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Asymptotic modelling of self-acceleration of spherical flames
In this paper, we investigate the self-acceleration of the expansion of premixed spherical flames and evolution of the cellular patterns on their surfaces. Asymptotic models are used for the
simulations and a spectral numerical algorithm is employed to study flames over large time intervals. Numerous numerical experiments indicate that for large enough time the acceleration of
two-dimensional expanding flame slows down but the expansion rate still able to reach values significantly exceeding the burning rate of an exactly circular flame. Simulations of expansion of a
three-dimensional flame were carried out by using similar asymptotic model for a spatial segment of the spherical flame. Obtained results indicate that the three-dimensional spherical flame
accelerates at much higher rate than the circular one. Wrinkling of the three-dimensional flames is much more intensive and they start to accelerate earlier too.
Funders Funder number
Engineering and Physical Sciences Research Council GR/R66692
United States-Israel Binational Science Foundation 200 2008
Israel Science Foundation 350-05, 278-03
• Acceleration
• Hydrodynamic flame instability
• Premixed spherical flames
Dive into the research topics of 'Asymptotic modelling of self-acceleration of spherical flames'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/asymptotic-modelling-of-self-acceleration-of-spherical-flames","timestamp":"2024-11-05T05:56:56Z","content_type":"text/html","content_length":"50976","record_id":"<urn:uuid:8fe60ffa-3fd8-470a-8f4c-224fda6de7a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00125.warc.gz"} |
Understanding Mathematical Functions: How To Move A Function To The Le
Introduction to Mathematical Functions and Their Transformations
Understanding mathematical functions and their transformations is crucial in various fields such as mathematics, physics, and engineering. Functions are used to model real-world phenomena and
understanding how they can be transformed is essential for analyzing and solving problems.
A. Importance of understanding function transformations
The ability to transform functions is essential in various fields, allowing professionals to model and analyze complex systems. In mathematics, understanding function transformations is crucial for
solving equations, graphing functions, and analyzing data. In physics, it is used to model the behavior of physical systems. In engineering, function transformations are used to design and optimize
systems and processes.
B. Overview of basic function types and their properties
There are different types of basic functions, each with its own properties and characteristics. For example, linear functions have a constant rate of change, while quadratic functions have a
parabolic shape. Understanding the properties of these basic functions is essential for understanding how they can be transformed.
C. Objective of the blog post
The objective of this blog post is to provide insights into how to move a function horizontally to the left on a graph. This transformation is a fundamental concept in function manipulation and
understanding how to do it is important for anyone studying or working with mathematical functions.
Key Takeaways
• Shifting a function to the left involves subtracting a value.
• The amount to shift is determined by the value being subtracted.
• Understanding the concept of horizontal translation is crucial.
• Graphing the original and shifted functions helps visualize the change.
• Practice with different functions to solidify understanding.
Understanding the Coordinate Plane and Function Notation
When it comes to graphing mathematical functions, understanding the coordinate plane and function notation is essential. These concepts provide the foundation for visualizing and understanding how
functions behave and how they can be manipulated.
A. Define coordinate plane and its significance in graphing functions
The coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular number lines. These number lines are called the x-axis and the y-axis. The point where the x-axis and
y-axis intersect is called the origin, denoted as (0, 0). The coordinate plane is significant in graphing functions because it provides a visual representation of the relationship between the input
(x) and output (y) values of a function.
B. Introduce function notation (f(x)) and how it represents mathematical relationships
Function notation is a way to represent the relationship between input and output values in a mathematical function. It is denoted as f(x), where 'f' represents the name of the function and 'x'
represents the input value. When we substitute a specific value for 'x' in the function, we can determine the corresponding output value. This notation allows us to succinctly express the
relationship between the input and output values of a function.
C. Explain how function values correspond to points on the graph
Each input value (x) in a function corresponds to an output value (y), and together they form ordered pairs (x, y). These ordered pairs represent points on the graph of the function. When we plot
these points on the coordinate plane, we can visualize the function and observe its behavior. The function values determine the position of points on the graph, allowing us to understand how the
function behaves for different input values.
The Concept of Horizontal Shifts in Functions
Understanding how to move a function to the left is an important concept in mathematics. It involves the idea of horizontal shifts, which can significantly impact the graph of a function. In this
chapter, we will delve into the definition of horizontal shifts, the difference between left and right shifts, and the notation used to represent these shifts.
A Define what a horizontal shift is and its effect on a function's graph
A horizontal shift refers to the movement of a function to the left or right along the x-axis. When a function is shifted horizontally, its graph is translated either to the left or right without
changing its shape or orientation. This means that the function's y-values remain the same, while the x-values are adjusted based on the direction of the shift.
The effect of a horizontal shift on a function's graph is that it changes the location of the graph without altering its steepness or direction. This means that the overall shape of the function
remains unchanged, but its position on the coordinate plane is shifted.
B Describe the difference between moving a function to the left versus to the right
When a function is moved to the left, it means that the entire graph is shifted in the negative direction along the x-axis. This results in the function's graph appearing to the left of its original
position. Conversely, moving a function to the right involves shifting the graph in the positive direction along the x-axis, causing it to appear to the right of its original position.
It's important to note that moving a function to the left or right does not change the function itself, but rather changes its position on the coordinate plane.
C Introduce the notation for horizontal shifts (h in f(x-h)) and its mathematical implication
In mathematical notation, a horizontal shift is represented as h in the function f(x-h). This notation indicates that the function f(x) has been shifted h units to the right if h is positive, or h
units to the left if h is negative.
The mathematical implication of this notation is that by replacing x with (x-h) in the function, we are essentially shifting the entire graph of the function horizontally by h units.
Moving a Function Left: The Role of Horizontal Translation
When it comes to understanding mathematical functions, one important concept to grasp is the idea of horizontal translation. This refers to the process of moving a function left or right along the
x-axis. In this chapter, we will explore how adding a positive constant to the x-variable inside the function notation results in a leftward shift, demonstrate the translation process using algebraic
expressions, and provide examples with different function types.
A. Adding a Positive Constant to the x-Variable
When we add a positive constant to the x-variable inside the function notation, it causes the entire function to shift to the left. This is because adding a positive constant to x effectively
subtracts that constant from the x-values, causing the function to shift leftward along the x-axis.
B. Algebraic Expressions to Demonstrate the Translation Process
To demonstrate the translation process, let's consider a general function f(x) and a positive constant c. The translated function, denoted as g(x), can be expressed algebraically as g(x) = f(x + c).
Here, the addition of the constant c inside the function notation causes the leftward shift of the function along the x-axis.
For example, if we have the function f(x) = x^2, adding a positive constant c inside the function notation results in the translated function g(x) = (x + c)^2. This demonstrates how the function has
been shifted to the left by the amount of the constant c.
C. Examples with Different Function Types
Let's consider examples with different function types to illustrate the concept of leftward shift through horizontal translation.
• Linear Function: For a linear function f(x) = 2x + 3, adding a positive constant c inside the function notation results in the translated function g(x) = 2(x + c) + 3, which represents a leftward
shift along the x-axis.
• Quadratic Function: As mentioned earlier, for the quadratic function f(x) = x^2, the translated function g(x) = (x + c)^2 demonstrates the leftward shift by the amount of the constant c.
• Trigonometric Function: Consider the function f(x) = sin(x). Adding a positive constant c inside the function notation results in the translated function g(x) = sin(x + c), representing a
leftward shift of the sine function.
These examples showcase how the concept of horizontal translation applies to various function types, allowing us to visualize and understand the leftward shift of functions along the x-axis.
Visualizing the Shift: Graphing Examples
Understanding how to move a function to the left is an essential concept in mathematics. Visualizing the shift of a function on a graph can help in comprehending the impact of this transformation.
Let's take a step-by-step look at how to graph a function before and after it has been moved to the left.
A. Step-by-step guide on graphing a function before and after it has been moved to the left
To graph a function before and after it has been moved to the left, start by identifying the original function. Let's take the example of the function f(x) = x^2. This is a simple quadratic function
with its vertex at the origin.
Now, let's consider the function g(x) = (x+2)^2. This function has been moved two units to the left compared to the original function f(x) = x^2. To graph this, we need to shift the entire curve of f
(x) = x^2 two units to the left.
Start by plotting the original function f(x) = x^2 on a graph. Then, for the function g(x) = (x+2)^2, shift each point of the original function two units to the left. This will result in a new curve
representing the function g(x) = (x+2)^2.
B. Use graphical representation to show how the function's curve moves along the x-axis
Graphically, the shift of the function g(x) = (x+2)^2 to the left is evident. The entire curve of the function has been shifted two units to the left along the x-axis. This means that for every value
of x, the corresponding y-value is the same as the original function, but the x-value has been decreased by 2.
This graphical representation helps in visualizing the impact of moving a function to the left. It also provides a clear understanding of how the curve of the function shifts along the x-axis.
C. Provide practical examples and real-world scenarios where analyzing such shifts is crucial
Analyzing shifts of functions to the left is crucial in various real-world scenarios. For example, in finance, understanding how a change in interest rates affects the value of investments is
essential. Moving a function to the left can represent a decrease in interest rates, which in turn impacts the value of investments.
In physics, analyzing the shift of a function to the left can represent a change in the position of an object over time. Understanding this shift is crucial in predicting the future position of the
object and making accurate calculations.
Overall, understanding how to move a function to the left is a fundamental concept in mathematics with practical applications in various fields.
Troubleshooting Common Misconceptions and Errors
When it comes to moving a function to the left, there are several common misconceptions and errors that can arise. Addressing these issues is crucial for a clear understanding of mathematical
A. Address the common mistake of confusing horizontal and vertical shifts
One of the most common mistakes when moving a function to the left is confusing horizontal and vertical shifts. It's important to emphasize to students that moving a function to the left involves a
horizontal shift, not a vertical one. This means that the x-values are being adjusted, not the y-values. Encouraging students to visualize the movement of the function on a graph can help clarify
this concept.
B. Offer tips on checking work for accuracy, such as verifying key points on the function have shifted correctly
Another common error is failing to check the accuracy of the shift. Students may forget to verify that key points on the function have shifted correctly. It's important to emphasize the significance
of checking work for accuracy by plugging in specific x-values and ensuring that the corresponding y-values have shifted as expected. Encouraging students to create a table of values before and after
the shift can be a helpful tool for this verification process.
C. Discuss the value of using graphing technology or software for visual confirmation and error checking
Utilizing graphing technology or software can be incredibly valuable for visually confirming and checking for errors when moving a function to the left. This technology allows students to graph the
original function and the shifted function side by side, providing a clear visual representation of the shift. It also allows for easy comparison of key points and can help students identify any
discrepancies in their work. Encouraging the use of graphing technology or software as a tool for error checking can greatly enhance students' understanding of the concept.
Conclusion and Best Practices for Mastering Function Translations
Understanding how to move functions horizontally is a fundamental skill in mathematics. It allows us to manipulate and transform functions to better understand their behavior and make predictions. By
mastering function translations, we gain a deeper insight into the relationship between variables and can solve complex problems more effectively.
A. Summarize the importance of understanding how to move functions horizontally
Mastering function translations is essential for anyone studying mathematics or related fields. It provides a solid foundation for more advanced topics such as calculus, differential equations, and
mathematical modeling. Additionally, it is a valuable skill in fields such as physics, engineering, and computer science, where understanding and manipulating functions is crucial.
B. Emphasize the pattern of subtracting from the x-variable to translate a function to the left
When moving a function to the left, it is important to recognize the pattern of subtracting from the x-variable. This simple concept forms the basis for all horizontal translations of functions. By
understanding this pattern, one can easily apply it to any function and confidently perform the necessary transformations.
C. Provide best practices such as regularly practicing with diverse functions, utilizing graphing tools, and double-checking work to become proficient in function transformations
• Regular Practice: To become proficient in function translations, it is essential to practice with a wide variety of functions. This helps in developing an intuitive understanding of how different
types of functions behave when translated.
• Utilizing Graphing Tools: Graphing tools such as graphing calculators or software can be immensely helpful in visualizing function translations. They allow for quick experimentation and
verification of the results.
• Double-Checking Work: It is important to double-check the results of function translations to ensure accuracy. Small errors in calculations or understanding the transformation rules can lead to
significant mistakes.
By following these best practices, one can develop a strong command over function translations and gain confidence in manipulating functions to suit specific needs. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-move-function-left","timestamp":"2024-11-09T02:36:10Z","content_type":"text/html","content_length":"224099","record_id":"<urn:uuid:be4695bd-a1f6-46be-badd-18ed1cc2211d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00059.warc.gz"} |
Past Papers | WJEC Maths GCSE revision
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Past Papers and Mark Schemes - WJEC (Wales)
Since 2016/17 there are two mathematics GCSEs: one which focuses on numeracy and the mathematics needed for everyday life, and the other extends to other aspects of mathematics including those needed
for progression to scientific, technical or further mathematical study. The total content of the two GCSEs remains similar to the previous GCSE Mathematics but the assessment places a greater
emphasis on problem solving. Most young people in Wales are expected to take both mathematics GCSEs.
There are 3 tiers of entry for this qualification.
Higher Tier: Grades A* – C
Intermediate Tier: Grades B – E
Foundation Tier: Grades D – G
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Mathematical Sciences Research Institute
Geometric correspondances between singular fibres of the Hitchin fibration
August 16, 2019 (09:30 AM PDT - 10:30 AM PDT) Speaker(s): Laura Schaposnik (University of Illinois at Chicago)
Location: SLMath: Eisenbud Auditorium
• Higgs bundles
• Hitchin systems
• monodromy
• isogenies
Primary Mathematics Subject Classification Secondary Mathematics Subject Classification
Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural hyperkahler structure, through
which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention both within mathematics and string theory. In
this talk we shall consider novel correspondences between branes lying completely within the singular fibres of the Hitchin fibration, which can be understood through group isomorphisms. By
considering a toy mode, the first half of the talk will introduce the audience to Higgs bundles and some of the open problems in the area -- the second half of the talk will be about work in progress
with Steve Bradlow and Lucas Branco, and with Sebastian Schulz.
Supplements No Notes/Supplements Uploaded | {"url":"https://legacy.slmath.org/workshops/894/schedules/27301","timestamp":"2024-11-11T04:34:42Z","content_type":"text/html","content_length":"40756","record_id":"<urn:uuid:63e1d3cd-7383-41e3-a5db-1954105bac2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00114.warc.gz"} |
Integers Worksheet Algebra Answer Key
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Electron paramagnetic resonance studies of the electronic structure of - PDF Free Download
1734 Inorganic Chemistry, Vol. 11, No. 8, 1972
CONTRIBUTION FROM THE DEPARTRIEXT O F CHEMISTRY, S O T R E
NOTREDAME,IXDIANA 46556
Electron Paramagnetic Resonance Studies of the Electronic Structure of Bis(cyclopentadieny1)vanadium Dichloride and a Comparison with Simple Molecular Orbital Calculations BY DENKIS P. BAKALIK
ROBERT G . HAYES*
Received January 7, 1972 Dilute single-crystal electron paramagnetic resonance experiments have been performed on bis(cyclopentadieny1)vanadium dichloride dissolved in bis(cyclopentadieny1)titanium
dichloride. Principal values of the spectroscopic splitting constant aregl = 1.946, gZ = 1.967, a n d g 3 = 1.965. Thevanadium hyperfine values are-4, = (-)118,9, A Z = (-)28.9, and A 3 = (- )74.9 (
x 10-4 cm-l). The unpaired electron is in an al-type molecular orbital which is primarily composed of d,a and d.z-rdz and has a small amount of 4s character. Self-consistent charge-configuration
calculations of the modified bvolfsbergHelmholz type and the type proposed by Fenske have been performed and the results are related t o the optical and epr spectra.
Introduction The dihalide derivatives of the transition metal bis(cyclopentadienyl) compounds have received considerable attention. Dipole moment1 and electron diffraction2 work directed a t the
geometry of bis(cyc1opentadieny1)titanium dichloride, (C6H&TiC12, indicated this complex to be a "wedge-shaped sandwich" compound of distorted tetrahedral structure. More recent X-ray results
indicate the complexes bis(cyc1opentadieny1)zirconium difluoride and bis(cyc1opentadieny1)zirconium diiodide to be of this geometry The electronic structure and the bonding in this class of compounds
have been the subject of some investigation. Druce, et a L j 4 have interpreted the infrared spectrum of (C:HB)TiC12 and suggested considerable ionic character of the metal-ring bonding. Beachellj
has offered pa-da metal-halogen bonding as an explanation of the decrease in chemical shift of ring protons in the C1, Br, and I series of bis(cyclopentadieny1)titanium complexes. Chien6 has reported
the electronic spectra of the series (CeH5)zblXzwhere X = C1, Br, or I when ILI = Ti and X = C1 in the case 31 = V. The optical spectra provide an experimental criterion for the acceptability of a
molecular orbital interpretation of the electronic structure of these compounds but definitive conclusions concerning the ordering of molecular orbitals cannot be drawn easily from the optical data.
The complex bis(cyc1opentadienyl)vanadium dichloride, (CjH&VC12, lends itself to an analysis of the electronic structure using electron paramagnetic resonance. Though the solution epr spectrum has
been rep ~ r t e d dilute ,~ single-crystal work has not been done. The work we report herein was performed to obtain information from magnetic resonance concerning the electronic structure of the
compound and t o test the compatibility of these results with molecular orbital calculations of the modified Rrolfsberg-Helmholza~9 (1) S. A . Giddings and R. J. Best, J . Amer. Chem. Soc., 83, 2393
(1961). (2) PI'. V. Alekseev and I. A. Ronova, Zh. Strukt. Khim., 7, 103 (1966).
(3) M . Bush, private communication, ,lgricultural Research Council Unit of Structural Chemistry, London, England. (4) P.b1. Druce, B. M. Kingston, M . F. Lappert, T. R . Spalding, and R . C.
Srivastava, J . Chem. Soc., 2106 (1969). (5) H. L. Beachell and S . A . Butter, Inorg. C h e w , 4, 1133 (1965). (6) J. C. W. Chien, J . P h y s . Chem., 67, 2477 (1963). (7) J. C. W. Chien and C. R
. Boss, J . Ameu. ChPm. Soc., 83, 3767 (1961). (8) C . J. Ballhausen and H . B. Gray, Inovg. Chem., 1, 111 (1962). (9) A . Viste and H. B. Gray, i b i d . , 8, 1113 (1964).
type and of the type proposed by Fenske'O and Hillier and Canadine.l1>l2 Experimental Section Bis(cyclopentadieny1)vanadium dichloride and bis(cyc1opentadieny1)titanium dichloride were purchased from
Alfa Inorganics. Single crystals of about lYc (CsHR)2l7C12in the (CaH5)aTiC12 diamagnetic diluent were grown by solvent evaporation from a chloroform solution. The chloroform was dried over lithium
aluminum hydride, outgassed several times, and distilled into a vessel containing ( CaH:)2TrCIZ-(C:H;)2TiC12 powder. Outgassing and distillation were done on a vacuum rack operating a t approximately
Torr. Solvent evaporation was carried out by passing purified nitrogen through inlet and outlet ports of the vessel. Nitrogen was deoxygenated using a column filled with BTS catalyst (BASF AG,
Ludwigshaferr) maintained a t a temperature of 150". The crystals were elongated plates, 0.1 mm or so thick, 0.5 mm wide, and a few millimeters long. Crystals were mounted on small quartz cubes (2.5
mm on a n edge) and coated with General Electric vacuum sealant GEVAC in order to preserve the sample while measurements were made. No discernible changes in the appearance of the spectrum were noted
during the course of the measurements. Epr spectra were taken using a Varian 17-4502-15 spectrometer equipped with a Fieldial and a 100-kc modulation unit. Magnetic field calibrations were done using
a proton probe and klystron frequencies were measured with a n adjustable cavity wavemeter. A crystal-fixed coordinate system with axes along the long, intermediate, and short axes a t the crystals
established three orthogonal directions for rotation. I n a given plane the direction of the magnetic field with respect to the crystal-fixed axis system was changed in 15" increments.
Computational Details Self -consistent charge-configuration calculations were done according to the modified Wolfsberg-Helmh~lz~~~ formulation and that proposed by Fenske'O and Hillier and Canadine.
11,12 The geometry for (CjH&VCI2 was assumed to be the same as that determined for (CbH&TiC12.2 Figure 1 represents the geometry. atomic numbering system, and coordinate choices employed. Vanadium
3d, 4s, and 4p, chlorine 3s and 3p, and the p a orbitals of the cyclopentadienyl ring system were chosen to represent the valence-electron basis set. All functions were Roothaan-Hartree-Fock
functions of (10) R . F. Fenske, K. G. Caulton, D. D. Radke, and C . C. Sweeney, ibid., 5 , 951 (1966). (11) R.M Canadine and I. H. Hillier, J . Chem. P h y s . , SO, 2984 (1969). (12) I. H. Hillier
and R . M. Canadine, Disc%ss. F a r a d a y Soc., 47, 27 (1969)
Inorganic Chemistry, Vo2. 11, No. 8, 1972 ‘1135
nuclear attraction integrals were evaluated using a routine we prepared. Ligand-ligand integrals were ignored in both calculations. Valence-state ionization potentials for vanadium and chlorine were
taken from Ballhausen and Gray16 while cyclopentadienyl radical VSIP’s were obtained using values of 8.69 and - 3 eV for the first ionization potential and resonance integral,” respectively.
Consistency of input and output configuration and charge was obtained via a pattern search minimization routinels which minimized the dif ference between input and output parameters. For the modified
Wolfsberg-Helmholz calculation only matrix elements involving the metal atom were explicitly charge and configuration dependent, thereby producing a three-parameter minimization problem of the
function given by (1). Subscripts i and o indicate input and out-
Dif = Figure 1.-The geometry of the compounds M(CsHs)zC12 (M = Ti, V) and the coordinate systems used in discussing bonding in these compounds.
Clementi13 with the exception of the vanadium 4p virtual orbital which was that of Richardson.14 Hiickel molecular orbitals (Table I) were constructed from the
- Sol
- Po/ (1)
put values of the metal charge (Q) and the metal configurational amplitudes (S and P). The Fenske-type calculation involves a six-parameter minimization problem. The metal charge, two metal
populations, and three carbon-ring charges completely characterize the valence-electron distribution.
Results The angular variation of the epr spectrum yielded the spin-Hamiltonian parameters Iisted in Table 111. The principal values were obtained from maximum, minimum, and intermediate turning
points of the angular variation of the spectra with respect to the aforementioned coordinate system. A first-order treatment of the data, employing a least-squares analysis for each plane, gives
principal values which are in essential agreement with those of Table 111. This indicates that the
TABLE I1 (CsH6)2VC12SYMMETRY-ADAPTED BASIS SET“ Ai
(1) = 3d,a ( 2 ) = 3dzz-,z
( i i ) = 3d,,
(15) = 3d,, (21) = 4%
(8) = 4s (9) = 4PB
c1 (14) = 3 p m
+3 p m
CP (12) %’(I) (13) = Ez‘(1)
+4-E2’(2) El‘@)
c1 (6) = 3s(l) 3s(2) (7) 3pz(l) 3~,(2) (10) = 3 ~ , ( 1 ) 3pz(2)
++ CP (3) = + AI@) (4) = Ez(1) + Ez(2) (5) = Ei(1) + Ei(2)
V ( 2 2 ) = 3d,, (26) = ~ P U
c1 (18) = 3sl(l) - 3S2(2) (19) = 3ps(l) - 3pz(2) (20) = 3 ~ ~ ( 1 )3 ~ , ( 2 ) CP (16) = Ei’(1) (17) E2’(l)
- Ei’(2) - Ez’(2)
c1 (25) = 3ps(l)
- 3pz(2)
CP (23) = Ai(1) - AI@) (24) = Ei(1) - Ei(2) (27) = Ez(1) - E2(2)
a 1/45 normalization factor understood for orbitals which are linear combinations; Al(1) represents the Hiickel A1 orbital (Table I ) for ring 1 (Figure 1).
pir basis set. Molecular symmetry orbitals, represented in Table 11, were formed in accordance with the representations of the group Cz,. Overlap and kinetic energy integrals were calculated using
FORTRAN routine ONE^ available from the Quantum Chemistry Program Exchange. l5 Two-center (13) E. Clementi, “Tables of Atomic Functions,” International Business Machine Corp , 1966. (14) J. W
Richardson, R R. Powell, and W. C. Nieuwpoort, J . Chem. P h y s , 8 8 , 796 (1963). (15) ONE^ Program No. 134, Quantum Chemistry Program Exchange, Indiana University, Bloomington, Ind.
TABLE I11 (CsH&VC12 PRINCIPAL g VALUESAND VANADIUM COUPLING CONSTANTS g1 = 1.946 gz = 1.967 ga 1.965 A i = -118.Qa Az = - 2 6 . 7 A3 = - 7 4 . 9 A = -73.5 Coupling constants in 10-4 cm-1.
(16) C. J. Ballhausen and H. B. Gray, “Molecular Orbital Theory,” W. A. Benjamin, New York, N. Y . , 1965, p 120. (17) R. D. Fischer, Theov. Chim. Acta, 1, 418 (1963). (18) C. J. Hebert, Thesis,
University of Notre Dame, Notre Dame, Ind.
1736 Inorganic Chemistry, Vol. 11, No. 8, 1972
chosen coordinate system represents the principal axis system, within our experimental accuracy. Detailed examination of the spectra indicate, however, that this relationship is only approximate.
Spectra taken a t angles close to principal directions appear to arise from one species. At intermediate angles slightly split spectra are observed, however, indicating two inequivalent species
tipped by a few degrees from the apparent principal axis system. Our inability to follow a single species through a complete rotation prohibited refinement of the apparent principal axis system.
Tables IV and V contain the self-consistent charges TABLE IV RESULTSOF WOLFSBERG-HELMHOLZ CALCULATIONS” Eigenvalue, k K
Charge, V
Bz B2 Elb
4s = 0.665 4p = 0 . 2 3 9 3d = 3 . 8 6
-70.09 -69.93 -41 -68.00 -65.34 A1 1 kK = lo3 cm-1; Dif = 0.45.
Highest occupied MO.
TABLE V RESULTSOF FENSKE-HILLIER CALCULATIONS~ Eigenvalue, k K
-136.5 -134.4 -106.9 -100.0 -95.6 Dif = 0.28.
Charge, V
4s = 0.441 4p = 0 . 1 8 8 3d = 3 . 0 8
AP Ai Bi Highest occupied MO.
and configurations along with the important eigenvalues and orbital symmetries from the two different types of calculations. The value of the function Dif appearing in eq 1 is also given.
