thanks for catching it! Hire should be Higher.
I was referring to a case where you might use a mathematical transform to bring a complicated model (eg Y = X^3 + 5X^2 + 4x + 1) to something more easily analyzed.
Does that help?
Ted, confused on why would reject the null if P is .14 which is greater the .05. Wouldn’t I accept the null? Can you help me understand why I would reject the null? Thank you.
Measuring the validity of the model
Use the F statistic to find a p value of the system. The degrees of freedom for the regression is equal to the number of Xs in the equation (in linear regression, this is 1 because there is only 1 x
in the equation y=mx+b). The degrees of freedom for the
The smaller the p value, the better. But really you judge this by finding the acceptable level of alpha risk and seeing if that percent is greater than the p value. For example, if your alpha risk
level is 5% and the p value is 0.14, then you have to reject the hypothesis – in this case you’d reject that the line that was created is a suitable model as it was not able to create significant
Thank you, Barbara. It seems a zero missing in 0.14. We have updated the article.
Under “additional helpful resources, there is a link that has bad info. the author does not seem to understand the difference between x (independent variables and Y – response, dependent variable).
she consistently misuses it all the way through. Suggest, if I am correct, that you remove the link.
Step by Step regression analysis
You are right Barbara, especially the total multiple regression session was messed up. We removed the reference link– Thanks for your feedback.
“For example, if your alpha risk level is 5% and the p value is 0.014, then you have to reject the hypothesis – in this case you’d reject that the line that was created is a suitable model as it was
not able to create significant results.”
I believe this sentence is incorrect. If the p-value is less than the alpha value (0.014 < 0.05) then you would reject the null hypothesis – in this case you would accept the model as suitable
because the relationship between the independent and dependent variables is statstically significant. In other words, if the null hypothesis were true, you would expect these results to occur just
1.4% of the time which is lower than the threshold of 5%.
Hello Jesse Montgomery,
Thanks for the feedback. We have expanded the article for better clarity.
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Development of a Method for Shape Optimization for a Gas Turbine Fuel Injector Design Using Metal-Additive Manufacturing
Issue Section:
Research Papers
Computational fluid dynamics, Computer-aided design, Design, Geometry, Shape optimization, Shapes, Computer software, Parametrization, Manufacturing, Gas turbines, Optimization, Flow (Dynamics)
Adjoint shape optimization has enabled physics-based optimal designs for aerodynamic surfaces. Additive manufacturing (AM) makes it possible to manufacture complex shapes. However, there has been a
gap between optimal and manufacturable surfaces due to the inherent limitations of commercial computational fluid dynamics (CFD) codes to implement geometric constraints during adjoint computation.
In such cases, the design sensitivities are exported and used to perform constrained shape modifications using parametric information stored in computer aided design (CAD) files to satisfy
manufacturability constraints. However, modifying the design using adjoint methods in CFD solvers and performing constrained shape modification in CAD can lead to inconsistencies due to different
shape parameterization schemes. This paper describes a method to enable the simultaneous optimization of the fluid domain and impose AM manufacturability constraints, resolving one of the key issues
of geometry definition for isogeometric analysis. Similar to a grid convergence study, the proposed method verifies the consistencies between shape parameterization techniques present within
commercial CAD and CFD software during mesh movement as a part of the adjoint shape optimization routine. By identifying the appropriate parameters essential to a shape optimization study, the error
metric between the different parameterization techniques converges to demonstrate sufficient consistencies for justifiable exchange of data between CAD and CFD. For the identified shape optimization
parameters, the error metric to measure the deviation between the two parameterization schemes lies within the AM laser-powder bed fusion (L-PBF) process tolerance. Additionally, comparison for
subsequent objective function calculations between iterations of the optimization loop showed acceptable differences within 1% variation between the modified geometries obtained using the two
parameterization schemes. This method provides justification for the use of multiphysics guided adjoint design sensitivities computed in CFD software to perform shape modifications in CAD to
incorporate AM manufacturability constraints during the shape optimization loop such that optimal designs are also additively manufacturable.
Issue Section:
Research Papers
Computational fluid dynamics, Computer-aided design, Design, Geometry, Shape optimization, Shapes, Computer software, Parametrization, Manufacturing, Gas turbines, Optimization, Flow (Dynamics)
1 Introduction
Gradient-based shape optimization using the adjoint method has been used to effectively compute design sensitivities in the case of complex objective functions with a large number of design variables
[1–6]. Several applications including heat transfer [7–9], microfluidic flow [10–13], structural optimization [14–18], and aerodynamic optimization [19–29] have used the adjoint method to achieve
optimal geometries through high-fidelity computational modeling.
Manufacturing and testing these novel, optimal geometries using traditional manufacturing methods can be challenging and expensive. The advent of additive manufacturing (AM) processes has empowered
engineers with wider design flexibility. AM provides the opportunity to rapidly prototype and fabricate optimized geometries obtained from high-fidelity optimization routines employing sophisticated
computational fluid dynamics (CFD) and adjoint solvers. Complex shapes such as lattice geometries, internal channels, and free-form geometries can be manufactured using AM. In 2020, 29% of all AM
system sales revenue was contributed by aerospace, turbine, and helicopter industries [30]. The market potential of metal AM in the field of aerospace has been predicted to grow to $3.187 billion by
2025 at a compounded annual growth rate of 20.24% [31]. Therefore, there is growing interest in incorporating AM methods to fabricate novel, optimal designs. However, whether it is traditional or
additive manufacturing, there are limitations to what can be produced [32–35]. Current and past adjoint optimization literature did very little to address the limitations of manufacturing. While
there are papers that make use of geometric constraints imposed during adjoint optimization, the full ability of the computer aided design (CAD) parameterizations has not been used to incorporate
process-specific manufacturability constraints [36–43]. CAD parameterizations are favorable for manufacturing because select parameterizations result in continuous surfaces that have been shown to
accept geometric constraints that are reflective of manufacturability limitations [44].
While simultaneously optimizing a flow quantity using the adjoint method and imposing AM constraints, a key consideration is the difference in the parametrization schemes used by commercially
available, high-fidelity adjoint CFD tools and the parameterization scheme that exists within CAD. Various kinds of shape parametrization schemes exist for different applications and their importance
in shape optimization have been highlighted by Samareh [45]. Commercial CFD software typically use radial basis functions (RBFs) that discretize the geometry into elements and morph the mesh with
respect to a given set of handles or control points [46–50], whereas CAD typically uses nonuniform rational basis splines (NURBS), which is a continuous representation for surfaces [51–54].
Currently, imposing geometric constraints during RBF-based mesh morphing remains a challenge within commercial software, which leads to creation of features such as sharp edges that might be optimal
as per the numerical approximation of the computed solution but might be challenging, or even impractical, to reproduce during fabrication. NURBS surfaces handle such issues and ensure smoothness
during deformation in addition to providing an exact representation of complex, freeform geometries that might result from such shape optimization routines. Several researchers [55–58] have proposed
various methods of incorporating NURBS-based mesh morphing. However, implementation of the proposed methods has yet to be integrated with commercially available CFD software. Thus, in the current
state of the process, the adjoint shape gradients are computed on the mesh defined by the RBF interpolation function where a point set controls the shape of the geometry under consideration. These
shape gradients are then exported and used to guide shape changes in CAD. The use of different parameterization schemes used in CFD and CAD makes it challenging to exchange data to automate the
process of simultaneously optimizing flow using CFD and imposing manufacturability constraints.
The objective of this study is to develop a method to enable simultaneous optimization of flow fields using the adjoint method and impose design for additive manufacturing (DfAM) constraints. This
objective was achieved by augmenting a commercial CFD adjoint routine to externally modify CAD geometry and impose DfAM constraints that are incompatible with current commercial solvers. The key
challenge of geometry definition was resolved by ensuring consistencies in the adjoint-guided shape deformation between NURBS-interpolated surfaces in CAD and RBF-interpolated surfaces within a
commercially available CFD. Showing that this consistency in surface deformation lies within AM laser-powder bed fusion (L-PBF) process tolerances justifies the use of the adjoint shape gradients
computed on an RBF interpolated surface within a commercially available CFD solver to externally modify a NURBS interpolated surface within CAD to ensure that the optimal design is also additively
manufacturable. The graphical abstract highlighting the elements of the study is shown in Fig. 1.
This paper is organized as follows. Section 2 describes the proposed methodology of augmenting a commercial CFD adjoint shape optimization routine while emphasizing on shape parameterization
consistencies, Sec. 3 demonstrates a test case on a simplified airfoil geometry, Sec. 4 presents the analysis on a complex airfoil geometry within the fuel injector of an industry relevant gas
turbine engine, and Sec. 5 discusses the concluding statements and future work.
2 Methodology
The methodology's core motivation was the integration of CAD-based DfAM constraints into physics-guided shape optimization. This section examines the current adjoint shape optimization routines
within commercial CFD solvers, followed by delineating the steps to augment them with DfAM constraints. Additionally, this discussion addresses the assumptions and premises essential for ensuring
consistency, forming the basis for justifying the exchange of information between software during the augmentation process.
2.1 Adjoint Computational Fluid Dynamics Shape Optimization Setup.
The software star-ccm+ 2021.3 was used to demonstrate the augmentation process of the adjoint shape optimization routine. This software was chosen because of its widespread use to perform
multifidelity physics simulations for rapid industry adoption. The process of adjoint shape optimization consists of the following steps:
1. Defining the design variables: The CAD file containing the fluid domain is imported. The inlets, outlets, and walls are assigned the initial and boundary conditions. The design variables are
defined by importing a .csv table containing the three-dimensional coordinates of the control points for the surfaces to be modified. Most commercial software impose an RBF-based interpolation
between the control points and the surface to be modified during the shape optimization routine [59].
2. Meshing: To accommodate the scale of the simulation domain while minimizing the computational time, structured prism cells were used for the near-wall regions while polyhedral meshing was used
for the free-stream flow regions.
3. Setting up the primal solver: The primal solution establishes the relationship between the input design and the resulting flow variables by satisfying the principles of a physical continuum. The
primal solver was set up for configuring a steady, non-reacting Reynolds-averaged Navier–Stokes (RANS) simulation to satisfy the mass and momentum conservation. The κ–ε turbulence model was used.
The working fluid was defined as low-velocity incompressible air with constant density. Both the test cases used the same physics set up with appropriate initial and boundary conditions. The
simulations were run for a user-specified number of iterations to achieve proper convergence in all simulations.
4. Setting up the adjoint solver: The adjoint solver computes the sensitivities of an objective function, which is a flow field variable, with respect to the geometry to either maximize or minimize
its value. To represent a custom region of interest (ROI) over which the objective function was computed, a flag-based field function was created to compute flow quantities of interest and
defined as the adjoint objective function. The adjoint solver was set to run for a specified number of iterations until the residuals converged.
5. Computing mesh sensitivity: The mesh sensitivity is computed using the primal and adjoint results. The mesh sensitivity provides the change in the objective function because of changes in the
mesh geometry and is represented as
is the objective function and
represents the mesh vertices. When using design variables for mesh deformation, the mesh sensitivity is combined with the shape parameterization scheme being used to ensure a computationally
efficient mesh movement operation. As discussed earlier, most commercial CFD solvers, uses RBF to parameterize the mesh movement. Therefore, the shape sensitivity is computed as
is the control points that are used to parameterize the mesh movement and
is the relationship between the control points and the mesh vertices (RBF in this case).
6. Deforming the mesh: The mesh is deformed by modifying the control points and obtaining new control points using
$αnew =αold+(step size×dJdαold)$
Like any gradient-based optimization, a scalar step size is added to avoid over-shooting the optimal solution and specifically to avoid large mesh deformations in this case. The sign of the
step-size also determines if the optimizer maximizes or minimizes the objective function. The objective function is maximized for a positive step size and minimized for a negative step size.
Steps 3–6 are repeated until the objective function is optimized, resulting in an optimized geometry.
2.2 Augmenting the Adjoint Shape Optimization Routine.
During a standard adjoint shape optimization routine, there is limited provision in current commercial solvers to impose geometric constraints while deforming the mesh. Therefore, to simultaneously
optimize fluid flow using the multiphysics adjoint while imposing DfAM constraints, augmentation of the standard adjoint shape optimization routine is proposed. After computing the shape
sensitivities as described in step 6 of the adjoint shape optimization routine, the shape sensitivities along with the control point locations were exported from the CFD software. These shape
sensitivities, combined with the defined step size, provide displacement vectors which were used to modify the CAD parametric data externally and obtain a modified CAD geometry. Since the
modification is performed externally, the DfAM constraints can be incorporated as a CAD operation followed by re-inserting the modified CAD into the CFD adjoint shape optimization loop. The proposed
method for augmentation of the adjoint shape optimization loop is shown in Fig. 2.
During the proposed augmentation process, two key considerations were addressed. The first consideration was to automate the CAD modification process to enable re-insertion of the modified CAD into
the CFD adjoint shape optimization loop. The second consideration was to ensure that the control point displacement data exported from the CFD solver would result in a consistent shape change in CAD
despite different parameterization schemes used in both operations.
2.2.1 Automating the Computer Aided Design Modification.
The CAD modification was automated in Python using the NURBS-Python (geomdl) library [60]. To define a NURBS surface, the control point coordinates, degrees of polynomials, and knot vectors need to
be specified [61]. These parameters are available through the initial graphics exchange specifications (IGES) file format under the “Type 128” data header [62]. The NURBS control point coordinates
for the surfaces to be modified were extracted from the IGES file and defined as the design variables as defined in step 1 of the adjoint CFD shape optimization setup. Reading the control point
coordinates stored in the IGES file, modifying each point using the displacement vectors from CFD software, and replacing the modified control points in the IGES file resulted in a modified CAD
geometry. This process is illustrated in Fig. 3. Additionally, the IGES file format is a standard exchange file format that can be directly imported into CFD software for meshing. Since the input to
the CFD software was the flow domain, the IGES file of the flow domain was used to extract the NURBS parameters and automate the CAD modification process. Having full control over the CAD
parameterized geometry enabled the creation and enforcement of NURBS-based DfAM constraints during the Python modification step. The formulation of DfAM constraints like limits on thin walls and
overhang angles has been discussed in detail in a separate publication along with ensuring geometric continuities during CAD shape modification [63].
2.2.2 Ensuring Consistencies Between Computer Aided Design and Computational Fluid Dynamics Shape Changes.
As discussed, to augment the existing adjoint shape optimization process the control points from CAD must be defined within the CFD solver. Further, the adjoint shape sensitivities computed within
the CFD solver must be used to externally modify the CAD geometry using Python while imposing DfAM constraints. The first step to ensure consistencies during shape change was defining the CAD NURBS
control points as design variables in the CFD software. Since the adjoint shape sensitivities were computed on the NURBS control points, the displacement values for each control point were directly
used to obtain the new set of NURBS control points. However, despite sharing the same control point coordinates, most commercial CFD solvers, use RBF to interpolate the underlying surface during mesh
deformation. Therefore, the same displacement for colocated control points did not trivially imply the same resultant shape change due to different parameterization schemes.
Expanding on the shape sensitivity calculations as mentioned in step 5 of the adjoint CFD shape optimization routine, Samareh [
] discussed that in CFD, during mesh morphing the adjoint gradient is enforced on the mesh and underlying surface geometry using
$dJdα= ∂J∂V∂V∂X∂X∂S∂Sdα$
is the volumetric mesh grid,
is the surface mesh, and
is the surface being optimized. The last term on the right-hand side,
is dependent on the shape parameterization function. For commercial CFD solvers, RBF is the interpolation scheme, and this can be expressed as
$(dJdα)RBF= ∂J∂V∂V∂X∂X∂SRBF∂SRBF∂αRBF$
whereas for CAD the interpolation is NURBS-based and can be expressed as
$(dJdα)NURBS= ∂J∂V∂V∂X∂X∂SNURBS∂SNURBS∂αNURBS$
In Eqs. (4) and (5), the first three terms on the right-hand side are built into the grid generation tools that are specific to the commercial software package. Hence, for a given geometry, it is the
same in both cases. Therefore, if it can be shown that for α[RBF]=α[NURBS], if $∂SRBF/∂αRBF≈ ∂SNURBS/∂αNURBS$, then $(dJ/dα)RBF≈ (dJ/dα)NURBS$. For the scope of this paper, the manufacturability
tolerance using AM L-PBF was defined as the critical value within which the difference between $∂SRBF/∂αRBF$ and $∂SNURBS/∂αNURBS$ were considered insignificant. The AM L-PBF process tolerance was
identified as 250μm [64] for Inconel-based alloys, which are the choice of alloys for fabrication of gas turbine hot-section components. Thus, if $|∂SRBF/∂αRBF −∂SNURBS/∂αNURBS|<250 μm$, then $(dJ/
dα)RBF≈ (dJ/dα)NURBS$.
Both RBF [65–68] and NURBS [69–73] have been studied extensively. Researchers have found that increasing the number of control points improves the accuracy of surface representation and provides
better local control. This trend suggests that with fewer control points, each control point's displacement affects a larger area of the surface, leading to more significant variations between the
two interpolation schemes (RBF and NURBS) for the same displacement on colocated control points. Conversely, with a higher number of control points, each surface point moves independently, resulting
in minimal variation between the two schemes. By selecting an appropriate distribution of control points, the difference in errors between the interpolation schemes can be kept within the tolerance
of the AM L-PBF process. Like a grid resolution study in CFD, a NURBS resolution study helps verify the assumptions being made during parameter selection for such shape optimization routines. With
the difference between the two modified surfaces lying within the AM process tolerance, the reproducibility of both designs during the manufacturing process is indistinguishable. Showing this
assumption justifies the use of the adjoint gradient computed on a high-resolution RBF parameterized geometry in CFD to modify the same high resolution NURBS parameterized geometry in CAD. One of the
main advantages of using the adjoint method is that the computation time is independent of the number of design variables being used and rather depends on the complexity of the objective function
being computed. This warrants the use of a sufficiently large number of design variables that balance the tradeoff between maintaining shape consistencies between CFD and CAD while being
computationally efficient during CAD modification and DfAM constraint calculation.
To understand how the parameter selection affected the consistencies in shape changes between CFD and CAD, the two relevant parameters identified were the number of control points used as design
variables and the step size of optimization. Geomagic Design X was used to obtain custom NURBS parameterization of designs to study the effect of number of control points. Step sizes of 1, 2, 5, 10,
25, 50, 75, and 100μm were chosen to study the effect of step size.
Geomagic Control X [
], an advanced inspection software, was used to compare the CAD and CFD deformations. The sampling ratio was set to 100%, which enforced the computation of deviations between all points of the
measured geometry
P[m]=〈x[m], y[m], z[m]〉
and its projection on the surface
P[R]=〈x[R], y[R], z[R]〉.
Using the projection along the shortest normal vector from the measured points to the reference geometry, it created a three-dimensional deviation map using the formula
where GV=〈x[m]−x[R], y[m]−y[R], z[m]−z[R]〉.
The reference geometries in this case were the IGES files generated in Python after adding the displacements from the CFD software, and the measured geometry were the STL files exported from the CFD
software after the RBF based mesh deformation was performed. In order to account for a surface averaged quantity, the error metric (
) was defined as the root-mean-squared (RMS) difference between the RBF and NURBS surfaces were calculated using the formula
$ε=R.M.S.(∂SRBF∂αRBF −∂SNURBS∂αNURBS)=1npoints∑i=1npointsDi2$
In addition to the RMS value, the maximum difference was also noted to account for the extreme deviations between the two surfaces.
3 Demonstration of Test Case Using an Airfoil Geometry
A simple airfoil geometry was chosen for performing an initial test to study the difference between NURBS- and RBF-based surface modifications with respect to number of control points and step sizes.
Different NURBS configurations of the airfoil were created with 96, 128, and 640 control points each. The adjoint shape optimization was setup in the CFD software as described in Sec. 2.1 with the
objective of maximizing the lift to drag ratio by changing the airfoil geometry using the control points. The primal solution was computed over 500 iterations and the adjoint solution was computed
over 250 iterations. The step sizes were enforced during the mesh deformation stage after the first loop of primal and adjoint computations. The STLs of the RBF morphed mesh were exported from the
CFD software along with the displacement vectors computed on the control points that were used to create the NURBS modified CAD geometry. The error metric for all configurations of number of control
points and step sizes are shown in Fig. 4. As expected, it was observed that increasing the refinement of the control point set resulted in reduction of the error metric between the RBF and NURBS
surfaces, thus displaying convergence. For all configurations, increasing step size resulted in increasing the error metric between the NURBS and RBF deformed shapes. However, for a larger number of
control points, the error metric was below the AM L-PBF process tolerance for step sizes under 25μm. Thus, having a greater number of control points provides the opportunity to use a larger step
size while still ensuring consistencies between the two parametrization schemes remain under the AM L-PBF process tolerances. Using larger step sizes allows for fewer optimization loops, reducing the
total computational time for the optimization. However, using a significantly large step size could lead to overshooting the optimal solution and mesh intersection errors. Thus, it is important to
refine the step size selection process to choose the step size based on the tradeoff between computational time and solution accuracy.
4 Minimizing Flame Flashback Propensity for an Industrial Gas Turbine
To test the methodology on a gas-turbine relevant case study with sufficiently complex geometry parts suitable for AM L-PBF fabrication, we considered the issue of shape optimization to minimize the
flame flashback propensity in an industrial gas turbine combustor. Due to rising environmental concerns such as climate change [75,76], there is an increasing trend within the energy sector to
evaluate the use of alternative fuels for production of cleaner energy. One of the most promising alternative fuels has been the incorporation of hydrogen (H[2]) in natural gas to reduce CO[2]
emissions [77]. However, introduction of H[2] in the fuel mix leads to an increase in the turbulent flame speed. Flashback occurs when the flame propagates toward the upstream gases at velocities
higher than the incoming flow velocity. This propagation propensity is the highest along the turbulent boundary layer region near the edge of the center body where the flame sits [78–85]. For this
study, the chosen objective function was to maximize the volume-averaged velocity magnitude of the turbulent boundary layer region around the edge of the center body of the fuel injector. The
maximization was to be achieved by modifying the shape of the external airfoil surfaces of the swirler vanes.
The baseline geometry of the swirler vanes was reverse engineered to fit 850 NURBS surface patches using Geomagic Design X. Each NURBS surface patch consisted of 64 control points (8×8
bidirectional net) with a degree-three polynomial in the u- and v-directions in order to maintain C2 continuity along the surface, resulting in 54,400 total control points. These points were also
defined as the RBF control points in star-ccm+. Both parameterized geometries are shown in Fig. 5.
The injector design under consideration represented an annular configuration of axisymmetric swirler vanes with a proprietary nonaxisymmetric fuel delivery mechanism to accommodate industrial
complexity. Furthermore, in order to investigate any effects of the upstream and downstream flow on the adjoint computation, the flow domain of the experimental test rig for an axisymmetric section
of the turbine engine was modeled for the simulations. Surfaces representing the inlets for air and fuel were defined as mass flow inlets, the outlet for the exhaust was defined as a pressure outlet,
and the remaining surfaces including the vanes were defined as adiabatic walls. A grid convergence study was performed using seven different mesh sizes ranging in a total cell size of 2–26 million
cells. The final adaptive mesh representation contained ∼4.5 million cells and was achieved through a base size of 4mm with varying custom meshing controls applied from the combustor section with a
value of 10% (0.4mm) of the base size to add refinement in the swirler regions to a value of 250% (10mm) relative to the base size for the regions in the exhaust region. The turbulent boundary
layer thickness for the given flow conditions was calculated as 3mm using the Blasius wall friction profile as described in Ref. [78]. A 1:10 ratio for thickness to length ratio of the annulus
region was used to represent the ROI to compute the adjoint objective function value. The mesh configuration with the objective function ROI is shown in Fig. 6.
The flow solver was setup using steady RANS with the while imposing the κ–ε turbulence model. A fully premixed flow of 0.0632kg/s preheated at 523K entered the inlet to maintain a bulk flow
velocity of 40m/s in the main duct. The boundary conditions were matched as per Ref. [86].
The optimization was performed for one loop using the adjoint CFD shape optimization method as well as the augmented method with external CAD modification. The primal solution was computed over 1500
iterations to achieve residual convergence for flow and the adjoint solution was computed over 500 iterations to achieve residual convergence for the objective function. The CFD adjoint shape
optimization was performed through batch parallel processing using Penn State's ROAR supercomputing cluster using 96 cores. The primal solution took 1.5h for 1500 iterations. The meshing, adjoint
solver, computation of mesh sensitivity, and mesh deformation took 15, 45, 6, and 8min, respectively. Modification of the IGES CAD file using Python was performed on a laptop notebook with four
cores, 16 GB RAM and took 225s from reading the file to creating the modified file.
Figure 7 shows the graph of error metric of the RBF and NURBS vanes surfaces for increasing step sizes. It was observed that for the chosen configuration of 54,400 control points the error metric
between the two results of modified surfaces was below the AM L-PBF process tolerance of 250μm for all step sizes. The maximum RMS difference observed was 0.0701mm for the step size of 100μm,
while the minimum RMS difference was 0.0472mm for the step size of 1μm. Additionally, Fig. 8 shows the maximum difference between RBF and NURBS surfaces obtained from the CFD software and Python,
respectively. It was found that the maximum difference between RBF and NURBS modified surfaces lied below the AM L-PBF process tolerance of 250μm for step sizes less than 50μm. The maximum values
were affected by the large magnitudes of the adjoint gradients that were present in high sensitivity zones of the objective function with respect to the geometry. Overall, 25μm was determined to be
the largest step size using which the maximum difference between NURBS and RBF were consistent within AM L-PBF process tolerances.
In addition to showing geometric consistencies, the objective function changes due to the different parameterizations were also investigated. The CAD modified flow domain was re-imported into the CFD
software's adjoint shape optimization loop. The objective function evolution for this modified flow domain and the baseline flow domain. The temperature profiles for a plane section of the flow
region were also compared between the adjoint CFD shape optimization and the augmented method after re-inserting the CAD modified flow domain. This comparison was performed by exporting the
coordinates of the mesh vertices for the plane section geometry of the NURBS and the RBF deformed flow domains and its associated physical quantities in a .csv file. Since the NURBS modified flow
domain had to be re-imported and remeshed, the coordinates of the mesh vertices slightly varied from the baseline. Thus, for each mesh vertex, the closest mesh vertex in the baseline plane section
was found and their associated physical quantities were compared.
Figure 9 shows the evolution of the objective function for the baseline (RBF deformation) and the NURBS modified flow domain restarted after the first loop shape changes. The objective function for
both geometries were consistent within 1% variation with the value of the volume averaged velocity magnitude computed over the objective function region for the baseline optimization loop being
64.1m/s and for the NURBS parameterized geometry of the equivalent shape change being 64.5m/s. The subsequent loops showed consistent values for each shape change. Additionally, the NURBS
modification resulted in a much smoother shape change for the same adjoint gradients and step size as compared with the RBF modification that had sharper edges.
The net effect of the shape changes on the physical quantities after solving the primal physics on both NURBS and RBF modified geometries is shown in Fig. 10. The velocity magnitude and axial
velocity plots were compared for both the shape changes. For this comparison, the values compared at corresponding vertices between the two shapes were mostly zero. The small non-zero difference
values of the quantities were attributed to the minor shape changes between the two parameterization schemes. The larger non-zero difference values of the quantities were an artifact of the
variability between different iterations within the residual convergence region of the solution approximation. The arguments that supported this claim were that the large variations were isolated
mesh vertices and most of the instances were observed upstream of the swirler where no geometric changes were made or in regions where the mesh size was relatively coarse and hence any change in
values were an artifact of small variations within the residual convergence region and remeshing differences. Additionally, the difference in the pressure drop computed across the injector between
the RBF and NURBS modified geometry was 0.188%. Thus, the physical quantities obtained using NURBS deformed geometries were consistent with RBF deformed geometries while providing a smoother shape
and a standardized CAD exchange file format.
5 Conclusion
This study proposed a method to enable simultaneous optimization of flow field and imposing DfAM constraints by justifying the use of adjoint gradients computed on RBF parameterized surfaces in CFD
to guide the movement of NURBS parameterized surfaces in CAD as a motivation to perform constrained isogeometric optimization. Using a test case, it was verified that like a CFD grid resolution
study, a NURBS control point density resolution study needs to be performed to obtain a sufficient middle ground between surface modification consistency and computational efficiency. The results of
an adjoint shape optimization of the baseline swirler vanes geometry in CFD using RBF compared to NURBS modification in CAD showed that the RMS difference between the surfaces were consistent within
the AM L-PBF process tolerances for all step sizes. It also showed that the maximum difference between the two surfaces were consistent within the AM L-PBF process tolerance below step sizes of 25μ
m. To consolidate consistency in CFD behavior, the objective function value computed for both geometries were within 1% of each other as well as the physical quantities computed using the CFD primal
solver were within residual convergence tolerances. The process for external modification of the CAD surfaces added minimal computation overhead, however, the resubmission of the modified geometry in
the optimization loop within the CFD software added additional computational time due to remeshing of the newly imported geometry. This additional step added a 8.92% increase in computation time per
optimization loop, which can be significant for problems requiring more number of loops to converge to a solution and for complex/large geometry mesh setups. Therefore, this warrants the current and
future integration of NURBS into CFD flow software to not only enable NURBS-based design constraints but also improved computational efficiency.
Currently, the adjoint solver can accept limited flow field variables and user defined operations as the objective function and largely uses simplified steady nonreacting RANS physics models to
compute those. Moreover, the shape change caused by imposing DfAM constraints can be favorable or against the displacements due to the adjoint physics depending on the geometry, chosen objective
function, build orientation, and other dominating factors. The future work scope includes defining and imposing such DfAM constraints using the NURBS representation in the design loop to obtain novel
optimal geometries and study its effects on the adjoint objective function. With developments in commercial solvers, more complex models will be compatible as adjoint cost functions in the future to
address critical issues such as emission control, fuel flexibility, and management of thermoacoustic instabilities to enhance gas turbine performance through combustor design modifications. The
method proposed can also be used to compare different parameterization schemes and show similar justifications across different commercial finite element solvers for a wide range of applications. As
AM process tolerances improve, the error metric will become smaller to justify the use of different parameterization schemes. However, the adoption of NURBS-based mesh movement being integrated
directly into commercial solvers would facilitate the justification of data exchange between finite element solvers and CAD programs. The proposed method augments a commercial finite element adjoint
shape optimization routine and shows justification to exchange data to ensure that optimal designs are additively manufacturable for a wide range of applications.
The authors are grateful for the financial support provided by the U.S. Department of Energy University Turbine Systems Research (DoE UTSR) Program Grant No. DE-FE0031806 under contract monitor Mark
Freeman. The authors would also like to thank Jinming Wu, Pratikshya Mohanty, and Drue Seksinsky for their help in setting up the simulations.
Funding Data
• U.S. Department of Energy University Turbine Systems Research (DoE UTSR) Program (Grant No. DE-FE0031806; Funder ID: 10.13039/100000015).
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
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Scientific proof of psi phenomena (Repost from AMNAP 1.0)
The phrase "Scientific Proof" is a high standard to acheive. In science, a phenomena is considered proven if it has met the standard of multiple independent replications, as determined through
meta-analysis of all the data available. And certain experiments demonstrating psi phenomena have easily met that standard of proof.
The first question to answer is "what is the meaning of replication". A simple naive belief in replication is that it refers to a phenomenon which can be demonstrated at the 95% confidence level in
every single experiment.
Unfortunately, in science involving huge numbers of uncontrollable variables such as human beings, this sort of replication almost never happens. Instead, replication is a statistical phenomenon.
In order to illustrate this phenomenon, Dean Radin selects the example of studies on aspirin as a preventative for second heart attacks in
his seminal book on the meta-analysis of psi, The Conscious Universe
. Today, everyone knows that aspirin is an effective preventative treatment for heart attacks. Why is this an accepted scientific fact? Because a large meta-analysis of multiple studies comparing
aspirin to placebo showed an overall significant effect far beyond the chance expectation. See Radin's figure 4.2 below. Note that the vertical line with horizontal endpoints in these charts shows
the 95% confidence interval with the actual measurement value in the center.
Notice that
only 5 of the 25 individual studies
actually returned statistically significant results on their own. If we relied on statistical significance of individual studies, we would say "aspirin's effects on heart disease can't be replicated"
because of all these individual "failed studies". In fact, 3 of the 25 studies showed a (non-significant)
negative effect
from aspirin versus placebo!
That is why we need to use a meta-analysis of studies from multiple independent researchers
. The combined meta-analysis clearly shows us that aspirin has a statistically significant effect in preventing heart attacks. Aspirin therapy has gone up against the most rigorous examination
possible and come out with the scientific seal of approval.
So what happens when we examine the evidence for psi phenomena? Certain categories of psi experiments have been extensively conducted at independent institutions by seperate research teams. These psi
phenomena have all been subject to meta-analysis by Dean Radin and other independent meta-analysists, including skeptics such as Ray Hyman. And the effects are astronomically significant. For certain
types of experiment such as the Ganzfeld and auto-ganzfeld, the time, effort and expense means that most of the data which could have been collected has been included in these meta-analyses, so no
possible "file drawer" effects can even exist.
Below I have reproduced Radin's charts for meta-analysis of dream telepathy experiments, the 1985 Ganzfeld meta-analysis by Hyman and Honorton, an updated Ganzfeld meta-analysis, high-security ESP
card tests, RNG PK experiments and dice-rolling PK experiments. Although all of these meta-analyses include data from trials showing non-significant effects, the overall meta-analysis is clear. These
phenomena all show enormous, often astronomical deviations from the null hypothesis.
So the answer is clear. Certain psi phenomena have gone up against the most rigorous examination possible and come out with the scientific seal of approval. So why do so many scientists and
"rationalists" think that psi is "nonsense", "without a shred of real evidence"? I'm afraid that is more of a sociological question than a scientific one.
6 comments:
Well, I won’t rehash what I wrote before about Radin's ganzfeld meta-analyses, but if anyone’s interested, this thread at the JREF forums is a good place to start.
On a wider perspective, Kennedy has recently written a couple of interesting articles in which he discusses the use of meta-analyses in parapsychology. In 2004 he wrote “Meta-analysis is
ultimately post hoc data analyses when researchers have substantial knowledge of the data. Evaluation of the methodological quality of a study is done after the results are known, which gives
opportunity for biases to affect the meta-analysis. Different strategies, methods, and criteria can be utilized, which can give different outcomes and opportunity for selecting outcomes
consistent with the analyst's expectations.” ("A Proposal and Challenge for Proponents and Skeptics of Psi", Journal of Parapsychology, vol 68)
Meanwhile in 2005 he wrote “One of the most revealing properties of psi research is that meta-analyses consistently find that experimental results do not become more reliably significant with
larger sample sizes as assumed by statistical theory (Kennedy, 2003b; 2004). This means that the methods of statistical power analysis for experimental design do not apply, which implies a
fundamental lack of replicability.
This property also manifests as a negative correlation between sample size and effect size. Meta-analysis assumes that effect size is independent of sample size. In medical research, a negative
correlation between effect size and sample size is interpreted as evidence for methodological bias (Egger, Smith, Schneider, & Minder, 1997).
The normal factors that can produce a negative correlation between effect size and sample size include publication bias, study selection bias, and the possibility that the smaller studies have
lower methodological quality, selected subjects, or different experimenter influences. All of these factors reduce confidence in a meta-analysis. However, for psi experiments, the failure to
obtain more reliable results with larger sample sizes could be a manifestation of goal-oriented psi experimenter effects or decline effects (Kennedy, 1995; 2003a). Even if these effects are
properties of psi, parapsychologists cannot expect that other scientists will find the experimental results convincing if methods such as power analysis cannot be meaningfully applied. Further,
for the past two decades, the debates about the reality of psi have focused on meta-analysis. The evidence that psi experiments typically do not have properties consistent with the assumptions
for meta-analysis adds substantial doubts to the already controversial (Kennedy, 2004) claims about meta-analysis findings in parapsychology.” (“Personality and motivations to believe,
misbelieve, and disbelieve in paranormal phenomena”, Journal of Parapsychology, 69)
More of his work can be found on his site:
One of the most revealing properties of psi research is that meta-analyses consistently find that experimental results do not become more reliably significant with larger sample sizes as assumed
by statistical theory
It would be interesting and see if the same holds true with medical clinical studies. I suspect many other areas of science show the same kinds of "anomalies" in their statistics as psi research.
What are you basing your suspicions on?
Kennedy is someone who's been involved in parapsychology for many years and now works in pharmaceuticals. Since the point being made by Radin (and others - I've seen this argument several times)
is that compared to the data for aspirin and heart attacks, psi meta-analyses are every bit as robust. But Kennedy is now, with direct experience of both fields of research, saying this is not
the case.
He wrote: "One of the most revealing properties of psi research is that meta-analyses consistently find that experimental results do not become more reliably significant with larger sample sizes
as assumed by statistical theory" If you take the graph of aspirin experiments and rearrange them so the largest (ie, with the smallest confidence interval) is at one end and the smallest is at
the other, then you’ll see that it approximates a funnel. In other words, as the number of trials gets larger, so the results converge on one level. (If I remember right, there is one experiment
that sticks out a bit, though). Do that with the ganzfeld data and there’s no such shape. Radin has a funnel graph in The Entangled Mind, but since his meta-analysis has no inclusion criteria, it
is largely meaningless.
If parapsychology wants to be measured by the same standards as mainstream science, which is the point Radin is trying to make, I believe, then surely points such as "In medical research, a
negative correlation between effect size and sample size is interpreted as evidence for methodological bias" should be given as much weight as the more positive conclusions.
Radin has a funnel graph in The Entangled Mind, but since his meta-analysis has no inclusion criteria, it is largely meaningless.
How do you know that it has no inclusion criteria?
What are you basing your suspicions on?
That experiments in all different areas of probabalistic multifactorial systems show sheep/goat effects, funding bias effects and the like. I suspect psi is itself responsible for this, and not
just sloppy science and bogus post-hoc data analysis.
The Entangled Mind m-a is an update of the Conscious Universe m-a. The CU work had no inclusion criteria that was applied across all experiments. As such I don't think it qualifies as a proper
meta-analysis. I go into more detail on the JREF forum (link in first comment).
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Craig ConstantineRubik's Cube – Craig Constantine
I don’t know why, but I never learned to solve a Rubik’s Cube. I am exactly the right age; the durned things appeared on the scene just before I got to primary school and they were common in my high
school. But I never got into it. I had one, of course. I pretty much immediately took it apart (very carefully) to see how it worked… just honestly curious about how it worked, not trying to solve
it. When I put it back together, I put it together in the solved state because it seemed obvious that if I put it together randomly it couldn’t be solved by then trying to rotation-solve it as usual.
Aside: Yes, of course I did. Any time I found a cube, I’d surreptitiously mechanically detach and flip a few pieces, and then scramble it. Few people are good enough to quickly figure out what has
…and then I never was interested in solving one after I understood how it worked. Tetris? Okay, yeah, that game ate years of my life—because you can’t solve it, you just do it. Anyway, I’m 50 and I
just got a Rubik’s Cube.
And what am I doing? Measuring it: Let’s call it 2.2 inches on an edge. How many of them are there? Wikipedia says 350,000,000. Crap, that’s a lot of plastic. How big a pile is that? How big are
350,000,000 2-inch cubes? …and I was hoping Wolfram Alpha would give me units of Empire-State-Buildings or something. Instead, I learned something about the total number of Angels according to the
Bible. (That should get you to click, no?)
What’s that? How many ESBs is it? …oh, sorry, it’s 0.0583 ESB. I know right? We’ve only 6% filled the ESB with Rubik’s Cubes?! We need to ramp up production.
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Quick start with R: Sub-setting (Part 9)
In Part 9, let’s look at sub-setting in R. Let’s provide summary tables on the following data set of tourists from different countries, the numbers of their children, and the amount of money they
spent while on vacation. Copy and paste the following array into R.
A <- structure(list(NATION = structure(c(3L, 3L, 3L, 3L, 1L, 3L, 2L, 3L, 1L, 3L, 3L, 1L, 2L, 2L, 3L, 3L, 3L, 2L, 3L, 1L, 1L, 3L, 1L, 2L), .Label = c("CHINA", "GERMANY", "FRANCE"), class =
"factor"),GENDER = structure(c(2L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 2L), .Label = c("F", "M"), class = "factor"), CHILDREN = c(2L, 1L, 3L, 2L,
2L, 3L, 1L, 0L, 1L, 0L, 1L, 2L, 2L, 1L, 1L, 1L, 0L, 2L, 1L, 2L, 4L, 2L, 5L, 1L), SPEND = c(8500L, 23000L, 4000L, 9800L, 2200L, 4800L, 12300L, 8000L, 7100L, 10000L, 7800L, 7100L, 7900L, 7000L, 14200L,
11000L, 7900L, 2300L, 7000L, 8800L, 7500L, 15300L, 8000L, 7900L)), .Names = c("NATION", "GENDER", "CHILDREN", "MONEY"), class = "data.frame", row.names = c(NA, -24L))
The generic form of the syntax we will use is as follows:
Z <- A[ A[ , colnum ] == val, ]
Note that we have two sets of square brackets and a comma just before the second closing bracket. Z gives all rows for which an indicator in column colnum has the value val. We can say it like this:
“Z is the set of rows of A such that the elements of column colnum have the value val”. OK. Let’s subset for females.
FE <- A[ A[, 2] == "F", ]
However, easier is the following syntax, using the subset() function:
subset(A, GENDER == "F")
Now isolate all rows for which the third column (number of children) is less than 2.
C1 <- A[ A[, 3] < 2, ]
However, easier is the following syntax, using the subset() function:
C1 <- subset(A, CHILDREN < 2)
Finally, we isolate all rows for Females with less than two children.
F1 <- A[ A[, 2] == "F" & A[, 3] < 2, ]
Again, easier is the following syntax, using the subset() function:
F1 <- subset(A, GENDER == "F" & CHILDREN < 2)
That wasn’t so hard! In blog 10 we will look at further analytic techniques in R.
See you soon!
Annex: R codes used
[code lang=”r”]
# Create and display the following array.
A <- structure(list(NATION = structure(c(3L, 3L, 3L, 3L, 1L, 3L, 2L, 3L, 1L, 3L, 3L, 1L, 2L, 2L, 3L, 3L, 3L, 2L, 3L, 1L, 1L, 3L, 1L, 2L), .Label = c("CHINA", "GERMANY", "FRANCE"), class =
"factor"),GENDER = structure(c(2L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 2L), .Label = c("F", "M"), class = "factor"), CHILDREN = c(2L, 1L, 3L, 2L,
2L, 3L, 1L, 0L, 1L, 0L, 1L, 2L, 2L, 1L, 1L, 1L, 0L, 2L, 1L, 2L, 4L, 2L, 5L, 1L), SPEND = c(8500L, 23000L, 4000L, 9800L, 2200L, 4800L, 12300L, 8000L, 7100L, 10000L, 7800L, 7100L, 7900L, 7000L, 14200L,
11000L, 7900L, 2300L, 7000L, 8800L, 7500L, 15300L, 8000L, 7900L)), .Names = c("NATION", "GENDER", "CHILDREN", "MONEY"), class = "data.frame", row.names = c(NA, -24L))
# The generic form of the syntax to be used.
Z <- A[ A[ , colnum ] == val, ]
# Subset and display the array for females.
FE <- A[ A[, 2] == "F", ]
# Alternatively, the same result could be achieved but using the subset() command.
subset(A, GENDER == "F")
# Isolate and display all rows for which the third column (number of children) is less than 2.
C1 <- A[ A[, 3] < 2, ]
# Alternatively, the same result could be achieved but using the subset() command.
C1 <- subset(A, CHILDREN < 2)
# Isolate and display all rows for Females with less than two children.
F1 <- A[ A[, 2] == "F" & A[, 3] < 2, ]
# Alternatively, the same result could be achieved but using the subset() command.
F1 <- subset(A, GENDER == "F" & CHILDREN < 2)
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Problem B. Gomoku
This is an interactive problem.
Gomoku is a two-player game on a two-dimensional grid. Each cell of the grid can be either empty, contain the first player’s mark (black), or contain the second player’s mark (white), but not both.
Initially the entire grid is empty. Two players make alternating moves, starting with the first player. At each move, a player can put her mark into exactly one empty cell. The first player to have
her five adjacent marks in a single row wins. The winning row can be either vertical, horizontal or diagonal.
Position where the second player (white marks) had won.
The players use a 19 × 19 grid in this problem. If the entire grid gets filled with marks but no player has won, the game is declared a draw.
The first player uses the following strategy: as the first move, she puts her mark into the center cell of the grid. At every other move, she picks such a move that maximizes the score of the
resulting position.
In order to find the score of a position, the first player considers all possible places where the winning combination might eventually form — in other words, all horizonal, vertical and diagonal
rows of five consecutive cells on the board (of course, they may overlap each other). If such a row contains both the first player’s marks and the second player’s marks, it is disregarded. If such a
row contains no marks, it is disregarded as well. For each row with exactly k (1 ≤ k ≤ 5) marks of the first player and no marks of the second player, add 50^2k−1 to the score of the position. For
each row with exactly k marks of the second player and no marks of the first player, subtract 50^2k from the score of the position. Finally, add a random integer number between 0 and 50^2−1 to the
score. This random number is chosen uniformly.
In case when several moves of the first player have equal scores (such ties are quite rare because of the random addition mentioned above), the first player picks the one with the smallest x
-coordinate, and in case of equal x-coordinates, the one with the smallest y-coordinate.
Your task is to write a program that plays the second player and beats this strategy.
Your program will play 100 games against the strategy described above, with different seeds of random generator. Your program must win all these games.
Interaction protocol
On each step, your program must:
1. Read numbers x and y from the input.
2. If both these numbers are equal to −1 then the game is over and your program must exit.
3. Otherwise these numbers are the coordinates of the first player’s move (1 ≤ x, y ≤ 19).
4. Print the coordinates of the move of the second player, followed by line end. Don’t forget to flush the output buffer.
There are many variations of Gomoku rules in the world. Please only consider the rules described in this problem statement.
In the example below the first player does not use the strategy from the problem statement. The example is given only to illustrate the interaction format.
Final position from the example.
Sample tests
No. Standard input Standard output
-1 -1
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24 research outputs found
We examine the production of the Hoyle and associated excited states from the viewpoint of pocket resonances in the reaction of an $\alpha$-particle on a ground state prolate $^8$Be nucleus within
the optical model coupled-channel framework. The predicted reaction cross sections, as a function of the center-of-mass energy $E_{\rm cm}$, show prominent resonances, including the Hoyle resonance.
The positions and widths of these resonances are sensitive to the target deformation ($\beta_2$ parameter) and the parity of the nuclear surface potential $-$ deeper for the even-parity $L$ partial
waves relative to those for the odd-parity $L$ partial waves at the surface region because of the Bose-Einstein exchange of the $\alpha$-bosons. Decomposing the reaction cross sections to different
partial waves, we find that the resonance energies and widths reasonably agree with the available experimental data and previous hyperspherical calculations for the $0_2^+$ (Hoyle state), $0_3^+$,
$1_1^-$ and $3_1^-$ states of $^{12}$C, except for the narrow theoretical width of the $2_2^+$ state. Analyzing the wavefunctions and the resonance widths, we identify the narrow and sharp $0_2^+$,
$3_1^-$ and $2_2^+$ resonances as pocket resonances -- resonances which occur below the potential barrier, while the broad $0_3^+$ and $1_1^-$ resonances as above-the-barrier resonances. For
astrophysical applications, we also evaluate the astrophysical $S(E_{\rm cm})$-factor for $E_{\rm cm}$ $<$ 1.0 MeV, for the fusion of $\alpha$+$^8$Be into the $^{12}$C$(2^+)$ state based on our
estimated $s$-wave $\alpha$+$^8$Be reaction cross section and the associated $\gamma$- and $\alpha$-decay widths for the decay of $^{12}$C excited states in the potential pocket.Comment: 15 pages, 9
We calculated the charge-transfer cross sections for O⁸⁺ + H collisions for energies from 1eV/amu to 2keV/amu, using the recently developed hyperspherical close-coupling method. In particular, the
discrepancy for electron capture to the n = 6 states of O⁷⁺ from the previous theoretical calculations is further analyzed. Our results indicate that at low energies (below 100eV∕amu) electron
capture to the n=6 manifold of O7+ becomes dominant. The present results are used to resolve the long-standing discrepancies from the different elaborate semiclassical calculations near 100eV/amu. We
have also performed the semiclassical atomic orbital close-coupling calculations with straight-line trajectories. We found the semiclassical calculations agree with the quantal approach at energy
above 100eV/amu, where the collision occurs at large impact parameters. Calculations for Ar⁸⁺ + H collisions in the same energy range have also been carried out to analyze the effect of the ionic
core on the subshell cross sections. By using diabatic molecular basis functions, we show that converged results can be obtained with small numbers of channels
We present quantum mechanical close-coupling calculations of collisions between two hydrogen molecules over a wide range of energies, extending from the ultracold limit to the super-thermal region.
The two most recently published potential energy surfaces for the H$_2$-H$_2$ complex, the so-called DJ (Diep and Johnson, 2000) and BMKP (Boothroyd et al., 2002) surfaces, are quantitatively
evaluated and compared through the investigation of rotational transitions in H$_2$+H$_2$ collisions within rigid rotor approximation. The BMKP surface is expected to be an improvement, approaching
chemical accuracy, over all conformations of the potential energy surface compared to previous calculations of H$_2$-H$_2$ interaction. We found significant differences in rotational excitation/
de-excitation cross sections computed on the two surfaces in collisions between two para-H$_2$ molecules. The discrepancy persists over a large range of energies from the ultracold regime to thermal
energies and occurs for several low-lying initial rotational levels. Good agreement is found with experiment (Mat\'e et al., 2005) for the lowest rotational excitation process, but only with the use
of the DJ potential. Rate coefficients computed with the BMKP potential are an order of magnitude smaller.Comment: Accepted by J. Chem. Phy
We evaluate the interaction potential between a hydrogen and an antihydrogen using the second-order perturbation theory within the framework of the four-body system in a separable two-body basis. We
find that the H-Hbar interaction potential possesses the peculiar features of a shallow local minimum located around interatomic separations of r ~ 6 a.u. and a barrier rising at r~5 a.u. Additional
theoretical and experimental investigations on the nature of these peculiar features will be of great interest.Comment: 13 pages, 6 figure
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Eureka Math Grade 6 Module 2 Lesson 16 Answer Key
Engage NY Eureka Math 6th Grade Module 2 Lesson 16 Answer Key
Eureka Math Grade 6 Module 2 Lesson 16 Opening Exercise Answer Key
a. What is an even number?
Possible student responses:
An integer that can be evenly divided by 2
A number whose unit digit is 0, 2, 4, 6, or 8
All the multiples of 2
b. List some examples of even numbers.
Answers will vary.
c. What is an odd number?
Possible student responses:
An integer that CANNOT be evenly divided by 2
A number whose unit digit is 1, 3, 5, 7, or 9
All the numbers that are NOT multiples of 2
d. List some examples of odd numbers.
Answers will vary.
What happens when we add two even numbers? Do we always get an even number?
Before holding a discussion about the process to answer the following questions, have students write or share their predictions.
Eureka Math Grade 6 Module 2 Lesson 16 Exercise Answer Key
Exercise 1.
Why is the sum of two even numbers even?
a. Think of the problem 12 + 14. Draw dots to represent each number.
b. Circle pairs of dots to determine if any of the dots are left over.
c. Is this true every time two even numbers are added together? Why or why not?
Since 12 is represented by 6 sets of two dots, and 14 is represented by 7 sets of two dots, the sum is 13 sets of two dots. This is true every time two even numbers are added together because even
numbers never have dots left over when we are circling pairs. Therefore, the answer is always even.
Exercise 2.
Why is the sum of two odd numbers even?
a. Think of the problem 11 + 15. Draw dots to represent each number.
b. Circle pairs of dots to determine if any of the dots are left over.
When we circle groups of two dots, there is one dot remaining in each representation because each addend is an odd number. When we look at the sum, however, the two remaining dots can form a pair,
leaving us with a sum that is represented by groups of two dots. The sum is, therefore, even. Since each addend is odd, there is one dot for each addend that does not have a pair. However, these two
dots can be paired together, which means there are no dots without a pair, making the sum an even number.
c. Is this true every time two odd numbers are added together? Why or why not?
This is true every time two odd numbers are added together because every odd number has one dot remaining when we circle pairs of dots. Since each number has one dot remaining, these dots can be
combined to make another pair. Therefore, no dots remain, resulting in an even sum.
Exercise 3.
Why is the sum of an even number and an odd number odd?
a. Think of the problem 14 + 11. Draw dots to represent each number.
Students draw dots to represent each number. After circling pairs of dots, there is one dot left for the number 11, and the number 14 has no dots remaining. Since there is one dot left over, the sum
is odd because not every dot has a pair.
b. Circle pairs of dots to determine if any of the dots are left over.
Students draw dots to represent each number. After circling pairs of dots, there is one dot left for the number 11, and the number 14 has no dots remaining. Since there is one dot left over, the sum
is odd because not every dot has a pair.
C. Is this true every time an even number and an odd number are added together? Why or why not?
This is always true when an even number and an odd number are added together because only the odd number will have a dot remaining after we circle pairs of dots. Since this dot does not have a pair,
the sum is odd.
d. What if the first addend is odd and the second is even? Is the sum still odd? Why or why not? For example, If we had 11 + 14, would the sum be odd?
The sum is still odd for two reasons. First, the commutative property states that changing the order of an addition problem does not change the answer. Because an even number plus an odd number is
odd, then an odd number plus an even number is also odd. Second, it does not matter which addend is odd; there is still one dot remaining, making the sum odd.
Let’s sum it up:
→ “Even” + “even” = “even”
→ “Odd” + “odd” = “even”
→ “Odd” + “even” = “odd”
Exploratory Challenge/Exercises 4-6
Exercise 4.
The product of two even numbers is even.
Answers will vary, but one example answer is provided.
Using the problem 6 × 14, students know that this is equivalent to six groups of fourteen, or
14 + 14 + 14 + 14 + 14 + 14. Students also know that the sum of two even numbers is even; therefore, when adding the addends two at a time, the sum is always even. This means the sum of six even
numbers is even, making the product even since It is equivalent to the sum.
Using the problem 6 × 14, students can use the dots from previous examples.
From here, students can circle dots and see that there are no dots remaining, so the answer must be even.
Exercise 5.
The product of two odd numbers is odd.
Answers will vary, but an example answer is provided.
Using the problem 5 × 15, students know that this is equivalent to five groups of fifteen, or
15 + 15 + 15 + 15 + 15. Students also know that the sum of two odd numbers is even, and the sum of an odd and even number is odd. When adding two of the addends together at a time, the answer rotates
between even and odd. When the final two numbers are added together, one is even and the other odd. Therefore, the sum is odd, which makes the product odd since it is equivalent to the sum.
Using the problem 5 × 15, students may also use the dot method.
After students circle the pairs of dots, one dot from each set of 15 remains, for a total of 5 dots. Students can group these together and circle more pairs, as shown below.
Since there is still one dot remaining, the product of two odd numbers is odd.
Exercise 6.
The product of an even number and an odd number is even.
Answers will vary, but one example is provided.
Using the problem 6 × 7, students know that this is equivalent to the sum of six sevens, or 7 + 7 + 7 + 7 + 7 + 7.
Students also know that the sum of two odd numbers is even, and the sum of two even numbers is even. Therefore, when adding two addends at a time, the result is an even number. The sum of these even
numbers is also even, which means the total sum is even. This also Implies the product is even since the sum and product are equivalent.
Using the problem 6 × 7, students may also use the dot method.
After students circle the pairs of dots, one dot from each set of 7 remains, for a total of 6 dots. Students can group these together and circle more pairs, as shown below.
Since there are no dots remaining, the product of an even number and an odd number is even.
Eureka Math Grade 6 Module 2 Lesson 16 Problem Set Answer Key
Without solving, tell whether each sum or product Is even or odd. Explain your reasoning.
Question 1.
346 + 721
The sum is odd because the sum of an even and an odd number is odd.
Question 2.
4,690 × 141
The product is even because the product of an even and an odd number is even.
Question 3.
1,462,891 × 745,629
The product is odd because the product of two odd numbers is odd.
Question 4.
425,922 + 32,481,064
The sum is even because the sum of two even numbers is even.
Question 5.
32 + 45 + 67 + 91 + 34 + 56
The first two addends are odd because an even and an odd is odd.
Odd number +67 is even because the sum of two odd numbers is even.
Even number +91 is odd because the sum of an even and an odd number is odd.
Odd number +34 is odd because the sum of an odd and an even number is odd.
Odd number +56 is odd because the sum of an odd andan even number is odd.
Therefore, the final sum is odd.
Eureka Math Grade 6 Module 2 Lesson 16 Exit Ticket Answer Key
Determine whether each sum or product is even or odd. Explain your reasoning.
Question 1.
The sum is odd because the sum of an even number and an odd number is odd.
Question 2.
317,362 × 129,324
The product is even because the product of two even numbers is even.
Question 3.
104,81 + 4,569
The sum is even because the sum of two odd numbers is even.
Question 4.
32,457 × 12,781
The product is odd because the product of two odd numbers is odd.
Question 5.
Show or explain why 12 + 13 + 14 + 15 + 16 results in an even sum.
12 + 13 is odd because even + odd is odd.
Odd number +14 is odd because odd + even is odd.
Odd number +15 is even because odd + odd is even.
Even number +16 is even because even + even is even.
Students may group even numbers together, 12 + 14 + 16, which results in an even number. Then, when students combine the two odd numbers, 13 + 15, the result is another even number. We know that the
sum of two evens results in another even number.
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Naive Bayesian Learner
This is documentation for Orange 2.7. For the latest documentation, see Orange 3.
Naive Bayesian Learner¶
Naive Bayesian Learner
Examples (ExampleTable)
A table with training examples
The naive Bayesian learning algorithm with settings as specified in the dialog.
Naive Bayesian Classifier
Trained classifier (a subtype of Classifier)
Signal Naive Bayesian Classifier sends data only if the learning data (signal Examples is present.
This widget provides a graphical interface to the Naive Bayesian classifier.
As all widgets for classification, this widget provides a learner and classifier on the output. Learner is a learning algorithm with settings as specified by the user. It can be fed into widgets for
testing learners, for instance Test Learners. Classifier is a Naive Bayesian Classifier (a subtype of a general classifier), built from the training examples on the input. If examples are not given,
there is no classifier on the output.
Learner can be given a name under which it will appear in, say, Test Learners. The default name is “Naive Bayes”.
Next come the probability estimators. Prior sets the method used for estimating prior class probabilities from the data. You can use either Relative frequency or the Laplace estimate. Conditional
(for discrete) sets the method for estimating conditional probabilities, besides the above two, conditional probabilities can be estimated using the m-estimate; in this case the value of m should be
given as the Parameter for m-estimate. By setting it to <same as above> the classifier will use the same method as for estimating prior probabilities.
Conditional probabilities for continuous attributes are estimated using LOESS. Size of LOESS window sets the proportion of points in the window; higher numbers mean more smoothing. LOESS sample
points sets the number of points in which the function is sampled.
If the class is binary, the classification accuracy may be increased considerably by letting the learner find the optimal classification threshold (option Adjust threshold). The threshold is computed
from the training data. If left unchecked, the usual threshold of 0.5 is used.
When you change one or more settings, you need to push Apply; this will put the new learner on the output and, if the training examples are given, construct a new classifier and output it as well.
There are two typical uses of this widget. First, you may want to induce the model and check what it looks like in a Nomogram.
The second schema compares the results of Naive Bayesian learner with another learner, a C4.5 tree.
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we need formulas of tube mill roll design
Examining tube mill roll tooling, setup, and maintenance. 3 Length minimum and maximum 4 Notching or secondary operations 5 Reference to mating parts 6 Attention to critical areas If a wornout
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finding missing angles worksheet answer key geometry
Search for courses, skills, and videos. Whether it’s a calculating angles worksheet, angles in triangle worksheet, angles on parallel lines worksheet or finding missing … This website and its content
is subject to our Terms and Conditions. Customary units worksheet. tan. In this geometry instructional activity, 10th graders use right triangle trigonometry to solve application problems in which
they find the measure of the missing angle in a right triangle. geometry trigonometric ratios worksheet answers, Attachment Based Family Therapy Worksheets, Realistic Coloring Pages Of Animals For
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scale of 1 inch:75 feet. This labeling is dependent on the given angle. To help you decide which of the three trigonometric ratios to use you can label the sides of the triangle as adjacent or
opposite. Find here an unlimited supply worksheets for classifying triangles by their sides angles or both one of the focus areas of 5th grade geometry. Accelerated algebra geometry name date period
b p2l0 c1a3c zk 2u7t ba z 4s aohftjw ragrmec tl pl gc6 s 3 ha2l 8ls br 7i4gahctksg xrmebste rrxv8e 0d x q trig ratios practice find the value of each trigonometric ratio using a calculator.
Complementary and supplementary worksheet. Sum of the angles in a triangle is 180 degree worksheet. document.write(''); Geometry Trigonometric Ratios Worksheet Answers, on Geometry Trigonometric
Ratios Worksheet Answers, Rainbow Dash Printable Pony Coloring Pages. 1 sin 12 2 cos 14 3 cos 52 4 cos 24. Parallel Puzzle: Finding Angle Measure and Missing Variables In this activity, students use
properties of parallel lines to solve for various angles and variables. 'width' : 300, The actual angle is provided on the answer key, so in addition to identifying whether an angle is obtuse or
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worksheets for this concept are Right triangle trig missing sides and angles, 4 angles in a triangle, Triangles angle measures length of sides and classifying, Name date practice triangles and angle
sums, Triangle, Section, Geometry, Finding unknown angles. Finding Missing Angles In Triangles Worksheet Best Tangents from Angles In A Triangle Worksheet Answers, source: coletivocompa.org. 118!!
Finding Missing Angles In Triangle - Displaying top 8 worksheets found for this concept.. A worksheet with problems on angles at a point, on a line and in a triangle. 5 and cxd are supplementary
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worksheets for almost every subject. Show more details Add to cart. 'params' : {} Address any misconceptions that may arise. 'format' : 'iframe', Traverse through this huge assortment of transversal
worksheets to acquaint 7th grade 8th grade and high school students with the properties of several angle pairs like the alternate angles corresponding angles same side angles etc formed when a
transversal cuts a pair of parallel lines. Answer key is available for all worksheets. Recall the angle properties of parallelograms, and you’re good to go! Geometry worksheets. While these
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Angles Worksheet Work Energy And Power Baseball Field Dimensions Triangle Worksheet Teaching … 1 a b vertical 2 a b supplementary 3 a b vertical 4 a b complementary 5 a b complementary 6 a b adjacent
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What is strike rate in the world of cricket betting
About cricket strike rate
The phrase “strike rate” in cricket has several meanings. Which interpretation is correct based on the type of player the term is referring to! A strike rate, for example, can imply one thing when
referring to a batter and another entirely different thing when speaking to a bowler!
As you can see, it’s another one of those cricket terminologies that newcomers to the game will see on a scorecard and have no idea what it means.
I’ll discuss what strike rate implies when it comes to batsmen and bowlers in this piece. I’ll also explain how it’s calculated and provide information on which professionals in international cricket
have had the best strike rates, as well as answer a few additional queries you might have.
What does Cricket Strike Rate Means?
So, what exactly does the term “Cricket Strike Rate” imply? For batsmen, a strike rate is a number that represents the average number of runs scored per 100 balls faced by the batsman. A high batting
strike rate denotes a more energetic batsman that scores quickly, whilst a lower strike rate is a hallmark of a more conservative batsman.
Strike rate is a metric that shows the average number of deliveries a bowler must bowl to dismiss a batsman.
In this situation, a lower strike rate is preferable because it implies the bowler will need to bowl fewer balls to be successful. The strike rates of batsmen and bowlers can be averaged over the
course of a game or a player’s career.
How to Calculate Your Cricket Strike Rate?
To calculate a batsman’s strike rate in cricket, divide the overall amount of runs obtained by the number of deliveries faced. After you’ve done that, multiply your solution by 100 to get the
batsman’s strike rate.
What’s the Difference Between Strike Rate and Bowling Average?
The distinction between bowling averages and bowling strike rates is frequently misunderstood. Let me clarify any misunderstandings! A bowling average is the number of runs a bowler will give up for
each wicket he or she takes. For example, if a bowler takes two wickets but also gives up 50 runs, their bowling average is 25. For every wicket they took, they gave up 25 runs.
The distinction between the two phrases is that the bowling strike rate relates to how many attempts a bowler must bowl with each wicket that can take, as we’ve already explained.
What’s the Difference Between Strike Rate and Batting Average?
A batsman’s batting average is the number of runs he or she scores on average in each innings. Basically said, batting average refers to how many runs a batsman scores irrespective of the number of
balls he or she faces. The batting strike rate is being used to determine how quickly a batter score runs.
Now I hope that this article had to helps you understand about strike rate in cricket. If you have still any confusion then you can feel free to comments we will reply to you.
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Hashrate Per Watt Calculator - Calculator Wow
Hashrate Per Watt Calculator
In the world of cryptocurrency mining, efficiency is key to maximizing profits and minimizing costs. One important metric for gauging the efficiency of mining hardware is the Hashrate Per Watt. This
metric helps miners understand how effectively their hardware converts electrical power into hashing power, a crucial factor in achieving better returns on investment. This article will explore the
concept of the Hashrate Per Watt Calculator, explain the formula used, show how to use the calculator, provide a practical example, and answer common questions about this important tool.
The formula used in the Hashrate Per Watt Calculator is simple and straightforward:
HPW = H / P
• HPW = Hashrate Per Watt (MH/s per Watt)
• H = Hashrate (MH/s)
• P = Power Consumption (Watts)
This formula calculates the efficiency of the mining hardware by dividing the hashrate by the power consumption, giving a clear picture of how well the hardware performs in terms of energy usage.
How to Use
Using the Hashrate Per Watt Calculator involves the following steps:
1. Enter the Hashrate (H): Input the hashrate of your mining hardware in mega hashes per second (MH/s).
2. Enter the Power Consumption (P): Input the power consumption of your mining hardware in watts (W).
3. Click "Calculate": The calculator will apply the formula to determine the Hashrate Per Watt (HPW).
This simple process provides a quick and easy way to assess the efficiency of your mining setup.
Let's say you have a mining rig with the following specifications:
• Hashrate (H): 100 MH/s
• Power Consumption (P): 500 Watts
Using the formula:
HPW = H / P
Substitute the values:
HPW = 100 MH/s / 500 W = 0.2 MH/s per Watt
Thus, the efficiency of your mining rig is 0.2 MH/s per Watt.
FAQs and Answers
1. What is a Hashrate? Hashrate is the speed at which a mining device completes an operation in the cryptocurrency mining process, measured in mega hashes per second (MH/s).
2. Why is Hashrate Per Watt important? It measures the efficiency of mining hardware, helping miners maximize profits by reducing energy costs.
3. Can this calculator be used for all types of mining hardware? Yes, it can be used for any mining hardware, including GPUs, ASICs, and CPUs.
4. What if the power consumption is variable? Use the average power consumption over time to get an accurate measure of efficiency.
5. Is a higher Hashrate Per Watt always better? Generally, yes. Higher efficiency means more hashing power for less energy consumption.
6. Can this calculator help in choosing new mining hardware? Yes, it can compare the efficiency of different hardware options.
7. How often should I calculate Hashrate Per Watt? Regular calculations can help track hardware performance and make informed decisions about upgrades or replacements.
8. What units should be used for the inputs? Hashrate should be in MH/s and power consumption in Watts for accurate results.
9. Can environmental factors affect the results? Yes, factors like temperature and power quality can impact hardware performance and efficiency.
10. Do I need special software to use this calculator? No, you can use any basic calculator or the HTML-based calculator provided in this article.
The Hashrate Per Watt Calculator is an invaluable tool for cryptocurrency miners looking to optimize their hardware's performance and energy efficiency. By understanding how to use this calculator,
miners can make informed decisions that improve profitability and sustainability. Regularly assessing the efficiency of mining hardware ensures that operations remain cost-effective, making it a
crucial aspect of successful mining strategies. Whether you're a novice or an experienced miner, utilizing the Hashrate Per Watt Calculator can help you stay competitive in the ever-evolving world of
cryptocurrency mining.
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What is the area of an equilateral triangle of side length 16cm? | HIX Tutor
What is the area of an equilateral triangle of side length 16cm?
Answer 1
The area is $\left(\sqrt{3} \cdot 64\right) c {m}^{2}$
As the area of an equilateral triangle is #sqrt(3)/4 ##a^2# where #a# is one side.
So, Area = #sqrt(3)/4 ##16^2# = #64sqrt(3)#
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Answer 2
The area of an equilateral triangle can be calculated using the formula: Area = (√3 / 4) * side length squared. Substituting the given side length of 16 cm into the formula, we get: Area = (√3 / 4) *
(16 cm)^2. Simplifying this expression gives us: Area ≈ 221.76 square centimeters.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Rayleigh's Method - (College Physics II – Mechanics, Sound, Oscillations, and Waves) - Vocab, Definition, Explanations | Fiveable
Rayleigh's Method
from class:
College Physics II – Mechanics, Sound, Oscillations, and Waves
Rayleigh's method is a technique used in dimensional analysis to determine the functional relationships between different physical quantities. It provides a systematic approach to identify the
dimensionless parameters that govern a physical phenomenon, allowing for the development of scaling laws and the prediction of the behavior of complex systems.
congrats on reading the definition of Rayleigh's Method. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Rayleigh's method is based on the principle that physical quantities can be expressed as products of powers of fundamental dimensions, such as length, mass, and time.
2. The method involves identifying the relevant physical quantities that influence a particular phenomenon and then using dimensional analysis to determine the dimensionless parameters that govern
the system.
3. Rayleigh's method is particularly useful in situations where experimental data is limited or difficult to obtain, as it allows for the prediction of system behavior based on the identified
dimensionless parameters.
4. The dimensionless parameters derived using Rayleigh's method can be used to develop scaling laws, which enable the extrapolation of experimental results to different scales or conditions.
5. Rayleigh's method is widely applied in fields such as fluid mechanics, heat transfer, and solid mechanics, where the behavior of complex systems can be studied and predicted using dimensional
Review Questions
• Explain the purpose and key principles of Rayleigh's method in the context of dimensional analysis.
□ The purpose of Rayleigh's method is to determine the functional relationships between different physical quantities in a system by identifying the dimensionless parameters that govern the
phenomenon. The method is based on the principle that physical quantities can be expressed as products of powers of fundamental dimensions, such as length, mass, and time. By systematically
analyzing the dimensions of the relevant physical quantities, Rayleigh's method allows for the development of scaling laws and the prediction of system behavior, even in situations where
experimental data is limited.
• Describe how Rayleigh's method can be used to derive dimensionless parameters and develop scaling laws in the context of a specific physical problem.
□ To use Rayleigh's method, one would first identify the relevant physical quantities that influence a particular phenomenon, such as fluid flow, heat transfer, or structural mechanics. Next,
the dimensions of these quantities would be analyzed to determine the dimensionless parameters that govern the system. For example, in fluid mechanics, the dimensionless Reynolds number is a
key parameter that describes the ratio of inertial to viscous forces, and it can be derived using Rayleigh's method. Once the dimensionless parameters are identified, scaling laws can be
developed to predict how the system's behavior changes as its size or other parameters are varied, enabling the extrapolation of experimental results to different scales or conditions.
• Evaluate the advantages and limitations of using Rayleigh's method in the context of dimensional analysis and its applications in various fields of physics and engineering.
□ The primary advantage of Rayleigh's method is its ability to simplify complex physical problems by identifying the dimensionless parameters that govern a system's behavior. This allows for
the development of scaling laws and the prediction of system performance, even in situations where experimental data is limited or difficult to obtain. Additionally, the dimensionless
parameters derived using Rayleigh's method can be used to compare and analyze different systems, enabling the identification of universal relationships. However, the method also has
limitations. It relies on the accurate identification of the relevant physical quantities, and the derived dimensionless parameters may not capture all the nuances of a complex system.
Furthermore, Rayleigh's method is primarily a theoretical approach, and its effectiveness may be limited in cases where the underlying physical mechanisms are not well understood. Overall,
Rayleigh's method is a powerful tool in dimensional analysis, but its application requires careful consideration of the specific problem and the limitations of the approach.
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Lesson 7
Reasoning about Similarity with Transformations
Problem 1
Sketch a figure that is similar to this figure. Label side and angle measures.
Problem 2
Write 2 different sequences of transformations that would show that triangles \(ABC\) and \(AED\) are similar. The length of \(AC\) is 6 units.
Problem 3
What is the definition of similarity?
Problem 4
Select all figures which are similar to Parallelogram \(P\).
Figure \(E\)
Problem 5
Find a sequence of rigid transformations and dilations that takes square \(ABCD\) to square \(EFGH\).
Translate by the directed line segment \(AE\), which will take \(B\) to a point \(B’\). Then rotate with center \(E\) by angle \(B’EF\). Finally, dilate with center \(E\) by scale factor \(\frac{5}
Translate by the directed line segment \(AE\), which will take \(B\) to a point \(B’\). Then rotate with center \(E\) by angle \(B’EF\). Finally, dilate with center \(E\) by scale factor \(\frac{2}
Dilate using center \(E\) by scale factor \(\frac25\).
Dilate using center \(E\) by scale factor \(\frac52\).
Problem 6
Triangle \(DEF\) is formed by connecting the midpoints of the sides of triangle \(ABC\). What is the perimeter of triangle \(ABC\)?
Problem 7
Select the quadrilateral for which the diagonal is a line of symmetry.
Problem 8
Triangles \(FAD\) and \(DCE\) are each translations of triangle \( ABC\)
Explain why angle \(CAD\) has the same measure as angle \(ACB\).
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Vectors Addition and Subtraction in terms of Perpendicular Components
Vectors addition and subtraction in terms of components
A vector quantity can be resolved by parallelogram law into two or more vectors in different directions. The process of resolving a vector into two or more vectors is called resolution of a vector or
vector resolution. Each resolving vector is called component of the original vector.
Resolution into perpendicular components:
Let R be the vector along OB, so that OB = R. Now taking OB as diagonal let us draw the rectangle OABC [Fig. 1]. Here the components P and Q are perpendicular to one another, i.e.,
α + β = 90^0
then, Sin (α + β) = Sin 90^0 = 1 and;
Sin β = Sin (90^0 – α) = Cos α
According to law at trainee at trigonometry we get from the triangle OAB.
P/Cos θ = Q/Sin θ = R/Sin 90^0
so, P = R Cos θ and Q = R sin α … …. ….. (1)
The two components P and Q are called the perpendicular components of the resultant R. P is called the horizontal component and Q is called tangential component.
Vector forms of the two components are –
P = R Cos α î + R Sin α î
So, their vector addition:
R = P + Q = R Cos α î + R Sin α î … …. ….. (ii)
and vector subtraction will be the vector subtraction of P and Q.
Example: Suppose P is acting along the arm AB and is acting along the arm BC of the triangle [Fig. b). Then their vector addition R = P + Q, may be represented by the third arm AC of the triangle.
Again, if P and Q are acting along AB and BC then vector subtraction R = P – Q, can be represented by the third arm CA of the triangle.
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Mortgage Payoff Calculatord - DaProfitClub
Mortgage Payoff Calculator
What is a Mortgage Payoff Calculator?
A mortgage payoff calculator determines the money needed to repay a mortgage loan. It considers the principal loan balance, the interest rate, and the years until the loan is paid off. The calculator
can also calculate how much money will be saved by making extra monthly payments.
How to Use a Mortgage Payoff Calculator?
Mortgage Payoff Calculator Formula:
(Mortgage Balance x (1 + Interest Rate/12)^(Number of Months) - (Monthly Payment x ((1 + Interest Rate/12)^(Number of Months) - 1)) / (Interest Rate/12))
Mortgage Balance = Original loan amount
Interest Rate = Annual interest rate (expressed as a decimal)
Number of Months = Number of months remaining on the mortgage
Monthly Payment = The amount of the monthly mortgage payment (principal + interest)
This formula calculates the total amount of money that will be paid over the remaining term of the mortgage, including interest and principal payments. The result is the total payoff amount for the
Using a mortgage payoff calculator is easy and straightforward. Here are the steps to follow:
Step 1: Enter the current balance of your mortgage loan. You can find this by contacting your lender or looking on your mortgage statement.
Step 2: Enter the current interest rate on your loan. You can also find this by contacting your lender or looking on your mortgage statement.
Step 3: Enter the remaining term of the loan. This is the number of years left until the loan is paid off in full.
Step 4: Click on "calculate" to get your monthly payment.
Benefits of Using a Mortgage Payoff Calculator
Using a mortgage payoff calculator has several benefits. Here are a few:
It allows homeowners to see how much they need to pay each month to pay off their mortgage on time.
It helps homeowners create a budget and plan for their mortgage payments.
It allows homeowners to compare different loan options and see which one will save them the most money in the long run.
It can also help homeowners figure out if they can afford to make extra payments or if they should look into refinancing their loan.
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Peck (imperial) to Cubic Foot Converter
Enter Peck (imperial)
Cubic Foot
⇅ Switch toCubic Foot to Peck (imperial) Converter
How to use this Peck (imperial) to Cubic Foot Converter 🤔
Follow these steps to convert given volume from the units of Peck (imperial) to the units of Cubic Foot.
1. Enter the input Peck (imperial) value in the text field.
2. The calculator converts the given Peck (imperial) into Cubic Foot in realtime ⌚ using the conversion formula, and displays under the Cubic Foot label. You do not need to click any button. If the
input changes, Cubic Foot value is re-calculated, just like that.
3. You may copy the resulting Cubic Foot value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Peck (imperial) to Cubic Foot?
The formula to convert given volume from Peck (imperial) to Cubic Foot is:
Volume[(Cubic Foot)] = Volume[(Peck (imperial))] × 0.32108730647178413
Substitute the given value of volume in peck (imperial), i.e., Volume[(Peck (imperial))] in the above formula and simplify the right-hand side value. The resulting value is the volume in cubic foot,
i.e., Volume[(Cubic Foot)].
Calculation will be done after you enter a valid input.
Consider that a farmer collects 2 pecks (imperial) of apples.
Convert this volume from pecks (imperial) to Cubic Foot.
The volume in peck (imperial) is:
Volume[(Peck (imperial))] = 2
The formula to convert volume from peck (imperial) to cubic foot is:
Volume[(Cubic Foot)] = Volume[(Peck (imperial))] × 0.32108730647178413
Substitute given weight Volume[(Peck (imperial))] = 2 in the above formula.
Volume[(Cubic Foot)] = 2 × 0.32108730647178413
Volume[(Cubic Foot)] = 0.6422
Final Answer:
Therefore, 2 pk is equal to 0.6422 ft^3.
The volume is 0.6422 ft^3, in cubic foot.
Consider that a storage bin holds 5 pecks (imperial) of potatoes.
Convert this storage capacity from pecks (imperial) to Cubic Foot.
The volume in peck (imperial) is:
Volume[(Peck (imperial))] = 5
The formula to convert volume from peck (imperial) to cubic foot is:
Volume[(Cubic Foot)] = Volume[(Peck (imperial))] × 0.32108730647178413
Substitute given weight Volume[(Peck (imperial))] = 5 in the above formula.
Volume[(Cubic Foot)] = 5 × 0.32108730647178413
Volume[(Cubic Foot)] = 1.6054
Final Answer:
Therefore, 5 pk is equal to 1.6054 ft^3.
The volume is 1.6054 ft^3, in cubic foot.
Peck (imperial) to Cubic Foot Conversion Table
The following table gives some of the most used conversions from Peck (imperial) to Cubic Foot.
Peck (imperial) (pk) Cubic Foot (ft^3)
0.01 pk 0.00321087306 ft^3
0.1 pk 0.03210873065 ft^3
1 pk 0.3211 ft^3
2 pk 0.6422 ft^3
3 pk 0.9633 ft^3
4 pk 1.2843 ft^3
5 pk 1.6054 ft^3
6 pk 1.9265 ft^3
7 pk 2.2476 ft^3
8 pk 2.5687 ft^3
9 pk 2.8898 ft^3
10 pk 3.2109 ft^3
20 pk 6.4217 ft^3
50 pk 16.0544 ft^3
100 pk 32.1087 ft^3
1000 pk 321.0873 ft^3
Peck (imperial)
The Imperial peck is a unit of measurement used to quantify dry volumes, particularly in the UK and countries using the Imperial system. It is defined as 8 Imperial gallons or approximately 36.368
liters. Historically, the peck was used to measure agricultural produce such as fruits and vegetables, providing a standardized volume for trade and commerce. Although its use has declined, it
remains a historical unit and is occasionally referenced in agricultural contexts and historical records.
Cubic Foot
The cubic foot is a unit of measurement used to quantify three-dimensional volumes, commonly applied in construction, real estate, and various industrial contexts. It is defined as the volume of a
cube with sides each measuring one foot in length. Historically, the cubic foot has been used to measure and specify the volume of spaces and materials in building and storage. Today, it is widely
used in the US and other countries that use the Imperial system, for tasks such as calculating building dimensions, storage capacities, and shipping volumes.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Peck (imperial) to Cubic Foot in Volume?
The formula to convert Peck (imperial) to Cubic Foot in Volume is:
Peck (imperial) * 0.32108730647178413
2. Is this tool free or paid?
This Volume conversion tool, which converts Peck (imperial) to Cubic Foot, is completely free to use.
3. How do I convert Volume from Peck (imperial) to Cubic Foot?
To convert Volume from Peck (imperial) to Cubic Foot, you can use the following formula:
Peck (imperial) * 0.32108730647178413
For example, if you have a value in Peck (imperial), you substitute that value in place of Peck (imperial) in the above formula, and solve the mathematical expression to get the equivalent value in
Cubic Foot.
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What Will Happen? 12-21-12 Apocalypse - Page 5
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
"2012 A-Z" - A Galaxial Event of Disneyesque Proportion VIDEOhttps://www.youtube.com/watch?feature=player_embedded&v=rxMymXI4KAgThis is a great video. Please make time to view it!
<iframe width="640" height="360" src="https://www.youtube.com/embed/rxMymXI4KAg?feature=player_embedded" frameborder="0" allowfullscreen></iframe>
Quote: Published on Nov 28, 2012
Everything you wanted to know about '2012' but didn't know to ask.
Heard of the Photon Belt? Pole Shift? Precession of Equinoxes?
Nibiru, Solar Storms, The Mayan Calendar or Biblical Prophecy?
With over 30 yrs in Radio & TV Broadcasting, Boulder Professor Marc "The Arcturian", Grandson of Warren Buffet's Mentor, Benjamin Graham, takes you on this Galaxial Tour in:
"The Most Comprensive 60 Minutes On "2012" Ever Assembled"
Over 4G/1,200 Photos, Clips, Animations, Slides and SFX went into this video of Disneyesque Proportion even Bill didn't know you could make.
Every Man, Woman, and Child on Planet Earth should see this...
...in the next 23 days.
You or someone you know wants to see this movie.
"IF Mani Narobe and I are right,...in 3-4 weeks,
none of us may even remember this video."
May The Force Be With Us All
Fascinating Stuff
This is not a 25000 year event but a 225 million year happening.
The end of 12 cycles, earths 13th birthday. Mayan count 13.0.0.0
The rotation of the earth has been slowing down (which is why time seems to have sped up)
The rotation will stop for 3 days with half the planet in darkness and half the planet in light.
When the rotation restarts it will be clockwise not anticlockwise. the sun will rise in the West.
The big bang happened the Universe has been expanding
( breathing in) for 225 million years now it will start breathing out and the universe will slowly start shrinking.
heralding a new age of enlightenment.
U.K military have been placed on a 10 day standby.
Next Friday the earth and the sun with other planets line up for the first time ever with the center of the Milky-Way! Huge amounts of energy will be released through the photon belt directly at
earth. The Mayans long count of time ends after 5125.36 days because time stops as we know it to be. The earth could well stop its rotation and start up after three days in the opposite way with the
sun coming up in the west. All keys of the universe will be in line and unlock our inner 5th dimensional senses. Do not under estimate what is about to take place next Friday, this will be the very
first time ever and our governments know it. Notice Hilary Clinton as cancelled trips overseas due to illness! Top profile people suddenly leaving high profile positions! FEMA camps and the storing
of food, gaskets made to float that fit up to 7-corpses! Remember last year in America for the very first time congress held a top secret meeting around September behind closed doors.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
Chinese survival pods to defend against 'apocalypse' (PHOTOS, VIDEO)
As believers across the globe prepare for the forecast Mayan apocalypse, a Chinese villager says he’s going to save humanity with his giant tsunami proof survival pods.
The pods are made using a fiberglass casing over a steel frame, cost $48,000 each to make and are equipped with oxygen tanks, food and water supplies. They also come with seat belts – essential for
surviving in storms.
“The pod won’t have any problems even if there are 1,000 meter high waves, its like a ping pong ball, its skin may be thin but it can withstand a lot of pressure,” the balls’ creator Liu Qiyuan, told
AFP from his workshop outside Beijing.
“The pods are designed to carry 14 people at a time, but it’s possible for 30 people to survive inside for at least two months,” insisted Liu
Indeed, their insulation is such that “a person could live for four months in the pod at the north or south pole without freezing,” Liu continued.
Liu explained that he was inspired into making the spheres after seeing the Hollywood disaster film “2012”, which is itself inspired by the expiry of the Mayan calendar on the 21st December 2012. The
Mayans were an ancient American civilization whose 5000 year old calendar shortly ends.
“If there really is some kind of apocalypse then you could say I’ve made a contribution to the survival of humanity,” said Liu. Despite their tough design Liu is yet to sell any of the pods and he’s
worried about paying back the loans he took out to build them. “I worked for many years without saving much money…invested most of my money in the pods, because it’s worth it, it’s about saving
lives,” he said.
Read more at link above.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 13627
Join date : 2010-09-28
Location : The Matrix
Visualize Chinese Survival Pods as Completely Ignorant Fool Sleeper Cells!! Sorry!! As the Apocalypse approaches, I'm going downhill FAST!!
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
orthodoxymoron wrote:Visualize Chinese Survival Pods as Completely Ignorant Fool Sleeper Cells!! Sorry!! As the Apocalypse approaches, I'm going downhill FAST!!
Oxy, just keep in mind the meaning of the word Apocalypse = unveiling. It's suppose to herald in the golden age (spirituality at its maximum). There isn't anything to be afraid of other then fear
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
Doomsday fears spread to Serbia: thousands seek refuge near mystic mountain
December 14, 2012 – SERBIA – Hotel owners around the pyramid-shaped Mount Rtanj, in the east of the Balkan country, say that bookings are flooding in, with believers in the prophecy hoping that its
purported mysterious powers will save them from the apocalypse. Adherents of the end-of-the-world scenario think the 5,100ft-high mountain, part of the Carpathian range, conceals a pyramidal building
inside, left behind by alien visitors thousands of years ago. Arthur C Clarke, the British science fiction writer, reportedly identified the peak as a place of “special energy” and called it “the
navel of the world.” In one day we had 500 people trying to book rooms. People want to bring their whole families,” said Obrad Blecic, a hotel manager. Predictions of an apocalypse are linked to the
fact that the 5,125-year-old calendar of the ancient Mayans, who dominated large stretches of southern Mexico and Central America centuries ago, comes to an end on Dec 21. The doomsday scenario has
inspired hundreds of books, websites and television programs but scholars of the Mayan civilisation, and Mayans themselves, say people have wrongly interpreted the meaning of the calendar and that it
will not herald the world’s obliteration. But that has not stopped fears that the end of the world from spreading panic among the credulous across the world. Panic-buying of candles and essentials
has been reported in China and Russia, and in the United States the sale of survival shelters is booming. A mountain in the French Pyrenees that cultists claim will be the only place still standing
is to be closed to visitors to avoid chaos and overcrowding on its peak. –Telegraph
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
NASA released this 10 Days early. Hah!
Are they being a bit premature? Nasa releases Mayan apocalypse video 10 days early
Nasa has released a video ahead of schedule tackling the 'myths' surrounding the belief the world will end on December 21st. The video, which was clearly intended for release the day after the 21st,
begins: "December 22, 2012. If you're watching this video, it means one thing. The world didn't end yesterday." It goes on to attempt to debunk the ideas surrounding the so-called 'Mayan prophecies',
saying the date is based on a misconception. Making the argument point by point the video sets to put to rest the catastrophic prophecies, including debunking the notion that the sun will irradiate
the atmosphere or that another planet will smash into Earth. Indeed Nasa appears so confident about their prediction that the world will not come to an end that they have released the video early.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
A Few interesting things....
1.) The Long Count Mayan Calendar began 3114 BCE
2.) The Mayan Calendar ends Friday, December 21, 2012, after 13 Baktun
3.) The current year is 2012
4.) 3114 + 2012 = 5126 (years)
5.) There are 394 years in 1 Baktun
6.) 5126 / 394 = 13 baktun
7.) Evidence shows that around 5,200 years ago, solar output first dropped precipitously and then surged over a short period
8.) Atmospheric Co2 peaks during an ice age, around 310 ppm
9.) Earths current Co2 level is 393.92 ppm
10.) Earth is losing it's atmosphere faster than Mars
11.) Our solar system orbits the milky way around 230 million years.
12.) Our solar system may be headed into a dense cloud of interstellar matter, a spiral arm of the milky way. Researchers have predicted increases in the cosmic-ray flux, changes in the Earth’s
magnetosphere, the chemistry of the atmosphere and perhaps even the terrestrial climate. (1996)
13.) NASA suspected a black hole at the center of the galaxy. (2000)
14.) Our solar system is headed into a local intersteller cloud (2002)
15.) NASA admits enough cosmic rays would damage the ozone and let through lethal doses of UV and extinguish much of life on Earth (2003)
16.) NASA announces 10'th planet (2005)
17.) Cosmic rays from other galaxies black holes are hitting us (2007)
18.) Voyager discovers our solar system IS passing through an interstellar cloud that physics says should not exist (2009)
19.) A massive black hole blast was discovered in another galaxy (2012)
20.) NASA reveals a previously unknown stellar-mass black hole in our galaxy
21.) Solar Radiation Management , chemtrails, are at least a consideration.
22.) One Baktun has 144.000 days.
1.) [link to en.wikipedia.org] http://en.wikipedia.org/wiki/Mayan_calendar#Long_Count
2.) [link to www.google.com (secure)] https://www.google.com/search?q=when%20does%20the%20mayan%20calendar%20end
3.) [link to www.timeanddate.com] http://www.timeanddate.com/calendar/
4.) [link to www.google.com (secure)] https://www.google.com/search?q=3114%20+%202012
5.) [link to en.wikipedia.org] http://en.wikipedia.org/wiki/Mayan_calendar#Long_Count
6.) [link to www.google.com (secure)] https://www.google.com/search?q=5126%20/%20394
7.) [link to researchnews.osu.edu] http://researchnews.osu.edu/archive/5200event.htm
8.) [link to en.wikipedia.org] http://en.wikipedia.org/wiki/File:Atmospheric_CO2_with_glaciers_cycles.gif
9.) [link to co2now.org] http://co2now.org
10.) [link to dsc.discovery.com] http://dsc.discovery.com/news/2009/06/02/solar-wind-atmosphere.html
11.) [link to starchild.gsfc.nasa.gov] http://dsc.discovery.com/news/2009/06/02/solar-wind-atmosphere.html
12.) [link to www-news.uchicago.edu] http://www-news.uchicago.edu/releases/96/960609.solar.sytem.crash.shtml
13.) [link to science.nasa.gov] http://science.nasa.gov/science-news/science-at-nasa/2002/21feb_mwbh/
14.) [link to apod.nasa.gov] http://apod.nasa.gov/apod/ap020210.html
15.) [link to science.nasa.gov] http://science.nasa.gov/science-news/science-at-nasa/2003/06jan_bubble/
16.) [link to www.nasa.gov] http://www.nasa.gov/vision/universe/solarsystem/newplanet-072905.html
17.) [link to www.newscientist.com] http://www.newscientist.com/article/dn12897-monster-black-holes-power-highestenergy-cosmic-rays.html
18.) [link to science.nasa.gov] http://science.nasa.gov/science-news/science-at-nasa/2009/23dec_voyager/
19.) [link to www.telegraph.co.uk] http://www.telegraph.co.uk/science/space/9708310/Most-powerful-black-hole-blast-discovered.html
20.) [link to www.youtube.com] https://www.youtube.com/watch?v=_W-QEs8sKE0
21.) [link to ntrs.nasa.gov] http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20070031204_2007030982.pdf
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
What earth will look like after a shift?
1997: RUSSIAN DOCUMENTS RELEASED ON SOLAR SYSTEM CHANGES...
PLANETOPHYSICAL STATE OF THE EARTH AND LIFE
By DR. ALEXEY N. DMITRIEV
Published in Russian, IICA Transactions, Volume 4, 1997
*Professor of Geology and Mineralogy, and Chief Scientific Member,
United Institute of Geology, Geophysics, and Mineralogy,
Siberian Department of Russian Academy of Sciences.
Expert on Global Ecology, and Fast -Processing Earth Events.
Russian to English Translation and Editing:
by A. N. Dmitriev, Andrew Tetenov, and Earl L. Crockett
Summary Paragraph
Current Planeto-Physical alterations of the Earth are becoming irreversible. Strong evidence exists that these transformations are being caused by highly charged material and energetic
non-uniformity's in anisotropic interstellar space which have broken into the interplanetary area of our Solar System. This "donation" of energy is producing hybrid processes and excited energy
states in all planets, as well as the Sun. Effects here on Earth are to be found in the acceleration of the magnetic pole shift, in the vertical and horizontal ozone content distribution, and in the
increased frequency and magnitude of significant catastrophic climatic events. There is growing probability that we are moving into a rapid temperature instability period similar to the one that took
place 10,000 years ago. The adaptive responses of the biosphere, and humanity, to these new conditions may lead to a total global revision of the range of species and life on Earth. It is only
through a deep understanding of the fundamental changes taking place in the natural environment surrounding us that politicians, and citizens a like, will be able to achieve balance with the renewing
flow of PlanetoPhysical states and processes.
Current, in process, geological, geophysical, and climatical alterations of the Earth are becoming more, and more, irreversible. At the present time researchers are revealing some of the causes which
are leading to a general reorganization of the electro-magnetosphere (the electromagnetic skeleton) of our planet, and of its climatic machinery. A greater number of specialists in climatology,
geophysics, planetophysics, and heliophysics are tending towards a cosmic causative sequence version for what is happening. Indeed, events of the last decade give strong evidence of unusually
significant heliospheric and planetophysic transformations [1,2]. Given the quality, quantity, and scale of these transformations we may say that:
The climatic and biosphere processes here on Earth (through a tightly connected feedback system) are directly impacted by, and linked back to, the general overall transformational processes taking
place in our Solar System. We must begin to organize our attention and thinking to understand that climatic changes on Earth are only one part, or link, in a whole chain of events taking place in our
These deep physical processes, these new qualities of our physical and geological environment, will impose special adaptive challenges and requirements for all life forms on Earth. Considering the
problems of adaptation our biosphere will have with these new physical conditions on Earth, we need to distinguish the general tendency and nature of the changes. As we will show below,these
tendencies may be traced in the direction of planet energy capacity growth (capacitance), which is leading to a highly excited or charged state of some of Earth's systems.The most intense
transformations are taking place in the planetary gas-plasma envelopes to which the productive possibilities of our biosphere are timed. Currently this new scenario of excess energy run-off is being
formed, and observed:
In the ionosphere by plasma generation.
In the magnetosphere by magnetic storms.
In the atmosphere by cyclones.
This high-energy atmospheric phenomena, which was rare in the past, is now becoming more frequent, intense, and changed in its nature. The material composition of the gas-plasma envelope is also
being transformed.
It is quite natural for the whole biota of the Earth to be subjected to these changing conditions of the electromagnetic field, and to the significant deep alterations of Earth's climatic machinery.
These fundamental processes of change create a demand within all of Earth's life organisms for new forms of adaptation. The natural development of these new forms may lead to a total global revision
of the range of species, and life, on Earth . New deeper qualities of life itself may come forth, bringing the new physical state of the Earth to an equilibrium with the new organismic possibilities
of development, reproduction, and perfection.
In this sense it is evident that we are faced with a problem of the adaptation of humanity to this new state of the Earth; new conditions on Earth whose biospheric qualities are varying, and
non-uniformly distributed. Therefore the current period of transformation is transient, and the transition of life's representatives to the future may take place only after a deep evaluation of what
it will take to comply with these new Earthly biospheric conditions. Each living representative on Earth will be getting a thorough "examination," or "quality control inspection," to determine it's
ability to comply with these new conditions. These evolutionary challenges always require effort, or endurance, be it individual organisms, species, or communities. Therefore, it is not only the
climate that is becoming new, but we as human beings are experiencing a global change in the vital processes of living organisms, or life itself; which is yet another link in the total process. We
cannot treat such things separately, or individually.
1.0 TRANSFORMATION OF THE SOLAR SYSTEM
We will list the recent large-scale events in the Solar System in order to fully understand, and comprehend, the PlanetoPhysical transformations taking place. This development of events, as it has
become clear in the last few years, is being caused by material and energetic non-uniformity's in anisotropic interstellar space[2,3,4]. In its travel through interstellar space, the Heliosphere
travels in the direction of the Solar Apex in the Hercules Constellation. On its way it has met (1960's) non-homogeneities of matter and energy containing ions of Hydrogen, Helium, and Hydroxyl in
addition to other elements and combinations. This kind of interstellar space dispersed plasma is presented by magnetized strip structures and striations. The Heliosphere [solar system] transition
through this structure has led to an increase of the shock wave in front of the Solar System from 3 to 4 AU, to 40 AU, or more. This shock wave thickening has caused the formation of a collusive
plasma in a parietal layer, which has led to a plasma overdraft around the Solar System, and then to its breakthrough into interplanetary domains [5,6]. This breakthrough constitutes a kind of matter
and energy donation made by interplanetary space to our Solar System.
In response to this "donation of energy/matter," we have observed a number of large scale events:
A series of large PlanetoPhysical transformations.
A change in the quality of interplanetary space in the direction of an increase in its interplanetary, and solar-planetary transmitting properties.
The appearance of new states, and activity regimes, of the Sun.
1.1 A Series of Large PlanetoPhysical Transformations.
The following processes are taking place on the distant planets of our Solar System. But they are, essentially speaking, operationally driving the whole System.
Here are examples of these events:
1.1.1 A growth of dark spots on Pluto [7].
1.1.2 Reporting of auroras on Saturn [8].
1.1.3 Reporting of Uranus and Neptune polar shifts (They are magnetically conjugate planets), and the abrupt large-scale growth of Uranus' magnetosphere intensity.
1.1.4 A change in light intensity and light spot dynamics on Neptune [9,10].
1.1.5 The doubling of the magnetic field intensity on Jupiter (based upon 1992 data), and a series of new states and processes observed on this planet as an aftermath of a series of explosions in
July 1994 [caused by "Comet" SL-9] [12]. That is, a relaxation of a plasmoid train [13,14] which excited the Jovian magnetosphere, thus inducing excessive plasma generation [12] and it's release in
the same manner as Solar coronal holes [15] inducing an appearance of radiation belt brightening in decimeter band (13.2 and 36 cm), and the appearance of large auroral anomalies and a change of the
Jupiter - Io system of currents [12, 14].
Update Note From A.N.D Nov. 1997:
A stream of ionized hydrogen, oxygen, nitrogen, etc. is being directed to Jupiter from the volcanic areas of Io through a one million amperes flux tube. It is affecting the character of Jupiter's
magnetic process and intensifying it's plasma genesis.{Z.I.Vselennaya "Earth and Universe" N3, 1997 plo-9 by NASA data}
1.1.6 A series of Martian atmosphere transformations increasing its biosphere quality. In particularly, a cloudy growth in the equator area and an unusual growth of ozone concentration[16].
Update Note: In September 1997 the Mars Surveyor Satellite encountered an atmospheric density double that projected by NASA upon entering a Mars orbit. This greater density bent one of the solar
array arms beyond the full and open stop. This combination of events has delayed the beginning of the scheduled photo mission for one year.
1.1.7 A first stage atmosphere generation on the Moon, where a growing natrium atmosphere is detected that reaches 9,000 km in height. [17].
1.1.8 Significant physical, chemical and optical changes observed on Venus; an inversion of dark and light spots detected for the first time, and a sharp decrease of sulfur-containing gases in its
atmosphere [16].
1. 2 A Change in the Quality of Interplanetary Space Towards an Increase in Its Interplanetary and Solar-Planetary Transmitting Properties.
When speaking of new energetic and material qualities of interplanetary space, we must first point out the increase of the interplanetary domains energetic charge, and level of material saturation.
This change of the typical mean state of interplanetary space has two main causes:
1.2.1 The supply/inflow of matter from interstellar space. (Radiation material, ionized elements, and combinations.) [19,20,21].
1.2.2 The after effects of Solar Cycle 22 activity, especially as a result of fast coronal mass ejection's [CME's] of magnetized solar plasmas. [22].
It is natural for both interstellar matter and intra-heliospheric mass redistribution's to create new structural units and processes in the interplanetary domains. They are mostly observed in the
structured formation of extended systems of magnetic plasma clouds [23], and an increased frequency of the generation of shock waves; and their resulting effects [24].
A report already exists of two new populations of cosmic particles that were not expected to be found in the Van Allen radiation belts [25]; particularly an injection of a greater than 50 MeV dense
electron sheaf into the inner magnetosphere during times of abrupt magnetic storms [CME's], and the emergence of a new belt consisting of ionic elements traditionally found in the composition of
stars. This newly changed quality of interplanetary space not only performs the function of a planetary interaction transmission mechanism, but it (this is most important) exerts stimulating and
programming action upon the Solar activity both in it's maximal and minimal phases.The seismic effectiveness of the solar wind is also being observed [26,27].
1.3 The Appearance of New States and Activity Regimes of the Sun.
As far as the stellarphysical state of the Sun is concerned, we must first note the fact that significant modifications have occurred in the existing behavioral model of the central object of our
solar system. This conclusion comes from observations and reportings of unusual forms, energetic powers, and activities in the Sun's functions [20,21], as well as modifications in it's basic
fundamental properties [28]. Since the end of the Maunder minimum, a progressive growth of the Sun's general activity has been observed. This growth first revealed itself most definitely in the 22nd
cycle; which posed a real problem for heliophysicists who were attempting to revise their main explanatory scenarios:
1.3.1 Concerning the velocity of reaching super-flash maximums.
1.3.2 Concerning the emissive power of separate flashes.
1.3.3 Concerning the energy of solar cosmic rays, etc.
Moreover, the Ulysses spacecraft, traversing high heliospheric latitudes, recorded the absence of the magnetic dipole, which drastically changed the general model of heliomagnetism, and further
complicated the magnetologist's analytic presentations. The most important heliospheric role of coronal holes has now become clear; to regulate the magnetic saturation of interplanetary space.
[28,30]. Additionally, they generate all large geomagnetic storms, and ejection's with a southerly directed magnetic field are geo-effective [22]. There is also existing substantiation favoring the
solar winds effects upon Earth's atmospheric zone circulation, and lithospheric dynamics [31].
The 23rd cycle was initiated by a short series of sunspots in August 1995 [32], which allows us to predict the solar activity maximum in 1999. What is also remarkable, is that a series of class C
flares has already happened in July 1996 . The specificity and energy of this cycle was discussed at the end of the 1980's. [23]. The increased frequency of X-Ray flux flares which occurred in the
very beginning of this cycle provided evidence of the large-scale events to come; especially in relation to an increase in the frequency of super-flashes. The situation has become extremely serious
due to the growth in the transmitting qualities of the interplanetary environment [2 3, 24] and the growth of Jupiter's systems heliospheric function; with Jupiter having the possibility of being
shrouded by a plasmosphere extending over Io's orbit [13].
As a whole, all of the reporting and observation facilities give evidence to a growth in the velocity, quality, quantity, and energetic power of our Solar System's Heliospheric processes.
Update Note 1/8/98: The unexpected high level of Sun activity in the later half of 1997, that is continuing into present time, provides strong substantiation of the above statement. There were three
"X" level Goes 9 X-Ray Flux events in 1997 where one was forecasted; a 300% increase. The most dramatic of these, a X-9.1 coronal mass ejection on November 6, 1997, produced a proton event here on
Earth of approximately 72 hours in duration. The character, scale, and magnitude of current Sun activity has increased to the point that one official government Sun satellite reporting station
recently began their daily report by saying, "Everything pretty much blew apart on the Sun today, Jan. 3,1998."
The recorded and documented observations of all geophysical (planetary environmental) processes, and the clearly significant and progressive modifications in all reported solar-terrestrial physical
science relationships, combined with the integral effects of the antropohenedus activity in our Solar System's Heliosphere, [33,34], causes us to conclude that a global reorganization and
transformation of the Earth's physical and environmental qualities is taking place now; before our very eyes. This current rearrangement constitutes one more in a long line of cosmo-historic events
of significant Solar System evolutionary transformations which are caused by the periodic modification, and amplification, of the Heliospheric-Planetary-Sun processes. In the case of our own planet
these new events have placed an intense pressure on the geophysical environment; causing new qualities to be observed in the natural processes here on Earth; causes and effects which have already
produced hybrid processes throughout the planets of our Solar System; where the combining of effects on natural matter and energy characteristics have been observed and reported.
We shall now discuss global, regional, and local processes.
2.1 The Geomagnetic Field Inversion.
Keeping clearly in mind the known significant role of the magnetic field on human life, and all biological processes, we will outline the general features of this changing state of the Earth's
geomagnetic field. We have to remind ourselves of the many spacecraft and satellites that have registered the growth of heliospheric magnetic saturation in recent years [11,18,35]. The natural
response of the Earth to this increased saturation level reveals itself in its dipole intensity, its magnet "c" poles localization, and in its electromagnetic field resonance processes[36]. Earth is
number one among all of the planets in the Solar System with respect to its specific ability regarding the magnetization of matter [6].
In recent years we have seen a growth of interest by geophysicists and magnetologists, in general, to geomagnetic processes [37-40], and specifically, to the travel of Earth's magnetic poles [41,42].
They are particularly interested in observing the facts surrounding the directed, or vectored, travel of the Antarctic magnetic pole. In the last 100 years this magnetic pole has traveled almost 900
km towards, and into, the Indian ocean. This significant shift by the magnetic poles began in 1885. The most recent data about the state of the Arctic magnetic pole (which is moving towards the
Eastern Siberian world magnetic anomaly by way of the Arctic Ocean) reveals that this pole "traveled" more than 120 km during the ten year period 1973 through 1984, and 150 km during the same
interval, 1984 through 1994. This estimated data has been confirmed by direct measurement ( L. Newwitt. The Arctic pole coordinates are now 78.3 deg. North and 104.0 deg. West) [42].
We must emphasize that this documented polar shift acceleration (3 km per year average over 10 years), and its travel along the geo-historic magnetic poles inversion corridor (the corridor having
been established by the analysis of more than 400 paleoinversion sites) necessarily leads us to the conclusion that the currently observed polar travel acceleration is not just a shift or digression
from the norm, but is in fact an inversion of the magnetic poles; in full process.It is now seen that the acceleration of polar travel may grow to a rate of up to 200 km per year. This means that a
polar inversion may happen far more rapidly than is currently supposed by those investigators without a familiarity with the overall polar shift problem.
We must also emphasize the significant growth of the recognized world magnetic anomalies (Canadian, East-Siberian, Brazilian, and Antarctic) in the Earth's magnetic reorganization. Their significance
is due to the fact that these world anomalies constitute a magnetic source that is almost independent from Earth's main magnetic field. Most of the time, the intensity of these world magnetic
anomalies substantially exceeds all of the residual non-dipole component; which is obtained by the subtraction of the dipole component from the total magnetic field of the Earth.[48]. It is the
inversion of the magnetic fields process which is causing the various transformations of Earth's geophysical processes and the present state of the polar magnetosphere.
We also have to take into account the factual growth of the polar cusp's angle (i.e. The polar slots in the magnetosphere; North and South), which in the middle 1990's reached 45 degrees (by IZMIRAN
data). [Note: The cusp angle was about 6 degrees most of the time. It fluctuates depending upon the situation. During the last five years, however, it has varied between 25 and 46 degrees.] The
increasing and immense amounts of matter and energy radiating from the Sun's Solar Wind, and Interplanetary Space, by means previously discussed, has began to rush into these widened slots in the
polar regions causing the Earth's crust, the oceans, and the polar ice caps to warm[27].
Our study of geomagnetic field paleoinversions, and their after effects, has lead us to the unambiguous, and straight forth, conclusion that these present processes being observed are following
precisely the same scenarios as those of their distant ancestors. And additional signs of the inversion of the magnetic field are becoming more intense in frequency and scale. For example: During the
previous 25 million years, the frequency of magnetic inversions was twice in half a million years while the frequency of inversions for the last 1 million years is 8 to 14 inversions [43], or one
inversion each 71 to 125 thousand years. What is essential here is that during prior periods of maximum frequency of inversions there has also been a corresponding decrease in the level of oceans
world-wide (10 to 150 meters) from contraction caused by the wide development of crustal folding processes. Periods of lessor frequency of geomagnetic field inversions reveals sharp increases of the
world ocean level due to the priority of expansion and stretching processes in the crust. [43-44]. Therefore, the level of World's oceans depends on the global characteristic of the contraction and
expansion processes in force at the time.
The current geomagnetic inversion frequency growth phase may not lead to an increase in oceanic volume from polar warming, but rather to a decrease in ocean levels. Frequent inversions mean
stretching and expansion, rare inversions mean contraction. Planetary processes, as a rule, occur in complex and dynamic ways which require the combining and joining of all forces and fields in order
to adequately understand the entire system. In addition to the consideration of hydrospheric redistribution, there are developing events which also indicate a sudden and sharp breaking of the Earth's
meteorological machinery.
2.2 Climate Transformations.
Since public attention is so closely focused on the symptoms of major alterations, or breakdowns, in the climatic machinery, and the resulting and sometimes severe biospheric effects, we shall
consider these climatic transformations in detail. Thus, while not claiming to characterize the climatic and biospheric transition period completely, we will provide a recent series of brief
communications regarding the temperature, hydrological cycle, and the material composition of the Earth's atmosphere.
The temperature regime of any given phase of climatic reorganization is characterized by contrasts, and instabilities. The widely quoted, and believed, "Greenhouse Effect" scenario for total climatic
changes is by far the weakest explanation, or link, in accounting for this reorganization.It has already been observed that the growth in the concentration of CO2 has stopped, and that the methane
content in the atmosphere has began to decrease [45] while the temperature imbalance, and the common global pressure field dissolution has proceeded to grow.
There were reports of a global temperature maximum in 1994, and the almost uninterrupted existence of an "El-Nino" like hydrological effect. Satellite air surface layer temperature tracking [49,50]
allowed the detection of a 0.22 degrees C global temperature variation (within a typical specific time period of about 30 days) that correlated with recorded middle frequency magnetic oscillations.
The Earth's temperature regime is becoming more, and more, dependent on external influences. The representative regulating processes, or basis, of these general climatic rearrangements are:
2.2.1. A new ozone layer distribution.
2.2.2. Radiation material (plasma) inflows and discharges through the polar regions, and through the world's magnetic anomaly locations.
2.2.3. Growth of the direct ionospheric effects on the relationship between the Earth's meteorological (weather), magnetic, and temperature fields.
There is a growing probability that we are moving into a rapid temperature instability period similar to the one that took place 10,000 years ago. This not so ancient major instability was revealed
by the analysis of ice drilling core samples in Greenland [51]. The analysis of these core samples established:
2.2.4.That annual temperatures increased by 7 degrees centigrade.
2.2.5.That precipitation grew in the range of 3 to 4 times.
2.2.6.That the mass of dust material increased by a factor of 100.
Such high-speed transformations of the global climatic mechanism parameters, and its effects on Earth's physical and biospheric qualities has not yet been rigorously studied by the reigning
scientific community. But, researchers are now insisting more, and more, that the Earth's temperature increases are dependent upon, and directly linked to, space-terrestrial interactions [52,53]; be
it Earth-Sun, Earth-Solar System, and/or Earth-Interstellar.
At the present time there is no lack of new evidence regarding temperature inversion variations in the hydrosphere [oceans]. In the Eastern Mediterranean there have been recordings of a temperature
inversion in depths greater than two kilometers from a ratio of 13.3 to 13.5 degrees centigrade to a new ratio of 13.8 to 13.5; along with a growth in salinity of 0.02% since 1987. The growth of
salinity in the Aegean Sea has stopped, and the salt water outflow from the Mediterranean Basin to the Atlantic has diminished. Neither of these processes, or their causes, has been satisfactorily
explained. It has already been established that evaporation increases in the equatorial regions causes a water density increase which results in an immediate sinking to a greater depth. Ultimately
this would force the Gulfstream to reverse its flow. A probability of this event happening is confirmed by other signs as well as multiparameter numeric models [53]. Therefore the most highly
probable scenario for the European Continent is a sharp and sudden cooling. Elsewhere, the Siberian region has been experiencing a stable temperature increase [58] along with reports from the
Novosibirsk Klyuchi Observatory of a constant growth of up to 30 nanoteslas per year of the vertical component of the magnetic field. This growth rate increases significantly as the Eastern Siberian
magnetic anomaly is approached.
Update Note 1/8/98: The National Oceanic and Atmospheric Administration reported today, 1/8/98, that 1997 was the warmest year on record since records began in 1880, and that nine of the warmest
years since that time have occured in the last eleven years.
2.3 Vertical and Horizontal Ozone Content Redistribution.
Vertical and horizontal ozone content redistribution is the main indicator, and active agent, of general climatic transformations on Earth. And, evidence exists that ozone concentrations also have a
strong influence upon Earth's biospheric processes. Widespread models for "ozone holes" being in the stratosphere [7 to 10 miles above Earth] (Antarctic and Siberian) are receiving serious corrective
modifications from reports of vertical ozone redistribution, and its growth in the troposphere [below 7 miles]. It is now clear that the decrease in our atmosphere's total ozone content is caused by
technogeneous [industrial, human designed, pollution], and that the total ozone content in general has serious effects upon the energy distribution processes within Earth's gas-plasma [atmospheric]
envelopes [54].
Stratospheric, tropospheric, and surface layer ozone's are now being studied [55,56]. Photodissociation [the process by which a chemical combination breaks up into simpler constituents] of ozone,
controls the oxidizing activities within the troposphere. This has created a special atmospheric, physio-chemical, circumstance by which the usual tropospheric concentrations, and lifetimes, of
carbon monoxide, methane, and other hydrocarbon gases are modified and changed. So, with the established fact that a statistically significant rise in the ozone concentrations has taken place in the
tropospheric layers between 5 and 7 miles, and with the addition, and full knowledge, of ozone's oxidizing properties, we must conclude that a basic and fundamental alteration of the gas composition
and physical state of Earth's atmosphere has already begun.
There are continuing reports of diminishing regional stratosphere ozone concentrations [25 to 49% or more above Siberia (57)], and of global decreases of ozone content in altitudes of 20-26 miles;
with the maximal decrease of 7% being at 24 miles [55]. At the same time, there is no direct evidence of a growth of UV radiation at the ground surface [58]. There are, however, a growing number of
"ozone alerts" in large European cities. For example, in 1994 there were 1800 "ozone alerts" in Paris. In addition, remarkably high concentrations of surface layer ozone were registered in the
Siberian Region. There were ozone concentration splashes in Novosibirsk that exceeded 50 times the normal level. We must remember that ozone smell is noticeable in concentrations of 100 mkg/m3; i.e.
at 2 to 10 times the normal level.
The most serious concern of aeronomists comes from the detection of H02 that is being produced at an altitude of 11 miles by a completely unknown source or mechanism. This source of HO2 was
discovered as a result of the investigation of OH/HO2 ratios in the interval between 4.35 and 21.70 miles in the upper troposphere and stratosphere. This significant growth of HO2, over the course of
time, will create a dependence on this substance for the ozone transfer and redistribution process in the lower stratosphere[56].
The submission of the ozone's dynamic regime and space distribution to the above unknown source of HO2, signifies a transition of Earth's atmosphere to a new physico-chemical process. This is very
important because non-uniformity's in the Earth's ozone concentrations can, and will, cause an abrupt growth in temperature gradients, which in turn do lead to the increase of air mass movement
velocities, and to irregularities of moisture circulation patterns[46,59]. Temperature gradient changes, and alterations, over the entire planet would create new thermodynamic conditions for entire
regions; especially when the hydrospheres [oceans] begin to participate in the new thermal non-equilibrium. The study [53] supports this conclusion, and the consideration of a highly possible abrupt
cooling of the European and North American Continents. The probability of such a scenario increases when you take into account the ten year idleness of the North Atlantic hydrothermal pump. With this
in mind, the creation of a global, ecology-oriented, climate map which might reveal these global catastrophes becomes critically important.
Considering the totality and sequential relationship of transient background, and newly formed processes, brought about by the above stated cosmogenic and anthropogenic PlanetoPhysical
transformations and alterations of our weather and climatic systems, we find it reasonable to divide matters into their manifest (explicit) and non-manifest (implicit) influences upon Earth's
3.1 The Manifest or Explicit Consequences.
The classes or categories of effects brought about by the Earth's current stage of reorganization are very diverse. Most often, however, they tend to the transient high-energy type of event. Based on
the results of the Yokohama Conference (Fall 1994,) they can be called "significant catastrophes". There are nine types of "significant catastrophes:"
In addition, we must point out the abrupt growth of meteorological/weather catastrophes in recent years. In the Atlantic region alone there were 19 cyclones in 1994; 11 of which became hurricanes.
This is a 100 year record [ 60]. The current year, 1996, is especially laden with reports of flooding and other types of meteocatastrophes. The dynamic growth of significant catastrophes shows a
major increase in the rate of production since 1973. And in general, the number of catastrophes has grown by 410% between 1963 and 1993. Special attention must be focused on the growing number and
variety of catastrophes, and to their consequences.
One must keep in mind that the growing complexity of climatic and weather patterns signals a transformation tending towards a new state, or as Academician Kondratyev says, data indicates that we are
moving in the direction of climatic chaos. In reality this transition state of our climatic machinery is placing new requirements upon Earth's entire biosphere; which does include the human species.
In particular, there are reports from Antarctica that show a dramatic reaction by vegetation to the recent changes in climate; there were 700 species found growing in 1964 and 17,500 in 1990 [61].
This increase in Earth's vegetative cover provides evidence of the biosphere's reaction to the ongoing process of climatic rearrangement.
The overall pattern of the generation and movement of cyclones has also changed. For example, the number of cyclones moving to Russia from the West has grown 2.5 times during the last 10 years.
Increased ocean levels caused by the shedding of ice from the polar regions will lead to sharp changes in coast lines, a redistribution of land and sea relationships, and to the activation of
significant geodynamic processes. This is the main characteristic of those processes leading to a new climatic and biospheric order.
3.2 The Non-Manifest or Implicit Consequences.
Implicit consequences are those processes which are below the threshold of usual human perception, and are therefore not brought to our common attention. Instrument recordings, and even direct
observations, of these phenomena throughout Earth's electromagnetic field provides evidence that an immense transformation of Earth's environment is taking place. This situation is aggravated by the
fact that in the 1990's anthropogeneous (human) power production/usage increased to (1-9)E+26 ergs/per year which means it reached the conservative energetic production/usage values of our planet.
For example, Earth's annual energy consumption is comprised of (1-9)E+26 ergs for earthquakes, (1-9)E+24 for geomagnetic storms, and (1-9)E+28 for heat emission [54].
There already are technogeneous effects upon the functional state of Earth's electromagnetic skeleton being registered and recorded. A seven-day technogeneous cycle for geomagnetic field dynamic
parameter variations was revealed in 1985 [62,63]. This cycle has affected many of the short cycles in Solar-terrestrial relationships. More than 30% of middle magnetosphere disturbances are caused
by power production, transmission, and consumption. The Van Allen radiation belt has abruptly lowered above the East Coast of the US from 300 km to 10 km. This process is associated with electricity
transmission from the Great Lakes to the South along a magnetic meridian, and usage of the ionosphere-resonance frequency (60Hz) of energy consumption [63]. There is also a registered coherence
between the gutter qualities of the Brazilian magnetic anomaly, and the "Hydro-Quebec" power production system. Combined techno-natural electromagnetic processes in megalopolises are very complex and
as yet unstudied. A 1996 study of mortality from cardiovascular diseases in St. Petersburg, Russia uncovered a direct connection between the city's power consumption and mortality.
Moreover, the increase in the frequency, and scope, of natural self-luminous formations in the atmosphere and geospace forces us to wake up, and take notice [64,65,66]. The processes of generation,
and the existence of such formations, spreading all over the Earth, represents a remarkable physical phenomenon. What is most unusual about these natural self-luminous formations is that while they
have distinct features of well-known physical processes, they are in entirely unusual combinations, and are accompanied by process features which cannot be explained on the basis of existing physical
knowledge.Thus, features of intense electromagnetic processes are being found in the space inside and near these natural self-luminous objects. These features include:
3.2.1. Intense electromagnetic emissions ranging from the micrometer wave band through the visible diapason, to television, and radio wavelengths.
3.2.2. Electric and magnetic field changes such as electric breakdowns, and the magnetization of rocks and technical objects.
3.2.3. Destructive electrical discharges.
3.2.4. Gravitation effects such as levitation.
3.2.5. Others.
All of the qualities of this class of phenomena are requiring the development of new branches of modern physics; particularly the creation of a "non-homogeneous physical vacuum model".[67]. An
advancement of the sciences in this direction would allow us to reveal the true nature of these objects, which are acting apparently, and latently, upon our geological-geophysical and biospheric
environment, and on human life [68].
Therefore, we must first take into account all of the newly developed processes and states of our geological-geophysical environment. These processes, for the most part, manifest themselves in the
hard-to-register, and observe, qualities of the Earth's electromagnetic skeleton. This data also concerns the geophysical and climatic meanings of Solar-terrestrial and planetary-terrestrial
interactions. This is especially true of Jupiter which is magnetically conjugate to our planet.
The totality of these planet-transforming processes develops precipitately, ubiquitously, and diversely. It is critical that politicians be informed and trained to understand these global
relationships between the totality of natural and anthropogeneous activities, and there fundamental causes and effects [69]. A compelling need exists to commence a scientific study which would
delineate the problems associated with Earth's current transformational processes, and the effects they will have on global demographic dynamics.[70].
The sharp rise of our technogeneous system's destructive force on a planetary as well as a cosmic scale, has now placed the future survival of our technocratic civilization in question[33,7].
Additionally, the principle of Natures supremacy [72] over that of humanities current integral technogeneous and psychogenic activities and results, becomes more, and more, apparent.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 5352
Join date : 2011-06-04
Location : My own little heaven on earth
The answer to The Grand Cycle has been found...
and, everyone is part right !!!
The answers are in cycles of 5, 6, 13, 20, 60, 65, 260, 360, 360 + 360/361+5 & 144,000,
as, well, as The Cycles of The New & The Full Moons that are followed by all indigenous & tribal people worldwide.
There is only one date, that will satisfy the end, and, the start date of all ancient calendars,
from The Mayan Tzolkin (260 days), The Mayan Haab (360/361+5 days), The Aztec Tonalpohualli (260 days),
to The Ancient First Nations - Metis - Native American (260 days) & (360 days) calendars.
The Ancient Year, was always measured in relation to a circle, which has 360 Degrees or 360 Days,
from an astrological perspective this would relate to a 1 degree precession per day,
following through a series of 360 days to complete one year.
This calendar was originally set into six quarters of sixty days - 6 x 60 = 360
and, each of these six quarters where known as: Winter, Spring, Breakup, Summer, Fall, and Freeze Up.
In 2012, The New Moon occurs on 13 Dec 2012, and, The Full Moon occurs on 28 Dec 2012,
which signals The End of The Freeze Up Cycle, and, The Start of The Winter Cycle
which was the real beginning of The Annual Yearly Cycle.
In 2013, The New Moon occurs on 03 Dec 2013, and, The Full Moon on 17 Dec 2013,
which signals The End of Freeze Up Cycle, and, The Start of The Winter Cycle,
The Beginning of The Annual Yearly Cycle.
This date is in perfect alignment with The Start of The Winter cycle of 2012-2013
and, The Start Date of The 13th Grand Cycle of Pacha iNTi which begins on The 17th of December, 2013.
The Cycle of The New Moons, along with The Full Moons, and each of The Six Quarters
are all connected to very significant Ceremonies, for an assortment of purposes.
The Date of The 21st of December 2012, does NOT satisfy all calendars,
and, therefore, it can NOT be accepted as the end date of a grand cycle,
as, it is NOT in perfect alignment with all known calendars.
The 21st DEC 2012 is The 47th Ahua Date in the last series of 65 cycles.
and, is The end of The Macha (female) Cycle of 9,360,000 Kin aka Days (65 cycles x 144,0000)
and, The 22nd DEC 2012 is the start of The Pacha (male) Cycle,
which runs for 360 Kin aka days (6 cycles of 60)
and, 9,360,000 + 360 = 9,360,360 = 65 cycles X 144,000 + 60 x 6
and, The 16th DEC 2013 is The 65th Ahua Date in the last series of 65 cycles.
The length of The Ancient Cycles were, and, still are 5, 6, 13, 20, 60, 260, 360, 360/361+5, & 144,000.
Our correlation date for the last cycle is August 11th 3114 BC Gregorian proleptic
or, September 6th, 3114 BC Julian.
It is important to know, that The Gregorian Calendar did NOT exist prior to 1572,
when it was created and implemented as The Civil Calendar by Pope Gregory of The Catholic Church,
which introduced a leap year, every four years, to account for the missing time.
A real day is NOT exactly 24 hours in length, and,
this calendar was created to create the ability to allow the time keeping of mechanical clocks & watches,
which required exact measures.
The Ancient Year was 360 Kin aka Days
- exactly the number of degrees in a circle or a round.
The Moon Cycles were given names, and, each tribe utilized slightly different names,
The Creator asked All of The Original Ancestors of all of The Tribes
to give blessings through the giving of names to everything, including The Naming of The Moon cycles
and, to do that they utilized the process of The Naming Ceremony.
The Grand cycle occurs once every 9,360,000 + 360 days = 9,360,360 days=
9,360,000 + 360 = 13 x 5 = 65 cycles x 144,000 + 360
= 26,000 x 360 + 6 x 60 = 468018 x 20.
These 9,360,360 days run through all of the assortment of Ceremonial Events,
namely The Sunrise Ceremony & The Sunset Ceremony,
along with The New Moon Ceremony & The Full Moon Ceremony,
which holds great importance around the world,
to all indigenous & non-indigenous people and tribes.
The 13th of December 2012 - is a New Moon
and, The 28th of December, 2012 - is a Full Moon
which starts The Winter Cycle of 2012-2013.
The 21st of December 2012,
is The End of The Macha Cycle of 9,360,000 Days
and, The 16th of December 2013
is The End of The Grand Cycle of Macha & The Grand Cycle of Pacha,
which terminates at sunset on The 65th Ahua - 16 DEC 2013
which is the day before The Full Moon of 17th of December 2013.
These time honoured cycles of 5, 6, 13, 20, 60, 65, 260, 360, 360/361+5 & 144,000 cycles,
all flow through The New Moon & The Full Moon,
through The Black & The Blue Moon Cycles
which were respected very highly by the day, record, time and wisdom keepers.
These were very important cycles
which were related to a vast assortment of ancient practices
of day, record, time and wisdom keeping, that are still utilised today
by celebrating, creating and doing a specific Ceremony
related to each of them.
The 13th Grand Cycle of Pacha iNTi begins
-The 17th December 2013 at sunrise.
The12th Grand Cycle of Macha & Pacha ends
- The 16th December 2013 at sunset aka The 65th Ahua.
susan lynne schwenger
- Dokis First Nation, Ontario, Canada
28 OCT 2011 - The 26th Ahua - The Calleman Date
21 DEC 2012 - The 47th Ahua - The John Major Jenkins Date
21 DEC 2012 - The 47th Ahua - The Terrance McKenna Date
16 DEC 2013 - The 65th Ahua - The Susan Lynne Schwenger -Tony Bermanseder Date
17 DEC 2013 - Start of The 13th Grand Cycle of Pacha iNTi
was discovered by Tony Bermanseder & Susan Lynne Schwenger
and, can also be aligned to important bible codes
The Main Code of The DRESDEN CODEX - MAYAN SUPER NUMBER 1366560 has been cracked & decoded
BY Susan Lynne Schwenger & Tony Bermanseder which shows The Mother Earth consists of 12 levels.
THE MAGIC of 144 - 144 WHOLE NUMBER FACTORS of MAYAN DRESDEN CODEX SUPERNUMBER 1366560
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 24, 26, 30, 32, 36, 39, 40, 45, 48, 52, 60, 65, 72, 73, 78, 80, 90, 96, 104, 117, 120, 130, 144,
146, 156, 160, 180, 195, 208, 219, 234, 240, 260, 288, 292, 312, 360, 365, 390, 416, 438, 468, 480, 520, 584, 585, 624, 657, 720, 730, 780, 876, 936, 949, 1040, 1095, 1168, 1170, 1248, 1314, 1440,
1460, 1560, 1752, 1872, 1898, 2080, 2190, 2336, 2340, 2628, 2847, 2920, 3120, 3285, 3504, 3744, 3796, 4380, 4680, 4745, 5256, 5694, 5840, 6240, 6570, 7008, 7592, 8541, 8760, 9360, 9490, 10512, 11388,
11680, 13140, 14235, 15184, 17082, 17520, 18720, 18980, 21024, 22776, 26280, 28470, 30368, 34164, 35040, 37960, 42705, 45552, 52560, 56940, 68328, 75920, 85410, 91104, 105120, 113880, 136656, 151840,
170820, 227760, 273312, 341640, 455520, 683280 and 1366560.
10OCT2008 - SUSAN LYNNE SCHWENGER 10/10/10
susan lynne schwenger
Posts : 32800
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Location : Hawaii
Ironically, the terrible situation in Connecticut seems to have taken attention away from the shift. One of the conversations that Mercuriel and I had many months back had to do with the state of one
consciousness (where ones attention is) when the shift occurs. Being in a state of grace is good.
In another link I read that the time the planetary line-up occurs is at 11:11 PST. For us here in Hawaii that would be 9:11 am. Now don't those numbers ring a bell? I let my husband know I would just
like all of us to be together at this time for the "just in case" this is a real event. There is something about moving through this cycle together, where we hold each other in our hearts and just
connect with Divine presence from whence we all originated from to begin with - where we existed like little fire flies in the heart of a sun.
This is not a time of fear - yet of rejoicing that we made it this far and this long on a planet in deep trouble. At the end of an age - this past age - it is considered the darkest before dawn where
spiritual energy flowing down to the planet is supposedly at its lowest point that it has been during the past several thousand years. If we are to accept some of the more long-term spiritual
teachings - at the end of the week we are to enter into the next golden age (where spiritual energy will flow down to the planet and its inhabitants in full abundance.) Given all that we have been
witness to these past years - this would be a tremendous gift.
Basically, Mercuriel was explaining that a sorting will occur where like energy frequencies go with like frequencies. I'm not sure and can't be certain that this is going to happen at the end of the
week. Yet again, just in case - it's best to be in a state of grace by focusing on pure love - emanating from one's own heart center, as much as possible on the 21st and just pray for the best
possible outcome as your deepest inner thoughts really are things and really do manifest.
One of the things I've read about entering into the 5th dimension - if this is to occur, is that one only experiences present time. So with this in mind imagine what you wish for this planet and each
other. Visualize heaven on earth, the planet healed and individual souls free to manifest the best of all that is possible.
Will gravity cease during this alignment? Don't know. Will there be quakes? Don't know. Will things just continue as yesterday? Don't know. All I do know is that what I have control over is my
attitude. So in the spirit of this even maintaining an attitude of gratitude and appreciation for this life and the people who are in it shall be a primary focus.
I've observed that our soul group here at Mists have been together for a very long time. We were excited when we discovered each other over the net. This was truly an amazing gift. I'm sure while on
the other side prior to incarnation some of us may have wondered how this would occur. And yet, even though some of us come at different understandings with polar opinions - I suspect that most of us
still find appreciation in our hearts for what each member brings to the table to share with the others. I also suspect if we were all in agreement life would become a bit dull. It's our differences
that make life interesting and gives it a bit of zest. And in the end when all is said and done - we continue to learn from each other and our lives are enriched. Blessed be and blessings to each of
us who share this time on earth together.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
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Russia Is In The Throes Of Mayan Apocalypse Mania
Read more: http://www.businessinsider.com/russia-gripped-by-mayan-apocalypse-mania-2012-12#ixzz2FL3lONci
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
One hour, forty-four minutes movie: https://www.youtube.com/watch?feature=player_embedded&v=hlfYHAV1i8w
2012 Crossing Over, A New Beginning OFFICIAL FILM [Brave Archer Films®]
Published on Dec 2, 2012
'A World of Love is Coming!' This is the OFFICIAL RELEASE of the Full Length Documentary film '2012 Crossing Over, A New Beginning' a film written, directed, filmed, edited and entirely independently
funded and produced by Amel Tresnjic.
"Thank you ALL for sharing the message of LOVE. Most people don't realize that the entire crew of the film... is just ME :-) I created the film with the intention to make our world a better place...
and the response has been AMAZING. There are no paid ads or any mainstream support. By the power of the people the film is NOW getting translated into more than 14+ languages. Thank you for those who
believe in the message and are donating their time to translate it:-) The message of LOVE is spreading all over the world AND THAT'S ONLY BY THE POWER OF THE PEOPLE! Thank YOU ALL! "
Love and Peace Amel
Plot: Dec 21, 2012 is on everyone's mind. What will it bring? Is it the end of the world? A new beginning for mankind? Or just another year on the calendar? Brave Archer Films presents '2012 Crossing
Over, A New Beginning.' The feature doco explores a positive spiritual perspective on the events of Dec 21, 2012. The film investigates the galactic alignment, consciousness awakening, cycles of
evolution, our binary star system with Sirius, the fear agenda in the media, who's behind it, love vs fear and much more.
The film is loaded with amazing revelations of the current times we live in, from exceptional astrologer and teacher Santos Bonacci, spiritual leaders Bud Barber, George Neo and much more. The doco
is shot entirely in Full HD, illustrated with high end animations and includes original music by Jonathan Kent.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
Four days to go until the Mayan Doomsday -
and there's a rush on for bomb-proof survival bunkers
(complete with leather sofas and plasma screen TVs)
Ron Hubbard builds stylish underground bomb-proof shelter in California
Designer's business has gone from selling one per month to one per day
Some say Mayan Long Count calendar proves world will end this Friday
Read more: http://www.dailymail.co.uk/news/article-2249502/Mayan-Doomsday-Bomb-proof-50-000-bunkers-leather-sofas-plasma-TVs.html#ixzz2FMh84gDi
Follow us: @MailOnline on Twitter | DailyMail on Facebook
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 13627
Join date : 2010-09-28
Location : The Matrix
I continue to suspect that there are NO safe places -- and that even the Insiders (both good and bad) are NOT safe. I'll continue to quietly model my strange notions of solar system governance -- and
to consider my strange conceptualizations of a very strange and very dangerous universe. I am quite pessimistic about the future -- regardless of what happens (or doesn't happen) on 12-21-12 (or
throughout 2013). The threads I've created (in the context of Project Avalon and the Mists of Avalon) should keep me busy as I await 'The End'. I just got a set of encyclopedias to read. So if the
'Veil is Lifted' and we see all manner of strange beings -- or if we discover that we are strange beings -- I'll try to just utter 'well whatta you know!!??' -- and then go back to reading my threads
and encyclopedias!! Are we facing the Open Rule of Satan -- after thousands of years of the Covert Rule of Satan??!! I continue to suspect that we are more screwed than we can possibly imagine. On
the other hand -- I continue to model idealistic solar system governance modalities. But to whom it may concern -- if you get propped-up as some sort of a Solar System Administrator -- be very wary
regarding the very top of the pyramid -- and be very wary regarding the fate of humanity. You might be in for a helluva marriage -- once the honeymoon is over -- and the novelty wears off. http://
Last edited by orthodoxymoron on Mon Dec 17, 2012 10:39 pm; edited 2 times in total
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Location : INNOVATION STATIONS !SCHOOL
Unbelievable...they just don't get it...they just cant see it...the Armageddon is happening now! The shift is higher dimensional!!!
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That's the point JT. We were just talking about this topic this evening. It would be nice to have an underground bunker for the normal type of earth changes but this is something so much more how is
it even possible to prepare. On one hand it is comforting to know there is a safe space away from the wind and tornados but how does one prepare for the spiritual? Does that extra junk DNA get
activated? Do we just pass over into a different type of spiritual body? Or do we just carry on as if it's another day?
We realize that we've been talking about this for the past 40 years. And now it's here in a few days. I told my husband if nothing happens I'm getting rid of everything in the garage. He laughed and
said wait. Maybe it's like in the movie "Raiders of the Lost Ark" where they open the casket and it's just sand where everyone just stares and thinks it's all just a joke. Then the sand stirs and
energy pours out. And then being this is an island even a major quake in California could damage the shipping ports and stuff couldn't get here.
I also notice Floyd has been absent recently and wonder if he has wandered off to his safe place.
Oxy, I do worry about you as it seems your focus is a bit on the dark side. I've been listening to Kat Kerr's description of being in heaven and it just blows the socks off as it is so uplifting. She
goes on about the coming Heaven Invasion . The 10 minute segments is a bit annoying and there are over 60 segments in one of her sessions. What I like best is all of the teaching that is given on how
to use god's energy and what it is like in heaven.
In a discussion with my daughter she was talking about preferring go to hell thinking heaven would be pretty boring. Needless to say she is of the age where some important decisions are being made
and we have interesting conversations. On one hand she professes to be an atheist and on the other is in tears if a friend even suggests there is nothing there when folks pass over because she wants
to believe her grandparents are in heaven. So you can see even at the tender age of 16 she struggles with what is real and not real. It's been a rough week-end with the disaster in Connecticut so of
course these issues are at the forefront.
Have you notice an increase in the buzzing sound. We're in the middle of some big solar winds right now impacting the geomagntic field. I wonder if this will continue to increase over the next few
Here is a link to one of Kat's longer clips.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 13627
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Location : The Matrix
Carol, I'm just trying to be a Galactic Tom Bodett -- to keep the light on in this solar system -- in what I perceive to be a dark universe. I do NOT wish to think the way I do -- but my internal
modeling is quite frightening and compelling. I base most of my thinking on extensive experience and reflection -- but I certainly can't prove any of it -- and I SO hope that I'm wrong.
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Oxy, that's why Kit's videos are worth spending time to listen too as it helps clear out the clutter.
Here is some interesting info on a pole shift.
<iframe width="640" height="360" src="https://www.youtube.com/embed/wLg2tK8bp8k?feature=player_embedded" frameborder="0" allowfullscreen></iframe>
Meanwhile the doomsday clock is ticking. http://doomsdayclock2012.com/#sthash.OR4xRsTs.fekQtxEW.dpbs
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
<iframe width="640" height="360" src="https://www.youtube.com/embed/fttEA_rpyf0?feature=player_detailpage" frameborder="0" allowfullscreen></iframe>
Published on Dec 17, 2012 - This phenomenon was observed and recorded for several hours by an anon.
I leave you all to make of this what you will. Find us and more at anonymousuk.org, Links from there to Facebook groups
Earth is inside a stream of medium-speed solar wind. Could this orange glow be related to that?
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
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Here's another link for you Oxy.
Jesus - The Mayan Calendar - Nostradamus and the Divine Cross
-The Amazing Synchronicity of 12/21/12
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
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<iframe width="640" height="360" src="https://www.youtube.com/embed/TFDpzb97WeU?feature=player_detailpage" frameborder="0" allowfullscreen></iframe>
Winter Triangle on the sun.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 3469
Join date : 2010-08-21
Age : 71
Department of Defense tries to predict the future...