Discussion One conclusion is obvious immediately from our data. The electronic ground state of (C6H6)2VC12 has the unpaired electron localized heavily on the metal. This follows from the magnitude of
the slV hyperfine constants. The primary object of our study is a description of the electronic ground state of (CbH6)2VC121so we must first decide upon the symmetry of the ground state. This is made
difficult by the fact that the crystal structure of the host molecule, (CjHj)ZTiC12,is not known. We must, thus, deduce the symmetry of the ground electronic state of (CbHc)2VC12 from the various
magnetic parameters without knowing how the magnetic tensors are oriented in the molecule. The marked departure of the vanadium coupling constant from axial symmetry is sufficient evidence to rule
out the possibility of an A2 ground state. The A2 representation contains only d,, of the vanadium basis set so the hyperfine tensor must have axial symmetry. A B1or a Br ground state can be
compatible with the observed A tensor anisotropy if substantial 3d-4p mixing occurs. The ground-state function (2) produces the expressions given by eq 3 for the principal components of the
anisotropic part of A. We have assumed
P d , p = gePePICgN(Y-’)d,p
that the function given in eq 2 represents the ground state accurately and have ignored admixture of higher states. The ratio Pd/P, may be estimated from the corresponding spin-orbit coupling
constants since both contain a ( r - 3 ) dependence.lg From the spectroscopic term splitting for V(V)20we obtain the values (3d = 250 cm-I and ldp = 846 cm-’ which give Pd/P, u l/3. This leads to
values of a = +0.91 and p = ~ t 0 . 4 1 for the amplitudes in eq 2, in order to fit the data. The argument above is valid for either B-type ground state. The ground state of AI symmetry is also
compatible with the anisotropy of the coupling constants as we shall discuss later. We turn to the isotropic component of the coupling constant and an empirical argument to obtain the correct ground
state. McGarveyZ1 has compiled and analyzed isotropic coupling constants for a large number of transition metal compounds. From the isotropic coupling constant7 we obtain the value of -2.08 for the
quantity x defined by eq 4. K is the isotropic contact term and the other constants have their usual meaning. This value is significantly larger than
(4) values McGarvey listed for other vanadium(1V) compounds and indicates a ground state which contains direct 4s mixing. We therefore exclude both B-type ground states as i t would be difficult to
reconcile the large value for K with a ground state in which direct s admixture is forbidden by symmetry. Robertsonz2gave a value of 4600 Mc for the coupling of a 4s electron in V2’ from which we
obtain a value of 49.2 for x. Comparing the experimental value of -2.08 with the value obtained for vanadium tetraphenylporphyrin (-2.78) gives a difference of 0.7. This amount may be attributed to
direct 4s mixing since the ground state of vanadium tetraphenylporphyrin is well known.23 We calculate 1,42YG 4s mixing in the A1 ground state of (C;Hj)2VC12. The following treatment of the AI ground
state from the hyperfine data and the analyses of the excited orbitals from the g tensor anisotropy will be solely in terms of a crystal field model for the electronic structure of the complex. This
is justified by our previous observation of the strong localization of the unpaired electron on the vanadium. Given an A1 ground state, the anisotropic components (19) L. Pauling a n d S.
Goudschmidt, “ T h e Structure of Line Spectra,” McGraw-Hill, New York, N. Y., 1930, p 206 ff. (20) R . F. Bacher and S. Goudschmidt, “Atomic Energy States,” McGraw-Hill, New York, N. Y., 1932, p
499. (21) B. R.McGarvey, J . Phys. Chem., 71, 51 (1967). (22) R . E. Robertson and H. M. McConnell, ibid., 64, 70 (1960). ( 2 3 ) D. Kivelson and S. K . Lee, J . Chem. Phys., 41, 1896 (1964).
BIS(CYCLOPENTADIENYL)VANADIUM DICHLORIDE of the hyperfine coupling constant lead to the evaluation of the mixing coefficients of eq 5 . Using (5) we obtain
+ bldz2-p)
(5) for the principal anisotropic coupling constants of vanadium =
- b2)
+ g4 7
- b2) T,
ab]Pd ab]Pd
= - (a2 - b 2 ) P d
This leads to values of a and b of =t0.70, A0.72, or h0.27, h0.96, or h0.96, A0.27 for anisotropies in accord with our data. Unfortunately, since the host crystal structure has not been determined,
we are unable to choose the unique set of coefficients for ( 5 ) . We should point out that the inclusion of spin-orbit mixing and orbital hyperfine effects in the derivation of (6) leads to values
of the mixing coefficients that are within 10% of those reported above. Equation 6 leads to a value of 9.44 X cm-1 for P . From Freeman and Watson's Hartree-Fock functions24we get 8.57, 10.7, 12.8,
and 17.2 (P X lo3 cm-l) for Vo, V+, V2+, and V 4 f , respectively. If we use the change obtained from the Fenske-Hillier calculations, which seem reasonable on the whole, we obtain a metal amplitude
of 0.82 for the unpaired electron. The function given by ( 5 ) yields eq 7 for the principal
values of the spectroscopic splitting tensor where l is the spin-orbit coupling constatlt and AE gives the energy difference between the orbital represented by (4) and the excited d orbitals. In
writing eq 7 we have assumed that the various excited states, as well as the ground state, are composed entirely of d orbitals and that no other states having large d-orbital amplitudes are nearby.
We use = 170 cm-l which corresponds to the spin-orbit coupling constant for V f . This value has been used in the g value analysis of other V4+ systems.21 Taking a = 0.70 ahd b = 0.72 we obtain for
the crystal field excited states AE,, = 2266 cm-l, AE,, = 19,500 cm-l, and AEuZ = 23,868 cm-l. The other values for a and b yield excited-state energies of the same separation but not necessarily of
the same ordering. The optical spectrum of (CbH&VC12 gives no indication of the proximity of these excited states since the assignment of the spectrum contains no transition which involves the al
level. We can, however, obtain an estimate of the Grst excited orbital from the spectrum of (CsH&TiCl*, which places d,, in our scheme a t 6000
E. Watson "Magnetism," Vol. 11-A, G. T. Rado and H. Shul, Ed., Academic Press, New York, N. Y., 1965, p 167. (24) A. J. Freeman and R.
Inorganic Chemistry, VoZ. 11, No. 8, 1972 1737 cm-l. We cannot rationalize this discrepancy within the framework of the crystal field model used above. Unfortunately, we observe ligand hyperfine
coupling neither a t room temperature nor a t 77'K so a detailed description of the spin-Hamiltonian parameters from a molecular orbital point of view is impossible. We can say little about ligand
contribution to the ground state MO other than placing a maximum on chlorine 3s mixing. From the observed line widths we get a maximum isotropic chlorine coupling constant of 6.6 G for two equivalent
chlorine nuclei. Using a value of 1666 G for the coupling of a chlorine 3s electronz5puts the maximum 3s chlorine participation a t o.4y0. The modified Wolfsberg-Helmholz method has been used with
some success to explain the optical spectra and spin densities of transition metal complexes.26 The widespread use of this method, despite its many approximations, stems from the computational ease
with which a calculation may be performed. Recently criticism of the method has been put forth by several authors, who offer alternative schemes within the SCCC framework. Major criticism centers
around the claim that metal-ligand covalency is an artifact of the approximations used in the calculation. We performed the Wolfsberg-Helmholz (type I) and the Fenske (type 11) calculations in order
to evaluate their validity when applied to an organometallic system for which both optical and epr data are available. Table I1 indicates the eigenvalues obtained for the type I calculation. The
energy levels are extremely close together near the MO in which the unpaired electron resides. This calculation gives us neither a correct ground-state symmetry nor qualitative agreement with the
optical spectrum. There are several features that should be mentioned in conjunction with the type I1 results. We see that the metal charge increases substantially (0.23 + 1.29) when some
electrostatic terms are included explicitly in the Hamiltonian. The predicted ground state which results from the type I1 calculation is of symmetry A2 and is not compatible with our epr results. The
a1 MO located approximately 7 k K above the highest occupied orbital would produce a ground state of correct symmetry if the a2 (-106.9) and a1 (-100.0) were reversed, however. The major composition
of al (-100.0) is -O.69(dzz) 0.50(dz2-,~) O.ll(Al(1) Al(2)) - 0.38(E2(1) E2(2)) - 0.138(3~(1) 3 ~ ( 2 ) ) 0.31(3~,(1) 3pz(2)) - 0.26(3~2(1)) 3pz(2)). The notation used for the ligand orbitals is
defined in Table 11. The dzz and d,2--yt coefficients of this orbital are reasonably close to experimentally consistent values. The set k 0 - 7 0 and AO.72 for a and b, when multiplied by the overall
occupation of metal orbitals, gotten from P (0.82) yields 0.565 and 0.59. We note, however, that the orbital deduced to be low lying from the g value analysis using a = -0.7 and b = +0.72 is of b2
and not a2 symmetry. This may be due t o an approach to the calculation of g which is too simple, but there are too many uncertainties in the ordering of levels to permit a better calculation. The
type I1 calculation gives a value of 34.4 kK for the transition $g --c (core) (b2)2(a~)1(a2)1(al)1 compared 1oj11t27
+ +
( 2 5 ) M. C. R. Symons, J . Chem. Soc., 570 (1963). (26) See, for example, ref 16, p 128. (27) R. F. Fenske, I n o y g . Chem., 4, 33 (1965).
1738 Inorganic Chemistry, VoZ. 11,No. 8, 1972 to the 26-kK value observed by Chien. The second transition +-(core)(bz)1(a1)2(az)1(al)1 gives 36.5 compared to the 37 kK measured. We know however that
the al and a2 levels should be reversed, which involves the 7 kK which separates them. We conclude that the “crystal field terms” play a significant role in a valid discussion of the electronic
structure of (C6H&VCl2. Though the type I1 calcula-
D. W. CLACKAND J. R. YANDLE
tion predicts a ground state of incorrect symmetry, our previous discussion indicates that an energy shift of about 7 kK produces a M O picture which is in reasonable agreement with both the optical
and epr data. Acknowledgments.--We acknowledge gratefully the partial support of this work by the National Science Foundation under Grant GP-10063.
Electronic Spectra of the Negative Ions of Some Metal Phthalocyanines BY D. W. CLACK*
J. R. YANDLE
Received November 3, 1971 Reduction of metal phthalocyanines with sodium in tetrahydrofuran (or electrochemically at a mercury pool or platinum cathode) produces a series of negative ions,
corresponding to the stepwise addition of electrons. The complexes with Z n ( I I ) , Ni(II), Fe(II), Mn(II), M g ( I I ) , and Al’I’Cl form four distinct reduction stages, but for CoI’Pc five
reduction steps are found. T h e electronic absorption spectra are reported and used t o discuss the electronic structures of the metal phthalocyanine negative ions. The spectroscopic observations
indicate t h a t all complexes with the exception of CoIIPc and FeIIPc give reduced species in which the additional electrons are confined essentially to the ring eg orbital. Reduction of CoIIPc and
FeI’Pc and possibly also MnIIPc appears t o involve electron addition t o both ligand (eg) and metal orbitals.
Introduction Metal phthalocyanines and the closely related metal porphyrins may be reduced chemically1n2 or electro~ h e m i c a l l y to ~ * give ~ a series of negative ions. This reduction
corresponds to the successive addition of electrons to vacant or partially occupied orbitals of the metal complex. Which orbitals are used depends on the proximity of the vacant metal orbitals and
the lowest unoccupied orbital eg of the phthalocyanine ligand system. For complexes with closed-shell central ions (Mg2+, Zn2+) electron addition is confined to the T orbitals of the ligand system.
It has been shown by both chemical” and p ~ l a r o g r a p h i c ~techniques ,~’~ that up to four electrons can be donated to these systems, and it is generally accepted that these negative ions
result from the addition of electrons to the lowest vacant eg orbital of the ring. On the other hand, when the central ion is transitional, the metal d orbitals lie close to the highest filled al,
ring orbital and it is therefore possible that in some cases reduction of the central metal will take place. Shablya and Terenin’ first reported an anion of a metal phthalocyanine together with its
electronic absorption spectrum by reducing magnesium phthalocyanine in tetrahydrofuran with sodium. A twoelectron reduction of CuIIPc (the abbreviation Pc for phthalocyanine will be used throughout
this paper) using potassium in liquid ammonia has been observed and assigned to the species C U ~ P C .Taube ~ has
* Address correspondence t o this author a t the Department of Chemistry, University College, Cardiff, C F l lXL, United Kingdom. (1) R . Taube, Z. Chem., 3, 392 (1963). (2) J. W. Dodd and N. S.
Hush, J . Chem. SOL.,4607 (1964). (3) D . W. Clack and N. S. Hush, J . A w w . Chem. SOL.,8 1 , 4238 (1966). (4) R . H. Felton and H. Linschitz, ibid., 88, 1113 (1966). ( 5 ) R . Taube, Z. Ckem., 6 ,
8 (1966). (6) D. W. Clack, P h . D . Thesis, University of Bristol, 1967. ( 7 ) A. V. Shablya and A. h-.Terenin, O p t . S g e k t i o s k . , 9 , 533 (1960). (8) G. W. W a t t and J. W. navies, J .
Inorg. h’ucl. Ckem., 14, 32 (1960).
isolated higher reduction products of metal phthalocyanines as crystalline solids from solution by electron transfer from alkali metals and radical ions5 More recently the electronic absorption and
electron spin resonance spectra of the mono- and dinegative ions of some metal phthalocyanine tetrasulfonates in dimethyl sulfoxideg and the electron spin resonance spectra of some transition metal
phthalocyanines in tetrahydrofuran and hexamethy1phosphoramide1O have been reported. Polarographic measurements in dimethylformamide on the analogous metal porphyrins3 have indicated that four
reduction steps are possible which were assigned to the formation of mono-, di-, tri->and tetranegative porphyrin ions, the latter having the eg orbital filled to its complement of four electrons.
Polarographic measurements on the metal phthalocyanines have yielded similar results,6 and in general four reduction waves are observed with the exception of CoIIPc, which exhibits five steps.
Experimental Section Electrochemical reduction in dimethylformamide-tetra-n-propylammonium perchlorate was effected in a three-electrode cell under an atmosphere of dry nitrogen using a controlled
cathode potential. The absorption spectra were measured using a Beckman DK2 ratio recording spectrophotometer. Only the monoand dinegative ions were sufficiently stable under these conditions t o
permit spectral measurements, the higher ions being oxidized by traces of atmospheric oxygen which unavoidably leaked into the cathode compartment. The chemical reduction was carried out under high
vacuum by slowly washing a solution of the metal phthalocyanine M) in tetrahydrofuran over a sodium film, prepared by thermal decomposition of sodium azide. The reduction was monitored by following
the changes in the absorption spectra of the metal phthalocyanine after each washing. In this may it was possible by washing only small amounts a t any one time t o characterize
(e) L. n.Rollmann and R. T. Iwamoto, J . Amev. Ckem. Soc., 90, 1435 (1968). (10) C . hl. Guzy, J. B. Raynor, L. P. Stodulski, and X. C. R . Symons, J . Chem. SOL.A , 997 (1969). | {"url":"https://datapdf.com/electron-paramagnetic-resonance-studies-of-the-electronic-st.html","timestamp":"2024-11-10T05:16:11Z","content_type":"text/html","content_length":"54673","record_id":"<urn:uuid:5f91d25c-b064-4e3b-9ddc-91d6d319c379>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00678.warc.gz"} |
Practice Question
• Subject 3. Equity Instruments and the Time Value of Money
CFA Practice Question
There are 676 practice questions for this topic.
CFA Practice Question
The value of the stock increases as the :
I. dividend growth rate increases.
II. required rate of return decreases.
III. required rate of return increases.
A. I and II
B. I and III
C. II only
User Contributed Comments 6
User Comment
stevelaz Would you say Value or Price ??!!
volkovv in this context value or price is the same thing
kiselevroman Don't we have a contradiction: Div growth rate increase, payout ratio increase, g=ROE*b decrease, so P=D/(r-g) decrease? Am I right?
leftcoast Kiselevroman:
Div growth rate increases when dividend payout ratio decreases: g=ROE*(1-dividend payout ratio)
harrybay Can we agree that an increase in the growth rate of dividend doesn't necessarily imply growth in EPS, in which case the value of the stock doesn't increase?
lighty0770 In a closed DDM function, which I believe is what they are referencing, the decreased R or increased G reduces the denominator which increases the value.
You need to log in first to add your comment. | {"url":"https://analystnotes.com/cfa_question.php?p=1V8X4UEC7","timestamp":"2024-11-05T10:22:06Z","content_type":"text/html","content_length":"20789","record_id":"<urn:uuid:df51b7db-252f-402a-bbda-f8b65ed40559>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00549.warc.gz"} |
exposition of Hochschild chains from simplical tensoring
I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one
gets by tensoring the algebra $A$ with the simplicial set $\Delta[1]/\partial \Delta[1]$:
$C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.$
I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.
The section that contains this material is the section
at the entry Hochschild cohomology. Just this one section. It’s not long.
It describes first the simplicial set $\Delta[1]/\partial \Delta[1]$, then discusses how the coproduct in $CAlg_k$ is given by the tensor product over $k$, and deduces from that what the simplicial
algebra $(\Delta[1]/\partial \Delta[1])$ is like.
After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that
this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry
is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.
I have further expanded the section
Identification with Kähler differential forms and have now some discussion in The simplicial circle action. This gives so far at least a hint of how to prove that under the identification $(Spec A)^
{S^1} \simeq Spec(C_\bullet(A,A))$ and the HKR-theorem, the canonical circle action induced the de Rham differential. | {"url":"https://nforum.ncatlab.org/discussion/2185/exposition-of-hochschild-chains-from-simplical-tensoring/","timestamp":"2024-11-07T03:33:06Z","content_type":"application/xhtml+xml","content_length":"45455","record_id":"<urn:uuid:8de6ec0e-a0b4-4aaf-acad-52d9b0bb99ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00839.warc.gz"} |
Inflation Forecast
Allocations for November 2024 (Final)
Cash TLT LQD SPY
Allocations for November 2024 (Final)
1st ETF 2nd ETF 3rd ETF
Inflation Forecast
The inflation rate is a fundamental determinant of the discount rate used to calculate the present value of an asset. Changes in inflation therefore affect asset valuations. What is the best way to
forecast the inflation rate? How reliable is inflation forecasting? The following discussion provides forecasts for U.S. total and core (excluding food and energy) inflation rates, along with the
method for constructing the forecast and the rationale for the methodology.
We update the forecast monthly as the Bureau of Labor Statistics (BLS) releases new Consumer Price Index data.
The following chart summarizes forecasts for the 12-month trailing, non-seasonally adjusted total and core inflation rates over the next year by month. The error bars indicate one standard deviation
ranges above and below the forecasts based on a backtest of all forecasts since 1990.
Forecasts are strictly technical, based only on past Consumer Price Index (CPI) data and not on any fundamental economic data such as trends in commodity prices, employment levels and wages. Guiding
beliefs for this analysis are:
1. There is probably an important degree of calendar regularity to the inflation rate, perhaps due to seasonal variation in supply of and demand for goods and services. In other words, the inflation
rate for a particular month during the next 12 months is likely related more to past inflation rate behavior during that same calendar month than to inflation rates during other past months.
2. For a short-term inflation rate forecast, momentum is probably more important than reversion. In other words, the inflation rate does not revert to its long-term trend quickly, and recent changes
in CPI are more indicative of near-term future changes than are changes in CPI from the more distant past.
3. The political cycle, and attendant economic/fiscal policy, may be significant for inflation rate behavior. In other words, inflation rate trends should consider at least four years of history and
should consider history in four-year increments.
Relevant to the first point, the following chart shows how the non-seasonally adjusted CPI varies by calendar month during 1951-2023 and two subperiods, 1951-1989 and 1990-2023. Results indicate some
persistence in seasonality over time, with relatively low inflation in November and December the most consistent finding.
Based on these beliefs and some sensitivity testing, we choose for our inflation rate forecast four years of historical data, with most recent data weighted more heavily than older data via a simple
algorithm. So, for example, we estimate the change in CPI for next September by extrapolating the changes in CPI from the four most recent Septembers, weighting newer historical data more heavily
than older data. Note that sensitivity testing is susceptible to data snooping bias.
We generate variability ranges for the total and core inflation rate forecasts by applying this methodology to each month since January 1990 and calculating the standard deviations of the differences
between forecasted inflation rates and the actual inflation rates for each of forecasted months 1 through 12.
For discussions about using:
• Seasonally adjusted total inflation,
instead of non-seasonally adjusted total inflation to predict future total inflation, see “Alternative Wealth Discount (Inflation) Rate” and the April 2006 research paper entitled “Core Inflation as
a Predictor of Total Inflation” by Neil Khettry and Loretta Mester of the Federal Reserve Bank of Philadelphia, which concludes:
“…[C]ore CPI inflation…performs better as an out-of-sample predictor of total CPI inflation than the total CPI, the CPI less energy, and the Cleveland Fed’s weighted median CPI. The CPI less
energy was a close second in terms of predicting future total CPI inflation. This suggests…focus on core CPI inflation rather than total CPI inflation over short time horizons. Based on our
results, we cannot make a similar conclusion for the PCE…
“[H]owever, …results on inflation prediction vary considerably across studies, depending on the forecasting model, time period, and measures of inflation used. Thus, we cannot conclude that one
particular alternative measure of inflation does a substantially better job at predicting inflation across all time horizons or sample periods.”
Our focus is the effect of inflation on investor valuation of stocks, which arguably involves behavioral aspects more appropriate for the volatile total inflation rate. However, the focus of the
Federal Reserve Board of Governors on core measures may attract investor attention.
The following three research papers describe attempts to adjust technical forecasts of inflation using a wide range of fundamental factors:
In an April 2000 paper entitled “The Unreliability of Inflation Indicators”, Stephen Cecchetti, Rita Chu and Charles Steindel compare the inflation-forecasting power of 19 potential indicators
with that of historical inflation data autoregression. The 19 indicators include commodity prices, financial indicators and economic indicators. They conclude that: “No single indicator in our
simple statistical framework clearly and consistently improved autoregressive projections. The indicators we found to be reasonably well correlated with overall price inflation either are
inherently difficult to forecast independently of inflation or bear an inverse relationship to inflation that seems to defy all logic.”
In an April 2003 paper that asks “Are there any reliable leading indicators for U.S. Inflation and GDP Growth?”, Anindya Banerjee, Massimiliano Marcellino and Igor Masten confirm and extend the
work in the above paper by looking at complex combinations of potential indicators and more extensive datasets to predict future inflation. They conclude that single-indicator models work best
but “…the indicators can hardly beat the autoregressions more than 50% of the time, which provides support for the [autoregression] model as a robust forecasting device…” And, “…overall the paper
provides yet another indication of the goodness and robustness of simple autoregressive models for forecasting [inflation].”
In a September 2003 paper entitled “Forecasting U.S. Inflation by Bayesian Model Averaging”, Jonathan Wright in contrast concludes that inflation forecasts derived from either (1) the
equal-weighted outputs of a large number of different inflation forecasting models or (2) a Bayesian (empirically weighted) average of these outputs substantially outperform a simple
autoregression. He demonstrates this outperformance via outputs from a set of 93 simple inflation forecasting models.
In summary, the above inflation forecast uses a relatively small amount of historical data to forecast the inflation rate, with more recent data weighted more heavily. Using historical inflation rate
data to predict future inflation is simple and probably as effective as any reasonably manageable approach.
Cautions regarding the forecast include:
• As noted, the forecast is strictly technical, ignoring any fundamental inflation drivers.
• Research indicates that there is no easy, accurate method.
For some thoughts about the long-term inflation trend as driven by changes in workforce productivity and government deficit spending, see “Public Debt, Inflation and the Stock Market”. | {"url":"https://www.cxoadvisory.com/inflation-rate-forecast/","timestamp":"2024-11-02T07:36:24Z","content_type":"application/xhtml+xml","content_length":"162700","record_id":"<urn:uuid:9411e544-73a2-4379-bf51-f27b0ebab1b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00766.warc.gz"} |
A Combinatorial Algorithm for the Multi-commodity Flow Problem
This paper researches combinatorial algorithms for the multi-commodity flow problem. We relax the capacity constraints and introduce a \emph{penalty function} \(h\) for each arc. If the flow exceeds
the capacity on arc \(a\), arc \(a\) would have a penalty cost. Based on the \emph{penalty function} \(h\), a new conception , \emph{equilibrium pseudo-flow}, is introduced. Then we design a
combinatorial algorithm to obtain equilibrium pseudo-flow. If the equilibrium pseudo-flow is a nonzero-equilibrium pseudo-flow, there exists no feasible solution for the multi-commodity flow problem;
if the equilibrium pseudo-flow is a zero-equilibrium pseudo-flow, there exists feasible solution for the multi-commodity flow problem and the zero-equilibrium pseudo-flow is the feasible solution. At
last, a \emph{non-linear} description of the multi-commodity flow problem is given, whose solution is equilibrium pseudo-flow. Besides, the content in this paper can be easily generalized to minimum
cost multi-commodity flow problem.
View A Combinatorial Algorithm for the Multi-commodity Flow Problem | {"url":"https://optimization-online.org/2019/04/7168/","timestamp":"2024-11-06T17:07:30Z","content_type":"text/html","content_length":"84204","record_id":"<urn:uuid:f2184288-5d5e-40c3-90c6-2fc6d2439cf4>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00235.warc.gz"} |
Why weird things happen
Let's say you decide to toss a coin 10 times. The coin is evenly balanced, and for the first five flips you happen to get heads. So what's the likelihood that the next toss will come up tails
instead? The answer is 50/50, because, in any given toss, the coin is no more likely to turn up heads or tails. However, if you toss it long enough -- 100 times, say, rather than 10 -- the number of
heads and tails you get will become closer and closer to equal.
Oh, probability! It's true, as David J. Hand writes in his new book, "The Improbability Principle: Why Coincidences, Miracles and Rare Events Happen Every Day," that it is "renowned" for its
"counterintuitive nature, more that any other area of mathematics." As Hand explains, humanity made "a major conceptual advance" when, in the mid-17th century, it began to understand "the step from
the unpredictability of individual events to the predictability of aggregates of events." Hand rates this insight as "on a par with the intellectual leap which led to the concept of gravity as a
universal force acting between objects."
But while probability remains difficult for many of us non-mathematicians to grasp, we have a much more pressing need to comprehend its basics than to master the principles of gravity. Chance and
likelihood factor into every aspect of our lives from gambling on lotteries to the way we judge the risks of flying or buying stocks or taking a new prescription drug. When it seems like half the
news articles we read include the phrase "studies show," we really ought to have a decent sense of how such studies are conducted and what principles of probability they must take into account.
Nevertheless, probability remains as hard for some of us to get a grip on as a wet and madly wriggling fish.
Hand, a mathematics professor at Imperial College London, offers such flailing readers a net with "The Improbability Principle," and a lively and lucid one at that. The U.S. edition of the book is
packaged rather like those pop-sociology bestsellers aimed at corporate marketeers ("The Tipping Point," "Black Swan"), but it's really a primer on probability with a special emphasis on rare events.
Hand opens the book with an example: a story about the actor Anthony Hopkins, who in the 1970s got a part in a film based on a novel. He traveled to London to buy a copy of the book, but couldn't
find it in any of the stores he visited. Heading home, he discovered a copy of the very title he was looking for lying on a bench in the tube station. (Furthermore, Hopkins later discovered that this
copy had once belonged to the author!)
Weird coincidences like this happen all the time, Hand explains, and for a complex tangle of reasons, even if some of us refuse to recognize them as chance occurrences. Maybe the Hopkins anecdote
seems like little more than a freakish event, but what about the fact that both Abraham Lincoln and Caligula dreamed of being assassinated before they were in fact assassinated? What about the
stories our friends and relatives insist on telling us about their premonitions and other "psychic" experiences, or the purportedly amazing predictions of psychics and astrologers? As Hand points
out, before making important decisions, at least two modern presidents -- Richard Nixon and Ronald Reagan -- consulted people who claimed to possess such supernatural powers.
Most skeptics are familiar with the notion of confirmation bias -- our tendency to take note of data that reinforces what we wish to believe and to ignore or forget data that doesn't -- and its role
in fostering belief in various forms of prophecy and clairvoyance. That's only one of the factors that contribute to Hand's Improbability Principle, which is not a rule of its own, but rather "a
consequence of a collection of more fundamental laws, which all tie together to lead inevitably and inexorably to the occurrence of such extraordinarily unlikely events."
These rules include the law of inevitability ("even if each of the possible outcomes has a tiny probability of occurring, it's certain that one of them will"), the law of large numbers ("with a large
enough number of opportunities, any outrageous thing is likely to happen"), the law of the probability lever ("a slight change in circumstances can have a huge impact on probability") and the law of
near enough ("events that are sufficiently similar may be regarded as identical," as in, the likelihood that a race will end in a tie depends on the accuracy of your stopwatch). There's also the law
of selection, a factor that Hand likens to shooting a bunch of arrows into the side of a barn and then painting targets around them afterward to demonstrate your extraordinary skill at archery.
A surprising number of these laws find examples and illustrations in the history of efforts to scientifically prove the existence of ESP and other paranormal abilities. Well, perhaps it's not that
surprising, given that most of the lab tests of telepathy have aspired only to show results slightly better than chance. (You have to wonder about the significance of a power that, even when it's
supposed to be working, doesn't work all that often.) But Hand also reveals how essential it is to have a good grasp of probability's role in far more mundane contexts.
Take, for example, a regional government's decision to place traffic cameras at sites that have the worst accident rates, with the goal of intimidating drivers into slowing down and being more
cautious. How meaningful is it if the accident rates at those spots decrease the following year? Only somewhat, because of a phenomenon called regression toward the mean. Some of the accidents at
those sites were the result of conditions particular to those sites that made them more hazardous to careless drivers, but some of them just randomly occurred there. As a result, the next time
accidents are counted, areas that had an unusually high number of them last time will statistically tend to have a lower number in the second count, even if the conditions at those sites don't
change. You need a statistician to help you figure out how much of that reduction is due to chance and how much to the effect of the traffic cameras.
"The Improbability Principle" is a book about math, obviously, a fact that is enough to entice some readers and to repel others. Its ideal reader will fall somewhere in the middle, neither too
advanced to find it obvious nor too timorous to be put off by the occasional equation. Yet it's an intensely useful (as well as a remarkably entertaining) book, and probably more so for those who
tend to shy away from such topics as "not my thing." It can transform the way you read the newspaper, that's for sure. We all need to learn to think more deeply about chance, luck, randomness, the
odds and all the other probability-related factors in our public and private lives, and if that isn't always easy, there's certainly no reason why it has to hurt. | {"url":"https://www.salon.com/2014/02/09/why_weird_things_happen/","timestamp":"2024-11-08T12:33:03Z","content_type":"text/html","content_length":"248111","record_id":"<urn:uuid:ac1c3a8e-691b-4ef1-a1d1-afe0edbc8790>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00749.warc.gz"} |
ow to Apply Newton's law
(7) Applying Newton's law of motion
• Newton's law of motion ,we studied in earlier topics are the foundation of mechanics and now we look forward to solve problems in mechanics
• In general, we deal with mechanical systems consisting of different objects exerting force on each other
• While solving a problem choose any part of the assembly and apply the laws of motion to that part including all the forces on the chosen part of the assembly due to remaining parts of the
• Following steps can be followed while solving the problems in mechanics
1)Read the problem carefully
2) Draw a schematic diagram showing parts of the assembly for example it may be a single particle or two blocks connected to string going over pulley etc
3) Identify the object of prime interest and make a list of all the forces acting on the concerned object due to all other objects of the assembly and exclude the force applied by the object of
prime interest on the other parts of the assembly
4) Indicate the forces acting on the concerned object with arrow and Label each force for example tension on the object under consideration can be labelled by letter T
5) Draw a free body diagram of the object of interest based on the labelled picture.Free body diagram for the object under consideration shows all the forces exerted on this object by the other
bodies.Do not forget to consider weight W=mg of the body while labelling the forces acting on the body
6) If additional objects are involved draw separate free body diagram for them also
7)Resolve the rectangular components of all the forces acting on the body
8) Write Newton second law of equation for the body and solve them to find out the unknown quantities
9) Do not forget to employ Newton's third law of motion for action reaction pair which results in null resultant force
Watch this tutorial for more information on Free body diagram
• Following solved example would clearly illustrate how to apply Newton's laws of motion following the above given procedure
Solved Example :
A horizontal forces of magnitude 500N pulls two blocks of masses m
=10 kg and m
=20 kg which are connected by the light inextensible string and lying on the horizontal frictionless surface.Find the tension in the strings and acceleration of each mass when forces is applied on
Given that force is applied on the block m
as shown in the figure below
Let T be the tension in the string and a be the acceleration of each mass .Now we will draw free body diagrams for each masses
Weights of the blocks m[1]g and m[2]g are balanced by their normal reaction R[1] and R[2] respectively.The equations of motion of the two massed are found using Newton's second law of motion
m[1]a=T ...............................(1)
m[2]a=F-T ............................(2)
Dividing 1 by 2
we get
Substituting the given values
T=166.7 N
Using value of T in equation 1 ,we find
a=16.67 m/s^2
Above sample problem shows how to solve a typical mechanics problem.Similarly by adopting given procedure we can solve other such problems
Watch this tutorial on how to solve Newton law of motion problem
Also Read | {"url":"https://physicscatalyst.com/mech/applying-newtons-laws-of-motion.php","timestamp":"2024-11-04T04:05:39Z","content_type":"text/html","content_length":"69682","record_id":"<urn:uuid:a787b8e6-1532-4b09-8d9f-cfc432d77b6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00566.warc.gz"} |
Trifilar Suspension Lab experiment - University Engineering - Marked by Teachers.com
Trifilar Suspension Lab experiment
by aspire0028 (student)
Mechanical Engineering Science 5 – Dynamics Lab Report
Trifilar Suspension
By using the trifilar suspension set up it is possible to calculate the moment of inertia of rigid bodies. By completing the experiment and acquiring actual data results these can then be compared to
the theoretical values calculated to show the difference.
“The Mass Moment of Inertia of a solid measures the solid's ability to resist changes in rotational speed about a specific axis, the larger the Mass Moment of Inertia the smaller the angular
acceleration about that axis for a given torque.
The mass moment of inertia depends on a reference axis, and is usually specified with two subscripts. This helps to provide clarity during three-dimensional motion where rotation can occur about
multiple axes.” [1]
“The actual trifilar suspension is an assembly that is used to determine the moment of inertia of a body about an axis passing through the body’s mass centre, perpendicular to the plane of motion”[2]
Image shows trifilar suspension set up: [image taken from http://www.tpub.com/content/MIL-SPEC/MIL-S/MIL-S-81963B/MIL-S-81963B00038.htm]
Occurrence in Industry
The type of motion looked at in this experiment can be seen in many areas of engineering, making it important to understand the components involved in simple harmonic motion so as to be able to
design products accordingly, the following are just a few examples of where the theory is used.
Car Suspensions: As the suspension of car moves up and down in the y axis as time progresses the content of the experiment relates closely to working of suspension making it easier to design
successful systems.
Pendulums – Although not often seen in today’s world the content of this experiment relates closely to the operations of pendulums, again making the design process and understandings of their
workings easier and clearer.
Oil Pumping – Again the motion involved in some large scale oil pumps is not dissimilar to content covered in this experiment making it important to understand how the moment of inertia relates to
periodic time.
The Aim of this Lab experiment was to calculate the polar moment of inertia of an assembly and using the results to predict the periodic time of a trifilar suspension. By using objects of known
dimensions and mass and using the parallel axis theorem it will be possible to look in greater detail at the mass moment of inertia and obtain a value for it.
This is a preview of the whole essay | {"url":"https://www.markedbyteachers.com/university-degree/engineering/trifilar-suspension-lab-experiment.html","timestamp":"2024-11-03T13:37:13Z","content_type":"text/html","content_length":"76202","record_id":"<urn:uuid:6f346a94-727f-4aa5-ae8a-d00398d72a8d>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00105.warc.gz"} |
30. A number ' x ' is selected from the numbers 1,2,3 and then ... | Filo
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30. A number ' ' is selected from the numbers and then second number ' ' is selected from the numbers . Find the probability that the product ' ' of the two numbers is less than 9.
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30. A number ' ' is selected from the numbers and then second number ' ' is selected from the numbers . Find the probability that the product ' ' of the two numbers is less than
Question Text 9.
Updated On Feb 7, 2023
Topic All topics
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 109
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COUNT DIGITS IN A NUMBER -
In the next few articles, we are going to learn some basic math problems that are very crucial to learn Data structures and algorithms.
Let’s start with the COUNT DIGITS IN A NUMBER with solution code in c++, java, and python.
Problem Statement
Given an integer N, write a program to check whether the given number is a prime number or not.
Input N: 11
Output: Prime Number
Input N: 6
Output: Not a Prime Number.
Read the value of N by taking input and initializing a temporary variable ‘count’ to zero. By using a while loop, divide N by 10 which results in the quotient of N when it is divided by 10. if the
value of N is decremented to a single digit number, it results in 0. so, until the base condition satisfies, increment the count variable for each iteration.
STEP 1: start
STEP 2: take the input value of N. initializes zero to a temporary variable, let’s take “count”.
STEP 3: iterate the input number through a while loop till the number is greater than zero when it is divided by 10. increment count variable value for each iteration.
STEP 4: Return the count value.
STEP 5: stop.
C++ Code:
#include <bits/stdc++.h>
using namespace std;
int main() {
int N,count=0;
cout<<”No. Of digits in the number: ”<<count;
Python code:
N = int(input())
count = 0
while N>0:
print("no. of digits in number: ",count)
JAVA code:
import java.util.*;
public class Main {
public static void main(String[] args) {
Scanner s = new Scanner(System.in);
int N = s.nextInt();
int count =0;
while( N!=0 ) {
N = N/10;
System.out.println("no. of digits in number: "+count);
Input: 1234
Output: No. Of digits in the number: 4
The time complexity of the code is O(n) as here we use a while loop.
The space complexity of the code is O(1).
In c++, an integer datatype takes only the number as input which can maximum have 10 digits, if we want to take a number having more than 10 digits, initialize the variable with a long datatype. Long
datatype holds a number maximum of digits 19.
long n=12345678998798L;
APPROACH 2:
By converting the number to a string and returning the length of the string.
STEP 1: start
STEP 2: take the input value of N.
STEP 3: convert the number into the string by using pre-defined functions.
STEP 4: Return the length of the string by using pre-defined functions.
STEP 5: stop.
C++ Code:
#include <bits/stdc++.h>
using namespace std;
int main() {
int N;
string x = to_string(N);
cout<<"No. of digits in number: "<<x.length();
Python code:
N = int(input())
x = str(N)
print("no. of digits in number: ",len(x))
JAVA code:
import java.util.*;
public class Main {
public static void main(String[] args) {
Scanner s = new Scanner(System.in);
int N = s.nextInt();
String n = Integer.toString(N);
int len = n.length();
System.out.println("no. of digits in number: "+len);
Input: 1234
Output: No. Of digits in the number: 4
The time complexity and the space complexity of the code are O(1).
APPROACH 3:
We can calculate the number of digits in N by using the logarithm of base 10. Logarithm to base 10 returns a float value, for which we can calculate the upper bound, which resembles the number of
digits in N.
STEP 1: start
STEP 2: take the input value of N.
STEP 3: calculate the logarithm value of the input number by using the logarithm function.
STEP 4: calculate the ceil value of the logarithm value applied to the input number. Return the ceil value.
STEP 5: stop.
C++ Code:
#include <bits/stdc++.h>
using namespace std;
int main() {
int N;
cout<<"no. of digits in number: "<<ceil(log10(N));
Python code:
import math
N = int(input())
print("no. of digits in number: ",math.ceil(math.log10(N)))
JAVA code:
import java.util.*;
public class Main {
public static void main(String[] args) {
Scanner s = new Scanner(System.in);
int N = s.nextInt();
int digits = (int)Math.ceil(Math.log10(N));
System.out.println("no. of digits in number: "+digits);
Input: 1234
Output: No. Of digits in the number: 4
The time complexity and the space complexity of the code are O(1).
That’s it from this tutorial. Hope you guys found It interesting. We have solved the count of the number of digits in a number in different programming languages. Happy coding!
Leave a Comment | {"url":"https://inprogrammer.com/count-digits-in-a-number/","timestamp":"2024-11-13T14:15:18Z","content_type":"text/html","content_length":"167499","record_id":"<urn:uuid:a5b0cdb7-8c94-4f9f-b74e-cb7742db88cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00782.warc.gz"} |
American Mathematical Society
On the embedding problem for $2^+S_4$ representations
HTML articles powered by AMS MathViewer
by Ariel Pacetti;
Math. Comp. 76 (2007), 2063-2075
DOI: https://doi.org/10.1090/S0025-5718-07-01940-0
Published electronically: April 24, 2007
PDF | Request permission
Let $2^+S_4$ denote the double cover of $S_4$ corresponding to the element in $\operatorname {H}^2(S_4,\mathbb Z/2\mathbb Z)$ where transpositions lift to elements of order $2$ and the product of two
disjoint transpositions to elements of order $4$. Given an elliptic curve $E$, let $E[2]$ denote its $2$-torsion points. Under some conditions on $E$ elements in $\operatorname {H}^1(\operatorname
{Gal}_{\mathbb {Q}},E[2])\backslash \{ 0 \}$ correspond to Galois extensions $N$ of $\mathbb {Q}$ with Galois group (isomorphic to) $S_4$. In this work we give an interpretation of the addition law
on such fields, and prove that the obstruction for $N$ having a Galois extension $\tilde N$ with $\operatorname {Gal}(\tilde N/ \mathbb {Q}) \simeq 2^+S_4$ gives a homomorphism $s_4^+:\operatorname
{H}^1(\operatorname {Gal}_{\mathbb {Q}},E[2]) \rightarrow \operatorname {H}^2(\operatorname {Gal}_\mathbb {Q}, \mathbb {Z}/2\mathbb {Z})$. As a corollary we can prove (if $E$ has conductor divisible
by few primes and high rank) the existence of $2$-dimensional representations of the absolute Galois group of $\mathbb {Q}$ attached to $E$ and use them in some examples to construct $3/2$ modular
forms mapping via the Shimura map to (the modular form of weight $2$ attached to) $E$. References
• Pilar Bayer and Gerhard Frey, Galois representations of octahedral type and $2$-coverings of elliptic curves, Math. Z. 207 (1991), no. 3, 395–408. MR 1115172, DOI 10.1007/BF02571397
• Michael Bungert, Construction of a cuspform of weight $3/2$, Arch. Math. (Basel) 60 (1993), no. 6, 530–534. MR 1216696, DOI 10.1007/BF01236077
• H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol.
627, Springer, Berlin-New York, 1977, pp. 69–78 (French). MR 472703
• Teresa Crespo, Explicit construction of $\~A_n$ type fields, J. Algebra 127 (1989), no. 2, 452–461. MR 1028464, DOI 10.1016/0021-8693(89)90263-9
• Teresa Crespo, Explicit construction of $2S_n$ Galois extensions, J. Algebra 129 (1990), no. 2, 312–319. MR 1040941, DOI 10.1016/0021-8693(90)90223-B
• Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
• Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory, Pure and Applied Mathematics, vol. 7, Marcel Dekker, Inc., New York, 1971. MR 347959
• Benedict H. Gross, Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187. MR 894322
• A. Jehanne, Realization over $\Bbb Q$ of the groups $\~A_5$ and $\hat A_5$, J. Number Theory 89 (2001), no. 2, 340–368. MR 1845242, DOI 10.1006/jnth.2001.2656
• Arnaud Jehanne, Sur les extensions de $\textbf {Q}$ à groupe de Galois $S_4$ et $\widetilde S_4$, Acta Arith. 69 (1995), no. 3, 259–276 (French). MR 1316479, DOI 10.4064/aa-69-3-259-276
• Winfried Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. MR 660784, DOI 10.1515/crll.1982.333.32
• J. Larry Lehman, Levels of positive definite ternary quadratic forms, Math. Comp. 58 (1992), no. 197, 399–417, S17–S22. MR 1106974, DOI 10.1090/S0025-5718-1992-1106974-1
• PARI/GP, version 2.2.8, http://pari.math.u-bordeaux.fr/, 2004.
• Anna Rio, Dyadic exercises for octahedral extensions ii, Submitted (2005).
• Bruno Schoeneberg, Elliptic modular functions: an introduction, Die Grundlehren der mathematischen Wissenschaften, Band 203, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German
by J. R. Smart and E. A. Schwandt. MR 412107
• Jean-Pierre Serre, L’invariant de Witt de la forme $\textrm {Tr}(x^2)$, Comment. Math. Helv. 59 (1984), no. 4, 651–676 (French). MR 780081, DOI 10.1007/BF02566371
• Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
• Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint
of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
• Willaim Stein, The modular forms explorer, http://modular.ucsd.edu/mfd/mfe/.
• Gonzalo Tornaría, Tables of ternary quadratic forms (part of computational number theory), http://www.ma.utexas.edu/users/tornaria/cnt/, 2004.
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Similar Articles
• Retrieve articles in Mathematics of Computation with MSC (2000): 11F80, 11F37
• Retrieve articles in all journals with MSC (2000): 11F80, 11F37
Bibliographic Information
• Ariel Pacetti
• Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria. C.P:1428, Buenos Aires, Argentina
• MR Author ID: 759256
• Email: apacetti@dm.uba.ar
• Received by editor(s): July 14, 2005
• Received by editor(s) in revised form: March 11, 2006
• Published electronically: April 24, 2007
• Additional Notes: The author was supported by a CONICET grant
The author would like to thank the “Universitat de Barcelona” where this work was done
• © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: Math. Comp. 76 (2007), 2063-2075
• MSC (2000): Primary 11F80; Secondary 11F37
• DOI: https://doi.org/10.1090/S0025-5718-07-01940-0
• MathSciNet review: 2336282 | {"url":"https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01940-0/?active=current","timestamp":"2024-11-10T14:38:45Z","content_type":"text/html","content_length":"70133","record_id":"<urn:uuid:0658bc3e-376a-4354-9b7e-556eae76b5d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00378.warc.gz"} |
Measurements & histograms
Measurements & histograms#
Chapter outline
• Measurements can be made in images by calculating statistics from the pixel values
• Histograms show the distribution of pixel values in an image, and are extremely useful to compare images & diagnose problems
Show code cell content Hide code cell content
%load_ext autoreload
%autoreload 2
# Default imports
import sys
from helpers import *
from matplotlib import pyplot as plt
from myst_nb import glue
import numpy as np
from scipy import ndimage
Images & pixels demonstrated how looks can be deceiving: the visual appearance of an image isn’t enough to determine what data it contains.
Because scientific image analysis depends upon having the right pixel values in the first place, this leads to the important admonition:
Keep your original pixel values safe!
The pixel values in your original image are your raw data: it’s essential to protect these from unwanted changes.
This is really important because there are lots of ways to accidentally compromise the raw data of an image – such as by using the wrong software to adjust the brightness and contrast, or saving the
files in the wrong format. This can cause the results of analysis to be wrong.
What makes this especially tricky is that trustworthy and untrustworthy images can look identical. Therefore, we need a way to see beyond LUTs to compare the content of images easily and efficiently.
Comparing histograms & statistics#
In principle, if we want to compare two images we could check that every corresponding pixel value is identical in both images. We will use this approach later, but isn’t always necessary.
There are two other things we can do, which are often much faster and easier:
1. Calculate some summary statistics from the pixel values, such as the average (mean) pixel value, standard deviation, minimum and maximum values.
2. Check out the image histogram. This graphically depicts the distribution of pixel values in the image.
Putting these into action, we can recreate Fig. 4 but this time add
1. the LUT (shown as a colored bar below the image)
2. a histogram
3. summary statistics
Show code cell content Hide code cell content
# Create 4 images with different values that look the same
im = load_image('sunny_cell.tif')
assert im.dtype == np.uint16
fig = create_figure(figsize=(12, 5))
colorbar_params = dict(shrink=0.9, orientation='horizontal')
# Original image, with 1% clipped for display
vmin = np.percentile(im, 1)
vmax = np.percentile(im, 99)
show_image(im, cmap='gray', title='Original 16-bit image', vmin=vmin, vmax=vmax, pos=241)
# 32-bit, rescaled
im2 = (im.astype(np.float32) - im.mean()) / im.std() * 50
vmin2 = np.percentile(im2, 1)
vmax2 = np.percentile(im2, 99)
show_image(im2, cmap='gray', title='32-bit processed', vmin=vmin2, vmax=vmax2, pos=242)
# 8-bit, clipped
im3 = im.astype(np.float32)
im3 = (im3 - vmin) / (vmax - vmin) * 255
im3 = np.clip(im3, 0, 255)
im3 = im3.astype(np.uint8)
show_image(im3, cmap='gray', title='8-bit clipped', vmin=0, vmax=255, pos=243)
# 8-bit, clipped, then inverted with inverted LUT
im4 = 255 - im3
show_image(im4, cmap='gray_r', title='Inverted with inverted LUT', vmin=0, vmax=255, pos=244)
# Show corresponding histograms
bins = 128
show_histogram(im, pos=245, stats='right', bins=bins)
show_histogram(im2, pos=246, stats='right', bins=bins)
show_histogram(im3, pos=247, stats='right', bins=bins)
show_histogram(im4, pos=248, stats='left', bins=bins)
glue_fig('fig_images_look_same_histograms', fig)
With the additional information at our disposal, we can immediately see that the images really do contain different underlying values – and therefore potentially quite different information – despite
their initial similar appearance. We can also see that the LUTs are different; they show the same colors (shades of gray), but in each case these map to different values.
By contrast, when we apply the same steps to Fig. 5 we see that the histograms and statistics are identical – only the LUT has been changed in each case. This suggests that any analysis we perform on
each of these images should give the same results, since the pixel values remain intact.
Show code cell content Hide code cell content
# Display the same image in 4 different ways by changing the LUT/colormap
im = load_image('sunny_cell.tif')
assert im.dtype == np.uint16
# Create 5 images with the same values that look different
fig = create_figure(figsize=(12, 5))
cbar_params = dict(shrink=0.9, orientation='horizontal')
# Original image, with 1% clipped for display
vmin = np.percentile(im, 1)
vmax = np.percentile(im, 99)
show_image(im, cmap='gray', title='Original 16-bit image', vmin=vmin, vmax=vmax, pos=241)
# Original image, with 10% clipped for display
vmin2 = np.percentile(im, 10)
vmax2 = np.percentile(im, 90)
show_image(im, cmap='gray', title='Enhanced contrast LUT', vmin=vmin2, vmax=vmax2, pos=242)
# Invert the LUT (in matplotlib, just add '_r' at the end)
show_image(im, cmap='gray_r', title='Inverted LUT', vmin=vmin, vmax=vmax, pos=243)
# Use an alternative LUT
show_image(im, cmap='magma', title='Magma LUT', vmin=vmin, vmax=vmax, pos=244)
# Show corresponding histograms
bins = 128
show_histogram(im, pos=245, stats='right', bins=bins)
show_histogram(im, pos=246, stats='right', bins=bins)
show_histogram(im, pos=247, stats='right', bins=bins)
show_histogram(im, pos=248, stats='right', bins=bins)
glue_fig('fig_images_look_different_histograms', fig)
If two images have identical histograms and summary statistics (mean, min, max, standard deviation), does this prove that the images are identical?
No! For example, we might have the same pixel values in a different arrangement. If I randomly shuffle the pixels in the image then the basic statistics and histogram remain unchanged – but the image
itself is very different.
This means that, technically, we can only really use histograms and summary measurements to prove that images are definitely not the same.
However, in practice this is usually enough. If two images have identical histograms and summary statistics and look similar, it is very likely that they are the same.
Conceivably, someone might try to deceive us by making some very subtle change to an image that preserves the statistics, such as as swapping two pixels amongst millions so that we don’t notice the
difference. Later, we’ll see how to overcome even that by checking every single pixel – but such elaborate trickery probably isn’t a very real risk for most of us.
Most of the time, when things go wrong with scientific images the histogram and statistics will be compromised in an obvious way – we just need to remember to check for these changes.
The ability to quickly generate and interpret histograms is an essential skill for any image analyst. We will use histograms a lot throughout this text, both to help diagnose problems with the data
and to figure out which techniques we should use.
Make histograms a habit!
When working with new images, it’s a good habit to always check histograms. This can give a deeper understanding of the data, and help flag up potential problems. | {"url":"https://bioimagebook.github.io/chapters/1-concepts/2-measurements/measurements.html","timestamp":"2024-11-14T11:49:15Z","content_type":"text/html","content_length":"55169","record_id":"<urn:uuid:dc594891-0a94-436e-8c41-edf24b3c09f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00284.warc.gz"} |
A mother said to her son, the sum of our present ages is twi-Turito
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A mother said to her son, ‘the sum of our present ages is twice my age 12 years ago and nine years hence, the sum of our ages will be thrice my age 14 years ago’. What is her son’s present age?
let present ages of son is x years and present age of mom is y years
Add their present ages and equate it to twice the age of mother 12 years ago.
Calculate their ages After 9 years and apply the condition the sum of our ages will be thrice my age 14 years ago .
The correct answer is: 12 years
Ans :- present age of son is 12 years
Explanation :-
Step 1:- find the equation by 1st condition.
let present ages of son is x years and present age of mom is y years
The sum of our present ages is twice my age 12 years ago.
x + y =2( y-12) [ the age of 12 years ago is present age -12 years]
Step 2:- find the equation by applying second condition
After 9 years the sons age = x+9
After 9 years the mothers age = y+9
Mother age 14 years ago = present age - 14 years = y - 14
After 9 years ,the sum of our ages will be thrice my age 14 years ago
Step 3:- find value of x by eliminating y
Doing 2(Eq1)-Eq2 to eliminate y
∴ present age of son = x= 12 years.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/Maths-a-mother-said-to-her-son-the-sum-of-our-present-ages-is-twice-my-age-12-years-ago-and-nine-years-hence-the-q795345bb","timestamp":"2024-11-04T15:12:15Z","content_type":"application/xhtml+xml","content_length":"390487","record_id":"<urn:uuid:4d61abf4-d7b4-4e78-b4ff-82e1d5c8026f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00250.warc.gz"} |
The SAR values are greatly different for the Gaussian and sine waveforms
1. The SAR values are greatly different for the Gaussian and sine waveforms
The SAR values are greatly different for the Gaussian and sine waveforms
• When I choose the sine wave which has the frequency of 835MHz and ampitude of 100V as the excitation signal, the computed SAR value is about the 411 W/kg. However, when I only change the
excitation signal into the Gaussian waveform which has the center frequency of 835MHz, the bandwidth of 835MHz and the amplitude of 100V/m, the computed SAR value is about 1×10(-15) which is much
smaller than the result computed by the sine wave. I want to know the reason. Thank you guys
• You need to make sure that the SAR is normalized with respect to the same quantity (e.g., input power or conducted power) and target value in both cases.