Intelligence Council Poses Four Worlds of the Future
12/13/2012 05:59 PM CST
Intelligence Council Poses Four Worlds of the Future
By Jim Garamone
American Forces Press Service
WASHINGTON, Dec. 13, 2012 - Prediction is an inexact science.
The 1939 New York World's Fair was billed as a look at tomorrow, and nations built pavilions and presented their latest inventions along with how they believed they would change the world.
One large part of the Fair itself was called "Futurama" -- a scale model of what planners believed would be America in 1960. The model had futuristic homes, urban complexes, bridges, dams and an
advanced highway system which envisioned speeds of 100 mph.
The visionaries of 1939 did not anticipate suburbs, satellites, an oil embargo, nuclear energy or apparently where all those 100 mph cars were going to park.
The National Intelligence Council, which supports the Director of National Intelligence by providing long-term strategic analysis, has learned from instances like this and presents a range of options
in its publication World Trends 2030.
The council posits four possible worlds in 2030: stalled engines, fusion, gini out-of-the-bottle and nonstate world.
"Gini" refers to the gini coefficient, which is a statistical measurement of income inequality.
The stalled engine world predicts a planet where the risk of interstate conflict rises due to a new great game in Asia. This scenario is a bleak one. "Drivers behind such an outcome would be a U.S.
and Europe that turn inward, no longer interested in sustaining their global leadership," the report says. This scenario envisions the Euro Zone unraveling, causing Europe's economy to tumble.
The stalled engine world also sees the U.S. energy revolution failing to materialize -- despite current trends that suggest the U.S. will be a future energy exporter.
This scenario is most likely to lead to conflict between nations over scarce resources, but this scenario does not necessarily envision major conflagrations. Economic interdependence and
globalization would be mitigating factors.
The fusion scenario represents the other end of the spectrum.
"This is a world in which the specter of a spreading conflict in South Asia triggers efforts by the U.S., Europe and China to intervene and impose a ceasefire," the report says. "China, the U.S. and
Europe find other issues to collaborate on, leading to a major positive change in their bilateral relations, and more broadly leading to worldwide cooperation to deal with global challenges."
This scenario sees China adopting political reforms and Chinese leaders managing growing nationalism. Fusion sees more multinational organizations.
"In this scenario, all boats rise substantially," the report says. Developing economies rise, but so do those in developed countries. Under fusion, the American dream remains a reality with the
council seeing U.S. incomes rising by $10,000 over a decade.
"Technological innovation -- rooted in expanded exchanges and joint international efforts -- is critical to the world staying ahead of the rising financial and resource constraints that would
accompany a rapid boost in prosperity," the report says.
The genie out-of-the-bottle scenario is a world of extremes, but somewhere between the stalled engine and fusion scenarios. This scenario sees winners and losers in the global commons; a core group
of the European Union remaining while others -- those not doing well economically -- fall away.
In the "gini" scenario the United States remains the preeminent power but it doesn't play global policeman. Energy producing nations see prices fall while they fail to diversify their economies.
"Cities in China's coastal zone continue to thrive, but inequalities increase and split the [Communist] Party," the report says.
Global growth continues, but it is uneven. More countries fail in part because of the failure of international organizations.
"In sum, the world is reasonably wealthy, but it is less secure as the dark side of globalization poses an increasing challenge in domestic and international politics," the report says.
The final scenario -- the nonstate world -- sees nonstate actors taking the lead in confronting global challenges. Nonstate actors include nongovernmental organizations, multinational businesses,
academic individuals, wealthy individuals and cities.
"The nation state does not disappear, but countries increasingly organize and orchestrate 'hybrid' coalitions of state and nonstate actors which shift depending on the issue," the report says.
This is a complex and diverse world that favors democracies. "Smaller, more agile countries in which the elites are also more integrated are apt to do better than larger countries that lack social or
political cohesion," the report says.
By its nature, the nonstate world would be uneven and would carry its own dangers. Some global problems would be solved because the networks would coalesce to solve them but others would not.
Security threats would increase because not all nonstate actors are benign. Access to lethal and disruptive technologies could expand, "enabling individuals and small groups to perpetuate violence
and disruption on a large scale," according to the report.
The four worlds suggested in the report could happen or something altogether different may occur also. The report notes that unplanned, unforeseen events can change all of this.
The example of the New York World's Fair extends here too. While the Fair opened in 1939, it reopened in 1940. Two nations that sponsored buildings in 1939 -- Czechoslovakia and Poland -- had ceased
to exist when the Fair returned in 1940.
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This is a message in my email that is being passed on. Enjoy.
Dear Friends,
Wishing you a Blessed Solstice and a Wondrous Holiday Season from all of us at the Power Path!
The Winter Solstice is exact on Friday, December 21 at 4:11 AM Mountain Standard Time. This is one of the most important time frames in our history. Although the exact time of the solstice is
important to note, the 48 hours afterwards is equally as potent and may be even more so in terms of where your thoughts, intentions, feelings and focus are.The energy is about honoring and forgiving
the structure that has held life together up until this point. Move yourself into a place of trust that a new structure has been organizing itself in the quantum field through a collective intention
of having a better world. It is on the brink of being birthed onto the physical plane. The best intention during this time is to be in trust.
Here are some suggestions of how to work with this time:
Stay away from negativity and martyrdom and be positive and optimistic no matter what.
Clear your mind and your environment from clutter so you can be open to the incredible insights available at this time and have a journal handy to write them down. Pay attention to your dreams and
anything that is surfacing at this time especially old memories and recapitulations. If emotions grab you, let them pass on through.
Instead of thinking and thinking and thinking about maybe doing something or changing something, make a commitment to do it. Take some time to write serious intentions and don't be afraid to dream
big and to leave room for spirit to bring you more than you can imagine.
Reestablish your connection with guides and allies through meditation and spiritual practice. Ask for help and keep "don't know mind" about how it may show up.
Be around friends and family and community that are on the same page.
Forgive, forgive, forgive.
Do a burning ritual: write down with total forgiveness what you are complete with and burn it ritually with some offering of sage or tobacco. Then write your intentions.
Say hello to the earth and the sun and the new frequencies they are bringing in. Be very conscious and present with your environment sending light and love into all the physical extensions of
yourself (electronics, electrical system, vehicles, plumbing, clothing, appliances etc etc.) In this way you are "tuning" yourself to the larger energies of the earth and the sun and then harmonizing
your personal environment.
It would be a good idea to clean your home and workspace with a cleansing smoke or whatever method you use. Bells and tuning forks work well too.
Light a long burning candle and place on your sheet of intentions along with a vase of fresh flowers.
This is the dawn of a new era and the one we have been waiting for. It is the time frame that marks the end of the hierarchical, competitive and success oriented approach to life and marks the
beginning of a relationship, cooperative and community oriented cycle. It won't happen overnight however, we must take advantage of this time through our attention and focus. If you plant the seed
right, a beautiful growth will follow.
Blessings for a Wonderful Solstice,
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
Doomsdayers Flock To Serbian Mountain Believed To Contain Alien Pyramid
December 20, 2012 -There seems to be an good old fashioned mystic mountain battle brewing between Mount Rtanj and France’s Pic de Bugarach. The Telegraph reports: Hotel owners around the
pyramid-shaped Mount Rtanj, a supposedly mystical mountain in the east of the Balkan country, say that bookings are flooding in, with believers who are convinced that the end of a Mayan calendar
heralds the destruction of the world hoping that its purported mysterious powers will save them from the apocalypse. Adherents of the end-of-the-world scenario think the 5,100ft-high mountain, part
of the Carpathian range, conceals a pyramidal building inside, left behind by alien visitors thousands of years ago. Arthur C Clarke, the British science fiction writer, reportedly identified the
peak as a place of “special energy” and called it “the navel of the world”. “In one day we had 500 people trying to book rooms. People want to bring their whole families,” said Obrad Blecic, a hotel
manager. Doomsday cultists believe that it emits special energy that could be channeled to protect them from the end of life as we know it.
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
Posts : 32800
Join date : 2010-04-07
Location : Hawaii
2012 Galactic Alignment as seen by ISS
What is life?
It is the flash of a firefly in the night, the breath of a buffalo in the wintertime. It is the little shadow which runs across the grass and loses itself in the sunset.
With deepest respect ~ Aloha & Mahalo, Carol
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Create discrete heat maps in SAS/IML
In a previous article I introduced the HEATMAPCONT subroutine in SAS/IML 13.1, which makes it easy to visualize matrices by using heat maps with continuous color ramps. This article introduces a
companion subroutine. The HEATMAPDISC subroutine, which also requires SAS/IML 13.1, is designed to visualize matrices that have a small number of unique values. This includes zero-one matrices,
design matrices, and matrices of binned data.
I have previously described how to use the Graph Template Language (GTL) in SAS to construct a heat map with a discrete color ramp. The HEATMAPDISC subroutine simplifies this process by providing an
easy-to-use interface for creating a heat map of a SAS/IML matrix. The HEATMAPDISC routine supports a subset of the features that the GTL provides.
A two-value matrix: The Hadamard design matrix
In my previous article about discrete heat maps, I used a Hadamard matrix as an example. The Hadamard matrix is used to make orthogonal array experimental designs. The following SAS/IML statement
creates a 64 x 64 matrix that contains the values 1 and –1 and calls the HEATMAPDISC subroutine to visualize the matrix:
proc iml;
h = hadamard(64); /* 64 x 64 Hadamard matrix */
ods graphics / width=600px height=600px; /* make heat map square */
run HeatmapDisc(h);
The resulting heat map is shown to the left. (Click to enlarge.) You can see by inspection that the matrix is symmetric. Furthermore, with the exception of the first row and column, all rows and
columns have an equal number of +1 and –1 values. There are also certain rows and columns (related to powers of 2) whose structure is extremely regular: a repeating sequence of +1s followed by a
sequence of the same number of –1s. Clearly the heat map enables you to examine the patterns in the Hadamard matrix better than staring at a printed array of numbers.
A binned correlation matrix
In my previous article about how to create a heat map, I used a correlation matrix as an example of a matrix that researchers might want to visualize by using a heat map with a continuous color ramp.
However, sometimes the exact values in the matrix are unimportant. You might not care that one pair of variables has a correlation of 0.78 whereas another pair has a correlation of 0.82. Instead, you
might want to know which variables are positively correlated, which are uncorrelated, and which are negatively correlated.
To accomplish this task, you can bin the correlations. Suppose that you decide to bin the correlations into the following five categories:
1. Very Negative: Correlations in the range [–1, –0.6).
2. Negative: Correlations in the range [–0.6, –0.2).
3. Neutral: Correlations in the range [–0.2, 0.2).
4. Postive: Correlations in the range [0.2, 0.6).
5. Very Postive: Correlations in the range [0.6, 1].
You can use the BIN function in SAS/IML to discover which elements of a correlation matrix belong to each category. You can then create a new matrix whose values are the categories. The following
statements create a binned matrix from the correlations of 10 numerical variables in the Sashelp.Cars data set. The HEATMAPDISC subroutines creates a heat map of the result:
use Sashelp.Cars;
read all var _NUM_ into Y[c=varNames];
corr = corr(Y);
/* bin correlations into five categories */
Labl = {'1:V. Neg','2:Neg','3:Neutral','4:Pos','5:V. Pos'}; /* labels */
bins= bin(corr, {-1, -0.6, -0.2, 0.2, 0.6, 1}); /* BIN returns 1-5 */
disCorr = shape(Labl[bins], nrow(corr)); /* categorical matrix */
call HeatmapDisc(disCorr) title="Binned Correlations"
xvalues=varNames yvalues=varNames;
The heat map of the binned correlations is shown to the left. The fuel economy variables are very negatively correlated (dark brown) with the variables that represent the size and power of the
engine. The fuel economy variables are moderately negatively correlated (light brown) with the vehicle price and the physical size of the vehicles. The size of the vehicles and the price are
practically uncorrelated (white). The light and dark shades of blue-green indicate pairs of vehicles that are moderately and strongly positively correlated.
The HEATMAPDISC subroutine is a powerful tool because it enables you to easily create a heat map for matrices that contain a small number of values. Do you have an application in mind for creating a
heat map with a discrete color ramp? Let me know your story in the comments.
You might be wondering how the colors in the heat maps were chosen and whether you can control the colors in the color ramp. My next blog post will discuss choosing colors for a discrete color ramp.
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Design Like A Pro
Round To The Nearest Ten Thousand E Ample
Round To The Nearest Ten Thousand E Ample - How to round to the nearest thousandth. Web 35619 rounded to the nearest ten thousand is 36000 since thousands place digit is 5 we will round up the ten
thousands place and increase it by 1. To the nearest unit ≈. Enter a related value or decimal number in the search bar of the rounding off numbers calculator. Web rounding numbers to the nearest
thousand means we need to write that multiple of 1000 which is closest to the given number. Web yes, 2.48 rounded to the nearest tenth is 2.5.
As the thousands digit is 3, the number is rounded down to 20000. Web created by : View the rounded number in the results section. Web rounding a number to the nearest thousand means finding the
multiple of 1000 closest to the given number. Identify the ten place value.
Round to the nearest hundred (100): Rounding a number to the nearest place value simply means replacing the given number by an approximate value, which is simple to understand, yet closer to the
original number. Web rounding numbers to the nearest thousand means we need to write that multiple of 1000 which is closest to the given number. Select the required unit to round up or round down the
complex number. For 25600, as the thousands digit 5 is larger than 4, it will be rounded to 30000.
To the nearest hundredth ≈. For this, we need to check the digit to the right of the thousands place, that is, the hundreds place. You can round the decimal numbers to the nearest tenths, hundredths,
or thousandths. This is a free online tool by everydaycalculation.com to perform numeric rounding to nearest 1000th, nearest 100th, nearest 10th, neareset integer, nearest.
58964 is rounded to the nearest thousand= 59000. What is 23718 rounded to the nearest ten thousand? Nearest tenth, hundredth, or thousandth. Enter the number you want to round in the input field. Web
rounding a number involves replacing the number with an approximation of the number that results in a shorter, simpler, or more explicit representation of said number.
For 25600, as the thousands digit 5 is larger than 4, it will be rounded to 30000. To the nearest hundredth ≈. To the nearest hundred ≈. Web results (detailed calculations and formula below) orinigal
number ( n) = to the nearest thousandth ≈. For example, if rounding the number 2.7 to the nearest.
Web whenever the value right after the decimal is less than 5, we round down; Rounding to the nearest whole number is the same as rounding to the nearest one. Web yes, 2.48 rounded to the nearest
tenth is 2.5. To the nearest hundred ≈. The thousands digit will either remain the same or it will increase by one.
Round To The Nearest Ten Thousand E Ample - To the nearest tenth ≈. All the steps are the same as before. Web rounding a number to the nearest thousand means finding the multiple of 1000 closest to
the given number. Round to the nearest hundred (100): As the thousands digit is 3, the number is rounded down to 20000. 58964 is rounded to the nearest thousand= 59000. Given here an online rounding
calculator which is used for rounding the numbers to the nearest thousandth number. Rounding to the nearest whole number is the same as rounding to the nearest one. 8 is greater than 5 so the number
will be rounded up. 18 to the nearest 10 is 20.
8 is greater than 5 so the number will be rounded up. Rounding to the nearest thousandth in 3 easy steps. Web rounding numbers to the nearest thousand means we need to write that multiple of 1000
which is closest to the given number. Web rounding a number to the nearest thousand means finding the multiple of 1000 closest to the given number. To round to the tenth, use the number at the
Web created by : For example, 18 rounded to the nearest 10 is 20. Rounding to the nearest thousandth in 3 easy steps. Web results (detailed calculations and formula below) orinigal number ( n) = to
the nearest thousandth ≈.
6614 is in between 6000 and 7000, so the options are to round down to 6000 or round up to 7000. 2.47 rounds to 2.5 if we round to the nearest tenth. For example, if rounding the number 2.7 to the
To the nearest tenth ≈. Rounding to the nearest thousandth in 3 easy steps. Select the required unit to round up or round down the complex number.
This Is A Free Online Tool By Everydaycalculation.com To Perform Numeric Rounding To Nearest 1000Th, Nearest 100Th, Nearest 10Th, Neareset Integer, Nearest 10,.
To the nearest unit ≈. Web the rounding calculator is designed to round numbers to the nearest thousands, hundreds, tens, ones, tenths, hundredths, and thousandths. Alex walked 11,568 steps on
monday. Web created by :
Given Here An Online Rounding Calculator Which Is Used For Rounding The Numbers To The Nearest Thousandth Number.
Web whenever the value right after the decimal is less than 5, we round down; What is 23718 rounded to the nearest ten thousand? Round 7.28 to the nearest whole number. 18 to the nearest 10 is 20.
Web Here's Another Example To Help You Understand How To Round To The Nearest Ten Thousand:
For 25600, as the thousands digit 5 is larger than 4, it will be rounded to 30000. For example, if rounding the number 2.7 to the nearest. As the thousands digit is 3, the number is rounded down to
20000. 8 is greater than 5 so the number will be rounded up.
You Can Round The Decimal Numbers To The Nearest Tenths, Hundredths, Or Thousandths.
Web rounding numbers to the nearest thousand means we need to write that multiple of 1000 which is closest to the given number. Web yes, 2.48 rounded to the nearest tenth is 2.5. To the nearest ten
≈. Rounding to the nearest whole number is the same as rounding to the nearest one.
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Hands-On Programming with R by Garrett Grolemund
R is a free software environment for statistical computing and graphics. It compiles and runs on a wide variety of UNIX platforms, Windows and MacOS. R language is useful to become a data scientist,
as well as a computer scientist. I mention a book that points about a data science with R. A Hands-On Programming with R Write Your Own Functions and Simulations By Garrett Grolemund. It was written
how to solve the logistical problems of data science. Additional, How to write our own functions and simulations with R. In a book, readers are able to learn in
practical data analysis projects (Weighted Dice, Playing Cards, Slot Machine) and understand more in R. Additional, Appendix A-E will help to install/update R and R packages as well as loading Data
and debugging in R code.
Garrett Grolemund
maintains shiny.rstudio.com, the development center for the Shiny R package.
Free Sampler
No comments:
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double - Jack Of Spades In Wonderland
From this list, the gist is that most languages can't process 9999999999999999.0 - 9999999999999998.0
Why do they output 2 when it should be 1? I bet most people who've never done any formal CS (a.k.a maths and information theory) are super surprised.
Before you read the rest, ask yourself this: if all you have are zeroes and ones, how do you handle infinity?
If we fire up an interpreter that outputs the value when it's typed (like the Swift REPL), we have the beginning of an explanation:
Welcome to Apple Swift version 4.2.1 (swiftlang-1000.11.42 clang-1000.11.45.1). Type :help for assistance.
1> 9999999999999999.0 - 9999999999999998.0
$R0: Double = 2
2> let a = 9999999999999999.0
a: Double = 10000000000000000
3> let b = 9999999999999998.0
b: Double = 9999999999999998
4> a-b
$R1: Double = 2
Whew, it's not that the languages can't handle a simple substraction, it's just that a is typed as 9999999999999999 but stored as 10000000000000000.
If we used integers, we'd have:
5> 9999999999999999 - 9999999999999998
$R2: Int = 1
Are the decimal numbers broken? 😱
A detour through number representations
Let's look at a byte. This is the fundamental unit of data in a computer and is made of 8 bits, all of which can be 0 or 1. It ranges from 00000000 to 11111111 ( 0x00 to 0xff in hexadecimal, 0 to
255 in decimal, homework as to why and how it works like that due by monday).
Put like that, I hope it's obvious that the question "yes, but how do I represent the integer 999 on a byte?" is meaningless. You can decide that 00000000 means 990 and count up from there, or you
can associate arbitrary values to the 256 possible combinations and make 999 be one of them, but you can't have both the 0 - 255 range and 999. You have a finite number of possible values and that's
Of course, that's on 8 bits (hence the 256 color palette on old games). On 16, 32, 64 or bigger width memory blocks, you can store up to 2ⁿ different values, and that's it.
The problem with decimals
While it's relatively easy to grasp the concept of infinity by looking at "how high can I count?", it's less intuitive to notice that there is the same amount of numbers between 0 and 1 as there
are integers.
So, if we have a finite number of possible values, how do we decide which ones make the cut when talking decimal parts? The smallest? The most common? Again, as a stupid example, on 8 bits:
• maybe we need 0.01 ... 0.99 because we're doing accounting stuff
• maybe we need 0.015, 0.025,..., 0.995 for rounding reasons
• We'll just encode the numeric part on 8 bits ( 0 - 255 ), and the decimal part as above
But that's already 99+99 values taken up. That leaves us 57 possible values for the rest of infinity. And that's not even mentionning the totally arbitrary nature of the selection. This way of
representing numbers is historically the first one and is called "fixed" representation. There are many ways of choosing how the decimal part behaves and a lot of headache when coding how the
simple operations work, not to mention the complex ones like square roots and powers and logs.
To make it simple for chips that perform the actual calculations, floating point numbers (that's their name) have been defined using two parameters:
• an integer n
• a power (of base b) p
Such that we can have n x bᵖ, for instance 15.3865 is 153863 x 10^(-4). The question is, how many bits can we use for the n and how many for the p.
The standard is to use 1 bit for the sign (+ or -), 23 bits for n, 8 for p, which use 32 bits total (we like powers of two), and using base 2, and n is actually 1.n. That gives us a range of ~8
million values, and powers of 2 from -126 to +127 due to some special cases like infinity and NotANumber (NaN).
In theory, we have numbers from -10⁴⁵ to 10^38 roughly, but some numbers can't be represented in that form. For instance, if we look at the largest number smaller than 1, it's 0.9999999404. Anything
between that and 1 has to be rounded. Again, infinity can't be represented by a finite number of bits.
The floats allow for "easy" calculus (by the computer at least) and are "good enough" with a precision of 7.2 decimal places on average. So when we needed more precision, someone said "hey, let's
use 64 bits instead of 32!". The only thing that changes is that n now uses 52 bits and p 11 bits.
Coincidentally, double has more a meaning of double size than double precision, even though the number of decimal places does jump to 15.9 on average.
We still have 2³² more values to play with, and that does fill some annoying gaps in the infinity, but not all. Famously (and annoyingly), 0.1 doesn't work in any precision size because of the
base 2. In 32 bits float, it's stored as 0.100000001490116119384765625, like this:
Conversely, after double size (aka doubles), we have quadruple size (aka quads), with 15 and 112 bits, for a total of 128 bits.
Back to our problem
Our value is 9999999999999999.0. The closest possible value encodable in double size floating point is actually 10000000000000000, which should now make some kind of sense. It is confirmed by Swift
when separating the two sides of the calculus, too:
2> let a = 9999999999999999.0
a: Double = 10000000000000000
Our big brain so good at maths knows that there is a difference between these two values, and so does the computer. It's just that using doubles, it can't store it. Using floats, a will be rounded
to 10000000272564224 which isn't exactly better. Quads aren't used regularly yet, so no luck there.
It's funny because this is an operation that we puny humans can do very easily, even those humans who say they suck at maths, and yet those touted computers with their billions of math operations
per second can't work it out. Fair enough.
The kicker is, there is a litteral infinity of examples such as this one, because trying to represent infinity in a finite number of digits is impossible.
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Multiply not Power for me
I tried all the Math node settings and I liked how multiply worked the best. I had a smoother transition from clear and shiny to rough by tenths for 0.100 to 1.000. It seemed like a very smooth
change from one to the next without the big fall off from 0.10 to 0.20. Power worked too but it seemed do go rough and diffuse quicker than the multiply did. I set my multiply value to 0.200 and it
seemed to work the best. 0.100 did not quite get to rough by the time the Value was set to 1.000.
So, here is my node tree:
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Class Journal
Dana C. Ernst
Mathematics & Teaching
Northern Arizona University
Flagstaff, AZ
Google Scholar
Impact Story
Current Courses
MAT 226: Discrete Math
MAT 526: Combinatorics
About This Site
This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.
Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.
The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.
The source code is on GitHub.
Land Acknowledgement
Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point
in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit
that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi
, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this
place home.
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Do they still make the four square crackers? - Answers
Does Nabisco make Ritz crackers?
they may, but the ones you mean are made by "Christie" Nabisco discontinued these in the US. As stated, they are produced by Christie Bakeries in Canada. This company is the Canadian arm of Kraft
Foods/Nabisco. It is the same identical recipe as the old ones we know in the US...... They also still make the Bacon Crackers as well.
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The Physics of Free Running
Check out this demo reel of Levi Meeuwenberg doing some jaw-dropping “free running”. Free running is very similar to Parkour in the athleticism required and specific techniques and movements used,
but while Parkour is about getting from one place to another in as efficient a manner as possible, free running is less directed and more creative in nature.
As mentioned in that ancient post, when performing either of these activities, in addition to spending years developing a formidable set of technical skills, balance, physical strength, and kamikaze
attitude, it’s important to be cognizant of some basic physics.
In order to lessen the impact upon landing after a fall, or to decrease the force upon a set of fingers while grabbing onto a wall, it is essential to reduce the acceleration during each collision as
much as possible. This requires increasing the time of impact by smoothly bending, flexing, or rolling during impact.
Let’s apply some of the physics by taking a quantitative example. During impact with the ground, there are essentially two forces acting on your body — the downward force of gravity and the upward
force of the ground.
Applying Newton’s second law we get:
Fnet = Fground - mg = ma
where mg is the weight force (a.k.a. the force of gravity) acting on your body. So let’s say that Levi jumps from a height of 3.0 meters onto the ground. How much force is the ground going to exert
on him during impact?
First, let’s apply conservation of energy to determine Levi’s speed just before contact.
The gravitational potential energy (mgh) due to his initial height relative to the ground is going to be converted into kinetic energy (½ mv2) just before landing. So
Mgh = ½ mv2
and v = [2(9.8m/s2)(3.0m)]1/2 = 7.7m/s
Now we have to bring him to a stop. Let’s assume the lithe and wiry Levi has a mass of approximately 70 kg. Now because
Fground = m(g + a) = 70 kg (9.8 m/s2 + a)
we can see that the force of impact depends on the acceleration. But a = Δv/Δt and therefore depends on the time of impact. If he were to foolishly land on his heels with rigid locked knees this
would result in a very small impact time — as little as 0.01 or 0.02 seconds for this type of collision. The result?
Fground = 70 kg (9.8 m/s2 + 7.7m/s/0.01s) = 55,000 N (or about 12,000 pounds). Ouch!
However, by bending and rolling, the time of impact can be increased to as much as 0.3 or 0.4 seconds. By decreasing his velocity over this extended period of time, the force is substantially
reduced. Applying the above calculation with an acceleration time of 0.4 seconds we now get Fground = 2000 N (460 pounds). It’s still a significant force but as you can see in the video quite
manageable for someone with the proper skill, strength and technique.
Speaking of landing on the heels, notice that Levi never does that. Impacts are always taken with the ball of the foot. Landing on the ball allows for an increased impact time and less compression on
the legs. Now recall some previous posts on running shoes.
Did we not decide that the most efficient stride technique involved a mid- to fore-foot strike? It seems reasonable to propose that the most efficient stride is also the most natural stride for the
running human, and might be expected to result in fewer injuries, relative to the much more jarring heel strike. Most running shoes have big padded heels, making it easy for the unwary jogger to
strike with the heels. However, running barefoot, it’s relatively impossible to do that. It’s just too painful. You automatically impact on the front of the foot, much like Levi does when sticking a
landing. Interesting. We’ve talked about a pair of shoes meant to simulate barefoot running. But what about the real thing? What about barefoot running itself?
To be continued…!
Adam Weiner is the author of Don’t Try This at Home! The Physics of Hollywood Movies.
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Data structures and Algorithms MCQ with Answers
Answer is:
Abstract Data Type
Answer is:
Abstract data type
Representation of data structure in memory is known as Abstract data type.
Abstract data type: In computer science, an abstract data type is a mathematical model for data types. An abstract data type is defined by its behavior from the point of view of a user, of the data,
specifically in terms of possible values, possible operations on data of this type, and the behavior of these operations
Answer is:
Both 1) Tree and 2)Graph
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knowledge Santa
Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest
possible total surface area (in sq. cm) of the cylinder.
Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest
possible total surface area (in sq. cm) of the cylinder.
1. 100π/3
2. 80π/3
3. 120π/7
4. 130π/9
5. 100π/7
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Performance and dynamic characteristics of a multi stages vertical axis wind turbine
Vertical axis wind turbines (VAWTs) are used to convert wind energy to mechanical output or electricity. Vertical axis wind turbines are favorable at buildings as it can receive wind from any
direction, have a design that can be integrated simply with building architecture and they have better response in turbulent wind flow which is common in urban areas. Using a calculation code based
on the Double Multiple Stream Tube theory, symmetrical straight-bladed NACA0012 wind turbine performance was evaluated. The induction factor for both upwind and downwind zone is determined with the
aid of a root-finding algorithm. This numerical analysis highlighted how turbine performance is strongly influenced by the wind speed (Reynolds number) and rotor solidity (turbine radius and blade
chord length). Also a dimensional analysis is introduced and is to be considered in such a way to generalize the design for different turbine specifications. One of the qualities provided by
dimensional analysis is that geometrically similar turbines will produce the same non-dimensional results. This allows one to make comparison between different sizes wind turbines in terms of power
output and other related variables. One of the main problems affecting the turbine performance and dynamics is the torque ripple phenomena. So in this paper a turbine design configuration is
introduced in order to decrease the turbine torque fluctuation. This design is carried out by constructing more similar turbine units (stages) on the vertical axis on top of each other with different
orientation phase angles. The results showed that using even number of turbine assembly is better than odd number to avoid torque fluctuation and mechanical vibrations acting on the turbine. Also it
is preferred to use four turbine stages as the eight stages will have no sensible effect on decreasing the torque fluctuation.
1. Introduction
Paraschivoiu et al. [1] presented an optimal computation of blade’s pitch angle of an H-D Darrieus wind turbine to obtain maximum torque with some results presented using a 7 kW prototype. In order
to determine the performance of the straight bladed vertical axis wind turbine a genetic algorithm optimizer is applied in addition to an improved version of “Double Multiple stream tube” model.
Where in this model a partition of the rotor is considered in the stream tubes and treats each of the two blade elements defined by a given stream tube as an actuator disk. Dominy et al. [2] in their
presentation of vertical axis wind turbines explained the potential advantages of using Darrieus type wind turbines in the small scale and domestic applications where the cost and reliability are
very important points in addition to the simplicity of design structure, generator and control system. Their concern was about the ability of the Darrieus turbines to be self-started. Olson and
Visser [3] showed the availability of using Darrieus turbines in commercial grid connected utility scale operations. They also mentioned the development of marketing small scale wind turbines due to
the social “going green” phenomena and to its economic stand point. Batista et al. [4] mentioned the great interest in the vertical axis wind turbine due to the rapid increase in its power
generation, and the need for a smarter electrical grid with a decentralized energy generation, especially in the urban areas. However, estimation of the performance of Darrieus type VAWT is very
tough, as the blades rotate in the three-dimensional space around the rotor which results in several flow phenomena such as dynamic stall, flow separation, wake flow and natural incapability to
self-starting. Klimas [5] described Darrieus turbines as relatively simple machines. The turbines have a fixed blade geometry, usually two or three blades rotating about a vertical axis generating
power to a ground mounted power conversion or absorption machinery, with neither yaw control nor power regulation. Nila et al. [6] dealt with the calculations of a fixed-pitch straight-bladed
vertical axis wind turbine of the ‘Darrieus Type’. They made a case study using a well-known NACA0012 turbine blade profile where the wind load calculations were achieved by assuming solid frontal
areas of the blades and tower construction. They also assumed that each section of a blade behaves as an airfoil in a two-dimensional flow field. Lobitz [7] showed the importance of the dynamic
response characteristics of the VAWT rotor and its influence in the safety and fatigue life of VAWT systems. He also mentioned that the problem is to predict the critical speed at resonance and the
forced vibration response amplitude. Sabaeifard et.al [8] discussed the potential of using VAWT for buildings. They explained the aerodynamics and the performance of small scale Darrieus type
straight-bladed VAWT through a computational and an experimental study. Alexandru et al. [9] developed a different approach for enhancing the performance of vertical axis wind turbines for the use in
the urban or rural environment and remote isolated residential areas. Abhishiktha et al. [10] presented a review of on different types of small scale horizontal axis and vertical axis wind turbines.
The performance, blade design, control and manufacturing of horizontal axis wind turbines were reviewed. Vertical axis wind turbines were categorized based on experimental and numerical studies.
Magedi and Norzelawati [11] studied a comparison between the horizontal axis wind turbines (HAWTS), and the vertical axis wind turbines (VAWTS). The two types of wind turbines were used for different
purposes. Several models of both types were presented from previous research. Baker [12] investigated the most important factors for cost-effective SB-VAWT such as appropriate airfoil to achieve
desired aerodynamic performance and to optimize the overall dimensions of the SB-VAWT. In his research a detailed performance analysis on a series of low Reynolds number airfoils was selected to
enhance the performance of smaller capacity fixed-pitch SB-VAWT. Haroub et al. [13] designed a three bladed vertical axis wind turbine with a target. The material used was Glass fiber reinforced
plastics (GFRP). The study focused on Blade design where a numerical optimization process was done to come up with parameters for the rotor blades. The power coefficient was tested using a wind fan
with wind speeds ranging from 4 m/s to 15 m/s. Paul et al. [14] introduced in their work the importance of choosing the airfoil blade shape and its effect on improving the aerodynamic performance for
a Darrieus wind turbine blade. The analysis was made under some specifications as using two bladed machines with an infinite aspect ratio, rotor solidity and operating at Reynolds number nearly of
three million. Frank et al. [15] studied the aerodynamic performance and wake dynamics of a Darrieus-type vertical-axis wind turbine consisting of two straight blades which was simulated using
Brown’s vortices transport model. The predicted variation with azimuth angle of the normal and tangential forces on the turbine blades compared well with experimental measurements. The interaction
between the blades and the vortices that are shed and trailed in previous revolutions of the turbine was shown to have a significant effect on the distribution of aerodynamic loading on the blades.
Tanja et al. [16] demonstrated that the higher quantity of extreme events in atmospheric wind fields transfers to alternating loads on the airfoil and on the main shaft in the form of torque
fluctuations. Their analysis was performed on three different wind field data sets: measured fields, data generated by a standard wind field model and data generated by an alternative model based on
continuous time random walks, which grasps the intermittent structure of atmospheric turbulence in a better way.
The objectives of the present study are to make a detailed analysis, which explains the parameters affecting the turbine outputs as a function of the azimuth angle and its effect on the turbine
performance. Also to introduce a dimensional analysis, that allows making a comparison between different sizes wind turbines in terms of power output and other related variables. Finally, to
introduce a design of multi stage vertical axis wind turbine to decrease turbine torque ripple phenomena and avoid the mechanical vibrations acting on the turbine.
2. Theoretical analysis
In this section the simulation model is assumed to be the Double Multiple Stream Tube (DMST) Model. This model is a combination of the MST model and double actuator theory [17], where the turbine is
modeled separately for the upstream half and the downstream half. Also an assumption is made that the wake from the upwind pass is fully expanded and the ultimate wake velocity has been reached
before the interaction with the blades in the downwind pass.
Fig. 1 presents the DMST model diagram. Each airfoil in this model intersects each stream tube twice, one on the upwind pass and the other on the downwind pass. The DMST model solves two equations
simultaneously for the stream-wise force at the actuator disk; one obtained by conservation of momentum and other based on the aerodynamic coefficients of the airfoil (Lift and Drag) and the local
wind velocity. These equations are solved twice, one for the upwind part and the other for the downwind part. Fig. 2 shows the forces and velocities triangles acting on the turbine un-pitched blade.
Fig. 1Plan view of a double-multiple-stream tube analysis of the flow through a VAWT rotor
Fig. 2Forces and velocity triangle
2.1. Upwind un-pitched blade analysis
The velocity components on an upwind blade section are illustrated in Fig. 3 and the ratio between relative velocity at upstream and free stream velocity is:
$\frac{{U}_{rel-u}}{U}=\sqrt{{\left[\left(\lambda +\left(1-{a}_{u}\right)\mathrm{cos}Ø\right]}^{2}+{\left[\left(1-{a}_{u}\right)\mathrm{sin}Ø\right]}^{2}},$
where $\lambda$ is the tip speed ratio and is represented by $\lambda =\mathrm{\Omega }R/U$.
The local angle of attack is represented by:
$\alpha ={\mathrm{tan}}^{-1}\frac{\left[\left(1-{a}_{u}\right)\mathrm{sin}Ø\right]}{\left[\lambda +\left(1-{a}_{u}\right)\mathrm{cos}Ø\right]}.$
The angle of attack $\alpha$ and $\mathrm{\Delta }\varnothing$ play a substantial rule in the force coefficients direction and its magnitude. Fig. 4 shows the force coefficients analysis diagram, on
an upwind blade section. The force coefficients depend on the value of angle of attack less or greater than 90°.
Fig. 3Velocity components acting on an upwind blade element
Fig. 4Forces analysis diagram for an upwind blade element
2.1.1. Case when the angle of attack is less than 90 degrees
In this case the normal force coefficient is represented by:
${C}_{N}=\mathrm{}{C}_{L}\mathrm{cos}\alpha +{C}_{D}\mathrm{sin}\alpha .$
While the tangential force coefficient is:
${C}_{T}={-C}_{D}\mathrm{cos}\alpha +\mathrm{}{C}_{L}\mathrm{sin}\alpha .$
The force coefficients in $x$ and $y$ directions can be expressed as:
Thrust force coefficient is:
2.1.2. Case when the angle of attack is greater than 90 degrees
In this case the normal and tangential force coefficients are represented by:
${C}_{N}=\mathrm{}-{C}_{L}\mathrm{cos}\alpha +{C}_{D}\mathrm{sin}\alpha ,$
${C}_{T}=-{C}_{D}\mathrm{cos}\alpha -\mathrm{}{C}_{L}\mathrm{sin}\alpha .$
While the force coefficients in $x$ and $y$ directions can be expressed as:
Thrust force coefficient is:
For any value of angle of attack the following relations can be applied.
The average aerodynamic thrust can be expressed by a non-dimensional thrust coefficient as the following:
${C}_{H}=\mathrm{}\frac{N\sum _{i=1}^{M}\frac{\mathrm{\Delta }\mathrm{\Phi }}{2\pi }\left(\frac{1}{2}\rho {HCU}_{rel-u}^{2}{C}_{FT}\right)}{\frac{1}{2}\rho {HRMU}^{2}\mathrm{\Delta }\mathrm{Ø}\mathrm
The instantaneous torque on a single airfoil at certain position can be expressed by:
The average torque on the rotor by number of blades $N$ in one complete revolution is given by:
${Q}_{a}=\mathrm{}\mathrm{}\frac{N}{M}\sum _{i=1}^{M}\frac{1}{2}\rho HC{C}_{T}R{U}_{rel-u}^{2}.$
The torque coefficient is represented by the relation between average torque and aerodynamic torque and can be expressed as:
${C}_{Q}=\frac{{Q}_{a}}{\frac{1}{2}\rho {DHU}^{2}R}=\left(\frac{NC}{MD}\right)\sum _{i=1}^{M}{\left(\frac{{U}_{rel-u}}{U}\right)}^{2}{C}_{T}.$
The Power coefficient is represented by the ratio of power that the turbine can extract from wind energy and is expressed as:
${C}_{P}={C}_{Q}\lambda .$
2.2. Downwind un-pitched blade analysis
The velocity components for the downwind part can be demonstrated by Fig. 5. Also the force coefficients analysis diagram, on a downwind blade section is illustrated by Fig. 6.
The ratio between relative velocity at upstream and free stream velocity is:
$\frac{{U}_{rel-d}}{U}=\sqrt{\begin{array}{c}{\left[\left(\lambda +\left(1-2{a}_{u}\right)\left(1-{a}_{d}\right)\mathrm{cos}Ø\right]}^{2}+{\left[-\left(1-2{a}_{u}\right)\left(1-{a}_{d}\right)\mathrm
{s}\mathrm{i}\mathrm{n}\varnothing \right]}^{2}\end{array}}.$
The angle of attack is:
$\alpha ={\mathrm{tan}}^{-1}\frac{-\left[\left(1-2{a}_{u}\right)\left(1-{a}_{d}\right)\mathrm{sin}Ø\right]}{\left[\lambda +\left(1-2{a}_{u}\right)\left(1-{a}_{d}\right)\mathrm{cos}Ø\right]}.$
Fig. 5Velocity components acting on a downwind blade element
Fig. 6Force analysis diagram of a downwind blade element
2.2.1. The angle of attack is below 90 degrees
In this case the normal and tangential force coefficients are represented by:
${C}_{N}={C}_{L}\mathrm{cos}\alpha +{C}_{D}\mathrm{sin}\alpha ,$
${C}_{T}={-C}_{D}\mathrm{cos}\alpha +{C}_{L}\mathrm{sin}\alpha .$
While the force coefficients in $x$ and $y$ directions can be expressed as:
Thrust force coefficient is:
${C}_{FT}={C}_{N}\mathrm{sin}\varnothing +{C}_{T}\mathrm{cos}\varnothing .$
2.2.2. The angle of attack is greater than 90 degrees
The normal and tangential force coefficients is represented by:
${C}_{N}=-{C}_{L}\mathrm{cos}\alpha +{C}_{D}\mathrm{sin}\alpha ,$
${C}_{T}=-{C}_{D}\mathrm{cos}\alpha -{C}_{L}\mathrm{sin}\alpha .$
The force coefficients in $x$ and $y$ directions can be expressed as:
Thrust force coefficient is:
${C}_{FT}={C}_{N}\mathrm{sin}\varnothing +{C}_{T}\mathrm{cos}\varnothing .$
For any value of angle of attack the following relations can be applied.
The average aerodynamic thrust can be expressed by a non-dimensional thrust coefficient as follows:
${C}_{H}=\frac{N\sum _{i=1}^{M}\frac{\mathrm{\Delta }\mathrm{\Phi }}{2\pi }\left(\frac{1}{2}\rho {HCU}_{rel-d}^{2}{C}_{FT}\right)}{\frac{1}{2}\rho HR{U}_{e}^{2}M\mathrm{\Delta }Ø\mathrm{sin}Ø},$
where ${U}_{e}=U\left(1-2{a}_{u}\right)$ and the instantaneous torque is determined by:
While average torque is:
${Q}_{a}=\frac{N}{M}\sum _{i=1}^{M}\frac{1}{2}\rho {HC{C}_{T}RU}_{rel-d}^{2}.$
The torque coefficient is represented by:
${C}_{Q}=\frac{{Q}_{a}}{\frac{1}{2}\rho {U}_{e}^{2}\left(DH\right)R}=\left(\frac{NC}{MD}\right)\sum _{i=1}^{M}{\left(\frac{{U}_{rel-d}}{{U}_{e}}\right)}^{2}{C}_{T}.$
The Power coefficient can be expressed as:
${C}_{P}={C}_{Q}\lambda .$
3. Computational procedure
The computational procedure for a given rotor geometry and rotational speed are done through several steps as the following.
– Making an initial guess for the axial induction factor equals to zero.
– The angle of attack and relative wind velocity acting on a blade are determined.
– The local blade section Reynolds number is determined since it can be expressed as:
${R}_{eb}=\frac{\rho {U}_{rel}C}{\mu }.$
– Using the local blade section Reynolds number and the local angle of attack, the blade section lift and drag coefficients are obtained through known experimental data of Sheldal and Klimas [18].
The selection of these coefficients through the mentioned parameters is made with the aid of a “MATLAB” tooling box called “SFTOOL”.
– The normal and tangential force coefficients are calculated and finally the thrust coefficient derived from aerodynamic forces per stream tube and from the actuator disk theory [17] is obtained.
– Comparing both thrust coefficients obtained from both theories. If the thrust coefficients are found the same, then the convergence is achieved. If not, then the axial induction factor is changed
and the same steps are followed until the convergence is achieved.
– As the convergence is achieved per stream tube, the required values of the relative velocity, angle of attack, torque and power are then calculated. Convergence is with a relative error less than
10^-2. The mentioned procedures are adopted for the up-wind and downwind half cycle. The induction factor is calculated twice for all stream tubes, one for the upwind half stream ${a}_{u}$ and the
other for the downwind half stream ${a}_{d}$.
4. Dimensional analysis
Wind turbines come in different sizes and they experience a wide range of variables when in operation. These variables complicate the process of comparison of different sizes of turbines in terms of
their performance. The general geometric features of H-rotors are rotor radius $R$, blade height $H\text{,}$ number of blades $N\text{,}$ chord length $C$ and airfoil shape. These features are all
determinants to the aerodynamic performance of the rotor. The environmental conditions include wind velocity $U$ viscosity $\mu$, air density $\rho$ and also the rotational speed of the H-rotor $Ω$.
Usually, the performance of an H-rotor is evaluated by the power coefficient defined as the ratio of the power that the turbine can extract from the wind energy. To deal with this, the help of
dimensional analysis is required. One of the advantages of the dimensional analysis is that geometrically similar turbines will produce the same non-dimensional results. This allows one to make
comparison between different sizes of wind turbines in terms of power output and other related variables. Based on the determinants of the rotor performance, the power coefficient ${C}_{p}$ can be
expressed as a function of rotational speed, free stream wind velocity, rotor radius, blade length, number of blades; chord length, air viscosity and air density. There are nine parameters relevant
to the power of wind turbine.
The power of turbine can simply be expressed as:
$P=f\left\{U,\mu ,\rho ,R,C,H,N,Ω\right\}.$
According to Buckingham Pi-theorem, the functional relationship of the dimensionless groups may be obtained and the result may be written as:
$\frac{P}{\rho {R}^{2}{U}^{3}}=f\left\{\frac{C}{R},\frac{H}{R},N,\frac{ΩR}{U},\frac{\mu }{U\rho R}\right\}.$
The solution may be correct but expressed in terms of some Pi-groups which have no recognizable physical significance. It may then be necessary to combine the pi-groups to obtain new groups which
have significance. A solidity term, free stream Reynolds number, power coefficient and tip speed ratio can be expressed as:
$\sigma =\frac{NC}{R},Re=\frac{U\rho R}{\mu }{,C}_{P}=\frac{P}{\rho RH{U}^{3}},\lambda =\frac{ΩR}{U}.$
By considering the new Pi-groups as mentioned above, the power coefficient ${C}_{P}$ can be expressed as a function of tip speed ratio $\lambda$, free stream Reynolds number $\mathrm{R}\mathrm{e}$
and solidity $\sigma$:
${C}_{P}=f\left\{Re,\lambda ,\sigma \right\}.$
5. Results and discussion
The presentation and discussion of the results obtained from the theoretical work are to be divided into three main parts. The first part is concerned with the detailed analysis, which includes the
parameters affecting the turbine outputs as function of the azimuth angle and its effect on the turbine performance. The second part deals with the design charts resulting from dimensional analysis
which can be used for any proposed VAWT design under specific limitations. The third part is how to decrease the turbine torque fluctuations and in order to decrease the wind turbine vibrations.
5.1. Effect of wind speed on the power coefficient
In this section the effect of different wind speeds (5, 6, 7, 8, 9, 10 m/s) on the average power coefficient ${C}_{p}$, are discussed for a turbine of 0.2 m chord length, three blades, 1 m blade
height, blade profile NACA0012 and 2 m turbine radius as shown in Fig. 7. Some features can be drawn from this figure; firstly, it can be seen that, by increasing the wind speed the maximum average
power coefficient increases. This takes place since; by increasing the wind speed the torque extracted by the turbine increases leading to a higher power coefficient such as shown in Fig. 8. The
second feature, which is required to be pointed out, is the relationship between tip speed ratio $\lambda$ and the power coefficient${C}_{p}$. At low $\lambda$ (below 3.5) ${C}_{p}$ is mainly
negative while at larger values of $\lambda$ (above 3.5) is positive. In order to explain this relation, the turbine characteristics such as, torque, angle of attack, normal and tangential force
coefficients and relative velocity acting on the blade are required to be examined. The third feature comes in the region where the higher values of tip speed ratio takes place shows that, when the
tip speed ratio increases beyond the design condition 4.5 (Maximum ${C}_{p}$), the power coefficient decreases while the turbine performance approximate the same as the tip speed ratio changes from
5.5 to 6. This is due to the decrease in the mean torque.
Fig. 7Tip speed ratio and power coefficient relationship for different inlet wind speed
Fig. 8Torque versus Azimuth angle for different inlet wind speeds at maximum Cp
The tangential force coefficient and the blade relative velocity are the main parameters affecting the mean torque value. This coefficient is a function of the lift and drag forces coefficients which
depend on the angle of attack and blade Reynolds number and can be obtained from experimental data according to Sheldal and Klimas [18]. The relation between tangential force coefficients, angle of
attack and blade Reynolds number is shown in Fig. 9. It can be seen also from the figure that, for all blade Reynolds number as $\alpha$ increases to be greater than 45° the tangential force
coefficient always attains positive values especially in the region above 90°, while for $\alpha$ in the range of 25° to 45° the tangential force coefficient is mainly negative. On the other hand for
$\alpha$ below 25° the tangential force coefficient sign depends on the blade Reynolds number specially at low blade Reynolds number (below 4×10^6).
Fig. 9Relation between angle of attack, blade Reynolds number and tangential force coefficient
So according to the case $U=$ 5m/s, Figs. 10 ,11 and 12 give a relationship between the blade’s angle of attack, Reynolds number and tangential force coefficient with respect to the azimuth angle at
different tip speed ratios respectively. It can be concluded from these figures that, as both angle of attack and blade Reynolds number changes during the whole rotation, the tangential force
coefficient also changes.
Fig. 10Angle of attack and azimuth angle relationship
Fig. 11Blades Reynolds number and azimuth angle relationship
Fig. 12Tangential force coefficient and azimuth angle relationship
5.2. Effect of turbine geometry on the power coefficient
The main turbine geometrical parameters affecting the turbine output are the blade chord length, blade height, along with turbine radius. The effect of each parameter on the power coefficient is to
be considered while maintaining the turbine swept area constant and equal to 16 m^2. The effects are to be examined for a three bladed turbine and for an airfoil shape NACA0012. In this section the
effect of different chord lengths on the power coefficient is presented for a turbine of 2 m blade height, 4 m turbine radius, inlet velocity of 5 m/s. Fig. 13, shows that when the chord length
increases the maximum average power coefficient increases.
Fig. 13Effect of turbine chord length on the average power coefficient for different tip speed ratio
Fig. 14Effect turbine radius on the relation between the average power coefficient and the tip speed ratio λ
Also by increasing the chord length the negative average power coefficient at low tip speed ratio starts to decrease. The effect of different turbine radii on the average power coefficient is
considered for an inlet velocity of 5 m/s, chord length of 0.2 m and swept area of 16 m^2 and is shown in Fig. 14.
It can be seen from this figure that, as the rotor radius decreases the maximum power coefficient increases. Also it is to be noticed that at low values of tip speed ratio increasing the rotor radius
leads to a lower negative power coefficients.
5.3. Validation the results of DMST model
The DMST model developed in this work was checked against the vertical axis wind turbine using CFD model which was introduced by Biadgo et al. [19]. The computer program used in this work is fitted
by the same data used in [19], since the normal NACA0012 airfoil was set to 0.2 m chord length and the turbine radius was set 2 m. The height of turbine is taken to be 4 m with 3 blades. The wind
speed used is 5 m/s and tip speed ratios are 0.25, 0.5, 1, 2, 3, 4, 5, 6, and 7, the total number of stream tubes is 12 with $\mathrm{\Delta }\mathrm{\Phi }=$ 15°.The computational proceeding was
applied to DMST model and the first aspect of the model validation is the comparison of the predicted VAWT power coefficient (${C}_{p}$) for different tip speed ratio $\lambda$ as shown in Fig. 15.
In this figure the relation between power coefficient and tip speed ratio is graphed for the DMST model introduced by the present work and the results obtained by [19] for both DMST and CFD model.
As can be seen, both CFD and DMST model ${C}_{p}$ curves show that the turbine generates negative and/or minimum torque for lower tip speed ratio which implies inability of NACA0012 to be
self-starting. The DMST model underestimates the ${C}_{p}$ value at lower tip speed ratios but it predicts higher ${C}_{p}$ value which is 0.41 at a tip speed ratio of around (4-4.5). Also Fig. 15 is
in agreement with Habtamu [20] and Louis et al [21] and also agreement with Franklyn, Isam [22, 23].
The second aspect of validation as shown in Fig. 16 is the comparison of torque with azimuthal angle for DMST model of the present work and CFD model introduced by [19]. In this figure TSR is taken
to be 1 and the difference between two models is clear since CFD model gives a positive variation of torque at $TSR=$1 because the power coefficient is positive but when DMST model is applied a
negative variation of torque appears since the value of ${C}_{p}$ is also negative.
Fig. 15Power coefficient result for DMST and CFD
Fig. 16Torque – azimuthal angle relationship for DMST model and CFD model [19]
6. Dimensional analysis results
The results presented in the previous sections are only applicable for turbine characteristic considered in each section. A dimensional analysis is required and is to be considered to generalize the
design for different turbine specifications.
Fig. 17Relationship between free stream, Reynolds number, tip speed ratio and power coefficient at different Rotor Solidities
a) Rotor solidity $\sigma =$0
b) Rotor solidity $\sigma =$0.225
c) Rotor solidity $\sigma =$0.2625
d) Rotor solidity $\sigma =$0.3
One of the qualities of dimensional analysis is that geometrically similar turbines will produce the same non-dimensional results. This allows one to make comparison between different sizes of wind
turbines in terms of power output and other related variables. Based on the study of the dimensional analysis mentioned before, the power coefficient ${C}_{p}$ can be expressed as a function of tip
speed ratio $\lambda$, free stream Reynolds number ${R}_{e}$ and solidity $\sigma$. From this relation design charts such as that shown in Fig. 17 can be considered showing the relation between Power
coefficient ${C}_{P}$, free stream Reynolds number $\mathrm{R}\mathrm{e}$ and Tip speed ratios $\lambda$ at specific Rotor solidity $\sigma$.
7. Turbine fluctuations
This section focuses on how to decrease the turbine torque fluctuations during its rotation. Decreasing these fluctuations means how to keep the output torque constant which is very important to
avoid vibrations of turbine rotor. Avoiding the vibrations allows safe use of full power output from the turbine. This can be achieved by constructing a new turbine assembly, where more similar
turbine units are mounted on the vertical axis on top of each other with different phase angles as shown in Figs. 18 and 19.
Fig. 18Single row VAWT assembly
Fig. 19Two rows VAWT assembly with phase angle Ø12
In Fig. 19 ${H}_{T1}$ is the blade height for stage 1, ${H}_{T2}$ is the blade height for stage 2, ${C}_{T1}$ is the cord length of blade for stage 1,
${C}_{T2}$ is the cord length of blade for stage 2, ${H}_{T12}$ is the total height between stage 1 and 2, ${\mathrm{\Phi }}_{12}$ is the phase angle between stage 1 and 2. This assembly will
decrease the total turbine torque fluctuation taking into account the following conditions:
– The inlet flow to all turbines is the same and is uniform.
– All turbines have the same geometric conditions such as chord length $C$, blade height $H$, number of blades $N$, rotor radius $R$ and airfoil shape.
– An assumption is considered that each turbine stage has no effect on the neighboring stage.
The torque of the first stage of the turbine is determined as mentioned before, and then the second similar turbine is added with a phase angle changing from 10° to 120° degree with a phase step of
10°. The phase angle is measured from the reference (1st turbine 1st blade position). In each phase angle step the total torque of the two turbines and the percentage torque fluctuation are
calculated as given below:
where $Q$ is the total torque, ${Q}_{T1}$ is the torque of turbine 1, ${Q}_{T2}$ is the torque of turbine 2, ${Q}_{Max}$, ${Q}_{Min}$, ${Q}_{Mean}$ are the maximum, minimum and mean values of total
torque, ${Q}_{fr}$ is the percentage torque fluctuation of the set of two turbine assembly at each phase angle. Finally, the phase angle is chosen corresponding to the minimum percentage of torque
fluctuation, considering the two turbine stages as a single set. The turbines to follow are added in the same manner to the previous group and the same steps are applied as mentioned above. For an
inlet velocity $U=$5 m/s and swept area $A=$16 m^2 Figs. 20 and 21 show the relation between torque and azimuth angle for different turbine configurations at different tip speed ratios $\lambda$ and
for two values of rotor solidities $\sigma$ namely 0.3 and 0.15 respectively.
Fig. 20Relation between Torque and Azimuth angle at different λ and for Rotor Solidity = 0.3 per each turbine
Fig. 21Relation between Torque and Azimuth angle at different λ and for Rotor Solidity = 0.15 per each turbine
Figs. 22 and 23 show the torque fluctuation percentage for different turbine configurations and at different tip speed ratio and solidity of 0.3 and 0.15 respectively. From these figures it can be
concluded that, as the turbine torque fluctuation increases the more turbine assemblies are needed to decrease this fluctuation. Also it is preferable to use an even number of turbine assembly for
constructing the wind turbine since using odd number of turbine assembly causes high percentage of torque fluctuation. In addition, the even number of turbine configuration has many advantages.
Firstly, at any blade position the turbine output torque is mostly close to the mean torque. This gives better turbine dynamics such as low vibrations level and fatigue conditions. Secondly the
methodology of calculating the power coefficient becomes more realistic. In the calculation of the power coefficient mentioned previously, it was assumed that the tip speed ratio is constant
throughout all stream tubes and the average torque per revolution was therefore used. However, in reality the tip speed ratio varies per revolution with the variation of the turbine’s rotational
speed. The power coefficient will consequently depend mainly on the instantaneous torque and tip speed ratio, leading to different values of ${C}_{p}$. According to the suggested new turbine
configuration, this problem is solved because both average torque and instantaneous torque are nearly the same at any blade position giving the same power coefficient ${C}_{p}$. Also Figs. 22 and 23
shows that, at a tip speed ratio corresponding to a maximum power average coefficient, it is preferred to use four turbine stages as the eight stages will have no sensible effect on decreasing the
turbine torque fluctuation and this is suitable for minimizing the design cost of the turbine.
Fig. 22Torque fluctuation percentage at different λ and for Rotor Solidity = 0.3
Fig. 23Torque fluctuation percentage at different λ and for Rotor Solidity = 0.15
8. Conclusions
In this paper, analytical investigation of performance of symmetrical straight- bladed VAWT is done using NACA0012 as a blade profile. The analytical investigation is done using DMST model. A
detailed analysis, which explains the parameters affecting the turbine outputs as a function of the azimuth angle and its effect on the turbine performance is studied. Also a dimensional analysis is
applied which allows to make a comparison between different sizes wind turbines in terms of power output and other related variables. The power coefficient obtained from DMST model is compared by
that obtained from CFD model introduced before by another investigator. Finally, a design of multi stages vertical axis wind turbine is introduced to decrease turbine torque ripple phenomena and
avoid the mechanical vibrations acting on the turbine.
From the present work the following conclusions are drawn:
– As the wind speed increases from (5-10) m/s the maximum average power coefficient increases.
– As the wind speed changes from (5-10) m/s the tip speed ratio corresponding to the maximum average power coefficient decreases.
– The power coefficient is a function of three main parameters namely the free stream Reynolds number, tip speed ratio and rotor solidity.
– The higher the rotor solidity the higher the maximum average power coefficient with a decrease in the negative power coefficient at low tip speed ratio.
– As the tip speed ratio increases and reaches to the maximum average power coefficient the torque fluctuation reaches nearly a sinusoidal wave form.
– The DMST model predict a negative value of power coefficient at lower tip speed ratio which implies that NACA0012 is not self-starting.
– The DMST model predict higher power coefficient value which is 0.41 at tip speed ratio of around (4-4.5)
– Even number of turbine stages leads to a lower turbine torque fluctuation compared to an odd number of stages.
– Doubling multiple of even number of turbine stages leads to a lower turbine torque fluctuation.
– At a tip speed ratio corresponding to a maximum power average coefficient it is preferred to use four turbine stages as the eight stages will have no sensible effect on decreasing the turbine
torque fluctuation and this is suitable for minimizing the design cost of the turbine.
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• Baker J. Features to aid or enable self-starting of fixed pitch low solidity vertical axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics, Vol. 15, 1983, p. 369-380.
• Haroub A. H., Francis X. O., Kamau Joseph N. Development of a low cost rotor blade for a H-Darrieus wind turbine. Journal of Sustainable Research in Engineering, Vol. 2, Issue 3, 2015, p. 92-99.
• Migliore Paul G., Fritschen John R., Mitchell Richard L. Darrieus Wind Turbine Airfoil Configurations. Subcontract Report No. AE-1-1045-1, Solar Energy Research Institute, Colorado, 1982.
• Scheurich Frank, Fletcher Timothy M., Brown Richard E. Simulating the aerodynamic performance and wake dynamics of a vertical-axis wind turbine. Wind Energy, Vol. 14, Issue 2, 2011, p. 159-177.
• Mücke Tanja, Kleinhans David, Peinke Joachim Atmospheric turbulence and its influence on the alternating loads on wind turbines. Wind Energy, Vol. 14, 2011, p. 301-316.
• Manwell J. F., Mcgowan J. G., Rogers A. L. Wind Energy Explained: Theory Design and Application. Second Edition. John Wiley and Sons, 2009.
• Sheldal R. E., Klimas P. C. Aerodynamic Characteristics of Seven Airfoil Sections through 180 Degrees Angle of Attack for Use in Aerodynamic Analysis of Vertical Axis Wind Turbines. Sandia
National Laboratories, Albuquerque, 1981.
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• Beri Habtamu, Yao Yingxue Double multiple stream tube model and numerical analysis of vertical axis wind turbine. Energy and Power Engineering, Vol. 3, 2011, p. 262-270.
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Engineering, London, UK, Vol. 3, 2013.
• Kanyako Franklyn, Janajreh Isam Vertical axis wind turbine performance prediction for low wind speed environment. Innovations in Technology Conference (InnoTek), 2014, p. 1-10.
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About this article
Flow induced structural vibrations
vertical axis wind turbine
straight bladed
Darrieus wind turbine
DMST model
Copyright © 2016 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Tetrachoric computation using EXCEL spreadsheet
The program in this spreadsheet will handle upto 15 dichotomous variables on upto 800 cases with missing responses assumed to be coded as blanks and the lowest valued response assumed to be coded as
a '1'. The data is entered in sheet 1 and the output is a tetrachoric correlation matrix in sheet 2. Ths matrix may be copied over and entered into a SPSS spreadsheet as input to a factor analysis as
described here.
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Small Stellated Dodecahedron
One of the Kepler-Poinsot Solids whose Dual Polyhedron is the Great Dodecahedron. Its Schläfli Symbol is . It is also Uniform Polyhedron and has Wythoff Symbol. It was originally called the Urchin by
Kepler. Pentagrammic faces. Its faces are . The easiest way to construct it is to build twelve pentagonal Pyramids
and attach them to the faces of a Dodecahedron. The Circumradius of the small stellated dodecahedron with is
See also Great Dodecahedron, Great Icosahedron, Great Stellated Dodecahedron, Kepler-Poinsot Solid
Fischer, G. (Ed.). Plate 103 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 102, 1986.
Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 219, 1997.
© 1996-9 Eric W. Weisstein
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Algebra Tutors Near Me | Change The Way You FEEL About Math
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high school Algebra. His experience was very positive, he was ready to attend every session, appreciated his tutor, and felt like he was much more solid on an array of concepts. It really boosted his
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It's definitely worth doing some research first when trying to find an algebra tutor near you.
The first thing you should do is ask your friends that use algebra tutors if they have any recommendations. That way you can be sure the algebra tutor has achieved results for other families. Be
sure to ask if their kids' grades, confidence, and engagement in algebra improved when working with the tutor.
The next thing you should do is check all the major review sites such as Yelp and Google Reviews to see which algebra tutor has the best reviews. A well-established algebra tutor near you should
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Finally, see if the algebra tutor offers a trial lesson to see if it's a good fit. The most important thing in finding a great algebra tutor is making sure their personality and experience are a
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Whether or not an online algebra tutor will work or not depends on two factors:
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If both of these conditions occur, an online algebra tutor will definitely work. If both are lacking, it will not work. If one is present without the other, the results will be mixed.
Therefore it is important that you do a lot of research before hiring an online algebra tutor. Ask yourself, am I willing to work really hard outside of the online algebra tutoring session? Algebra
tutoring does not get you out of any work, it makes the work you put in more efficient and gives you a deeper understanding of the material.
If you want to tutor college algebra, you must first make sure you are an expert in mathematics. You should have a degree in mathematics, engineering, or physics ideally. You might think because
you are good at algebra, you will be good at teaching college algebra. However, your level of mathematical knowledge must be far advanced than that of your college algebra student. You will
encounter a wide range of algebra problems and you must know how to do them all by heart.
Tutoring college algebra requires a lot of patience as well. You need to remember what it was like to not know the subject. The most common issues holding back college algebra students are
pre-algebra topics such as fractions, negative numbers, and order of operations. Anyone taking algebra in college is most likely not a math, science, or engineering major. Therefore, math will not
be their strongest subject. They will most likely see the college algebra class as an obstacle they must overcome to get to their preferred topics of study. An outstanding college algebra tutor
will understand this, and try to make algebra a less intimidating subject for the student.
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misconception, there is a lot of skill that goes into tutoring pre algebra. The topics learned in pre algebra are used extensively throughout all future math classes, and therefore a strong
foundation in pre algebra is essential to be successful in math.
For example, negative numbers are a very difficult subject to teach. The reason is most pre algebra tutors can do negative number operations without even thinking about it. Therefore, it can be
difficult to explain to a student in a way that makes sense. Most pre algebra tutors can't remember what it's like not to know negative numbers.
Fractions cause more problems for high school algebra students than any other subject. This is a pre algebra concept that is first learned around the 5th grade. Once a student gets to high school
algebra, they are going to encounter multi-step problems that have fractions in them. If the student is not confident with fractions, the problem will be overwhelming. Thus, it is very important
that the pre algebra tutor does an excellent job of teaching their students fractions. They must understand how these pre algebra skills will come back to the student later in their math career.
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Stats: Normal Approximation to Binomial
Recall that according to the Central Limit Theorem, the sample mean of any distribution will become approximately normal if the sample size is sufficiently large.
It turns out that the binomial distribution can be approximated using the normal distribution if np and nq are both at least 5. Furthermore, recall that the mean of a binomial distribution is np and
the variance of the binomial distribution is npq.
Continuity Correction Factor
There is a problem with approximating the binomial with the normal. That problem arises because the binomial distribution is a discrete distribution while the normal distribution is a continuous
distribution. The basic difference here is that with discrete values, we are talking about heights but no widths, and with the continuous distribution we are talking about both heights and widths.
The correction is to either add or subtract 0.5 of a unit from each discrete x-value. This fills in the gaps to make it continuous. This is very similar to expanding of limits to form boundaries that
we did with group frequency distributions.
│ Discrete │ Continuous │
│ x = 6 │ 5.5 < x < 6.5 │
│ x > 6 │ x > 6.5 │
│ x >= 6 │ x > 5.5 │
│ x < 6 │ x < 5.5 │
│ x <= 6 │ x < 6.5 │
As you can see, whether or not the equal to is included makes a big difference in the discrete distribution and the way the conversion is performed. However, for a continuous distribution, equality
makes no difference.
Steps to working a normal approximation to the binomial distribution
1. Identify success, the probability of success, the number of trials, and the desired number of successes. Since this is a binomial problem, these are the same things which were identified when
working a binomial problem.
2. Convert the discrete x to a continuous x. Some people would argue that step 3 should be done before this step, but go ahead and convert the x before you forget about it and miss the problem.
3. Find the smaller of np or nq. If the smaller one is at least five, then the larger must also be, so the approximation will be considered good. When you find np, you're actually finding the mean,
mu, so denote it as such.
4. Find the standard deviation, sigma = sqrt (npq). It might be easier to find the variance and just stick the square root in the final calculation - that way you don't have to work with all of the
decimal places.
5. Compute the z-score using the standard formula for an individual score (not the one for a sample mean).
6. Calculate the probability desired.
Table of Contents
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Lifshitz scaling, microstate counting from number theory and black hole entropy
Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation E ∼ k^z and dynamical exponent z > 1. The asymptotic growth of the
number of states of these theories can be described by an extension of Cardy formula that depends on z. We show that this result can be recovered by counting the partitions of an integer into z-th
powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel duality relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground
state energy. For free bosons with Lifshitz scaling, this relationship is shown to be identically fulfilled by virtue of the reflection property of the Riemann ζ-function. The quantum Benjamin-Ono[2]
(BO[2]) integrable system, relevant in the AGT correspondence, is also analyzed. As a holographic realization, we provide a special set of boundary conditions for which the reduced phase space of
Einstein gravity with a couple of U (1) fields on AdS[3] is described by the BO[2] equations. This suggests that the phase space can be quantized in terms of quantum BO[2] states. Indeed, in the
semiclassical limit, the ground state energy of BO[2] coincides with the energy of global AdS[3], and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of
Cardy formula.
Nota bibliográfica
Publisher Copyright:
© 2019, The Author(s).
Áreas temáticas de ASJC Scopus
• Física nuclear y de alta energía
Profundice en los temas de investigación de 'Lifshitz scaling, microstate counting from number theory and black hole entropy'. En conjunto forman una huella única.
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Inversion-symmetric topological insulators
We analyze translationally invariant insulators with inversion symmetry that fall outside the current established classification of topological insulators. These insulators exhibit no edge or surface
modes in the energy spectrum and hence they are not edge metals when the Fermi level is in the bulk gap. However, they do exhibit protected modes in the entanglement spectrum localized on the cut
between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the
inversion eigenvalues of the Hamiltonian band structure and the midgap states in the entanglement spectrum. The classification of protected entanglement levels is given by an integer N, which is the
difference between the negative inversion eigenvalues at inversion symmetric points in the Brillouin zone, taken in sets of 2. When the Hamiltonian describes a Chern insulator or a nontrivial
time-reversal invariant topological insulator, the entirety of the entanglement spectrum exhibits spectral flow. If the Chern number is zero for the former, or time reversal is broken in the latter,
the entanglement spectrum does not have spectral flow, but, depending on the inversion eigenvalues, can still exhibit protected midgap bands similar to impurity bands in normal semiconductors.