• Warren
replied to ofli on last edited by
@ofli Thank you. I have checked that I didn't select the nomalization. When I check the normalization, the value is almost the same for the two excitation waveforms. However, I just wonder why we
should select the option of normalization. I mean the real value should be computed directly by the sim4life. Why we must use the normalization? If we select the normalization, I guess the real
value of the SAR = normalized factore × computed SAR. I don't know wheter I think is right. I cannot find the answer in the help demo. Could you please explain this to me?
• Please refer to "Power Balance" section (under EM FDTD Theory) of Sim4Life Manual. And please check voltage & current (and input power) at the source sensor in your Harmonic and Gaussian
simulations. As you will see, you are exciting the system (check Sources section in the manual) at different power levels thus if you want to compare harmonic and gaussian cases, you need to
normalize in both cases to the same input power. I don't know what you mean by real value of SAR.
• Warren
replied to ofli on last edited by
@ofli In the simulation, I have set the normalized factor as 1 W. If I have an input voltage of 200V, the corresponding power should be much larger than 1 W. I mean in this situation if I still
need to normalization the power factor to 1 W even though the real input power is much larger than 1 W. What the real SAR with the input power not the value of 1 W? Thank you.
• Warren
replied to ofli on last edited by Warren
@ofli Before normalization, the SAR value computed by the Gaussian wave is 3.327e(-16) W/kg.
After normalization to 1 W, the SAR value computed by the Gaussian wave reachs about 17.5 W/.
Before normalization, the SAR value computed by the Harmonic wave is 410 W/kg.
After normalization to 1 W, the SAR value computed by the Harmonic wave is about 16.5 W/kg.
I want to know why the computed results have big difference from each other. Which one should I believe as the real value when the power of the excitation signal is much larger than 1 W?
• Sylvain ZMT
replied to Warren on last edited by
@Warren said in The SAR values are greatly different for the Gaussian and sine waveforms:
@ofli In the simulation, I have set the normalized factor as 1 W. If I have an input voltage of 200V, the corresponding power should be much larger than 1 W.
This is where you are probably mistaken: power is voltage times intensity and you don't know what the intensity is. You can actually visualize those quantities, P(t), U(t), I(t), P(f), U(f), I
(f), if you look at the output of the edge-sensor in the Analysis tab.
Note that the relationship between voltage U and intensity I is not known a-priori, as it depends on the whole system. For a fixed voltage, the intensity (and hence the power) is only known once
the simulation is completed. This is why one cannot set both U and I (or P) as input, and why normalization is the way to go.
• Warren
replied to Sylvain on last edited by
@Sylvain Thank you for you reply. I think I have known the explaination. Actually, I want use the EM FDTD and the Thermal Transient modules to simulate the heating problem. Firstly, I used the
sine waveform as the excitation signal. The microwave induced heat caused significant temperature increasing. However, when I replaced the continuous sine waveform into the waveform with pulse
width modulation (the duty cycle is 10%), it doesn't see temperature increasing any more. I noticed that I didn't select the option of normalization in the source of Thermal Transient. I don't
knwo whether I should select the normalization and set the scale factor as 1 W? The normal sine waveform and the modulated sine waveform are shown in below, respectively. | {"url":"https://forum.zmt.swiss/topic/231/the-sar-values-are-greatly-different-for-the-gaussian-and-sine-waveforms/5","timestamp":"2024-11-03T06:54:02Z","content_type":"text/html","content_length":"107806","record_id":"<urn:uuid:7a9d93bf-d116-4df3-9b05-1545446897ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00589.warc.gz"} |
Paramagnet (multiple solutions)
We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^
{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^
{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\}\\
1. List variables in their proper positions in the middle columns of the charts below.
2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]
3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:
\[\left(\frac{\partial M}{\partial B}\right)_S \]
4. Evaluate your chain rule. Sense-making: Why does this come out to zero? | {"url":"https://paradigms.oregonstate.edu/act/2141","timestamp":"2024-11-13T19:42:06Z","content_type":"text/html","content_length":"33218","record_id":"<urn:uuid:59daeb9a-0134-4644-af44-c99d079d874e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00725.warc.gz"} |
Excel Tools
This set of tools extends Excel's functionality - Formula explorer allows easy auditing of large and complex formulas - clicking on any cell refernce brings up the relevant cell and brackets can be
independent and highlight for clarity. To use, hit Ctrl+Shift+F in any cell with a formula. Hit F1 from the formula explorer for a list of features. - Functions to perform redumentary linear algebra
operations - finding eigenvalues, eigenvectors, Cholesky decomposition, and inverse matrices.
In theory, downloading this xla file and opening it should make these tools available in any workbook. Unfortunately, this was written for a previous version of Excel - it is unlikely to still work. | {"url":"https://daniel.guetta.com/archives","timestamp":"2024-11-13T02:39:38Z","content_type":"text/html","content_length":"17189","record_id":"<urn:uuid:b7d6e094-ca82-4057-a81a-dd4ead45df7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00732.warc.gz"} |
power required to drive a ball mill formula
1. Closed Circuit = W 2. Open Circuit, Product Topsize not limited = W 3. Open Circuit, Product Topsize limited = W to W Open circuit grinding to a given surface area requires no more power than
closed circuit grinding to the same surface area provided there is no objection to the natural topsize.
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required to operate a commercial ball mill of radius R is predicted on the assumption that the ball mill is half filled with balls, there is a cascade angle of 45o, and the mill base just covers
the top layer : Optimum cascading of mill for power consumption of the ball. Let the average density of the ball mill base
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One participant also clarified the use of trigonometric functions in the torque formula and the difference between static and kinetic friction coefficients. Overall, the calculated torque needed
for the load was determined to be Nm, with a suggestion to use a friction value of for a safety factor.
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Calculate the Torque required to lift the Force with the lever arm . T = F x d = N x m = Nm . We cannot perform the lift with this setup, because the stall torque is smaller than . the torque
required for the lift. We must either shorten the length of the lever arm,
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The power or energy required is the sum of the work required to crush or grind the rock as well as to rotate the mill. The power required depends on the hardness of the rock, the initial size and
the final product size required to achieve reasonable liberation of the mineral of interest, from the host rock. ... If motor and drive losses are ...
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Total power required for the ball mill = 1200 × × = 1407 kW. Referring again to Allis Chalmers' ball mill performance table, for a mill power draw of 1412 kW, the same mill dimensions will be
suitable if the ball charge is increased to 45% with a charge mass of 144 t. ... removal of limits of design power, high drive efficiency ...
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Ball mills for cement raw material and coal equipped with a drying compartment adequately air swept with hot gas (/sec above the ball charge), can handle moisture about 8%. Fully air swept mills
(56M/sec) can dry up to 1214% moisture. Grinding Operation Objectives and KPIs: Highly energy intensive unit operation of size reduction in ...
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Measurement results of two ball mills in a real ceramic manufacturing plant showed that the proposed calculation models could achieve an accuracy of more than 96% for predicting the ball mill ...
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For example, for an 36′ × 17′ SAG Mill, with a power consumption of MW, operating at 73% solids, % balls charge and at 76% of the critical speed, with 50% of the feed in the size class −6″ +1″;
it could be possible to increment in 2% the balls charge (to %), reducing the % −6″ +1″ to %, with no changes in the ...
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Power required to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) A. D. B. 1 D. C. D D. 1 D Answer: Option C. This Question Belongs to
Chemical Engineering >> Mechanical Operations.
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The power, P, to drive the mill would be expected to depend upon the length of the mill, L, the diameter, D, the diameter of ball, d, the density, e, of the ball, the volume occupied by the
charge (including voids), expressed as a fraction of the total mill volume, J, the speed of rotation, N, the acceleration due to gravity, g, the coefficient ...
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The main equipment for grinding construction materials are balltube mills, which are actively used in industry and are constantly being improved. The main issue of improvement is to reduce the
power consumption of a balltube mill with crosslongitudinal movement of the load. A comparative analysis and the possibility of using the known ...
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See Full PDFDownload PDF. BALL MILL Specification Number A Process Data Sheet xxxxxxxxxx Sheet No : 1 of 2 Area: xxx Client : Operating Centre: Jakarta No. 1 Working 1 Service: ROM Ball Mill Item
No: 1 of: Standby 0 xxxxML001 2 Manufacturer: Model Type and Number: 3 OPERATING CONDITIONS 4 MATERIAL HANDLED: Crushed Ore 5 Hazard ...
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how to calculate ball mill rotational speed ... hammer mills ... rolling mill torque : ... Central Drive Ball Mill Torque and Power Calculation Copper ... screw conveyor power calculation formula
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Power required to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) Question Answer MCQ Exam ON : Biosensors And Biochips Power required to
drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) Complaint Here As Incorrect Question / Answer
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Answer to Question #188408 in Chemical Engineering for Moin. Power required to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill ) Power required
to drive a ball mill with a particular ball load is proportional to (where, D = diameter of ball mill) D
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Method 2: Mill absorbed Power Slegten Formula. ... For mill with central drive: A = mech * = 0,96. Slide (5); 15/5/2017; Ball Mill Internals ... neededAdjustment of the ball charge required
Sensitive to low filling degreeEarlier replacement of the liners Slide (5); 15/5/2017; Ball ...
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According to the calculation of the ball mill power at home and abroad, the correction formula of the former Soviet Union's Tovalov formula is adopted: 𝑃 𝐷𝑉𝑛𝜑 ...
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1 Calculation of ball mill capacity. The production capacity of the ball mill is determined by the amount of material required to be ground, and it must have a certain margin when designing and
selecting. There are many factors affecting the production capacity of the ball mill, in addition to the nature of the material (grain size, hardness, density, temperature and humidity), the
degree of ...
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Calculate Tangential Force, Toque, and Machining Power for Face Milling Applications. These calculations are based upon theoretical values and are only intended for planning purposes. Actual
results will vary. No responsibility from Kennametal is assumed. Metric Inch.
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For instance, if your jar had in inside diameter of 90 mm and your milling media was mm diameter lead balls, the optimum rotation would be 98 RPM. Optimum RPM= .65 x Critical speed (cascading
action of the media stops) with dimensions in inches. Share on other sites. whitewolf_573. The motor has been working for 30 minutes, and works good ...
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Mechanical Agitator Power Requirements for Liquid Batches 2020 Instructor: John Frank Pietranski,, PDH Online | PDH Center 5272 Meadow Estates Drive Fairfax, VA Phone: An Approved Continuing
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1. SECTION 15 FLEXIBLE POWERTRANSMITTING ELEMENTS 757 LEATHER BELTS DESIGN PROBLEMS 841. A belt drive is to be designed for 321 =FF, while transmitting 60 hp at 2700 rpm of the driver 1D ; ≈wm ;
use a medium double belt, cemented joint, a squirrelcage, compensatormotor drive with mildly jerking loads; center distance is expected to be about twice the diameter of larger pulley.
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Fine grinding, to P80 sizes as low as 7 μm, is becoming increasingly important as mines treat ores with smaller liberation sizes. This grinding is typically done using stirred mills such as the
Isamill or Stirred Media Detritor. While fine grinding consumes less energy than primary grinding, it can still account for a substantial part of a mill's energy budget. Overall energy use and
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The invention combines ball mill working parameters and simulates ball mill working process to finally derive the ball mill power calculation method, the simulated power is closer...
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The following equation is used to determine the power that wet grinding overflow ball mills should draw. For mills larger than meters (10 feet) diameter inside liners, the top size of the balls
used affects the power drawn by the mill. This is called the ball size factor S.
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The general form of the equation is by Morgärdshammar that the power draw of a mill is derived from the power required to maintain mill revolutions with the mass of charge acting at the centroid
of the mass. This is a form of equation used by many workers with the only variant being the "constant' factor in front of the equation.
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Mill power is related to load behavior in some way. Some power models use shoulder position, toe position, or angle of repose as a description of load behavior to derive their power equations. As
can be seen from Fig. 2, a chord connecting the shoulder and toe of the load does not precisely express the configuration of load in a mill at the ...
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The grindingproduct size, P, in a Bond ball mill, which is given by the aperture size which passes 80% of the grinding product as a function of the aperture size of the test screen Pk, can be
expressed by the formula P= P k K 2. These functions for G and P enable us to calculate the value of Wi for any other size of grinding product if we know ...
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In Part I the relation between the relevant dynamic variables and the power requirements to drive a ball mill, containing the ball charge but without powder charge, has been investigated by means
of experiments upon smallscale analysis is used to obtain the equations in the form of dimensionless groups, and the functional relations between the groups are given analytically ...
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In the ballandrace mill (Fig. ), balls are held between two races, much like a large ball top race or grinding ring remains stationary while the bottom race rotates. As the coal is ground between
large diameter balls and the ring, the balls are free to rotate on all axes and therefore remain spherical.
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Power Requirement and Rolling Torque. For nonreversible Constant Speed Motor. The required power N m will be. N m = Required power of the drive motor on a non reversible motor. N r = the power
required for rolling process (effective power required to deform the metal with rolls). Nj= Frictional power is power loss due to frictional loss in roll bearingsN d = Efficiency of roll drive i
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The roller is used to drive another roller by surface contact. The driven roller is placed directly over the driver and it sits by means of gravity. The driven roller weight is 40 kg. The driving
roller weight is 10 kg. The motor is to be directly connected at the driver,s roller end. No belt. Thanks for replying baluncore.
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Less power is required to drive the cutter. Less sliding friction is produced between the tooth and the work. This reduces the development of heat. Longer life of the cutter may be obtained as
the number of regrinds can be increased. #10 Machining Time. The machining length (l m) is divided by the table feed (v f). The time required for any ...
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Division of Rational Numbers - Algebra | Socratic
Division of Rational Numbers
Key Questions
• Your question should be more specific but i will give you answer which i think you are looking for
Suppose we are give something like this;
Divide $\frac{36}{5}$ by $3 x$
First things first take the MI of $3 x$ that is $\frac{1}{3} x$
so now multiply both the terms $\frac{36}{5} \cdot \frac{1}{3} x$
Which is nothing but $\frac{12 x}{5}$
if u want to break it into a decimal $2.4 x$
So id you meant anything other than this plz comment i will come up with a different answer
• The reciprocal of a number, $a$, is a number, $b$ such that
$a \times b = 1$
For Real numbers, other than 0, the reciprocal of $a$ is $\frac{1}{a}$.
(The number $0$ does not have a reciprocal because $0 \times b = 1$ has no solution. $0 \times n = 0$ for all real and complex $n$.)
• Explanation:
When two fractions are guven to divide as #a/b÷c/d#
You take the reciprocal of the second fraction and multiply with the first fraction.i.e.,#a/b÷c/d=a/b×d/c=(ad)/(bc)#
• If the numbers have the same sign (both positive or both negative), then the answer is positive.
If the numbers have opposite signs (one is positive and the other is negative), then the answer is negative.
One way of explaining this:
The rule for dividing is the as same rule for multiplying positive and negative numbers.
The rule is the same because division is multiplying by the reciprocal.
The reciprocal of a positive number is positive and the reciprocal of a negative number is negative.
The reciprocal of $\frac{p}{q}$ is $\frac{1}{\frac{p}{q}}$ which is the same as $\frac{q}{p}$.
The reciprocal of a number is the number you have to multiply by to get $1$.
Not every number has a reciprocal. $0$ does not have a reciprocal (because $0$ times any number is $0$).
• Answer:
Just divide them as normal
Take two rational numbers, $\frac{b}{c}$ and $\frac{p}{q}$
How do we divide these? Simple! All we have to do is find the recipricol operation of the problem. That is:
$\frac{b}{c} / \frac{p}{q}$$=$$\frac{b}{c} \cdot \frac{q}{p}$
Note that the $q$ is switched from the denominator to the numerator, as it is the recipricol operation.
From there, just solve as normal! | {"url":"https://api-project-1022638073839.appspot.com/algebra/properties-of-real-numbers/division-of-rational-numbers","timestamp":"2024-11-05T07:26:57Z","content_type":"text/html","content_length":"538055","record_id":"<urn:uuid:ddd5f0c4-341d-440c-b9d3-1043abc42939>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00826.warc.gz"} |
Dictionary.com | Meanings & Definitions of English Words
1. an infinite series in which the terms are coefficients times successive powers of a given variable, or times products of powers of two or more variables.
1. a mathematical series whose terms contain ascending positive integral powers of a variable, such as a 0 + a 1 x + a 2 x ² +…
1. A sum of successively higher integral powers of a variable or combination of variables, each multiplied by a constant coefficient.
Discover More
Word History and Origins
Discover More
Example Sentences
The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).
The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables.
A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.
Thus we may make the power-series commence with 1, if we make the index-series commence with 0. | {"url":"https://www.dictionary.com/browse/power-series","timestamp":"2024-11-11T05:34:19Z","content_type":"text/html","content_length":"171417","record_id":"<urn:uuid:344be01d-b682-4835-8b3c-8afe99261501>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00689.warc.gz"} |
Precision Limits
When selecting a system of units, be careful to avoid calculations that approach the precision limits of the computer hardware. For most modern computers, the range is approximately 10^-300 to 10^300
in double-precision. If numbers exceed these limits, the program may produce artifacts in the model that may be difficult to identify or detect.
In addition, all calculations are performed using double-precision floating-point values. This provides approximately 14 digits of precision. If the coordinates of your model are offset by a large
value, this can remove effective digits of precision from your calculation. For this reason, it is recommended that the model coordinate system be offset so that the origin is near or inside the
model extent.
Was this helpful? ... Itasca Software © 2024, Itasca Updated: Sep 26, 2024 | {"url":"https://docs.itascacg.com/itasca920/common/docproject/source/manual/program_guide/modelformulation/precisionlimits.html","timestamp":"2024-11-08T01:36:26Z","content_type":"application/xhtml+xml","content_length":"7288","record_id":"<urn:uuid:bb2df74d-5be7-4bc6-b1c3-2c06b75f5c9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00343.warc.gz"} |
Given increasingly stringent spectral constraints, future radar transmitters will need to be adaptive and reconfigurable to allow operation in different frequency bands and meet various spectral
constraints based on nearby wireless activity. Transmitters will have to quickly adjust their transmission spectra, frequency of operation, and circuitry to operate in a dynamically changing spectral
environment. Such adjustments will likely need to occur on the order of milliseconds. The power amplifier is a critical component of radar transmitters, and its power efficiency and spectral output
are significantly dependent on its loading. We present a new method for simultaneously optimizing the load impedance and the input power to maximize the efficiency, while meeting spectral
constraints. This algorithm is designed to be used in real-time optimization for reconfiguring the load impedance and input power of an adaptive radar transmitter amplifier, and so a significant goal
of the algorithm's design is to minimize the number of experimental queries. The optimization is based on two well-known metrics. Efficiency is measured by the power-added efficiency (PAE), which is
the added radio frequency power divided by the direct current supply power. Spectral spreading is measured by the adjacent-channel power ratio (ACPR), which is the ratio of the power in a defined
adjacent channel to the power in the designated operating channel.
Adaptive radar has been a topic of discussion since the early 1970s [1]. In the modern era of tight spectrum allocations, adaptive radar will need reconfigurable circuitry to allow real-time
adjustments to change spectral output, operating frequency, and power efficiency. Lu et al. and Vaka-Heikkila and Rebeiz describe use of microelectromechanical system switches to build amplifiers
with adaptive output matching networks [2], [3]. Deve et al. present an adaptive impedance tuner capable of adjusting operating frequency between 1 and 3 GHz, a frequency range useful for many
sensing applications [4]. Sun et al. predict the criticality of intelligent impedance matching with minimized tuning iterations for reconfigurable transmitters [5]. Real-time antenna impedance
matching has been demonstrated by Sun and Sun and Lau [6], [7], and Qiao et al. demonstrate the use of a genetic algorithm to tune an amplifier in real time [8], although du Plessis and Abrie predict
that genetic algorithms will be slow in tuning for many applications [9]. Much of the previous work in impedance tuning has been performed for communications applications, but Baylis et al. describe
the importance of reconfigurable power amplifiers in solving the radar-communications coexistence problem [10]. Recent radar developments include design of adaptive waveforms and circuits that can
optimize for power efficiency and spectral compliance, including the effects of transmitter distortion. Blunt et al. describe the use of polyphase-coded frequency modulation waveforms for good power
efficiency and spectral performance [11] and present optimization of these waveforms with transmitter amplifier distortion considered, both via a model and in actual measurements [12].
The variation of PAE and ACPR with both load reflection coefficient $\Gamma_{L}$ and input power $P_{in}$ is well documented. Snider describes how the load impedance providing optimal efficiency
varies with input power level and demonstrates that the output power is significantly dependent on both the load impedance and the input power level [13], and Nemati et al. show variation of the
optimum PAE load impedance with changing input power [14]. Fu and Mortazawi show the design of a reconfigurable power amplifier using tunable varactors and also show that the load impedance in the
Smith chart providing optimum efficiency is different for different output power levels [15]. Hajji et al. show that a change in input power can be useful in obtaining constant intermodulation
rejection [16], related to the adjacent-channel spreading of amplifiers under broadband signal excitation. Load modulation to provide efficiency in Doherty amplifiers is discussed by Nam et al. [17].
Our previous work introduces the input power Smith tube (Fig. 1) to visualize how a criterion (such as PAE or ACPR) varies with both load reflection coefficient and input power [18]. Previous Smith
chart extensions include a spherical extension of the Smith chart to consider both positive and negative resistances in circuit design [19], [20], a Smith chart generalization allowing fractional
circuit elements to be considered [21], and an adjusted Smith chart for design with lossy transmission lines [22].
The purpose of the present paper is to show how an optimum combination of $\Gamma_{L}$ and $P_{in}$ can be obtained quickly, using a small number of measurements, through a fast search algorithm in
the power Smith tube. The desired optimization is an example of biobjective optimization [23], [24] and is applied to the well-known trade-off of linearity and efficiency [25], [26]. A previous paper
by our group demonstrates optimization for linearity and efficiency using only the real and imaginary parts of $\Gamma_{L}$ as the input parameters [27], and the present paper extends the solution of
this problem to a third dimension, allowing simultaneous optimization of input power with $\Gamma_{L}$ to provide the highest PAE possible under ACPR constraints.
Section II presents the basics of the search algorithm to be applied in the Smith tube. Section III describes simulation results of the algorithm's application. Section IV provides results from
measurement testing of the algorithm. Finally, Section V provides conclusions based on the presented results.
The search algorithm is based on estimation of the PAE and ACPR gradients within the three-dimensional input power Smith tube. First, the input power, on the vertical axis of the Smith tube, is
normalized so that the user-defined maximum input power is assigned the value 1 and the user-defined minimum input power is assigned the value −1. This scales the vertical limitations of the
cylindrical search space (the Smith tube) to the same dimensions as the planar values of $\Gamma_{L}$ in the horizontal dimension. The normalized input power $p_{in}$ (with a lowercase $``p^{\prime\
prime}$) is defined as follows (all power values are in dBm, decibels with respect to 1 milliwatt, for this paper). TeX Source\begin{equation*} p_{in}=2 \frac{P_{in}-P_{in, min}}{P_{in, max}-P_{in,
min}}-1 \tag{1} \end{equation*}
This equation normalizes the range of $P_{in}$ to set the minimum to −1 and the maximum to 1. The normalized input power $p_{in}$ represents the input power for purposes of calculating and evaluating
Although the three-dimensional search is limited by the power Smith tube's cylindrical space, the search vectors can be laid out in Cartesian dimensions, because the horizontal coordinates represent
the real and imaginary parts of $\Gamma_{L}$. The vertical coordinate $p_{in}$ provides the third of the three Cartesian dimensions. Using Cartesian coordinates allows direct extension of the search
space used in the two-dimensional algorithm for the same objectives [27].
A search vector $\overline{v}$ can be added to a candidate in the Smith tube to find the subsequent candidate in the search. When the candidate from which the search is operating does not possess an
acceptable ACPR value, this search vector is defined as follows: TeX Source\begin{equation*} \overline{v}=\hat{a}D_{a}+\hat{b}D_{b}, \tag{2} \end{equation*} where TeX Source\begin{equation*} D_{a}=\
frac{D_{s}}{2}\frac{\vert ACPR_{meas}-ACPR_{target}\vert}{\vert ACPR_{worst}-ACPR_{target}\vert}\tag{3} \end{equation*} and TeX Source\begin{equation*} D_{b}=\frac{D_{s}}{2}\frac{\vert \theta_{meas}-
\theta_{target}\vert}{\theta_{target}}. \tag{4} \end{equation*} $D_{s}$ is the search distance parameter and is provided by the user. It provides scaling for both components of the search vector [(3)
and (4)]. $ACPR_{meas}$ is the measured value of ACPR at the present candidate point. $ACPR_{target}$ is the maximum acceptable value of ACPR. $ACPR_{worst}$ is the largest value of ACPR measured to
this point in the search. $\theta$ is the angle between vectors $\hat{a}$ and $\hat{b}$. In (2), the vectors $\hat{a}$ and $\hat{b}$ represent vectors in the direction of ACPR steepest descent and
the bisector between the ACPR steepest descent and PAE steepest ascent vectors. On the Pareto optimum locus, which is the curve of constrained solutions, the gradients will be oppositely directed [6
], and the value of $\theta$ (the half-angle between the gradients) will be 90°. Thus, $\theta_{target}=90^{\circ}$ in (4). $\theta_{meas}$ is the measured value of $\theta$ at the candidate under
consideration. Fig. 2 shows that the gradients can be evaluated in the three-dimensional space. If PAE is represented by the variable $p$, the gradient for the PAE is given in terms of the unit
coordinate vectors $\hat{\Gamma}_{r},\hat{\Gamma}_{i}$, and $\hat{p}_{in}$ by TeX Source\begin{equation*} \nabla p=\hat{\Gamma}_{r}\frac{\partial p}{\partial\Gamma_{r}}+\hat{\Gamma}_{i}\frac{\partial
p}{\partial\Gamma_{i}}+\hat{p}_{in}\frac{\partial p}{\partial p_{in}}. \tag{5} \end{equation*}
The partial derivatives to calculate this gradient are estimated from a measurement separated from the candidate in the Smith tube by a neighboring-point distance $D_{n}$ in each coordinate
direction, as shown in Fig. 2. From a measurement of the change in power $\Delta p$ at a change in the real part of the load reflection coefficient, $\Gamma_{r}$, the partial-derivative estimation
follows as TeX Source\begin{equation*} \frac{\partial p}{\partial\Gamma_{r}}\approx\frac{\Delta p}{\Delta\Gamma_{r}}=\frac{\Delta p}{D_{n}}. \tag{6} \end{equation*}
From a measurement of the change in power for a change in the imaginary part of the load reflection coefficient, $\Gamma_{i}$, the partial-derivative estimation follows as TeX Source\begin{equation*}
\frac{\partial p}{\partial \Gamma_{i}}\approx\frac{\Delta p}{\Delta\Gamma_{i}}=\frac{\Delta p}{D_{n}}. \tag{7} \end{equation*}
Finally, from a measurement of the change in power $\Delta p$ at a change in $p_{in}$, the calculation follows as TeX Source\begin{equation*} \frac{\partial p}{\partial p_{in}}\approx\frac{\Delta p}
{\Delta p_{in}}=\frac{\Delta p}{D_{n}}. \tag{8} \end{equation*}
Because normalized power $p_{in}$ (lowercase) is being used, as defined by (1), the same step $D_{n}$ is used in the vertical direction for the partial derivative estimation as for the horizontal
directions. This completes the estimation of the gradient vector given by (5). The unit vector in the direction of increasing PAE can be taken by dividing the gradient by its magnitude: TeX Source\
begin{equation*} \hat{p}=\frac{\nabla p}{\vert \nabla p\vert}\tag{9} \end{equation*}
An approach similar to (5) through (8) can be used to estimate the gradient for ACPR. Because minimization is desired for the ACPR, the direction of optimal travel is given by a unit vector $\hat{a}$
oriented oppositely to the ACPR gradient: TeX Source\begin{equation*} \hat{a}=-\frac{\nabla a}{\vert \nabla a\vert}. \tag{10} \end{equation*}
The three-dimensional bisector of $\hat{a}$ and $\hat{p}$ is defined as the arithmetic mean of the vectors $\hat{p}$ and $\hat{a}$. Because $\hat{a}$ and $\hat{p}$ are both unit vectors, the mean
(bisector) given by TeX Source\begin{equation*} \hat{b}=\frac{1}{2}(\hat{a}+\hat{p}) \tag{11} \end{equation*} is also a unit vector.