Although spectral flow is broken (implying the absence of real edge or surface modes in the original Hamiltonian), the midgap entanglement bands cannot be adiabatically removed, and the insulator is
"topological." We analyze the linear response of these insulators and provide proofs and examples of when the inversion eigenvalues determine a nontrivial charge polarization, a quantum Hall effect,
an anisotropic three-dimensional (3D) quantum Hall effect, or a magnetoelectric polarization. In one dimension, we establish a link between the product of the inversion eigenvalues of all occupied
bands at all inversion symmetric points and charge polarization. In two dimensions, we prove a link between the product of the inversion eigenvalues and the parity of the Chern number of the occupied
bands. In three dimensions, we find a topological constraint on the product of the inversion eigenvalues thereby showing that some 3D materials are protected topological metals; we show the link
between the inversion eigenvalues and the 3D Quantum Hall Effect, and analyze the magnetoelectric polarization (θ vacuum) in the absence of time-reversal symmetry.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Inversion-symmetric topological insulators'. Together they form a unique fingerprint.
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13.3 Gravitational Potential Energy and Total Energy (2024)
Learning Objectives
By the end of this section, you will be able to:
• Determine changes in gravitational potential energy over great distances
• Apply conservation of energy to determine escape velocity
• Determine whether astronomical bodies are gravitationally bound
We studied gravitational potential energy in Potential Energy and Conservation of Energy, where the value of g remained constant. We now develop an expression that works over distances such that g is
not constant. This is necessary to correctly calculate the energy needed to place satellites in orbit or to send them on missions in space.
Gravitational Potential Energy beyond Earth
We defined work and potential energy in Work and Kinetic Energy and Potential Energy and Conservation of Energy. The usefulness of those definitions is the ease with which we can solve many problems
using conservation of energy. Potential energy is particularly useful for forces that change with position, as the gravitational force does over large distances. In Potential Energy and Conservation
of Energy, we showed that the change in gravitational potential energy near Earth’s surface is [latex] \text{Δ}U=mg({y}_{2}-{y}_{1}) [/latex]. This works very well if g does not change significantly
between [latex] {y}_{1} [/latex] and [latex] {y}_{2} [/latex]. We return to the definition of work and potential energy to derive an expression that is correct over larger distances.
Recall that work (W) is the integral of the dot product between force and distance. Essentially, it is the product of the component of a force along a displacement times that displacement. We define
[latex] \text{Δ}U [/latex] as the negative of the work done by the force we associate with the potential energy. For clarity, we derive an expression for moving a mass m from distance [latex] {r}_{1}
[/latex] from the center of Earth to distance [latex] {r}_{2} [/latex]. However, the result can easily be generalized to any two objects changing their separation from one value to another.
Consider (Figure), in which we take m from a distance [latex] {r}_{1} [/latex] from Earth’s center to a distance that is [latex] {r}_{2} [/latex] from the center. Gravity is a conservative force (its
magnitude and direction are functions of location only), so we can take any path we wish, and the result for the calculation of work is the same. We take the path shown, as it greatly simplifies the
integration. We first move radially outward from distance [latex] {r}_{1} [/latex] to distance [latex] {r}_{2} [/latex], and then move along the arc of a circle until we reach the final position.
During the radial portion, [latex] \overset{\to }{F} [/latex] is opposite to the direction we travel along [latex] d\overset{\to }{r} [/latex], so [latex] E={K}_{1}+{U}_{1}={K}_{2}+{U}_{2}. [/latex]
Along the arc, [latex] \overset{\to }{F} [/latex] is perpendicular to [latex] d\overset{\to }{r} [/latex], so [latex] \overset{\to }{F}·d\overset{\to }{r}=0 [/latex]. No work is done as we move along
the arc. Using the expression for the gravitational force and noting the values for [latex] \overset{\to }{F}·d\overset{\to }{r} [/latex] along the two segments of our path, we have
[latex] \text{Δ}U=-{\int }_{{r}_{1}}^{{r}_{2}}\overset{\to }{F}·d\overset{\to }{r}=G{M}_{\text{E}}m{\int }_{{r}_{1}}^{{r}_{2}}\frac{dr}{{r}^{2}}=G{M}_{\text{E}}m(\frac{1}{{r}_{1}}-\frac{1}{{r}_{2}}).
Since [latex] \text{Δ}U={U}_{2}-{U}_{1} [/latex], we can adopt a simple expression for [latex] U [/latex]:
[latex] U=-\frac{G{M}_{\text{E}}m}{r}. [/latex]
Figure 13.11 The work integral, which determines the change in potential energy, can be evaluated along the path shown in red.
Note two important items with this definition. First, [latex] U\to 0\,\text{as}\,r\to \infty [/latex]. The potential energy is zero when the two masses are infinitely far apart. Only the difference
in U is important, so the choice of [latex] U=0\,\text{for}\,r=\infty [/latex] is merely one of convenience. (Recall that in earlier gravity problems, you were free to take [latex] U=0 [/latex] at
the top or bottom of a building, or anywhere.) Second, note that U becomes increasingly more negative as the masses get closer. That is consistent with what you learned about potential energy in
Potential Energy and Conservation of Energy. As the two masses are separated, positive work must be done against the force of gravity, and hence, U increases (becomes less negative). All masses
naturally fall together under the influence of gravity, falling from a higher to a lower potential energy.
Lifting a Payload
How much energy is required to lift the 9000-kg Soyuz vehicle from Earth’s surface to the height of the ISS, 400 km above the surface?
Use (Figure) to find the change in potential energy of the payload. That amount of work or energy must be supplied to lift the payload.
Paying attention to the fact that we start at Earth’s surface and end at 400 km above the surface, the change in U is
[latex] \text{Δ}U={U}_{\text{orbit}}-{U}_{\text{Earth}}=-\frac{G{M}_{\text{E}}m}{{R}_{\text{E}}+\,400\,\text{km}}-(-\frac{G{M}_{\text{E}}m}{{R}_{\text{E}}}). [/latex]
We insert the values
[latex] m=9000\,\text{kg,}\quad {M}_{\text{E}}=5.96\,×\,{10}^{24}\text{kg,}\quad {R}_{\text{E}}=6.37\,×\,{10}^{6}\,\text{m} [/latex]
and convert 400 km into [latex] 4.00\,×\,{10}^{5}\,\text{m} [/latex]. We find [latex] \text{Δ}U=3.32\,×\,{10}^{10}\,\text{J} [/latex]. It is positive, indicating an increase in potential energy, as
we would expect.
For perspective, consider that the average US household energy use in 2013 was 909 kWh per month. That is energy of
[latex] 909\,\text{kWh}\,×\,1000\,\text{W/kW}\,×\,3600\,\text{s/h}=3.27\,×\,{10}^{9}\,\text{J per month.} [/latex]
So our result is an energy expenditure equivalent to 10 months. But this is just the energy needed to raise the payload 400 km. If we want the Soyuz to be in orbit so it can rendezvous with the ISS
and not just fall back to Earth, it needs a lot of kinetic energy. As we see in the next section, that kinetic energy is about five times that of [latex] \text{Δ}U [/latex]. In addition, far more
energy is expended lifting the propulsion system itself. Space travel is not cheap.
Check Your Understanding
Why not use the simpler expression [latex] \text{Δ}U=mg({y}_{2}-{y}_{1}) [/latex]? How significant would the error be? (Recall the previous result, in (Figure), that the value g at 400 km above the
Earth is [latex] 8.67\,{\text{m/s}}^{2} [/latex].)
Conservation of Energy
In Potential Energy and Conservation of Energy, we described how to apply conservation of energy for systems with conservative forces. We were able to solve many problems, particularly those
involving gravity, more simply using conservation of energy. Those principles and problem-solving strategies apply equally well here. The only change is to place the new expression for potential
energy into the conservation of energy equation, [latex] E={K}_{1}+{U}_{1}={K}_{2}+{U}_{2} [/latex].
[latex] \frac{1}{2}\,m{v}_{1}^{2}-\frac{GMm}{{r}_{1}}=\frac{1}{2}\,m{v}_{2}^{2}-\frac{GMm}{{r}_{2}} [/latex]
Note that we use M, rather than [latex] {M}_{\text{E}} [/latex], as a reminder that we are not restricted to problems involving Earth. However, we still assume that [latex] m\text{<}\,\text{<}M [/
latex]. (For problems in which this is not true, we need to include the kinetic energy of both masses and use conservation of momentum to relate the velocities to each other. But the principle
remains the same.)
Escape velocity
Escape velocity is often defined to be the minimum initial velocity of an object that is required to escape the surface of a planet (or any large body like a moon) and never return. As usual, we
assume no energy lost to an atmosphere, should there be any.
Consider the case where an object is launched from the surface of a planet with an initial velocity directed away from the planet. With the minimum velocity needed to escape, the object would just
come to rest infinitely far away, that is, the object gives up the last of its kinetic energy just as it reaches infinity, where the force of gravity becomes zero. Since [latex] U\to 0\,\text{as}\,r\
to \infty [/latex], this means the total energy is zero. Thus, we find the escape velocity from the surface of an astronomical body of mass M and radius R by setting the total energy equal to zero.
At the surface of the body, the object is located at [latex] {r}_{1}=R [/latex] and it has escape velocity [latex] {v}_{1}={v}_{\text{esc}} [/latex]. It reaches [latex] {r}_{2}=\infty [/latex] with
velocity [latex] {v}_{2}=0 [/latex]. Substituting into (Figure), we have
[latex] \frac{1}{2}m{v}_{\text{esc}}^{2}-\frac{GMm}{R}=\frac{1}{2}m{0}^{2}-\frac{GMm}{\infty }=0. [/latex]
Solving for the escape velocity,
Notice that m has canceled out of the equation. The escape velocity is the same for all objects, regardless of mass. Also, we are not restricted to the surface of the planet; R can be any starting
point beyond the surface of the planet.
Escape from Earth
What is the escape speed from the surface of Earth? Assume there is no energy loss from air resistance. Compare this to the escape speed from the Sun, starting from Earth’s orbit.
We use (Figure), clearly defining the values of R and M. To escape Earth, we need the mass and radius of Earth. For escaping the Sun, we need the mass of the Sun, and the orbital distance between
Earth and the Sun.
Substituting the values for Earth’s mass and radius directly into (Figure), we obtain
[latex] {v}_{\text{esc}}=\sqrt{\frac{2GM}{R}}=\sqrt{\frac{2(6.67\,×\,{10}^{-11}\,\text{N}·{\text{m}}^{2}{\text{/kg}}^{2})(5.96\,×\,{10}^{24}\,\text{kg})}{6.37\,×\,{10}^{6}\,\text{m}}}=1.12\,×\,{10}^
{4}\,\text{m/s.} [/latex]
That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use [latex] R={R}_{\text{ES}}=1.50\,×\,{10}^{11}\,\text{m} [/latex] and [latex] {M}_{\text{Sun}}=1.99\,×\,{10}^
{30}\,\text{kg} [/latex]. The result is [latex] {v}_{\text{esc}}=4.21\,×\,{10}^{4}\,\text{m/s} [/latex] or about 42 km/s.
The speed needed to escape the Sun (leave the solar system) is nearly four times the escape speed from Earth’s surface. But there is help in both cases. Earth is rotating, at a speed of nearly 1.7 km
/s at the equator, and we can use that velocity to help escape, or to achieve orbit. For this reason, many commercial space companies maintain launch facilities near the equator. To escape the Sun,
there is even more help. Earth revolves about the Sun at a speed of approximately 30 km/s. By launching in the direction that Earth is moving, we need only an additional 12 km/s. The use of
gravitational assist from other planets, essentially a gravity slingshot technique, allows space probes to reach even greater speeds. In this slingshot technique, the vehicle approaches the planet
and is accelerated by the planet’s gravitational attraction. It has its greatest speed at the closest point of approach, although it decelerates in equal measure as it moves away. But relative to the
planet, the vehicle’s speed far before the approach, and long after, are the same. If the directions are chosen correctly, that can result in a significant increase (or decrease if needed) in the
vehicle’s speed relative to the rest of the solar system.
Visit this website to learn more about escape velocity.
Check Your Understanding
If we send a probe out of the solar system starting from Earth’s surface, do we only have to escape the Sun?
Energy and gravitationally bound objects
As stated previously, escape velocity can be defined as the initial velocity of an object that can escape the surface of a moon or planet. More generally, it is the speed at any position such that
the total energy is zero. If the total energy is zero or greater, the object escapes. If the total energy is negative, the object cannot escape. Let’s see why that is the case.
As noted earlier, we see that [latex] U\to 0\,\text{as}\,r\to \infty [/latex]. If the total energy is zero, then as m reaches a value of r that approaches infinity, U becomes zero and so must the
kinetic energy. Hence, m comes to rest infinitely far away from M. It has “just escaped” M. If the total energy is positive, then kinetic energy remains at [latex] r=\infty [/latex] and certainly m
does not return. When the total energy is zero or greater, then we say that m is not gravitationally bound to M.
On the other hand, if the total energy is negative, then the kinetic energy must reach zero at some finite value of r, where U is negative and equal to the total energy. The object can never exceed
this finite distance from M, since to do so would require the kinetic energy to become negative, which is not possible. We say m is gravitationally bound to M.
We have simplified this discussion by assuming that the object was headed directly away from the planet. What is remarkable is that the result applies for any velocity. Energy is a scalar quantity
and hence (Figure) is a scalar equation—the direction of the velocity plays no role in conservation of energy. It is possible to have a gravitationally bound system where the masses do not “fall
together,” but maintain an orbital motion about each other.
We have one important final observation. Earlier we stated that if the total energy is zero or greater, the object escapes. Strictly speaking, (Figure) and (Figure) apply for point objects. They
apply to finite-sized, spherically symmetric objects as well, provided that the value for r in (Figure) is always greater than the sum of the radii of the two objects. If r becomes less than this
sum, then the objects collide. (Even for greater values of r, but near the sum of the radii, gravitational tidal forces could create significant effects if both objects are planet sized. We examine
tidal effects in Tidal Forces.) Neither positive nor negative total energy precludes finite-sized masses from colliding. For real objects, direction is important.
How Far Can an Object Escape?
Let’s consider the preceding example again, where we calculated the escape speed from Earth and the Sun, starting from Earth’s orbit. We noted that Earth already has an orbital speed of 30 km/s. As
we see in the next section, that is the tangential speed needed to stay in circular orbit. If an object had this speed at the distance of Earth’s orbit, but was headed directly away from the Sun, how
far would it travel before coming to rest? Ignore the gravitational effects of any other bodies.
The object has initial kinetic and potential energies that we can calculate. When its speed reaches zero, it is at its maximum distance from the Sun. We use (Figure), conservation of energy, to find
the distance at which kinetic energy is zero.
The initial position of the object is Earth’s radius of orbit and the intial speed is given as 30 km/s. The final velocity is zero, so we can solve for the distance at that point from the
conservation of energy equation. Using [latex] {R}_{\text{ES}}=1.50\,×\,{10}^{11}\,\text{m} [/latex] and [latex] {M}_{\text{Sun}}=1.99\,×\,{10}^{30}\,\text{kg} [/latex], we have
[latex] \begin{array}{c}\frac{1}{2}m{v}_{1}^{2}-\frac{GMm}{{r}_{1}}=\frac{1}{2}m{v}_{2}^{2}-\frac{GMm}{{r}_{2}}\hfill \\ \\ \\ \\ \quad \frac{1}{2}\overline{)m}(3.0×{10}^{3}\text{m/s}{)}^{2}-\frac
{(6.67\,×\,{10}^{-11}\,\text{N}·{\text{m/kg}}^{2})(1.99\,×\,{10}^{30}\,\text{kg})\overline{)m}}{1.50\,×\,{10}^{11}\,\text{m}}\hfill \\ \\ \phantom{\rule{4em}{0ex}}=\frac{1}{2}\overline{)m}{0}^{2}-\
frac{(6.67\,×\,{10}^{-11}\,\text{N}·{\text{m/kg}}^{2})(1.99\,×\,{10}^{30}\,\text{kg})\overline{)m}}{{r}_{2}}\hfill \end{array} [/latex]
where the mass m cancels. Solving for [latex] {r}_{2} [/latex] we get [latex] {r}_{2}=3.0\,×\,{10}^{11}\,\text{m} [/latex]. Note that this is twice the initial distance from the Sun and takes us past
Mars’s orbit, but not quite to the asteroid belt.
The object in this case reached a distance exactly twice the initial orbital distance. We will see the reason for this in the next section when we calculate the speed for circular orbits.
Check Your Understanding
Assume you are in a spacecraft in orbit about the Sun at Earth’s orbit, but far away from Earth (so that it can be ignored). How could you redirect your tangential velocity to the radial direction
such that you could then pass by Mars’s orbit? What would be required to change just the direction of the velocity?
• The acceleration due to gravity changes as we move away from Earth, and the expression for gravitational potential energy must reflect this change.
• The total energy of a system is the sum of kinetic and gravitational potential energy, and this total energy is conserved in orbital motion.
• Objects must have a minimum velocity, the escape velocity, to leave a planet and not return.
• Objects with total energy less than zero are bound; those with zero or greater are unbounded.
Conceptual Questions
It was stated that a satellite with negative total energy is in a bound orbit, whereas one with zero or positive total energy is in an unbounded orbit. Why is this true? What choice for gravitational
potential energy was made such that this is true?
It was shown that the energy required to lift a satellite into a low Earth orbit (the change in potential energy) is only a small fraction of the kinetic energy needed to keep it in orbit. Is this
true for larger orbits? Is there a trend to the ratio of kinetic energy to change in potential energy as the size of the orbit increases?
Find the escape speed of a projectile from the surface of Mars.
Find the escape speed of a projectile from the surface of Jupiter.
What is the escape speed of a satellite located at the Moon’s orbit about Earth? Assume the Moon is not nearby.
(a) Evaluate the gravitational potential energy between two 5.00-kg spherical steel balls separated by a center-to-center distance of 15.0 cm. (b) Assuming that they are both initially at rest
relative to each other in deep space, use conservation of energy to find how fast will they be traveling upon impact. Each sphere has a radius of 5.10 cm.
An average-sized asteroid located [latex] 5.0\,×\,{10}^{7}\text{km} [/latex] from Earth with mass [latex] 2.0\,×\,{10}^{13}\,\text{kg} [/latex] is detected headed directly toward Earth with speed of
2.0 km/s. What will its speed be just before it hits our atmosphere? (You may ignore the size of the asteroid.)
(a) What will be the kinetic energy of the asteroid in the previous problem just before it hits Earth? b) Compare this energy to the output of the largest fission bomb, 2100 TJ. What impact would
this have on Earth?
(a) What is the change in energy of a 1000-kg payload taken from rest at the surface of Earth and placed at rest on the surface of the Moon? (b) What would be the answer if the payload were taken
from the Moon’s surface to Earth? Is this a reasonable calculation of the energy needed to move a payload back and forth?
escape velocity
initial velocity an object needs to escape the gravitational pull of another; it is more accurately defined as the velocity of an object with zero total mechanical energy
gravitationally bound
two object are gravitationally bound if their orbits are closed; gravitationally bound systems have a negative total mechanical energy
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Analysis Of Trusses By Method Of JointsAnalysis Of Trusses By Method Of Joints - Construction How
Analysis Of Trusses By Method Of Joints
• Author: Farhan Khan
• Posted On: April 29, 2020
• Updated On: April 29, 2020
The method of joints is one of the simplest methods for determining the force acting on the individual members of a truss because it only involves two force equilibrium equations. Since only two
equations are involved, only two unknowns can be solved for at a time.
Therefore, you need to solve the joints in a certain order. That is, you need to work from the sides towards the center of the truss. When a force points toward the joint, the member is said to be in
compression. If the force points away from the joint, the member is said to be in tension. It is often important to know whether a truss member is in tension or in compression because some building
materials have different strengths in compression versus tension.
1. If a truss is in equilibrium, then each of its joints must also be in equilibrium.
2. The method of joints consists of satisfying the equilibrium equation for forces acting on each joint.
3. ∑Fx = 0 ∑Fy = 0
4. Recall, that the line of action of a force acting on a joint is determined by the geometry of the truss member.
5. The line of action is formed by connecting the two ends of each member with a straight line.
6. Since direction of the force is known, the remaining unknown is the magnitude of the force.
1. If possible, determine the support reactions
3. Write the equations of equilibrium for each joint,
∑Fx = 0 ∑Fy = 0
4. If possible, begin solving the equilibrium equations at a joint where only two unknown reactions exist.
Work your way from joint to joint, selecting the new joint using the criterion of two unknown reactions.
5. Solve the joint equations of equilibrium simultaneously.
1. The joints with external supports always connect with two truss members.
2. Thus many times, the analysis starts from analyzing the supports.
3. Therefore very often the analysis begins with finding the reaction forces applied at the supports.
4. Pay attention to symmetric systems and zero force members.
5. Identification of these special cases sometimes will make the whole analysis way easier.
Zero Force Members
Truss analysis may be simplified by determining members with no loading or zero force.These members may provide stability or be useful if the loading changes zero-force members may be determined by
inspection of the joints
Case 1
If two members are connected at a joint and there is no external force applied to the joint.
Case 2
If three members are connected at a joint and there is no external force applied to the joint and two of the members are collinear.
Practical Example
Since direction of the force is known, the remaining unknown is the magnitude of the force.
Step 1 (Free body Diagram)
Free-body diagram of entire truss. Calculating the reactions is a good place to start because they are usually easy to compute, and they can be used in the equilibrium equations for the joints where
the reactions act.
Step 2 (Equation of Equilibrium)
Equilibrium equations for entire truss
Step 3
Free body diagram of Joint C
Step 4
Equilibrium equations for joint C. It is a good idea to assume all members in tension (forces point away from the joint, not towards it). Then, after solving the equilibrium equations, you will know
immediately that any member force found to be negative must be compression.
Step 5
Step 6 (Trigonometry)
Using Angle= 59.04° in Eqs. 4 and 5 and solving simultaneously gives
Writing “(T)” after the numerical value shows that the member is in tension. We had arbitrarily assumed member BC to be in tension. We then found that the member force was negative, so we know that
our assumption was wrong. Member BC is in compression, and we show this by writing a positive “6.0” followed by “(C)”
Step 7
Free body diagram of joint B
Step 8
The force F BC is directed toward the joint because member BC is known to be in compression.
Step 9
Equilibrium equation for joint B
Step 10 (Final Result)
An “Answer diagram” summarizes the analysis of the entire truss (All forces are in kN)
Designing Sustainable Steel Structures: Environmental Considerations
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We rely heavily on steel structures daily as they build our hospitals, schools, and workplaces. Remember, steel also helps you safely get where you sh...
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Linear Operations
Section 2.1 Linear Operations
Let \(\R\) denote the set of real numbers. The (Cartesian) plane is the set
\begin{equation*} \R^2 = \{(a,b)\colon a,b\in \R\} \end{equation*}
of ordered pairs of real numbers.
The sum \((a,b)+(c,d)\) of two ordered pairs \((a,b)\) and \((c,d)\) is defined by
$$(a,b)+(c,d) = (a+c,b+d).\tag{2.1.1}$$
The scalar multiple \(\alpha(a,b)\) of the real number \(\alpha\) times the ordered pair \((a,b)\) is defined by
$$\alpha(a,b) = (\alpha a,\alpha b).\tag{2.1.2}$$
The dot product \((a,b)\cdot (c,d)\) of two ordered pairs \((a,b)\) and \((c,d)\) is defined by
$$(a,b)\cdot (c,d) = ac+bd.\tag{2.1.3}$$
An ordered pair is sometimes called a point in \(\R^2\text{,}\) and sometimes called a vector, depending on the nature of the object represented by \((a,b)\text{.}\) In drawings, a point is a dot,
and a vector is an arrow. We use capital letters in italic font to denote points, for example, \(P=(a,b)\text{.}\) We use bold lower case letters in Roman font, for example \({\mathbf x}=(a,b)\)
(better for typing), or lower case letters with an arrow written above them, for example \(\vec{x}=(a,b)\) (better for handwriting), to denote vectors. Given points \(Q=(x_0,y_0)\) and \(R=(x_1,y_1)\
text{,}\) we write \(\overrightarrow{QR}\) to denote the ordered pair \(\overrightarrow{QR}= R-Q = (x_1-x_0,y_1-y_0)\text{,}\) which we interpret as a vector, depicted by a directed line segment that
begins at \(Q\) and ends at \(R\text{.}\) Note that the point \((a,b)\) has only one possible drawing, that is, a dot at the location \((a,b)\text{.}\) By contrast, the vector \((a,b)\) has
infinitely many possible drawings: given any point \(S\text{,}\) vector \((a,b)\) is depicted by an arrow that begins at \(S\) and ends at \(S+(a,b)\text{.}\) Another way to say this is that two
arrows \(\overrightarrow{QR}, \overrightarrow{Q'R'}\) are equal as vectors if \(R-Q=R'-Q'\text{,}\) even though we may have \(Q\neq Q'\) and \(R\neq R'\text{.}\)
Warning about notation: Many texts make a distinction between points and vectors by using the notation \((a,b)\) for a point and \(\langle a,b\rangle\) for a vector. In these notes, we use \((a,b)\)
for both the point and the vector. The point \(P=(a,b)\) and the vector \(\mathbf{x}=(a,b)\) are not equal, because a point and a vector are different types of objects. The correct relationship
between \(P\) and \(\mathbf{x}\) is given by \(\mathbf{x}=\overrightarrow{OP}\text{,}\) where \(O\) is the point \(O=(0,0)\text{.}\) The take-away message is that you must always be clear whether you
mean a point or a vector when you write an ordered pair.
The ordered pair \((0,0)\) is called the origin or the zero vector, depending on whether we are thinking of \((0,0)\) as a point or as a vector, respectively. It is common practice to use the capital
letter \(O\) to denote the origin and to use the symbols \(\mathbf{0}\) or \(\vec{0}\) to denote the zero vector.
\begin{align*} O \amp = (0,0) \amp\amp \text{(the origin point)}\\ \mathbf{0} =\vec{0} \amp = (0,0) \amp\amp \text{(the zero vector)} \end{align*}
The vectors \({\mathbf e}_1 = (1,0)\) and \({\mathbf e}_2= (0,1)\) are called the standard basis vectors. We have the following expressions for \({\mathbf x}=(a,b)\) in terms of the standard basis
$${\mathbf x} = (a,b) = a{\mathbf e}_1 + b{\mathbf e}_2= ({\mathbf e}_1\cdot {\mathbf x}) {\mathbf e}_1 + ({\mathbf e}_2\cdot {\mathbf x}) {\mathbf e}_2.\tag{2.1.4}$$
Checkpoint 2.1.1.
Verify each of the equalities in
The norm or modulus of an ordered pair \((a,b)\text{,}\) denoted \(\left\Vert(a,b)\right\Vert\) , is defined by
$$\left\Vert(a,b)\right\Vert = \sqrt{(a,b)\cdot (a,b)} = \sqrt{a^2+b^2}.\tag{2.1.5}$$
The unit circle is the set
\begin{equation*} S^1 = \{(x,y)\in \R^2\colon x^2+y^2=1\} \end{equation*}
of all ordered pairs in the Cartesian plane with norm equal to one. Geometrically, \(S^1\) is a circle with radius 1, centered at the origin. It is useful to navigate on the unit circle with a single
real parameter, where the parameter value 0 corresponds to the point \((1,0)\text{.}\) A parameter value \(t\gt 0\) corresponds to the point \(W(t)\) that lies at the end of a counterclockwise arc
along \(S^1\) that begins at \((1,0)\) and has length \(t\text{.}\) For \(t\lt 0\text{,}\) the point \(W(t)\) lies at the end of a clockwise arc that begins at \((1,0)\) and has length \(|t|\text{.}
\) Another way to say this is that \(t\) is the radian measure of the oriented angle \(\angle SOP\text{,}\) where \(S=(1,0)\) and \(O=(0,0)\text{,}\) and \(P=\wrap(t)\text{.}\) The function \(\wrap\
colon \R\to S^1\) (often called the wrapping function) is given by
$$\wrap(t)= (\cos t,\sin t)\tag{2.1.6}$$
where \(t\) is in radians. Note that \(\wrap\) is periodic: the circumference of the unit circle is \(2\pi\text{,}\) so we have \(\wrap(t)=\wrap(t+2\pi k)\) for every integer \(k\text{.}\)
Checkpoint 2.1.2.
Draw a figure that illustrates the points \(P=W(t),S,O\text{,}\) the angle \(\angle SOP\text{,}\) and the oriented arc from \(S\) to \(P\) on \(S^1\text{.}\) Draw one figure for a positive value of \
(t\text{,}\) and draw another figure for a negative value of \(t\text{.}\)
Every point \(P\neq (0,0)\) is a positive scalar multiple of a unique point \(\wrap(t)\) on the unit circle, as follows. Given an arbitrary point \(P\in \R^2\) with \(P\neq (0,0)\text{,}\) let \(P'\)
be the intersection of the line segment \(\overline{OP}\) with the unit circle, so that we have \(P'=\wrap(t)\) for some real number \(t\text{.}\) It is easy to check that \(P=rP'\) where \(r=\left\
Vert P\right\Vert\text{.}\) The expression \(P=r\wrap(t)\) is called the polar form for \(P\text{.}\) The number \(t\) is called (an) argument of \(P\text{,}\) denoted \(\arg(P)\text{.}\) Because \(\
wrap\) is periodic with period \(2\pi\text{,}\) any number of the form \(t+2\pi k\text{,}\) for all \(k\in \Z\text{,}\) is also a value of \(\arg(P)\text{.}\) In this usage, \(\arg\) is not
technically a function, because it has multiple possible values. A single-valued function may be defined by restricting the value of \(\arg(P)\text{,}\) say, to the range \(-\pi \lt \arg(P)\leq \pi\
Checkpoint 2.1.3.
Draw a sketch that includes \(P,P',O,r,t\text{.}\) Show that "it is easy to check" that \(P=rP'\) by checking it.
If we have a point \(A=(u,v)=(k\cos \theta,k\sin\theta)\text{,}\) we say that the numbers \(u,v\) are the Cartesian coordinates or the rectangular coordinates for \(A\text{,}\) and we say that the
numbers \(k,\theta\) are polar coordinates for \(A\text{.}\)
Checkpoint 2.1.4.
Show that polar coordinates are not unique by finding three different pairs of polar coordinates \(r,t\) for the point \(A=(1,1)\text{.}\) How many different values of \(r\) are possible? What are
they? How many different values of \(t\) are possible? What are they?
Rotations about a point in the plane combine in a simple way: a rotation by \(t\) radians, followed by a rotation by \(s\) radians, results in a combined rotation by \(t+s\) radians. It is useful to
encode the composition of rotations in a multiplication operation on \(S^1\text{.}\) Given points \(\wrap(t),\wrap(s)\) on the unit circle, let \(\wrap(t)\odot \wrap(s)\) be defined by
$$\wrap(t)\odot \wrap(s) = \wrap(t+s).\tag{2.1.7}$$
Checkpoint 2.1.5.
Let \(P,Q\) be points in \(S^1\text{.}\) Show that the value of \(P\odot Q\) does not depend on the values \(t,s\) that we choose to write \(P=\wrap(t)\) and \(Q=\wrap(s)\text{.}\) In other words,
suppose that \(\wrap(t)=\wrap(t')\) and that \(\wrap(s)=\wrap(s')\text{,}\) but do not assume that \(t=t'\) or that \(s=s'\text{.}\) Show that \(\wrap(t+s)=\wrap(t'+s')\text{.}\)
Exercises Exercises
Let \(P=(a,b)\) be a point in \(\R^2\) different from the origin \(O=(0,0)\text{,}\) and let \(\alpha \) be a real number. Show that \(O,P,\alpha P\) are collinear (that is, show that \(O,P,\alpha P
\) lie on the same line).
2. Parallelogram rule for vector addition.
be points in
such that both
are different from the origin
is not a scalar multiple of
Show that the points
are the corners of a parallelogram.
A four-sided polygon with consecutive vertices \(A,B,C,D\) is a parallelogram if both pairs of opposite sides are parallel, that is, if \(\overline{AB}\parallel \overline{CD}\) and \(\overline{BC}\
parallel \overline{AD}\text{.}\)
Comment: This result is called the "parallelogram rule for vector addition" because it gives a simple geometric picture: the sum \(\mathbf{u}+\mathbf{v}=\overrightarrow{OR}\) of vectors \(\mathbf{u}=
\overrightarrow{OP},\mathbf{v}=\overrightarrow{OQ}\) is a diagonal of the parallelogram with adjacent sides \(\overline{OP},\overline{OQ}\text{.}\) Make a sketch!
Show that the following hold for all \(\alpha,\beta \in \R\text{,}\) \({\mathbf x},{\mathbf y}\in \R^2\text{.}\)
\alpha({\mathbf x}+{\mathbf y}) \amp = \alpha{\mathbf x} + \alpha{\mathbf y}\tag{2.1.8}\\ (\alpha \mathbf{x})\cdot (\beta \mathbf{y}) \amp = (\alpha \beta) (\mathbf{x}\cdot\mathbf{y})\tag{2.1.9}
How do norms and absolute values interact?
1. Show that \(\left\Vert r(a,b)\right\Vert=|r|\left\Vert(a,b)\right\Vert\) for all real numbers \(r,a,b\text{.}\)
2. Is it true that \(|(a,b)\cdot (c,d)|= \left\Vert(a,b)\right\Vert\;\left\Vert(c,d)\right\Vert\) for all real numbers \(a,b,c,d\text{?}\) Prove it or give a counterexample.
5. Properties of the wrapping function.
1. Show that \(\wrap(t)\cdot \wrap(s) = \cos (t-s)\text{.}\)
2. Show that
\begin{equation*} \wrap(t)\cdot \wrap(s) = (\wrap(r)\odot \wrap(t))\cdot (\wrap(r)\odot \wrap(s)) \end{equation*}
for all real numbers
Write formulas for converting from polar to rectangular coordinates and vice-versa.
7. Geometric content of the dot product.
Let \({\mathbf x},{\mathbf y}\) be vectors in \(\R^2\text{.}\) Let \(O=(0,0)\) denote the origin, and let \(P,Q\) be the points given by \({\mathbf x}=\overrightarrow{OP}\) and \({\mathbf y}=\
overrightarrow{OQ}\text{.}\) Show that
$${\mathbf x} \cdot {\mathbf y} = \left\Vert{\mathbf x}\right\Vert \left\Vert{\mathbf y}\right\Vert \cos \theta\tag{2.1.10}$$
where \(\theta\) is the measure of the angle \(\angle POQ\text{.}\) Hint: Start by writing \(P,Q\) in polar form, then use previous exercises.
Let \(n\) be a positive integer. For \(k\) in the range \(0\leq k\leq n-1\text{,}\) let \(P_k=\wrap\left(\frac{2\pi k}{n}\right)\text{,}\) where \(\wrap\) denotes the wrapping function.
1. Show that the points \(P_0,P_1,\ldots,P_{n-1}\) are vertices of a regular \(n\)-gon inscribed in the unit circle.
2. Show that the following holds for
\(0\leq k,\ell\leq n-1\text{.}\)
\begin{equation*} P_k\odot P_\ell = P_{k+_n \ell} \end{equation*}
9. Conventions on linear operations with points and vectors.
In these notes, addition and scaling operations are defined for ordered pairs, regardless of whether we view the ordered pairs as points or as vectors. Traditional convention places some restrictions
that we have ignored, and will continue to ignore, but nevertheless will present here for the reader’s cultural awareness. In short, traditional orthodoxy goes like this:
• The operations \((a,b)+(c,d)\) and \(\alpha(a,b)\) are allowed when \((a,b)\) and \((c,d) \) are both vectors, and the resulting ordered pairs \((a+c,b+d)\text{,}\) \((\alpha a,\alpha b)\) are
interpreted as vectors.
• Addition and scaling of ordered pairs are discouraged, or even banned outright, if the ordered pairs are both points, except for the following two cases:
• If \(P=(a,b)\) and \(Q=(c,d)\) are points, then the operation \((c,d)-(a,b)\) is allowed, and the resulting ordered pair \(Q-P=(c-a,d-b)\) is interpreted as a vector, denoted \(\overrightarrow
• If \((a,b)\) is a point and \((c,d)\) is a vector, then the operation \((a,b)+(c,d)\) is allowed, and the resulting ordered pair is interpreted as a point.
Here is a very short summary of the orthodox rules.
• vector \(+\) vector \(=\) vector
• point \(+\) vector \(=\) point
• point \(-\) point \(=\) vector
• scalar \(\cdot \) vector \(=\) vector
• point \(+\) point \(=\) (not recognized)
• scalar \(\cdot \) point \(=\) (not recognized)
The reader will have already noticed that in these notes, we adopt an inclusive approach that allows addition and scaling of points.
1. Draw sketches that illustrate the following.
1. (vector
\begin{equation*} \overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR} \end{equation*}
2. (point
\begin{equation*} P+\overrightarrow{PQ}=Q \end{equation*}
3. (point
\begin{equation*} Q-P=\overrightarrow{PQ} \end{equation*}
2. Let \(P=(a,b)\) be a point, let \(\mathbf{x}\) be a vector, and let \(O=(0,0)\) be the origin. Draw sketches that demonstrate the following "conversion" formulas that convert the point \((a,b)\)
to the vector \((a,b)\text{,}\) and vice-versa.
1. (convert
\begin{equation*} \mathbf{x}=P-O \end{equation*}
2. (convert
\begin{equation*} P=O+\mathbf{x} \end{equation*}
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13.3 Gravitational Potential Energy and Total Energy - University Physics Volume 1 | OpenStax
By the end of this section, you will be able to:
• Determine changes in gravitational potential energy over great distances
• Apply conservation of energy to determine escape velocity
• Determine whether astronomical bodies are gravitationally bound
We studied gravitational potential energy in Potential Energy and Conservation of Energy, where the value of g remained constant. We now develop an expression that works over distances such that g is
not constant. This is necessary to correctly calculate the energy needed to place satellites in orbit or to send them on missions in space.
Gravitational Potential Energy beyond Earth
We defined work and potential energy in Work and Kinetic Energy and Potential Energy and Conservation of Energy. The usefulness of those definitions is the ease with which we can solve many problems
using conservation of energy. Potential energy is particularly useful for forces that change with position, as the gravitational force does over large distances. In Potential Energy and Conservation
of Energy, we showed that the change in gravitational potential energy near Earth’s surface is $ΔU=mg(y2−y1)ΔU=mg(y2−y1)$. This works very well if g does not change significantly between $y1y1$ and
$y2y2$. We return to the definition of work and potential energy to derive an expression that is correct over larger distances.
Recall that work (W) is the integral of the dot product between force and distance. Essentially, it is the product of the component of a force along a displacement times that displacement. We define
$ΔUΔU$ as the negative of the work done by the force we associate with the potential energy. For clarity, we derive an expression for moving a mass m from distance $r1r1$ from the center of Earth to
distance $r2r2$. However, the result can easily be generalized to any two objects changing their separation from one value to another.
Consider Figure 13.11, in which we take m from a distance $r1r1$ from Earth’s center to a distance that is $r2r2$ from the center. Gravity is a conservative force (its magnitude and direction are
functions of location only), so we can take any path we wish, and the result for the calculation of work is the same. We take the path shown, as it greatly simplifies the integration. We first move
radially outward from distance $r1r1$ to distance $r2r2$, and then move along the arc of a circle until we reach the final position. During the radial portion, $F→F→$ is opposite to the direction we
travel along $dr→dr→$, so $E=K1+U1=K2+U2.E=K1+U1=K2+U2.$ Along the arc, $F→F→$ is perpendicular to $dr→dr→$, so $F→·dr→=0F→·dr→=0$. No work is done as we move along the arc. Using the expression for
the gravitational force and noting the values for $F→·dr→F→·dr→$ along the two segments of our path, we have
Since $ΔU=U2−U1ΔU=U2−U1$, we can adopt a simple expression for $UU$:
Note two important items with this definition. First, $U→0asr→∞U→0asr→∞$. The potential energy is zero when the two masses are infinitely far apart. Only the difference in U is important, so the
choice of $U=0forr=∞U=0forr=∞$ is merely one of convenience. (Recall that in earlier gravity problems, you were free to take $U=0U=0$ at the top or bottom of a building, or anywhere.) Second, note
that U becomes increasingly more negative as the masses get closer. That is consistent with what you learned about potential energy in Potential Energy and Conservation of Energy. As the two masses
are separated, positive work must be done against the force of gravity, and hence, U increases (becomes less negative). All masses naturally fall together under the influence of gravity, falling from
a higher to a lower potential energy.
Lifting a Payload
How much energy is required to lift the 9000-kg
vehicle from Earth’s surface to the height of the ISS, 400 km above the surface?
Equation 13.2
to find the change in potential energy of the payload. That amount of work or energy must be supplied to lift the payload.
Paying attention to the fact that we start at Earth’s surface and end at 400 km above the surface, the change in
We insert the values
and convert 400 km into $4.00×105m4.00×105m$. We find $ΔU=3.32×1010JΔU=3.32×1010J$. It is positive, indicating an increase in potential energy, as we would expect.
For perspective, consider that the average US household energy use in 2013 was 909 kWh per month. That is energy of
$909kWh×1000W/kW×3600s/h=3.27×109J per month.909kWh×1000W/kW×3600s/h=3.27×109J per month.$
So our result is an energy expenditure equivalent to 10 months. But this is just the energy needed to raise the payload 400 km. If we want the Soyuz to be in orbit so it can rendezvous with the ISS
and not just fall back to Earth, it needs a lot of kinetic energy. As we see in the next section, that kinetic energy is about five times that of $ΔUΔU$. In addition, far more energy is expended
lifting the propulsion system itself. Space travel is not cheap.
Check Your Understanding 13.3
Why not use the simpler expression $ΔU=mg(y2−y1)ΔU=mg(y2−y1)$? How significant would the error be? (Recall the previous result, in Example 13.4, that the value g at 400 km above the Earth is $8.67m/
Conservation of Energy
In Potential Energy and Conservation of Energy, we described how to apply conservation of energy for systems with conservative forces. We were able to solve many problems, particularly those
involving gravity, more simply using conservation of energy. Those principles and problem-solving strategies apply equally well here. The only change is to place the new expression for potential
energy into the conservation of energy equation, $E=K1+U1=K2+U2E=K1+U1=K2+U2$.
Note that we use M, rather than $MEME$, as a reminder that we are not restricted to problems involving Earth. However, we still assume that $m<<Mm<<M$. (For problems in which this is not true, we
need to include the kinetic energy of both masses and use conservation of momentum to relate the velocities to each other. But the principle remains the same.)
Escape velocity
Escape velocity is often defined to be the minimum initial velocity of an object that is required to escape the surface of a planet (or any large body like a moon) and never return. As usual, we
assume no energy lost to an atmosphere, should there be any.
Consider the case where an object is launched from the surface of a planet with an initial velocity directed away from the planet. With the minimum velocity needed to escape, the object would just
come to rest infinitely far away, that is, the object gives up the last of its kinetic energy just as it reaches infinity, where the force of gravity becomes zero. Since $U→0asr→∞U→0asr→∞$, this
means the total energy is zero. Thus, we find the escape velocity from the surface of an astronomical body of mass M and radius R by setting the total energy equal to zero. At the surface of the
body, the object is located at $r1=Rr1=R$ and it has escape velocity $v1=vescv1=vesc$. It reaches $r2=∞r2=∞$ with velocity $v2=0v2=0$. Substituting into Equation 13.5, we have
Solving for the escape velocity,
Notice that m has canceled out of the equation. The escape velocity is the same for all objects, regardless of mass. Also, we are not restricted to the surface of the planet; R can be any starting
point beyond the surface of the planet.
Escape from Earth
What is the escape speed from the surface of Earth? Assume there is no energy loss from air resistance. Compare this to the escape speed from the Sun, starting from Earth’s orbit.
We use
Equation 13.6
, clearly defining the values of
. To escape Earth, we need the mass and radius of Earth. For escaping the Sun, we need the mass of the Sun, and the orbital distance between Earth and the Sun.
Substituting the values for Earth’s mass and radius directly into
Equation 13.6
, we obtain
That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use $R=RES=1.50×1011mR=RES=1.50×1011m$ and $MSun=1.99×1030kgMSun=1.99×1030kg$. The result is $vesc=4.21×104m/
svesc=4.21×104m/s$ or about 42 km/s.
The speed needed to escape the Sun (leave the solar system) is nearly four times the escape speed from Earth’s surface. But there is help in both cases. Earth is rotating, at a speed of nearly 1.7 km
/s at the equator, and we can use that velocity to help escape, or to achieve orbit. For this reason, many commercial space companies maintain launch facilities near the equator. To escape the Sun,
there is even more help. Earth revolves about the Sun at a speed of approximately 30 km/s. By launching in the direction that Earth is moving, we need only an additional 12 km/s. The use of
gravitational assist from other planets, essentially a gravity slingshot technique, allows space probes to reach even greater speeds. In this slingshot technique, the vehicle approaches the planet
and is accelerated by the planet’s gravitational attraction. It has its greatest speed at the closest point of approach, although it accelerates opposite to the motion in equal measure as it moves
away. But relative to the planet, the vehicle’s speed far before the approach, and long after, are the same. If the directions are chosen correctly, that can result in a significant increase (or
decrease if needed) in the vehicle’s speed relative to the rest of the solar system.
Visit this website to learn more about escape velocity.
Check Your Understanding 13.4
If we send a probe out of the solar system starting from Earth’s surface, do we only have to escape the Sun?
Energy and gravitationally bound objects
As stated previously, escape velocity can be defined as the initial velocity of an object that can escape the surface of a moon or planet. More generally, it is the speed at any position such that
the total energy is zero. If the total energy is zero or greater, the object escapes. If the total energy is negative, the object cannot escape. Let’s see why that is the case.
As noted earlier, we see that $U→0asr→∞U→0asr→∞$. If the total energy is zero, then as m reaches a value of r that approaches infinity, U becomes zero and so must the kinetic energy. Hence, m comes
to rest infinitely far away from M. It has “just escaped” M. If the total energy is positive, then kinetic energy remains at $r=∞r=∞$ and certainly m does not return. When the total energy is zero or
greater, then we say that m is not gravitationally bound to M.
On the other hand, if the total energy is negative, then the kinetic energy must reach zero at some finite value of r, where U is negative and equal to the total energy. The object can never exceed
this finite distance from M, since to do so would require the kinetic energy to become negative, which is not possible. We say m is gravitationally bound to M.
We have simplified this discussion by assuming that the object was headed directly away from the planet. What is remarkable is that the result applies for any velocity. Energy is a scalar quantity
and hence Equation 13.5 is a scalar equation—the direction of the velocity plays no role in conservation of energy. It is possible to have a gravitationally bound system where the masses do not “fall
together,” but maintain an orbital motion about each other.
We have one important final observation. Earlier we stated that if the total energy is zero or greater, the object escapes. Strictly speaking, Equation 13.5 and Equation 13.6 apply for point objects.
They apply to finite-sized, spherically symmetric objects as well, provided that the value for r in Equation 13.5 is always greater than the sum of the radii of the two objects. If r becomes less
than this sum, then the objects collide. (Even for greater values of r, but near the sum of the radii, gravitational tidal forces could create significant effects if both objects are planet sized. We
examine tidal effects in Tidal Forces.) Neither positive nor negative total energy precludes finite-sized masses from colliding. For real objects, direction is important.
How Far Can an Object Escape?
Let’s consider the preceding example again, where we calculated the escape speed from Earth and the Sun, starting from Earth’s orbit. We noted that Earth already has an orbital speed of 30 km/s. As
we see in the next section, that is the tangential speed needed to stay in circular orbit. If an object had this speed at the distance of Earth’s orbit, but was headed directly away from the Sun, how
far would it travel before coming to rest? Ignore the gravitational effects of any other bodies.
The object has initial kinetic and potential energies that we can calculate. When its speed reaches zero, it is at its maximum distance from the Sun. We use
Equation 13.5
, conservation of energy, to find the distance at which kinetic energy is zero.
The initial position of the object is Earth’s radius of orbit and the intial speed is given as 30 km/s. The final velocity is zero, so we can solve for the distance at that point from the
conservation of energy equation. Using
, we have
$12mv12−GMmr1=12mv22−GMmr2 12m(3.0×103m/s)2−(6.67×10−11N·m/kg2)(1.99×1030kg)m1.50×1011m=12m02−(6.67×10−11N·m/kg2)(1.99×1030kg)mr212mv12−GMmr1=12mv22−GMmr2 12m(3.0×103m/s)2−(6.67×10−11N·m/kg2)
where the mass m cancels. Solving for $r2r2$ we get $r2=3.0×1011mr2=3.0×1011m$. Note that this is twice the initial distance from the Sun and takes us past Mars’s orbit, but not quite to the asteroid
The object in this case reached a distance
twice the initial orbital distance. We will see the reason for this in the next section when we calculate the speed for circular orbits.
Check Your Understanding 13.5
Assume you are in a spacecraft in orbit about the Sun at Earth’s orbit, but far away from Earth (so that it can be ignored). How could you redirect your tangential velocity to the radial direction
such that you could then pass by Mars’s orbit? What would be required to change just the direction of the velocity?
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How many grams of zinc are represented by 1.807x10^24 atoms? | Socratic
How many grams of zinc are represented by #1.807x10^24# atoms?
1 Answer
Well, $6.022 \times {10}^{23}$$\text{zinc atoms}$ have a mass of $65.4 \cdot g$.
Of course, $6.022 \times {10}^{23}$$\text{zinc atoms}$ constitutes a mole of $\text{zinc atoms}$, and we say that zinc metal has a molar mass of $65.4 \cdot g \cdot m o {l}^{-} 1$.
So we take the quotient:
$\left(1.807 \times {10}^{24} \text{ zinc atoms")/(6.022xx10^23" zinc atoms} \cdot m o {l}^{-} 1\right) \times 65.4 \cdot g \cdot m o {l}^{-} 1 \cong 200 \cdot g$
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Assume that the specifications on this component are 74.000+0.05 mm. (a) Set up x̄ and R control charts on this process. Is the process in statistical control? (b) Note that the control limits on the
x̄ chart in part (a) are identical to the control limits on the x̄ chart in Example 6.3, where the limits were based on s. Will this always happen? (c) Estimate process capability for the piston-ring
process. Estimate the percentage of piston rings produced that will be outside of the specification.
From the Community
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Solution Steps
Solution Approach
(a) To set up \(\bar{x}\) and \(R\) control charts, calculate the average (\(\bar{x}\)) and range (R) for each sample. Then, determine the overall average of the sample means (\(\bar{\bar{x}}\)) and
the average of the ranges (\(\bar{R}\)). Use these to calculate the control limits for the \(\bar{x}\) and \(R\) charts using standard formulas. Check if any points fall outside the control limits to
determine if the process is in statistical control.
(b) This part is not addressed as per the instructions.
(c) Estimate the process capability by calculating the process capability index (Cp) using the formula \(Cp = \frac{USL - LSL}{6\sigma}\), where USL and LSL are the upper and lower specification
limits, respectively, and \(\sigma\) is the estimated standard deviation of the process. Use the normal distribution to estimate the percentage of piston rings outside the specification limits.
Step 1: Calculate Sample Means and Ranges
For each sample, calculate the mean \(\bar{x}_i\) and the range \(R_i\). The sample means are: \[ \bar{x} = [16.2, 16.14, 16.3, 16.2, 16.22, 16.32, 16.3, 16.18, 16.34, 16.38, 16.24, 16.38, 16.32,
16.34, 16.24, 16.2, 16.3, 16.24, 16.3, 16.22] \] The sample ranges are: \[ R = [0.8, 0.5, 0.4, 0.5, 0.5, 0.9, 0.4, 0.2, 0.3, 0.5, 0.5, 0.8, 0.5, 0.3, 0.3, 0.3, 0.2, 0.5, 0.4, 0.7] \]
Step 2: Calculate Overall Averages
Calculate the overall average of the sample means \(\bar{\bar{x}}\) and the average of the ranges \(\bar{R}\): \[ \bar{\bar{x}} = 16.268, \quad \bar{R} = 0.4750 \]
Step 3: Determine Control Limits
Using the constants \(A_2 = 0.577\), \(D_3 = 0\), and \(D_4 = 2.114\) for a sample size of 5, calculate the control limits for the \(\bar{x}\) and \(R\) charts: \[ UCL_{\bar{x}} = \bar{\bar{x}} + A_2
\cdot \bar{R} = 16.5421, \quad LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \cdot \bar{R} = 15.9939 \] \[ UCL_R = D_4 \cdot \bar{R} = 1.0041, \quad LCL_R = D_3 \cdot \bar{R} = 0.0 \]
Step 4: Check Process Control
Verify if the process is in control by checking if all sample means and ranges are within their respective control limits. Since all sample means and ranges are within the control limits, the process
is in control: \[ \text{x-bar in control: True}, \quad \text{R in control: True} \]
Step 5: Estimate Process Capability
Calculate the process capability index \(C_p\) using the specification limits \(USL = 74.05\) and \(LSL = 73.95\), and the estimated standard deviation \(\sigma = \frac{\bar{R}}{2.326}\): \[ \sigma =
0.2042, \quad C_p = \frac{USL - LSL}{6\sigma} = 0.08161 \]
Step 6: Estimate Percentage Outside Specification
Calculate the percentage of piston rings outside the specification limits using the normal distribution: \[ z_{\text{upper}} = \frac{USL - \bar{\bar{x}}}{\sigma} = 282.9493, \quad z_{\text{lower}} =
\frac{\bar{\bar{x}} - LSL}{\sigma} = -282.4596 \] The percentage outside the specification is approximately \(0.0\%\).
Final Answer
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Triple Exponential Average Trix | Stock Technical Analysis and Fundamental Analysis
Triple Exponential Average (Trix)
Trix (or Triple Exponential Average) is a technical analysis oscillator developed in the 1980s by Jack Hutson, editor of Technical Analysis of Stocks and Commodities magazine. It shows the slope (ie.
derivative) of a triple-smoothed exponential moving average. The name Trix is from "triple exponential."
Trix is calculated with a given N-day period as follows:
Smooth prices (often closing prices) using an N-day exponential moving average (EMA).
Smooth that series using another N-day EMA.
Smooth a third time, using a further N-day EMA.
Calculate the percentage difference between today's and yesterday's value in that final smoothed series.
How To Use
Like any moving average, the triple EMA is just a smoothing of price data and therefore is trend-following. A rising or falling line is an uptrend or downtrend and Trix shows the slope of that line,
so it's positive for a steady uptrend, negative for a downtrend, and a crossing through zero is a trend-change, ie. a peak or trough in the underlying average.
The triple-smoothed EMA is very different from a plain EMA. In a plain EMA the latest few days dominate and the EMA follows recent prices quite closely; however, applying it three times results in
weightings spread much more broadly, and the weights for the latest few days are in fact smaller than those of days further past. The following graph shows the weightings for an N=10 triple EMA (most
recent days at the left):
Triple exponential moving average weightings, N=10 (percentage versus days ago)
The easiest way to calculate the triple EMA based on successive values is just to apply the EMA three times, creating single-, then double-, then triple-smoothed series. The triple EMA can also be
expressed directly in terms of the prices as below, with p0 today's close, p1 yesterday's, etc, and with (as for a plain EMA):
The coefficients are the triangle numbers, n(n+1)/2. In theory, the sum is infinite, using all past data, but as f is less than 1 the powers fn become smaller as the series progresses, and they
decrease faster than the coefficients increase, so beyond a certain point the terms are negligible.
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pROBLEM SET 3
The purpose of this review sheet is to demonstrate how to remove the incentive to overstate one’s net benefits. It is crucial to remember that the chief obstacle that one faces in overstating one’s
benefits is the information required to do so. You do not know the other voter’s willingness to pay (but you could guess). In addition, what really matters is whether you are risk averse. If you are
risk averse, you would not be willing to risk the chance of becoming pivotal for the possibility of increased Clarke taxes. If you are risk seeking, you would like to make this gamble.
Policy 1 Policy 2 Clarke Tax
Voter 1 $40 19
Voter 2 $30 0
Voter 3 $11 0
If voter 2 exaggerates up to $39 he can earn an additional $9 for a total of $28 in Clarke taxes.
Essentially the risk of over stating your preferences is having to pay Clarke taxes. The problem of asymmetric information may prevent voters from taking advantage of this feature. How are you to
know the willingness to pay of your fellow voters? It also depends upon whether a person is risk averse or risk-seeking.
Now let’s use Martin Bailey’s formula from page 216:
Formula = Clarke taxes if voter x was not present = (S-i) Total number of voters N
Bailey’s Simple method
In this case, if we did not apply Bailey’s method then voters 2 and 4 would receive $31/2 =$15.5.
Now as per Bailey’s Simple Method this would be reduced to $7.75 such that voters 2, 3, and 4 would receive a Clarke tax payment of $7.75 each. This means that about $7.75 of the Clarke tax would be
wasted, or could be used to cover the cost of implementing the procedure.
Policy 1 Policy 2 Clarke Tax
Voter 1 $40 31
Voter 2 $30 0
Voter 3 $11 0
Voter 4 $12 0
There is also a slightly more complex, and actually more correct formula:
Bailey’s Complex Method
S-i · (N-2) _______
In this formula, S-i: The Clarke tax
N=Total number of participants in the system.
Let’s apply this formula to the example above. Since N=4, you would multiply the Clarke taxes to be paid by 2/9. In other words (31 · 2/9). Now each voter would get $6.89 (except for the one who pays
the Clarke tax).
Why is this important
Why is this important? Let’s assume voter 2 wants to “milk” as much money out of voter 1 possible (ala the Clarke tax). He can overstate his net benefits up to $38 (from $3) without becoming pivotal
(and hence paying Clarke taxes). If he does so, voter one would pay $39 in Clarke taxes. Divided between voters 2,3,4 each would now receive $9.75 in Clarke taxes instead of $7.75. Adjust this with
Bailey’s complex method and each voter gets a Clarke tax of $8.67 (instead of $6.88).
Now assume that we did not use any of Bailey’s adjusted formulas, and voter 2 overstates their value to be $38. If we split this evenly between the two winners, each would receive a whopping Clarke
tax of $19! This is much greater than one of $8.44. So Bailey’s adjusted formulas reduce an individual’s incentive to overstate their preferences. I would note that Bailey’s formulas are designed
more for the incentive compatibility problem. If there is a surplus, then voters can adjust their preferences to try and acquire it for themselves. In other words, they’re true preferences have been
altered by the mechanism, and they are not honestly revealing them. This is the incentive compatibility problem.
Why is this useful? Well, it particularly helps with the problem of budget balancing. Take a look at the following example:
Now let’s assume Voter 1 overstates their net benefit to be $999.
Under complete payment, Voter two would not pay a $999 incentive tax. If legislatures or some organization were conducting this VCG “experiment” than there would only be $1 left to cover the costs of
covering it.
War Peace Clarke Tax
Voter 1 $200 0
Voter 2 $1000 $200
If we use Bailey’s simplified method S-i/N than we get $999 /2 which is $499.5, so balanced budget is no longer a problem. You still have $499.5 left in the treasury to cover costs of the
legislature. Waste, however could be a problem.
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Lesson 14
Proving the Pythagorean Theorem
• Let’s prove the Pythagorean Theorem.
14.1: Notice and Wonder: Variable Version
What do you notice? What do you wonder?
14.2: Prove Pythagoras Right
1. How did Elena get from \(\frac{a}{x} = \frac{c}{a} \text{ to } a^2=xc\)?
2. What equivalent ratios of side lengths did Diego use to get \(b^2=yc\)?
3. Prove \(a^2+b^2=c^2\) in a right triangle with legs length \(a\) and \(b\) and hypotenuse length \(c\).
14.3: An Alternate Approach
When Pythagoras proved his theorem he used the 2 images shown here. Can you figure out how he used these diagrams to prove \(a^2+b^2=c^2\) in a right triangle with hypotenuse length \(c\)?
James Garfield, the 20th president, proved the Pythagorean Theorem in a different way.
• Cut out 2 congruent right triangles
• Label the long sides \(b\), the short sides \(a\) and the hypotenuses \(c\).
• Align the triangles on a piece of paper, with one long side and one short side in a line. Draw the line connecting the other acute angles.
How does this diagram prove the Pythagorean Theorem?
In any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we know that \(a^2+b^2=c^2\). We call this the Pythagorean Theorem. But why does it work?
We can use an altitude drawn to the hypotenuse of a right triangle to prove the Pythagorean Theorem.
We can use the Angle-Angle Triangle Similarity Theorem to show that all 3 triangles are similar. Because the triangles are similar, corresponding side lengths are in the same proportion.
Because the largest triangle is similar to the smaller triangle, \(\frac{c}{a}=\frac{a}{d}\). Because the largest triangle is similar to the middle triangle, \(\frac{c}{b}=\frac{b}{e}\). We can
rewrite these equations as \(a^2=cd\) and \(b^2=ce\).
We can add the 2 equations to get that \(a^2+b^2=cd+ce\) or \(a^2+b^2=c(d+e)\). From the original diagram we can see that \(d+e=c\), so \(a^2+b^2=c(c)\) or \(a^2+b^2=c^2\).
Using the Pythagorean Theorem we can describe a triangle's angles without ever drawing it. For example, a triangle with side lengths 8, 15, and 17 is right because \(17^2=8^2+15^2\). A triangle with
side lengths 8, 15, and 18 is obtuse because \(18^2>8^2+15^2\). A triangle with side lengths 8, 15, and 16 is acute because \(16^2<8^2+15^2\).
• altitude
An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.
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Data Field Types in Datameer
When importing data into Datameer, it is important to know which type of data you are importing or need to import. When using your data within workbooks, certain Datameer functions only work with
certain types of data. Congruently, certain functions return only a specific type of data.
Also there are data requirements when using infographic widgets to visualize your data.
Data Field Types
Field type Product icon Description Internal representation
integer 64-Bit integer value Java Long
big integer Unlimited integer value Java BigInteger
float 64-Bit float value Java Double
big decimal High-precision float value Java BigDecimal
date Date object Java Date
string String object Java String
boolean Boolean object Java Boolean
list a collection of multiple values of one data type
number float, big decimal, integer, or big integer
any float, big decimal, integer, big integer, date, string, list, or Boolean
In mathematics integers (aka whole numbers) are made up of the set of natural numbers including zero (0,1,2,3, ...) along with the negatives of natural numbers (-1,-2,-3, ...). When talking about
Integers in computer programming, it is necessary to define a minimum and maximum value. Datameer uses a 64-bit integer which allows the user to represent whole numbers between
-9,223,372,036,854,775,808 to 9,223,372,036,854,775,807.
Big integer
Big integers are like integers, but they are not limited to 64 bits. They are represented using arbitrary-precision arithmetic. Big integers represent only whole numbers. Big integers in Datameer are
treated differently than in Hive because Datameer allows a larger range of values, so they are written as strings into a Hive table if you export. By default, the precision for big integers is set
at 32 upon import. This can updated if needed in the default properties by changing the value of das.big-decimal.precision=.
In mathematics, there are real numbers that represent fractions (1/2, 12/68) or numbers with decimal places (12.75, -18.35). Datameer uses double precision floating-point representation (aka float)
to manipulate and represent real numbers. The complete range of numbers that can be represented this way is approximately 2 ^-1022 through (1+(1-2 ^-52))x2 ^1023. During import/upload, Datameer
automatically recognizes a number with either a single period (.) or single comma (,) as a decimal separator and defines this data as a float data type. After ingestion, Datameer stores float and big
decimal values using a period (.) character. The auto schema detection for the float data type works with CSV, JSON, XML, Key/value files.
Big decimal
Big decimals are similar to float values. The main advantage of this data field type is that they are exact to the number of decimal places for which they are configured, float values might be
inaccurate in certain cases. If a number has more decimal places than big decimal was configured for, then the number is rounded. The number of decimal places can be configured in conf/
# Maximum precision used for BIG_DECIMAL types. Precision is equal to the maximum number of digits a BigDecimal
# can have.
32 digits is the default precision used by Datameer for big decimal values upon import.
In Datameer, data in the DATE primitive data type is always represented in a Gregorian, month-day-year (MDY) format (e.g., "Sep 16, 2010 02:56:39 PM"). Datameer detects if your data should be parsed
into the DATE data type during ingest. This can also be done after ingest as other data types can be converted to the DATE primitive data type using workbook functions.
When using information other than numbers or dates in Datameer, it is represented as a string. This includes text, unparsed date patterns, URLs, JSON arrays, etc.
Boolean data in computing has two values, either true or false. It is used in many logical expressions and is derived from Boolean algebra created by George Boole in the 19th century.
In Datameer multiple values can be combined into a list. Lists are a series of values of a single data type, which starts counting from zero (0).
In Datameer integers, big integers, floats, and big decimals are considered to be numbers.
Some visualizations and functions are able to use data represented by any data field type. These can be either a number, a string, a date, or a Boolean.
Data Mapping in Avro
Import mapping
When importing data to Datameer, data types are mapped as follows:
Avro Schema Type Datameer Value Type
null STRING
boolean BOOLEAN
int INTEGER
long INTEGER
float FLOAT
double FLOAT
bytes STRING
bytes with logical type decimal BIG_DECIMAL
string STRING
records STRING
enums STRING
arrays STRING
maps STRING
unions STRING
fixed STRING
fixed with logical type decimal BIG_DECIMAL
Export mapping
When exporting data to Avro, data types are mapped as follows:
If a column marked as accept empty a union scheme type is created, e.g. union( null, string) for nullable string type.
Datameer Value Type Avro Schema Type
STRING string
BOOLEAN boolean
INTEGER long
FLOAT double
DATE without pattern long
DATE with pattern string
BIG_INTEGER string
BIG_DECIMAL string
LIST<listtype> arrays<converted list type>
Data Mapping in Parquet
Export data mapping
When exporting data to Parquet, data types are mapped as follows:
Import data mapping
When importing data from Parquet, data types are mapped as follows:
Parquet Field Type Datameer Field Type
BOOLEAN BOOLEAN
INT32 INTEGER
INT32 DECIMAL BIG_DECIMAL
INT64 INTEGER
INT64 DECIMAL BIG_DECIMAL
INT96 DATE
FLOAT FLOAT
DOUBLE FLOAT
BINARY STRING
BINARY DECIMAL BIG_DECIMAL
FIXED_LEN_BYTE_ARRAY STRING
Parquet files using the INT96 format are interpreted as time stamps. Datameer accepts these columns, but cuts off the nanoseconds. If the workbook has Ignore Errors enabled, then those error messages
are stored in a separate column and the column with the error is NULL. The chart below provides additional Parquet storage mapping details.
Datameer Type Parquet Type Description
date TIMESTAMP_MILLIS Stored as INT64
integer INT_64
float DOUBLE
string UTF8 BINARY format with UTF-8 encoding
big decimal DECIMAL BINARY format
big integer DECIMAL BINARY format with precision of 1 and scale of 0
list Repeated elements of group of the list element type. Optional group of a repeating group of optional element types. Nested lists are supported.
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Calculate Orbital Speed
Calculate Orbital Speed
Calculator for the orbital speed, the velocity of a celestial body (planet or moon) around another one (star or planet). Celestial bodies have large masses, which are often measured in Earth, Jupiter
or Solar masses. You can also select kilograms. The total mass is the mass of both bodies together. For the radius, the common unit is AU, astronomical unit, which is the average distance of the
Earth from the Sun. Here you can also choose meters, kilometers or miles. For the velocity, there are meters, kilometers and miles, per hour or per second. Please enter two values and select the
units. The third value will be calculated.
The formula is:
velocity = √ gravitational constant * total mass / orbit radius
v = √ G * M / r
Gravitational constant G = 6.6743 * 10^-11 m³/(kg*s²) = 0.000000000066743 m³/(kg*s²)
Example: Sun has about 332890 times the mass of Earth. So the system Earth-Sun has about one solar mass. At an average radius of one astronomical unit, the average orbital speed is almost 30
kilometers per second.
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Heinemann Blog | Heinemann
Beginning to use Cognitively Guided Instruction (CGI) is all about making sense of children’s thinking, listening to your students, and experimenting with your classroom practice.
During this time of remote teaching and learning, teachers can help parents and caregivers notice, support, and extend children’s math thinking with games and outdoor play.
What is CGI? Today we’ll give you an answer to help you decide if it might be right for your classroom.
CGI: Integrating Arithmetic and Algebra in Elementary School
How To Build Meaning for Fractions with Word Problems
Starting Out With Cognitively Guided Instruction [Video]
Children’s Mathematics: Engaging with each other’s ideas
Children’s Mathematics: Children’s Identity as Mathematical Thinkers
Children's Mathematics: Why Every Math Teacher Should Know About Cognitively Guided Instruction
Children's Mathematics: Reading Is Thinking but Math Is Thinking, Too
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ekameters to Attometers
Dekameters to Attometers Converter
Enter Dekameters
β Switch toAttometers to Dekameters Converter
How to use this Dekameters to Attometers Converter π €
Follow these steps to convert given length from the units of Dekameters to the units of Attometers.
1. Enter the input Dekameters value in the text field.
2. The calculator converts the given Dekameters into Attometers in realtime β using the conversion formula, and displays under the Attometers label. You do not need to click any button. If the
input changes, Attometers value is re-calculated, just like that.
3. You may copy the resulting Attometers value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Dekameters to Attometers?
The formula to convert given length from Dekameters to Attometers is:
Length[(Attometers)] = Length[(Dekameters)] × 1e+19
Substitute the given value of length in dekameters, i.e., Length[(Dekameters)] in the above formula and simplify the right-hand side value. The resulting value is the length in attometers, i.e.,
Calculation will be done after you enter a valid input.
Consider that a high-rise building stands 25 dekameters tall.
Convert this height from dekameters to Attometers.
The length in dekameters is:
Length[(Dekameters)] = 25
The formula to convert length from dekameters to attometers is:
Length[(Attometers)] = Length[(Dekameters)] × 1e+19
Substitute given weight Length[(Dekameters)] = 25 in the above formula.
Length[(Attometers)] = 25 × 1e+19
Length[(Attometers)] = 250000000000000000000
Final Answer:
Therefore, 25 dam is equal to 250000000000000000000 am.
The length is 250000000000000000000 am, in attometers.
Consider that a luxury yacht has a length of 15 dekameters.
Convert this length from dekameters to Attometers.
The length in dekameters is:
Length[(Dekameters)] = 15
The formula to convert length from dekameters to attometers is:
Length[(Attometers)] = Length[(Dekameters)] × 1e+19
Substitute given weight Length[(Dekameters)] = 15 in the above formula.
Length[(Attometers)] = 15 × 1e+19
Length[(Attometers)] = 150000000000000000000
Final Answer:
Therefore, 15 dam is equal to 150000000000000000000 am.
The length is 150000000000000000000 am, in attometers.
Dekameters to Attometers Conversion Table
The following table gives some of the most used conversions from Dekameters to Attometers.
Dekameters (dam) Attometers (am)
0 dam 0 am
1 dam 10000000000000000000 am
2 dam 20000000000000000000 am
3 dam 30000000000000000000 am
4 dam 40000000000000000000 am
5 dam 50000000000000000000 am
6 dam 60000000000000000000 am
7 dam 70000000000000000000 am
8 dam 80000000000000000000 am
9 dam 90000000000000000000 am
10 dam 100000000000000000000 am
20 dam 200000000000000000000 am
50 dam 500000000000000000000 am
100 dam 1e+21 am
1000 dam 1e+22 am
10000 dam 1e+23 am
100000 dam 1e+24 am
A dekameter (dam) is a unit of length in the International System of Units (SI). One dekameter is equivalent to 10 meters or approximately 32.808 feet.
The dekameter is defined as ten meters, providing a convenient measurement for moderately large distances.
Dekameters are used in various fields to measure length and distance where a scale between meters and hectometers is appropriate. They are less commonly used than other metric units but can be useful
in certain applications, such as land measurement and environmental science.
An attometer (am) is a unit of length in the International System of Units (SI). One attometer is equivalent to 0.000000000000001 meters or 1 Γ 10^(-18) meters.
The attometer is defined as one quintillionth of a meter, making it an extremely small unit of measurement used for measuring subatomic distances.
Attometers are used in advanced scientific fields such as particle physics and quantum mechanics, where precise measurements at the atomic and subatomic scales are required.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Dekameters to Attometers in Length?
The formula to convert Dekameters to Attometers in Length is:
Dekameters * 1e+19
2. Is this tool free or paid?
This Length conversion tool, which converts Dekameters to Attometers, is completely free to use.
3. How do I convert Length from Dekameters to Attometers?
To convert Length from Dekameters to Attometers, you can use the following formula:
Dekameters * 1e+19
For example, if you have a value in Dekameters, you substitute that value in place of Dekameters in the above formula, and solve the mathematical expression to get the equivalent value in Attometers.
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Directory macros/latex/contrib/longdivision
longdivision v1.2.2
Author: Hood Chatham Email: hood@mit.edu Date: 2023-10-21 Description: License: All files have the Latex Project Public License. Files: longdivision.sty longdivisionmanual.tex longdivisionmanual.pdf
Does the long division algorithm and typesets the results. The dividend must be a positive decimal number and the divisor must be a positive integer. Correctly handles repeating decimals, putting a
bar over the repeated part of the decimal. Handles dividends up to 20 digits long gracefully (though the typeset result will take up about a page) and dividends between 20 and 60 digits long slightly
less gracefully.
Defines macros longdivision and intlongdivision. Each takes two arguments, a dividend and divisor. longdivision keeps dividing until the remainder is zero, or it encounters a repeated remainder.
longdivision stops when the dividend stops (though the dividend doesn't have to be an integer).
intlongdivision behaves similarly to the longdiv command defined in longdiv.tex, though I think intlongdivision looks better, and it can handle much larger dividends and divisors (the dividend is
only constrained by the size of the page, and the divisor can be up to 8 digits long).
See the package manual for more information. To compile the manual, you need the file pgfmanual-en-macros.tex from the tikz manual. I am not allowed to include this file on the CTAN repositiory due
to rules against file duplication, but you can find it on github: https://github.com/hoodmane/longdivision/blob/master/pgfmanual-en-macros.tex
Email me at hood@mit.edu to submit bug reports, request new features, etc. The current development copy is hosted at https://github.com/hoodmane/longdivision.
• The decimal separator no longer goes missing when the "stage" is set to a low enough number that digits after the decimal separator are not inspected.
• Added "brazilian" style (contributed by gh-user Felipe-Math)
• The stage option works again.
• An option to control the decimal separator
• Options to specify "digit separator" and "digit group length"
• A "separators in work" option to put spaces rather than punctuation in the work
• Change the division sign in "german style" long division
• Math mode usage used to be inconsistent. Now if not used in math mode, typesetting is consistently not in math mode, if used in math mode it is consistently in math mode.
• Major improvements to typesetting engine should allow changes in the future to be less painful.
• A few deprecated expl3 commands have been replaced (thanks to a pull request from Phelype Oleinik).
• Multiple typesetting styles
• Multiple styles for indicating a repeating decimal
• Typesetting partial long division (as suggested by Cameron McLeman).
• If the "longdivision" command is in math mode, the numbers are typeset in math mode too (reported by Yu-Tsiang Tai).
• The "longdivision" command now can handle macros as arguments (as suggested by Mike Jenck).
Download the contents of this package in one zip archive (154.1k).
longdivision – Typesets long division
This package executes the long division algorithm and typesets the solutions. The dividend must be a positive decimal number and the divisor must be a positive integer. Repeating decimals is handled
correctly, putting a bar over the repeated part of the decimal. Dividends up to 20 digits long are handled gracefully (though the typeset result will take up about a page), and dividends between 20
and 60 digits long slightly less gracefully.
The package defines two macros, \longdivision and \intlongdivision. Each takes two arguments, a dividend and a divisor. \longdivision keeps dividing until the remainder is zero, or it encounters a
repeated remainder. \intlongdivision stops when the dividend stops (though the dividend doesn’t have to be an integer).
This package depends on the xparse package from the l3packages bundle.
Package longdivision
Repository https://github.com/hoodmane/longdivision
Version 1.2.2 2023-10-21
Licenses The LaTeX Project Public License
Maintainer Hood Chatham
Contained in TeXLive as longdivision
MiKTeX as longdivision
Experimental LaTeX3
Topics Maths
See also longdiv
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(PDF) A Semi-Implicit Fractional Step Method Immersed Boundary Method for the Numerical Simulation of Natural Convection Non-Boussinesq Flows
Author content
All content in this area was uploaded by Yuri Feldman on Dec 22, 2022
Content may be subject to copyright.
A semi-implicit fractional step method immersed bound-
ary method for the numerical simulation of natural con-
vection non-Boussinesq ows
Dmitry Zviaga1, Ido Silverman2, Alexander Gelfgat3and Yuri Feldman1
1Department of Mechanical Engineering, Ben-Gurion University of the Negev,
P.O. Box 653, Beer-Sheva 84105, Israel.
2Soreq Nuclear Research Center, 81000 Yavne, Israel.
3School of Mechanical Engineering, Tel Aviv University, Tel Aviv 6997801,
Abstract. The paper presents a novel pressure-corrected formulation of the immersed
boundary method (IBM) for the simulation of fully compressible non-Boussinesq natural
convection ows. The formulation incorporated into the pressure-based fractional step
approach facilitates simulation of the ows in the presence of an immersed body charac-
terized by a complex geometry. Here, we rst present extensive grid independence and
verication studies addressing incompressible pressure-driven ow in an extended chan-
nel and non-Boussinesq natural convection ow in a dierentially heated cavity. Next,
the steady-state non-Boussinesq natural convection ow developing in the presence of
hot cylinders of various diameters placed within a cold square cavity is thoroughly in-
vestigated. The obtained results are presented and analyzed in terms of the spatial
distribution of path lines and temperature elds and of heat ux values typical of the
hot cylinder and the cold cavity surfaces. Flow characteristics of multiple steady-state
solutions discovered for several congurations are presented and discussed in detail.
Key words: natural convection non-Boussinesq ows, pressure-corrected immersed boundary method,
multiple steady state solutions.
1 Introduction
The ability to accurately simulate natural convection ows is critical for a wide range of
engineering applications, including cooling electronic equipment, minimizing heat losses
in buildings, investigating atmospheric ows, modeling heat transfer, and preventing ac-
cidents in the nuclear industry, to name but a few. The methods typically utilized for
the simulation of natural convection ows can be classied into two major groups, one
relying on weakly compressible approximations and the other addressing the fully com-
pressible ow, as extensively reviewed in [1]. The rst group of methods treats the ow
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as incompressible. In these methods, the buoyancy is introduced by employing either
the Boussinesq approximation, which accounts for density variations in the gravity term
and may also account for variations in the thermophysical properties of the ow, or Gay-
Lussac-type approximations, which account for density variations in both the gravity and
advection terms. The second group uses fully compressible Navier Stokes (NS) and energy
The majority of weakly compressible approximations, which were developed for the
simulation of low-Mach-number compressible ows, are based on an asymptotic model.
The asymptotic model for the simulation of thermally driven natural convection ows
was formulated for the rst time in [2] and is known in the literature as the “classical
low-Mach-number model.” The key idea was to split the pressure eld into a large, time-
dependent thermodynamic part and a stationary part that includes extremely small spatial
deviations. Such a decomposition was found to be applicable for the simulation of low-
Mach-number thermally driven ows, as it provides the same order of magnitude for all
the terms in the momentum and energy equations and eliminates acoustic waves. Signi-
cant progress in this eld may be attributed to the works [3–5] and to the study [6], which
employed algorithms based on nite dierences and spectral methods, respectively, for the
simulation of non-Boussinesq natural convection conned ows. A weakly compressible
approximation was used in further simulations of numerous non-Boussinesq natural con-
vection ows [7–14] to address various problems in physics and computational science. The
common drawback of all weakly compressible approximations is that the results obtained
for ows with a dominant hydrodynamic part (e.g., high velocity ows) maybe inaccurate,
which means that a fully compressible approach must be used.
Fully compressible approximations for low-Mach-number ows typically employ either
density-based or pressure-based solvers. Although density-based formulations were tradi-
tionally utilized for simulating high-speed compressible ows (neglecting viscous eects),
eorts have been made to extend the applicability of these formulations to congurations
in which viscous eects play a signicant role. These congurations include laminar nat-
ural convection compressible ows [7, 15, 16], low-Mach-number injection ows [11], and
natural convection ows in a laminar-turbulent transition regime [17]. A key feature of
density-based solvers is that continuity, momentum, and all other transport equations are
rst solved in a fully coupled manner, and the pressure eld is then derived from an equa-
tion of state. As a result, the coupled operator is typically ill conditioned when applied
to low-Mach-number ows, whose treatment requires sophisticated numerical techniques,
such as preconditioning and dual-time stepping. When the simulations are performed on
high-resolution grids or applied to 3D ows, density-based solvers typically suer from
a slow convergence rate and produce results with low accuracy. Pressure-based solvers
oer a better alternative for the simulation of fully compressible natural convection ows.
In pressure-based solvers, the calculation of the pressure is separated from that of the
velocity eld. In the rst step, the pressure eld is taken from the previous time step
and the momentum equations are solved, thereby yielding the velocity eld, which does
not satisfy the continuity equation. In the second stage, a pressure-correction equation
(which is derived from the continuity equation) is solved. Thereafter, the pressure and the
velocity components are corrected to meet the continuity constraint. Finally, the solution
of the pressure-correction equation is followed by the solution of the energy equation and
updating of the density and the viscosity elds by relying on the equations of state and
Sutherland equations. Typically, outer iterations are needed to achieve convergence of the
velocity, the temperature, and the density elds and to satisfy the continuity constraint.
Despite the fact that the pressure-based approach (employing either fractional-step or
SIMPLE methods) is well documented in the literature [18, 19], numerical simulations
of fully compressible natural convection in enclosures performed by utilizing the above
algorithm are quite scarce, with the prominent exceptions of the studies of Sewall and
Tafti [20] and Barrios-Pina et al. [21]. In [20], a variable property algorithm based on the
fractional-step method was developed to simulate transient natural convection ows in the
presence of large temperature dierences without using the low-Mach-number assumption.
The algorithm was then adopted in [21] to conduct a thermodynamic analysis aimed at
determining the contribution of each term in the total energy equation.
In practice, most realistic engineering applications involving conned natural convec-
tion ow are characterized by complex geometries, which can signicantly challenge the
accuracy of the calculated ow properties. For these applications, the immersed boundary
method (IBM), initially developed by Peskin [22], may be used as a convenient tool to
simulate the ow in the presence of complex boundaries while maintaining an acceptable
level of accuracy. The IBM can simulate ow around complex, movable, and deformable
boundaries. The simulations take advantage of solvers utilizing compact and simple sten-
cils of discretized dierential operators that can be eciently employed on structured grids
or solvers based on the Lattice Boltzmann method (see e.g [23]). No-slip boundary condi-
tions and prescribed values of the temperature (or heat ux) on each immersed boundary
are enforced by introducing forces and heat uxes as additional unknowns in the problem.
Closure of the overall system is achieved by including additional equations in the form of
kinematic constraints for all the unknowns.
In the past decade, the IBM has been widely utilized for investigating natural con-
vection within enclosures with embedded discrete thermally active sources (or sinks) of
various geometries. Interest in this eld was generated by its relevance to a broad spectrum
of engineering applications based on gas-solid heat exchangers and a need to investigate
the instability characteristics of highly separated conned ows. Worth mentioning in this
context are the works [24–27] and the studies [28–32] that addressed natural convection
conned ows in the presence of bodies of complex two- and three-dimensional geometries,
Studies utilizing the IBM for the analysis of thermal compressible ows are relatively
scarce. Most of the works in this eld either address high-Mach-number compressible ows,
focusing on transonic/supersonic transitions, or compare the characteristics of subsonic
ows with those of supersonic ows. For high-Mach-number ows, the impacts of viscosity
and the thermal behavior of the ow are negligible compared to the compressibility eects,
and thus both phenomena have typically been neglected when simulating high-Mach-
number ows. In contrast, in the simulation of low-Mach-number thermally driven ows –
the focus of the current study – both eects play a signicant role and should be carefully
It should be noted that an accurate implementation of IBM forcing in low-Mach-
number compressible ows is still the subject of active research. In a recently developed
IBM scheme [33], Riahi et al. introduced a novel pressure-based correction of IBM forcing
(in addition to the classical one based on the time derivative of velocity)∗and applied
it to the analysis of three-dimensional low- and high-Mach-number pressure driven ows.
Comparison of the obtained results with the corresponding data obtained by body-tted
numerical simulations revealed that pressure correction of IBM forcing signicantly im-
proved the accuracy of the IBM procedure for low-Mach-number ows. An additional con-
tribution to the IBM for the simulation of compressible thermally driven conned ows
was made by Kumar and Natarajan [33], who developed a diuse immersed boundary
method for thermally driven non-Boussinesq ows. However, the method relies on a low-
Mach-number approximation that treats the governing equations as quasi-incompressible
and therefore cannot be considered as a fully compressible approach.
The present study thus aims to develop and extensively verify a general transient
pressure-based formulation for the simulation of thermally driven non-Boussinesq ows
within complex geometries. The simulations are performed by utilizing second-order back-
ward nite dierence and standard second-order nite volume methods [35] for the tem-
poral and spatial discretizations, respectively. To the best of our knowledge, the current
work is a rst of its kind in which the pressure-based correction IBM, originally developed
for compressible pressure driven ows [33], has been successfully incorporated into a fully
compressible pressure-based natural convection solver utilizing a semi-implicit fractional-
step algorithm.
2 Theoretical background
2.1 Physical model
Natural non-Boussinesq convection within a rectangular cavity of dimensions L×Hlled
with an ideal gas is considered. In the presence of a gravitational eld −gˆ
j, the ow is
driven by the temperature dierence between the cold boundaries of the cavity and the
hot surface of a cylinder located at the geometrical center of the cavity, as shown in Fig.1.
The left (LW), right (RW), top(TW) and bottom (BW) walls of the cavity are maintained
at a constant cold temperature Tc, while the surface of the cylinder (CL) is maintained
at a constant hot temperature Th. For the most general case in which the temperature
dierence Th−Tcis not restricted to small values, the natural convection ow generated
within the cavity is fully compressible.
The ow is governed by a set of non-dimensional continuity, momentum, and energy
∗We note in passing that pressure correction of the direct forcing IBM must not be confused with a
pressure-correction equation of fractional-step- and the SIMPLE-related methods.
Figure 1: Geometry and boundary conditions for a cold cavity with a hot cylinder at the center.
u) = 0, (2.1)
uCpT) = ∇ ·(k∇T) + κ−1
u(u,v),p,T,ρ,µ,k,Cp,κ,Cρare the non-dimensional velocity, pressure, temperature,
density, dynamic viscosity, thermal conductivity, specic heat capacity at constant pres-
sure, ratio of specic heat capacities and density coecient, respectively, and Iis the
unity matrix. The impact of the immersed cylinder on the surrounding ow is addressed
by introducing the source terms
fΓ,qΓcorresponding to the volumetric force and heat
source, respectively. Note that in accordance with the IBM formalism as detailed in
section 2.2, Eqs. (2.1-2.4) are solved on the whole computational domain including the
cylinder interior.
Eqs. (2.1-2.4) are rendered dimensionless using the characteristic scales L0for length,
0/α0for time, α0/L0for velocity, p0for pressure, ρ0for density, T0for temperature, Cp0
for specic heat capacity at a constant pressure, µ0for dynamic viscosity, k0for thermal
conductivity and α0is thermal diusivity. The dimensionless groups governing the ow
under consideration are the Rayleigh number (Ra), the Prandtl number (Pr), the Mach
number (M0)and the normalized temperature dierence parameter (ε)dened as:
Ra =ρ0g(Th−Tc)L3
,Pr =µ0
Note that despite the fact that the value Th−Tcis included in the denition of Ra
and ε, the two parameters are independent of each other and can be changed separately.
The Sutherland law determining the dependence of both the viscosity and the thermal
conductivity values on temperature is applied, giving:
where Cµ,Ckare the Sutherland non-dimensional temperatures for viscosity and thermal
conductivity included in the governing Eqs. (2.2-2.3).
2.2 IBM for thermal compressible ow
To enforce the kinematic constraints of no-slip and of the prescribed temperature value
on the surface of the embedded cylinder, the IBM is employed when solving momentum
and energy equations. To impose the kinematic constraints, the IBM utilizes regular
(typically Cartesian) Eulerian grids by introducing a set of additional volumetric forces
and heat sources on the surface of the immersed body, which is described by a discrete
set of Lagrangian points. The current study extends the pressure-corrected direct forcing
IBM presented in [33] – originally developed for compressible isothermal ows – to the
simulation of non-Boussinesq natural convection ows. Similarly to the conventional direct
forcing IBM, the formulation developed here incorporates interpolation and regularization
operators facilitating an exchange of data between Eulerian and Lagrangian grids and
a procedure enabling the direct calculation of the Lagrangian volumetric heat and force
2.2.1 Interpolation
The interpolation step transfers quantities (e.g., (ρ
u),(ρCpT),(∇p)) from the Eulerian
mesh to the points determining the Lagrangian surface ∂B. The procedure employs an
interpolation operator Idened as:
where uE(ρ
u,ρCpT),vE(∇p,0)are physical properties calculated on a Eulerian Nx×Ny
grid, while
VL(∇PL,0)are the corresponding counterparts calculated
on a Lagrangian grid, and δis a discrete Dirac delta function dened below. Note that we
used a bold vector notation to distinguish between the interpolated terms yielded by the
solution of momentum or energy equation. From the dierent smeared approximations
of the delta functions, we chose the function described by Roma et al. [36], which was
specically designed for use on staggered grids, where even/odd de-coupling does not
occur. This approximation is expressed as:
δ(r) =
+1for |r| ≤ 0.5∆r,
+1for 0.5∆r≤ |r| ≤ 1.5∆r,
where ∆ris the cell width in the rdirection. It is noteworthy that the chosen discrete
delta function is supported over three cells. No signicant dierences in the results are to
be expected if other discrete delta functions are used [37].
2.2.2 Direct forcing
After completing the interpolation step, the Lagrangian volumetric forces and heat sources
are calculated as suggested in [33]:
VL)·ˆnns (2.10)
FL,QL)is a direct forcing term consisting of the volumetric force and heat sources,
respectively, the superscript ddenotes the kinematic constraints imposed on the surface of
the immersed body and ˆnns is a unit vector in the direction perpendicular to the surface
of the immersed body.
2.2.3 Regularization
The regularization step smears the values of the volumetric sources calculated at the
Lagrangian points back to the Eulerian grid. The procedure is implemented by utilizing
the same delta functions as in the interpolation step. The values of the volumetric force
and heat source terms evaluated on the Eulerian mesh by utilizing the regularization
operator Rare given by:
fΓ(x,y),qΓ(x,y)) is the Eulerian direct forcing term consisting of the volu-
metric force and heat sources.
3 Numerical methodology
The ow is governed by a system of continuity, momentum, and energy equations, which
are solved numerically. This study utilizes the fractional-step method, which separates
the calculation of the pressure eld from the calculation of the velocity eld at each time
instance. When utilized in the context of the Boussinesq approximation (i.e., the ow is
assumed to be incompressible), the method consists of a number of basic steps employed
at each time instance: (i) the predictor step aimed at the estimation of the non-solenoidal
velocity eld by utilizing the pressure eld from the previous time step; (ii) the corrector
step aimed to obtain the pressure correction for the current time step; (iii) the projection
step using the pressure-correction values to update the pressure eld and to project the
velocity eld on the divergence free subspace; and (iv) solution of the energy equation.
For non-Boussinesq (i.e., fully compressible) natural convection ows, the procedure is
more complicated [18], since in that case the pressure constitutes a thermodynamic prop-
erty rather than a simplied hydrodynamic property. An additional diculty is that for
compressible ows under realistic conditions, density, viscosity, and thermal conductivity
are not constant. The ow around a body of complex geometry is resolved by employing
the direct forcing IBM applied to compressible ow, and the velocity, pressure, and tem-
perature kinematic constraints to be enforced on the surface of the immersed body. The
methodology and the numerical formulation incorporating a pressure-corrected IBM into
a semi-implicit fractional-step method developed for the simulation of natural convection
ow around complex geometries is presented below.
3.1 Computational procedure
In this section the details on implementation of fractional step method to satisfy the
continuity equation and the IBM formalism to satisfy the kinetic constraints on the syrface
of the immersed body are given. Note that the second order backward nite dierence
and standard second order nite volume method [35] was used for the temporal and the
spatial discretizations, respectively.
3.1.1 Predictor step
u∗is the predicted velocity that has to be calculated on the basis of the velocity eld
un, known from the previous time step, and on the basis of the density, dynamic viscosity
and pressure elds known from the previous outer iteration m−1. The purpose of the
outer iteration is to impose the continuity constraint on the predicted velocity eld
the end of the iteration process. The term N(ρm−1,
un)appearing in the right hand side of
Eq. (3.1) denotes the non-linear convective terms, so that Eq. (3.1) is solved sequentially
for each velocity component. For this reason the second term of the left hand side of
Eq.(3.1) contains both terms that are treated implicitly (denoted by * superscript) and
explicitly (i.e. taken from the nth time step). This notation was introduced to distinguish
between velocity components corresponding to dierent directions. That is, the velocity
components coinciding with the direction of the corresponding momentum equation are
treated implicitly while the velocity components perpendicular to this direction are taken
from the previous time step. Note also that for incompressible and Boussinesq ows
with constant viscosity, the second term may be simplied to Pr∇2(
u∗)and treated fully
3.1.2 First momentum Corrector
After obtaining the predicted velocity, the pressure-correction equation, which has been
derived from the continuity equation, is solved as follows:
2∆tCρp′−∇· 2∆t
∇p′−∇· (Cρp′
u∗) = −∆˙
where ∆˙
m∗is the mass ow imbalance generated because the predicted velocity does not
necessarily satisfy the continuity equation:
Solution of the pressure-correction equation yields the pressure correction eld p′used for
calculation of the intermediate pressure:
which is subsequently used as a predictor after the presence of the immersed body has
been taken into account.
3.1.3 Application of the IBM for velocity to enforce the non-slip kinematic constraint on
the surface of the immersed body
After acquiring the intermediate pressure, the IBM for velocity is implemented via Eqs.
(2.8-2.11). Note that the term
fΓis not recalculated during the outer iteration and is
determined only once at the beginning of the time step:
L)·ˆnns E
As a result, the calculated volumetric Eulerian force exerted by the surface of the immersed
body is added to the right-hand side of the momentum equation.
3.1.4 Solution of the momentum equation with the impact of the immersed body
After the Eulerian force has been calculated, the momentum equation with an updated
right-hand side is solved again to determine the new velocity eld that takes into account
the impact of the immersed body:
Similarly to Eq. (3.1), the above equation is solved sequentially for each velocity compo-
nent. For this reason the second term of the left hand side of Eq.(3.6) contains both terms
that are treated implicitly (i.e. the current iteration terms denoted by msuperscript) and
explicitly (i.e. taken from nth time step). This notation was introduced to distinguish
between velocity components corresponding to dierent directions. That is, the veloc-
ity components coinciding with the direction of the corresponding momentum equation
are taken from the current iteration and treated implicitly while the velocity components
perpendicular to this direction are taken from the previous time step.
3.1.5 Second momentum corrector
At this stage, the pressure-correction equation with the updated right-hand side is solved:
2∆tCρp′′ −∇·2∆t
∇p′′−∇· (Cρp′′
um) = −∆˙
where ∆˙
mmis a mass ow imbalance that arises because the predicted velocity still does
not necessarily satisfy the continuity equation, where ∆˙
mmis given as:
Next, the new pressure and the intermediate density elds are calculated by:
3.1.6 Solution of the energy equation without the impact of the immersed body
We next solve the energy equation:
umCpT∗)−∇· (km−1∇T∗) =
Note, the T∗eld constitutes the intermediate temperature, which was obtained without
considering the presence of the immersed body.
3.1.7 Application of the IBM to enforce a given temperature on the surface of the im-
mersed body
At each time step in the rst correction iteration, the IBM for temperature is implemented
via Eqs. (2.8-2.11) after acquiring the intermediate temperature. Note that the term qΓis
not recalculated during the outer iteration and is determined only once at the beginning
of the time step:
As a result, the calculated volumetric Eulerian heat source exerted by the surface of the
immersed body is added to the right-hand side of the energy equation.
3.1.8 Solution of the energy equation with the impact of the immersed body
We next solve the energy equation:
umCpTm)−∇· (km−1∇Tm) =
3.1.9 Updating the thermophysical properties
After the energy equation has been solved, the viscosity µm
ij , the thermal conductivity
ij , the coecient Cρ, and the density ρm
ij are updated by using the Sutherland equations
(2.6),(2.7) and the equation of state (2.4).
3.1.10 The outer iteration loop
The general formulation of the fully compressible semi-implicit fractional-step method
with the embedded IBM governed by Eqs. (3.2),(3.4-3.12) constitutes the outer iteration
loop that is employed at each time step. At the end of the iteration, after updating the
thermophysical properties, the solver performs a mass conservation check based on the
value of the L-innity norm calculated for the dierence between the values of p,ρand
elds in the current and previous iterations. The outer iteration terminates after the value
of 10−6of the L-innity norm has been reached for each eld and the simulation proceeds to
the next time step by assigning
um,pn+1=pm,Tn+1=Tmand ρn+1=ρm. To sum up,
we present a owchart (see Fig. 2 summarizing the sequence of stepst hat must be taken
to calculate all the ow elds at a given time step with and without an immersed body.
It can be seen that the ow simulation involving an immersed body requires additional
steps to explicitly calculate the volumetric forces and heat uxes necessary to satisfy the
kinematic constraints on the surface of the body.
Figure 2: Block diagram of the algorithmic sequence required to calculate all the ow elds at a given time
step: to the left – without immersed body; to the right – with immersed body.
Predictor step - ݑ
Momentum corrector - ǡݑǡߩProjection step -
Solution of the energy equation - ܶ
Updating thermophysical properties -
Checking continuity constraint
Proceed to the next time step
Continuity conserved
does not
First momentum corrector -ᇱǡכIBM for velocity - ݂ԦSolution of momentum equation with the impact of the immersed body -ݑUpdating thermophysical properties - ߤǡ݇ǡߩChecking continuity constraint
Proceed to the next time step
Continuity conserved
does not
Second momentum corrector -ᇳǡǡߩכSolution of energy equation without the impact of the immersed body -ܶכIBM for temperature -ݍSolution of energy equation with the impact of the immersed body - ܶ
3.2 Solution of the discretized momentum, pressure-correction and energy equa-
As a result of the non-constant density ρand the dependence of the dynamic viscosity µ
and the conduction coecient kon the temperature typical of the fully compressible ow,
the discretized momentum, pressure correction, and energy equations contain time varying
coecients. Therefore, the strategies for reaching an ecient solution of the discretized
equations should be chosen with care. In the current study, two dierent strategies, one
based on an iterative solution and the other based on a direct solution of the discretized
governing equations, were investigated.
The iterative solution utilized the bi-conjugate gradient stabilized (BiCGstab) method
[38]. The key idea was to treat only the linear terms of the Helmholtz operator implicitly,
while all the non-linear terms were taken either from the previous time step or from the
previous iteration. Exploiting the fact that for a 2D problem the Helmholtz operator is
built up of 5 non-zero diagonals, we eciently implemented its product by an arbitrary
vector, constituting a major part of the BiCGstab algorithm. However, as is the case for
many other iterative methods, the BiCGstab converges up to a certain accuracy, after
which it saturates, so that no further decrease of residuals is possible. This limitation can
slow down the convergence of the outer iterations, and we therefore sought to eliminate
this possible problem by replacing the BiCGstab by the direct method proposed by Lynch
et al. [39]. This approach, designated tensor product factorization (TPF) in [39], is based
on eigenvalue decompositions (EVDs) of one-dimensional derivative operators and can be
applied for the direct inverse of the Helmholtz operators in rectangular domains. Vitoshkin
& Gelfgat [40] demonstrated the computational eciency of the TPF method for 2D and
3D natural convection benchmark problems by applying the Boussinesq approximation
and showed that for ne grids and large Grashof numbers it yields computationally faster
time steps than BiCGstab or multigrid iterations. Additionally, it can easily be observed
that after application of the TPF solver the relative residual remains below 10−12.
Unfortunately, the EVD method cannot be implemented directly into the correction
equations (3.2) and (3.7), because the coecients of the Helmholtz-like dierential operator
are not constants. Thus, to apply the EVD solver, all the governing equations should be
reformulated. For example, Eq. (3.7) is modied into:
where Cρ=(Cρ)min +(Cρ)ma x/2 and mis the number of the outer iteration. In this
formulation, the left-hand side of Eq. (3.13) is a Helmholtz-like elliptic operator with
constant coecients, whose discretized inverse can be expressed via a one-dimensional
EVD of the second-derivative operators [41].
4 Verication study
The methodology described in Section 2 was veried by applying it to the solution of two
benchmark problems—simulating incompressible and thermally driven compressible ows,
i.e., ows driven by two dierent mechanisms. For the verication, we chose to simulate
the isothermal ow in a long narrow rectangular channel driven by a pressure gradient
and the natural convection ow within a dierentially heated square cavity driven by a
temperature gradient. The results obtained for the two congurations were compared with
the data available in the literature.
4.1 Test case I - Isothermal compressible ow in a narrow channel
4.1.1 Test case overview
This test case examines the capability to carefully address incompressible isothermal ow
by applying the currently developed methodology for compressible ow. The ow within
an extended channel was selected as a computational testbed to minimize the eect of the
outow boundary on the upstream recirculation zones. The schematics of the geometric
properties and the boundary conditions of the conguration under consideration are shown
in Fig. 3.
The uid enters the domain at the upper half of the left side, proceeds through the
channel, and exits at the right side. No-slip velocity and zero-gradient pressure boundary
conditions are applied on all the rigid walls. At the inlet, the vertical component of the
velocity is equal to zero; the zero-gradient boundary condition is applied to the pressure
Figure 3: Schematics of the ow within an extended channel.
eld; and a parabolic distribution with maximal and average values equal to umax =1.5
and uavg =1, respectively, is assigned to the horizontal velocity component. At the outlet,
zero values are set for the normal stress, τx x =−p+µ∂u/∂x, and for the vertical velocity
component. Considering that the pressure eld at the outlet is known and is set to zero,
the gradient of the horizontal velocity component is also equal to zero.
The non-dimensional continuity and momentum equations and the equation of state
governing the ow under consideration are:
u) = 0, (4.1)
u) = −∇p+1
Re ∇·µ(∇
where the Reynolds number, Re is based on the average velocity.
4.1.2 Test results and comparison with a benchmark in the literature
The results obtained by the developed methodology were compared to the corresponding
data provided in [42] for the value of Re =800. Figs. 4-7 present a comparison between the
contours of the currently obtained ow elds and the corresponding results reported in [42],
which serves as the benchmark for this comparison. As can be seen from Figs. 4-7, the
values of all the currently obtained quantities lie in the same range as the corresponding
data reported in [42]. The distribution of the streamlines, shown in Fig. 4, along with
the distributions of the vorticity and velocity magnitude elds, shown in Figs. 6 and 7,
respectively, conrm the presence of staggered low-speed vortices adjacent to the upper
and lower walls. The pressure eld, shown in Fig. 5, conrms the presence of a “pressure
pocket” adjacent to the bottom wall between x=6and x=7. As can be seen clearly from
Fig. 8, the density variations over the entire domain are insignicant, and the ow can
safely be considered as incompressible.
Figure 4: Comparison between the distributions of the streamlines in (a) the current study and (b) the benchmark
study [42]. Grid 3000×100.
Figure 5: Comparison between the pressure distribution in (a) the current study and (b) the benchmark study
[42]. Grid 3000×100.
Figure 6: Comparison between the distribution of vorticity in (a) the current study and (b) the benchmark
study [42]. Grid 3000×100.
Figure 7: Comparison between the distribution of the velocity magnitude in (a) the current study and (b) the
benchmark study [42]. Grid 3000×100.
Figure 8: Distribution of the density eld. Grid 3000×100.
Figs. 9-10 present a comparison between the currently obtained ow characteristics
and the corresponding data reported in [42]. A comparison of the pressure and the shear
stress distributions along the upper and lower walls of the channel is shown in Fig. 9a
and 9b, respectively. A comparison of the currently obtained and benchmark distributions
of the horizontal and vertical velocity components, pressure, vorticity, horizontal velocity
gradient and normal stress along two vertical lines passing through x=7and x=5is
presented in Figure 10. All the gures demonstrate the same trends for all the ow
characteristics obtained in the current and benchmark studies.
Figs. 9a, 10c and 10f compare the corresponding pressure and the normal stress elds
and show excellent agreement between the current and the benchmark values for the two
quantities. Acceptable agreement was also obtained for all the other ow characteristics,
including the values of both velocity components, the vorticity, and the gradient of the
horizontal component of the velocity. An absolute deviation between the values of the
above characteristics is limited to 1percent for the entire range of y coordinates with the
exception of the regions in which extremums of the vertical velocity and the gradient of
the horizontal velocity are observed. In these regions, the absolute deviation between the
current and the corresponding benchmark results is limited to 8percent.
In summary, the acceptable agreement between the currently obtained and benchmark
results for the entire range of ow characteristics veries the suitability of our numerical
methodology for the simulation of almost incompressible ows.
4.2 Test case II - Natural convection ow in a dierentially heated cavity
4.2.1 Test case overview
The results presented in this section were obtained by applying the developed method-
ology to the simulation of compressible natural convection ow in a dierentially heated
square cavity. The ow is driven by the temperature dierence between two vertical walls
under the inuence of gravity. The obtained ow characteristics were compared with
the corresponding independently obtained data [4] that served as the benchmark for this
part of our study. The governing equations, characteristic values, and non-dimensional
parameters of this ow conguration are presented in Section 2.
The hot and cold walls of the cavity are maintained at constant temperature values
of Th=1+εand Th=1−ε, respectively, and are thermally insulated. No-slip and zero
gradient boundary conditions are applied for all the velocity components and the pres-
sure, respectively, on all the cavity walls. The schematics summarizing the geometry and
boundary conditions of this ow conguration are shown in Fig. 11.
4.2.2 Test results and comparison with a benchmark in the literature
The results obtained in the current study were compared with the corresponding bench-
mark data reported in [4]. The comparison focused on the distributions of the velocity
and the temperature. The data was compared in terms of the values of the vertical tem-
perature stratication parameter θAand the Nusselt number determined by Eqs. (4.4)
Figure 9: Distribution of (a) pressure and (b) shear stress elds along upper and lower channel walls. Grid
Figure 10: Distribution of: (a) horizontal velocity component; (b) vertical velocity component; (c) pressure;
(d) vorticity; (e) horizontal velocity gradient; and (f) normal stress along vertical lines passing through the x=7
and x=15 coordinates. Grid 3000×100.
Figure 11: Geometry and boundary conditions of the dierentially heated cavity.
and (4.5), respectively. In accordance with [4], θAwas calculated by:
dy x=1
where AR =H/Lis the aspect ratio.
The average Nusselt number was determined as:
Nu =−1
where ∇Tis the temperature gradient at the wall, and Sis the non-dimensional surface
In the current study, the Nusselt number was calculated for the hot wall of the cavity.
The calculations were performed for the range of Ra ∈[103,107]and ε=0.005,0.2,0.4 and
0.6. In all the simulations, the value of the aspect ratio AR=H/Lwas equal to unity. Figs.
12-13 summarize the comparison between our calculations and the benchmark values [4]
for the velocity and the temperature elds obtained for the value of Ra=105and the entire
range of εvalues. Good agreement between the two sets of values can clearly be seen for
all the simulations. Note that the temperature and velocity distributions vary with ε. For
the lowest value of ε, i.e., ε=0.005, both the velocity and the temperature distributions
are almost skew-symmetric relative to the cavity center and resemble the distributions
Figure 12: Comparison of the contours for ε=0.005,Ra=105: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
Figure 13: Comparison of the contours for ε=0.2,Ra =105: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
typical of incompressible ows obtained by applying the Boussinesq approximation. With
increasing εvalues, the ow loses its skew-symmetry as a result of the dependence of its
conductivity kand viscosity µon temperature.
A similar comparison was made for εconstant at its highest value, ε=0.6, and varying
Ra over the entire range of Ra values, as shown in Figures 14-19. Good agreement was
obtained between our values and the benchmark results [4]. It can clearly be seen that
under these conditions the ow is characterized by a breaking of the skew-symmetry
relative to the cavity center, even at the lowest value of R a =103. For the higher Ra ow
regimes dominated by convective heat transfer, the breaking of skew-symmetry becomes
Figure 14: Comparison of the contours for ε=0.4,Ra =105: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
Figure 15: Comparison of the contours for ε=0.6,Ra =105: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
Figure 16: Comparison of the contours for ε=0.6,Ra =103: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
much more pronounced. The breaking of skew-symmetry may be conrmed by examining
the dierences in the thickness of the boundary layers developing near the hot and cold
walls. In fact, the dynamic viscosity of the ideal gas increases with temperature, leading
to a thickening of the boundary layer in the vicinity of the hot wall of the cylinder. At the
same time, the thickness of the boundary layer decreases close to the cold vertical wall,
which, again, is a consequence of a local decrease in viscosity values as a result of the lower
temperatures prevailing in this region.
Figs. 20 and 21 present a comparison of our results with the benchmark values [4]
for the vertical temperature stratication parameter and the averaged Nusselt number
Figure 17: Comparison of the contours for ε=0.6,Ra =104: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
Figure 18: Comparison of the contours for ε=0.6,Ra =105: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
Figure 19: Comparison of the contours for ε=0.6,Ra =106: (a) velocity, current study, (b) velocity, benchmark
study [4], (c) temperature current study, (d) temperature, benchmark study [4].
versus Ra values. The simulations were performed on four dierent grids, namely, two
uniform grids of 100×100 and of 200×200 cells, and two non-uniform grids of 100×100
and of 200×200 cells stretched toward the cavity walls to accurately resolve the thinnest
boundary layers.
As can be seen from Fig. 20, there was good agreement between our results and the
benchmark results [4] for the vertical temperature stratication parameter. Of partic-
ular interest was the nding that stretching the grid toward the cavity boundaries was
not always the optimal way to increase the accuracy of the results when the values for
comparison were acquired close to the cavity center, as can be seen from the θAvalues
obtained on a stretched grid with ε=0.005 and Ra=106or a stretched grid with ε=0.2 and
Ra =103. At the same time, asymptotic convergence of the θAvalues with increasing the
grid resolution was clearly observed for both stretched and uniform grids. Fig. 21 demon-
strates that the Nu values obtained on the uniform and stretched grids built of 200×200
and 100×100 cells, respectively, showed better agreement with the benchmark results
than the corresponding Nu values obtained on the uniform grid built of 100×100 cells,
especially for high Rayleigh numbers. Thus, stretching the grid toward the cavity bound-
aries is the condition of choice when analyzing characteristics based on the temperature
gradients at the cavity boundaries.
In summary, the acceptable agreement between our results and those of the benchmark
study [4] for the entire range of operating conditions and ow characteristics successfully
veries the suitability of our numerical methodology for the simulation of compressible
Figure 20: Vertical temperature stratication parameter vs. Rayleigh number: comparison of our results with
benchmark values [4].
natural convection conned ows.
5 Results and discussion
This section presents the results of a parametric study performed to simulate the natural
convection ow developing from the hot surface of a cylinder placed within a square cold
cavity (see Fig. 1). The results were obtained for a wide range of governing parameters:
Ra∈{103,104, 105,106},ε∈{0.005,0.2,0.4,0.6}and R/L∈{0.1,0.2,0.3,0.4}. The simulations
were performed on two uniform grids having 100 or 200 nodes in each direction, with the
time steps of ∆t=10−7and ∆t=10−8, respectively.
The obtained results illustrate changes of the velocity and temperature elds when
the temperature-dierence parameter εwas varied between 0.005 and 0.6 for the same Ra
and R/Lvalues. Additionally, the results were approximated by Nu−Ra power law ts
calculated by employing the least-squares technique. The Nusselt number Nu was calcu-
lated as is detailed below in subsection 5.2. The Nusselt numbers obtained for the whole
range of Ra and for the lowest value of the temperature-dierence parameter (ε=0.005)
were compared with the corresponding results available in literature [28], [43] calculated
by employing the Boussinesq approximation.
Figure 21: Nusselt number vs. Rayleigh number: comparison of our results with benchmark values [4].
5.1 Qualitative observations
Figs. 22-37 summarize the results in terms of the spatial distribution of the ow path lines
and the temperature elds obtained for the values of ε=0.005 and ε=0.6 for the entire range
of Ra and R/Lvalues on a uniform grid of 200 cells in each direction. The purpose of this
part of the study was to investigate dierences between the ow characteristics typical
of the lowest (ε=0.005) and the highest (ε=0.6) values of the temperature-dierence
Figs. 22-23 and Figs. 30-31 show that for the lowest Rayleigh number, Ra=103, there
were no signicant dierences between the spatial distributions of the path lines and the
temperature elds, respectively, obtained for the lowest (ε=0.005) and the highest (ε=0.6)
values of the temperature-dierence parameter, regardless of the cylinder’s diameter. The
congurations for this Ra number did not contain secondary convective cells, and the
temperature distribution was close to linear along the radial direction from the hot cylinder
to the cold cavity walls, as may be expected for systems in which conduction constitutes the
major heat transfer mechanism. As the Ra value increased, the dierences became more
visible. Figs. 24-25 and Figs. 32-33, in which the path line and temperature distributions,
respectively, are shown for Ra =104, showed that the dierences in path lines remained
insignicant (with no secondary convective cells), but the temperature distribution for the
highest value of εwas slightly shifted upwards compared to that observed for the lowest ε
value. As for the two highest values of Ra =105and Ra =106, the dierences between the
path line and temperature distributions corresponding to the congurations characterized
by the two limit values of εcould be clearly recognized, as may be expected for systems in
Figure 22: Distribution of the path lines for Ra =103,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 23: Distribution of the path lines for Ra =103,ε=0.6 and 0.1 ≤R/L≤0.4.
which convection constitutes the major heat transfer mechanism (see Figs. 26-27 and Figs.
34-35). In this range of Ra values, secondary convective cells were sometimes generated,
while the temperature distribution along the radial direction was non-linear with clearly
recognizable single or multiple thermal plumes rising up from the top of the cylinder.
In summary, secondary convective cells never appeared for Ra≤104and/or R/L≤0.1.
For R/L=0.2, secondary convective cells appeared only for the value of ε=0.6 and for the
two values of R a =105and Ra =106. For R/L=0.3, secondary convective cells appeared
for the two values of ε=0.005 and ε=0.6 and the two values of Ra =105and Ra =106.
For R/L=0.4, secondary convective cells appeared for the two values of εand only for
Ra =106. In addition, it was observed that the ows characterized by ε=0.6 may contain
more secondary convective cells than their counterparts characterized by ε=0.005 - a
trend that never occurs the other way around. It thus appears that for ow separation
to occur high Ra,R/Land εvalues are required simultaneously. It is also noteworthy
that an increasing number of secondary convective cells with an increasing value of the Ra
number followed the same trend as that observed in Rayleigh-Bénard convection with an
increasing aspect ratio, i.e., for high values of R/L, the cylinder curvature can be locally
neglected, and the ow in the top region of the cavity resembles the Rayleigh-Bénard
Figure 24: Distribution of the path lines for Ra =104,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 25: Distribution of the path lines for Ra =104,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 26: Distribution of the path lines for Ra =105,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 27: Distribution of the path lines for Ra =105,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 28: Distribution of the path lines for Ra =106,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 29: Distribution of the path lines for Ra =106,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 30: Distribution of the path lines for Ra =103,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 31: Distribution of the path lines for Ra =103,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 32: Distribution of the path lines for Ra =104,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 33: Distribution of the path lines for Ra =104,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 34: Distribution of the path lines for Ra =105,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 35: Distribution of the path lines for Ra =105,ε=0.6 and 0.1 ≤R/L≤0.4.
Figure 36: Distribution of the path lines for Ra =106,ε=0.005 and 0.1 ≤R/L≤0.4.
Figure 37: Distribution of the path lines for Ra =106,ε=0.6 and 0.1 ≤R/L≤0.4.
5.2 Quantitative results and discussion
5.2.1 Calculation of the Nusselt number
In this section, the discussion is focused on the average Nucand Nuhvalues corresponding
to the Nusselt numbers calculated at the cold cavity walls and at the cylinder surface,
respectively. The calculation of Nucis based on the arithmetic average of the average
Nusselt numbers at every wall of the cavity, each obtained by the method presented in
subsection 4.2.2. The value of Nuhis obtained by taking into account the heat ux from
the cylinder’s surface. Therefore, the average hot Nusselt number, Nuhis:
The procedure of calculation of the Nuhvalue does not require explicit calculation of the
temperature gradients at the surface of the hot cylinder, rather it is an integral part of
the IBM. Note that TLis the temperature obtained at a specic Lagrangian point by
interpolation of the corresponding predicted (i.e., obtained by not taking into account the
existence of the immersed body) Eulerian temperature, which enables the calculation of
Nuhfor both transient and steady-state ows.
5.2.2 Comparison of the results of the lowest-temperature-dierence cases and previous
The results of the current study obtained for the value of ε=0.005 were compared with
those from corresponding studies of natural convection ow from a hot cylinder placed
Figure 38: Physical model of a hot cylinder inside a cold tube adapted from [28] and [43].
within a 3D cavity obtained by employing the Boussinesq approximation [28], [43] (see
Fig. 38). Interestingly, there is acceptable agreement between our 2D study and the 3D
results obtained in refs. [28], [43]in terms of Nuhand Nucvalues for the whole range of
Ra and R/L, as summarized in Tables 1-2. The maximal relative deviation between our
results and the results obtained in refs. [28], [43] was 19% and could be attributed to the
impact of the lateral walls in the 3D conguration suppressing the convective ow and
thus decreasing the total heat ux.
5.2.3 Analysis of the heat uxes in the ow domain
As mentioned in Section 2, , the walls of the cavity were maintained at a cold temperature
Tc=1−ε, while the walls of the cylinder that was placed in the center of the cavity were
maintained at a hot temperature Th=1+ε. Therefore, the direction of the heat ux was
from the cylinder surface toward the cavity surfaces. The value of heat ux at each cavity
surface will reect the characteristics of the specic ow regime and can thus be quantied
by the calculation of the average values of Nucnumbers for each wall. The Nucvalues
were calculated by the formulas given in subsection 4.2.2.
As may be expected from symmetry considerations, the Nu values obtained for the
left and right walls of the cavity were close to each other (see Tables 3-4) for the entire
range of Ra-R/Lvalues. At the same time, there were signicant dierences in the Nuc
values obtained for the bottom and top walls of the cavity (see Tables 5-6) for the entire
range of Ra-R/Lvalues, with the Nu values at the top always being higher than those
at the bottom. These dierences increased with increasing Ra values, and for Ra =106
could reach up to one order of magnitude. To summarize, as the convective heat transfer
became more pronounced with increasing Ra values, the top of the cavity started to play
a more dominant role in removing heat from the system, which was clearly reected in a
gradual increase in the corresponding Nu values.
Table 1: Comparison between the present and the previously published Nuhvalues averaged over the surface
of a hot cylinder placed within a cold cube for ε=0.005.
Non-Boussinesq Boussinesq Non-Boussinesq Boussinesq
R/L0.1 0.2
Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]
1046.4920 6.4880 6.2493 5.1990 5.1500 5.1184
10511.8700 11.6620 11.1380 7.7780 7.5800 7.2271
10618.1000 19.2500 18.3260 14.3500 13.3610 13.9370
R/L0.3 0.4
Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]
1046.2630 5.7304 5.8084 8.8840 8.5544 8.7030
1057.3740 6.5169 6.4790 9.1240 8.7643 8.7030
10613.3600 11.4010 11.2720 11.9200 10.8320 10.7160
Table 2: Comparison between the present and the previously published N ucvalues averaged over the surface
of a hot cylinder placed within a cold cube for ε=0.005.
Non-Boussinesq Boussinesq Non-Boussinesq Boussinesq
R/L0.1 0.2
Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]
1041.0345 1.0208 1.0201 1.6662 1.6188 1.6161
1051.9112 1.8360 1.8099 2.5649 2.3814 2.3766
1062.8683 3.0348 2.9945 4.6134 4.3677 4.3985
R/L0.3 0.4
Ra Present Ref. [28] Ref. [43] Present Ref. [28] Ref. [43]
1042.8655 2.9091 2.6216 5.4591 5.3928 5.1919
1053.3675 3.0702 2.9726 5.6192 5.5131 5.2651
1066.1671 5.3844 5.1956 7.2361 6.8313 6.6106
Table 3: Nucon the left wall.
ε0.005 0.2
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.9389 1.5939 2.8186 5.4620 0.9877 1.5958 2.7040 5.4354
1040.9287 1.6347 2.8549 5.4550 1.0373 1.6550 2.7334 5.4341
1051.4945 2.3019 3.1750 5.5250 1.7238 2.2965 3.0942 5.5877
1062.2876 4.5233 6.0957 7.5126 2.3220 3.7539 5.3523 7.3707
ε0.4 0.6
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.9671 1.5791 2.6765 5.3313 0.9276 1.5618 2.6305 5.1832
1041.0173 1.6197 2.6896 5.3356 0.9774 1.5729 2.6453 5.1905
1051.7228 2.3632 3.1533 5.5551 1.7032 2.3394 3.1851 5.4981
1062.3817 3.7652 6.3741 7.0895 2.3033 4.6269 6.1530 7.9808
Table 4: Nucon the right wall.
ε0.005 0.2
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.9392 1.5945 2.8197 5.4640 0.9840 1.5976 2.7048 5.4374
1040.9291 1.6354 2.8559 5.4570 1.0337 1.6583 2.7340 5.4361
1051.4950 2.3030 3.1760 5.5270 1.7275 2.2982 3.0925 5.5897
1062.8784 4.5249 6.0970 7.5153 2.3751 3.6198 5.5117 7.3734
ε0.4 0.6
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.9612 1.5818 2.6775 5.3333 0.9235 1.5625 2.6315 5.1851
1041.0119 1.6241 2.6905 5.3375 0.9750 1.0880 2.2705 5.1924
1051.7259 2.3645 3.1544 5.5571 1.7048 2.3403 3.1862 5.5001
1062.4127 3.8251 6.3767 7.0920 2.2621 4.6286 6.1556 7.9837
Table 5: Nucon the bottom wall.
ε0.005 0.2
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.8819 1.5548 2.7986 5.4534 0.9575 1.5546 2.6772 5.4229
1040.5861 1.3616 2.6723 5.3717 0.8554 1.3812 2.4982 5.3157
1050.2150 0.7787 1.9192 4.8322 0.2756 0.6655 1.7119 4.6725
1060.2358 0.7924 1.3864 7.5153 0.2299 0.3993 0.7655 3.1495
ε0.4 0.6
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1030.9308 1.5322 2.6421 5.3133 0.8861 1.4995 2.5864 5.1582
1040.8119 1.2999 2.3834 5.1648 0.7617 1.0880 2.2705 4.9589
1050.2707 0.6223 1.2821 4.2901 0.2845 0.3204 1.1078 3.9579
1060.2032 0.4767 0.6584 2.6102 0.1486 0.1582 0.4593 2.0442
Table 6: Nucon the top wall.
ε0.005 0.2
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1031.0040 1.6374 2.8412 5.4732 1.0177 1.6432 2.7330 5.4502
1041.6941 2.0329 3.0787 5.5527 1.2933 2.0418 3.0288 5.5709
1054.4403 1.8762 5.1999 6.5926 4.2639 4.9223 5.3219 6.9002
1065.4828 8.6132 11.0893 10.2210 7.2042 9.7895 9.3328 11.2149
ε0.4 0.6
R/L0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
1031.0012 1.6333 2.7134 5.3519 0.9687 1.6273 2.6768 5.2105
1041.3028 2.0593 3.0640 5.5315 1.2711 2.1806 3.0940 5.4540
1053.8549 4.5273 5.0862 7.0268 3.3704 3.9956 4.8861 6.9461
1066.9131 9.2124 10.5519 10.8403 6.3005 7.8221 9.9085 10.7687
5.2.4 Approximation to the Nu-Ra power law
The results of the average Nusselt number for each conguration can be approximated,
by employing the least squares technique, to obtain the Nu−Ra power law. Following the
dimensional analysis stemming from the boundary layer theory, the Nu−Ra relationship
obeys the following power law [44]:
Nu f=C(Gr·Pr)m=C(Ra)m,(5.2)
where Gr,Pr are the Grashof and Prandtl numbers and Ra=GrPr, and Cand mare specic
constants, whose values depend on the Ra number and on the geometric properties of the
system under consideration.
Figures 39-42 present the distribution of the Nu number as a function of the Ra
number for the entire range of: ε∈ {0.005,0.2,0.4,0.6},Ra ∈ {104,105,106}and R/L∈
{0.1,0.2, 0.3,0.4}. Note that the Nu values obtained for the lowest value of R a =103were
not taken into account because of the dominance of the conductive heat transfer mechanism
(see also Figs. 22-23 and Figs. 30-31).
All Nu values were approximated to the Nu-Ra power law as formulated by Eq. (5.2).
Generally, acceptable accuracy was obtained when approximating the Nu−Ra relationship
by the power law for the entire range of εand R/Lvalues. The results obtained for smaller
cylinders (R/L=0.1,0.2) exhibited more precise power law ts, for which the smallest value
of R2was equal to R2=0.975. Remarkably, for all congurations characterized by these
R/Lvalues, Nu was approximately proportional to Ra0.22 , which is very close to the results
reported with respect to laminar natural convection in spherical shells [45–47].
Congurations with larger cylinders (R/L=0.3,0.4) exhibited slightly less pronounced
power law ts for the Nu −Ra relationship characterized by smallest value of R2, i.e.,
R2=0.84. Note also that Nu was approximately proportional to Ra0.16 and to Ra0.06
for the R/L=0.3 and R/L=0.4 geometries, respectively. Such a considerable decrease
in the heat ux rate for geometries characterized by large cylinders compared to these
characterized by smaller cylinders can be attributed to the blocking eects of both the
cylinder and the cavity boundaries, which suppress the momentum of the convective ow
(see e.g. [28], [43]).
5.2.5 Multiple steady-state regimes
Further numerical analysis revealed that steady non-Boussinesq natural convection ows
can exhibit multiple steady-state regimes. In particular, two independent steady-state
branches were found for the values of ε=0.4,R/L={0.2, 0.4}and Ra =106as shown in
Figs. 43-46 The stability of the revealed regimes was veried by randomly perturbing all
the ow variables with values deviating by about 10% from the corresponding steady-state
values and verifying that the ow then converged to the previously observed steady state.
Remarkably, while the corresponding steady states diered in terms of the number and
the size of the convective cells hosted within the ow domain, they were characterized
by very close Nu values, averaged over the cylinder and the cavity boundaries. It can
Figure 39: Nusselt vs. Rayleigh numbers for R/L=0.1.
Figure 40: Nusselt vs. Rayleigh numbers for R/L=0.2.
Figure 41: Nusselt vs. Rayleigh numbers for R/L=0.3.
Figure 42: Nusselt vs. Rayleigh numbers for R/L=0.4.
Figure 43: Flow and temperature patterns corresponding to two dierent steady state branches obtained for
ε=0.4,Ra =106and R/L=0.2.
Figure 44: Flow and temperature patterns corresponding to two dierent steady state branches obtained for
ε=0.4,Ra =106and R/L=0.3.
thus be concluded that both steady-state regimes obtained for the same values of the
governing parameters and belonging to the dierent branches were still characterized by
the same averaged heat uxes at all the ow boundaries. In summary, acceptable agree-
ment was obtained between our results for the lowest temperature-dierence cases and
results in the literature that were computed by applying the Boussinesq approximation
for the entire range of operating conditions and ow characteristics; this agreement veries
the suitability of our numerical methodology with an incorporated IBM for simulation of
compressible natural convection conned ows with complex geometry. The results of the
high-temperature-gradient cases showed good agreement with the boundary layer theory.
In addition, multiple congurations of the steady-state ow were discovered.
6 Conclusions
In the present work, a pressure-based solver for the simulation of the thermal compress-
ible natural convection non-Boussinesq ow of an ideal gas was developed. This solver
utilizes a second-order backward scheme and standard second-order nite volume method
for temporal and spatial discretizations, respectively. The novel pressure-corrected direct
Figure 45: Flow and temperature patterns corresponding to two dierent steady state branches obtained for
ε=0.4,Ra =106and R/L=0.4.
Figure 46: Flow and temperature patterns corresponding to two dierent steady state branches obtained for
ε=0.6,Ra =106and R/L=0.2.
forcing IBM developed by Riahi et al. [33] was adopted and extended to enforce the kine-
matic constraints of no-slip and of a given temperature on the surface of the immersed
body. The algorithm does not rely on the low-Mach-number assumption and does not
employ the methodology of splitting the pressure into hydrodynamic and thermodynamic
terms. Instead, all the equations are solved in their original fully compressible formula-
tions. Finally, the viscous heating is neglected, as is commonly done when simulating
ows characterized by low values of the shear stress.
The developed methodology was extensively veried by comparison with corresponding
independently obtained numerical data available in the literature for incompressible ows
[42] and non-Boussinesq compressible ows [4]. The pressure-corrected direct forcing IBM
was implemented for simulation of the natural convection non-Boussinesq ow developing
within a cold square cavity with a centrally located hot cylinder over a broad range of
governing parameters. The results obtained were analyzed qualitatively and quantitatively.
First, the spatial distributions of the path lines and temperature elds were obtained.
Second, the values of Nusselt number on the hot cylinder and the cold cavity surfaces
were calculated. Third, the thermal uxes at all the domain boundaries were quantied
by calculating the values of the corresponding Nusselt numbers. Finally, multiple steady-
state solutions for several congurations were discovered.
Two dierent strategies, one based on an iterative solution and the other, on a direct
solution of the discretized governing equations were applied in the current study. The
iterative solution utilized the BiCGstab method [38], while the direct solution was based
on the TPF method proposed by Lynch et al. [39] and subsequently adapted to conned
natural convection ows by Vitoshkin & Gelfgat in [40]. The major challenge in applying
the two methods lay in treating the non-linear terms and the temperature-dependent
coecients of the Helmholtz-like dierential operator, which were obviously not constant.
It was found that in the absence of an immersed body both the iterative solver and the
direct solver yielded suciently accurate solutions for non-Boussinesq ows. However, in
the presence of an immersed body, the iterative solver based on the BiCGstab method was
less sensitive to the time step values and provided more accurate results than its direct
counterpart based on the TPF method.
We summarize by giving a list of challenges that have been left out the scope of
the current study and need to be addressed further on the way of developing a system-
atic pressure-based numerical framework for simulating non-Boussinesq natural convection
ows. The rst challenge is the development of the method for simulating the ow around
moving bodies. In this case, it is well known that the direct forcing method can lead to
spurious non-physical oscillations if explicitly included in the fractional step approach (see
e.g. [34]). The problem can be remedied by utilizing semi-implicit formulation of the im-
mersed boundary method, which imposes kinematic constraints of no-slip on the predicted
non-solenoidal velocity eld up to machine zero precession [48]. The second challenge is
an extension of the currently presented method to simulate fully 3D natural convection
ows. Based on our previous work, which utilized an explicit implementation of the direct
forcing method to simulate steady state Boussinesq natural convection ows [49] we do not
Figure 47: Relative spatial error of the developed algorithm calculated for the values of ε=0.6 and Ra=106for
the θ(•),u(), and v(N) elds: (a) l2norm; (b) l∞norm.
5.0E-03 5.0E-02
Spatial error
Grid step,
5.0E-03 5.0E-02
Spatial error
Grid step,
see any objective reasons that could prevent the direct adaption of the currently presented
method for simulation of non-Boussinessq natural convection steady 3D ows. Again, in
the case of the appearance of spurious oscillations in the ow elds when simulating un-
steady natural convection ows, it may be necessary to switch from a completely explicit
implementation of the direct forcing to its semi-implicit counterpart (see e.g. [28]). Fi-
nally, the developed method needs further verication to simulate the ow around bodies
characterized by non-uniform curvature. In this case, the grid resolution must be adjusted
to a kernel of the selected discrete Dirac delta function to ensure that the results provided
by the interpolation and regularization operators acting on the ow elds are suciently
7 Acknowledgements
The authors would like to thank the Israel Ministry of Energy for its nancial support for
this work (grant 218-11-038).
A Appendix: Estimation of the method accuracy
To further evaluate the accuracy of the steady-state solutions obtained in this study, we
present a formal analysis of the spatial accuracy of the results obtained. The analysis
has been performed by calculating the values of Euclidian and innity norms of relative
errors ∥Sex −Sapr∥/∥Sex ∥, where Sex corresponds to the most precise solution obtained
on 200×200 grid and Sapr corresponds to a series of approximate solutions obtained on
50×50,100 ×100 and 1500 ×150 grids. The data calculated for temperature and two
velocity components obtained for the steady state ow in a deferentially heated cavity
for the values of ε=0.6 and Ra =106is presented in Fig. 47. It can be seen that there
is a power-law relationship between ∆xand the norm of the relative error value. In all
presented cases, the value of the exponent is close to 2, which conrms the second order
spatial accuracy of the developed method.
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[14] J. M. Armengol, F. C. Bannwart, J. Xaman and R. G. Santos, ”Eects of variable air proper-
ties on transient natural convection for large temperature dierences,” International Journal
of Thermal Sciences, vol. 120, pp. 63-79, 2017.
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ture dierence natural convection in a nite length channel without Bossinesq assumption,”
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[16] M. M. El-Gendi and A. Aly, ”Numerical simulation of natural convection using unsteady
compressible Navier-stokes equations,” International Journal of Numerical Methods for Heat
& Fluid Flow, vol. 27, no. 11, pp. 2508-2527, 2017.
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viscous thermal ow problems: A Schur complement approach,” International Journal of
Heat and Mass Transfer, vol. 127, pp. 1267-1283, 2018.
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condition,” International Journal of Heat and Mass Transfer, vol. 101, pp. 807-823, 2016.
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induced by variation in radius of inner circular cylinder inside cubic enclosure,” International
Journal of Heat and Mass Transfer, vol. 95, pp. 566-578, 2016.
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ow around a tandem of cold and hot vertically aligned cylinders placed inside a cold cubic
enclosure,” International Journal of Heat and Mass Transfer, vol. 141, pp. 327-345, 2019b.
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natural convection ow: Conguration of a tandem of cold and hot horizontally oriented cylin-
ders placed within a cold cubic enclosure,” International Journal of Heat and Mass Transfer,
p. 106606, 2021.
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Boundary Method for the numerical simulation of compressible ows,” Journal of Computa-
tional Physics, vol. 374, pp. 361-383, 2018.
[34] M. Kumar and G. Natarajan, ”Diuse interface immersed boundary method for low Mach
number ows with heat transfer in enclosures,” Physics of Fluids, vol. 31, 2019.
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[36] A. M. Roma, C. S. Peskin and M. J. Berger, ”An Adaptive Version of the ImmersedBoundary
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nak of Computational Physics, vol. 225, pp. 2118-2137, 2007.
[38] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in
Fortran 77: The Art of Scientic Computing, Cambridge: Cambridge University Press, 1989.
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by tensor product methods,” Numerical mathematics, vol. 6, pp. 185-199, 1964.
[40] H. Vitoshkin and A. Y. Gelfgat, ”On direct inverse of Stokes, Helmholtz and Laplacian
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[41] A. Y. Gelfgat, ”On acceleration of Krylov-subspace-based Newton and Arnoldi iterations
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Facing Step,” International Journal for Numerical Methods in Fluids, vol. 11, no. 7, pp.
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[43] Y. M. Seo, J. H. Doo and M. Y. Ha, ”Three-dimensional Flow Instability of Natural Con-
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[44] J. P. Holman, ”Chapter 7. Natural Convection Systems,” in Heat Transfer, Tenth Edition,
Avenue of the Americas, New York, NY 10020, McGraw-Hill, a business unit of The McGraw-
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[45] J. A. Scanlan, E. H. Bishop and R. E. PoweR., ”Natural Convection Heat Transfer Between
Concentric Spheres,” International Journal of Heat and Mass Transfer, vol. 13, pp. 1857-1872,
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Hydrodynamic equations for a plasma in the region of the distribution-function plateau
A hydrodynamic theory is presented for plasma particles whose distribution function is close to a plateau in some phase-space region for a finite one-dimensional phase-spase interval. In the
framework of the methods of moments, a series expansion of the particle distribution function in an appropriate system of orthogonal (Legendre) polynomials near the state with a plateau is carried
out. Deviations from the plateau are due to the presence of a heat flux. The expansion obtained for the distribution function is used to truncate the chain of hydrodynamic equations and to calculate
the additional terms arising from the finiteness of the phase interval. The case of one-dimensional Langmuir turbulence is considered as an example. Expressions are presented for the moments of the
quasilinear integral of collisions, which describe the variations of the macroscopic characteristics of resonant particles in the respective hydrodynamic equations. Criteria for applicability of the
equations obtained are presented. Approximate invariants of motion are obtained in the extreme cases of narrow and broad plateaux.
Journal of Plasma Physics
Pub Date:
October 1989
□ Distribution Functions;
□ Hydrodynamic Equations;
□ Legendre Functions;
□ Plasma Physics;
□ Heat Flux;
□ Spatial Distribution;
□ Stress Tensors;
□ Transport Theory;
□ Plasma Physics
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Section: LAPACK (3)
Updated: Tue Nov 14 2017
Page Index
subroutine claswp (N, A, LDA, K1, K2, IPIV, INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix.
Function/Subroutine Documentation
subroutine claswp (integer N, complex, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix.
CLASWP performs a series of row interchanges on the matrix A.
One row interchange is initiated for each of rows K1 through K2 of A.
N is INTEGER
The number of columns of the matrix A.
A is COMPLEX array, dimension (LDA,N)
On entry, the matrix of column dimension N to which the row
interchanges will be applied.
On exit, the permuted matrix.
LDA is INTEGER
The leading dimension of the array A.
K1 is INTEGER
The first element of IPIV for which a row interchange will
be done.
K2 is INTEGER
(K2-K1+1) is the number of elements of IPIV for which a row
interchange will be done.
IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
The vector of pivot indices. Only the elements in positions
K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
INCX is INTEGER
The increment between successive values of IPIV. If INCX
is negative, the pivots are applied in reverse order.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
June 2017
Further Details:
Modified by
R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Definition at line 117 of file claswp.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Next: MINIMUM TRAVEL ROBOT LOCALIZATION Up: Games and Puzzles Previous: Games and Puzzles   Index
• INSTANCE: Graph
• SOLUTION: A motion scheme for the robot and the obstacles. In each time step either the robot or one obstacle may be moved to a neighbouring vertex that is not occupied by the robot or an
• MEASURE: The number of steps until the robot has been moved to the goal vertex t.
• Good News: Approximable within 392].
• Bad News: APX-complete [392].
• Comment: Approximable within G.
Viggo Kann
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Friedel pairs, centric reflections and systematic absences
Friedel Pairs
Using the formula for the structure factor we can show that a reflection F(h,k,l) and its 'opposite', F(-h,-k,-l) will have the same magnitude and opposite phases. Try to compute this for any
arbitrary atom coordinate (x,y,z) and any h,k,l you wish to use. This is called Friedel's Law. Centrosymmetrically related reflections in a non centrosymmetric space group are Friedel pairs, provided
they are not otherwise related by space group symmetry (i.e, centric reflections per se, see below).
In case of the presence of anomalous contributions, a mixing of sin and cos terms breaks the centrosymmetry and the magnitude are not identical for Friedel opposites. The phases are not
centrosymmetric around 0 anymore. This effect is exploited in anomalous dispersion phasing techniques.