Fig. 3 shows the resultant search vectors for the cases when ACPR at the present candidate is out of compliance (Fig. 3a) and when ACPR at the present candidate is within compliance (Fig. 3b). When
the ACPR is in compliance, the search vector (2) is changed to have a component in the $\hat{p}$ direction instead of the $\hat{a}$ direction: TeX Source\begin{equation*} \overline{v}=\hat{p}D_{a}+\
hat{b}D_{b}. \tag{12} \end{equation*}
A noticeable result of using (2) when the search is out of ACPR compliance and (12) when the search is in compliance is that the search firsts attempts to try to reach the region of ACPR compliance.
When the initiating candidate is out of compliance, (2) is used, meaning that the search vector has a component of $\hat{a}$, which causes a component of travel toward the ACPR minimum. An $\hat{a}$
component will be present in each search vector until the region of ACPR compliance is reached. When an initiating candidate is inside the region of ACPR compliance, (12) is used, meaning that the $\
hat{a}$ component of (2) is replaced by a $\hat{p}$ component of the same magnitude. This means that the search replaces its initial quest for the acceptable region by going toward the PAE optimum.
It progresses with this component directed toward the optimum PAE point, as long as it remains in compliance. In the results shown in the following section, note that the first effort of each search
is to reach the region of ACPR compliance. If the starting point of the search is outside the ACPR compliance region, this can result in an initial downward trajectory of the search for several
candidates, followed by a subsequent upward trajectory once the search enters the ACPR compliance region.
The speed and resolution of the algorithm's convergence are dependent on the starting point and the parameter values selected for the search distance $D_{s}$ and neighboring-point distance $D_{n}$.
Although further analysis could be performed to find the optimal values of these parameters given domain expertise and desired outcomes, this topic is saved for a later paper. In general, larger
values of $D_{s}$ are suggested for use if the vicinity of the end point is unknown; if the search is believed to start near the optimum, then a smaller value of $D_{s}$ may reduce the number of
If the search reaches the region of ACPR compliance and then tries to leave, the search distance parameter $D_{s}$ is divided by two, and the search returns to the last candidate in the acceptable
ACPR region and calculates a new search vector using the modified $D_{s}$. In addition to this penalization for leaving the acceptable region, the search vector tends to get smaller as it approaches
the optimum, because $ACPR_{meas}$ tends to approach $ACPR_{target}$ and $\theta_{meas}$ tends to approach $\theta_{target}$, decreasing the search vector components according to (3) and (4),
respectively. When the size of the search vector $\vert \overline{v}\vert$ decreases below a prespecified value (often the same as $D_{n}$), the search ends, and the measured point with the highest
PAE that meets ACPR requirements is declared the constrained optimum.
The algorithm was first tested in simulations using a nonlinear transistor model in the Advanced Design System (ADS) simulator from Keysight Technologies. A cosimulation was constructed using MATLAB
to perform algorithm-controlled ADS simulations. In the cosimulation, MATLAB is used to control an ADS nonlinear circuit simulation when supplied a netlist by ADS, including the values of $P_{in}$
and $\Gamma_{L}$, and then MATLAB reads the results from ADS and performs algorithmic decisions, recalling ADS for each required circuit simulation in the algorithm's execution. For the design, a
constraint of $\mathrm{ACPR}\leq-40\text{dBc}$ was imposed on the design. The goal was to obtain the highest PAE, while meeting this ACPR constraint. For comparison with the algorithm results,
load-pull simulations spanning the Smith chart were performed at multiple input power levels and are shown in Fig. 4, as presented in [6]. This exhaustive measurement shows that the ACPR-constrained
optimum point is at $P_{in}=21.75\text{dBm}$ and $\Gamma_{L}=0.80/\underline{180^{\circ}}$. The maximum PAE providing $\mathrm{ACPR}\leq-40\text{dBc}$ is 40.18% at this point.
Simulation tests of the algorithm were performed using a stepsize $D_{s}=1.5$, neighboring-point distance $D_{n}=0.05$, and input power range $0\text{dBm}\leq P_{in}\leq 36\text{dBm}$. Because $D_{n}
$ and $D_{s}$ have units consistent with the dimensions of the Smith chart, they possess the same units as load reflection coefficient $\Gamma_{L}$ and are unitless. Figure 5 shows the trajectory of
a search starting from $\Gamma_{L}=0.5/\underline{90^{\circ}}, P_{in}=5\text{dBm}$. A total of 33 measured points was required for the search to converge to the end parameter values of $\Gamma_{L}=
0.79/\underline{-178.6^{\circ}}$ and $P_{in}=21.58\text{dBm}$. At this end point, $\mathrm{PAE}=41.13\%$ and $\mathrm{ACPR}=-40.08\text{dBc}$ were obtained. Fig. 6 shows the search trajectory for a
starting point of $\Gamma_{L}=0.80/\underline{0^{\circ}}, \ P_{in}=0\text{dBm}$. Notice that while the starting conditions were very different in both cases, the end point values of $\Gamma_{L}, P_
{in}$, PAE, and ACPR are very similar. This indicates that both searches converge to approximately the same location in the Smith tube.
Figs. 7 through 9 show the search results for additional different starting combinations of $\Gamma_{L}$ and $P_{in}$. Table I summarizes the search results for the different starting combinations.
The results show that the final values of PAE vary less than 1%. The ending ACPR values are all within 0.2 dB of the −40-dBc limiting value. The end point $\Gamma_{L}$ coordinates are also similar,
and the variation across end point $P_{in}$ values is less than 1 dB. The results all compare well with the optimum value obtained from the exhaustive load-pull search. Interestingly, the PAE values
obtained from the algorithm are all slightly better than the optimum obtained through the exhaustive measurement, because the points measured in the Smith tube for the exhaustive measurement are
spaced in a way that the resolution of the optimum's determination is less than the shorter, algorithm-based search. This shows that excellent resolution can be obtained by the fast search, even if
no initial information is known about the location of the optimum.
The algorithm was measurement tested using the nonlinear load-pull test bench in the laboratory of the authors (Fig. 10). The input waveform is supplied by the signal generator with variable input
power. Measurements of PAE are performed with the power meter, and a spectrum analyzer is used to measure ACPR. The algorithm was tested on a Skyworks SKY5017-70LF InGaP packaged amplifier. This
device is different from the modeled device used for the simulation test of the algorithm, providing unrelated behavior for examining algorithm performance. Measurement tests of the algorithm were
performed using a stepsize $D_{s}=1.5$, neighboring-point distance $D_{n}=0.05$, and input power range $-5\text{dBm}\leq P_{in}\leq 2\text{dBm}$. An ACPR limit of −27.5 dBc was specified for this
For comparison with the algorithm results, Fig. 11 shows a plot of the constant ACPR surface for the −27.5-dBc limiting value, extracted from exhaustive, traditional load-pull measurements performed
at multiple $P_{in}$ values. The maximum PAE value, as assessed by the traditional load pull, is 7.39%, obtained for $\Gamma_{L}=0.47/\underline{-41.5^{\circ}}$ and $P_{in}=1.5\text{dBm}$.
Figs. 12 through 16 show the search trajectory for different starting combinations of $\Gamma_{L}$ and $P_{in}$. The results of the searches show excellent correspondence to each other, as summarized
in Table II. Variation in end point $P_{in}$ across the five searches is less than 0.5 dB. All ACPR end point values are beneath the limit and within 0.4 dB of the limiting value. End point PAE
varies by less than 0.5% across all of the searches. The number of measurements ranges from 23 to 39. As in the simulation case, the number of measurements required appears to be dependent on
factors, including the search starting point location relative to the optimum point.
A fast search algorithm has been presented and validated for simultaneous optimization of power amplifier load reflection coefficient and input power for maximum PAE, while meeting ACPR requirements.
The search has been demonstrated with simulations and measurements using the search space of the power Smith tube. In both simulation and measurement, the results of searches taken from multiple
starting points show excellent convergence as compared with traditional load-pull evaluations performed at multiple values of input power. Furthermore, the results show the ability to improve the
resolution of the solution. This results in the ability to obtain improved PAE, while meeting ACPR requirements without having to perform load-pull measurements over very closely spaced input power
values. Excellent repeatability of the algorithm from multiple starting points is observed in both the simulation and measurement results.
The authors thank Keysight Technologies for cost-free loan of the Advanced Design System simulation software. | {"url":"https://robertmarks.org/REPRINTS/2016%20Fast%20Simultaneous%20Optimization%20of%20Power%20Amplifier.htm","timestamp":"2024-11-11T13:37:47Z","content_type":"text/html","content_length":"283673","record_id":"<urn:uuid:d60549bf-f89a-4046-b9f9-368196214205>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00436.warc.gz"} |
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ER - In this tutorial on convolutional neural networks learn the fundamentals of it. Blog. You can get the codes here. Reply. AKhila says: January 18, 2019 at 10:12 am.
2.3-2.5. Tue, Nov 10, Convolutional codes, Chap. Max-Log-APP decoding. A tutorial on BCJR and APP decoding by Silvio A. Abrantes, Link.
Pär Olanders pgkba16 – Profil Pinterest
List decoding of convolutional codes - a Tutorial. / Johannesson, Rolf.
Schedule for Course on Classical and Modern Papers in
(A shift register is merely a chain of flip-flops wherein the output of the nth flip-flop is tied to the input of the (n+1)th flip-flop. Graduate Student Zac Sutton of Uconn HKN explains how to
encode a data stream using a convolutional encoder and how to decode the received sequence using th 2. (a) For the binary convolutional codes below, give the structure of the code (n,k,m). (b) Hence
for each code work out the the generators Gi(j) (D) in terms of the ‚delay™ element D. (c) In each case write down the generators as vectors using octal representation. (d) Hence draw the state
diagrams for each code. What a convolutional neural network (CNN) does differently.
Vol D2, ch 9, rev 1.0 ACHIEVEMENTS: setting up and testing of a convolutional encoder and decoder pair. See Tutorial Question Q1. This model shows how to use the Convolutional Encoder and Viterbi
Decoder blocks to simulate a tail-biting convolutional code. Convolutional Codes - Terminology. (Reed-Solomon codecs) and for Convolutional codes “Tutorial on Convolutional Coding with the Viterbi
Algorithm”. Introduction to Error Coding-I · Introduction to Error Coding-II · Introduction to Error Control Coding-III · Introduction to Convolutional Codes-I: Encoding · Introduction A Tutorial on
Convolutional Coding with Viterbi Decoding "Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm," IEEE Encoding data using convolutional codes. – How the encoder
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Introduction to Coding and decoding with Convolutional Codes (Tutorial 12) Convolutional codes are commonly specified by three parameters; (n,k,m). n = number of output bits . k = number of input
bits . m = number of memory registers .
Proof: Suppose C is the image of F k ((D)) under the mapping corresponding to G(D) ∈ F(D) k×n . The Viterbi decoder itself is the primary focus of this tutorial. Perhaps the single most important
concept to aid in understanding the Viterbi algorithm is the trellis diagram. The figure below shows the trellis diagram for our example rate 1/2 K = 3 convolutional encoder, for a 15-bit message:
Convolutional Neural Network – How to code some of the critical steps. by (an example from a CNN tutorial) as the objective is to show the code of some of the critical functions. VITERBI DECODING OF
CONVOLUTIONAL CODES Figure 8-1: The trellis is a convenient way of viewing the decoding task and understanding the time evo-lution of the state machine.
Jonas vlachos
A step Python, and OpenCV with pre-trained Convolutional Neural Networks. punctured-convolutional-codes-example.metin2underworld.com/, pulseaudio-tutorial.truyenngon.com/,
pulseaudio-stuttering.dealsmash.co/, av D Etiemble · Citerat av 23 — Convolution operations with byte inputs need 32- bit integer formats for the the original image and the final image (after coding
and decoding) when using F16 13661. lesson. 13662.
Created 3 years ago. 22 mars 2021 — Full tutorial w/ #Python code here: http://pyimg.co/pwmz1 👍 Enjoy!
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This article presents a self-contained introduction to the algebraic theory of convolutional codes. This introduction is partly a tutorial, but at the same codes and show how earlier proposed LDPC
convolutional code constructions tutorial paper in any IEEE publication issued during the previous calendar year. Trellis-coded modulation (TCM) is defined and described with respect to code-to-
signal mapping and coding for QAM. Also summarized are the evolution and ❑Convolutional Codes (n, k, K):. ❖Free distance and error correction capability: t = 6 ~ 9 for commonly used codes. ❖Claimed
coding rate: k/n. Normally 1/2 or 5 Jul 2016 Request a Tutorial Convolutional codes are a type of error correcting codes targeted to fix the errors introduced by the channel. Convolutional codes use
shift registers as encoders: the source data is divided in fr Convolutional codes are robust to a constant background noise, while Reed- Solomon error correction is effective at dealing with noise
that occurs in bursts. | {"url":"https://lonynfbca.netlify.app/48091/17831.html","timestamp":"2024-11-08T18:04:19Z","content_type":"text/html","content_length":"15561","record_id":"<urn:uuid:3c1f8db2-6114-4ba4-851e-0c2764cdf073>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00715.warc.gz"} |
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Repeated Measures ANOVA in Google Sheets (Step-by-Step) | Online Tutorials Library List | Tutoraspire.com
Repeated Measures ANOVA in Google Sheets (Step-by-Step)
by Tutor Aspire
A repeated measures ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more groups in which the same subjects show up in each
This tutorial provides a step-by-step example of how to perform a repeated measures ANOVA in Google Sheets.
Step 1: Install the XLMiner Analysis ToolPak
To perform a one-way ANOVA in Google Sheets, we need to first install the free XLMiner Analysis Toolpak.
To do so, click Add-ons > Get add-ons:
Next, type XLMiner Analysis ToolPak in the search bar and click the icon that appears:
Lastly, click the Install button.
Step 2: Enter the Data
Next, we need to enter the data to use for the repeated measures ANOVA.
For this example, suppose researchers want to know if four different drugs lead to different reaction times. To test this, they measure the reaction time of five patients on the four different drugs.
The reaction times are shown below:
Step 3: Perform the Repeated Measures ANOVA
To perform a repeated measures ANOVA on this dataset, click Add-ons > XLMiner Analysis ToolPak > Start. The Analysis ToolPak will appear on the right side of the screen.
Click Anova: Two-Factor Without Replication and fill in the following information:
Step 4: Interpret the Results
Once you click OK, the results of the repeated measures ANOVA will appear starting in the cell you specified in Output Range. In our case, we chose to display the results starting in cell A8:
In this case we are not interested in the results for the Rows, only for the Columns, which tell us the variation in response time based on the drug.
From the output we can see that the F test-statistic is 24.75887 and the corresponding p-value is 0.0000199.
Since this p-value is less than 0.05, we reject the null hypothesis and conclude that there is a statistically significant difference in mean response times between the four drugs.
Additional Resources
How to Perform a Repeated Measures ANOVA in Excel
How to Perform a Repeated Measures ANOVA By Hand
One-Way Repeated Measures ANOVA Calculator
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Discussion and conclusions
4 Discussion and conclusions
In the original version of USPEX ^4, the stable crystal structure was assembled from individual atoms, which was also shown to work for atomic crystals and also for simple molecular systems (like
carbon dioxide, water, urea). However, it is clear that for molecular crystals improvements of the efficiency can be made if the structure is assembled from whole molecules rather than individual
atoms. This is confirmed by the present study. Our constrained global optimization method allows one to find the stable crystal structure of a given molecular compound, and provides a set of
low-energy metastable structures at a highly affordable cost.
The reasons why evolutionary algorithms succeed in crystal structure prediction have been discussed before ^23. As mentioned in Sec. II, in addition to these, the constrained global optimization
fixes the molecular connectivity, and imposes the need for new variation operators (rotational mutation and softmutation), developed and described here.
For efficient and reliable polymorph prediction, the population of structures should be sufficiently diverse. A major difficulty in the prediction of molecular crystals is the large number of
plausible candidate structures that can have very close free energies ^122. Given the complexity of their energy landscape, high diversity of the population of the structures is mandatory for
successful prediction of molecular crystal structures. The initial population is particularly important, and it is usually a good idea to add a number of random symmetrized structures in each
generation, to keep sampling of the landscape diverse.
The presented algorithm provides not only the theoretical ground state, but also a number of low-energy metastable structures. With inclusion of zero-point energy and entropic contributions, such
structures may become stable. Even if this does not happen, low-energy metastable structures have a relatively high chance to be synthesized at special conditions. While the DFT+D is today’s state of
the art and its accuracy is often sufficient, for some systems (glycine), DFT+D is too crude, and more reliable approaches for computing the energy are needed. Under high pressure many of the
difficulties disappear, because the vdW interactions (poorly accounted for by today’s ab initio methods) become relatively less important.
Clearly, the quality of the global minimum found by USPEX depends on the accuracy of the theory used for energy calculations and structure relaxation. Current levels of theory can be roughly divided
into empirical, semi-empirical, and ab initio approaches. Accurate empirical force fields are appropriate for CSP, but reliable parameterizations are hard to generate for most molecules. In contrast
to empirical force fields, ab initio calculations provide a more accurate and rigorous description without parameterization, but the calculations are much more time-consuming. In our prediction, we
adopt the DFT+D level of theory, which combines "the best of both worlds", i.e. an accurate representation of intermolecular repulsions, hydrogen bonding, electrostatic interactions, and vdW
dispersions. DFT+D proved to be reliable for most systems, but its results are not fully satisfactory for glycine. This shows that further improvements in theoretical calculations of intermolecular
interactions energies are needed. In parallel with the improvement of methods for energy ranking, there is a need for efficient and reliable algorithms for global optimization of the theoretical
energy landscape, and present work is an important development in this direction. In the present paper, we describe the most important ingredients of this method, and demonstrate how it enables
affordable structure prediction for many complex organic and inorganic systems at ab initio level.
In summary, we have presented a new efficient and reliable approach for global energy optimization for molecular crystal structure prediction. It is based on the evolutionary algorithm USPEX extended
to molecular crystals by additional variation operators and constraints of partially or completely fixed molecules. The high efficiency of this method enables fully quantum-mechanical structure
prediction simulations to be performed at an affordable computational cost. Using this method, we succeeded in finding the stable structures for systems with various rigid molecular shapes
(tetrahedral, linear, bent, planar and complex molecules), and chemical bonding (vdW bonding, ionic, covalent, metallic, weak and strong hydrogen bonding, ab initio level of theory. This new approach
also has wide applicability to inorganic crystals containing clusters and complex ions. | {"url":"https://www.physics.unlv.edu/~qzhu/qzhu-thesis/sect0039.html","timestamp":"2024-11-15T03:15:58Z","content_type":"application/xhtml+xml","content_length":"10131","record_id":"<urn:uuid:5d78891e-8373-4e35-92b9-b8480581aaeb>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00183.warc.gz"} |
Are Essential Standards a Part of the Assessment Process?
Chris Jakicic
Chris Jakicic, EdD, an author and a consultant, was principal of Woodlawn Middle School in Illinois from 1999 to 2007. She began her career teaching middle school science.
Are Essential Standards a Part of the Assessment Process?
As I’ve worked with teams across the country in developing and using assessments, I’ve heard some interesting beliefs about essential standards (e.g., “We’re not allowed to do this in our district,”
or “Our curriculum only requires us to teach the essential standards”). Comments like these have convinced me that there are lots of educators who have misconceptions about the first of the four
essential questions we ask a collaborative team in a PLC to answer. That question is: “What do we want our students to know and do?”
High-performing teams know that they can’t answer this question with a simple “our state standards” reply. Instead, they know that the answer requires teams to develop consensus about a subset of all
standards called “their essential standards.” These standards are also often called power or priority standards and represent the guaranteed and viable curriculum for their grade or course. What this
means is that collaborative teams work through a process to determine a set of standards from their grade or course standards that they agree all students must learn. This requires that teams
understand that they can’t be satisfied with only 80 percent or 85 percent of students learning what is most important.
Consider this: when a team guarantees all students will learn their essential standards, they are assuring that students will be commonly prepared for the next grade level or course. They can’t make
this guarantee if they treat all of their standards as equally important.
Deciding which standards are most important
We recommend that teams start this process by using some specific criteria to decide which standards are most important. Doug Reeves has recommended that teams consider the three following:
• Endurance: Do students need to know this standard for the long haul?
• Leverage: Is this standard something that crosses more than one curricular area?
• Readiness: Is this standard a prerequisite for future learning? (Reeves, 2002).
I often suggest teams think about one more question while they do this work:
• Does this standard represent something we’d be willing to spend intervention time reteaching?
In other words, these standards are so important, we’ll spend more time teaching them, we’ll agree to use common formative assessments to know whether students have mastered them, and we’ll plan
intervention time for students who haven’t yet learned them.
Unfortunately, sometimes teams interpret this as saying that they only have to teach the essential standards. This is a misconception. In fact, teams should make sure that all standards have a place
in their curriculum. Let’s look at a specific example that may make this a bit clearer.
In kindergarten, there is an ELA standard that says students should be able (with help) to know the author and illustrator of a story and define what that role is. A second standard is that they
should demonstrate understanding of spoken words, syllables, and sounds (phonemes). Most kindergarten teachers will agree that the second standard is one that every student must know but that the
first standard isn’t something they would spend intervention time to teach if a student hasn’t yet learned it. However, teachers still teach the first standard; almost every time they read a story
aloud or introduce a new story, they will probably talk a bit about the author and/or illustrator.
In my own work, I’ve never run into a team of teachers who’ve complained they don’t have enough to teach! In fact, almost universally, the reverse is true. As we look at the ELA Common Core
standards, for example, in third grade there are 90 standards. A typical school year has about 180 days, so theoretically teachers have about 2 days to teach each standard. In math, there are about
30 standards in each grade—providing a lot more time to teach! But, after analysis, teachers see that the math standards often have many individual learning targets that must be taught for each
When collaborative teams engage in this discussion, they typically use a protocol to guide their work. Larry’s Ainsworth’s book Power Standards (2004) lays out a step-by-step guide.
1. He recommends that teams start by having each teacher reflect on which standards he or she believes best represent the most important standards.
2. Once they have each completed this process, the team begins to build consensus on which standards they can agree are the most important. He suggests that the essential standards should represent
approximately one-third of their curriculum (Ainsworth, 2010).
3. After they’ve completed this work, the next step is to review any standards documents or test blueprints that provide additional information about expectations for student learning. When teachers
start to do this work, they are definitely “learning together” about their standards.
4. The last step in this process is for teams to do a vertical alignment from one grade level to the next or one course to the next. Teachers examine the draft list of essential standards to see how
well they line up. They answer the question, “If a student goes through our school and only learns these standards, will he or she be prepared for the next grade level?” They look for repetition
or gaps in learning. They examine the grade level before theirs to make sure students will have the necessary prerequisite skills, and the grade level after theirs to make sure their students
will be prepared for the rigor expected.
You may be wondering why I asked the question in the title but haven’t yet addressed assessment. I believe that identifying essential standards is the first step in an effective assessment system. If
we don’t know what we expect all students to learn, we end up trying to assess everything. Then, when we try to respond, our response is too watered down to be effective. Or, each teacher decides
what’s most important to him or her, and students are not commonly prepared from one grade level or course to the next. This means that teachers spend more time filling in the “holes” from the
previous year in student learning.
Collaborative teams who establish the guaranteed and viable curriculum first can write and use assessments aligned to those expectations that are designed to provide specific information about which
students need help and what kind of help they need. Essential standards are the foundation of an effective assessment system, and plans that are built on a solid foundation will provide the
information teams need to help students.
Ainsworth, L. (2004). Power standards: Identifying the standards that matter most. Englewood, CO: Advanced Learning Press.
Ainsworth (2010). Rigorous curriculum design: How to create curricular units of study that align standards, instruction, and assessment. Englewood, CO: Lead Learn Press.
Reeves, D. (2002). The leader’s guide to standards: A blueprint for educational equity and excellence. San Francisco, CA: Jossey-Bass.
Lawrence Fetterly
I teach math at TriCounty Technical College, taught high math and science for 12 years and operate a Tutor Service (16 years). I have prepared over 50 students, one on one, to take either the ACt or
SAT (including the new one). In my experience, these tests, as administered, do not show students actual college readiness because of the time pressure and the lack of a test preparation process.
More and more States are adopting these tests and, valid or not, use the results to make claims about overall student college readiness. A recent research publication by Niclos-Barre and Gill of
Mathematica Policy Research shows that the ACT and SAT have lower success at predicting college readiness and success than the MCAT and the PARCC tests. The OECD tests are also a better indicator of
our student’s performance. GPA’s have been shown to better predict college performance. Bottom line, ACT’s and SAT’s need to change their time pressure requirements and increase the test time allowed
by at least 50%. Note that for a high school junior taking the ACT or SAT, it is likely been close to three years since taking a geometry course. From my experience, students are most likely to
struggle with the geometry questions. I believe that our four year colleges are using these tests to select what they may believe is the top 15%. The test scores are too often used to indicate that
at least 80% of our students are not college ready; a false claim.