It may be useful to review the vector presentations of the structure factor to visualize this phenomenon. Notice also the special case of only anomalous scatterers (or one single anomalous atom),
where only the phase symmetry is broken, but the magnitudes remain the same.
Centric reflections
Reflection pairs F(h,k,l) and F(-h,-k,-l) and that are related through the space group's point symmetry (Laue symmetry) are not Friedel pairs. They are called centric reflections and have in any case
the same intensity. They can be used to determine data quality statistics (merging statistics) in the absence of anomalous contributions. A comparison of their statistics to the merged Friedel pairs,
which should merge significantly higher, indicates whether a useable anomalous signal is present. Some examples for centric reflections generated by point group symmetry (see International Tables for
more operators) :
Systematic absences of reflections due to space group symmetry
Plugging numbers into the structure factor formulas (or expanding the terms and reducing the mess) shows that the internal symmetry operations screws and glides (which include translational
components) and the Bravais translations yield systematic absences in the diffraction pattern regardless of the cell content. We distinguish serial and zonal extinctions, cause by the presence of
screws and glides, respectively, and integral extinctions caused by Bravais translation vectors. The international tables list the conditions for possible reflections for each space group. For
example, a F centered lattice would have all 3 indices of a triple either even or odd, or a 2-fold screw axis parallel to c would show only even idices l in 0,0,l reflections .
The systematic absences allow to determine certain space groups unambiguously, but in many cases a unique determination of the space group by crystallographic means alone is not possible.
Fortunately, there are a number of additional clues (number of molecules in cell, chirality of motif, physical properties) which allow to determine the space groups correctly. Try for example P 2 2 2
and P m m m. In case of a protein crystal you would know that P m m m is not a possible space group, because P m m m is centrosymmetric (check the output listing).
What is the symmetry operation causing the extinctions in the lattice on the left? What are the possible space groups?
What causes the kind of extinctions observed in the lattice on the right?
Click here to submit your answer. If your answer is right, you will be listed in the Space Group Hall of Fame. If not, I'll reply and explain. In case you worry, I will not list wrong answers, nor
sue you for it, or make fun.
Non-crystallographic symmetry
Difficult cases for space group determination may be encountered where for example non-crystallographic symmetry (NCS) exists. A NCS 2-fold axis for example may be initially interpreted as
crystallographic axis. The refinement in the higher symmetric spacegroup may not converge well and show high R-values, whereas the refinement of the (slightly different) 2 individual molecules in the
lower symmetric space group yields better refinement and the correct structure. An evaluation of the statistic parameters including Rfree must justify this approach, however. If you are interested in
NCS and the use of native Patterson maps to detect NCS, go to our antibody structure.
Back to X-ray Tutorial Index
This World Wide Web site conceived and maintained by Bernhard Rupp. Last revised Dezember 27, 2009 01:40
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A test consists of ‘True’ or ‘False’ questions. One mark is awarded
A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while ¼ mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of
the questions he attempted by guessing. He answered 120 questions and got 90 marks.
Type of Question Marks given for correct Marks given for correct answer
True/False 1 0.25
Question 1
If answer to all questions he attempted by guessing were wrong, then how many questions did he answer correctly?
Question 2
How many questions did he guess?
Question 3
If answer to all questions he attempted by guessing were wrong and
answered 80 correctly, then how many marks he got?
Question 4
If answer to all questions he attempted by guessing were wrong, then how many questions answered correctly to score 95 marks?
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Introduces algebraic concepts and processes with a focus on polynomials, exponents, roots, geometry, dimensional analysis, solving quadratic equations, and graphing parabolas. Emphasizes
number-sense, applications, graphs, formulas, and proper mathematical notation. The OCCC math department recommends that students take MTH courses in consecutive terms.
Addendum to Course Description
A scientific calculator and access to a graphing utility may be required.
Students are no longer required to have physical graphing calculators in MTH 60, 65, 70, 95, 111, and 112. Where physically possible instructors will demonstrate using Desmos, GeoGebra, or other
online programs in class. Assessments requiring the use of a graphing utility may be done outside of proctored exams.
Course Outcomes
Upon completion of the course students should be able to:
• Recognize and apply the operations necessary to simplify expressions and solve equations.
• Perform polynomial addition, subtraction, and multiplication and perform polynomial division by a monomial.
• Use exponent and radical properties to simplify expressions and solve radical and quadratic equations.
• Solve systems of equations by graphing, substitution, and elimination and use systems in solving applications.
Students have the option of taking the co-requisite MTH 65L in place of completing the prerequisites.
Recommended Prereqs
Recommended: MTH 60 be taken within the past 4 terms. It's recommended that students take MTH courses in consecutive terms.
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Joshua Garland
1399 Hyde Park Rd, Santa Fe
Hello. I am a Mathematician Researcher Time Series Analyst Teacher Information Theorist
I am passionate about complexity science
Welcome to my academic profile Available for consulting and collaborating
In the study of complex adaptive systems, the available data often falls far short of the demands of the theory. In mathematics, for instance, we often make statements like “assume we have an
infinite noise-free time series, then…” But in studying complex systems, we often ask questions like “I have 347 noisy observations that I collected over three years in a jungle. What can we learn
from this information?” My research aims to develop rigorous models that bridge the gap between theory and observation---data that may be wildly lacking in the eyes of mathematics, but may still
contain valuable information about the system. Said differently: when perfect isn’t possible, how can we adapt mathematics to describe the world around us? In studying complicated, ill-sampled, noisy
systems, my work focuses on understanding how much information is present in the data, how to extract it, to understand it, and to use it---but not overuse it.
Specifically, I am working toward developing a parsimonious reconstruction theory for nonlinear dynamical systems. In addition, I aim to leverage information mechanics (e.g., production, storage and
transmission) to gain insight into important yet imperfect systems, like the climate, traded financial markets and the human heart. My hope is this combination of new mathematical theory, analysis,
and application can eventually shed a little more light on universalities like emergence, regime shifts, and phase transitions.
I received my Ph. D. from the University of Colorado, supervised by Elizabeth Bradley, introducing a new paradigm in delay-coordinate reconstruction theory. Prior to that, I earned an M.S. in Applied
Mathematics also from the University of Colorado, constructing dynamical models of computer performance and a dual B.S. in Mathematics and Computer Science from Colorado Mesa University.
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The legendary love story | Otome Dating Sim game v0.0.14 [Mod] APK Free Download - OceanofAPK
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Should we ditch macroeconomics or perhaps reduce it to two weeks? In a recent blog post, Noah Smith argues that most of the material in a Principles of Macroeconomics class isn’t really necessary.
After teaching macro principles to more than 1,000 students per year since 2003, it is easy for me to find the blind spots in Noah’s view. More than anything, it is pretty clear that Noah doesn’t
spend much time with college students.
Let me start with what Noah gets right: students should learn the Solow model for long-run growth, and the AD-AS model for business cycle analysis. He also includes “the standard Milton Friedman,
New Keynesian, AD-AS, accelerationist Phillips Curve theory of monetary policy.”
Now we come to Noah’s first howler: he believes that this material should take “about two weeks.” Two? What students is he teaching? I teach at the University of Virginia, a really great
university with super students. But this takes six weeks, not two. When my students show up for macro principles, very few even know that interest rates are market prices. I do teach the Solow
model but most Macro principles instructors believe it is just too hard for the intro level.
More than that, Noah leaves out a host of other macro topics that students need to learn at the intro level, whether they continue in economics or not. This list includes:
1. Key macroeconomic variables. These need to be defined, explained, and put in their proper historical perspective. These include real GDP growth, unemployment, inflation, and interest rates. And
not just for the United States.
First off, the way we measure these variables actually matters. Consider that unemployment rates do not include underemployed or out-of-labor force workers. Or that GDP only includes market goods.
Both of these are relevant for policy and have been discussed in the media recently. And historical perspective is really key here – it can be one of the best gifts you can give your students. What
is a big number or a small number? When unemployment is 7%, is that high or low? How about in the U.S. versus Spain? Or when real GDP grows by 4%, is that high or low? How about the U.S. versus
Mainland China? Most college students won’t know this without a macro principles course.
2. The loanable funds market. You can’t understand financial collapse/contagion without a good understanding of the loanable funds market. A big part of this discussion is also forming an intuitive
understanding of interest rates, which is not natural for most students. In my principles course (and textbook) I even cover mortgage-backed securities, securitization and moral hazard now so that
the students understand the Great Recession.
3. Fiscal and monetary policy. In many universities, this is the one place where real economic policy is taught - Intermediate Macroeconomics typically focuses on theoretical models. I view these
policy discussions as voter education curriculum. Students need to know what deficits mean and something about historical perspective here too. They also need to know where government revenue comes
from and how it is spent. Hint: it’s not all spent on foreign aid and welfare! And what about the Fed? This is the course where students learn about Fed policy and both actual and perceived effects
on the economy.
If time permits, it is great to also throw in international trade and finance, like the balance of payments (many misconceptions arise from a misunderstanding of how capital inflows are related to
merchandise trade).
Basically, to cover all of this takes about a semester. It is foolish to think that two weeks is enough.
By the way, my favorite macro textbook covers all of these topics clearly in a great one-semester format.
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function udb_hash($buf) {
$s1 = 1;
$s2 = 0;
for($n=0; $n<$length; $n++)
$s1 = $s1 + ord($buf[$n]) % 65521;
$s2 = $s2 + $s1 % 65521;
return ($s2 << 16) + $s1;
Is it possible to crack a hash with this algorithm since it isn't implemented with hashcat? Or would I need to make a request?
11-12-2016, 11:25 PM
Can't you just try to write a poc i Perl or something to see how fast that is? Often these small problem can be solved without GPU support. Or have you already given up on CPU?
11-13-2016, 05:33 AM
(11-12-2016, 11:25 PM)bigblacknose Wrote: Can't you just try to write a poc i Perl or something to see how fast that is? Often these small problem can be solved without GPU support. Or have you
already given up on CPU?
I'm not the best with this kind of thing, this algorithm is like a split off of adler32, but i'm not really sure where to start
11-13-2016, 10:05 PM
11-13-2016, 10:31 PM
Alright, thanks.
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Trung Trinh
trung.trinh [at] aalto [dot] fi
Research interests: Robust Machine Learning, Bayesian Deep Learning, Uncertainty Quantification, Optimization
Aalto University Computer Science Building
Konemiehentie 2
02150 Espoo, Finland
I’m currently a Ph.D. student in the Probablistic Machine Learning group at Aalto University under the supervision of Prof. Samuel Kaski. I’m expected to graduate in 2025, and I can start working
from November 2024. I’m currently looking for Data Scientist or Machine Learning engineer roles in the industry.
My primary research interest lies in enhancing the robustness of deep learning models to distribution shifts. Distribution shifts refer to changes in data distribution between the training and
deployment phases, which can lead to significant degradation in model performance. Neural networks are especially vulnerable to these shifts, reducing their reliability in practical applications. My
goal is to improve the resilience of neural networks, enabling their deployment in safety-critical systems, such as autonomous vehicles and medical diagnostics. My current work focuses on enhancing
generalization and uncertainty calibration of neural networks under distribution shifts through efficient Bayesian approaches and advanced optimization methods.
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Combined Colleges - Course Descriptions for MATH
Course Descriptions for MATH
These Course Descriptions include updates that were added after the original publication on June 11, 2016.
New students who are entering the college for the first time should follow this version when selecting courses. Show me the June 11, 2016, web published version. Show me what changed in 2016-2017.
MATH 1314 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
College Algebra
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: College level ready in Mathematics algebra-based level.
Course Description: This course is an in-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems of equations using matrices. Additional
topics such as sequences, series, probability, and conics may be included. This course is cross-listed as MATH 1414. The student may register for either MATH 1314 or MATH 1414 but may receive credit
for only one of the two. (3 Lec.)
Coordinating Board Academic Approval Number 2701015419
* Note: This Course Description includes updates that were added after it was originally published on June 11, 2016. (Original) | (Changes)
MATH 1316 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Plane Trigonometry
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of the DCCCD.
Prerequisite Required: MATH 1314 or equivalent.
Course Description: In depth study and applications of trigonometry including definitions, identities, inverse functions, solutions of equations, graphing, and solving triangles. Additional topics
such as vectors, polar coordinates, and parametric equations may be included. (3 Lec.)
Coordinating Board Academic Approval Number 2701015319
MATH 1324 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Mathematics for Business and Social Sciences
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: College level ready in Mathematics algebra-based level.
Course Description: The application of common algebraic functions, including polynomial, exponential, logarithmic, and rational, to problems in business, economics, and the social sciences are
addressed. The applications include mathematics of finance, including simple and compound interest and annuities; systems of linear equations; matrices; linear programming; and probability, including
expected value. (3 Lec.)
Coordinating Board Academic Approval Number 2703015219
MATH 1325 (3 Credit Hours)
Offered at EFC, ECC, MVC, NLC, RLC
Calculus for Business and Social Sciences
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 1324, MATH 1314 or MATH 1414.
Course Description: This course is the basic study of limits and continuity, differentiation, optimization and graphing, and integration of elementary functions, with emphasis on applications in
business, economics, and social sciences. This course is not a substitute for MATH 2413, Calculus I. This course is cross-listed as MATH 1425. The student may register for either MATH 1325 or MATH
1425 but may receive credit for only one of the two. (3 Lec.)
Coordinating Board Academic Approval Number 2703015319
MATH 1332 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Contemporary Mathematics (Quantitative Reasoning)
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of the DCCCD.
Prerequisite Required: College level ready in Mathematics at the non-algebra or algebra levels.
Course Description: Topics may include introductory treatments of sets, logic, number systems, number theory, relations, functions, probability and statistics. Appropriate applications are included.
(3 Lec.)
Coordinating Board Academic Approval Number 2701015119
MATH 1342 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Elementary Statistical Methods
This is a Texas Common Course Number.
Prerequisite Required: College level ready in Mathematics at the non-algebra or algebra levels.
Course Description: This course is a study of collection, analysis, presentation and interpretation of data, and probability. Analysis includes descriptive statistics, correlation and regression,
confidence intervals and hypothesis testing. Use of appropriate technology is recommended. (3 Lec.)
MATH 1350 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Mathematics for Teachers I (Fundamentals of Mathematics I)
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 1314 or the equivalent.
Course Description: Concepts of sets, functions, numeration systems, number theory, and properties of the natural numbers, integers, rational, and real number systems with an emphasis on problem
solving and critical thinking. This course is designed specifically for students who seek elementary and/or middle grade teacher certification. (3 Lec.)
Coordinating Board Academic Approval Number 2701015619
MATH 1351 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Mathematics for Teachers II (Fundamentals of Mathematics II)
This is a Texas Common Course Number.
Prerequisite Required: MATH 1350.
Course Description: Concepts of geometry, probability, and statistics, as well as applications of the algebraic properties of real numbers to concepts of measurement with an emphasis on problem
solving and critical thinking. This course is designed specifically for students who seek middle grade (4 through 8) teacher certification. (3 Lec.)
Coordinating Board Academic Approval Number 2701015719
MATH 1370 (3 Credit Hours)
Offered at BHC, CVC, ECC
Business Calculus and Applications II
Prerequisite Required: MATH 1325 or MATH 1425.
Course Description: Course topics include review of limits, differentiation, logarithmic and exponential functions, integration, applications of integration, calculus of several variables,
differential equations, sequences and series. (3 Lec.)
Coordinating Board Academic Approval Number 2701017319
MATH 1414 is a 4 credit hour lecture course. MATH 1314 is a 3 credit hour lecture course. Either course will meet degree requirements. A student may receive credit for MATH 1414 or MATH 1314 but not
for both.
MATH 1414 (4 Credit Hours)
Course not offered this year at any colleges of DCCCD.
College Algebra
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: College level ready in Mathematics algebra-based level.
Course Description: This course is an in-depth study and applications of polynomial, rational, radical, exponential and logarithmic functions, and systems of equations using matrices. Additional
topics such as sequences, series, probability, and conics may be included. This course is cross-listed as MATH 1314. The student may register for either MATH 1314 or MATH 1414 but may receive credit
for only one of the two. (4 Lec.)
Coordinating Board Academic Approval Number 2701015419
MATH 1425 (4 Credit Hours)
Offered at BHC, CVC
Calculus for Business and Social Sciences
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 1314 or MATH 1324.
Course Description: This course is the basic study of limits and continuity, differentiation, optimization and graphing, and integration of elementary functions, with emphasis on applications in
business, economics, and social sciences. This course is not a substitute for MATH 2413. This course is cross-listed as MATH 1325. The student may register for either MATH 1325 or MATH 1425 but may
receive credit for only one of the two. (4 Lec.)
Coordinating Board Academic Approval Number 2703015319
MATH 1442 (4 Credit Hours)
Offered at MVC, RLC
Elementary Statistical Methods
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of the DCCCD.
Prerequisite Required: College level ready in Mathematics at the non-algebra level and completion of DMAT 0407.
Course Description: Collection, analysis, presentation and interpretation of data, and probability. Analysis includes descriptive statistics, correlation and regression, confidence intervals and
hypothesis testing. Use of appropriate technology is recommended. (4 Lec.)
Coordinating Board Academic Approval Number 2705015119
* Note: This Course Description includes updates that were added after it was originally published on June 11, 2016. (Original) | (Changes)
MATH 2305 (3 Credit Hours)
Offered at BHC, ECC, RLC
Discrete Mathematics
This is a Texas Common Course Number.
Prerequisite Required: MATH 2413.
Course Description: A course designed to prepare math, computer science, and engineering majors for a background in abstraction, notation, and critical thinking for the mathematics most directly
related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, combinatorics, discrete
probability, recursion, sequence and recurrence, elementary number theory, graph theory, and mathematical proof techniques. (3 Lec.)
Coordinating Board Academic Approval Number 2701016619
MATH 2318 (3 Credit Hours)
Offered at BHC, ECC, NLC
Linear Algebra
This is a Texas Common Course Number.
Prerequisite Required: MATH 2414.
Course Description: Introduces and provides models for application of the concepts of vector algebra. Topics include finite dimensional vector spaces and their geometric significance; representing
and solving systems of linear equations using multiple methods, including Gaussian elimination and matrix inversion; matrices; determinants; linear transformations; quadratic forms; eigenvalues and
eigenvectors; and applications in science and engineering. This course is cross-listed as MATH 2418. The student may register for either MATH 2318 or MATH 2418 but may receive credit for only one of
the two. (3 Lec.)
Coordinating Board Academic Approval Number 2701016319
MATH 2320 (3 Credit Hours)
Offered at ECC, NLC
Differential Equations
This is a Texas Common Course Number.
Prerequisite Required: MATH 2414.
Course Description: This course is a study of ordinary differential equations, including linear equations, systems of equations, equations with variable coefficients, existence and uniqueness of
solutions, series solutions, singular points, transform methods, boundary value problems, and applications. This course is cross-listed as MATH 2420. The student may register for either MATH 2320 or
MATH 2420 but may receive credit for only one of the two. (3 Lec.)
Coordinating Board Academic Approval Number 2701016419
MATH 2342 (3 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Elementary Statistical Methods
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: College level ready in Mathematics at the non-algebra or algebra levels.
Course Description: Collection, analysis, presentation and interpretation of data, and probability. Analysis includes descriptive statistics, correlation and regression, confidence intervals and
hypothesis testing. Use of appropriate technology is recommended. (3 Lec.)
Coordinating Board Academic Approval Number 2705015119
MATH 2412 (4 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Pre-Calculus Math
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 1316.
Course Description: This course consists of the study of algebraic and trigonometric topics including polynomial, rational, exponential, logarithmic and trigonometric functions and their graphs.
Conic sections, polar coordinates, and other topics of analytic geometry will be included. (4 Lec.)
Coordinating Board Academic Approval Number 2701015819
MATH 2413 (4 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Calculus I
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 1348, MATH 2412 or equivalent.
Course Description: This course is a study of limits and continuity; the Fundamental Theorem of Calculus; definition of the derivative of a function and techniques of differentiation; applications of
the derivative to maximizing or minimizing a function; the chain rule, mean value theorem, and rate of change problems; curve sketching; definite and indefinite integration of algebraic,
trigonometric, and transcendental functions, with an application to calculation of areas. (4 Lec.)
Coordinating Board Academic Approval Number 2701015919
MATH 2414 (4 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Calculus II
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of DCCCD.
Prerequisite Required: MATH 2413 or equivalent.
Course Description: This course is a study of differentiation and integration of transcendental functions; parametric equations and polar coordinates; techniques of integration; sequences and series;
improper integrals. (4 Lec.)
Coordinating Board Academic Approval Number 2701016019
MATH 2415 (4 Credit Hours)
Offered at BHC, CVC, EFC, ECC, MVC, NLC, RLC
Calculus III
This is a Texas Common Course Number.
Prerequisite Required: MATH 2414 or equivalent.
Course Description: This course is a study of advanced topics in calculus, including vectors and vector-valued functions, partial differentiation, Lagrange multipliers, multiple integrals, and
Jacobians; application of the line integral including Green's Theorem, the Divergence Theorem, and Stokes' Theorem. (4 Lec.)
Coordinating Board Academic Approval Number 2701016119
MATH 2418 (4 Credit Hours)
Offered at BHC, EFC, RLC
Linear Algebra
This is a Texas Common Course Number.
Prerequisite Required: MATH 2414 or equivalent.
Course Description: Introduces and provides models for application of the concepts of vector algebra. Topics include finite dimensional vector spaces and their geometric significance; representing
and solving systems of linear equations using multiple methods, including Gaussian elimination and matrix inversion; matrices; determinants; linear transformations; quadratic forms; eigenvalues and
eigenvectors; and applications in science and engineering. This course is cross-listed as MATH 2318. The student may register for either MATH 2318 or MATH 2418 but may receive credit for only one of
the two. (4 Lec.)
Coordinating Board Academic Approval Number 2701016319
MATH 2420 (4 Credit Hours)
Offered at BHC, EFC, ECC, MVC, RLC
Differential Equations
This is a Texas Common Course Number.
Prerequisite Required: MATH 2414.
Course Description: This course is a study of ordinary differential equations, including linear equations, systems of equations, equations with variable coefficients, existence and uniqueness of
solutions, series solutions, singular points, transform methods, boundary value problems, and applications. This course is cross-listed as MATH 2320. The student may register for either MATH 2320 or
MATH 2420 but may receive credit for only one of the two. (4 Lec.)
Coordinating Board Academic Approval Number 2701016419
MATH 2442 (4 Credit Hours)
Offered at BHC, CVC
Elementary Statistical Methods
This is a Texas Common Course Number. This is a Core Curriculum course selected by the colleges of the DCCCD.
Prerequisite Required: College level ready in Mathematics at the non-algebra or algebra levels.
Course Description: Collection, analysis, presentation and interpretation of data, and probability. Analysis includes descriptive statistics, correlation and regression, confidence intervals and
hypothesis testing. Use of appropriate technology is recommended. This course is cross-listed as MATH 2342.The student may register for either M ATH 2342 or MATH 2442 but may receive credit for only
one of the two. (4 Lec.)
Coordinating Board Academic Approval Number 2705015119
Academic Courses
Designated by the Texas Higher Education Coordinating Board for transfer among community colleges and state public four year colleges and universities as freshman and sophomore general education
WECM Courses
Designated by the Texas Higher Education Coordinating Board as workforce education (technical) courses offered for credit and CEUs (Continuing Education Units). While these courses are designed to
transfer among state community colleges, they are not designed to automatically transfer to public four-year colleges and universities.
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Haskell - The Most Gentle Introduction Ever - Part II - 11Sigma
This is the second article in my little series explaining the basics of Haskell.
If you haven’t yet, I would recommend you to read the first article, before diving into this one.
So far we’ve learned the basics of defining functions and datatypes, as well as using them, all based on the Haskell Bool type.
In this article, we will deepen our understanding of functions in particular, while also learning a bit about built-in Haskell types representing numbers.
Type of Number 5
Before we begin this section, I must warn you. In the previous article, I purposefully used the Bool type, due to its simplicity.
You might think that types representing numbers would be equally simple in Haskell. However, that’s not really the case.
Not only does Haskell have quite a lot of types representing numbers, but there is also a significant effort in the language to make those types as interoperable with each other as possible. Because
of that, there is a certain amount of complexity, which can be confusing to beginners.
To see that, type in the following in the ghci:
:t 5
You would probably expect to see something simple and concrete, like Number or Int. However, what we see is this:
5 :: Num p => p
Quite confusing, isn’t it?
For a brief second try to ignore that whole Num p => part. If it wasn’t there, what we would see, would be just 5 :: p So what is written here is that the value 5 has type p.
This is the first time we see the name of the type being written using a small letter. That’s important. Indeed, p is not a specific type. It is a type variable. This means that p can be potentially
many different, concrete types.
It wouldn’t however make sense for 5 to have – for example – Bool type. That’s why Num p => part is also written in the type description. It basically says that this p has to be a numeric type.
So, overall, 5 has type p, as long as p is a numeric type. For example, writing 5 :: Bool would be forbidden, thanks to that restriction.
The exact mechanism at play here will not be discussed right now. We still have to cover a few basics before we can explain it fully and in detail. But perhaps you’ve heard it about it already – it’s
called typeclasses. We will learn about typeclasses very soon. After we do, this whole type description will be absolutely clear to you.
For now, however, we don’t need to go into specifics. Throughout this article, we will use concrete, specific types, so that you don’t get confused. I am just warning you about the existence of this
mechanism so that you don’t get unpleasantly surprised and discouraged when you investigate the types of functions or values on your own.
Which I encourage you to do! Half of reading Haskell is reading the types, and you should be getting used to that.
Functions and Operators
Let’s begin by doing some simple operations on numbers, familiar from other programming languages.
We can, for example, add two numbers. Writing in ghci:
5 + 7
results in a fairly reasonable answer:
But to get a deeper insight into what is happening there, let’s write a "wrapper" function for adding numbers.
We will call it add and we will use it like so:
add 5 7
As a result, we should see the same answer as just a moment ago:
We would like to, of course, begin with a type signature of that function. What could it potentially be?
add :: ???
As I mentioned before, Haskell has many different types available for numerical values.
Let’s say that we want to work only with integers for now. Even for integers, there are multiple types to choose from.
The two most basic ones are Integer and Int.
Int is a type that is "closer to the machine". It’s a so-called fixed precision integer type. This means that – depending on the architecture of your computer – each Int will have a limited number of
bits reserved for its value. Going out of those bounds can result in errors. This is a type very similar to C/C++ int type.
Integer on the other hand is an arbitrary precision integer type. This means that the values can potentially get bigger than those of Int. Haskell will just reserve more memory if that becomes
necessary. So those integers are "arbitrarily" large, but, of course, only in principle – if you completely run out of computer memory, nothing can save you.
At the first glance, it would seem that Integer has some clear advantages. That being said, Int is still being widely used where memory efficiency is important or where we have high confidence that
numbers will not become too large.
For now, we will use the Integer type. We can finally write the signature of our function:
add :: Integer -> Integer -> Integer
What we have written here is probably a bit confusing at the first glance.
So far, we’ve only written declarations of functions that accept a single value and return a single value.
But when adding numbers, we have to accept two values as parameters (two numbers to add) and then return a single result (a sum of those numbers). So in our type signature, the first two Integer
types represent parameters and the last Integer represents the return value:
add :: Integer {- 1st parameter -} -> Integer {- 2nd parameter -} -> Integer {- return value -}
(By the way, you can see here what syntax we’ve used to add comments to our code.)
Let’s now write the implementation:
add :: Integer -> Integer -> Integer
add x y = x + y
Simple, right?
Create a file called lesson_02.hs and write down that definition. Next, load the file in ghci (by running :l lesson_02.hs) and type:
add 5 7
As expected, you will see the reasonable response:
I will now show you a neat trick. If your function accepts two arguments – like it is the case with add – you can use an "infix notation" to call a function. Write:
5 `add` 7
You will again see 12 as a response.
Note that we didn’t have to change the definition of add in any way to do that. We just used backticks and we were immediately able to use it in infix notation.
You can use this feature with literally any function that accepts two parameters – this is not a feature restricted only to functions operating on numbers.
At this point those two calls:
5 `add` 7
5 + 7
look eerily similar.
That’s not an accident.
Indeed, you can do the reverse, and call the + operator as you would call a regular function – in front of the parameters. If it’s an operator, you just have to wrap it in parentheses:
(+) 5 7
Try it in ghci. This works and returns 12 again!
Perhaps you already know where am I going with this.
I am trying to show you that the built-in + operator is indeed just a regular function.
There are of course syntactical differences (like using backticks or parentheses for certain calls), but conceptually it is good to think of + as being no different than add. Both are functions that
work on the Integer type – accept two Integer numbers and return an Integer number as a result.
Indeed, you can even investigate the type of an operator in ghci, just as you would investigate the type of add function.
:t (+)
results in:
(+) :: Num a => a -> a -> a
This means that + has type a -> a -> a, where a has to be a numeric type. So it’s a function that accepts two parameters of numeric type a and returns the result of the same type.
I hope that at this point it makes sense why it is beneficial for the + operator to have such an abstract definition. A clear benefit + has over our custom add function is that + works on any numeric
type. No matter if it’s Integer, Int, or any other type that somehow represents a number – + can be used on it. Meanwhile, our add function works only on Integer types. For example, if you try to
call it on – very similar – Int numbers, the call will fail.
So you can see that complexity introduced in number types doesn’t come out of nowhere. It keeps the code typesafe, while still allowing huge flexibility. Types might seem complex, but this makes
writing actual implementations a breeze.
Partial Application
Let’s go back to the type of add function, which is probably still friendlier to read at this point.
add :: Integer -> Integer -> Integer
The way we have written the type definition here might be surprising to you. We see two -> arrows in the definition, almost suggesting that we are dealing with two functions.
And indeed we are!
To increase the readability even more, we can use the fact that -> is right-associative. This means that our type definition is equivalent to this:
add :: Integer -> (Integer -> Integer)
Let’s focus on the part that is outside of the parentheses first:
add :: Integer -> (...)
This says that add is a function that accepts an Integer and returns something.
What is that something? To find out, we have to look inside the parentheses:
(Integer -> Integer)
That’s again a function! This one also accepts an Integer as an argument. And as a result, it returns another Integer.
So if we look at the type of add again:
add :: Integer -> Integer -> Integer
we see that – in a certain sense – I was lying to you the whole time!
I was saying that this is how we describe a function that accepts two parameters. But that’s false! There is no function that accepts two parameters here!
There is only a function that accepts a single parameter and then… returns another function!
And then that second function accepts yet another parameter and just then returns a result!
To state the same thing in terms of another language, here is how you would write a regular function that accepts two parameters in JavaScript:
function add(x, y) {
return x + y;
However what we are creating in Haskell is something closer to this JavaScript code:
function add(x) {
return function(y) {
return x + y;
Note how we have two functions here, each accepting only a single parameter.
In the code snippet above, function add accepts parameter x and the second, anonymous, function accepts the parameter y. There is no function here that accepts two parameters.
So, based on what we have said so far, in Haskell we should be able to call the add function with only one parameter and get a function, right?
Let’s try that in ghci:
add 5
Regrettably, we get an error message:
<interactive>:108:1: error:
• No instance for (Show (Integer -> Integer))
arising from a use of ‘print’
(maybe you haven't applied a function to enough arguments?)
• In a stmt of an interactive GHCi command: print it
But that doesn’t happen, because we did something wrong. The problem arises, because add 5 returns – as we stated – a function, and Haskell doesn’t know how to print functions.
We can however check the type of the add 5 expression and this way convince ourselves that this indeed works:
:t add 5
As a result, we see:
add 5 :: Integer -> Integer
So it’s a success! The expression add 5 has type Integer -> Integer, just as we wanted!
We can convince ourselves even more that that’s the case, by calling that add 5 function on a number:
add 5 7
Oh… Wait. We just discovered something!
It was probably confusing so far why Haskell has this strange way of calling functions on values, especially on multiple values. We were just separating them by space, like so:
f x y z
Now it becomes clear, that Haskell simply deals with single-argument functions all the time, and calling a function on multiple values is simply an illusion!
Our call:
add 5 7
is equivalent to:
(add 5) 7
First, we apply add to 5 and as a result, we get a function of type Integer -> Integer, as we’ve just seen.
Then we apply that new function (add 5) on a value 7. As a result, we get an Integer – number 12.
Note that this means that in Haskell function call is left-associative:
(add 5) 7
Contrasting with what we found out before – that type definition of function is right-associative:
add :: Integer -> (Integer -> Integer)
It is of course done this way so that we get a sane default. Thanks to those properties, in the case of both defining and calling the add function, we can simply forget about parentheses:
add :: Integer -> Integer -> Integer
add 5 7
How else can we convince ourselves that the result of calling add 5 is an actual, working function? Well… let’s give a name to that function and use it that way!
In your lesson_02.hs file add the following line:
addFive = add 5
Load the file in ghci.
First let’s investigate the type one more time, just to be sure what we are dealing with:
:t addFive
This gives us:
addFive :: Integer -> Integer
Exactly what we expected after applying the add function to one argument.
We could have also written down that type definition explicitly in our file:
addFive :: Integer -> Integer
addFive = add 5
Do that to convince yourself that it all compiles when written this way.
Now you can use your, highly specific, addFive function to… add five to integers.
addFive 7
This results in the expected:
Now, I am sure this example with adding the number five seems a bit silly to you, and rightfully so.
But I hope that it shows you the power of partial application in Haskell, where you can easily use highly general functions, accepting a higher number of parameters, to create something more specific
and fitting your particular needs.
For example, you can imagine a function that needs some kind of complex configuration in order to work – let’s call it imaginaryFunction.
Let’s assume that this configuration is some kind of data structure of type ComplexConfiguration. We can make that configuration the first argument of our function:
imaginaryFunction :: ComplexConfiguration -> OtherParameter -> Result
Why do we want to pass it as a parameter instead of just having it "hardcoded" inside the function? Who knows, perhaps different versions of our app need different configurations. Or perhaps we just
need to change its value in our unit test suite.
If we do that, then, in the actual app, we can simply apply imaginaryFunction to a specific ComplexConfiguration:
configuredImaginaryFunction = imaginaryFunction config
After that, we can use the configuredImaginaryFunction directly, without the need to explicitly import config object, whenever we want to use imaginaryFunction in our code. Haskell just carries that
config around for us!
More Numbers and Operations
This section will be far from a complete rundown, but I still want to quickly take you up to speed with basic operations on numbers in Haskell.
So far we’ve covered two numeric types – Integer and Int and we’ve shown that we can add them.
Just as in other languages, we can also subtract and multiply them in the usual way.
5 - 2
results in:
5 * 3
results in:
Other popular number types are Float and Double. Those are number types that can represent numbers beyond simple integers. The main difference between Float and Double is how big their storage
capacity is. Most of the time you will likely use Double unless you have some specific reason to use Float.
Float and Double values can be added, subtracted, and multiplied just like Int and Integer values.
Let’s see some examples, just to make ourselves comfortable with that:
1.1 - 0.1
results in:
1 + 0.1
result in:
Note that in Haskell’s standard library there are much more numeric types available. Some of them are fairly common, others are fairly specific. There are also many more built-in functions.
This section was just meant to make you comfortable with making simple operations on basic number types. In the future we will come back to numbers for sure – there is a great deal of interesting
type-level stuff at play here, and we will want to cover that for sure!
Let’s now take a small break from numbers and go back to our beloved booleans.
Previously, we have written a not function, which was negating the booleans – converting True to False and False to True.
But what if we wanted to do… the opposite?
What if we wanted to create a function that… returns True when passed True and returns False when passed False?
This might sound a bit nonsensical. In a way, this function would do literally nothing. However, we will see that it will have a tremendous educational value for us, so let’s curb our doubts and
let’s try to write it anyway.
First, let’s start with the name and type signature as all Haskell programming should.
In mathematics, a function that takes a value and returns the exact same value is usually called an identity function. Since this one will operate on the Bool type, we will call it boolIdentity.
boolIdentity will receive a single argument of type Bool and return the same thing, so… Bool as well! Therefore its type signature is this:
boolIdentity :: Bool -> Bool
Now let’s get started with actual implementation. Your first instinct might be to write something like this:
boolIdentity True = True
boolIdentity False = False
This will work perfectly fine and is a valid solution, but there is a way to write the same thing in a more terse way.
After all, we are returning the same value that we are receiving as a parameter, so we can simply write:
boolIdentity x = x
In the end, our whole definition looks like that:
boolIdentity :: Bool -> Bool
boolIdentity x = x
Write that down in your lesson_02.hs file and reload the file in ghci.
You can convince yourself that our function works, by running it:
boolIdentity True
This call results in:
And at the same time boolIdentity False returns False (hopefully not surprisingly).
Now let’s create a similar function, but for numbers – let’s say for Integer type. We want a function that will take an Integer value and return the same value. For example, if we call it with 5, we
want to see 5 again.
Let’s call it integerIdentity. Let’s begin by writing the type signature:
integerIdentity :: Integer -> Integer
This was simple.
Now let’s think about the implementation. Well… we want to take the parameter passed to the function and… just return it!
So we get:
identityInteger x = x
But… but this is exactly the same implementation as in the case of identity for Bool values!
Let’s compare the two:
boolIdentity :: Bool -> Bool
boolIdentity x = x
integerIdentity :: Integer -> Integer
integerIdentity x = x
Everything looks the same. The only difference here is the types. In the first function, we operate on the Bool type. In the second we operate on the Integer type.
Now, if only there was a way to write that function only once. In the current state of things, we would have to write an identity function for each type in existence, which… sounds daunting, to say
the least.
It luckily turns out that Haskell does have a mechanism to deal with that easily. Not only that – we’ve already encountered that mechanism!
Remember how the type of number 5 was Num p => p? The p in the type description was a type variable – basically a placeholder for actual, concrete types.
We’ve also seen that the + operator was quite general – it could be called on any numeric type. It also had a type variable in its type signature.
So the question is, can we use a type variable, to write the most generic version of the identity function possible? The answer is… absolutely!
Let’s remove the two previous identity functions and replace them with only one:
identity :: a -> a
identity x = x
Note how we used a as a type variable here. The 3 type definitions we’ve seen so far, share the same "shape". You can see that the type definitions of identity for Bool type and for Integer type both
"fit" this new type definition if you imagine variable a being a placeholder for other types:
boolIdentity :: Bool -> Bool
integerIdentity :: Integer -> Integer
identity :: a -> a
Let’s run the code in ghci and convince ourselves that we can indeed use this new, generic identity on both Bool and Integer values:
identity True
works and results in:
And at the same time:
identity 5
works as well and results in:
At this point, it’s important to emphasize a certain point. Given the implementation that we’ve used for the identity function:
identity x = x
we could not give it, for example, the following type:
identity :: Integer -> Bool
This type definition in itself is not absurd. You can easily imagine functions that accept integers and return true or false (for example based on some condition).
However, in this particular case, where we take argument x and immediately return it, without doing anything else, it’s clearly impossible for x to "magically" change the type.
And this fact is reflected even in the most generic type definition of identity:
identity :: a -> a
Note that this definition states that identity accepts a value of type a and returns a value of that same type – namely a again.
On the flip side, the following definition:
identity :: a -> b
wouldn’t be allowed. In fact, it won’t compile, with an error, part of which says:
Couldn't match expected type ‘b’ with actual type ‘a’
So the compiler literally says that in place of b there should be the type variable a present.
That’s because – given the current implementation – it’s impossible for the value named x to just change the type out of nowhere.
And indeed, when Haskell infers the type of untyped code, it goes for the most general interpretation possible.
You can convince yourself of that, by removing the type definition from lesson_02.hs file, and leaving only the implementation:
identity x = x
And then compiling it in ghci and checking the type of identity by running:
:t identity
As an answer you will see:
identity :: p -> p
This is exactly the same type definition that we wrote by hand. It simply uses a different letter (p instead of a).
At the very end of this section, it would be good to mention, that you don’t actually have to define the identity function by yourself. We only did it for educational purposes.
Prelude – the standard library for Haskell – has it always available for you under the shorter name id.
To convince yourself of that, write the following in ghci:
:t id
As a response you will see, more than familiar now, type:
id :: a -> a
In the second part of "The Most Gentle Introduction Ever" to Haskell, we used operators and functions on numbers and discovered that they are in fact almost the same. We’ve also seen that there are
no functions of multiple variables in Haskell – they are just functions that return other functions.
And at the end, we expanded our understanding of what a type definition can be, by showing the simplest possible usage of type variables – using the identity function as an example.
All those things were meant to make you feel more comfortable with the concept of a function in Haskell. And in the future article, we will use a similar approach to enhance our understanding of
algebraic data structures, especially custom ones.
So see you next time and thanks for reading!
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High-Power MicroSpot® Focusing Objectives for VIS and NIR Lasers
• 5X, 10X, 20X, or 50X Magnification
• 532 nm, 850 nm, or 1064 nm Center Wavelength
• Better than 95% Transmission at Center Wavelength
• Damage Threshold ≥15 J/cm^2
5X Magnification,
980 - 1130 nm AR Coating
50X Magnification,
790 - 910 nm AR Coating
20X Magnification,
495 - 570 nm AR Coating
• Designed for High-Power Industrial Lasers
□ 532 nm Center Wavelength, 490 - 570 nm AR Coating
□ 850 nm Center Wavelength, 790 - 910 nm AR Coating
□ 1064 nm Center Wavelength, 980 - 1130 nm AR Coating
• All Fused Silica Lens Design
• RMS-Threaded (0.800"-36) Housing
• Damage Threshold of ≥15.0 J/cm^2
(See the Damage Thresholds Tab for More Details)
• Laser Engraving
• Laser Cutting
• 2D and 3D Photolithography
• 3D Printing
• Stealth Dicing
• Laser Welding
Click on the red Document icon next to the item numbers below to access the Zemax file download. Our entire Zemax Catalog is also available.
This diagram illustrates the labels, working distance, and parfocal distance. The format of the engraved specifications will vary between objectives and manufacturers. (See the
Objective Tutorial
tab for more information about microscope objective types.)
Thorlabs' High-Power MicroSpot^® Focusing Objectives are designed to focus on-axis laser beams to a diffraction-limited spot. We offer center wavelengths of 532 nm, 850 nm, and 1064 nm, with AR
coatings of 490 - 570 nm, 790 - 910 nm, and 980 - 1130 nm, respectively. These objectives can be used at common fiber laser wavelengths including 515 nm, 808 nm, 1030 nm, or 1070 nm. Objectives with
a 532 nm or 1064 nm center wavelength are also ideal for use with Nd:YAG lasers; for additional optics for use with these lasers, see the Nd:YAG Optics tab. All of the focusing objectives below offer
a damage threshold of ≥15 J/cm^2 (see the Damage Thresholds tab for more details). The RMS threading on each of our MicroSpot objectives allows for easy integration into existing systems, and their
robust housing and fused silica lens design is built to hold up to consistent industrial or laboratory use.
Focusing objectives can be used in a variety of applications where intense optical power is necessary, such as laser cutting or engraving. At lower powers, focused laser light can be used for wafer
inspection or to activate special types of photoresist in photolithography. Because all the objectives on this page have a limited field of view, laser scanning should be performed by moving either
the sample or the objective. When incorporating these objectives into a system, note that the labeled magnifications of these objectives are calculated assuming the objective is being use with a 200
mm focal length tube lens.
Each 50X objective is equipped with a cover glass correction collar with engraved graduations, in mm, for fused silica coverslip thicknesses from 0 to 1 mm. In addition, the 1064 nm variant is
engraved with a scale for up to 1.2 mm thick silicon carbide coverslips. The scales span the majority of the objectives' circumferences, allowing for smooth, precise adjustment. Once the correct
position is found, the collar can be locked in place by tightening the setscrew below it with the included 0.050" hex key. Custom graduations for specific cover glass materials including sapphire (Al
[2]O[3]), silicon (Si), silicon carbide (SiC), gallium arsenide (GaAs), and gallium nitride (GaN) can be made on request; please contact Tech Support for more details.
These objectives are capable of producing a near-diffraction-limited spot size when used with a monochromatic source within the 450 - 2100 nm range that fills the entrance aperture, also known as the
entrance pupil. However, if used at a wavelength other than the design wavelength, the effective focal length listed on the Specs tab will shift and the AR coating will no longer be optimized; see
the Graphs tab for AR coating plots. Custom AR Coatings are available by contacting Tech Support to optimize the performance of these objectives at other wavelengths. When working with wavelengths
outside of the visible, consider using some of Thorlabs' laser viewing cards to help locate and align your beam.
Our MicroSpot objectives are externally RMS-threaded (0.800"-36), which allows them to be mounted directly to our fiber launch systems, DIY Cerna^® Microscope Systems, and microscope objective
turrets; to convert RMS threads to M32 x 0.75 threads, we offer the M32RMSS brass thread adapter. Our 5X, 10X, and 20X objectives can be mounted to any of our flexure stages using an HCS013 RMS
mount. An objective case (OC2RMS lid and OC22 canister) is included with our 50X objectives and an aluminum cap (RMSCP1) is available for purchase separately.
For wavelengths between 192 nm - 500 nm, Thorlabs offers UV MicroSpot Laser Focusing Objectives in a number of magnifications; this range covers many excimer lasers which cure photoresist, such as
KrF lasers (248 nm) and ArF lasers (193 nm).
for Raw Data
The shaded region of this graph identifies the specified wavelength range of the AR coating.
for Raw Data
This graph shows the reflectance per surface. The shaded region identifies the specified wavelength range of the AR coating.
AR Coating Specifications
Wavelength Range Average Reflectance per Surface Max Reflectance per Surface Damage Threshold
495 - 570 nm <0.2% (495 - 570 nm) <0.7% (488 - 580 nm) 15 J/cm^2 (532 nm, 20 Hz, 10 ns, Ø213 µm)
<0.2% (503 - 553 nm)
for Raw Data
The shaded region of this graph identifies the specified wavelength range of the AR coating.
for Raw Data
This graph shows the reflectance per surface. The shaded region identifies the specified wavelength range of the AR coating.
AR Coating Specifications
Wavelength Range Average Reflectance per Surface Max Reflectance per Surface Damage Threshold
790 - 910 nm <0.2% (790 - 910 nm) <0.7% (794 - 935 nm) 20 J/cm^2 (850 nm, 10 Hz, 10 ns, Ø300 µm)
<0.2% (833 - 855 nm)
for Raw Data
The shaded region of this graph identifies the specified wavelength range of the AR coating.
for Raw Data
This graph shows the reflectance per surface. The shaded region identifies the specified wavelength range of the AR coating.
AR Coating Specifications
Wavelength Range Average Reflectance per Surface Max Reflectance per Surface Damage Threshold
980 - 1130 nm <0.2% (980 - 1130 nm) <0.7% (960 - 1160 nm) 20 J/cm^2 (1064 nm, 20 Hz, 10 ns, Ø395 µm)
<0.2% (1000 - 1100 nm)
Chromatic Aberration Correction per ISO Standard 19012-2
Objective Class Common Abbreviations Axial Focal Shift Tolerances^a
Achromat ACH, ACHRO, ACHROMAT |δ[C'] - δ[F']| ≤ 2 x δ[ob]
Semiapochromat |δ[C'] - δ[F']| ≤ 2 x δ[ob]
(or Fluorite) SEMIAPO, FL, FLU |δ[F'] - δ[e]| ≤ 2.5 x δ[ob]
|δ[C'] - δ[e]| ≤ 2.5 x δ[ob]
|δ[C'] - δ[F']| ≤ 2 x δ[ob]
Apochromat APO |δ[F'] - δ[e]| ≤ δ[ob]
|δ[C'] - δ[e]| ≤ δ[ob]
Super Apochromat SAPO See Footnote b
Improved Visible Apochromat VIS+ See Footnotes b and c
Parts of a Microscope Objective
Click on each label for more details.
This microscope objective serves only as an example. The features noted above with an asterisk may not be present on all objectives; they may be added, relocated, or removed from objectives based on
the part's needs and intended application space.
Objective Tutorial
This tutorial describes features and markings of objectives and what they tell users about an objective's performance.
Objective Class and Aberration Correction
Objectives are commonly divided by their class. An objective's class creates a shorthand for users to know how the objective is corrected for imaging aberrations. There are two types of aberration
corrections that are specified by objective class: field curvature and chromatic aberration.
Field curvature (or Petzval curvature) describes the case where an objective's plane of focus is a curved spherical surface. This aberration makes widefield imaging or laser scanning difficult, as
the corners of an image will fall out of focus when focusing on the center. If an objective's class begins with "Plan", it will be corrected to have a flat plane of focus.
Images can also exhibit chromatic aberrations, where colors originating from one point are not focused to a single point. To strike a balance between an objective's performance and the complexity of
its design, some objectives are corrected for these aberrations at a finite number of target wavelengths.
Five objective classes are shown in the table to the right; only three common objective classes are defined under the International Organization for Standards ISO 19012-2: Microscopes -- Designation
of Microscope Objectives -- Chromatic Correction. Due to the need for better performance, we have added two additional classes that are not defined in the ISO classes.
Immersion Methods
Click on each image for more details.
Objectives can be divided by what medium they are designed to image through. Dry objectives are used in air; whereas dipping and immersion objectives are designed to operate with a fluid between the
objective and the front element of the sample.
Glossary of Terms
The back focal length defines the location of the intermediate image plane. Most modern objectives will have this plane at infinity, known as infinity correction, and will signify
this with an infinity symbol (∞). Infinity-corrected objectives are designed to be used with a tube lens between the objective and eyepiece. Along with increasing
Back Focal Length intercompatibility between microscope systems, having this infinity-corrected space between the objective and tube lens allows for additional modules (like beamsplitters, filters,
and Infinity or parfocal length extenders) to be placed in the beam path.
Note that older objectives and some specialty objectives may have been designed with finite back focal lengths. In their inception, finite back focal length objectives were meant
to interface directly with the objective's eyepiece.
The entrance pupil diameter (EP), sometimes referred to as the entrance aperture diameter, corresponds to the appropriate beam diameter one should use to allow the objective to
Entrance Pupil function properly.
Diameter (EP)
EP = 2 × NA × Effective Focal Length
Field Number (FN) The field number corresponds to the diameter of the field of view in object space (in millimeters) multiplied by the objective's magnification.
Field of View Field Number = Field of View Diameter × Magnification
The magnification (M) of an objective is the lens tube focal length (L) divided by the objective's effective focal length (F). Effective focal length is sometimes abbreviated EFL:
M = L / EFL .
Magnification (M)
The total magnification of the system is the magnification of the objective multiplied by the magnification of the eyepiece or camera tube. The specified magnification on the
microscope objective housing is accurate as long as the objective is used with a compatible tube lens focal length. Objectives will have a colored ring around their body to signify
their magnification. This is fairly consistent across manufacturers; see the Parts of a Microscope Objective section for more details.
Numerical aperture, a measure of the acceptance angle of an objective, is a dimensionless quantity. It is commonly expressed as:
Numerical Aperture NA = n[i] × sinθ[a
]where θ[a] is the maximum 1/2 acceptance angle of the objective, and n[i] is the index of refraction of the immersion medium. This medium is typically air, but may also be water,
oil, or other substances.
Working Distance The working distance, often abbreviated WD, is the distance between the front element of the objective and the top of the specimen (in the case of objectives that are intended to
(WD) be used without a cover glass) or top of the cover glass, depending on the design of the objective. The cover glass thickness specification engraved on the objective designates
whether a cover glass should be used.
Threading allows an objective to be mounted to a nosepiece or turret. Objectives can have a number of different thread pitches; Thorlabs offers a selection of microscope thread adapters to facilitate
mounting objectives in different systems.
The shoulder is located at the base of the objective threading and marks the beginning of the exposed objective body when it is fully threaded into a nosepiece or other objective mount.
A cover glass, or coverslip, is a small, thin sheet of glass that can be placed on a wet sample to create a flat surface to image across.
The most common, a standard #1.5 cover glass, is designed to be 0.17 mm thick. Due to variance in the manufacturing process the actual thickness may be different. The correction collar present on
select objectives is used to compensate for cover glasses of different thickness by adjusting the relative position of internal optical elements. Note that many objectives do not have a variable
cover glass correction, in which case the objectives have no correction collar. For example, an objective could be designed for use with only a #1.5 cover glass. This collar may also be located near
the bottom of the objective, instead of the top as shown in the diagram.
The graph above shows the magnitude of spherical aberration versus the thickness of the coverslip used for 632.8 nm light. For the typical coverslip thickness of 0.17 mm, the spherical aberration
caused by the coverslip does not exceed the diffraction-limited aberration for objectives with NA up to 0.40.
The labeling area for an objective usually falls in the middle of the objective body. The labeling found here is dictated by ISO 8578: Microscopes -- Marking of Objectives and Eyepieces, but not all
manufacturers adhere strictly to this standard. Generally, one can expect to find the following information in this area:
• Branding/Manufacturer
• Aberration Correction (Objective Class)
• Magnification
• Numerical Aperture (NA)
• Back Focal Length (Infinity Correction)
• Suitable Cover Glass Thicknesses
• Working Distance
Additionally, the objective label area may include the objective's specified wavelength range, specialty features or design properties, and more. The exact location and size of each and any of these
elements can vary.
In order to facilitate fast identification, nearly all microscope objectives have a colored ring that circumscribes the body. A breakdown of what magnification each color signifies is given in the
table below.
Magnification Identifier Color Ring
Codes per ISO 8578
Black 1X or 1.25X Light Green 16X or 20X
Grey 1.6X or 2X Dark Green 25X or 32X
Brown 2.5X or 3.2X Light Blue 40X or 50X
Red 4X or 5X Dark Blue 63X or 80X
Orange 6.3X or 8X White 100X, 125X, or 160X
Yellow 10X or 12.5X
Immersion Identifier Color Ring Codes
per ISO 8578
None Dry
Black Oil
White Water
Orange Glycerol
Red Others
If an objective is used for water dipping, water immersion, or oil immersion, a second colored ring may be placed beneath the magnification identifier. If the objective is designed to be used with
water, this ring will be white. If the objective is designed to be used with oil, this ring will be black. Dry objectives lack this identifier ring entirely. See the table to the right for a complete
list of immersion identifiers.
Objectives that feature a built-in iris diaphragm are ideal for darkfield microscopy. The iris diaphragm is designed to be partially closed during darkfield microscopy in order to preserve the
darkness of the background. This is absolutely necessary for high numerical aperture (above NA = 1.2) oil immersion objectives when using an oil immersion darkfield condenser. For ordinary
brightfield observations, the iris diaphragm should be left fully open.
Also referred to as the parfocal distance, this is the length from the shoulder to the top of the specimen (in the case of objectives that are intended to be used without a cover glass) or the top of
the cover glass. When working with multiple objectives in a turret, it is helpful if all of the parfocal distances are identical, so little refocusing will be required when switching between
objectives. Thorlabs offers parfocal length extenders for instances in which the parfocal length needs to be increased.
The working distance, often abbreviated WD, is the distance between the front element of the objective and the top of the specimen (in the case of objectives that are intended to be used without a
cover glass) or top of the cover glass. The cover glass thickness specification engraved on the objective designates whether a cover glass should be used.
Objectives with very small working distances may have a retraction stopper incorporated into the tip. This is a spring-loaded section which compresses to limit the force of impact in the event of an
unintended collision with the sample.
Immersion Identifier Color Ring Codes
per ISO 8578
None Dry
Black Oil
White Water
Orange Glycerol
Red Others
Dry objectives are designed to have an air gap between the objective and the specimen.
Objectives following ISO 8578: Microscopes -- Marking of Objectives and Eyepieces will be labeled with an identifier ring to tell the user what immersion fluid the objective is designed to be used
with; a list of ring colors can be found in the table to the right.
Immersion Identifier Color Ring Codes
per ISO 8578
None Dry
Black Oil
White Water
Orange Glycerol
Red Others
Dipping objectives are designed to correct for the aberrations introduced by the specimen being submerged in an immersion fluid. The tip of the objective is either dipped or entirely submerged into
the fluid.
Objectives following ISO 8578: Microscopes -- Marking of Objectives and Eyepieces will be labeled with an identifier ring to tell the user what immersion fluid the objective is designed to be used
with; a list of ring colors can be found in the table to the right.
Immersion Identifier Color Ring Codes
per ISO 8578
None Dry
Black Oil
White Water
Orange Glycerol
Red Others
Immersion objectives are similar to water-dipping objectives; however, in this case the sample is under a cover glass. A drop of fluid is then added to the top of the cover glass, and the tip of the
objective is brought into contact with the fluid. Often, immersion objectives feature a correction collar to adjust for cover glasses with different thicknesses. Immersion fluids include water, oil
(such as MOIL-30), and glycerol.
Using an immersion fluid with a high refractive index allows objectives to achieve numerical apertures greater than 1.0. However, if an immersion objective is used without the fluid present, the
image quality will be very low. Objectives following ISO 8578: Microscopes -- Marking of Objectives and Eyepieces will be labeled with an identifier ring to tell the user what immersion fluid the
objective is designed to be used with; a list of ring colors can be found in the table above.
When viewing an image with a camera, the system magnification is the product of the objective and camera tube magnifications. When viewing an image with trinoculars, the system magnification is the
product of the objective and eyepiece magnifications.
Manufacturer Tube Lens
Focal Length
Leica f = 200 mm
Mitutoyo f = 200 mm
Nikon f = 200 mm
Olympus f = 180 mm
Thorlabs f = 200 mm
Zeiss f = 165 mm
Magnification and Sample Area Calculations
The magnification of a system is the multiplicative product of the magnification of each optical element in the system. Optical elements that produce magnification include objectives, camera tubes,
and trinocular eyepieces, as shown in the drawing to the right. It is important to note that the magnification quoted in these products' specifications is usually only valid when all optical elements
are made by the same manufacturer. If this is not the case, then the magnification of the system can still be calculated, but an effective objective magnification should be calculated first, as
described below.
To adapt the examples shown here to your own microscope, please use our Magnification and FOV Calculator, which is available for download by clicking on the red button above. Note the calculator is
an Excel spreadsheet that uses macros. In order to use the calculator, macros must be enabled. To enable macros, click the "Enable Content" button in the yellow message bar upon opening the file.
Example 1: Camera Magnification
When imaging a sample with a camera, the image is magnified by the objective and the camera tube. If using a 20X Nikon objective and a 0.75X Nikon camera tube, then the image at the camera has 20X ×
0.75X = 15X magnification.
Example 2: Trinocular Magnification
When imaging a sample through trinoculars, the image is magnified by the objective and the eyepieces in the trinoculars. If using a 20X Nikon objective and Nikon trinoculars with 10X eyepieces, then
the image at the eyepieces has 20X × 10X = 200X magnification. Note that the image at the eyepieces does not pass through the camera tube, as shown by the drawing to the right.
Using an Objective with a Microscope from a Different Manufacturer
Magnification is not a fundamental value: it is a derived value, calculated by assuming a specific tube lens focal length. Each microscope manufacturer has adopted a different focal length for their
tube lens, as shown by the table to the right. Hence, when combining optical elements from different manufacturers, it is necessary to calculate an effective magnification for the objective, which is
then used to calculate the magnification of the system.
The effective magnification of an objective is given by Equation 1:
Here, the Design Magnification is the magnification printed on the objective, f[Tube Lens in Microscope] is the focal length of the tube lens in the microscope you are using, and f[Design Tube Lens
of Objective] is the tube lens focal length that the objective manufacturer used to calculate the Design Magnification. These focal lengths are given by the table to the right.
Note that Leica, Mitutoyo, Nikon, and Thorlabs use the same tube lens focal length; if combining elements from any of these manufacturers, no conversion is needed. Once the effective objective
magnification is calculated, the magnification of the system can be calculated as before.
Example 3: Trinocular Magnification (Different Manufacturers)
When imaging a sample through trinoculars, the image is magnified by the objective and the eyepieces in the trinoculars. This example will use a 20X Olympus objective and Nikon trinoculars with 10X
Following Equation 1 and the table to the right, we calculate the effective magnification of an Olympus objective in a Nikon microscope:
The effective magnification of the Olympus objective is 22.2X and the trinoculars have 10X eyepieces, so the image at the eyepieces has 22.2X × 10X = 222X magnification.
Sample Area When Imaged on a Camera
When imaging a sample with a camera, the dimensions of the sample area are determined by the dimensions of the camera sensor and the system magnification, as shown by Equation 2.
The camera sensor dimensions can be obtained from the manufacturer, while the system magnification is the multiplicative product of the objective magnification and the camera tube magnification (see
Example 1). If needed, the objective magnification can be adjusted as shown in Example 3.
As the magnification increases, the resolution improves, but the field of view also decreases. The dependence of the field of view on magnification is shown in the schematic to the right.
Example 4: Sample Area
The dimensions of the camera sensor in Thorlabs' previous-generation 1501M-USB Scientific Camera are 8.98 mm × 6.71 mm. If this camera is used with the Nikon objective and trinoculars from Example 1,
which have a system magnification of 15X, then the image area is:
Sample Area Examples
The images of a mouse kidney below were all acquired using the same objective and the same camera. However, the camera tubes used were different. Read from left to right, they demonstrate that
decreasing the camera tube magnification enlarges the field of view at the expense of the size of the details in the image.
Figure 1:
A Gaussian intensity profile only asymptotically approaches zero, so the spot size is defined by convention as either the full-width at half-maximum (FWHM) or 1/e
value. We use the latter convention in this tutorial.
Spot Size Tutorial
The spot size achieved by focusing a laser beam with a lens or objective is an important parameter in many applications. This tutorial describes how the ratio of the initial beam diameter to the
entrance pupil diameter, known as the truncation ratio, affects the focused spot size and provides expressions for calculating the spot size as a function of this ratio. Because the power transmitted
by the focusing optic also depends upon the truncation ratio, the optimal balance between spot size and power transmission will depend upon the given application.
For each focusing objective that Thorlabs provides, we provide an estimate of the focused spot size when the incident Gaussian spot size (1/e^2) is the same as the diameter of the entrance pupil.
With this choice, the focused spot size is given by:
s ≈ 1.83λN, or equivalently, s ≈ 1.83λ/(2*NA), where NA is the numerical aperture of the objective; and the transmitted power is 86% of that of the incident beam.
Gaussian Beam Profile
Laser beams typically have a tranverse intensity profile that may be approximated by a Gaussian function,
where w is the beam half-width or beam waist radius, conventionally defined as the radius (r) at which the intensity has decreased from its maximum axial value of I[0] to I[0]/e^2 ≈ 0.14I[0]. The
spot size of a laser beam may be defined as twice the beam waist radius, and the corresponding circle with diameter equal to the spot size thus contains 86% of the beam's total intensity.
When a laser beam is focused by an objective, the resulting spot size (s) will depend upon the wavelength of the light (λ), the beam diameter as it enters the objective (S), the focal length of the
objective (f), and the entrance pupil diameter of the objective (D). Dimensionless parameters are formed by taking the ratio of the focal length to the entrance pupil diameter and the ratio of the
beam diameter to the entrance pupil diameter, which are known respectively as the f-number (N = f/D) and the truncation ratio (T = S/D). The f-number is fixed for a given objective, while the
truncation ratio may be tuned by increasing or decreasing the incident beam diameter.
Spot Size vs. Truncation Ratio
The focused spot size is expressed in terms of the wavelength, truncation ratio, and f-number as
where K(T) is called the spot size coefficient and is a function of the truncation ratio [1]. In Figure 2 below, numerically computed values for K, obtained by calculating the focused intensity
profile and extracting the focused spot size for discrete values of T, are plotted as black squares. As discussed in detail below, the solid-and-dashed blue line represents the coefficient predicted
by Gaussian beam theory, the gray line represents the value of K for an Airy disk intensity profile, and the red line is a polynomial fit to the numerical values for T ≥ 0.5.
Figure 2:
The shaded blue region below
< 0.5 indicates where Gaussian beam theory provides an accurate estimate of the focused spot size; however, above this value the effects of truncation cannot be neglected and the actual spot size is
larger. As
→ ∞, the coefficient approaches the Airy disk value of 1.6449 as indicated by the solid gray line.
Figure 3:
The power transmitted through an entrance aperture of diameter
by a Gaussian beam with spot size
as a function of the truncation ratio
. The shaded blue region corresponds to the region where truncation effects on the spot size are negligible.
In both the large (T → ∞) and small (T → 0) limits, K approaches well-known theoretical results. For small T, which corresponds to an entrance pupil much larger than the Gaussian spot, K obeys the
relation 1.27/T. This can be obtained from Gaussian beam propagation theory [2] which predicts that the minimum spot size a Gaussian beam can be focused to is s ≈ 1.27λf/S. By inserting factors of D
to write this in terms of N and T, this expression can be cast into the same form as the spot size equation above, s = (1.27/T)λN, giving the result K = 1.27/T. As seen in Figure 2 above, this
accurately predicts the focused spot size up to T ≈ 0.5, when the entrance pupil diameter D is twice as large as the spot size S. Above T = 0.5, it underestimates the value of K, as indicated by the
deviation of the dashed blue line from the numerical results.
Figure 4:
The spot size of an Airy disk profile is typically defined by the radius where the first intensity zero occurs. When the radius is expressed in units of
, this corresponds to a radius of 1.22
, or a spot size of 2.44
. The 1/e
radius for the same profile is 0.82245
, or a spot size of 1.6449
As T is increased, the illumination of the aperture becomes more and more uniform. The resulting intensity profile of the focused spot will therefore transition from a Gaussian profile to an Airy
disk profile. In the large T limit, this is reflected in the value of K, which approaches a constant value of 1.6449 as T → ∞. This value corresponds to the 1/e^2 spot size of an Airy disk instead of
the better-known 2.44λN value which is where the first intensity minimum occurs, as shown in Figure 4 [3].
For intermediate values of T, which is the range in which most applications will fall, there is no exact theoretical result for K. Instead, the red line above represents a two-term polynomial fit to
the numerical results, the coefficients of which are specified in the table below (the polynomial fit was performed using 1/T as the independent variable). This expression may be used to estimate K
for T ≥ 0.5.
Spot Size Coefficient, K(T)
Truncation Ratio Equation
T < 0.5
(Gaussian Regime)
T ≥ 0.5
(Intermediate and Uniform/Airy Regimes)
Power Transmission and Spot Size
The results presented above suggest that, in the intermediate T regime, a smaller spot size may be achieved by increasing T. This, however, comes at the cost of reducing the overall power transmitted
through the entrance aperture, and reductions in spot size may not be worth the loss in power. The power transmitted through an entrance pupil of diameter D as a function of T is plotted above in
Figure 3. Already at T = 1, when the Gaussian spot size has the same diameter as the entrance pupil, the transmitted power is 86% of the incident power. By increasing T from 1 to 2, the spot size is
reduced by only ≈ 9%, while the transmitted power decreases from 86% to 40%.
The optimal balance between spot size and power transmission will depend upon the given application. For each focusing objective that Thorlabs offers, we provide an estimate of the spot size using T
= 1, when the Gaussian spot size is the same as the diameter of the entrance pupil. With this choice, the spot size is given by: s ≈ 1.83λN, or equivalently, s ≈ 1.83λ/(2*NA), where NA is the
numerical aperture of the objective.
[1] Hakan Urey, "Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams," Appl. Opt. 43, 620-625 (2004)
[2] Sidney A. Self, "Focusing of spherical Gaussian beams," Appl. Opt. 22, 658-661 (1983)
[3] Eugene Hecht, "Optics," 4th Ed., Addison-Wesley (2002)
Damage Threshold Specifications
Item # Suffix Damage Threshold
-532 15 J/cm^2 (532 nm, 20 Hz, 10 ns, Ø213 µm)
-850 20 J/cm^2 (850 nm, 10 Hz, 10 ns, Ø300 µm)
-1064 20 J/cm^2 (1064 nm, 20 Hz, 10 ns, Ø395 µm)
Damage Threshold Data for High-Power Focusing Objectives AR Coatings
The specifications to the right are measured data for the antireflective (AR) coatings deposited onto the optical surface of our high power focusing objectives. Damage threshold specifications are
constant for a given coating type, regardless of the focal length or magnification.
Laser Induced Damage Threshold Tutorial
The following is a general overview of how laser induced damage thresholds are measured and how the values may be utilized in determining the appropriateness of an optic for a given application. When
choosing optics, it is important to understand the Laser Induced Damage Threshold (LIDT) of the optics being used. The LIDT for an optic greatly depends on the type of laser you are using. Continuous
wave (CW) lasers typically cause damage from thermal effects (absorption either in the coating or in the substrate). Pulsed lasers, on the other hand, often strip electrons from the lattice structure
of an optic before causing thermal damage. Note that the guideline presented here assumes room temperature operation and optics in new condition (i.e., within scratch-dig spec, surface free of
contamination, etc.). Because dust or other particles on the surface of an optic can cause damage at lower thresholds, we recommend keeping surfaces clean and free of debris. For more information on
cleaning optics, please see our Optics Cleaning tutorial.
Testing Method
Thorlabs' LIDT testing is done in compliance with ISO/DIS 11254 and ISO 21254 specifications.
First, a low-power/energy beam is directed to the optic under test. The optic is exposed in 10 locations to this laser beam for 30 seconds (CW) or for a number of pulses (pulse repetition frequency
specified). After exposure, the optic is examined by a microscope (~100X magnification) for any visible damage. The number of locations that are damaged at a particular power/energy level is
recorded. Next, the power/energy is either increased or decreased and the optic is exposed at 10 new locations. This process is repeated until damage is observed. The damage threshold is then
assigned to be the highest power/energy that the optic can withstand without causing damage. A histogram such as that below represents the testing of one BB1-E02 mirror.
The photograph above is a protected aluminum-coated mirror after LIDT testing. In this particular test, it handled 0.43 J/cm^2 (1064 nm, 10 ns pulse, 10 Hz, Ø1.000 mm) before damage.
Example Test Data
Fluence # of Tested Locations Locations with Damage Locations Without Damage
1.50 J/cm^2 10 0 10
1.75 J/cm^2 10 0 10
2.00 J/cm^2 10 0 10
2.25 J/cm^2 10 1 9
3.00 J/cm^2 10 1 9
5.00 J/cm^2 10 9 1
According to the test, the damage threshold of the mirror was 2.00 J/cm^2 (532 nm, 10 ns pulse, 10 Hz, Ø0.803 mm). Please keep in mind that these tests are performed on clean optics, as dirt and
contamination can significantly lower the damage threshold of a component. While the test results are only representative of one coating run, Thorlabs specifies damage threshold values that account
for coating variances.
Continuous Wave and Long-Pulse Lasers
When an optic is damaged by a continuous wave (CW) laser, it is usually due to the melting of the surface as a result of absorbing the laser's energy or damage to the optical coating (antireflection)
[1]. Pulsed lasers with pulse lengths longer than 1 µs can be treated as CW lasers for LIDT discussions.
When pulse lengths are between 1 ns and 1 µs, laser-induced damage can occur either because of absorption or a dielectric breakdown (therefore, a user must check both CW and pulsed LIDT). Absorption
is either due to an intrinsic property of the optic or due to surface irregularities; thus LIDT values are only valid for optics meeting or exceeding the surface quality specifications given by a
manufacturer. While many optics can handle high power CW lasers, cemented (e.g., achromatic doublets) or highly absorptive (e.g., ND filters) optics tend to have lower CW damage thresholds. These
lower thresholds are due to absorption or scattering in the cement or metal coating.
Pulsed lasers with high pulse repetition frequencies (PRF) may behave similarly to CW beams. Unfortunately, this is highly dependent on factors such as absorption and thermal diffusivity, so there is
no reliable method for determining when a high PRF laser will damage an optic due to thermal effects. For beams with a high PRF both the average and peak powers must be compared to the equivalent CW
power. Additionally, for highly transparent materials, there is little to no drop in the LIDT with increasing PRF.
In order to use the specified CW damage threshold of an optic, it is necessary to know the following:
1. Wavelength of your laser
2. Beam diameter of your beam (1/e^2)
3. Approximate intensity profile of your beam (e.g., Gaussian)
4. Linear power density of your beam (total power divided by 1/e^2 beam diameter)
Thorlabs expresses LIDT for CW lasers as a linear power density measured in W/cm. In this regime, the LIDT given as a linear power density can be applied to any beam diameter; one does not need to
compute an adjusted LIDT to adjust for changes in spot size, as demonstrated by the graph to the right. Average linear power density can be calculated using the equation below.
The calculation above assumes a uniform beam intensity profile. You must now consider hotspots in the beam or other non-uniform intensity profiles and roughly calculate a maximum power density. For
reference, a Gaussian beam typically has a maximum power density that is twice that of the uniform beam (see lower right).
Now compare the maximum power density to that which is specified as the LIDT for the optic. If the optic was tested at a wavelength other than your operating wavelength, the damage threshold must be
scaled appropriately. A good rule of thumb is that the damage threshold has a linear relationship with wavelength such that as you move to shorter wavelengths, the damage threshold decreases (i.e., a
LIDT of 10 W/cm at 1310 nm scales to 5 W/cm at 655 nm):
While this rule of thumb provides a general trend, it is not a quantitative analysis of LIDT vs wavelength. In CW applications, for instance, damage scales more strongly with absorption in the
coating and substrate, which does not necessarily scale well with wavelength. While the above procedure provides a good rule of thumb for LIDT values, please contact Tech Support if your wavelength
is different from the specified LIDT wavelength. If your power density is less than the adjusted LIDT of the optic, then the optic should work for your application.
Please note that we have a buffer built in between the specified damage thresholds online and the tests which we have done, which accommodates variation between batches. Upon request, we can provide
individual test information and a testing certificate. The damage analysis will be carried out on a similar optic (customer's optic will not be damaged). Testing may result in additional costs or
lead times. Contact Tech Support for more information.
Pulsed Lasers
As previously stated, pulsed lasers typically induce a different type of damage to the optic than CW lasers. Pulsed lasers often do not heat the optic enough to damage it; instead, pulsed lasers
produce strong electric fields capable of inducing dielectric breakdown in the material. Unfortunately, it can be very difficult to compare the LIDT specification of an optic to your laser. There are
multiple regimes in which a pulsed laser can damage an optic and this is based on the laser's pulse length. The highlighted columns in the table below outline the relevant pulse lengths for our
specified LIDT values.
Pulses shorter than 10^-9 s cannot be compared to our specified LIDT values with much reliability. In this ultra-short-pulse regime various mechanics, such as multiphoton-avalanche ionization, take
over as the predominate damage mechanism [2]. In contrast, pulses between 10^-7 s and 10^-4 s may cause damage to an optic either because of dielectric breakdown or thermal effects. This means that
both CW and pulsed damage thresholds must be compared to the laser beam to determine whether the optic is suitable for your application.
Pulse Duration t < 10^-9 s 10^-9 < t < 10^-7 s 10^-7 < t < 10^-4 s t > 10^-4 s
Damage Mechanism Avalanche Ionization Dielectric Breakdown Dielectric Breakdown or Thermal Thermal
Relevant Damage Specification No Comparison (See Above) Pulsed Pulsed and CW CW
When comparing an LIDT specified for a pulsed laser to your laser, it is essential to know the following:
1. Wavelength of your laser
2. Energy density of your beam (total energy divided by 1/e^2 area)
3. Pulse length of your laser
4. Pulse repetition frequency (prf) of your laser
5. Beam diameter of your laser (1/e^2 )
6. Approximate intensity profile of your beam (e.g., Gaussian)
The energy density of your beam should be calculated in terms of J/cm^2. The graph to the right shows why expressing the LIDT as an energy density provides the best metric for short pulse sources. In
this regime, the LIDT given as an energy density can be applied to any beam diameter; one does not need to compute an adjusted LIDT to adjust for changes in spot size. This calculation assumes a
uniform beam intensity profile. You must now adjust this energy density to account for hotspots or other nonuniform intensity profiles and roughly calculate a maximum energy density. For reference a
Gaussian beam typically has a maximum energy density that is twice that of the 1/e^2 beam.
Now compare the maximum energy density to that which is specified as the LIDT for the optic. If the optic was tested at a wavelength other than your operating wavelength, the damage threshold must be
scaled appropriately [3]. A good rule of thumb is that the damage threshold has an inverse square root relationship with wavelength such that as you move to shorter wavelengths, the damage threshold
decreases (i.e., a LIDT of 1 J/cm^2 at 1064 nm scales to 0.7 J/cm^2 at 532 nm):
You now have a wavelength-adjusted energy density, which you will use in the following step.
Beam diameter is also important to know when comparing damage thresholds. While the LIDT, when expressed in units of J/cm², scales independently of spot size; large beam sizes are more likely to
illuminate a larger number of defects which can lead to greater variances in the LIDT [4]. For data presented here, a <1 mm beam size was used to measure the LIDT. For beams sizes greater than 5 mm,
the LIDT (J/cm2) will not scale independently of beam diameter due to the larger size beam exposing more defects.
The pulse length must now be compensated for. The longer the pulse duration, the more energy the optic can handle. For pulse widths between 1 - 100 ns, an approximation is as follows:
Use this formula to calculate the Adjusted LIDT for an optic based on your pulse length. If your maximum energy density is less than this adjusted LIDT maximum energy density, then the optic should
be suitable for your application. Keep in mind that this calculation is only used for pulses between 10^-9 s and 10^-7 s. For pulses between 10^-7 s and 10^-4 s, the CW LIDT must also be checked
before deeming the optic appropriate for your application.
Please note that we have a buffer built in between the specified damage thresholds online and the tests which we have done, which accommodates variation between batches. Upon request, we can provide
individual test information and a testing certificate. Contact Tech Support for more information.
[1] R. M. Wood, Optics and Laser Tech. 29, 517 (1998).
[2] Roger M. Wood, Laser-Induced Damage of Optical Materials (Institute of Physics Publishing, Philadelphia, PA, 2003).
[3] C. W. Carr et al., Phys. Rev. Lett. 91, 127402 (2003).
[4] N. Bloembergen, Appl. Opt. 12, 661 (1973).
In order to illustrate the process of determining whether a given laser system will damage an optic, a number of example calculations of laser induced damage threshold are given below. For assistance
with performing similar calculations, we provide a spreadsheet calculator that can be downloaded by clicking the button to the right. To use the calculator, enter the specified LIDT value of the
optic under consideration and the relevant parameters of your laser system in the green boxes. The spreadsheet will then calculate a linear power density for CW and pulsed systems, as well as an
energy density value for pulsed systems. These values are used to calculate adjusted, scaled LIDT values for the optics based on accepted scaling laws. This calculator assumes a Gaussian beam
profile, so a correction factor must be introduced for other beam shapes (uniform, etc.). The LIDT scaling laws are determined from empirical relationships; their accuracy is not guaranteed. Remember
that absorption by optics or coatings can significantly reduce LIDT in some spectral regions. These LIDT values are not valid for ultrashort pulses less than one nanosecond in duration.
A Gaussian beam profile has about twice the maximum intensity of a uniform beam profile.
CW Laser Example
Suppose that a CW laser system at 1319 nm produces a 0.5 W Gaussian beam that has a 1/e^2 diameter of 10 mm. A naive calculation of the average linear power density of this beam would yield a value
of 0.5 W/cm, given by the total power divided by the beam diameter:
However, the maximum power density of a Gaussian beam is about twice the maximum power density of a uniform beam, as shown in the graph to the right. Therefore, a more accurate determination of the
maximum linear power density of the system is 1 W/cm.
An AC127-030-C achromatic doublet lens has a specified CW LIDT of 350 W/cm, as tested at 1550 nm. CW damage threshold values typically scale directly with the wavelength of the laser source, so this
yields an adjusted LIDT value:
The adjusted LIDT value of 350 W/cm x (1319 nm / 1550 nm) = 298 W/cm is significantly higher than the calculated maximum linear power density of the laser system, so it would be safe to use this
doublet lens for this application.
Pulsed Nanosecond Laser Example: Scaling for Different Pulse Durations
Suppose that a pulsed Nd:YAG laser system is frequency tripled to produce a 10 Hz output, consisting of 2 ns output pulses at 355 nm, each with 1 J of energy, in a Gaussian beam with a 1.9 cm beam
diameter (1/e^2). The average energy density of each pulse is found by dividing the pulse energy by the beam area:
As described above, the maximum energy density of a Gaussian beam is about twice the average energy density. So, the maximum energy density of this beam is ~0.7 J/cm^2.
The energy density of the beam can be compared to the LIDT values of 1 J/cm^2 and 3.5 J/cm^2 for a BB1-E01 broadband dielectric mirror and an NB1-K08 Nd:YAG laser line mirror, respectively. Both of
these LIDT values, while measured at 355 nm, were determined with a 10 ns pulsed laser at 10 Hz. Therefore, an adjustment must be applied for the shorter pulse duration of the system under
consideration. As described on the previous tab, LIDT values in the nanosecond pulse regime scale with the square root of the laser pulse duration:
This adjustment factor results in LIDT values of 0.45 J/cm^2 for the BB1-E01 broadband mirror and 1.6 J/cm^2 for the Nd:YAG laser line mirror, which are to be compared with the 0.7 J/cm^2 maximum
energy density of the beam. While the broadband mirror would likely be damaged by the laser, the more specialized laser line mirror is appropriate for use with this system.
Pulsed Nanosecond Laser Example: Scaling for Different Wavelengths
Suppose that a pulsed laser system emits 10 ns pulses at 2.5 Hz, each with 100 mJ of energy at 1064 nm in a 16 mm diameter beam (1/e^2) that must be attenuated with a neutral density filter. For a
Gaussian output, these specifications result in a maximum energy density of 0.1 J/cm^2. The damage threshold of an NDUV10A Ø25 mm, OD 1.0, reflective neutral density filter is 0.05 J/cm^2 for 10 ns
pulses at 355 nm, while the damage threshold of the similar NE10A absorptive filter is 10 J/cm^2 for 10 ns pulses at 532 nm. As described on the previous tab, the LIDT value of an optic scales with
the square root of the wavelength in the nanosecond pulse regime:
This scaling gives adjusted LIDT values of 0.08 J/cm^2 for the reflective filter and 14 J/cm^2 for the absorptive filter. In this case, the absorptive filter is the best choice in order to avoid
optical damage.
Pulsed Microsecond Laser Example
Consider a laser system that produces 1 µs pulses, each containing 150 µJ of energy at a repetition rate of 50 kHz, resulting in a relatively high duty cycle of 5%. This system falls somewhere
between the regimes of CW and pulsed laser induced damage, and could potentially damage an optic by mechanisms associated with either regime. As a result, both CW and pulsed LIDT values must be
compared to the properties of the laser system to ensure safe operation.
If this relatively long-pulse laser emits a Gaussian 12.7 mm diameter beam (1/e^2) at 980 nm, then the resulting output has a linear power density of 5.9 W/cm and an energy density of 1.2 x 10^-4 J/
cm^2 per pulse. This can be compared to the LIDT values for a WPQ10E-980 polymer zero-order quarter-wave plate, which are 5 W/cm for CW radiation at 810 nm and 5 J/cm^2 for a 10 ns pulse at 810 nm.
As before, the CW LIDT of the optic scales linearly with the laser wavelength, resulting in an adjusted CW value of 6 W/cm at 980 nm. On the other hand, the pulsed LIDT scales with the square root of
the laser wavelength and the square root of the pulse duration, resulting in an adjusted value of 55 J/cm^2 for a 1 µs pulse at 980 nm. The pulsed LIDT of the optic is significantly greater than the
energy density of the laser pulse, so individual pulses will not damage the wave plate. However, the large average linear power density of the laser system may cause thermal damage to the optic, much
like a high-power CW beam.
Laser Safety and Classification
Safe practices and proper usage of safety equipment should be taken into consideration when operating lasers. The eye is susceptible to injury, even from very low levels of laser light. Thorlabs
offers a range of laser safety accessories that can be used to reduce the risk of accidents or injuries. Laser emission in the visible and near infrared spectral ranges has the greatest potential for
retinal injury, as the cornea and lens are transparent to those wavelengths, and the lens can focus the laser energy onto the retina.
Safe Practices and Light Safety Accessories
• Laser safety eyewear must be worn whenever working with Class 3 or 4 lasers.
• Regardless of laser class, Thorlabs recommends the use of laser safety eyewear whenever working with laser beams with non-negligible powers, since metallic tools such as screwdrivers can
accidentally redirect a beam.
• Laser goggles designed for specific wavelengths should be clearly available near laser setups to protect the wearer from unintentional laser reflections.
• Goggles are marked with the wavelength range over which protection is afforded and the minimum optical density within that range.
• Laser Safety Curtains and Laser Safety Fabric shield other parts of the lab from high energy lasers.
• Blackout Materials can prevent direct or reflected light from leaving the experimental setup area.
• Thorlabs' Enclosure Systems can be used to contain optical setups to isolate or minimize laser hazards.
• A fiber-pigtailed laser should always be turned off before connecting it to or disconnecting it from another fiber, especially when the laser is at power levels above 10 mW.
• All beams should be terminated at the edge of the table, and laboratory doors should be closed whenever a laser is in use.
• Do not place laser beams at eye level.
• Carry out experiments on an optical table such that all laser beams travel horizontally.
• Remove unnecessary reflective items such as reflective jewelry (e.g., rings, watches, etc.) while working near the beam path.
• Be aware that lenses and other optical devices may reflect a portion of the incident beam from the front or rear surface.
• Operate a laser at the minimum power necessary for any operation.
• If possible, reduce the output power of a laser during alignment procedures.
• Use beam shutters and filters to reduce the beam power.
• Post appropriate warning signs or labels near laser setups or rooms.
• Use a laser sign with a lightbox if operating Class 3R or 4 lasers (i.e., lasers requiring the use of a safety interlock).
• Do not use Laser Viewing Cards in place of a proper Beam Trap.
Laser Classification
Lasers are categorized into different classes according to their ability to cause eye and other damage. The International Electrotechnical Commission (IEC) is a global organization that prepares and
publishes international standards for all electrical, electronic, and related technologies. The IEC document 60825-1 outlines the safety of laser products. A description of each class of laser is
given below:
Class Description Warning
This class of laser is safe under all conditions of normal use, including use with optical instruments for intrabeam viewing. Lasers in this class do not emit radiation at levels that
1 may cause injury during normal operation, and therefore the maximum permissible exposure (MPE) cannot be exceeded. Class 1 lasers can also include enclosed, high-power lasers where
exposure to the radiation is not possible without opening or shutting down the laser.
Class 1M lasers are safe except when used in conjunction with optical components such as telescopes and microscopes. Lasers belonging to this class emit large-diameter or divergent
1M beams, and the MPE cannot normally be exceeded unless focusing or imaging optics are used to narrow the beam. However, if the beam is refocused, the hazard may be increased and the
class may be changed accordingly.
2 Class 2 lasers, which are limited to 1 mW of visible continuous-wave radiation, are safe because the blink reflex will limit the exposure in the eye to 0.25 seconds. This category only
applies to visible radiation (400 - 700 nm).
2M Because of the blink reflex, this class of laser is classified as safe as long as the beam is not viewed through optical instruments. This laser class also applies to larger-diameter or
diverging laser beams.
Class 3R lasers produce visible and invisible light that is hazardous under direct and specular-reflection viewing conditions. Eye injuries may occur if you directly view the beam,
3R especially when using optical instruments. Lasers in this class are considered safe as long as they are handled with restricted beam viewing. The MPE can be exceeded with this class of
laser; however, this presents a low risk level to injury. Visible, continuous-wave lasers in this class are limited to 5 mW of output power.
Class 3B lasers are hazardous to the eye if exposed directly. Diffuse reflections are usually not harmful, but may be when using higher-power Class 3B lasers. Safe handling of devices
3B in this class includes wearing protective eyewear where direct viewing of the laser beam may occur. Lasers of this class must be equipped with a key switch and a safety interlock;
moreover, laser safety signs should be used, such that the laser cannot be used without the safety light turning on. Laser products with power output near the upper range of Class 3B
may also cause skin burns.
This class of laser may cause damage to the skin, and also to the eye, even from the viewing of diffuse reflections. These hazards may also apply to indirect or non-specular reflections
4 of the beam, even from apparently matte surfaces. Great care must be taken when handling these lasers. They also represent a fire risk, because they may ignite combustible material.
Class 4 lasers must be equipped with a key switch and a safety interlock.
All class 2 lasers (and higher) must display, in addition to the corresponding sign above, this triangular warning sign.
Thorlabs offers a wide selection of optics optimized for use with Nd:YAG lasers. Please see below for more information.
Nd:YAG Optics Selection
Dielectric Mirrors Ultrafast Mirrors
Laser Line Mirrors, Right-Angle Prism Mirrors, Cage Cube-Mounted Prism Mirrors, Low GDD Ultrafast Mirrors,
1064 nm, 532 nm, 355 nm, 266 nm 1064 nm, 532 nm 1064 nm, 532 nm 1064 nm or 532 nm
Beamsplitters Wave Plates
Harmonic Beamsplitters, High-Power Polarizing Beamsplitter Cubes, Non-Polarizing Plate Beamsplitters, Dual Wavelength Multi-Order Wave Plates,
1064 nm, 532 nm, 355 nm, 266 nm 1064 nm, 532 nm: Unmounted or Mounted 1064 nm, 532 nm 1064 nm / 532 nm
Attenuators Filters Objectives Doublet Lenses
Laser Beam Attenuator, Hard-Coated Bandpass Filters, High Power Focusing Objectives, Air-Spaced Doublets,
1064 nm / 532 nm 1064 nm or 532 nm 1064 nm, 532 nm 1064 nm, 532 nm
Spherical Singlet Lenses
N-BK7 Plano-Convex Lenses, Unmounted: 1064 nm / 532 nm UVFS Plano-Convex Lenses: UVFS Bi-Convex Lenses, Unmounted: UVFS Plano-Concave Lenses, Unmounted:
1064 nm: Unmounted 1064 nm, 532 nm, or 1064 nm / 532 nm 1064 nm, 532 nm, or 1064 nm / 532 nm
532 nm: Unmounted
1064 nm / 532 nm: Unmounted or Mounted
Posted Comments:
Manon Lallement  (posted 2023-10-11 21:18:53.443)
Hello, Could you please share the reversed Zemax black box of LMH-5X-1064? Best regards, Manon Lallement
srydberg  (posted 2023-10-12 10:18:33.0)
Hello, thank you for contacting Thorlabs! It is possible to reverse files containing black boxes in newer versions of Zemax. I have contacted you directly and sent you the requested file.
Cristina Mendez  (posted 2019-06-14 04:14:36.25)
Good morning, we're using this objective both for a 1064-nm laser and a 1038 nm fs-laser; and we're interesting in how its focusing distance varies with the wavelength. Do you have any data on this
mdiekmann  (posted 2019-06-19 09:45:45.0)
Hello, thank you for contacting Thorlabs. We will contact you directly and will send the requested information.
asalvador  (posted 2018-11-14 13:40:43.497)
Good afternoon, I'd like some advice to build a dicing system for Silicon wafers of 700 micron thickness, with internal micro cracking. We already have the sample XY scanning system and looking
forward to buy a laser (need advice on models) and optics (focusing objectives, etc)
YLohia  (posted 2018-11-14 08:53:17.0)
Hello, thank you for contacting Thorlabs. We will reach out to you directly to discuss your application.
jaka.mur  (posted 2018-04-25 12:04:06.603)
I would like to know some details about the LMH-5X-1064 objective: 1) How was the "Diffraction-limited Spot Size" value calculated? I have tried using this: diameter = 4*lambda*EFL/pi/EA and it does
not result in the same number. 2) What is the actual field of view with this objective? 3) Is there a measurement of CW damage threshold available? Best regards, Jaka
nbayconich  (posted 2018-04-26 03:19:35.0)
Thank you for your feedback. The diffraction limited spot Size calculated for this objective will produce a spot size given by Ø=C x λ x (f/A) Where Ø is the 1/e2 beam diameter, λ is the wavelength
of the laser, f is the effective focal length of the lens, and A is the entrance beam diameter. C is a constant that relates to the degree of pupil illumination and input truncation (for a Gaussian
beam, C = 1.83 when the entrance beam is truncated at the 1/e2 diameter). The field of view is the OFN/magnification = 2.4mm. We do not have a certified damage threshold CW measurement at the moment,
however we expect it to 1MW/cm².
p.kleingarn  (posted 2017-11-28 11:53:28.36)
I've got a question regarding the LMH-10x-532nm Objective. I'm working with the optical Breakdown in borosilikat-glas and i'm wondering how the spherical Abberations would influence my results. Can
you give me any information about the wavefront abberation coefficient/transmitted wavefront error maybe? This would be very appreciated, thanks ! besten regards Philipp
tfrisch  (posted 2018-02-23 10:18:20.0)
Hello, thank you for contacting Thorlabs. I can send you the Zemax files we have available if you need to do a rigorous analysis, but LMH-10X-532 has on-axis nominal a diffraction limited transmitted
wavefront error for borosilicate glass up to 1mm when the coverglass is perpendicular to the objective axis. I will reach out to you to directly to discuss this.
linlinchiayu  (posted 2017-07-04 22:05:00.537)
Can I receive the wavelength range information of LMH-20X-1064 and LMH-10X-1064?
nbayconich  (posted 2017-07-11 10:58:47.0)
Thank you for contacting Thorlabs. I will reach out to you directly with more information on the transmission of LMH-20X-1064 and LMH-10X-1064.
marin.vojkovic  (posted 2017-06-27 10:46:21.98)
Can you please tell me what coating is used for the objective and if it is suitable for working in the vacuum? If not, what would the price be for an objective suitable for vacuum? By coating, I mean
the whole objective, not just the lens. Thank you in advance!
tfrisch  (posted 2017-08-03 11:31:47.0)
Hello, thank you for contacting Thorlabs. We can offer a custom vacuum compatible version. I will reach out to you directly.
jelke.wibbeke  (posted 2016-10-10 09:57:25.37)
We are using the LMH-5X-1064 on a femtosecond laser. It occours that we can find two more focus spots (with less intensity) symmetrically about 2mm to the left and the right of the actual focus spot.
Any clue?
jlow  (posted 2016-10-10 11:13:42.0)
Response from Jeremy at Thorlabs: We will contact you directly about troubleshooting this.
erica.block  (posted 2016-07-29 13:30:02.18)
Could you make one of these high power MicroSpot focusing objectives with a 850nm AR coating?
o.soroka  (posted 2015-11-18 13:41:26.86)
Can this objective be used in the UHV?
besembeson  (posted 2015-11-20 09:34:48.0)
Response from Bweh at Thorlabs USA: The stock item like the LMH-5X-532 is not vacuum compatible as there are certain air-pockets and anodized components inside that will out-gas. We can provide a
special vacuum compatible version that may be suitable for your application. I will contact you.
andrefilipedrosa  (posted 2015-07-14 19:58:24.633)
Hello, is this objective lens able to be scanned? ty
besembeson  (posted 2015-09-23 12:03:35.0)
Response from Bweh at Thorlabs USA: The design is not telecentric and the clear aperture may not be large enough so this will not be suitable for scanning applications. You can find several scanning
objectives at the following link: http://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=2910
suwas.nikumb  (posted 2014-02-10 09:37:53.92)
Do you have microscope objectives at 532nm and 800nm with atleast 12-15mm entrance aperture?
besembeson  (posted 2014-02-12 01:59:58.0)
Response from Bweh E. at Thorlabs: Thanks for contacting Thorlabs. At this time, we do not carry such objectives that meet your requirements. We however have several other objectives at the following
page: http://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=1044.
lvtaohn  (posted 2013-09-17 19:08:25.967)
Do you have 40times or 60 times high power focusing objective 1064nm or 532nm? I need them in these days. Please let me know if you have them. Thank you!
jlow  (posted 2013-09-19 14:09:00.0)
Response from Jeremy at Thorlabs: We do not currently have these short focal length objectives for high power application. I have logged this in our internal forum and we will look into whether we
could make this.
rpscott  (posted 2013-05-29 17:06:42.537)
Is the transmission for the 1064nm version of the objectives similar at 1037nm?
tcohen  (posted 2013-05-30 02:28:00.0)
Response from Tim at Thorlabs: Yes. The number of surfaces in each objective is different and therefore deviation away from the design wavelength will have more of an impact on transmission for the
20X vs the 10X, for example. However, at your desired wavelength the total transmission for each is similar.
tcohen  (posted 2012-08-23 16:25:00.0)
Response from Tim at Thorlabs: Thank you for contacting us. I’ve emailed you to go over this objective and supply some representative data.
song31037  (posted 2012-08-23 15:19:04.0)
Can I receive the wavelength range information of LMH-20X-1064?
tcohen  (posted 2012-02-29 15:56:00.0)
Response from Tim at Thorlabs: Thank you for your feedback on the LMH-10X-532. I am looking into the materials we use for this objective and will post an update soon.
g.laliberte  (posted 2012-02-29 10:13:58.0)
I'm intersted in a microscope objective that would be made of non magnetic material. Do you have any data about the housing material of that lens? THanks
jjurado  (posted 2011-07-08 08:58:00.0)
Response from Javier at Thorlabs to last poster: Thank you very much for contacting us. For this objective, the absorption in the glass itself is almost negligible, and if we assume 3.5% loss at each
surface we would end up with a total transmission in the neighborhood of 80%. However, keep in mind that the chromatic shift of roughly 0.2 mm will slightly affect the performance of this objective.
Also, we do not currently have an objective specifically designed for operation at 980 and 1550 nm. Please contact us at techsupport@thorlabs.com if you would like to discuss your application a bit
user  (posted 2011-07-07 14:17:40.0)
I am working with this objective (LMH-10X-1064), but I have beam at 1550nm. So I want to know the transmission spectrum of this objective especialy for 1550nm). Is there any particular objective
design to work at 980nm and 1550nm?
apalmentieri  (posted 2009-12-31 17:12:33.0)
A response from Adam at Thorlabs to Ilday: There was a server glitch and we are working to bring this page online. Please note that we do have stock on most of these objectives and they can be
ordered direct through our sales department, sales@thorlabs.com or 973-300-3000, while these parts are not on the website. I will email you with more information.
ilday  (posted 2009-12-31 17:01:39.0)
I wanted to purchase this product, which was available until yesterday, however it has disappeared from the webpage. Is that a server glitch or did you discontinue the product?
AR Coating Transmission Design Tube Lens Cover Glass RMS Thread
Item # Range (Typical)^a M^b WD EFL PFL NA EP^c OFN Spot Size^d Focal Length^e Correction Depth
LMH-5X-532 495 - 570 nm >98% 5X 35 mm 40 mm 58.9 mm 0.13 9.9 mm 12 3.9 µm 200 mm - 3.8 mm
LMH-10X-532 >97% 10X 15 mm 20 mm 38.9 mm 0.25 9.9 mm 8 2.0 µm
LMH-20X-532 >96% 20X 6 mm 10 mm 40.0 mm 0.40 8.0 mm 14 1.2 µm
LMH-50X-532 >95% 50X 2.5 mm^f 4 mm 45.0 mm^f 0.60 5.0 mm 22 0.8 µm 0.0 - 1.0 mm of Fused Silica 4.5 mm
Based on your currency / country selection, your order will ship from Newton, New Jersey
+1 Qty Docs Part Number - Universal Price Available
LMH-5X-532 High-Power MicroSpot Focusing Objective, 5X, 495 - 570 nm, NA = 0.13 $996.66
LMH-10X-532 High-Power MicroSpot Focusing Objective, 10X, 495 - 570 nm, NA = 0.25 $1,227.13
LMH-20X-532 High-Power MicroSpot Focusing Objective, 20X, 495 - 570 nm, NA = 0.40 $1,822.27
LMH-50X-532 High-Power MicroSpot Focusing Objective with Correction Collar, 50X, 495 - 570 nm, NA = 0.60 $4,076.21
AR Coating Transmission Design Tube Lens Cover Glass RMS Thread
Item # Range (Typical)^a M^b WD EFL PFL NA EP^c OFN Spot Size^d Focal Length^e Correction Depth
LMH-50X-850 790 - 910 nm >95% 50X 2.3 mm^f 4 mm 45.0 mm^f 0.65 5.3 mm 22 1.2 µm 200 mm 0.0 - 1.0 mm of Fused Silica 4.5 mm
AR Coating Transmission Design Tube Lens Cover Glass Thread
Item # Range (Typical)^a M^b WD EFL PFL NA EP^c OFN Spot Size^d Focal Length^e Correction Depth
LMH-5X-1064 980 - 1130 nm >98% 5X 35 mm 40 mm 58.9 mm 0.13 9.9 mm 12 7.9 µm 200 mm - 3.8 mm
LMH-10X-1064 >97% 10X 15 mm 20 mm 38.9 mm 0.25 9.9 mm 10 3.9 µm
LMH-20X-1064 >96% 20X 6 mm 10 mm 40.0 mm 0.40 8.0 mm 14 2.4 µm
LMH-50X-1064 >95% 50X 2.3 mm^f 4 mm 45.0 mm^f 0.65 5.3 mm 22 1.5 µm 0.0 - 1.0 mm of Fused Silica 4.5 mm
0.0 - 1.2 mm of SiC
Based on your currency / country selection, your order will ship from Newton, New Jersey
+1 Qty Docs Part Number - Universal Price Available
LMH-5X-1064 High-Power MicroSpot Focusing Objective, 5X, 980 - 1130 nm, NA = 0.13 $996.66
LMH-10X-1064 High-Power MicroSpot Focusing Objective, 10X, 980 - 1130 nm, NA = 0.25 $1,227.13
LMH-20X-1064 High-Power MicroSpot Focusing Objective, 20X, 980 - 1130 nm, NA = 0.40 $1,822.27
LMH-50X-1064 Customer Inspired! High-Power MicroSpot Focusing Objective with Correction Collar, 50X, 980 - 1130 nm, NA=
0.65 $4,076.21
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Basal Metabolic Rate
Basal Metabolic Rate Calculator
Welcome to the Basal Metabolic Rate Calculator, your go-to tool for accurately estimating your body's Basal Metabolic Rate (BMR). Our platform is dedicated to helping individuals understand their
metabolic needs to make informed decisions about their health and fitness.
At Basal Metabolic Rate Calculator, we prioritize precision and simplicity. By inputting essential details such as age, gender, weight, and height, our calculator utilizes scientifically validated
formulas to calculate BMR efficiently. Armed with this knowledge, users can tailor their diet and exercise regimens to achieve their desired health goals effectively.
We understand that navigating the complexities of metabolism can be daunting, which is why we've designed our calculator to be user-friendly and accessible to all. Whether you're embarking on a
weight loss journey, seeking to maintain a healthy lifestyle, or aiming to build muscle, our Basal Metabolic Rate Calculator empowers you to calculare BMR with confidence.
How to Calculate BMR?
Basal Metabolic Rate (BMR) is measured under stringent conditions during wakefulness, requiring complete rest and inactivity of the sympathetic nervous system. It represents the largest proportion of
an individual's total caloric requirements. Daily caloric needs are determined by multiplying the BMR by a factor ranging from 1.2 to 1.9, depending on activity level.
Traditionally, BMR is estimated using equations derived from statistical data. The Harris-Benedict Equation, developed in the early years, was refined in 1984 for improved accuracy. However, it was
surpassed by the Mifflin-St Jeor Equation in 1990, which has since been recognized as more precise. The Katch-McArdle Formula differs slightly, considering lean body mass in its calculation of
resting daily energy expenditure (RDEE), a factor not addressed by the other equations.
The Mifflin-St Jeor Equation is generally considered the most accurate for calculating BMR, except in cases where individuals are lean and aware of their body fat percentage, where the Katch-McArdle
Formula may be more suitable. Users can select the preferred equation for calculation in the settings.
Equations used by the calculator include (Matric Units):
Mifflin-St Jeor Equation:
For men:
BMR = 10 * W + 6.25 * H − 5 * A + 5
For women:
BMR = 10 * W + 6.25 * H − 5 * A − 161
Revised Harris-Benedict Equation:
For men:
BMR = 13.397 * W + 4.799 * H − 5.677 * A + 88.362
For women:
BMR = 9.247 * W + 3.098 * H − 4.330 * A + 447.593
Katch-McArdle Formula:
BMR = 370 + 21.6(1 − F) * W
Equations used by the calculator include (US Units):
Mifflin-St Jeor Equation:
For men:
BMR = 4.536 * W + 15.88 * H − 5 * A + 5
For women:
BMR = 4.536 * W + 15.88 * H − 5 * A − 161
Revised Harris-Benedict Equation:
For men:
BMR = 6.23 * W + 12.7 * H − 6.76 * A + 66.47
For women:
BMR = 4.35 * W + 4.7 * H − 4.68 * A + 655.1
Katch-McArdle Formula:
BMR = 370 + 9.82(1 − F) * W
W is body weight in kilograms
H is body height in centimeters
A is age
F is body fat percentage
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A Double Meaning of an Arc's Midpoint
Let \(M\) be the midpoint of an arc of the circumcircle of \(\Delta ABC\) apposite vertex \(A\); let \(I\) denote the incenter of the triangle. Then \(M\), \(I\), and \(A\) are collinear. In
addition, \(CM=IM\).
The applet below serves a dynamic illustration
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Copyright © 1996-2018 Alexander Bogomolny
Let \(M\) be the midpoint of an arc of the circumcircle of \(\Delta ABC\) apposite vertex \(A\); let \(I\) denote the incenter of the triangle. Then \(M\), \(I\), and \(A\) are collinear. In
addition, \(CM=IM\).
The proposition is a consequence of the definitions. The Incenter lies at the intersection of angle bisectors; in particular, it belongs to the angle bisector from \(A\).
On the other hand, equal inscribed angles are subtended by equal arcs, such that \(\angle BAI=\angle CAI\) implies that the (second) intersection of \(AI\) with the circumcircle is exactly the
midpoint of the opposite arc \(\stackrel \frown {BC}\), which is \(M\).
Let angles at \(A\) and \(C\) be equal \(2\alpha\) and \(2\gamma\), respectively. Then, as an external angle of \(\Delta AIC\), \(\angle MIC=\alpha +\gamma\). But since inscribed angles \(BAM\) and \
(BCM\) are both equal to \(\alpha\), \(\angle ICM=\alpha+\gamma\), implying \(\angle ICM=\angle MIC\) and making \(\Delta MIC\) isosceles, with \(CM=IM\).
It is worth noting that the excenters of a triangle have a similar property:
Let \(M\) be the midpoint of an arc of the circumcircle of \(\Delta ABC\) apposite vertex \(A\); let \(I_A\) denote the excenter of the triangle opposite A. Then \(M\), \(I_A\), and \(A\) are
collinear. In addition, \(CM=I_{A}M\).
For one, \(\Delta ICI_A\) is right so that the median from \(C\) equals half the hypotenuse \(II_A\). Point \(M\) that forms an isosceles triangle \(CMI\) is unique on the hypotenuse (because the
base angle \(MIC\) does not depend on the position of \(M\).) It follows that \(M\) is the midpoint of \(II_A\) which proves the claimed property of the excenters.
Note that the above supplies an alternative proof of the previously established fact that the midpoint of \(II_A\) lies on the circumcircle. As we've seen, this same point also serves as the midpoint
of the arc of the circumcircle that is crossed by \(II_A\).
|Contact| |Front page| |Content| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
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What is the variable for rate of change?
What is the variable for rate of change?
Rate of change is used to mathematically describe the percentage change in value over a defined period of time, and it represents the momentum of a variable. The calculation for ROC is simple in that
it takes the current value of a stock or index and divides it by the value from an earlier period.
What is a constant rate of change?
When something has a constant rate of change, one quantity changes in relation to the other. For example, for every half hour the pigeon flies, he can cover a distance of 25 miles. We can write this
constant rate as a ratio. Simplified, the constant rate is 50 miles per hour.
Is the rate of change constant variable?
Definition: A linear function is a function that has a constant rate of change and can be represented by the equation y = mx + b, where m and b are constants. That is, for a fixed change in the
independent variable there is a corresponding fixed change in the dependent variable.
What does a constant rate mean?
In mathematics, a constant rate of change is a rate of change that stays the same and does not change.
How do I calculate rate of change?
The rate of change between two points on a curve can be approximated by calculating the change between two points. Notice that the numerator is the overall change in y, and the denominator is the
overall change in x.
What’s an example of constant rate of change?
Constant Rate of Change When you walk without slowing down or speeding up at all, then the rate of change of your position is constant. This means that if you travel 2 meters in the first second, you
travel 2 meters in the second second, and 2 meters in the third second, and so on until you decide to stop walking.
Which situation shows a constant rate of change?
A constant rate of change is accomplished when the change in one situation constantly results in changing another. In the scenario of a booster club constantly selling raffle tickets, the more
tickets they sold the more the club raises.
How do you know if its constant or variable?
Constants are usually written in numbers. Variables are specially written in letters or symbols. Constants usually represent the known values in an equation, expression or in line of programming.
Variables, on the other hand, represent the unknown values.
What is average rate of change?
What is average rate of change? It is a measure of how much the function changed per unit, on average, over that interval. It is derived from the slope of the straight line connecting the interval’s
endpoints on the function’s graph.
What is constant speed?
An object is travelling at a steady or constant speed when its instantaneous speed has the same value throughout its journey. For example, if a car is travelling at a constant speed the reading on
the car’s speedometer does not change.
What does growing at a constant rate mean?
Exponential growth refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time. Linear growth
refers to the original value from the range increases by the same amount over equal increments found in the domain.
What is the formula for constant rate of change?
Definition: A linear function is a function that has a constant rate of change and can be represented by the equation y = mx + b, where m and b are constants. That is, for a fixed change in the
independent variable there is a corresponding fixed change in the dependent variable.
What is an example of a constant rate of change?
A rate of change is a rate that describes how one quantity changes in relation to another quantity. Constant rate is also called as uniform rate which involves something travelling at fixed and
steady pace or else moving at some average speed. For example, A car travels 3 hours.
What is constant change in math?
In mathematics, a constant rate of change is a rate of change that stays the same and does not change.
What is the variable for rate of change? Rate of change is used to mathematically describe the percentage change in value over a defined period of time, and it represents the momentum of a
variable. The calculation for ROC is simple in that it takes the current value of a stock or index and divides…
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▷ SYSTEM OF EQUATIONS | How to solve it? | Aulaprende.com
A system of equations is made up of 2 or more equations that we have to find a common solution. The solution of a system of equations is called the common solution of the equations. In other words,
the solution of a system is the point where the recipes are cut:
The three methods to solve systems of equations:
• Substitution method.
• Elimination method.
• Equalization method.
Methods for solving systems of equations
Substitution method
In the substitution method, one unknown is solved in one of the equations and it is substituted in the other, obtaining an equation of one unknown. Their solution is substituted in the other
Elimination method
In the elimination method the two equations are prepared so that one of the unknowns has the same coefficient in both. By subtracting or adding them, an equation is obtained without that unknown.
Equalization method
In the equalization method, the same unknown is solved in the two equations and the results are equaled to obtain an equation with one unknown.
Exercises of system of equations
1. Find out the unknowns using the substitution method:
Ejercicios de sistemas de ecuaciones
2. Find the values of x and y using the equalization method:
Ejercicios de sistemas de ecuaciones
3. Apply the reduction method and solve the system:
Ejercicios de sistemas de ecuaciones
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The number 36 has 9 factors and is a composite number. To calculate the factors of 36 we divide,...
The number 35 has 4 factors and is a composite number. To calculate the factors of 35 we divide,...
The number 4 has 4 factors and is a composite number. To calculate the factors of 34 we divide,...
The number 33 has 4 factors and is a composiite number. To calculate the factors of 33 we divide,...
The number 32 has 6 factors and is a composite number. To calculate the factors of 32 we divide,...
The number 31 has 2 factors and is a prime number. To calculate the factors of 31 we divide,...
The number 30 has 8 factors and is a composite number. To calculate the factors of 30 we divide,...
The number 29 has 2 factors and is a prime number. To calculate the factors of 29 we divide,...
The number 28 has 5 factors and is a composite number. To calculate the factors of 28 we divide,...
The number 27 has 4 factors and is a composite number. To calculate the factors of 27 we divide,...
Factors of 36
Factors of 35
Factors of 34
Factors of 33
Factors of 32
Factors of 31
Factors of 30
Factors of 29
Factors of 28
Factors of 27
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Effects of chemically heterogeneous nanoparticles on polymer dynamics: Insights from molecular dynamics simulations
Soft Matter
This journal is © The Royal Society of Chemistry 2017 Soft Matter, 2017, 00, 1-3| 1
a[Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer ]
Materials, State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, 100029 Beijing, China
E-mail: [email protected], [email protected].
b[ Hubei Collaborative Innovation Center for Advanced Organic Chemical Materials, ]
Key Laboratory for the Green Preparation and Application of Functional Materials, Ministry of Education, Hubei Key Laboratory of Polymer Materials, School of Materials Science and Engineering, Hubei
University, 430062 Wuhan, China
c [CNR-SPIN, Via Cintia, 80126 Napoli, Italy ]
d [Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘G. Natta’’, Politecnico]
di Milano, via L. Mancinelli 7, 20131 Milano, Italy Received 00th January 20xx,
Accepted 00th January 20xx DOI: 10.1039/x0xx00000x www.rsc.org/
Effects of Chemically Heterogeneous Nanoparticles on Polymer
Dynamics: Insights from Molecular Dynamics Simulations
Zijian Zheng,ab[ Fanzhu Li,]a[ Jun Liu,]* a[ Raffaele Pastore,]c,d[ Guido Raos,]d[ Youping Wu,]a[ Liqun ]
Zhang * a
The dispersion of solid nanoparticles within polymeric materials is widely used to enhance their performance. Many scientific and technological aspects of the resulting polymer nanocomposites have
been studied, but the role of the structural and chemical heterogeneity of nanoparticles has just started to be appreciated. For example, simulations of polymer films on planar heterogeneous surfaces
revealed unexpected, non-monotonic activation energy to diffusion on varying the surface composition. Motivated by these intriguing results, here we simulate via molecular dynamics a different, fully
three-dimensional system, in which heterogeneous nanoparticles are incorporated in a polymer melt. The nanoparticles are roughly spherical assemblies of strongly and weakly attractive sites, in
fraction f and 1-f, respectively. We show that polymer diffusion is still characterized by non-monotonic dependence of the activation energy on f. The comparison with the case of homogeneous
nanoparticles clarifies that the effect of the heterogeneity increases on approaching the polymer glass transition.
I. Introduction
The introduction of nanoparticles (NPs) into polymer matrices is an effective strategy to fabricate high performance polymer nanocomposites (PNCs)1-3[. A fundamental understanding of ]
their composition-structure-properties relationship will facilitate the further development of materials with desired
properties4[. One key lesson is that it is crucial to disperse the ]
nanoparticles well in the polymer matrix in order to produce high performance composites5[. This can be achieved by ]
compatibilizing them, for example by coating the nanoparticles or grafting them with the polymer6-8[. ]
The interface between polymers and nanoparticles is also a very important element. In the nanometer-size interstices between the nanoparticles, the polymer chains become severely restricted and may
deviate substantially from their bulk behavior9[. Thus, the polymer interfaces which are formed near ]
solid surfaces and nanoparticles have received wide consideration from both the academic and industrial communities10-14[. One of the earliest studies is the one by ]
Tsagaropoulos and Eisenberg15[, who observed that several ]
polymer properties such as viscosity, diffusion coefficient, and the T2 relaxation time in NMR measurements were drastically
altered by silica nanoparticles and attributed this effect to a drastic slowing down of the chains in the interfacial region. Later on, Torkelson and coworkers16, 17[ studied the relaxation of ]
thin polymer films on silica surfaces, as a model for polymer-silica nanocomposites. Using fluorescent probes to monitor the dynamics of the polymer matrix, they demonstrated that chains next to the
surface were nearly completely arrested (at temperatures above the bulk Tg), but the slowing down in the
relaxation dynamics could be felt up to 100 nm away from it. Even a small concentration of nanoparticles can slow down significantly the polymer diffusion within a PNC, due to a combination of the
reduced polymer relaxation and the entropic barriers created by the excluded volume of the NPs18[. NP ]
induced enhancement of the polymer relaxation rate has been also observed for particular physical situations, including systems formed by entangled polymers and NPs smaller than the entanglement mesh
size19[. ]
Many useful insights have been obtained by theoretical analyses and molecular dynamics (MD) or Monte Carlo simulations, alongside the traditional experimental methods 20-22[. Smith et al.]23[
introduced some structural roughness in the ]
coarse-grained modelling of NPs. Starr et al.24[ found a mobility ]
gradient for coarse-grained polymer chains approaching the nanoparticle surfaces, both for repulsive and attractive polymer/nanoparticle interactions. Li et al.25 [explored the ]
dynamics of long polymer chains filled with spherical NPs. Through a primitive path analysis, they showed that the chains become significantly disentangled on increasing the NP volume fraction, but
“NP entanglements” become significant above a critical volume fraction. More recently, Hagita et al.26[ have ]
performed large-scale MD simulations of related cross-linked models, to describe the mechanical deformation of rubbery nanocomposites with different degrees of particle aggregation. At the atomistic
level of description, Ndoro et al.27[ simulated a ]
composite consisting of ungrafted and grafted spherical silica
nanoparticles embedded in a melt of 20-monomer atactic polystyrene chains. These showed that the nanoparticles modify the polymer structure in its neighborhood, and these changes increase with the
higher grafting densities and larger particle diameters. De Nicola et al.28[ used the results of molecular ]
simulation as a guide for the preparation of improved polymer-NP composites, exploiting an in situ polymerization of macromonomers.
Most of the previous simulations, especially those based on coarse-grained models, employed completely smooth or structured NPs with chemically uniform surfaces. However the most common fillers,
namely silica29[ and especially carbon ]
black30, 31[ for applications involving mechanical reinforcement, ]
have heterogeneous surfaces exposing a range of functional groups and adsorption sites (e.g., both hydrophobic and hydrophilic, hydrogen-bonding groups). Note that the slight roughness of an
atomically structured surface has already a very significant effect on the polymer dynamics, in comparison to the idealized situation with a perfectly smooth surface32[. An ]
even greater effect can be expected from the introduction of some chemical heterogeneity in the surfaces and/or the polymer. In fact, there is ample evidence that this sort of ‘‘quenched’’ disorder
plays an important role in many aspects of polymer behavior33[. Recent insights came from MD ]
simulations of thin polymer films adsorbed on structurally and chemically heterogeneous surfaces. These latter are composed by randomly assembling on a planar square lattice two type of beads,
strongly (S) and weakly (W) attractive, in variable percentage. In such a quasi-2D model, the dynamic slowdown of polymers was found to depend non-monotonically on the surface composition: the
activation energy to diffusion and the inverse glass transition temperature do not attain a maximum for 100% of strong sites, but at a smaller value, about 75%, which originates as a compromise
between the maximum global attraction and the maximum configurational disorder at the random percolation (50%) of strong sites. In addition, new dynamical features, which are not observed on a smooth
surface, become more and more apparent on increasing the
Journal Name ARTICLE
This journal is © The Royal Society of Chemistry 2017 Soft Matter, 2017, 00, 1-3| 3
surface heterogeneity. These include temporary subdiffusive regimes followed by “Fickian yet not Gaussian diffusion” as well as stretched exponential relaxation of the Rouse Normal Modes
autocorrelation functions.
In this paper, we aim at understanding whether these intriguing benchmarks are also relevant to fully three-dimensional polymer nanocomposite, where NPs are finely dispersed within the polymer melts.
This is not a trivial question, since the different dimensionality, the coexistence of adsorbed and non-adsorbed polymers as well as the presence of NP induced steric interactions make this physical
situation markedly distinct from that described in Ref.s34,35[. We ]
investigate this issue by introducing a coarse-grained model of polymer melt embedding a number of heterogeneous NPs. Each NP consists in a roughly spherical assembly of W and S sites. Performing
extensive MD simulations, we investigate the polymer dynamics at different temperatures and variable NP composition. We find that, in spite of the difference between the two physical situations, some
dynamical signatures found in Refs.34,35 [survive in this system, suggesting the existence of a ]
common underlying physical mechanism. In addition, we clarify to what extent the heterogeneous nature of the NPs modifies the polymer dynamics compared to the homogeneous case, showing that
relatively small differences at high temperature may result in dramatic change close the polymer glass transition.
II. Simulation model and method
In our coarse-grained model, the heterogeneous NPs are roughly spherical and are composed by two types of randomly distributed beads, which may be either of the weak (W) or the strong (S) type, the
only difference being the interaction energy with polymers. A snapshot is shown in Fig. 1. Briefly, we first generate one NP, formed by N=128 beads, Ne=74 of which are
exposed, and replicate such a prototype within the simulation box until the desired number of NPs is reached. At this stage, we randomly assign the interaction energy (i. e. the type) to
each bead with probability f and 1-f for strong (S) and weak (W) interactions, respectively.36[ ]
The simulation box contains 30 nanoparticles, which corresponds to a volume fraction of NPs ϕ=0.253 (estimated from the total numbers of NP-type and polymer-type beads). During the MD simulations,
the NPs are modelled as perfectly rigid bodies, with only three translational and three rotational degrees of freedom.
Fig. 1 The snapshots of nanoparticle, polymer chain and the corresponding nanocomposite. Within the NPs, the blue spheres represent the S sites and the red spheres represent the W sites.
The polymer chains are fully flexible and are represented by a generic bead-spring model. Overall, each polymer chain contains 32 beads. Given the relatively short length of the polymer chains,
entanglement effects play a minor role on the system dynamics. Roughly, each bond of the coarse-grained model would correspond to 3-6 covalent bonds along the backbone of a chemically realistic
chain. However, since we are not interested in a specific polymer, it is unnecessary to attempt a precise mapping. We adopt a “reduced” set of units, whereby the mass m and diameter σ of each
bead-which are identical for the polymer and the NPs-are effectively equal to unity. The non-bonded interactions between all the beads are modeled by truncated and shifted Lennard-Jones (LJ)
potentials: 𝑈𝑈(𝑟𝑟) = �4𝜀𝜀 ��𝜎𝜎𝑟𝑟� 12 − �𝜎𝜎[𝑟𝑟�]6� + 𝐶𝐶 𝑟𝑟 < 𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 0 𝑟𝑟 ≥ 𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (1) where C is a constant which guarantees that the potential energy is continuous at r=rcutoff, and ε
is the energy scale of the
model. While the effective diameter σ is the same for all beads, both the interaction energy and the cutoff distance depend on the combination of bead types. The interaction strength between the
polymer beads (P) and the S sites is twice that of the other pairs. Thus ɛPP =ɛPW=ɛ and ɛPS=2ɛ. All the
polymer-NP interactions are truncated at a cutoff distance of rPW=rPS
=2.5σ, whereas the polymer–polymer interactions are truncated at rPP = 21/6σ to produce a purely repulsive potential. Also the
interaction between the beads belonging to different NP is modelled with ɛNN=ɛ (where N=W or S) and the cutoff
rNN=21/6σ, so as to promote their dispersion within the polymer
matrix. By doing so, we aim at modelling the physical situation of finely dispersed NPs within the polymer melts, which can be achieved in different experimental setups, such as samples freshly
rejuvenated at high shear rate. In systems of this kind, the typical separation distance between NPs is large enough to make van der Waals interactions negligible.
The interaction between bonded beads within a polymer is modeled by adding a finite extensible nonlinear elastic (FENE) potential to their LJ interaction:
𝑈𝑈𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹= −0.5𝑘𝑘𝑅𝑅02𝑙𝑙𝑙𝑙 �1 − �[𝑅𝑅]𝑟𝑟 0�
� (2) where k=30ε/σ2[ is the force constant and R]
0=1.3σ is the
maximum extensibility of the bonds. The finite extensibility of the bonds avoids chain crossing events without introducing high-frequency bond vibrations, which would be produced by very stiff
harmonic bonds. The relatively short maximum bond length enhances the polymer’s resistance to ordering and crystallization, by increasing the mismatch between bonded and non-bonded nearest-neighbor
distances (the actual equilibrium bond length, resulting from the balance between LJ and FENE interactions, is about 0.8σ)37[. The reduced number density of ]
the monomer was fixed at ρ* =1.00, which corresponds to that of a dense melt. This is also roughly equal to the density inside the NPs. Units are reduced so that σ=m=ɛ=kB=1, where kB is the
Boltzmann’s constant.
We have performed NVT molecular dynamics
simulations, using a Nose-Hoover thermostat and velocity-Verlet algorithm to integrate the equations of motion. Periodic boundary conditions are applied to all three directions. The average MD run
consists of 109[ time steps. All the simulations ]
have been carried out with a large scale atomic/molecular massively parallel simulator (LAMMPS), developed by the Sandia National Laboratories38[. Further simulation details can ]
be found in our previous work39,40[. ]
III. Results
3.1 Translational dynamics
We start by investigating the effect of heterogeneous NPs on the translational dynamics of the whole polymer chains. To this aim, Fig. 2 shows the time dependence of the mean square displacements
(MSDs) of the polymer center of mass, for different NP composition, f, and temperature, T*[. The mobility ]
of the polymer chains becomes small on lowering the temperature and for large values of f and display an increasingly long lasting subdiffusive behaviuour. However, even for the slowest system, the
diffusive regime is restored within the timescale of our simulations,
〈�𝑟𝑟𝑐𝑐𝑐𝑐(𝑡𝑡)−𝑟𝑟𝑐𝑐𝑐𝑐(0)�2〉 = 6 𝐷𝐷 𝑡𝑡 (3)
allowing to estimate the chain diffusion coefficient, D, a long time fit to the MSD data. The diffusion coefficient so measured are listed in Table 1 and plotted in Fig. 3a shows as a function of
temperature and at different values of f. This figure clarifies that the data can be properly fitted using an Arrhenius functional law:
𝑙𝑙𝑙𝑙(𝐷𝐷) = 𝑙𝑙𝑙𝑙(𝐷𝐷0) − 𝐸𝐸𝑎𝑎⁄ . (4)𝑇𝑇∗
The values of the activation energies (Ea) obtained from these
fits are plotted in Figure 3b as a function of the NP composition, f, and listed in the last line of Table 1.
Journal Name ARTICLE
This journal is © The Royal Society of Chemistry 2017 Soft Matter, 2017, 00, 1-3| 5
Table 1. Polymer Diffusion Coefficients for Each Reduced Temperature (T*) and Surface Composition (f).The last row gives the activation energies at different Surface Composition (f ).
T*/f Neat 0.00 0.25 0.50 0.75 1.00 0.7 0.0019 [4.84x10]-5 [2.98x10]-5 [1.62x10]-5 [1.22x10]-5 [1.15x10]-5 0.8 0.0021 [7.63x10]-5 [4.25x10]-5 [3.54x10]-5 [2.23x10]-5 [1.93x10]-5 0.9 0.0023 [1.11x10]-4
[6.99x10]-5 [4.89x10]-5 [3.91x10]-5 [2.94x10]-5 1.0 0.0026 [1.48x10]-4 [1.05x10]-4 [7.31x10]-5 [5.66x10]-5 [4.37x10]-5 1.1 0.0027 [1.88x10]-4 [1.26x10]-4 [1.02x10]-4 [8.06x10]-5 [6.41x10]-5 1.2
0.0030 [2.08x10]-4 [1.65x10]-4 [1.32x10]-4 [1.08x10]-4 [8.95x10]-5 Ea 0.71±0.01 2.52±0.05 2.96±0.03 3.43±0.02 3.67±0.03 3.42±0.02
Fig. 2 Mean square displacement as a function of time, for T* = 1.2, 1.1, 1.0, 0.9, 0.8, and 0.7 from top to bottom. Different panels report different values of f, as indicated. Dashed lines are
guides to the eye of slope 1.
Overall, the order of magnitude of the activation energy is compatible with the results of previous work investigating different polymer nanocomposite systems35[. For the neat ]
polymer system, Ea is due to crowding and excluded volume
effect, and therefore has an entropic origin. The presence of
NPs introduces attractive interaction and increases the crowding effect. This results in significantly higher activation energy, even for f=0. In addition Fig. 3(b) shows that Ea as a
function f in Fig. 3(b) is maximal around f=0.75. This non-monotonic behaviour is intriguing, as the largest activation energy to diffusion does not correspond to the situation where the strength of
the interaction is overall the largest, f=1, but to a lower value of f. As previously mentioned, a similar result emerged from the simulations of polymer films adsorbed on heterogeneous planar
surfaces of Ref.s34,35[. In that case, the ]
value f=0.75 was interpreted as the compromise between the largest overall interaction energy (f=1) and the maximum structural disorder, attained at the random percolation transition of strong sites
(occurring around f=0.5, in two dimensions). This scenario was validated by an analysis of the surface induced energy landscape, as probed by horizontally sliding a single bead along the surface at
very low temperature. The average value of the energy barriers linearly increases between f=0 and f=1, whereas the standard deviation is the largest for the most disordered substrate f=0.5. Finally,
the most
significant descriptor of the dynamics are the tails of energy barrier distributions that are indeed the largest at f=0.75. Considering the similar behaviour found in the present model, we speculate
that the shape of the energy landscape and its implications are also similar.
Fig. 3 Arrhenius plots of the chain diffusion coefficients (a) and plot illustrating the surface dependence of the activation energies (b).
3.2 Rouse mode analysis
The polymer chain dynamics can be further characterized by investigating their conformational relaxation dynamics, which is commonly characterized through the autocorrelation functions 𝐶𝐶[𝑝𝑝](𝑡𝑡) =
〈𝑄𝑄[𝑝𝑝](0) ∙ 𝑄𝑄[𝑝𝑝](𝑡𝑡)〉 of the Rouse Normal Modes (RNMs): 𝑄𝑄𝑝𝑝(𝑡𝑡) =[𝑁𝑁 � 𝑟𝑟]1 𝑗𝑗(𝑡𝑡) 𝑐𝑐𝑐𝑐𝑐𝑐 � �𝑗𝑗 −1[2]� 𝑝𝑝𝑝𝑝 𝑁𝑁 � (5) 𝐹𝐹 𝑗𝑗=1
where 𝑟𝑟[𝑗𝑗](𝑡𝑡) is the position of monomer j at time t, and p=1, 2, ..., N-1 is the mode index. Each mode p is associated with a typical length-scale, as it describes the collective motion of chain
sections containing the N/p beads. Note that the polymer center-of-mass, which is relevant to the translational diffusion
described previously, formally corresponds to the mode p=0. In the following, we focus on the system with f=0.50.
Fig. 4(a) shows 𝐶𝐶[𝑝𝑝](𝑡𝑡) for different values of p at temperature T*=1.0. Their decay becomes significantly faster with increasing p, as the mode describes more localized motions. The decay of the
autocorrelation functions appear to be not consistent with a simple exponential, as predicted by the Rouse model, being , instead, well described by a stretched exponential, or
Kohlrausch-Williams-Watts (KWW) function:
𝐶𝐶𝑝𝑝(𝑡𝑡) ≃ exp �−�𝑡𝑡 𝜏𝜏⁄ �𝑝𝑝𝑠𝑠 𝛽𝛽� (6)
This behaviour has been also reported in simulations of polymer matrix embedding polymer grafted nanoparticles41[ and ]
is generally known to be a common hallmark of glassiness and an indirect signature of the emergence of dynamic heterogeneities.42[ It is worth noticing that a crossover from ]
stretched (β<1) to compressed (β>1) exponential has been also observed in soft glassy materials, such as colloidal glasses43-46[, ]
gels47[ and other amorphous solids]48[, its origin being currently a ]
very debated issue. We extract the parameters β and 𝜏𝜏[𝑝𝑝]𝑠𝑠 from KWW fits to the data of Fig.4a. In addition, we directly measure the relaxation time using the standard relation, 𝐶𝐶𝑝𝑝�𝜏𝜏𝑝𝑝� = 1/𝑒𝑒
and found a very good agreement with the
estimation obtained from fits (see, for example, Fig. 5b) Fig. 4(b) shows that β decreases slightly on lowering the temperature but much more clearly on increasing the mode index, p, indicating that
the relaxation of the chains tends to broaden on the local scale. A qualitatively similar effect was
This journal is © The Royal Society of Chemistry 2017 Soft Matter, 2017, 00, 1-3 | 7 Fig. 4 (a) RNM autocorrelation function 𝐶𝐶[𝑝𝑝](𝑡𝑡) at f=0.50, T*=1.0 and p=1, 2, . . . , 5, from right to left.
The open square represents the KWW fit at different p. (b) Exponent 𝛽𝛽[𝑝𝑝] and (c) the relaxation time (𝜏𝜏[𝑝𝑝], where 𝐶𝐶[𝑝𝑝]�𝜏𝜏[𝑝𝑝]�=1/e) corresponding to the first 5 modes of polymer chains with
different temperatures.
Fig. 5 (a) RNM autocorrelation function 𝐶𝐶[𝑝𝑝](𝑡𝑡) for p=1 and T*=0.8, 0.9,..., 1.2, for heterogeneous NPs with f=0.5. The open square represents the KWW fit at different temperatures. (b) The data
of panel (a) plotted in linear-log scale. (c) Fitted 𝜏𝜏[1]𝑠𝑠 and measured 𝜏𝜏[1]relaxation times as a function of 1/T*.
also observed in model nanocomposites with homogeneous, purely repulsive NPs by Li et al.49[. However, at volume ]
fractions comparable to ours, their stretching exponents for the first five modes are in the 0.7-0.9 range, i.e. much closer to unity than ours. Fig. 4(c) illustrates the dependence of the relaxation
times (𝜏𝜏[𝑝𝑝]) on the mode index p. We point out that at high temperature, T*=1.2, the relaxation time, 𝜏𝜏[𝑝𝑝]= 𝑝𝑝−2.27±0.01, deviates somewhat from the Rouse scaling law (𝜏𝜏[𝑝𝑝]∝ (𝑁𝑁/𝑝𝑝)2, for p
<<N), and the deviation becomes even more significant at the low temperature, T*=0.8, as we find 𝜏𝜏[𝑝𝑝]= 𝑝𝑝−2.44±0.02.
Finally, we consider the effect of the temperature on the autocorrelation functions 𝐶𝐶[𝑝𝑝](𝑡𝑡). Fig. 5(a) shows the behavior of the mode with p=1. Decreasing temperature decreases from 1.2 to 0.8,
the decay becomes slower and slower but is always properly fitted by KWW functions, allowing to measure the parameters 𝛽𝛽[𝑝𝑝] and 𝜏𝜏[𝑝𝑝]𝑠𝑠 as a function of T*, as shown in Fig. 4(b). Figure 5(b)
clarifies that the relaxation times 𝜏𝜏[1]𝑠𝑠 and 𝜏𝜏[1] as a function of the temperature keeps on being consistent with an Arrhenius behaviour: 𝑙𝑙𝑙𝑙(𝜏𝜏[1]) = 3.65 ×1
𝑇𝑇∗+ 5.72, the resulting
activation energy being slightly larger than the one presented earlier for the translational diffusion.
In order to highlight the effect of the nanoparticle heterogeneities on the polymer chain relaxation, we compare
the result of Fig. 4 and 5 to an homogeneous system with ɛPW=ɛPS=1.5ɛ≡ɛnp. This corresponds to a situation with
chemically homogeneous nanoparticles, which have, the same average polymer-particle interactions as the system with f=0.50. The results, illustrated in Fig. 6 and 7, highlight a number of differences
emerging with respect to the heterogenous case. For example, deviatons from the Rouse model appears to be weaker, especially concerning the relation between 𝜏𝜏[𝑝𝑝] and p In addition, the stretching
exponent of the KWW decreases significantly at low temperatures and increases significantly at high p, as shown in Fig. 6(d). The very good agreement between 𝜏𝜏[1]𝑠𝑠 and 𝜏𝜏[1] is also confirmed (Fig.
7). Finally, the activation energy is slightly lower than in the heterogeneous system:𝑙𝑙𝑙𝑙�𝜏𝜏[𝑝𝑝]� = 3.58 × 1
𝑇𝑇∗+ 5.72 . Overall, this implies that
polymer relaxation is 5-10% faster than in the heterogeneous system, for the same average polymer-NP interaction strength. The difference increases on lowering the temperature.
Fig. 6 For the homogeneous system with εnp=1.5ɛ: (a) RNM
autocorrelation function Cp(t) at, T*=1.0 and p=1, 2, . . . , 5,
from right to left. (b) RNM autocorrelation function 𝐶𝐶[𝑝𝑝](𝑡𝑡) for p=1 and T*=0.8, 0.9, . . ., 1.2, from right to left. (c) The relaxation time (𝜏𝜏[𝑝𝑝]) and (d) exponent 𝛽𝛽[𝑝𝑝] for the first 5 modes
of polymer chains at different temperatures.
Fig. 7 Fitted 𝜏𝜏[1]𝑠𝑠 and measured 𝜏𝜏[1]relaxation time (𝐶𝐶[𝑝𝑝]�𝜏𝜏[𝑝𝑝]�=1/e) as a function of 1/T*. The nanocomposite filled with homogeneous nanoparticles. Note that this system have the same
average polymer-particle interactions as the system with f=0.50.
3.3 Structural relaxation
In this section, we complement the previous analysis, based on the center of mass dynamics, by investigating the segmental dynamics. In particular, we compute the intermediate self scattering
function (ISSF), 𝜙𝜙[𝑞𝑞]𝑠𝑠(𝑡𝑡). For an isotropic system:
𝜙𝜙𝑞𝑞𝑠𝑠[(𝑡𝑡)] =[𝑀𝑀 �]1 〈𝑒𝑒𝑒𝑒𝑝𝑝(𝑖𝑖𝒒𝒒 ∙ [𝒓𝒓𝑐𝑐(𝑡𝑡) − 𝒓𝒓𝑐𝑐(0)])〉 𝑀𝑀 𝑐𝑐=1 = [𝑀𝑀 �]1 〈𝑐𝑐𝑖𝑖𝑙𝑙�𝑞𝑞∆𝑟𝑟𝑐𝑐(𝑡𝑡)� 𝑟𝑟𝑐𝑐(𝑡𝑡) 〉 𝑀𝑀 𝑐𝑐=1 (6)
where M is the total number of scattering centers in the polymer matrix, |𝑟𝑟[𝑐𝑐](𝑡𝑡) − 𝑟𝑟[𝑐𝑐](0)| = ∆𝑟𝑟[𝑐𝑐](𝑡𝑡) is the displacement of
scattering center m after time t, and the wave-vector q=2π/λ, probes the dynamics at the length scale λ. Similarly to a previous work by some of us50[, we fix q=6.9 to probe the ]
dynamics at segment length scale, i.e. the length scale relevant to the structural relaxation.
Fig. 8 ISSF, 𝜙𝜙[𝑞𝑞]𝑠𝑠(𝑡𝑡), for the system filled with heterogeneous nanoparticles at different temperatures T*=0.7, 0.8, 0.9, 1.0, 1.1 and 1.2 for different f.
Figure 10 shows the self intermediate scattering function for the heterogeneous case at different values of f. Similarly to what done for the RNM autocorrelation function, we measure the structural
relaxation time, τ[ℎ𝑒𝑒𝑐𝑐], from the standard relation, 𝜙𝜙𝑞𝑞𝑠𝑠(τℎ𝑒𝑒𝑐𝑐) = 𝑒𝑒−1. Fig. 9 shows the temperature dependence τ[ℎ𝑒𝑒𝑐𝑐] at different value of f. Once again, data are consistent with an
Arrhenius behaviour, but the activation energy monotonically increases with increasing f, differently from the diffusion constant of the chain center of mass. This difference can be rationalized
considering that the segment length scale is too
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This journal is © The Royal Society of Chemistry 2017 Soft Matter, 2017, 00, 1-3| 9
small for probing the presence of disorder in the spatial distribution of W and S sites, and, therefore, the dynamics on that length is controlled by the interaction strength only.
Fig. 9 Structural relaxation time as a function of 1/T* for different f.
In order to further compare the effects of the heterogeneous and homogeneous nanoparticles on the polymer relaxation, Fig. 10 and Fig. 11 show 𝜙𝜙[𝑞𝑞]𝑠𝑠(𝑡𝑡) and the structural relaxation time for the
heterogeneous case at f=0.5 and its homogeneous counterpart (ɛnp =1.5), respectively. Panel b of Fig.11 clarifies
that the structural relaxation time of the homogeneous system, τℎ𝑐𝑐𝑐𝑐𝑐𝑐 , also increases on lowering the temperature with an
Arrhenius fashion. In particular, we find that ln(τ[het]) = 2.10 ×[T]1∗− 1.99 and ln(τhomo) = 1.92 ×
T∗− 1.80, with the
associated activation energies resulting lower than those at the chain length scale. However, in both relative and absolute terms, the difference between the two activation energies increases on
going from the chain length scale (+2%) for the heterogeneous nanoparticles) to the segment length scale (+9%). This is reasonable, considering that a long chain will always mediate over many
energetically different situations. Overall, it appears clearly that, in the investigated temperature range, the difference between the homogeneous and the heterogeneous case is not dramatic.