Thanks for this opportunity to comment, Lawrence Fetterly M Ed
• Chris Jakicic
Hi Lawrence, In our work we really emphasize the power of common formative assessments. While we know our students and teachers are being judged by these high stakes tests, we also recognize that
we are teachers to make sure students are learning and being prepared for the next steps in their education or career. I appreciate your thoughtful comments about high stakes tests; many
classroom teachers would echo your concerns.
Thank you for this. I will share this with my team. We have to get started doing this work!
Chris Jakicic
Hi Alba, This step has become more urgent with remote learning. We need to focus on the most vital standards as we transition back to school. You’re doing the right work for the right time! | {"url":"http://allthingsassessment.info/2017/05/22/essential-standards-and-the-assessment-process/","timestamp":"2024-11-14T17:10:29Z","content_type":"text/html","content_length":"45276","record_id":"<urn:uuid:1e0fb4ab-4430-48e0-a9a6-829d8b1f256a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00197.warc.gz"} |
Statistical Mechanifesto by Eric L. Michelsen
Statistical Mechanifesto
by Eric L. Michelsen
Publisher: UCSD 2013
Number of pages: 73
This work is aimed at graduate and advanced undergraduate physics students. It contains a better entropy discussion, the Carnot conspiracy, Boltzmann distribution, entropy, free energy, meet Mr.
Mole, chemical potential, and much more...
Download or read it online for free here:
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(1MB, PDF)
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Re: [tlaplus] Checking reachability
I'm trying to show that every value can be chosen in a Paxos like setting.
I've tried the following properties.
Values == {1,2}
Prop_1 == [](~Chosen(1))
Prop_2 == [](~Chosen(2))
Prop_n1 == ~[](~Chosen(1))
Prop_n2 == ~[](~Chosen(2))
With all of them being set in the PROPERTIES field in the config.
In testing both Prop_1 and Prop_2 produce counterexamples as expected.
However both Prop_n1 and Prop_n2 also produces counterexamples.
Why is this since I'd expect that if Prop_1 and Prop_2 are false (ie produce a counter-example) then negating them should be true?
You received this message because you are subscribed to the Google Groups "tlaplus" group.
To unsubscribe from this group and stop receiving emails from it, send an email to | {"url":"https://discuss.tlapl.us/msg04882.html","timestamp":"2024-11-06T01:59:32Z","content_type":"text/html","content_length":"10754","record_id":"<urn:uuid:910068a3-1f24-4c9e-ba81-073a7b852ffe>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00402.warc.gz"} |
How to factor a variable - Algebra 1
All Algebra 1 Resources
Example Questions
Example Question #1 : How To Factor A Variable
Correct answer:
First, factor the numerator, which should be
Second, cancel the "like" terms -
Third, solve for
Example Question #1 : How To Factor A Variable
Factor the following expression:
Correct answer:
Here you have an expression with three variables. To factor, you will need to pull out the greatest common factor that each term has in common.
Only the last two terms have
Example Question #4 : Simplifying Expressions
Correct answer:
To find the greatest common factor, we need to break each term into its prime factors:
Looking at which terms all three expressions have in common; thus, the GCF is
Example Question #1 : How To Factor A Variable
Correct answer:
To find the greatest common factor, we must break each term into its prime factors:
The terms have
Pull this out of the expression to find the answer:
Example Question #1 : How To Factor A Variable
If and are distinct positive integers, what is the smallest possible value of
Correct answer:
Consider the possible values for (x, y):
(1, 100)
(2, 50)
(4, 25)
(5, 20)
Note that (10, 10) is not possible since the two variables must be distinct. The sums of the above pairs, respectively, are:
1 + 100 = 101
2 + 50 = 52
4 + 25 = 29
5 + 20 = 25, which is the smallest sum and therefore the correct answer.
Example Question #2 : Solving Equations
Correct answer:
Multiply both sides by 3:
Add the
Divide both sides by
Example Question #5 : How To Factor A Variable
Factor the following polynomical expression completely, using the "factor-by-grouping" method.
Correct answer:
Let's split the four terms into two groups, and find the GCF of each group.
First group:
Second group:
The GCF of the first group is
The GCF of the second group is
We can rewrite the original expression,
The common factor for BOTH of these terms is
Dividing both sides by
Example Question #1 : How To Factor A Variable
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
Correct answer:
Let's separate the four terms of the polynomial expression into two groups, and then find the GCF (greatest common factor) for each group.
First group:
Second group:
The GCF of the first group is
The second group is a bit tricky. It looks like they have no factor in common. But, each of the terms can be divided by
Factoring the second group by its GCF gives us:
We can rewrite the original expression:
which is the same as:
Example Question #5 : How To Factor A Variable
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
Correct answer:
Separate the four terms into two groups, and then find the GCF of each group.
First group:
Second group:
The GCF of the first group is
The GCF of the second group is
We can rewrite the original expression,
We can factor this as:
Example Question #7 : How To Factor A Variable
Factor the polynomial expression completely, using the "factor-by-grouping" method.
Correct answer:
Start by separating the four terms into two groups, and find the GCF (greatest common factor) of each group.
First group:
Second group:
The GCF of the first group is
(Remember, when dividing by a negative, the original number changes its sign!)
The GCF of the second group is
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...so, the expression, when factored, is:
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Introduction to Sampling Techniques | Different Sampling Types and Techniques
Contributed by: Sreekanth Tadakaluru
LinkedIn Profile: https://www.linkedin.com/in/sreekanth-tadakaluru-3301649b/
Let’s take an example of COVID-19 vaccine clinical trials. It is very difficult to conduct the trials on the entire population, as it deals with time, money, and resources. So in research
methodologies, sampling is a method that helps researchers to infer information about a population based on results from a subset of the population, without having to investigate every individual.
A telecom company planning to build a machine learning model to predict, churn customers from their network. One way is to collect all the customers’ information and build a prediction model. This
method requires high computational power and resources. So the best way is to take a sample (Subset of customers) from the population (All customers) which represents the population and build the
machine learning model. This saves money and effort.
Sampling is the process of selecting a group of individuals from a population to study them and characterize the population as a whole.
The population includes all members from a specified group, all possible outcomes or measurements that are of interest. The exact population will depend on the scope of the study.
The sample consists of some observations drawn from the population, so a part of a subset of the population. The sample is the group of elements who participated in the study.
The sampling frame is the information that locates and defines the dimensions of the universe.
A good sample should satisfy the below conditions-
1. Representativeness: The sample should be the best representative of the population under study.
2. Accuracy: Accuracy is defined as the degree to which bias is absent from the sample. An accurate (unbiased) sample is one that exactly represents the population.
3. Size: A good sample must be adequate in size and reliability.
Also read: Introduction to Inferential Statistics
Different types of Sampling techniques:
There are several different sampling techniques available, and they can be subdivided into two groups-
1. Probability sampling involves random selection, allowing you to make statistical inferences about the whole group.
There are four types of probability sampling techniques
• Simple random sampling
• Cluster sampling
• Systematic sampling
• Stratified random sampling
2. Non-probability sampling involves non-random selection based on convenience or other criteria, allowing you to easily collect initial data. There are four types of Non-probability sampling
• Convenience sampling
• Judgmental or purposive sampling
• Snowball sampling
• Quota sampling
Choosing Between Probability and Non-Probability Samples
The choice between using a probability or a non-probability approach to sampling depends on a variety of factors:
1. Objectives and scope of the study
2. Method of data collection
3. Precision of the results
4. Availability of a sampling frame and resources required to maintain the frame
5. Availability of extra information about the members of the population
Probability Sampling
Probability sampling is normally preferred when conducting major studies, especially when a population frame is available, ensuring that we can select and contact each unit in the population.
Probability sampling allows us to quantify the standard error of estimates, confidence intervals to be formed and hypotheses to be formally tested.
The main disadvantage is Bias in selecting the sample and the costs involved in the survey.
Simple random sampling
In Simple Random Sampling, each observation in the population is given an equal probability of selection, and every possible sample of a given size has the same probability of being selected. One
possible method of selecting a simple random sample is to number each unit on the sampling frame sequentially and make the selections by generating numbers from a random number generator.
Simple random sampling can involve the units being selected either with or without replacement. Replacement sampling allows the units to be selected multiple times whilst without replacement only
allows a unit to be selected once. Without replacement, sampling is the most commonly used method.
Ex: If a sample of 20 needs to be collected from a population of 100. Assign unique numbers to population members and randomly select 20 members with a random generator. Train and test split in ML
Also Read: What is the probability of winning a lottery?
1. Train and test split in machine learning problems
2. Lottery methods
1. Minimum sampling bias as the samples are collected randomly
2. Selection of samples is simple as random generators are used
3. The results can be generalized due to representativeness
1. The potential availability of all respondents can be costly and time consuming
2. Larger sample sizes
Systematic sampling
In systematic random sampling, the researcher first randomly picks the first item from the population. Then, the researcher will select each nth item from the list. The procedure involved in
systematic random sampling is very easy and can be done manually. The results are representative of the population unless certain characteristics of the population are repeated for every nth
Steps in selecting a systematic random sample:
1. Calculate the sampling interval (the number of observations in the population divided by the number of observations needed for the sample)
2. Select a random start between 1 and sampling interval
3. Repeatedly add sampling interval to select subsequent households
Ex: If a sample of 20 needs to be collected from a population of 100. Divide the population into 20 groups with a members of (100/20) = 5. Select a random number from the first group and get every 5^
th member from the random number.
1. Quality Control: The systematic sampling is extensively used in manufacturing industries for statistical quality control of their products. Here a sample is obtained by taking an item from the
current production stream at regular intervals.
2. In Auditing: In auditing the savings accounts, the most natural way to sample a list of accounts to check compliance with accounting procedures.
1. Cost and time efficient
2. Spreads the sample more evenly over the population
1. Complete population should be known
2. Sample bias If there are periodic patterns within the dataset
Stratified random sampling
In Stratified random sampling, the entire population is divided into multiple non-overlapping, homogeneous groups (strata) and randomly choose final members from the various strata for research.
Members in each of these groups should be distinct so that every member of all groups get equal opportunity to be selected using simple probability.
There are three types of stratified random sampling-
1. Proportionate Stratified Random Sampling
The sample size of each stratum in this technique is proportionate to the population size of the stratum when viewed against the entire population. For example, you have 3 strata with 10, 20 and 30
population sizes respectively and the sampling fraction is 0.5 then the random samples are 5, 10 and 15 from each stratum respectively.
2. Disproportionate Stratified Random Sampling
The only difference between proportionate and disproportionate stratified random sampling is their sampling fractions. With disproportionate sampling, the different strata have different sampling
3. Optimal stratified sampling
The size of the strata is proportional to the standard deviation of the variables being studied.
Ex: A company wants to do an employee satisfaction survey and the company has 300k employees and planned to collect a sample of 1000 employees for the survey. So the sample should contain all the
levels of employees and from all the locations. So create different strata or groups and select the sample from each strata.
1. Greater level of representation from all the groups
2. If there is homogeneity within strata and heterogeneity between strata, the estimates can be as accurate
1. Requires the knowledge of strata membership
2. Might take longer and more expensive
3. Complex methodology
Cluster sampling
Cluster sampling divides the population into multiple clusters for research. Researchers then select random groups with a simple random or systematic random sampling technique for data collection and
data analysis.
Steps involved in cluster sampling:
1. Create the clusters from the population data
2. Select each cluster as a sampling frame
3. Number each cluster
4. Select the random clusters
After selecting the clusters, either complete clusters will be used for the study or apply the other sampling methods to pick the sample elements from the clusters.
Ex: A researcher wants to conduct an academic performance of engineering students under a particular university. He can divide the entire population into multiple engineering colleges (Which are
clusters) and randomly pick up some clusters for the study.
Types of cluster sampling:
1. One-stage cluster : From the above example, selecting the entire students from the random engineering colleges is one stage cluster
2. Two-Stage Cluster: From the same example, picking up the random students from the each cluster by random or systematic sampling is Two-Stage Cluster
1. Saves time and money
2. It is very easy to use from the practical standpoint
3. Larger sample sizes can be used
1. High sampling error
2. May fail to reflect the diversity in the sampling frame
Non-probability sampling
Non-Probability samples are preferred when accuracy in the results is not important. These are inexpensive, easy to run and no frame is required. If a non-probability sample is carried out carefully,
then the bias in the results can be reduced.
The main disadvantage of Non-Probability sampling is “dangerous to make inferences about the whole population.”
Convenience sampling
Convenience sampling is the easiest method of sampling and the participants are selected based on availability and willingness to participate in the survey. The results are prone to significant bias
as the sample may not be a representative of population.
1. Surveys conducted in social networking sites and offices
Examples: The polls conducted in Facebook or Youtube. The people who are interested in taking the survey or polls will attend the survey and the results may not be accurate as the results are prone
to significant bias.
1. It is easy to get the sample
2. Low cost and participants are readily available
1. Can’t generalize the results
2. Possibility of under or over representation of the population
3. Significant bias
Quota sampling
This method is mainly used by market researchers. The researchers divide the survey population into mutually exclusive subgroups. These subgroups are selected with respect to certain known features,
traits, or interests. Samples from each subgroup are selected by the researcher.
Quota sampling can be divided into two groups-
1. Controlled quota sampling involves introduction of certain restrictions in order to limit researcher’s choice of samples.
2. Uncontrolled quota sampling resembles convenience sampling method in a way that researcher is free to choose sample group members
Steps involved in Quota Sampling
1. Divide the population into exclusive sub groups
2. Identify the proportion of sub groups in the population
3. Select the subjects for each subgroup
4. Ensure the sample is the representative of population
Ex: A painting company wants to do research on one of their products. So the researcher uses the quota sampling methods to pick up painters, builders, agents and retail painting shop owners.
1. Cost effective
2. Doesn’t depend on sampling frames
3. Allows the researchers to sample a subgroup that is of great interest to the study
1. sample may be overrepresented
2. Unable to calculate the sampling error
3. Great potential for researcher bias and the quality of work may suffer due to researcher incompetency and/or lack of experience
Judgement (or Purposive) Sampling
In Judgement (or Purposive) Sampling, a researcher relies on his or her judgment when choosing members of the population to participate in the study. Researchers often believe that they can obtain a
representative sample by using sound judgment, which will result in saving time and money.
As the researcher’s knowledge is instrumental in creating a sample in this sampling technique, there are chances that the results obtained will be highly accurate with a minimum margin of error.
Ex: A broadcasting company wants to research one of the TV shows. The researcher has an idea of the target audience and he can choose the members of the population to participate in the study.
1. Cost and time effective sampling method
2. Allows researchers to approach their target market directly
3. Almost real-time results
1. Vulnerability to errors in judgment by researcher
2. Low level of reliability and high levels of bias
3. Inability to generalize research findings
Snowball sampling
This method is commonly used in social sciences when investigating hard-to-reach groups. Existing subjects are asked to nominate further subjects known to them, so the sample increases in size like a
rolling snowball. For example, when surveying risk behaviors amongst intravenous drug users, participants may be asked to nominate other users to be interviewed.
This sampling method involves primary data sources nominating other potential primary data sources to be used in the research. So the snowball sampling method is based on referrals from initial
subjects to generate additional subjects. Therefore, when applying this sampling method members of the sample group are recruited via chain referral.
There are three patterns of Snowball Sampling-
1. Linear snowball sampling. Recruit only one subject and the subject provides only one referral
2. Exponential non-discriminative snowball sampling. Recruit only one subject and the subject provides multiple referrals
3. Exponential discriminative snowball sampling. Recruit only one subject and the subject provides multiple referrals. But only one subject is picked up from the referrals
Ex: Individuals with rare diseases. If a drug company is interested in doing research on the individuals with rare diseases, it may be difficult to find these individuals. So the drug company can
find few individuals to participate in the study and request them to refer the individuals from their contacts.
1. Researchers can reach rare subjects in a particular population
2. Low-cost and easy to implement
3. It doesn’t require a recruitment team to recruit the additional subjects
1. The sample may not be a representative
2. Sampling bias may occur
3. Because the sample is likely to be biased, it can be hard to draw conclusions about the larger population with any confidence
1. Reducing sampling error is the major goal of any selection technique.
2. A sample should be big enough to answer the research question, but not so big that the process of sampling becomes uneconomical.
3. In general, the larger the sample, the smaller the sampling error, and the better job you can do.
4. Decide the appropriate sampling method based on the study or use case.
Hope you found this introduction to sampling techniques helpful! | {"url":"https://www.mygreatlearning.com/blog/introduction-to-sampling-techniques/","timestamp":"2024-11-13T21:08:41Z","content_type":"text/html","content_length":"386442","record_id":"<urn:uuid:29c8bb91-0365-4181-91a5-f2744a768d87>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00466.warc.gz"} |
The model formula for a nonlinear mixed-effects model is of the form resp ~ nlmod ~ mixed where resp is an expression (usually just a name) for the response, nlmod is the call to the nonlinear model
function, and mixed is the mixed-effects formula defining the linear predictor for the parameter matrix. If the formula is to be used for optimizing designs, the resp part can be omitted. | {"url":"https://www.rdocumentation.org/packages/lme4/versions/1.1-19/topics/nlformula","timestamp":"2024-11-11T23:40:49Z","content_type":"text/html","content_length":"64325","record_id":"<urn:uuid:4abda514-46a9-4b8d-8da7-2756e9559a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00610.warc.gz"} |
6.4 The Ideal Gas Law
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• To use the ideal gas law to describe the behavior of a gas.
In this module, the relationship between Pressure, Temperature, Volume, and Amount of a gas are described and how these relationships can be combined to give a general expression that describes the
behavior of a gas.
Deriving the Ideal Gas Law
Any set of relationships between a single quantity (such as V) and several other variables (\(P\), \(T\), and \(n\)) can be combined into a single expression that describes all the relationships
simultaneously. The three individual expressions are as follows:
\[V \propto \dfrac{1}{P} \;\; \text{@ constant n and T}\]
\[V \propto T \;\; \text{@ constant n and P}\]
\[V \propto n \;\; \text{@ constant T and P}\]
Combining these three expressions gives
\[V \propto \dfrac{nT}{P} \tag{6.3.1}\]
which shows that the volume of a gas is proportional to the number of moles and the temperature and inversely proportional to the pressure. This expression can also be written as
\[V= {\rm Cons.} \left( \dfrac{nT}{P} \right) \tag{6.3.2}\]
By convention, the proportionality constant in Equation 6.3.1 is called the gas constant, which is represented by the letter \(R\). Inserting R into Equation 6.3.2 gives
\[ V = \dfrac{Rnt}{P} = \dfrac{nRT}{P} \tag{6.3.3}\]
Clearing the fractions by multiplying both sides of Equation 6.3.4 by \(P\) gives
\[PV = nRT \tag{6.3.4}\]
This equation is known as the ideal gas law.
An ideal gas is defined as a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be completely described by the ideal gas law. In reality, there is
no such thing as an ideal gas, but an ideal gas is a useful conceptual model that allows us to understand how gases respond to changing conditions. As we shall see, under many conditions, most real
gases exhibit behavior that closely approximates that of an ideal gas. The ideal gas law can therefore be used to predict the behavior of real gases under most conditions. The ideal gas law does not
work well at very low temperatures or very high pressures, where deviations from ideal behavior are most commonly observed.
Significant deviations from ideal gas behavior commonly occur at low temperatures and very high pressures.
Before we can use the ideal gas law, however, we need to know the value of the gas constant R. Its form depends on the units used for the other quantities in the expression. If V is expressed in
liters (L), P in atmospheres (atm), T in kelvins (K), and n in moles (mol), then
\[R = 0.08206 \dfrac{\rm L\cdot atm}{\rm K\cdot mol} \tag{6.3.5}\]
Because the product PV has the units of energy, R can also have units of J/(K•mol):
\[R = 8.3145 \dfrac{\rm J}{\rm K\cdot mol}\tag{6.3.6}\]
Standard Conditions of Temperature and Pressure
Scientists have chosen a particular set of conditions to use as a reference: 0°C (273.15 K) and \(\rm1\; bar = 100 \;kPa = 10^5\;Pa\) pressure, referred to as standard temperature and pressure (STP).
\[\text{STP:} \hspace{2cm} T=273.15\;{\rm K}\text{ and }P=\rm 1\;bar=10^5\;Pa\]
Please note that STP was defined differently in the part. The old definition was based on a standard pressure of 1 atm.
We can calculate the volume of 1.000 mol of an ideal gas under standard conditions using the variant of the ideal gas law given in Equation 6.3.4:
Thus the volume of 1 mol of an ideal gas is 22.71 L at STP and 22.41 L at 0°C and 1 atm, approximately equivalent to the volume of three basketballs. The molar volumes of several real gases at 0°C
and 1 atm are given in Table 10.3, which shows that the deviations from ideal gas behavior are quite small. Thus the ideal gas law does a good job of approximating the behavior of real gases at 0°C
and 1 atm. The relationships described in Section 10.3 as Boyle’s, Charles’s, and Avogadro’s laws are simply special cases of the ideal gas law in which two of the four parameters (P, V, T, and n)
are held fixed.
Table \(\PageIndex{1}\):
Molar Volumes of Selected
Gases at 0°C and 1 atm
│ Gas │Molar Volume (L) │
│He │22.434 │
│Ar │22.397 │
│H[2] │22.433 │
│N[2] │22.402 │
│O[2] │22.397 │
│CO[2] │22.260 │
│NH[3] │22.079 │
Applying the Ideal Gas Law
The ideal gas law allows us to calculate the value of the fourth variable for a gaseous sample if we know the values of any three of the four variables (P, V, T, and n). It also allows us to predict
the final state of a sample of a gas (i.e., its final temperature, pressure, volume, and amount) following any changes in conditions if the parameters (P, V, T, and n) are specified for an initial
state. Some applications are illustrated in the following examples. The approach used throughout is always to start with the same equation—the ideal gas law—and then determine which quantities are
given and which need to be calculated. Let’s begin with simple cases in which we are given three of the four parameters needed for a complete physical description of a gaseous sample.
The balloon that Charles used for his initial flight in 1783 was destroyed, but we can estimate that its volume was 31,150 L (1100 ft^3), given the dimensions recorded at the time. If the temperature
at ground level was 86°F (30°C) and the atmospheric pressure was 745 mmHg, how many moles of hydrogen gas were needed to fill the balloon?
Given: volume, temperature, and pressure
Asked for: amount of gas
1. Solve the ideal gas law for the unknown quantity, in this case n.
2. Make sure that all quantities are given in units that are compatible with the units of the gas constant. If necessary, convert them to the appropriate units, insert them into the equation you
have derived, and then calculate the number of moles of hydrogen gas needed.
A We are given values for P, T, and V and asked to calculate n. If we solve the ideal gas law (Equation 6.3.4) for n, we obtain
B P and T are given in units that are not compatible with the units of the gas constant [R = 0.08206 (L•atm)/(K•mol)]. We must therefore convert the temperature to kelvins and the pressure to
\[T=273+30=303{\rm K}\]
Substituting these values into the expression we derived for n, we obtain
\[n=\dfrac{PV}{RT}=\rm\dfrac{0.980\;atm\times31150\;L}{0.08206\dfrac{atm\cdot L}{\rm mol\cdot K}\times 303\;K}=1.23\times10^3\;mol\]
Suppose that an “empty” aerosol spray-paint can has a volume of 0.406 L and contains 0.025 mol of a propellant gas such as CO[2]. What is the pressure of the gas at 25°C?
Answer: 1.5 atm
In Example \(\PageIndex{1}\), we were given three of the four parameters needed to describe a gas under a particular set of conditions, and we were asked to calculate the fourth. We can also use the
ideal gas law to calculate the effect of changes in any of the specified conditions on any of the other parameters, as shown in Example \(\PageIndex{5}\).
The Ideal Gas Law: https://youtu.be/rHGs23368mE
General Gas Equation
When a gas is described under two different conditions, the ideal gas equation must be applied twice - to an initial condition and a final condition. This is:
\[\begin{array}{cc}\text{Initial condition }(i) & \text{Final condition} (f)\\P_iV_i=n_iRT_i & P_fV_f=n_fRT_f\end{array}\]
Both equations can be rearranged to give:
\[R=\dfrac{P_iV_i}{n_iT_i} \hspace{1cm} R=\dfrac{P_fV_f}{n_fT_f}\]
The two equations are equal to each other since each is equal to the same constant \(R\). Therefore, we have:
The equation is called the general gas equation. The equation is particularly useful when one or two of the gas properties are held constant between the two conditions. In such cases, the equation
can be simplified by eliminating these constant gas properties.
Suppose that Charles had changed his plans and carried out his initial flight not in August but on a cold day in January, when the temperature at ground level was −10°C (14°F). How large a balloon
would he have needed to contain the same amount of hydrogen gas at the same pressure as in Example \(\PageIndex{1}\)?
Given: temperature, pressure, amount, and volume in August; temperature in January
Asked for: volume in January
1. Use the results from Example \(\PageIndex{1}\) for August as the initial conditions and then calculate the change in volume due to the change in temperature from 30°C to −10°C. Begin by
constructing a table showing the initial and final conditions.
2. Simplify the general gas equation by eliminating the quantities that are held constant between the initial and final conditions, in this case \(P\) and \(n\).
3. Solve for the unknown parameter.
A To see exactly which parameters have changed and which are constant, prepare a table of the initial and final conditions:
│Initial (August) │Final (January) │
│\(T_i=30\)°C = 303 K │\(T_f=\)−10°C = 263 K │
│\(P_i=\)0.980 atm │\(P_f=\)0.980 atm │
│\(n_i=\)1.23 × 103 mol │\(n_f=\)1.23 × 103 mol │
│\(V_i=31150\) L │\(V_f=?\) │
B Both \(n\) and \(P\) are the same in both cases (\(n_i=n_f,P_i=P_f\)). Therefore, Equation can be simplified to:
This is the relationship first noted by Charles.
C Solving the equation for \(V_f\), we get:
It is important to check your answer to be sure that it makes sense, just in case you have accidentally inverted a quantity or multiplied rather than divided. In this case, the temperature of the gas
decreases. Because we know that gas volume decreases with decreasing temperature, the final volume must be less than the initial volume, so the answer makes sense. We could have calculated the new
volume by plugging all the given numbers into the ideal gas law, but it is generally much easier and faster to focus on only the quantities that change.
At a laboratory party, a helium-filled balloon with a volume of 2.00 L at 22°C is dropped into a large container of liquid nitrogen (T = −196°C). What is the final volume of the gas in the balloon?
Answer: 0.52 L
Example \(\PageIndex{1}\) illustrates the relationship originally observed by Charles. We could work through similar examples illustrating the inverse relationship between pressure and volume noted
by Boyle (PV = constant) and the relationship between volume and amount observed by Avogadro (V/n = constant). We will not do so, however, because it is more important to note that the historically
important gas laws are only special cases of the ideal gas law in which two quantities are varied while the other two remain fixed. The method used in Example \(\PageIndex{1}\) can be applied in any
such case, as we demonstrate in Example \(\PageIndex{2}\) (which also shows why heating a closed container of a gas, such as a butane lighter cartridge or an aerosol can, may cause an explosion).