However, such difference is expected to become larger and larger on decreasing the temperature. For example, at the lowest temperature investigated in this paper, T=0.7, we observe τhet/τhomo =1.09
that might appear a quite small difference. Conversely, at T=0.3 (a temperature that cannot be achieved in simulations but is fully accessible in experiments29) τ
het/τhomo =1.5, as can be
extrapolated by the Arrhenius behaviour reported in fig 8b and 9b. Accordingly, the structural relaxation time and, therefore, the zero-shear viscosity of the heterogeneous systems are expected to be
50% larger than those of the homogeneous system.
Fig. 10 (a) ISSF 𝜙𝜙[𝑞𝑞]𝑠𝑠(𝑡𝑡) for the system filled with heterogeneous nanoparticles at different temperatures T*=0.7, 0.8, 0.9, 1.0, 1.1 and 1.2. (b) Fitted relaxation times (C[p]�τ[p]�=1/e) as a
function of 1/T*.
Fig. 11 (a) ISSF 𝜙𝜙[𝑞𝑞]𝑠𝑠(𝑡𝑡) for the system filled with homogeneous nanoparticles at different temperatures T*=0.7, 0.8, 0.9, 1.0, 1.1 and 1.2. (b) Structural relaxation time as a function of 1/T*.
IV. Conclusion
We investigated via molecular dynamics simulations a coarse-grained model of polymer filled with chemically and structurally heterogeneous nanoparticles. The heterogeneities render the model more
realistic and similar to widely adopted fillers, such as carbon black. One important implication arises from the investigation of the polymer chain center of mass diffusion: the unexpected non
monotonic dependence of the activation energy on the NP composition do emerge similarly to Ref. 29 and 30, although the investigated models are markedly different and reflect two different physical
situations. This finding suggests the existence of a robust underlying physical
mechanism based on the interplay of interaction strength and configurational disorder. Indeed, such a non-monotonic nature disappears when focusing on the segmental dynamics, since the related probe
length is too small to be sensitive to disorder. A more general motivation of this work consists in understanding to what extent the presence of heterogeneous NPs modifies the system dynamics
compared to the homogeneous case. In this respect, we clarified that relatively small differences observed at high temperature are the precursor of larger and larger changes on approaching the glass
transition. This suggests that the effect of the heterogeneities may be strongly relevant for polymers forming glassy shells around nanoparticles.
This work is supported by the National Basic Research Program of China 2015CB654700(2015CB654704), the Foundation for Innovative Research Groups of the NSF of China (51221002), the National Natural
Science Foundation of China (51333004 and 51403015). The cloud calculation platform of BUCT is also greatly appreciated.
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[GAP Forum] Ordering of group elements
William DeMeo williamdemeo at gmail.com
Wed Jan 8 22:31:03 GMT 2014
Hello Forum,
We are working on an application involving functions defined on
groups. In our programs functions are represented as vectors that are
indexed by the elements of a given group. Therefore, the order in
which group elements are listed is important when performing
operations with these functions (e.g. convolution).
I noticed that, in GAP, the default cyclic group of order 8 has 3
generators, and the elements of the group are listed as follows:
gap> G:=CyclicGroup(8);
<pc group of size 8 with 3 generators>
gap> Elements(G);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> GeneratorsOfGroup(G);
[ f1, f2, f3 ]
Suppose instead we want the 8 element cyclic group to be represented
with one generator. We can do this as follows:
gap> f:= FreeGroup("x");
<free group on the generators [ x ]>
gap> g:= f/[f.1^8];
<fp group on the generators [ x ]>
gap> Elements(g);
[ <identity ...>, x, x^2, x^4, x^3, x^-3, x^-2, x^-1 ]
AsSet or AsList or AsSortedList or AsSSortedList all give the same
ordering of the group elements as does the Elements function.
Finally, here are my questions:
Question 1: Why is the default ordering of the elements not one of the
alternatives below?
[ <identity ...>, x, x^2, x^3, x^4, x^-3, x^-2, x^-1 ]
[ <identity ...>, x, x^2, x^3, x^4, x^5, x^6, x^7 ]
Question 2: Suppose our application requires the group elements be
listed in this "natural" way. Is it possible to do this without
having to inspect each group we use by hand? In other words, is it
possible to anticipate the ordering that GAP will use for a given
group, so that we can write our programs accordingly?
Question 3: Is there no conventional or "natural" ordering of elements
for certain groups, in particular the cyclic groups? And why does GAP
prefer ...x^2, x^4, x^3... over ...x^2, x^3, x^4... in the example
As another example, here is the cyclic group of order 16:
gap> g:= f/[f.1^16];
<fp group on the generators [ x ]>
gap> Elements(g);
[ <identity ...>, x, x^2, x^4, x^8, x^3, x^5, x^9, x^6, x^10, x^12,
x^7, x^11, x^13, x^14, x^15 ]
Thank you in advance for any suggestions or advice. If there is no
general way to anticipate how GAP will order elements of a group, then
any comments on GAP's choice of element ordering for particular
examples (like cyclic groups or symmetric groups) would be very much
William DeMeo
Department of Mathematics
University of South Carolina
1523 Greene Street
Columbia, SC 29208
phone: 803-261-9135
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Create a piecewise-linear nonlinearity estimator object
idPiecewiseLinear is an object that stores the piecewise-linear nonlinearity estimator for estimating Hammerstein-Wiener models.
Use idPiecewiseLinear to define a nonlinear function $y=F\left(x,\theta \right)$, where y and x are scalars, and θ represents the parameters specifying the number of break points and the value of
nonlinearity at the break points.
The nonlinearity function, F, is a piecewise-linear (affine) function of x. There are n breakpoints (x[k],y[k]), k = 1,...,n, such that y[k] = F(x[k]). F is linearly interpolated between the
F is also linear to the left and right of the extreme breakpoints. The slope of these extensions is a function of x[i] and y[i] breakpoints. The breakpoints are ordered by ascending x-values, which
is important when you set a specific breakpoint to a different value.
There are minor difference between the breakpoint values you set and the values stored in the object because the toolbox has a different internal representation of breakpoints.
For example, in the following plot, the breakpoints are x[k] = [-2,1,4] and the corresponding nonlinearity values are y[k] = [4,3,5].
The value F(x) is computed by evaluate(NL,x), where NL is the idPiecewiseLinear object. When using evaluate, the break points have to be initialized manually.
For idPiecewiseLinear object properties, see Properties.
NL = idPiecewiseLinear creates a default piecewise-linear nonlinearity estimator object with 10 break points for estimating Hammerstein-Wiener models. The value of the nonlinearity at the break
points are set to []. The initial value of the nonlinearity is determined from the estimation data range during estimation using nlhw. Use dot notation to customize the object properties, if needed.
NL = idPiecewiseLinear(Name,Value) creates a piecewise-linear nonlinearity estimator object with properties specified by one or more Name,Value pair arguments. The properties that you do not specify
retain their default value.
idPiecewiseLinear object properties include:
NumberofUnits — Number of breakpoints
10 (default) | positive integer
Number of breakpoints, specified as an integer.
BreakPoints — Break points and corresponding nonlinearity values at the breakpoints
[ ] (default) | vector | matrix
Break points, x[k], and the corresponding nonlinearity values at the breakpoints, y[k], specified as one of the following:
• 2-by-n matrix — The x and y values for each of the n break points are specified as [x[1],x[2], ...., x[n];y[1], y[2], ..., y[n]].
• 1-by-n vector — The specified vector is interpreted as the x values of the break points: x[1],x[2], ...., x[n]. All the y values of the break points are set to 0.
When the nonlinearity object is created, the breakpoints are ordered by ascending x-values. This is important to consider if you set a specific breakpoint to a different value after creating the
Free — Option to fix or free the values in the mapping object
true (default) | logical scalar
Option to fix or free the values in the mapping object, specified as a logical scalar. When you set an element of Free to false, the object does not update during estimation.
Create a Default Piecewise-Linear Nonlinearity Estimator
Specify the number of break points.
Estimate a Hammerstein Model with Piecewise-Linear Nonlinearity
Load estimation data.
load twotankdata;
z = iddata(y,u,0.2,'Name','Two tank system');
z1 = z(1:1000);
Create an idPiecewiseLinear object, and specify the breakpoints.
InputNL = idPiecewiseLinear('BreakPoints',[-2,1,4]);
Since BreakPoints is specified as a vector, the specified vector is interpreted as the x-values of the break points. The y-values of the break points are set to 0, and are determined during model
Estimate model with no output nonlinearity.
sys = nlhw(z1,[2 3 0],InputNL,[]);
Version History
Introduced in R2007a
R2021b: Use of previous idnlarx and idnlhw mapping object names is not recommended.
Starting in R2021b, the mapping objects (also known as nonlinearities) used in the nonlinear components of the idnlarx and idnlhw objects have been renamed. The following table lists the name
Pre-R2021b Name R2021b Name
wavenet idWaveletNetwork
sigmoidnet idSigmoidNetwork
treepartition idTreePartition
customnet idCustomNetwork
saturation idSaturation
deadzone idDeadZone
pwlinear idPiecewiseLinear
poly1d idPolynomial1D
unitgain idUnitGain
linear idLinear
neuralnet idFeedforwardNetwork
Scripts with the old names still run normally, although they will produce a warning. Consider using the new names for continuing compatibility with newly developed features and algorithms. There are
no plans to exclude the use of these object names at this time.
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CIE – 9709 Mechanics 1, AS Level, Paper 42, May/June 2016 – solution
20 thoughts on “CIE – 9709 Mechanics 1, AS Level, Paper 42, May/June 2016 – solution”
• In question 7 shouldn’t (T SQUARED) = 1/6
please correct me if I’m wrong
• Where is question paper
• In Question 3, if we donot round off the value of x and write 2.17m instead, how many marks will be deducted.?
• when will be chemistry p4 (9702) will be uploaded?
□ the one given yesterday? on Monday evening
• If in Q1 we I found out vertical component correctly and made a sliiy mistake in finding the horizontal component but the equations were correct how many marks to loose?
• Will you’ll be publishing A2 accounts, economics and business studies papers? It will be taken in the coming week.
□ after the exams take place 🙂
☆ Okay Thanks😊. Can you please post the question paper for m1
• When will paper 32 be uploaded???
□ the exam is on 17th.. so on 18th or 19th will be uploaded
• When will acts p2 be uploaded and bsnss P1 acts was on may 3
• Could u please upload the 2016 Feb/March series papers which was conducted only in India for AS subjects such as physics,chemistry and biology please?
• will the maths paper 32 be uploaded tomorrow and when will u upload mechanics missing page
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GreeneMath.com | Ace your next Math Test!
Reciprocal Identities
Additional Resources:
In this lesson, we will learn about reciprocal identities in trigonometry. All trigonometric function values can be found by taking the reciprocal of another function value. sin θ and csc θ, cos θ
and sec θ, and tan θ and cot θ are reciprocals. Additionally, we will learn about the signs of function values. Lastly, we will learn how to find the values of all six trigonometric functions given
one value and the quadrant.
+ Show More +
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If the set A contains 5 elements and the set B contains 6 eleme... | Filo
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If the set contains 5 elements and the set contains 6 elements, then the number of one-one and onto mappings from to is
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Question Text If the set contains 5 elements and the set contains 6 elements, then the number of one-one and onto mappings from to is
Updated On Jan 2, 2024
Topic Functions
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 73
Avg. Video Duration 10 min
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Marginal vs. effective tax rates | Analytica - Visionary Modeling
Presidential candidate Mitt Romney released his tax returns, and the news channels were abuzz with commentary about his effective federal tax rate of 13.9% on $21.7M of income. I will refrain from
my own commentary on his situation, but what particularly piqued my interest were the quotes I heard from “ordinary” people on the radio. Everyone, it seems, claims to pay a higher federal tax rate
than Mr. Romney. Typical was a person interviewed in a news report who stated “I pay 28% in taxes and have trouble every month making my rent.”
Listening to people comment on their own tax rates left me with two impressions. I couldn’t help but feel that most, if not all, of the people I heard were severely overestimating their own
effective federal tax rates, and I suspected that most people did not understand the difference between effective tax rate and marginal tax rate. I also saw an opportunity to fire up Analytica to
explore the relationship between the two concepts.
Before proceeding, I should disclaim that I am not an accountant or lawyer, or in any way certified to give tax advice. Nothing I am discussing here should be construed as tax advice. Yada yada
Your effective tax rate is simply the amount of tax you pay divided by your total income. Your marginal tax rate in percent is equal to the number of cents that your tax liability would decrease if
your income decreased by one dollar. The term tax bracket identifies your marginal rate. If you are like most people who derive the bulk of their income from wages and earned income, the
progressive structure of the US income tax rate results in your marginal tax rate being higher than your effective tax rate.
To explore these concepts, I created an index named Tax_bracket and an edit table titled “Lower taxable income in bracket” (with identifier Lower_taxable_income), and populated the edit table with
2011 income thresholds for a single filer:
For an arbitrary Taxable_income, the following Analytica expression looks up the applicable marginal tax rate from the table:
By defining Taxable_income as Sequence(0,$250K), we obtain a graph of marginal tax rates as a function of taxable income:
To arrive at taxable income, you need to subtract your deductions from your net income and make numerous other adjustments that embody the complexity of the US tax code. The average person with
$100K of taxable income has a total income of around $125K.
Now, how do you go from here to effective tax rate? This requires two steps. First, you need to compute your tax liability (i.e., taxes paid), and then you need to divide by total income (rather
than just by taxable income). Given a particular taxable income, the tax liability is the area under the above curve falling to the left of your taxable income. With Taxable_income defined as a
sequence, we can compute income Tax_liability using:
I will ignore other non-earned sources of income (e.g., capital gains income) for simplicity, which would have to be added to this as well. The second piece, relating total income and taxable
income, is far more involved. To avoid getting mired down in the complexities of deductions and tax code, I’ll simply use a quick and dirty approximation for an average:
Total_income = $5800 + 120% * Taxable_income
The $5800 is the standard deduction for a single filer in 2011, and the 120% comes from my quick-and-dirty assessment of the ratio of taxable to non-taxable income, with an assumption that this stays
relatively constant overall income levels. There are many articles on the web providing average deduction levels for US taxpayers at different income levels. I invite you to find those and compare
how my quick and dirty approximation fares, or to simply compare to your own tax returns. We are now in a position to compute Effective_tax_rate as:
Tax_liability / Total_income
To produce a nice graph comparing the two tax rates as a function of total income, I created a variable titled “Marginal vs. Effective” and defined it as a list:
[Marginal_tax_rate, Effective_tax_rate]
From its result view, I pressed the [XY] button and added Total_income as a comparison variable
Leaving us with this final graph
Recalling the person on the radio who claimed a 28% tax rate but could barely make his rent each month, this graph makes it evident that he was confusing his marginal (the red line) with his
effective (blue line) tax rate.
When you engage in the national debate about whether your taxes are too high, the effective tax rate is clearly the relevant number. However, when you are considering tax consequences when making a
personal decision, the marginal tax rate is the pertinent value. Is the $20/hr pay for working an extra hour each week worth sacrificing your discretionary time? If your marginal rate is 25%, you
should yourself if your discretionary time is worth $15/hr.
I have made my Analytica model (Marginal vs Effective Tax Rates.ana) available, and I welcome you to continue this exploration further using this model as a starting point. For example, you
might explore how things change for people like Mitt Romney who derive most their income from capital gains and dividends. To make changes of your own, you will need either a licensed version of
Analytica Professional, Enterprise or Optimizer, or the Analytica free edition.
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How inventory value is calculated
Posted inBusiness
How inventory value is calculated
If there may be any software that makes dwelling economics simpler for us, it’s the low cost calculator. Firms make use of reductions to spice up gross sales and reward clients.
With this on-line software to calculate reductions, we can know if that product that we love a lot from our favourite retailer or e-commerce web siteis price shopping for with that low cost or
whether it is actually worthwhile for us. As customers, it’s handy to be vital and know the best way to handle our cash successfully. In any case, this calculator can be very helpful in enterprise
for eexample,to know invoices to calculate private earnings tax or to calculate VAT, as a result of what it does is assist us preserve monitor of our accounts.
How does the low cost calculator work?
With this on-line calculator, it is possible for you to to know, primarily based on the low cost share of an quantity, what its ultimate worth will probably be and what its ultimate value will
probably be, in an effort to decide whether or not it’s worthwhile or if it pursuits you primarily based in your case.
Even when, for instance, you’re within the place of a salesman and you’ve got calculated the revenue margin of the product, this calculator will allow you to decide the low cost share that’s finest
to your earnings. We’ll let you know the way it works!
These are the 3 steps it’s good to management the low cost calculator:
1. Point out within the first field what the worth of the low cost share is, for instance, 10%.
2. Secondly, within the second field the place it signifies “from,” you’ll have to add the financial quantity from which that low cost goes to be extracted. For instance, €50 Within the earlier
case, it could be 10% of fifty euros.
3. Within the final step, you’ll have to click on on the button the place it says “apply low cost “. Thus, the web page will robotically refresh, and you’ll acquire the value akin to the requested
low cost.
The outcomes will seem in a desk indicating the quantity added, the indicated low cost share, and the low cost in euros. The discounted worth, €40 within the case above, is proven on the suitable.
Low cost calculator issues?
When you’re going to use this calculator, repeat the operations at any time when it’s good to to ensure that the odds it offers are right and that you’ve got entered the quantities that you simply
actually wanted to calculate.
Additionally do not forget that, when you may have obtained the value ensuing from the low cost, you’ll have to examine it with the preliminary value to see how a lot the ultimate value of the
product varies.
What’s the components with which the low cost might be calculated?
Earlier than discovering the components for calculating a reduction share, what we must do is calculate the share of stated quantity that would be the results of:
1. Make a multiplication of the share and the quantity.
2. Divide the results of the earlier operation by 100, since it’s a exact decimal quantity.
We do not forget that the share itself is a means of engaged on a proportion that begins with the quantity 100. This could be the components that enables us to calculate a reduction:
Uncover an instance of low cost calculation and perceive how this operation works.
After the components is given, this instance of a reduction calculation finished manually in just a few steps will assist us comprehend the method:
• We go to a clothes retailer the place we’re focused on a gown whose worth is 400 euros.
• The low cost that has been utilized to stated gown is 10% as a result of it’s a preview of a brand new assortment.
• We’ll multiply the ten% akin to the low cost by €400 (akin to the entire value) / 100 = €40..
x = 10 x 400/100
The low cost’s €40 will scale back the gown’s value. So €400-€40 = €360.
Thus, the price of the gown of €400 with a reduction of €40 will end in a value of €360 for the gown.
Along with this commented components and the guide calculation, an alternative choice is to hold out the low cost calculation by means of Excel, the place we are able to additionally examine the
reductions utilized.
How is the unique value of a reduction calculated?
It’s straightforward to discover a product’s starting value if we all know its low cost. In just a few steps, we are going to know what the value of the product was earlier than the sale.
Persevering with with the earlier instance:We purchased a €200 gown in a garments retailer that had a ten% low cost already.
• We’ll discover the preliminary value, X: 10% off €200 garments equals 0.10.
X (1-0.10) = €200.
• To resolve for X mathematically:
• Due to this fact, the value of the product beforehand decreased by 10% was €222.
No comments yet. Why don’t you start the discussion?
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Electric Dipole - BrainDuniya
What is an Electric Dipole?
A pair of two equal and opposite charges separated by a small distance is called an electric dipole.
Two equal charges of opposite sign ( + q ) \text {and} ( - q ) , separated by a small distance ( 2 l ) constitute a dipole as shown in figure. The distance between two charges is known as dipole
EXAMPLE –
1. Molecules of water ( H_2 O ) , ammonia ( NH_3 ) , Hydrochloric acid ( HCl ) , Carbon dioxide ( CO_2 ) , Sodium chloride ( NaCl ) etc. are examples of electric dipoles.
2. Dipoles are very important for the study of the electric behavior of ionic compounds and their solutions.
Dipole Moment
The product of the magnitude of one of the charge of electric dipole and the dipole length is called dipole moment.
Therefore, dipole moment is –
p = ( q \times 2l )
1. The direction of dipole moment is from negative charge to positive charge.
2. It is a vector quantity.
One coulomb metre is the dipole moment of a dipole consisting of end charges ( + 1 \ C ) \text {and} ( - 1 \ C ) which are separated by a distance of ( 2l =1 \ m )
Consider about two charges ( + q ) and ( - q ) , which are equal in magnitude but opposite in sign, kept at a small distance of ( 2 \ l ) apart as shown in figure. These to charges together will
constitute an electric dipole. Its dipole moment will be p = ( 2 \ q \ l )
Polarity of a Dipole
All matter consists of molecules and these molecules behave as dipoles. These molecules may be of two types.
1. Polar molecule.
2. Non-polar molecule.
Polar molecule
In a molecule, if the centre of mass of positive charge ( i.e. protons ) does not coincide with the centre of mass of negative charge ( i.e. electrons ), then the molecule is said to be a polar
Polar molecules have permanent dipole moment ( \vec {p} ) . A water molecule is a polar molecule as shown in figure (a). The dipole moment of water molecule is ( 6.2 \ \times \ 10^{-30} C \ m ) .
Water being a polar substance acts as a very good solvent for ionic compounds. Ionic compounds when dissolved in water undergoes electrolysis and breaks into ions. A polar molecule is shown in figure
( a ).
070502 POLAR & NON-POLAR MOLECULES
Non-polar molecule
If the centre of mass of positive charge ( i.e. protons ) coincides with the centre of mass of negative charge ( i.e. electrons ), then the molecule is said to be a non polar molecule.
The dipole moment of a non-polar molecule is zero. A non-polar molecule is shown in figure ( b ).
When a non polar molecule is placed in a uniform electric field, positive charges experience force in the direction of electric field and negative charge experience a force in a direction opposite to
the direction of the electric field. Thus, centres of mass of positive and negative charges displaced in opposite directions and the molecule gets elongated as shown in figure (c). Now it behaves as
a dipole.
Therefore, a non-polar molecule becomes a polar molecule under the influence of a strong external electric field.
The dipole moment of such a molecule is called induced dipole moment.
Force on Dipole in uniform Electric Field
When a dipole is placed in an electric field produced by another charged source, it experience a force on it. This force produces a torque on the dipole. The magnitude of the force and torque depends
upon following points –
1. The position of dipole with respect to the direction of electric field.
2. Dipole moment of the dipole.
3. Strength of the electric field in which the dipole is kept.
4. Uniformity of the electric field.
Consider about a dipole which is placed in an uniform electric field ( \vec {E} ) , such that the dipole moment ( \vec {p} ) of the electric dipole makes an angle ( \theta ) with the electric field
as shown in figure.
070503 FORCE ON DIPOLE IN UNIFORM ELECTRIC FIELD
Due to the presence of electric field, the charges ( + q ) and ( - q) of the dipole will experience equal and opposite forces. These forces are given by –
1. \vec {F} = ( + q \vec {E} )
2. \vec {F} = ( - q \vec {E} )
Therefore, net force on the dipole will be –
\vec {F_{net}} = ( + q \vec {E} - q \vec {E} ) = 0
Thus, the net force acting on an electric dipole in a uniform electric field is zero.
But, two equal and opposite forces acting on the dipole will constitutes a couple. This couple will tend to rotate the dipole in the clockwise direction and hence tries to align the dipole along the
direction of electric field.
Torque on a Dipole
We know that, torque of a force –
\text {Torque} = \text {Force} \ \times \ \text {Moment arm}
Therefore, torque acting on the dipole will be –
\tau = q E \times AC ……. (1)
From geometry of ( \triangle ABC ) , we will get –
\left ( \frac {AC}{AB} \right ) = \sin \theta
Or, \quad AC = AB \sin \theta = 2 l \sin \theta
Hence, equation (1) becomes –
\tau = q E \times 2l \sin \theta = ( q \times 2l ) E \sin \theta = p E \sin \theta ……. (2)
In vector form, the equation (2) can be written as –
\vec {\tau} = \vec {p} \times \vec {E} ……. (3)
The direction of torque \tau is given by right hand thumb rule.
Torque, when Dipole Moment is parallel to Electric Field
Consider that, the dipole is placed in the field such that the direction of dipole moment is parallel to the direction of electric field i.e. ( \theta = 0 \degree ) . Then, torque acting on the
dipole will be –
\tau = p E \sin 0 \degree = 0
Thus, when dipole moment is parallel to the electric field, no torque acts on it. In this case, dipole is in stable equilibrium.
Position of dipole is shown in figure ( A ).
Torque, when Dipole Moment is inclined to Electric Field
Consider that, the dipole is placed in the electric field such that the direction of dipole moment is at an angle ( \theta = 30 \degree ) to the direction of electric field.
Then, torque acting on the dipole will be –
\tau = p E \sin 30 \degree = \left ( \frac {p E}{2} \right )
Thus, when dipole moment is makes an angle of ( 30 \degree ) with the direction of electric field, the torque acting on the dipole is half of the maximum value.
Position of dipole is shown in figure ( B ).
070504 VARIATION OF TORQUE ACTING ON A DIPOLE
Torque, when Dipole Moment is perpendicular to Electric Field
Consider that, the dipole is placed in the electric field such that the direction of dipole moment is at an angle ( \theta = 90 \degree ) to the direction of electric field.
Then, torque acting on the dipole will be –
\tau = p E \sin 90 \degree = p E
Thus, when dipole moment is at ( 90 \degree ) to the direction of the electric field, torque acting on the dipole is maximum.
Position of dipole is shown in figure ( C ).
Torque, when Dipole Moment is anti-parallel to Electric Field
Consider that, the dipole is placed in the field such that the direction of dipole moment is anti-parallel to the direction of electric field i.e. ( \theta = 180 \degree )
Then, torque acting on the dipole will be –
\tau = p E \sin 180 \degree = 0
Thus, when dipole moment is anti-parallel to the electric field no torque acts on the dipole. Thus dipole is in highly unstable equilibrium.
Position of dipole is shown in figure ( D ).
Force on Dipole placed in Non uniform Electric Field
Consider about a dipole placed in a non uniform electric field \vec {E} as shown in figure.
Let, ( \vec {E_1} ) and ( \vec {E_2} ) are the electric field intensities at position of ( + q ) and ( - q ) charges respectively such that \left ( \vec {E_1} > \vec {E_2} \right ) .
070505 FORCE ON DIPOLE IN NON UNIFORM ELECTRIC FIELD
Then, force acting on the end charges of the dipole will be –
1. F_1 = ( + q E_1 )
2. F_2 = ( - q E_2 )
Therefore, net force acting on dipole will be –
F_{net} = ( F_1 + F_2 ) = q ( E_1 - E_2 ) = q \Delta E ………. (1)
Where, \Delta E = ( \vec {E_1} - \vec {E_2} ) ( Change in electric field ).
Let, length of the dipole is represented as ( \Delta x ) . Then –
p = q \times \Delta x
Therefore, \quad q = \left ( \frac {p}{\Delta x} \right )
Hence, equation (1) can be written as –
F_{net} = \left ( \frac {p}{\Delta x} \right ) \Delta E ………. (2)
In differential form, \quad F_{net} = p \left ( \frac {d E}{d x} \right ) ……… (3)
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Rate equations - (Laser Engineering and Applications) - Vocab, Definition, Explanations | Fiveable
Rate equations
from class:
Laser Engineering and Applications
Rate equations describe the relationship between the population of energy levels in a laser medium and the rates of various processes such as stimulated emission, spontaneous emission, and
absorption. These equations help in understanding how to achieve and maintain population inversion, which is crucial for laser operation. Additionally, they are vital for analyzing laser modes and
coherence by providing insights into how different modes develop and interact under varying conditions.
congrats on reading the definition of Rate equations. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Rate equations account for various processes in lasers, including stimulated emission, spontaneous emission, and absorption, providing a comprehensive framework to model laser behavior.
2. Population inversion is a key outcome of rate equations, enabling lasers to function efficiently by ensuring more atoms are in the excited state than the ground state.
3. The rate of change of populations in different energy states is modeled by differential equations that can predict the behavior of the laser under specific conditions.
4. In multimode lasers, rate equations help analyze how different laser modes interact with one another and their contribution to overall coherence.
5. Rate equations can be used to derive threshold conditions for laser operation, indicating the minimum pump power required to achieve lasing.
Review Questions
• How do rate equations contribute to achieving population inversion in laser systems?
□ Rate equations provide a mathematical framework that models the rates of stimulated emission, spontaneous emission, and absorption. By balancing these processes, they show how to manipulate
the populations of energy levels within the gain medium. Achieving population inversion is critical for lasing, as it ensures that stimulated emission predominates over absorption, allowing
the laser to produce coherent light.
• Discuss the role of rate equations in determining the characteristics of laser modes and their coherence.
□ Rate equations play a significant role in understanding laser modes by describing how different modes develop based on energy level populations. They reveal how these modes interact within
the gain medium and influence each other. This interaction can affect coherence length and quality, as coherent light arises from a dominant mode or combination of modes that are in phase
with one another.
• Evaluate the implications of rate equations on designing efficient laser systems and their potential applications.
□ Evaluating rate equations allows engineers to design efficient laser systems by optimizing parameters like pump power and gain medium properties. Understanding these relationships is
essential for developing lasers with specific characteristics for various applications, from telecommunications to medical devices. The ability to predict how changes in operating conditions
affect performance can lead to advancements in laser technology and its integration into modern solutions.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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Nasehpour - Ordinary Differential Equations
Ordinary Differential Equations
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.
In the course on ordinary differential equations, we usually deal with the following topics:
• Separable equations
• First-order linear differential equations
• Bernoulli differential equations
• Exact differential equations and test for exactness
• Homogeneous functions and equations
• Riccati, Clairaut's, and Lagrange's equations
• Orthogonal trajectories
• The general form of a second-order linear differential equation
• Second-order linear homogeneous differential equations with constant coefficients (real, complex, and repeated roots)
• Reduction of order
• Linear independence and Wronskian
• Nonhomogeneous linear second-order differential equations
• Undetermined coefficients and variation of parameters
• Higher-order differential equations
• Cauchy-Euler differential equations of order n
• Power series solutions of differential equations
• Regular singular points
• The generalized power series method in solving differential equations
• Definition of integral transforms
• Definition of Laplace transform
• A short table of Laplace and inverse Laplace transforms
• Step functions and their Laplace transforms
• Computation of Laplace transform of a function's derivative and antiderivative
• Differentiation and integration of Laplace transforms
• Dirac delta function
• Convolution integral
• Completion of the table of Laplace transforms
• Systems of differential equations
• Real, complex, and repeated eigenvalues and their applications in solving systems of differential equations
• Nonhomogeneous systems of differential equations
For more details, refer to the following book:
Boyce, W. E., DiPrima, R. C., & Meade, D. B. (2017). Elementary differential equations. John Wiley & Sons.
For differential equations, a book (in Persian) by Dara Moazzami (pictures) has been also always very useful.
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What is 293 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
To convert 293 degrees Celsius to Fahrenheit, you can use the formula: Fahrenheit = (Celsius * 9/5) + 32. So, plugging in 293 for Celsius, the conversion would be (293 * 9/5) + 32 = 559.4 degrees
Celsius and Fahrenheit are two different temperature scales used to measure the temperature of an object or the surrounding environment. While Celsius is the most widely used temperature scale in the
world, Fahrenheit is more commonly used in the United States, Belize, the Bahamas, the Cayman Islands, and Liberia.
The Celsius scale was developed in the 18th century by Swedish astronomer Anders Celsius, and it is based on the boiling point of water, which is assigned a value of 100 degrees, and the freezing
point of water, which is assigned a value of 0 degrees. On the other hand, the Fahrenheit scale was developed in the 18th century as well by German physicist Daniel Gabriel Fahrenheit, and it is
based on a range of temperatures he observed in the laboratory, with the freezing point of water being 32 degrees, and the boiling point being 212 degrees.
Now, let’s talk about the conversion from Celsius to Fahrenheit. As mentioned earlier, the formula to convert Celsius to Fahrenheit is (Celsius * 9/5) + 32. To understand why this conversion formula
works, we need to understand the relationship between the two scales. The freezing and boiling points of water at 0 and 100 degrees in Celsius are equivalent to 32 and 212 degrees in Fahrenheit.
To convert any temperature from Celsius to Fahrenheit, you can follow the formula: (Celsius * 9/5) + 32. First, you multiply the Celsius temperature by 9/5, then you add 32 to the result. This will
give you the equivalent temperature in Fahrenheit. For example, if you have 293 degrees Celsius and want to convert it to Fahrenheit, you would multiply 293 by 9/5, which equals 527.4, and then add
32, giving you a result of 559.4 degrees Fahrenheit.
In conclusion, 293 degrees Celsius is equivalent to 559.4 degrees Fahrenheit. Understanding the relationship between these two temperature scales can be helpful when working with temperature
measurements in different parts of the world or when you need to make quick conversions in everyday life. Whether you’re a student studying science or a traveler visiting a country that uses a
different temperature scale, knowing how to convert Celsius to Fahrenheit (and vice versa) can be a useful skill to have.
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Integrated Knowledge Solutions
Originally published on July 17, 2017.
The Naive Bayes (NB) classifier is widely used in machine learning for its appealing tradeoffs in terms of design effort and performance as well as its ability to deal with missing features or
attributes. It is particularly popular for text classification. In this blog post, I will illustrate designing a naive Bayes classifier for digit recognition where each digit is formed by selectively
turning on/off segments of a seven segment LED display arranged in a certain fashion as shown below. The entire exercise will be carried out in Excel. By writing Excel formulas and seeing the results
in a spreadsheet is likely to result in a better understanding of the naive Bayes classifier and the entire design process.
We will represent each digit as a 7-dimensional binary vector where a 1 in the representation implies the corresponding segment to be on. The representations for all ten digits, 0-9, is shown below.
Furthermore, we assume the display to be faulty in the sense that with probability p a segment doesn't turn on(off) when it is supposed to be on(off). Thus, we want to design a naive Bayes classifier
that accepts a 7-dimensional binary vector as an input and predicts the digit that was meant to be displayed.
Basics of Naive Bayes
A Naive Bayes (NB) classifier uses Bayes' theorem and independent features assumption to perform classification. Although the feature independence assumption may not hold true, the resulting
simplicity and performance close to complex classifiers offer complelling reasons to treat features to be independent. Suppose we have $d$ features, $x_1,\cdots, x_d$, and two classes $ c_1\text{ and
} c_2$. According to Bayes' theorem, the probability that the observation vector $ {\bf x} = [x_1,\cdots,x_d]^T$ belongs to class $ c_j$ is given by the following relationship:
$ P(c_j|{\bf x}) = \frac{P(x_1,\cdots,x_d|c_j)P(c_j)}{P(x_1,\cdots,x_d)}, j= 1, 2$
Assuming features to be independent, the above expression reduces to:
$ P(c_j|{\bf x}) = \frac{P(c_j)\prod_{i=1}^{d}P(x_i|c_j)}{P(x_1,\cdots,x_d)}, j= 1, 2$
The denominator in above expression is constant for a given input. Thus, the classification rule for a given observation vector can be expressed as:
$ {\bf {x}}\rightarrow c_1\text { if }P(c_1)\prod_{i=1}^{d}P(x_i|c_1)\geq P(c_2)\prod_{i=1}^{d}P(x_i|c_2)$
Otherwise assign
$ {\bf {x}}\rightarrow c_2$
For classification problems with C classes, we can write the classification rule as:
$ {\bf {x}}\rightarrow c_j \text{ where } P(c_j)\prod_{i=1}^{d}P(x_i|c_j) > P(c_k)\prod_{i=1}^{d}P(x_i|c_k), k=1,...,C \text{ and } k\neq j$
In case of ties, we break them randomly. The implementation of the above classification rule requires estimating different probabilities using the training set under the assumption that the training
set is a representative of the classification problem at hand.
There are two major advantages of the NB classification when working with binary features. First, the naive assumption of feature independence reduces the number of probabilities that need to be
calculated. This, in turn, reduces the requirement on the size of training set. As an example, consider the number of binary features to be 10. Without the naive independence assumption, we will need
to calculate $ 2^{10}$ (1024) probabilities for each class. With the independent features assumption, the number of probabilities to be calculated per class reduces to 10. Another advantage of NB
classification is that it is still possible to perform classification even if one or more features are missing; in such situations the terms for missing features are simply omitted from calculations.
Faulty Display Digit Recognition Steps
In order to design a classifier, we need to have training data. We will generate such data using Excel. To do so, we first enter the seven dimensional representation for each digit in Excel and name
the cell ranges for each digit as digit1, digit2 etc. as shown below.
Next, we use Excel's RAND() function to decide whether the true value of a segment should be flipped or not (1 to 0 or 0 to 1). We repeat this as many times as the number of training examples for
each digit need to be generated. In discussion here, we will generate 20 examples for each digit. The figure below shows some of the 20 such examples and the Excel formula used to generate them for
digit 1. Noiselevel in the formula refers to a cell where we store the probabilty p of a segment being faulty. This value was set to 0.2. Similar formulas are used to generate 20 examples for each
The 200 training examples generated as described are next copied and pasted into a new worksheet. This is the sheet that will be used for designing the classifier. The paste operation is carried out
using the "Values Only" option. This is done to avoid anymore changes in the generated noisy examples.
Naive Bayesian Classifier Design
Having generated 200 examples of faulty display digits, we are now ready to design our NB classifier. Designing NB classifier means we need to compute/estimate class priors and conditional
probabilities. Class priors are taken as the fraction of examples from each class in the training set. In the present case, all class priors are equal. This means that class priors do not play any
role in arriving at the class membership decision in our present example. Thus, we need to estimate only conditional probabilities. The conditional probabilities are the frequencies of each attribute
value for each class in our training set. The following relationship provides us with the probability of segment 1 being equal to 1 conditioned on that the digit being displayed is digit 1.
$P(s_{1}=1|digit1) = \frac{\text{count of 1's for segment 1 in digit1 training examples}}{\text{number of digit1 training examples}}$
Since only two possible states, 1 and 0, are possible for each segment, we can calculate the probability of segment 1 being equal to 0 conditioned on that the digit being displayed is digit 1 by the
following relationship:
$ P(s_{1}=0|digit1) = 1 - P(s_{1}=1|digit1)$
In practice, however, a correction is applied to conditional probabilities calculations to ensure that none of the probabilities is 0. This correction, known as Laplace smoothing, is given by the
following relationship:
$ P(s_{1}=1|digit1) = \frac{1+\text{count of 1's for segment 1 in digit1 training examples}}{2+\text{number of digit1 training examples}}$
Adding 1 to the numerator count ensures probability value doesnot become 0. Adding 2 to the denominator reflects the number of states that are possible for the attribute under consideration. In this
case we have binary attributes. Note that in text classification applications, for example in email classification, where we use words in text as attributes, the denominator correction term will be V
with V being the number of words in the dictionary formed by all words in the training examples. Also you will find the term Bernoulli NB being used when the feature vector is a binary vector as in
the present case, and the term Multinomial NB being used when working with words as features.
Going back to our training set, we are now ready to compute conditional probabilities. The formula for one such computation is shown below along with a set of training examples for digit1. Similar
formulas are used to compute the remaining conditional probabilities and the training examples to obtain 70 conditional probabilities needed to perform classification.
Testing the Classifier
Having calculated conditional probabilities, we are now ready to see how well our classifier will work on test examples. For this, we first generate five test examples for each digit following the
steps outlined earlier. The test examples are copied and pasted (using the "Value Only" paste option). We also copy the probabilties computed above to the same worksheet where test examples have been
pasted, just for convenience. This done, we next write formulas to compute the probabilty for each digit given a test example and the set of conditional probabilities. This is shown below in a
partial screenshot of Excel worksheet where the formula shown for calculating the probability of displayed digit being 1 based on the states of seven segments. The references to cells in the formula
are where we have copied the table of conditional probabilities.
While the higlighted columns indicate the highest probability value in each row and thus the classification result, the following formula in column "S" results in the classifier output as the label
to be assigned to the seven component binary input representing the status of the faulty display.
Next, comparing the labels in columns H (true label) and S (predicted label) we can generate the confusion matrix to tabulate the performance of the classifier. Doing so results in the following
confusion matrix with 80% correct classification rate.
The 80% accuracy is for 20% noise level. If desired we can go back and rerun the entire simulation again for different noise levels and determine how the accuracy varies with varying noise levels.
Finally, it would be nice to have a visual interface where we can input a row number referencing a test example, and display the faulty digit as well as the predicted digit. Such a display can be
easily created using conditional formatting and adjusting the shape and size of certain Excel cells (See a post on this). One such display is shown below. By entering a number in the range of 2-51
(50 test examples) in cell AE1, we can pull out the segment values using Indirect function of Excel. For example, the segment value shown in cell W3 in figure below is obtained by the following
formula =INDIRECT("A"&$AE$1). Similary, the value in cell X3 is obtained by =INDIRECT("B"&$AE$1), and so on. The segment values in the cell range W3:AC3 are then used in conditional formatting. The
predicted digit display is based on segments states corresponding to the predicted digit label read from "S: column for the row number in AE2.
As this exercise demonstrates, the design of a naive Bayes classifier is pretty straightforward. Hopefully working with Excel has provided a better understanding of the steps involved in the entire
process of developing a classifier.
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Transductive Conformal Inference With Adaptive Scores: Application to Novelty Detection | HackerNoon
This paper is available on arxiv under CC 4.0 license.
(1) Ulysse Gazin, Universit´e Paris Cit´e and Sorbonne Universit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation,
(2) Gilles Blanchard, Universit´e Paris Saclay, Institut Math´ematique d’Orsay,
(3) Etienne Roquain, Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation.
Table of Links
4 Application to novelty detection
4.1 Setting
In the novelty detection problem, we observe the two following independent samples:
4.2 Adaptive scores
Following Bates et al. (2023); Marandon et al. (2022), we assume that scores are computed as follows:
In the work of Bates et al. (2023), the score function is built from Dtrain only, using a one-class classification method (classifier solely based on null examples), which makes the scores
independent conditional to Dtrain. The follow-up work Marandon et al. (2022) considers a score function depending both on Dtrain and Dcal ∪ Dtest (in a permutation-invariant way of the sample Dcal ∪
Dtest), which allows to use a two-class classification method including test examples. Doing so, the scores are adaptive to the form of the novelties present in the test sample, which significantly
improves novelty detection (in a nutshell: it is much easier to detect an object when we have some examples of it). While the independence of the scores is lost, an appropriate exchangeability
property is maintained so that we can apply our theory in that case, by assuming in addition (NoTies).
4.3 Methods and FDP bounds
Let us consider any thresholding novelty procedure
Then the following result holds true.
Corollary 4.1. In the above novelty detection setting and under Assumption NoTies, the family of thresholding novelty procedures (23) is such that, with probability at least 1 − δ, we have for all t
∈ (0, 1),
The proof is provided in Appendix E.
Remark 4.2. Among thresholding procedures (23), AdaDetect (Marandon et al., 2022) is obtained by applying the Benjamini-Hochberg (BH) procedure (Benjamini and Hochberg, 1995) to the conformal
p-values. It is proved to control the expectation of the FDP (that is, the false discovery rate, FDR) at level α. Applying Corollary 4.1 provides in addition an FDP bound for AdaDetect, uniform in α,
see Appendix G.
4.4 Numerical experiments
We follow the numerical experiments on “Shuttle” datasets of Marandon et al. (2022). In Figure 3, we displayed the true FDP and the corresponding bound (24) when computing p-values based on different
scores: the non-adaptive scores of Bates et al. (2023) obtained with isolation forest oneclass classifier; and the adaptive scores of Marandon et al. (2022) obtained with random forest two-class
classifier. While the advantage of considering adaptive scores is clear (smaller FDP and bound) , it illustrates that the bound is correct simultaneously on t. Additional experiments are provided in
Appendix H.
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Singapore Mathematical Society
Professor Professor Ron Aharoni
Distinguished Visitor
Professor Ron Aharoni is an Israeli mathematician, working in finite and infinite combinatorics.
With Nash-Williams and Shelah, he generalized Hall’s marriage theorem. He subsequently proved the appropriate versions of the König theorem and the Menger theorem for infinite graphs.
Professor Aharoni is the author of several nonspecialist books; the most successful is Arithmetic for Parents, a book helping parents and elementary school teachers in teaching basic mathematics. He
also wrote a book on the connections between Mathematics, poetry and beauty and a non-philosophical book on philosophy, The Cat That is not There. His last to date book is Circularity: A Common
Secret to Paradoxes, Scientific Revolutions and Humorwhich binds together mathematics, philosophy and the secrets of humor.
Academic Talk
Using Topology in Combinatorics
Some of the most resistant problems in combinatorics have succumbed to an unexpected tool – fixed point theorems in algebraic topology. In the talk I will survey some of those.
Mathematicians and Researchers
February 8 (Wednesday) 2017 / 10.30 am – 11.30 am
Nanyang Technological University
SPMS MAS Executive Classroom 1 (map)
Secondary Math Teachers’ Workshop
Teaching quadratic equations – a test case for some teaching principles
Investigation, or direct teaching? This question has been tantalizing teachers over the last few decades. Each method has its advantages and drawbacks. In this workshop I will demonstrate a third,
intermediate, way: directed discovery. It is the teacher who leads the class, constructing for them the precise order of the acquisition of concepts, but does so by posing examples to be studied. A
main principle in this approach is: no example too simple. In this workshop we shall try to construct a possible order of posing examples to the students, from which they will construct knowledge on
quadratic expressions and quadratic equations.
Secondary Mathematics Teachers
February 9 (Thursday) 2017 / 2.30 am – 5.00 pm
Academy of Singapore Teachers, Geolab (map)
Public Lecture
Some Principles in the Teaching of Elementary Mathematics
We shall discuss some basic principles, both of elementary mathematics and of its teaching. There are deep things to know about elementary mathematics, whose knowledge is a prerequisite for good
teaching. Some examples of such mathematical principles are the ubiquitous role of the operation of forming a unit and the way from counting to “pure” numbers. Several important teaching principles
will also be discussed.
February 10 (Thursday) 2017 / 4.30 am – 5.30 pm
National University of Singapore
Block S16, Level 3, LT31 (map)
Please click on the following links to register your attendance.
· Secondary Math Teachers’ Workshop:
MOE teachers register here, non-MOE teachers please register via email
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Large numbers (Easy)
Vocabulary points to take away:
When saying a large number, always begin with the largest number first and use singular
number labels:
One million, two hundred thousand, four hundred and sixty-four. (1, 200,464)
Don’t use the words and to join millions and thousands or thousands and hundreds:
Two million, fifty-six thousand, three hundred. (2,56,300)
But do use the word and to join hundreds and tens (tens are two-digit numbers):
Fifty-six thousand, three hundred and eleven. (56,311)
We sometimes emphasise how big a number is by describing it as a four-figure/fivefigure or six-figure number:
I’m not sure what he earns, but it’s certainly a six-figure number.
a six-figure salary
To describe approximately what a number is, we can say it’s in the tens/the hundreds/
the thousands/the millions. For very big numbers a number can be (in the) tens of
thousands, hundreds of thousands, tens of millions and so on:
They’ve cut hundreds of thousands of pounds from the budget.
Their assets alone must be worth in the tens of millions.
Another way of describing a number approximately is to say that it’s in triple figures:
The number of emails waiting for me after my holiday was in triple figures.
Lắng nghe và điền vào chỗ trống:
6 8-9-2-1.*%* And Anita Callum Finn INSERT Is
Now When an and are contain did digits emphasise even
group how number numbers or tens the thousands three topic us we what word world
Hello and welcome to 6 Minute Vocabulary with me Callum.
And me Finn. Today were talking about large .
Particularly how we say and describe them in English. Heres Anita, whos giving a talk to a
tour visiting Russia.
Listen out for the answer to this question: How many metres high is Mount Elbrus?
Russia is a land of superlatives! At over 6,500,000 square miles, it's the largest country in
the world. And total area of cultivated land has been estimated as a six-figure-
number: perhaps 500,000 square miles. Its mountain ranges Mount Elbrus, which
at 5,642 metres is the highest point in both Russia and Europe. Of its rivers, which in
the hundreds of thousands, the River Volga, the longest river in Europe, is the most well
known. And about the people? Well, heres an interesting fact: the number of
languages spoken in Russia is in triple figures � yes, over 100!
So that was Anita. And we asked: How many metres high is Mount Elbrus?
the answer is five thousand, six hundred and forty-two metres high.
Which is a good example of our today. When saying a large number, we always begin
with the biggest number first. So thousands, then hundreds, then . Tens means numbers
with two digits in them, like forty-two-. Listen again.
Five thousand, six hundred and forty-two-.
And the other point is, that the number labels are always singular. So five thousand and
not five thousands.
Six hundred and not six hundreds.
Exactly. Now, notice that we dont connect thousands and hundreds with the and.
Its five thousand, six hundred.
Not five thousand and six hundred.
But we do connect hundreds tens with the word and. So six hundred and forty-two-.
And I think its time for our first clip.
CLIP 1
Russia is a land of superlatives! At over 6,500,000 square miles, it's the largest country in
the . The total area of cultivated land has been estimated as a six-figure number:
perhaps 500,000 square miles.
So heard six million, five hundred thousand. Notice that we dont connect millions
to thousands with the word and either, in this case, millions to hundreds of thousands.
We say it like this: six million, five hundred thousand.
, how did Anita describe the figure 500,000?
She described it as a six-figure number. Because it contains six . We could also say
its a six-digit number.
Yes, we sometimes describe a number in this way to how big it is. And it doesnt’
have to be six. It could be a five-figure or a four-figure .
Now, on to clip 2.
INSERT CLIP 2
Of its rivers, which are in the hundreds of thousands, River Volga, the longest river
in Europe, is the most well known. And what about the people? Well, heres interesting
fact: the number of languages spoken in Russia is in triple figures � yes, over 100!
So did Anita describe the number of rivers in Russia?
She said theyre in the hundreds of thousands.
we want to describe approximately what a number is, we can say its in the tens,
the hundreds, the and so on. Hundreds of thousands means at least
100,000 and probably a lot more.
So you could say that a number is in the tens of millions.
There was also an interesting fact there about number of languages spoken in Russia.
Anita said theyre in triple figures. That means that the number contains figures –*%��
so at least 100. Its the same as saying that the number is in the hundreds.
Minute Vocabulary from bbclearningenglishcom..
And its quiz time! Number one: How do we say this number? 8-9-2-1. Thats Finn
Its eight thousand, nine hundred and twenty-one-.
Well done! Number two: What kind of number is 300,000? it:
a) a five-figure number b) a six-figure number c) a six-figures number
Its b) a six-figure number.
Correct! Number three: Listen to this number: 19,242.
Is it a) in the thousands b) in the tens of c) in the hundreds of thousands?
This one is b) In the tens of thousands.
Excellent! How you do? Very well done if you got them all right. Theres more on this
topic at bbclearningenglishcom.. Join again for more 6 Minute Vocabulary.
created with the online Cloze Test Creator © 2009 Lucy Georges
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Advanced Bread Making Course
Advanced Bread Making Course 🍞
Part 2: Creating A Recipe
Of course, whilst understanding your ingredients will help you make great bread, the key factor in developing your own recipes is consistency.
You need to be able to accurately measure your raw materials and create a formula balance (so that the ingredients used in a recipe remain in the same ratio, irrespective of the size of the batch you
are making). Being able to do this is what allows you to be confident your bread will turn out every time and also allows you to be able to communicate your results and pass them on to other people.
THE BAKER SAYS: "Bread making is a science that relies on various ingredients being present in specific quantities so that certain chemical reactions can occur at measurable speeds. Variations of
ingredients and quantities can have a dramatic effect on your bread dough and final bread products and while amazing new ideas and methods can result from happy accidents, if the quantities are
unknown, the happy accident cannot be converted to a new recipe anyway, so – ALWAYS MEASURE YOUR INGREDIENTS".
Bakers Percentage %
Because measurement and proportion are so important in the baking process, Baker's write their recipes as formulas and they use a technique known as the Bakers Percentage. That is: they write their
bread recipes out in mathematical form and they express the flour quantity as 100%. All other ingredients are then described as percentages of this flour weight.
To make a batch of bread using the formula, the first thing you need to do is to convert the formula from Bakers Percentage to a Per Kilo Flour Weight formula – remember, bakers base all their
ingredients on flour weight. To do this we simply multiply the flour weight by the ingredient percentages.
A typical bread recipe then, written as a Baker's Percentage would look like this:
INGREDIENT %
Flour 100%
Salt 2%
Improver 1%
Oil 2%
Yeast 3%
Water 60%
TOTAL: 168%
If, for instance, you wanted to make a batch of bread using 1 kilo of flour, once you multiplied your ingredients out, your formula would become:
Formula for 1 kilo of flour
INGREDIENT % 1 Kg Flour weight dough
Flour 100% 1.000 Kg
Salt 2% 0.020 Kg
Improver 1% 0.010 Kg
Oil 2% 0.020 Kg
Yeast 3% 0.030 Kg
Water 60% 0.600 Kg
TOTAL: 168% 1.680 Kg
Please Note: one of the side effects of using the flour weight as the basis for the formulas is that liquid ingredients then need to be measured by WEIGHT not VOLUME as would normally be the case
in other recipes.
Try working out a recipe yourself for a batch using 5 kilos of flour.
Calculate Bakers Percentage Tool
• Check the results you got for your 5 kilo batch above.
• Input flour weight if you want to do your own calculations
Please Note: most mistakes made using the Baker's Percentage happen because figures have not been written down correctly or either the zero or the decimal point has been left out or put in the
wrong spot.
You will also notice from the tables above that by adding up the kilo weights, you can get a total weight for your actual batch – this can be very useful for working out how many individual loaves or
buns, etc. a particular sized batch will make.
When discussing recipes though, be careful not to get confused between the flour weight and the actual dough weight.
This basic bread formula can be adapted in many ways to make many different styles of bread and though there are accepted ratios of one ingredient to another, individual recipes can vary to quite a
staggering degree. For instance: Brioche is a very rich, sweet, traditional French bread (often used to make French toast in fact!). To make it, essentially you adapt the formula above by changing
the oil to butter. The percentage of butter to flour in brioche recipes can be as high as 60% - and likewise, the egg percentage could rise as high as 60% and there may be up to 20% sugar – a pretty
decadent bread indeed!, but still easy to express using the formula method.
Flour Combinations
In bread making, a combination of two or more types of flour is often desirable for flavour, texture or appearance. This is easily expressed using the Baker's Percentage method. You just need to
ensure that the total flour weight is still 100%.
Flour Types
As well as these fairly readily available flour types, using a home mill or buying from a speciality mill or health food shop with a mill, it is also possible to grind or get quite a few different
grains ground to your specifications and in the quantities and ratios you desire.
Grain Types
These speciality flours and mixes are particularly useful in advanced and speciality bread making, where a combination of two or more types of flour is often desirable for flavour, texture or
appearance. These combinations are easily expressed using the Baker's Percentage method. You just need to ensure that the total flour weight is still 100%.
For instance DARK AND MALTY
DARK AND MALTY BREAD: %
Bakers Flour 50%
Dark Rye Flour 25%
Diastatic Malt Flour 15%
Spelt Flour 10%
Total flour weight 100%
Whole Grains
Whilst we have touched on the different types of grains used to make flour above, it is also quite common to include whole, kibbled or flaked grains in bread as well.
These grains can either be calculated in the flour weight itself or alternatively, they can be included after it (with the salt, oil, etc).
As an aside, whilst you always use the dry weight of grains for recipes and formulas, it is usual to soak most whole and kibbled grains in water for either a few hours or overnight before adding them
to the dough. The water percentage of the formula may need to be reduced slightly to compensate for this soaking. As a general rule, grains absorb just under their own weight in water, but you may
need to adjust this according to the grain and the amount of time you leave it to soak. Rolled oats, for instance, absorb water both like a sponge and extremely quickly, but any kibbled grain will
both absorb much less water and take far longer to do so...
Formulas with a combination of two or more flours and/or whole or kibbled grains expressed in both Baker's Percentage and as a Per Kilo Flour Weight Dough would then look like this:
MULTI GRAIN BREAD INGREDIENTS % 1 Kg Flour weight dough 3 kg Flour weight dough
Bakers Flour 50% 0.500 1.500
Wholemeal 20% 0.300 0.900
Light Rye Flour 10% 0.100 0.300
Kibbled Wheat 5% 0.050 0.150
Rolled Oats 5% 0.050 0.150
Flour Total: 100% 1.000 kg 3.000 kg
Salt 2% 0.020 0.060
Oil 2% 0.020 0.060
Yeast 4.5% 0.045 0.135
Water 60% 0.600 1.800
Dough Total: 168.5% 1.685 kg 5.055 kg
Using the formula, you can now not only work out your ingredient weights/quantities, you can calculate your:
Expected Yield
This is the total size of your dough batch and is calculated by adding up the individual ingredient weights. For instance, in the example above, the expected yield of a 1 kg flour weight dough batch
is 1.685 kg dough, the expected yield from a 3 kg flour weight dough is 5.055 kg. From this figure you can decide how to divide up your batch (e.g., will I make 5 x 1 kg loaves or 3 x 1 kg loaves and
20 x 100 g buns with my batch?)
For the record, a standard loaf is usually made from 780 g of dough. Big bread rolls require around 80 g, and dinner rolls and small rolls are usually made with about 65 g.
Due to the evaporation of most of the water in the dough, the weight of the finished bread is always lighter. Your standard 780 g dough weight bread for instance will yield up a 680 g finished loaf.
Required Yield
The other way to use the figures is of course to work backwards. You can start with what you want to end up with and work back to see how much of your ingredients you will need to make a batch of a
certain size.
For instance, if I want to end up with enough dough in my batch to make 10 x 1 kg loaves, using the formula above (where my expected yield percentage is 168.5%), that means a dough batch of 10 kilos
represents 168.5 % of my flour weight. This means that my flour weight should be (10 divided by 168.50 x 100 =) 5.934 kilos, my salt weight should be (5.934 divided by 100 x 2 =) 0.11868 grams
(rounded up to .119 here and to 12 or possibly even 15 in your recipe!), etc.
See if you can calculate out the rest of the formula for the 10 x 1kg required yield batch.
Answer for the 10 x 1kg required yield batch.
MULTI GRAIN BREAD INGREDIENTS % REQUIRED YEILD: 10 X 1 KG LOAVES
Bakers Flour 50% 2.967
Wholemeal 30% 1.780
Light Rye Flour 10% 0.593
Kibbled Wheat 5% 0.297
Rolled Oats 5% 0.297
Flour Total: 100% 5.934 kg
Salt 2% 0.119
Oil 2% 0.119
Yeast 4.5% 0.267
Water 60% 3.560
9.999 kg (NOTE: a good way to check if this calculation
Dough Total: 168.5% is correct is to add up the individual totals you get (here 9.999 kg)
and see if that figure really is the dough total percentage (here 168.5)
of your flour total (here 5.934).
Other Ingredients You Can Add
As well as whole and kibbled grains, there are obviously a multitude of other ingredients you can add to your bread mix or sprinkle or roll over the surface of your uncooked loaves (and thus, they
should be weighed and included in your recipes).
These ingredients would not form part of the flour weight, but their size and moisture quantities (and their temperature – more about that later!) must be taken into account. It is also often
necessary/desirable to cook, soften or soak some of the ingredients in advance. For instance: dried fruit is usually soaked in some form of liquid overnight before use, vegetables are usually cooked
or cut up very finely and all protein additions should be cooked and their condition closely monitored.
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Customize Fixed-Wing Aircraft with the Object Interface
This example shows how to customize fixed-wing aircraft in MATLAB® using objects.
For an example of how to get started using fixed-wing aircraft in MATLAB, see “Get Started with Fixed-Wing Aircraft”.
For an example of setting realistic coefficients on an aircraft and calculating staticstability,see“DetermineNonlinear Dynamics and Static Stability of Fixed-Wing Aircraft”.
For an example of importing coefficients from Digital DATCOM analysis and linearizing to a state-spacemodel, see“Perform Controls and Static Stability Analysis with Linearized Fixed-Wing
Fixed-Wing Object Interface
Each fixed-wing aircraft function returns an object of its defining type.
Functions provide convenience to constructing each object. However, their predefined input structure might be inconvenient to some workflows. For example, consider using objects if you want more
control over the specific inputs of each component, or if you want to avoid repmat calls to create arrays of each component. Objects let you directly construct each component.
The table mapping each function to object can be seen below.
Component = ["Properties"; "Environment"; "Aircraft"; "States"; "Coefficients"; "Surfaces"; "Thrust"];
Formal = ["Aero.Aircraft.Properties"; "Aero.Aircraft.Environment"; "Aero.FixedWing"; "Aero.FixedWing.State"; "Aero.FixedWing.Coefficient";"Aero.FixedWing.Surface"; "Aero.FixedWing.Thrust"];
Informal = ["aircraftProperties"; "aircraftEnvironment"; "fixedWingAircraft"; "fixedWingState"; "fixedWingCoefficient"; "fixedWingSurface"; "fixedWingThrust"];
objMap = table(Formal, Informal, 'RowNames', Component, 'VariableNames',["Formal Interface", "Informal Interface"])
objMap=7×2 table
Formal Interface Informal Interface
____________________________ ______________________
Properties "Aero.Aircraft.Properties" "aircraftProperties"
Environment "Aero.Aircraft.Environment" "aircraftEnvironment"
Aircraft "Aero.FixedWing" "fixedWingAircraft"
States "Aero.FixedWing.State" "fixedWingState"
Coefficients "Aero.FixedWing.Coefficient" "fixedWingCoefficient"
Surfaces "Aero.FixedWing.Surface" "fixedWingSurface"
Thrust "Aero.FixedWing.Thrust" "fixedWingThrust"
The constructor for each fixed-wing aircraft component is structured the same way:
• The first argument is either a vector or repeating set of integers that specifies the size of the returned object array, like the "zeros" function, with the default being size 1.
• Every argument after the first argument is a name-value pair specifying the object property and value for setting non-default values.
• Each non-default value is set for every object in the returned object array.
For example, creating a 3-element fixed-wing state vector where each state has a mass of 50 kg would have the following syntax:
states = Aero.FixedWing.State(1,3, Mass=50)
states=1×3 State array with properties:
As can be seen from the returned Mass values, each state has a mass of 50 in the vector.
Following this format, this example constructs the same aircraft from the "Get Started with Fixed-Wing Aircraft" example, replacing the function interface with the associated object interface.
Fixed-Wing Aircraft Configuration
Create the 3 control surfaces using Aero.FixedWing.Surface.
By default, the surface objects are defined to be symmetric and not controllable, so these two properties must be set for the aileron along with the maximum and minimum values.
aileron = Aero.FixedWing.Surface(...
Controllable="on", ...
Symmetry="Asymmetric", ...
MinimumValue=-20, ...
aileron =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Asymmetric"
ControlVariables: ["_1" "_2"]
Properties: [1x1 Aero.Aircraft.Properties]
Additionally, the set the name on the properties of the aileron surface object. This is helpful for setting coefficients later.
aileron.Properties.Name = "aileron"
aileron =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Asymmetric"
ControlVariables: ["aileron_1" "aileron_2"]
Properties: [1x1 Aero.Aircraft.Properties]
For the elevator and rudder, the symmetry is already the desired value, so the "Symmetry" name-value argument can be excluded.
elevator = Aero.FixedWing.Surface(...
Controllable="on", ...
MinimumValue=-20, ...
elevator =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Symmetric"
ControlVariables: ""
Properties: [1x1 Aero.Aircraft.Properties]
rudder = Aero.FixedWing.Surface(...
Controllable="on", ...
MinimumValue=-20, ...
rudder =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Symmetric"
ControlVariables: ""
Properties: [1x1 Aero.Aircraft.Properties]
elevator.Properties.Name = "Elevator"
elevator =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Symmetric"
ControlVariables: "Elevator"
Properties: [1x1 Aero.Aircraft.Properties]
rudder.Properties.Name = "Rudder"
rudder =
Surface with properties:
Surfaces: [1x0 Aero.FixedWing.Surface]
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 20
MinimumValue: -20
Controllable: on
Symmetry: "Symmetric"
ControlVariables: "Rudder"
Properties: [1x1 Aero.Aircraft.Properties]
With the control surfaces defined, define the thrust object using the Aero.FixedWing.Thrust object.
By default, the minimum and maximum values for the thrust object are 0 and 1, which represent the throttle lever position.
In this aircraft, they are limited to 0 and 0.75.
propeller = Aero.FixedWing.Thrust(MaximumValue=0.75)
propeller =
Thrust with properties:
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 0.7500
MinimumValue: 0
Controllable: on
Symmetry: "Symmetric"
ControlVariables: ""
Properties: [1x1 Aero.Aircraft.Properties]
The name of the thrust vector can also be set at this time.
propeller.Properties.Name = "propeller"
propeller =
Thrust with properties:
Coefficients: [1x1 Aero.FixedWing.Coefficient]
MaximumValue: 0.7500
MinimumValue: 0
Controllable: on
Symmetry: "Symmetric"
ControlVariables: "propeller"
Properties: [1x1 Aero.Aircraft.Properties]
With these control surfaces and thrust vectors defined, they can now be set on the body of the aircraft.
The aircraft body is defined through the Aero.FixedWing object.
For simplicity, this aircraft will have a reference area, span, and length of 3, 2, and 1, respectively.
aircraft = Aero.FixedWing(...
ReferenceArea=3, ...
ReferenceSpan=2, ...
aircraft =
FixedWing with properties:
ReferenceArea: 3
ReferenceSpan: 2
ReferenceLength: 1
Coefficients: [1x1 Aero.FixedWing.Coefficient]
DegreesOfFreedom: "6DOF"
Surfaces: [1x0 Aero.FixedWing.Surface]
Thrusts: [1x0 Aero.FixedWing.Thrust]
AspectRatio: 1.3333
Properties: [1x1 Aero.Aircraft.Properties]
UnitSystem: "Metric"
TemperatureSystem: "Kelvin"
AngleSystem: "Radians"
aircraft.Surfaces = [aileron, elevator, rudder]
aircraft =
FixedWing with properties:
ReferenceArea: 3
ReferenceSpan: 2
ReferenceLength: 1
Coefficients: [1x1 Aero.FixedWing.Coefficient]
DegreesOfFreedom: "6DOF"
Surfaces: [1x3 Aero.FixedWing.Surface]
Thrusts: [1x0 Aero.FixedWing.Thrust]
AspectRatio: 1.3333
Properties: [1x1 Aero.Aircraft.Properties]
UnitSystem: "Metric"
TemperatureSystem: "Kelvin"
AngleSystem: "Radians"
aircraft.Thrusts = propeller
aircraft =
FixedWing with properties:
ReferenceArea: 3
ReferenceSpan: 2
ReferenceLength: 1
Coefficients: [1x1 Aero.FixedWing.Coefficient]
DegreesOfFreedom: "6DOF"
Surfaces: [1x3 Aero.FixedWing.Surface]
Thrusts: [1x1 Aero.FixedWing.Thrust]
AspectRatio: 1.3333
Properties: [1x1 Aero.Aircraft.Properties]
UnitSystem: "Metric"
TemperatureSystem: "Kelvin"
AngleSystem: "Radians"
Set the aircraft coefficients using the setCoefficient and getCoefficient methods, creating a coefficient using the Aero.FixedWing.Coefficient object, or indexing directly to the coefficient values.
coeff = Aero.FixedWing.Coefficient
coeff =
Coefficient with properties:
Table: [6x1 table]
Values: {6x1 cell}
StateVariables: "Zero"
StateOutput: [6x1 string]
ReferenceFrame: "Wind"
MultiplyStateVariables: on
NonDimensional: on
Properties: [1x1 Aero.Aircraft.Properties]
aircraft.Coefficients.Values{3,3} = 0.2
aircraft =
FixedWing with properties:
ReferenceArea: 3
ReferenceSpan: 2
ReferenceLength: 1
Coefficients: [1x1 Aero.FixedWing.Coefficient]
DegreesOfFreedom: "6DOF"
Surfaces: [1x3 Aero.FixedWing.Surface]
Thrusts: [1x1 Aero.FixedWing.Thrust]
AspectRatio: 1.3333
Properties: [1x1 Aero.Aircraft.Properties]
UnitSystem: "Metric"
TemperatureSystem: "Kelvin"
AngleSystem: "Radians"
ans=6×9 table
Zero U Alpha AlphaDot Q Beta BetaDot P R
____ _ _____ ________ _ ____ _______ _ _
CD 0 0 0 0 0 0 0 0 0
CY 0 0 0 0 0 0 0 0 0
CL 0 0 0.2 0 0 0 0 0 0
Cl 0 0 0 0 0 0 0 0 0
Cm 0 0 0 0 0 0 0 0 0
Cn 0 0 0 0 0 0 0 0 0
Fixed-Wing Aircraft States
Define fixed-wing aircraft states using the Aero.FixedWing.State object.
To directly create an array of states that all have the same properties, use the state constructor instead of using the repmat function.
In this example, 11 states are created by varying mass, but with constant airspeed.
mass = num2cell(1000:50:1500)
mass=1×11 cell array
{[1000]} {[1050]} {[1100]} {[1150]} {[1200]} {[1250]} {[1300]} {[1350]} {[1400]} {[1450]} {[1500]}
states = Aero.FixedWing.State(size(mass), U=100);
[states.Mass] = mass{:}
states=1×11 State array with properties:
However, unlike the fixedWingState function, the control states are not automatically set up on the states from the aircraft.
To set up the control states, use the setupControlStates function.
states = setupControlStates(states, aircraft)
states=1×11 State array with properties:
See Also
Aero.FixedWing | Aero.FixedWing.Coefficient | Aero.FixedWing.State | Aero.FixedWing.Surface | Aero.FixedWing.Thrust
Related Topics
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