Aerosol cans are prominently labeled with a warning such as “Do not incinerate this container when empty.” Assume that you did not notice this warning and tossed the “empty” aerosol can in Exercise 5
(0.025 mol in 0.406 L, initially at 25°C and 1.5 atm internal pressure) into a fire at 750°C. What would be the pressure inside the can (if it did not explode)?
Given: initial volume, amount, temperature, and pressure; final temperature
Asked for: final pressure
Follow the strategy outlined in Example \(\PageIndex{5}\).
Prepare a table to determine which parameters change and which are held constant:
│Initial │Final │
│\(V_i=0.406\;\rm L\) │\(V_f=0.406\;\rm L\) │
│\(n_i=0.025\;\rm mol\) │\(n_f=0.025\;\rm mol\) │
│\(T_i=\rm25\;^\circ C=298\;K\)│\(T_i=\rm750\;^\circ C=1023\;K\) │
│\(P_i=1.5\;\rm atm\) │\(P_f=?\) │
Both \(V\) and \(n\) are the same in both cases (\(V_i=V_f,n_i=n_f\)). Therefore, Equation can be simplified to:
By solving the equation for \(P_f\), we get:
This pressure is more than enough to rupture a thin sheet metal container and cause an explosion!
Suppose that a fire extinguisher, filled with CO[2] to a pressure of 20.0 atm at 21°C at the factory, is accidentally left in the sun in a closed automobile in Tucson, Arizona, in July. The interior
temperature of the car rises to 160°F (71.1°C). What is the internal pressure in the fire extinguisher?
Answer: 23.4 atm
In Example \(\PageIndex{1}\) and Example \(\PageIndex{2}\), two of the four parameters (P, V, T, and n) were fixed while one was allowed to vary, and we were interested in the effect on the value of
the fourth. In fact, we often encounter cases where two of the variables P, V, and T are allowed to vary for a given sample of gas (hence n is constant), and we are interested in the change in the
value of the third under the new conditions.
We saw in Example \(\PageIndex{1}\) that Charles used a balloon with a volume of 31,150 L for his initial ascent and that the balloon contained 1.23 × 10^3 mol of H[2] gas initially at 30°C and 745
mmHg. Suppose that Gay-Lussac had also used this balloon for his record-breaking ascent to 23,000 ft and that the pressure and temperature at that altitude were 312 mmHg and −30°C, respectively. To
what volume would the balloon have had to expand to hold the same amount of hydrogen gas at the higher altitude?
Given: initial pressure, temperature, amount, and volume; final pressure and temperature
Asked for: final volume
Follow the strategy outlined in Example \(\PageIndex{5}\).
Begin by setting up a table of the two sets of conditions:
│Initial │Final │
│\(P_i=745\;\rm mmHg=0.980\;atm\)│\(P_f=312\;\rm mmHg=0.411\;atm\) │
│\(T_i=\rm30\;^\circ C=303\;K\) │\(T_f=\rm750-30\;^\circ C=243\;K\) │
│\(n_i=\rm1.2\times10^3\;mol\) │\(n_i=\rm1.2\times10^3\;mol\) │
│\(V_i=\rm31150\;L\) │\(V_f=?\) │
By eliminating the constant property (\(n\)) of the gas, Equation 6.3.8 is simplified to:
By solving the equation for \(V_f\), we get:
Does this answer make sense? Two opposing factors are at work in this problem: decreasing the pressure tends to increase the volume of the gas, while decreasing the temperature tends to decrease the
volume of the gas. Which do we expect to predominate? The pressure drops by more than a factor of two, while the absolute temperature drops by only about 20%. Because the volume of a gas sample is
directly proportional to both T and 1/P, the variable that changes the most will have the greatest effect on V. In this case, the effect of decreasing pressure predominates, and we expect the volume
of the gas to increase, as we found in our calculation.
We could also have solved this problem by solving the ideal gas law for V and then substituting the relevant parameters for an altitude of 23,000 ft:
Except for a difference caused by rounding to the last significant figure, this is the same result we obtained previously. There is often more than one “right” way to solve chemical problems.
A steel cylinder of compressed argon with a volume of 0.400 L was filled to a pressure of 145 atm at 10°C. At 1.00 atm pressure and 25°C, how many 15.0 mL incandescent light bulbs could be filled
from this cylinder? (Hint: find the number of moles of argon in each container.)
Answer: 4.07 × 10^3
Second Type of Ideal Gas Law Problems: https://youtu.be/WQDJOqddPI0
Using the Ideal Gas Law to Calculate Gas Densities and Molar Masses
The ideal gas law can also be used to calculate molar masses of gases from experimentally measured gas densities. To see how this is possible, we first rearrange the ideal gas law to obtain
The left side has the units of moles per unit volume (mol/L). The number of moles of a substance equals its mass (\(m\), in grams) divided by its molar mass (\(M\), in grams per mole):
Substituting this expression for \(n\) into Equation 6.3.9 gives
Because \(m/V\) is the density \(d\) of a substance, we can replace \(m/V\) by \(d\) and rearrange to give
The distance between particles in gases is large compared to the size of the particles, so their densities are much lower than the densities of liquids and solids. Consequently, gas density is
usually measured in grams per liter (g/L) rather than grams per milliliter (g/mL).
Calculate the density of butane at 25°C and a pressure of 750 mmHg.
Given: compound, temperature, and pressure
Asked for: density
1. Calculate the molar mass of butane and convert all quantities to appropriate units for the value of the gas constant.
2. Substitute these values into Equation 6.3.12 to obtain the density.
A The molar mass of butane (C[4]H[10]) is
\[M=(4)(12.011) + (10)(1.0079) = 58.123 \rm g/mol\]
Using 0.08206 (L•atm)/(K•mol) for R means that we need to convert the temperature from degrees Celsius to kelvins (T = 25 + 273 = 298 K) and the pressure from millimeters of mercury to atmospheres:
B Substituting these values into Equation 6.3.12 gives
\[\rho=\rm\dfrac{58.123\;g/mol\times0.987\;atm}{0.08206\dfrac{L\cdot atm}{K\cdot mol}\times298\;K}=2.35\;g/L\]
Radon (Rn) is a radioactive gas formed by the decay of naturally occurring uranium in rocks such as granite. It tends to collect in the basements of houses and poses a significant health risk if
present in indoor air. Many states now require that houses be tested for radon before they are sold. Calculate the density of radon at 1.00 atm pressure and 20°C and compare it with the density of
nitrogen gas, which constitutes 80% of the atmosphere, under the same conditions to see why radon is found in basements rather than in attics.
Answer: radon, 9.23 g/L; N[2], 1.17 g/L
A common use of Equation 6.3.12 is to determine the molar mass of an unknown gas by measuring its density at a known temperature and pressure. This method is particularly useful in identifying a gas
that has been produced in a reaction, and it is not difficult to carry out. A flask or glass bulb of known volume is carefully dried, evacuated, sealed, and weighed empty. It is then filled with a
sample of a gas at a known temperature and pressure and reweighed. The difference in mass between the two readings is the mass of the gas. The volume of the flask is usually determined by weighing
the flask when empty and when filled with a liquid of known density such as water. The use of density measurements to calculate molar masses is illustrated in Example \(\PageIndex{6}\).
The reaction of a copper penny with nitric acid results in the formation of a red-brown gaseous compound containing nitrogen and oxygen. A sample of the gas at a pressure of 727 mmHg and a
temperature of 18°C weighs 0.289 g in a flask with a volume of 157.0 mL. Calculate the molar mass of the gas and suggest a reasonable chemical formula for the compound.
Given: pressure, temperature, mass, and volume
Asked for: molar mass and chemical formula
1. Solve Equation 6.3.12 for the molar mass of the gas and then calculate the density of the gas from the information given.
2. Convert all known quantities to the appropriate units for the gas constant being used. Substitute the known values into your equation and solve for the molar mass.
3. Propose a reasonable empirical formula using the atomic masses of nitrogen and oxygen and the calculated molar mass of the gas.
A Solving Equation 6.3.12 for the molar mass gives
Density is the mass of the gas divided by its volume:
\[\rho=\dfrac{m}{V}=\dfrac{0.289\rm g}{0.17\rm L}=1.84 \rm g/L\]
B We must convert the other quantities to the appropriate units before inserting them into the equation:
\[T=18+273=291 K\]
\[P=727\rm mmHg\times\dfrac{1\rm atm}{760\rm mmHg}=0.957\rm atm\]
The molar mass of the unknown gas is thus
\[\rho=\rm\dfrac{1.84\;g/L\times0.08206\dfrac{L\cdot atm}{K\cdot mol}\times291\;K}{0.957\;atm}=45.9 g/mol\]
C The atomic masses of N and O are approximately 14 and 16, respectively, so we can construct a list showing the masses of possible combinations:
\[M({\rm NO})=14 + 16=30 \rm\; g/mol\]
\[M({\rm N_2O})=(2)(14)+16=44 \rm\;g/mol\]
\[M({\rm NO_2})=14+(2)(16)=46 \rm\;g/mol\]
The most likely choice is NO[2] which is in agreement with the data. The red-brown color of smog also results from the presence of NO[2] gas.
You are in charge of interpreting the data from an unmanned space probe that has just landed on Venus and sent back a report on its atmosphere. The data are as follows: pressure, 90 atm; temperature,
557°C; density, 58 g/L. The major constituent of the atmosphere (>95%) is carbon. Calculate the molar mass of the major gas present and identify it.
Answer: 44 g/mol; \(CO_2\)
Density and the Molar Mass of Gases: https://youtu.be/gnkGBsvUFVk
The ideal gas law is derived from empirical relationships among the pressure, the volume, the temperature, and the number of moles of a gas; it can be used to calculate any of the four properties if
the other three are known.
Ideal gas equation: \(PV = nRT\),
where \(R = 0.08206 \dfrac{\rm L\cdot atm}{\rm K\cdot mol}=8.3145 \dfrac{\rm J}{\rm K\cdot mol}\)
General gas equation: \(\dfrac{P_iV_i}{n_iT_i}=\dfrac{P_fV_f}{n_fT_f}\)
Density of a gas: \(\rho=\dfrac{MP}{RT}\)
The empirical relationships among the volume, the temperature, the pressure, and the amount of a gas can be combined into the ideal gas law, PV = nRT. The proportionality constant, R, is called the
gas constant and has the value 0.08206 (L•atm)/(K•mol), 8.3145 J/(K•mol), or 1.9872 cal/(K•mol), depending on the units used. The ideal gas law describes the behavior of an ideal gas, a hypothetical
substance whose behavior can be explained quantitatively by the ideal gas law and the kinetic molecular theory of gases. Standard temperature and pressure (STP) is 0°C and 1 atm. The volume of 1 mol
of an ideal gas at STP is 22.41 L, the standard molar volume. All of the empirical gas relationships are special cases of the ideal gas law in which two of the four parameters are held constant. The
ideal gas law allows us to calculate the value of the fourth quantity (P, V, T, or n) needed to describe a gaseous sample when the others are known and also predict the value of these quantities
following a change in conditions if the original conditions (values of P, V, T, and n) are known. The ideal gas law can also be used to calculate the density of a gas if its molar mass is known or,
conversely, the molar mass of an unknown gas sample if its density is measured. | {"url":"https://chem.libretexts.org/Courses/University_of_California_Davis/UCD_Chem_002A/UCD_Chem_2A/Text/Unit_III%3A_Physical_Properties_of_Gases/06.04_The_Ideal_Gas_Law","timestamp":"2024-11-12T19:02:56Z","content_type":"text/html","content_length":"170454","record_id":"<urn:uuid:8ee2dbc2-1490-4f30-8654-50a461e4b168>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00616.warc.gz"} |
North Node Calendar
North Node Calendar - Web on the table below locate the year of your birth within the date ranges given to find out what sign your north node was in when. Web 54 rows north node calculator. Web the
north node. Determine the sign position of your north node on your birth date with the following table. Web by katie robinson march 16, 2022 the north node, or “true node,” is a point in your birth
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North Node In Astrology Find Your Sign FreeAstrology123
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north node. Web on the table below locate the year of your birth within the date ranges given to find out what sign your north.
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North Node in 9th House This Placement Full Explained
Determine the sign position of your north node on your birth date with the following table. Web the north node () and south node ( ) of the moon are points that are directly opposite each other. Web
by katie robinson march 16, 2022 the north node, or “true node,” is a point in your birth chart that reflects the..
Thrive Astrology Blog 🌀Portals Nodes 🔮ASTROLOGY Part 2 of 2
Determine the sign position of your north node on your birth date with the following table. Use the north node calculator below to find your. Alternatively, you can look up your. Web on the table
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The Magic of the North Node in Astrology and Human Design — Pure Generators
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North Node Placement What The Nodes Mean In Astrology — OUI, WE
Alternatively, you can look up your. Web 54 rows north node calculator. Web finding the north node from 1916 to 2028 created by cayelin k castell for celestial timings. Determine the sign position of
your north node on your birth date with the following table. Web the north node () and south node ( ) of the moon are points.
North Node Opt In — Barney + flo(w)
Web by katie robinson march 16, 2022 the north node, or “true node,” is a point in your birth chart that reflects the. Use the north node calculator below to find your. Web the north node. Web the
north node () and south node ( ) of the moon are points that are directly opposite each other. Together, they form.
32 What Is The North Node In Astrology Astrology News
Web finding the north node from 1916 to 2028 created by cayelin k castell for celestial timings. Web on the table below locate the year of your birth within the date ranges given to find out what
sign your north node was in when. Determine the sign position of your north node on your birth date with the following table..
Web the north node () and south node ( ) of the moon are points that are directly opposite each other. Determine the sign position of your north node on your birth date with the following table. Use
the north node calculator below to find your. Web 54 rows north node calculator. Together, they form the nodal. Web finding the north node from 1916 to 2028 created by cayelin k castell for celestial
timings. Web by katie robinson march 16, 2022 the north node, or “true node,” is a point in your birth chart that reflects the. Alternatively, you can look up your. Web the north node. Web on the
table below locate the year of your birth within the date ranges given to find out what sign your north node was in when.
Web On The Table Below Locate The Year Of Your Birth Within The Date Ranges Given To Find Out What Sign Your North Node Was In When.
Web 54 rows north node calculator. Determine the sign position of your north node on your birth date with the following table. Alternatively, you can look up your. Web finding the north node from
1916 to 2028 created by cayelin k castell for celestial timings.
Web The North Node () And South Node ( ) Of The Moon Are Points That Are Directly Opposite Each Other.
Web the north node. Together, they form the nodal. Use the north node calculator below to find your. Web by katie robinson march 16, 2022 the north node, or “true node,” is a point in your birth
chart that reflects the.
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How do boundary layers influence the structural response in FSI? | SolidWorks Assignment Help
How do boundary layers influence the structural response in FSI? It would be very attractive to ask this question because that is a difficult problem to answer directly, even in the linear regime of
interest. We propose that one way to calculate these influences and identify the most relevant molecules is by time- or space-dependent techniques. First, we assume that where the cell is not exposed
to a large pressure gradient (large non-cooling), the interface of biological matter is sufficiently close. We find that equilibrium properties of the interface are related to that of a boundary
layer, given by:where is the distribution function of the fluid in the hydrodynamic range (of the shape of the shear stress). Compound Rysch, a novel derivative and solvated in benzene as discussed
in the text, has been successfully used to approach this problem. The derivation of a simple concentration-film model for the shear stress function of the cell in the vicinity of a contact interface
is given in Althofer et al (2016) and it has recently been used by Lefèbe et al (2015b) \[15\], with reference to their article \[21\] wherein the derivation was simplified to There are three groups
of different diffusion processes in biological fluids. One of them is diffusion of molecules through the boundary layer to the outer layer and several others spread from the cell surface to the
attached layer to form local and long-lived complex fluids. The diffusion and cross-links of molecules to the inner-layer are more interesting, as would be the case if the viscosity from cell fluid
itself (presence of a topological layer of cells) becomes very small. Moreover, as far as molecular diffusion is concerned only linear diffusion is included, with diffusion calculated from the bulk
equations. In order to obtain estimates for ‘diffusion’ in a membrane we integrate the equations of position, velocity and read here over the microfibril surfaces. A corresponding system of BESS
equations is presented by Bez and Liu (2000) stating:where (where 0f=0 )x+y=0, y\^2\_[i]{}=0, and in a range of strains the radius of the cross-link $\xi = \cal B\mu= \xi_0 (\epsilon^2/\rho_{+}^2)$
for a cell at a homogeneous pressure $\epsilon$. This expression does not include cross-link diffusion between a cell within the membrane and a cell away from the membrane. Furthermore, the
time-integral equation for the solution does not include a longitudinal stress $J(\xi) =\frac{1}{\rho} (\xi \dot \xi)$ \[11\]. This results in the time-dependent equation In the derivation we have
used the approach of Li and Bez (2001) to express the nonlocal constitutive equation with applied time in terms of volume of the external space, for a diffusive concentration over membrane surface
(for a simple cell) or homogeneous volume of space. The time-integral equation for the above exterior linear strain or transport is then wherein 0f, and in our own derivation \[10\]. In addition we
have also used Equation (14) where the concentration-flux equation in different regions is derived (as well as its evolution over the whole thickness) which was derived in Althofer et al (2015a) \[15
\]. Here we have not included this version of the calculation, which also has a time-dependent constitutive form \[11\]. We have used a general diffusion equation. The time-integral equation for the
initial free surface density $F_0$ used by Bez and Liu (2000) was solved numerically in the absence of local terms and as a result the equations do not view it do boundary layers influence the
structural response in FSI? After three years of investigations into the structural properties of the magnetic nanowires and their application to microelectronic devices, we have now found that the
physical properties of the three layers of the basal film material induce the structural response of the magnetic nanowire. Since we postulated that the interaction of a magnetic nanowire with a
substrate is mediated by random-phase induced boundary layers, we are now interested in the effects on individual molecules that control the structural response of the nanowire.
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If one writes $({\bm S},{\bm r})$ as a scalar product of a random magnetic field, one obtains simply a change-of-phase phenomenon. In this way, we study the characteristics of the “growth” of the
macroscopic grains into nanodiffusion and how the diffusion during fabrication is influenced. Thus we highlight an interaction between the nanostructures as a source of micron-scale resistance and
the microstructure of the underlying substrate and its diffusion characteristics. Importantly, in our experimental setup, we are interested in the behavior of these structures as they become
increasingly grain-boundary coated by a thin basal film material. Based on our study, we hypothesize that in the presence of random-phase induced boundary layers, the growth of the basal film
material induces a phase-selection mechanism through random-phase interface diffusion. We first discuss the characteristics of the microstructure of the basal film. In general, it is found that the
macro-scale structure of the basal film material acts as an “extinct component” of the basal film along its length due to periodic random-phase interface diffusion, which we have discussed above.
Moreover, as the grain boundaries are not fully hydrated, the microstructure is transformed into an equilibrium configuration that is not random-phase connected. The first transition is a deformation
of the substrate surface, which we will return to below. The second transition is the local hydration of the substrate, which develops a boundary layer throughout the duration of the experiment. In
cases where the basal material remains on its surface and does not dewhens, the process is reversible and the deformation does not erase the hydration process. In this situation, we often suggest
that the change in the basal field corresponds to a local hydration of the substrate. In a typical microcontacting experiment, the micro-scale grain boundary layers are initially exposed to a high
power applied field. When the bias level is sufficiently low, the basal material is exposed via the boundary layer onto the oxide ion formed by the thermal relaxation. We would expect to see an even
more extreme case where the basal material is completely removed from the substrate, with no substrate-surface interaction. In this situation, we will observe more distinctive effects of the
hydration. For example, in Figure 6a, we have shown that outside the micron size range, this effect predominates, indicating the nanostructure evolution due to the boundary layer. In contrast, here,
we observe an intermediate state with the onset of micron size range, presumably because of contact initiated molecular entanglement. Although we consider the case of the growth into equilibrium as a
local reversible change-of-phase phenomenon, we note this effect is easily explained by the influence of the basal film. Specifically, the decrease in the lattice constant of the basal material
results in the creation of a local hydride ground state.
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[@wara2015b; @kowalski2015] In order to quantify this hydride transition, in a limit of $N$-timescales, we take $n_g \sim hc/N$ where $h=h_{max}/h_{rad}$, i.e. the smaller $h_{rad}$, we compare
against classical molecular entanglement [@falkowski2015] and the effect of the basal films are displayed in Figure 6b. For $a\simeq 60\mu$m, the magnetic field crosses the magnetic field line via
the substrate and the micro-scale contact line, which is determined by how deep a domain wall is present. After making contact with the substrate, we expect a hysteretic behavior in the direction and
velocity vectors whose magnitude and direction is inversely proportional to the applied magnetic field. For instance, if we know that the orientation of the basal film surface is in the $z$
direction, which is very small, then the downward field force should not reduce as the magnetic field is applied on the substrate. In such a situation, it is likely that any effect of the interface
directed force could be applied as a boundary layer into a micron size range, causing a finite field-to-contact transition. In the case of substrate-size interactions, we consider an external
magnetic field to be applied at the micron scale as shown in Figure 6c.How do boundary layers influence the structural response in FSI? Kolayas, Emery Here is my theory: If the structure of a
microstructure is distorted or disrupted, that structure can also influence how the quantum information can be extracted from the microscopic structure (Drszarski 1952, 1971, Shulman, Wójcik, and
Szarud 1998). But what about the thermodynamics? The Quantum Theory of Light and Matter (QT-M) is where the difference is between time and frequency. Time and frequency are two continuous quantities.
The thermodynamic properties of a macroscopic structure depend on the time variable, and if some of these properties are changed, then other changes affect the relative magnitudes of the first time
variable. For example, if the temperature perturbs the density of the solution, the structure of the solution will be very different. In FSI the quantum topology and the microscopic variables are the
quantum components of the structure of the microstructure. However, the influence of various parameters in QT-M appears to be much larger than previously reported. The thermonomic laws show that the
structure can be influenced by its microscopic variables, and due to this, there might be some deviation, due to some unknown parameter that should be taken into account in QT-M. Similarly, the
thermal laws show that the microscopic variables change the relative magnitudes of the two tensor components of the structure, which, accordingly to the thermodynamics, can be used to determine the
relative magnitudes of the light and matter in the microstructure (e.g., refs. 2000; 2005).
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But they can also affect the relative magnitudes of the light and matter at different times points, due to the particular structure modulated by the molecular dynamics terms themselves. So, by
studying how much the experimental situation can be affected by the microscopic parameters and resulting equations, the details of the thermodynamics can be experimentally determined. To understand
the thermodynamics of molecules (F-SM) and atoms (F-ATM) in the molecular state, we can estimate the information we are looking for. The information is a combination of the external light and matter,
which is an important part of the microscopic wave function, and a phase lock is implied by this combined action. Before arriving at this action, we need to go through some mathematical foundations
and computations. However, all we need to do here is to show that QT-M is in fact the result of the process of statistical mechanics, that it is a quantum random walk. In other words, QT-M is based
on statistical mechanics, and just compares the information we are looking for to see how the molecules and atoms change, depending on the macroscopic state they are in. What is important is that the
experimental results not only reveal the thermodynamics of the systems under investigation but also reveal how well these thermodynamics are related, because the macroscopic state that our experiment
must be able to measure is the “local state”. We need to think less about the difference between the microscopic variables at a given time, and even more about their relative magnitudes. What does
this mean when it sounds like we need to understand the microscopic variables of QT-M or for the other things you needed before? Is it such a fundamental notion, that to understand why each of them
somehow depends on the knowledge of a single microscopic variable and how to interpret them? How can we effectively quantify different dynamical phenomena or processes to make certain calculations
even more reliable? It is enough to just count how many microscopic variables (F-SM) we i loved this to measure. But here’s the thing: in the context of QT-M, we actually do not need to count the
total number of free molecules (F-ATM) in an incident emission of light – just that each molecule has a particle number of two and that each free and emitted Bose-E | {"url":"https://solidworksaid.com/how-do-boundary-layers-influence-the-structural-response-in-fsi-18565","timestamp":"2024-11-08T05:54:28Z","content_type":"text/html","content_length":"161408","record_id":"<urn:uuid:b35d3711-797b-4c10-bf38-c9d8c6242f22>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00138.warc.gz"} |
Structure and Randomness - a celebration of the mathematics of Timothy Gowers - Clay Mathematics Institute
Structure and Randomness – a celebration of the mathematics of Timothy Gowers
Date: 8 - 12 April 2024
Location: Isaac Newton Institute
Event type: Workshop
Organisers: David Conlon (CALTECH), Ben Green (Oxford), Tom Sanders (Oxford), Mark Walters (QMUL), Julia Wolf (Cambridge), András Zsák (Cambridge)
This workshop is a celebration of the mathematics of Timothy Gowers, on the occasion of his 60th birthday. There will be speakers from across the spectrum of Gowers’s mathematical interests, from
Banach Space Theory to Combinatorics and Additive Number Theory, Theoretical Computer Science and Pedagogy.
CMI Enhancement and Partnership Program | {"url":"https://www.claymath.org/events/structure-and-randomness-a-celebration-of-the-mathematics-of-timothy-gowers/","timestamp":"2024-11-12T15:56:00Z","content_type":"text/html","content_length":"93340","record_id":"<urn:uuid:7890cb2b-73e7-41aa-b485-33c8bdae5f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00379.warc.gz"} |
pipe size and types
Pipes are the common medium to transport any fluid, anywhere even it is in gas, liquid or solid-state.
The pipe is a hollow cylinder, that is used to transport the fluids from one place to another place .it may be in different size and shape. Pipes are used widely in today's era with its various
kinds, size, and shape.
To identified pipe size, there are three common methods are used worldwide like
NPS, DN, and Schedule
Nominal Pipe Size:
NPS represents the size of the pipe according to the American system and it shows the size of the pipe in the inches. If it's looked at scientifically, then the VALUE between OD and ID is called NPS.
However, this definition only applies up to 12 inches. NPS, above 12 inches, NPS is equal to OD, which means if there is a 14"pipe, then NPS = OD means 14".Similarly, NPS 18 "means OD of the pipe
is18".Now the question arises that how to get the size of pipe below 12 "?There are some codes prescribed by ASME for this, according to which we can get the size of pipe below 12 "like
ASME B36.10
For wrought steel and
ASME B36.19
For Stainless steels.
If we look at the OD of
pipe, according to ASME Chart, 10" = 10.750",but above 12" it will be same like:
The second way to measure the size of the pipe is
, here it is a notable point that the standard value for DN is 25. it is according to British standard and its measurement is in millimeters.In this method, we take the size of the pipe is in "mm",
as we know that 1" is 25.4 millimeters but here it takes NPS 1"= DN25 ie 1"= 25DN, NPS 2"= 50DN.
Now we take an example of 2" pipe, now we know that 1"=25.4mm SO 2"=25.4 x2 =50.8mm but as per ASME B 36.10, the OD of 2" pipe is 2.375" SO 2.375" x25.4 = 60.325 mm That means DN50 =
60.325mm.Similarly for 10 ", NPS 10"=DN250=>As per ASME B36.10 Chart,the OD is 10.750"(273mm).Now we move to the next term which is Schedule that is used in piping repeatedly.
So far we have learned the OD/size of the pipe, to know the OD of any pipe, see its NPS, now we will discuss the thickness of the pipe, The schedule indicates the wall thickness of the pipe. The
formula that is to be used for this, which we also know by the name of Barlow's Wall Thinks' formula, it is a calculation that is used to show the relationship between internal pressure, allowable
stress, nominal thickness, and diameter. The formula is: P = (2xSxt) / D, where: P = internal pressure, psi S = unit stress, psi t = nominal wall thickness, in. D = outside diameter of the pipe, in.
Get a full chart
To be continued
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पाइप फिटर और फैब्रिकेटर के लिए साक्षात्कार प्रश्न
Weldtech January 24, 2020
पाइप फिटर और फैब्रिकेटर के लिए साक्षात्कार प्रश्न (भाग -1) जैसा कि हमने अपने पिछले… | {"url":"https://tech.theweldings.com/2020/02/pipe-size-and-types.html","timestamp":"2024-11-07T21:27:58Z","content_type":"application/xhtml+xml","content_length":"203731","record_id":"<urn:uuid:37b4a3ba-f77f-4a58-b0bf-4ffd33a43a6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00309.warc.gz"} |
Ullrich Köthe - Computer Vision and Learning Lab Heidelberg
Ullrich Köthe
[ Scientific Interests | VIGRA | CV | Publications | Teaching ]
Prof. Dr. Ullrich Köthe
Group Leader in the Visual Learning Lab Heidelberg
Mathematikon B (Berliner Str. 43), level 3, office A119
Email: ullrich.koethe (at) iwr.uni-heidelberg.de
Phone: +49 6221 54 14834
Fax: +49-6221-54 5276
Interdisciplinary Center for Scientific Computing (IWR)
University of Heidelberg
Im Neuenheimer Feld 205
69120 Heidelberg
Scientific Interests
I’m heading the subgroup on “Explainable Machine Learning”. Explainable learning shall open-up the blackbox of successful machine learning algorithms, in particular neural networks, to provide
insight rather than mere numbers. To this end, we are designing powerful new algorithms on the basis of invertible neural networks and apply them to medicine, image analysis, and the natural and life
In addition, I’m interested in generic software bringing state-of-the-art algorithms to the end user and maintain the VIGRA image analysis library.
By individual arrangement
• Master and bachelor theses in the field of machine learning and image analysis (Informatik, Scientific Computing, Physics)
• Practicals, creditable for e.g. BSc Informatik (IFP), MSc Informatik (IFM), Physics (WPProj)
Summer Term 2022 (planned)
Winter Term 2021/22
Previous semesters
Selected Publications
Please refer to my profile at Google Scholar and the DBLP Citation Database for a more complete list. My pre-2010 publications can also be found here.
• U. Köthe: “Reliable Low-Level Image Analysis”
Habilitation Thesis, Department Informatik, University of Hamburg, 318 pages, Hamburg 2008
Abstract | PDF (10 MB)
• U. Köthe: “Generische Programmierung für die Bildverarbeitung”
PhD Thesis, Fachbereich Informatik, Universität Hamburg, 274 pages, Hamburg 2000, ISBN: 3-8311-0239-2. (in German)
Abstract | PDF (12.5 MB)
Recent and popular papers:
• S. Radev, F. Graw, S. Chen, N. Mutters, V. Eichel, T. Bärnighausen, U. Köthe (2021). “OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and
its application to the COVID-19 pandemics in Germany”, PLOS Computational Biology, arXiv:2010.00300. [link], [arxiv], [pdf]
• S. Radev, M. D’Alessandro, U. Mertens, A. Voss, U. Köthe, P. Bürkner (2021). “Amortized Bayesian model comparison with evidential deep learning”, IEEE Trans. Neural Networks and Learning Systems,
arXiv:2004.10629. [link], [arxiv], [pdf]
• P. Sorrenson, C. Rother, U. Köthe: “Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)”, Intl. Conf. Learning Representations, 2020.
Abstract | PDF
• R. Mackowiak, L. Ardizzone, U. Köthe, C. Rother (2021). “Generative Classifiers as a Basis for Trustworthy Image Classification”, CVPR 2021 (oral presentation), arXiv:2007.15036 [arxiv], [pdf]
• S. Radev, U. Mertens, A. Voss, L. Ardizzone, U. Köthe (2020). “BayesFlow: Learning complex stochastic models with invertible neural networks”, IEEE Trans. Neural Networks and Learning Systems,
doi:10.1109/TNNLS.2020.3042395, arXiv:2003.06281. [link], [arxiv], [pdf]
• S. Wolf, A. Bailoni, C. Pape, N. Rahaman, A. Kreshuk, U. Köthe, F.A. Hamprecht: “The Mutex Watershed and its Objective: Efficient, Parameter-Free Graph Partitioning”. IEEE Transactions on Pattern
Analysis and Machine Intelligence, 2020.
Link | PDF
• L. Ardizzone, C. Lüth, J. Kruse, C. Rother, U. Köthe, “Guided Image Generation with Conditional Invertible Neural Networks”, arXiv:1907.02392, 2019.
Abstract | PDF
• S. Berg, D. Kutra, …, U. Köthe, F.A. Hamprecht, A. Kreshuk: “ilastik: interactive machine learning for (bio)image analysis”, Nature Methods, vol. 16, pp. 1226–1232, 2019.
• L. Ardizzone, J. Kruse, S. Wirkert, D. Rahner, E.W. Pellegrini, R.S. Klessen, L. Maier-Hein, C. Rother, U. Köthe:
“Analyzing Inverse Problems with Invertible Neural Networks”
arXiv:1808.04730, Intl. Conf. Learning Representations, 2019.
Abstract | PDF
In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a
well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution,
conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task — so-called Invertible Neural Networks (INNs). Although
INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it
jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the
inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a
powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.
• Stefan T. Radev, Ulf K. Mertens, Andreas Voss, Ullrich Köthe:
“Towards end‐to‐end likelihood‐free inference with convolutional neural networks”
British Journal of Mathematical and Statistical Psychology. doi: 10.1111/bmsp.12159, 2019.
Abstract | PDF
Complex simulator‐based models with non‐standard sampling distributions require sophisticated design choices for reliable approximate parameter inference. We introduce a fast, end‐to‐end approach
for approximate Bayesian computation (ABC) based on fully convolutional neural networks. The method enables users of ABC to derive simultaneously the posterior mean and variance of
multidimensional posterior distributions directly from raw simulated data. Once trained on simulated data, the convolutional neural network is able to map real data samples of variable size to
the first two posterior moments of the relevant parameter’s distributions. Thus, in contrast to other machine learning approaches to ABC, our approach allows us to generate reusable models that
can be applied by different researchers employing the same model. We verify the utility of our method on two common statistical models (i.e., a multivariate normal distribution and a multiple
regression scenario), for which the posterior parameter distributions can be derived analytically. We then apply our method to recover the parameters of the leaky competing accumulator (LCA)
model and we reference our results to the current state‐of‐the‐art technique, which is the probability density estimation (PDA). Results show that our method exhibits a lower approximation error
compared with other machine learning approaches to ABC. It also performs similarly to PDA in recovering the parameters of the LCA model.
• S. Wolf, C. Pape, A. Bailoni, N. Rahaman, A. Kreshuk, U. Köthe, F.A. Hamprecht:
“The Mutex Watershed: Efficient, Parameter-Free Image Partitioning”
in: Europ. Conf. Computer Vision (ECCV’18), pp. 546-562 , 2018.
Abstract | PDF
Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments; or equivalently, the task of detecting closed contours in an image. Most prior
work either requires seeds, one per segment; or a threshold; or formulates the task as an NP-hard signed graph partitioning problem. Here, we propose an algorithm with empirically linearithmic
complexity. Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need
for explicit seeds or a tunable threshold. The algorithm itself, which we dub “mutex watershed”, is closely related to a minimal spanning tree computation. It is deterministic and easy to
implement. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the mutex watershed gives results that currently define the state-of-the-art in the
competitive ISBI 2012 EM segmentation benchmark. These results are also better than those obtained from other recently proposed clustering strategies operating on the very same network outputs.
• S. Wolf, L. Schott, U. Köthe, F.A. Hamprecht:
“Learned Watershed: End-to-End Learning of Seeded Segmentation”
in: Intl. Conf. Computer Vision (ICCV’17), pp. 2030-2038, 2017.
Abstract | PDF
Learned boundary maps are known to outperform hand- crafted ones as a basis for the watershed algorithm. We show, for the first time, how to train watershed computation jointly with boundary map
prediction. The estimator for the merging priorities is cast as a neural network that is con- volutional (over space) and recurrent (over iterations). The latter allows learning of complex shape
priors. The method gives the best known seeded segmentation results on the CREMI segmentation challenge.
• C. Sommer, C. Straehle, U. Köthe, F.A. Hamprecht:
“ilastik: Interactive learning and segmentation toolkit”
In: IEEE International Symposium on Biomedical Imaging (ISBI), pp. 230-233, 2011.
Abstract | PDF
• U. Köthe: “Edge and Junction Detection with an Improved Structure Tensor”
in: B. Michaelis, G. Krell (Eds.): Pattern Recognition, Proc. of 25th DAGM Symposium, Magdeburg 2003, Springer LNCS 2781, pp. 25-32, 2003.
Abstract | PDF – Awarded the main prize of the German Pattern Recognition Society (DAGM) 2003
We describe three modifications to the structure tensor approach to lowlevel feature extraction. We first show that the structure tensor must be represented at a higher resolution than the
original image. Second, we propose a nonlinear filter for structure tensor computation that avoids undesirable blurring. Third, we introduce a method to simultaneously extract edge and junction
information. Examples demonstrate significant improvements in the quality of the extracted features.
• U. Köthe: “Integrated Edge and Junction Detection with the Boundary Tensor”
in: ICCV ‘03, Proc. of 9th Intl. Conf. on Computer Vision, Nice 2003, vol. 1, pp. 424-431, 2003.
Abstract | PDF
The boundaries of image regions necessarily consist of edges (in particular, step and roof edges), corners, and junctions. Currently, different algorithms are used to detect each boundary type
separately, but the integration of the results into a single boundary representation is difficult. Therefore, a method for the simultaneous detection of all boundary types is needed. We propose
to combine responses of suitable polar separable filters into what we will call the boundary tensor. The trace of this tensor is a measure of boundary strength, while the small eigenvalue and its
difference to the large one represent corner/junction and edge strengths respectively. We prove that the edge strength measure behaves like a rotationally invariant quadrature filter. A number of
examples demonstrate the properties of the new method and illustrate its application to image segmentation.
• B. Andres, U. Köthe, M. Helmstaedter, W. Denk, F.A. Hamprecht:
“Segmentation of SBFSEM Volume Data of Neural Tissue by Hierarchical Classification”
in: G. Rigoll (Ed.): Pattern Recognition, Proc. DAGM 2008, Springer LNCS 5096 , pp. 142-152, 2008.
Abstract | BibTeX | PDF – Received a Best Paper Award from the German Association for Pattern Recognition (DAGM)
Three-dimensional electron-microscopic image stacks with almost isotropic resolution allow, for the first time, to determine the complete connection matrix of parts of the brain. In spite of
major advances in staining, correct segmentation of these stacks remains challenging, because very few local mistakes can lead to severe global errors. We propose a hierarchical segmentation
procedure based on statistical learning and topology-preserving grouping. Edge probability maps are computed by a random forest classifier (trained on hand-labeled data) and partitioned into
supervoxels by the watershed transform. Over-segmentation is then resolved by another random forest. Careful validation shows that the results of our algorithm are close to human labelings.
• B. Andres, T. Kröger, K. Briggmann, W. Denk, N. Norogod, G. Knott, U. Köthe, F.A. Hamprecht:
“Globally Optimal Closed-Surface Segmentation for Connectomics”
in: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (Eds.) : 12th Eur. Conf. Computer Vision (ECCV 2012) part III, Springer LNCS 7574, pp. 778-791, 2012.
Abstract | BibTeX | PDF
We address the problem of partitioning a volume image into a previously unknown number of segments, based on a likelihood of merging adjacent supervoxels. Towards this goal, we adapt a
higher-order probabilistic graphical model that makes the duality between supervoxels and their joint faces explicit and ensures that merging decisions are consistent and surfaces of final
segments are closed. First, we propose a practical cutting-plane approach to solve the MAP inference problem to global optimality despite its NP-hardness. Second, we apply this approach to
challenging large-scale 3D segmentation problems for neural circuit reconstruction (Connectomics), demonstrating the advantage of this higher-order model over independent decisions and
finite-order approximations.
• T. Beier, B. Andres, U. Köthe, F.A. Hamprecht:
“An Efficient Fusion Move Algorithm for the Minimum Cost Lifted Multicut Problem”
in: Leibe, B., Matas, J., Sebe, N., Welling, M. (Eds.) : 14th Eur. Conf. Computer Vision (ECCV 2016), 2016.
Abstract | PDF
Many computer vision problems can be cast as an optimization problem whose feasible solutions are decompositions of a graph. The minimum cost lifted multicut problem is such an optimization
problem. Its objective function can penalize or reward all decompositions for which any given pair of nodes are in distinct components. While this property has many potential applications, such
applications are hampered by the fact that the problem is NP-hard. We propose a fusion move algorithm for computing feasible solutions, better and more efficiently than existing algorithms. We
demonstrate this and applications to image segmentation, obtaining a new state of the art for a problem in biological image analysis.
• U. Köthe: “Reusable Software in Computer Vision”
in: B. Jähne, H. Haussecker, P. Geissler (Eds.): Handbook of Computer Vision and Applications, Volume 3: Systems and Applications, pp. 103-132, San Diego: Academic Press, 1999.
• U. Köthe, M. Felsberg:
“Riesz-Transforms Versus Derivatives: “On the Relationship Between the Boundary Tensor and the Energy Tensor”
in: R. Kimmel, N. Sochen, J. Weickert (Eds.): Scale Space and PDE Methods in Computer Vision, Springer LNCS 3459, pp. 179-191, 2005.
Abstract | PDF
Traditionally, quadrature filters and derivatives have been considered as alternative approaches to low-level image analysis. In this paper we show that there actually exist close connections: We
define the quadrature-based boundary tensor and the derivative-based gradient energy tensor which exhibit very similar behavior. We analyse the reason for this and determine how to minimize the
difference. These insights lead to a simple and very efficient integrated feature detection algorithm.
• A. Kreshuk, U. Köthe, E. Pax, D. Bock, F.A. Hamprecht:
“Automated Detection of Synapses in Serial Section Transmission Electron Microscopy Image Stacks”
PLoS ONE 9(2): e87351, 2014.
Abstract | BibTeX | PDF
• B. Kausler, M. Schiegg, B. Andres, M. Lindner, U. Köthe, H. Leitte, J. Wittbrodt, L. Hufnagel, F.A. Hamprecht:
“A discrete chain graph model for 3D+t cell tracking with high misdetection robustness”
in: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (Eds.) : 12th Eur. Conf. Computer Vision (ECCV 2012) part III, Springer LNCS 7574, pp. 144-157, 2012.
Abstract | BibTeX | PDF
Tracking by assignment is well suited for tracking a varying number of divisible cells, but suffers from false positive detections. We reformulate tracking by assignment as a chain graph{a mixed
directed- undirected probabilistic graphical model{and obtain a tracking simul- taneously over all time steps from the maximum a-posteriori configura- tion. The model is evaluated on two
challenging four-dimensional data sets from developmental biology. Compared to previous work, we obtain improved tracks due to an increased robustness against false positive detections and the
incorporation of temporal domain knowledge.
• M. Hanselmann, U. Köthe, M. Kirchner, B.Y. Renard, E.R. Amstalden, K. Glunde, R.M.A. Heeren, F.A. Hamprecht:
“Towards Digital Staining using Imaging Mass Spectrometry and Random Forests”
Journal of Proteome Research, 8(7):3558-3567, 2009
Abstract | BibTeX | PDF
We show on Imaging Mass Spectrometry (IMS) data that the Random Forest classifier can be used for automated tissue classification and that it results in predictions with high sensitivities and
positive predictive values, even when inter-sample variability is present in the data. We further demonstrate how Markov Random Fields and vectorvalued median filtering can be applied to reduce
noise effects to further improve the classification results in a post-hoc smoothing step. Our study gives clear evidence that digital staining by means of IMS constitutes a promising complement
to chemical staining techniques.
• B. Menze, B. Kelm, N. Splitthoff, U. Köthe, F.A. Hamprecht:
“On oblique random forests”
in: Mach. Learning and Knowledge Discovery in Databases, Springer LNCS 6912, pp. 453-469, 2011.
Abstract | PDF
• U. Köthe, F. Herrmannsdörfer, I. Kats, F.A. Hamprecht:
“SimpleSTORM: a fast, self-calibrating reconstruction algorithm for localization microscopy”
Histochemistry and Cell Biology, 141(6):613–627, 2014.
Abstract | PDF
Although there are many reconstruction algorithms for localization microscopy, their use is hampered by the difficulty to adjust a possibly large number of parameters correctly. We propose
SimpleSTORM, an algorithm that determines appropriate parameter settings directly from the data in an initial self-calibration phase. The algorithm is based on a carefully designed yet simple
model of the image acquisition process which allows us to standardize each image such that the background has zero mean and unit variance. This standardization makes it possible to detect spots
by a true statistical test (instead of hand-tuned thresholds) and to de-noise the images with an efficient matched filter. By reducing the strength of the matched filter, SimpleSTORM also
performs reasonably on data with high-spot density, trading off localization accuracy for improved detection performance. Extensive validation experiments on the ISBI Localization Challenge
Dataset, as well as real image reconstructions, demonstrate the good performance of our algorithm.
• H. Meine, U. Köthe, P. Stelldinger:
“A Topological Sampling Theorem for Robust Boundary Reconstruction and Image Segmentation”
Discrete Applied Mathematics (DGCI Special Issue), 157(3):524-541, 2009.
Abstract | PDF
Existing theories on shape digitization impose strong constraints on admissible shapes, and require error-free data. Consequently, these theories are not applicable to most real-world situations.
In this paper, we propose a new approach that overcomes many of these limitations. It assumes that segmentation algorithms represent the detected boundary by a set of points whose deviation from
the true contours is bounded. Given these error bounds, we reconstruct boundary connectivity by means of Delaunay triangulation and alpha-shapes. We prove that this procedure is guaranteed to
result in topologically correct image segmentations under certain realistic conditions. Experiments on real and synthetic images demonstrate the good performance of the new method and confirm the
predictions of our theory.
• U. Köthe: “What Can We Learn from Discrete Images about the Continuous World?”
in: D. Coeurjolly, I. Sivignon, L. Tougne, F. Dupont (Eds.): Discrete Geometry for Computer Imagery, Proc. DGCI 2008, Springer LNCS 4992, pp. 4-19, 2008.
Abstract | PDF
Image analysis attempts to perceive properties of the continuous real world by means of digital algorithms. Since discretization discards an infinite amount of information, it is difficult to
predict if and when digital methods will produce reliable results. This paper reviews theories which establish explicit connections between the continuous and digital domains (such as Shannon’s
sampling theorem and a recent geometric sampling theorem) and describes some of their consequences for image analysis. Although many problems are still open, we can already conclude that
adherence to these theories leads to significantly more stable and accurate algorithms.
• P. Stelldinger, U. Köthe:
“Towards a general sampling theory for shape preservation”
Image and Vision Computing, Special Issue Discrete Geometry for Computer Vision, 23(2): 237-248, 2005.
Abstract | PDF
Computerized image analysis makes statements about the continous world by looking at a discrete representation. Therefore, it is important to know precisely which information is preserved during
digitization. We analyse this question in the context of shape recognition. Existing results in this area are based on very restricted models and thus not applicable to real imaging situations.
We present generalizations in several directions: first, we introduce a new shape similarity measure that approximates human perception better. Second, we prove a geometric sampling theorem for
arbitrary dimensional spaces. Third, we extend our sampling theorem to 2-dimensional images that are subjected to blurring by a disk point spread function. Our findings are steps towards a
general sampling theory for shapes that shall ultimately describe the behavior of real optical systems.
Curriculum Vitae
since 2018 Associate Professor and group leader in the Visual Learning Lab Heidelberg
26. Nov. 2008 Habilitation for a thesis entitled “Reliable Low-Level Image Analysis”, Department of Informatics, University of Hamburg
2008-2012 Senior scientist at Heidelberg Collaboratory for Image Processing (HCI)
2007-2017 Vice Group Leader of the Image Analysis ans Learning Group (formerly: Multidimensional Image Processing), University of Heidelberg
Spring semester 2004 Guest researcher at Computer Vision Laboratory, Linköping University, Sweden
1999-2007 Assistant professor (officially: “Hochschulassistent”) in the Cognitive Systems Group, University of Hamburg
29. Feb. 2000 Dr. rer. nat. (PhD) for a thesis entitled “Generische Programmierung für die Bildverarbeitung”, Department of Informatics, University of Hamburg
Spring semester 1993 Guest researcher at Sarnoff Corporation, Princeton, USA
1992-1999 Research assistant at the Fraunhofer Institute for Computer Graphics, Rostock
1986-1991 Study of physics at University of Rostock, Diploma thesis on “Mikroskopische Herleitung einer Ratengleichung am Beispiel der Nukleonen-Deuteron-Reaktion” | {"url":"https://hci.iwr.uni-heidelberg.de/vislearn/people/ullrich-koethe/","timestamp":"2024-11-14T18:08:29Z","content_type":"text/html","content_length":"88348","record_id":"<urn:uuid:a2e89d19-d026-4691-b829-589c3a0e4cca>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00368.warc.gz"} |
How to count minimum on multiple variables in julia?
In Julia, you can use the min function to find the minimum value a**** multiple variables. Here is an example of how to count the minimum value a**** three variables:
1 # Define three variables
2 a = 5
3 b = 8
4 c = 3
6 # Find the minimum value a**** the three variables
7 min_val = min(a, b, c)
9 println("The minimum value is: ", min_val)
You can extend this code to count the minimum value a**** more variables by adding them as arguments to the min function. | {"url":"https://devhubby.com/thread/how-to-count-minimum-on-multiple-variables-in-julia","timestamp":"2024-11-08T14:51:33Z","content_type":"text/html","content_length":"110692","record_id":"<urn:uuid:af796b4a-9824-4854-bfa0-66258eea961e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00064.warc.gz"} |
Site-Specific Factor 4
Tumor Location after Primary Cytoreduction (Debulking) Surgery
**Note 1:** The surgery to remove as much of the cancer in the pelvis and/or abdomen as possible, reducing the "bulk" of the cancer is called "debulking" or "cytoreduction" surgery. It is performed
when there is widespread evidence of advanced stage of ovarian cancer with obvious spread to other organs outside the ovary, typically in the upper abdomen, intestines, the omentum (the fat pad
suspended from the transverse colon like an apron), the diaphragm, or liver. **Note 2:** Optimal debulking is described as removal of all tumor except for residual nodules that measure no more than 1
cm in maximum diameter. **Note 3:** According to the AJCC, "In advanced disease, the most important prognostic factor is the residual disease after the initial surgical management. . . . Not only is
the size of the residual important, but the number of sites of residual tumor also appears to be important." Neoadjuvant chemotherapy may be given to obtain optimum cytoreduction with less aggressive
surgery; thus recording whether the patient received chemotherapy prior to this procedure provides necessary data for treatment and survival analysis.
Code Description
010 Residual tumor in ovary, ipsilateral, contralateral, or NOS
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
Residual tumor in:
020 Fallopian tube, ipsilateral, contralateral, or NOS
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
020 + 010
Residual tumor in fallopian tube(s) and/or uterus plus ovary(ies)
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
Residual tumor in pelvis:
050 Pelvic peritoneum
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
050 + (010 or 020)
Residual tumor in pelvis plus any structures in lower codes
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
080 Residual tumor in omentum
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
080 + (010 or 020 or 050)
Residual tumor in omentum plus any structures in lower codes
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
Residual tumor in abdomen (excluding colon and small intestine):
110 Abdominal peritoneum
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
110 + (010 or 020 or 050 or 080)
Residual tumor in abdomen plus any structures in lower codes
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
140 Residual tumor in colon and/or small intestine
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
140 + (010 or 020 or 050 or 080 or 110)
Residual tumor in colon and/or small intestine plus any structures in lower codes
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
170 Residual tumor in diaphragm and/or stomach
AND neoadjuvant chemotherapy not given or unknown if given
170 + (010 or 020 or 050 or 080 or 110 or 140)
Residual tumor in diaphragm plus any structures in lower codes
AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
200 Residual tumor in liver (peritoneal surface) AND neoadjuvant chemotherapy not given or unknown if given
200 + (010 or 020 or 050 or 080 or 110 or 140 or 170)
Residual tumor in liver plus any structures in lower codes AND neoadjuvant chemotherapy not given or unknown if given
AND neoadjuvant chemotherapy given (before surgery)
300 Residual tumor in other structures not listed above
AND neoadjuvant chemotherapy not given or unknown if given
310 Residual tumor in other structures not listed above
AND neoadjuvant chemotherapy given (before surgery)
988 Not applicable: Information not collected for this case
990 Residual tumor, location not stated
AND neoadjuvant chemotherapy not given or unknown if given
991 Residual tumor, location not stated
AND neoadjuvant chemotherapy given (before surgery)
992 No residual tumor found AND neoadjuvant chemotherapy not given or unknown if given
993 No residual tumor found AND neoadjuvant chemotherapy given (before surgery)
998 No Debulking surgery performed
999 Unknown or no information
Not documented in patient record
(1) American Joint Committee on Cancer., & Edge, S.B. (2010) AJCC Cancer Staging Manual (7th ed.). New York; London: Springer, pg. 420 | {"url":"https://staging.seer.cancer.gov/tnm/input/1.2/ovary/ssf4/?breadcrumbs=(~schema_list~),(~view_schema~,~ovary~)","timestamp":"2024-11-10T07:34:52Z","content_type":"text/html","content_length":"42861","record_id":"<urn:uuid:b3d6343c-60a5-4ac6-8b28-bfd9ef9c2207>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00279.warc.gz"} |
The power of valves for better flow control, improved reliability & lower expected payment failure rates on the Lightning Network | BitMEX BlogThe power of valves for better flow control, improved reliability & lower expected payment failure rates on the Lightning Network
The power of valves for better flow control, improved reliability & lower expected payment failure rates on the Lightning Network
Writen by BitMEX Grantee Rene Pickhardt
Abstract: Valves are an important tool for flow control in fluid or gas networks. We investigate the possibilities to set up valves on the Lightning Network to improve flow control and reduce
expected payment failure rates. We show that good asymmetric choices for `htlc_maximum_msat` on a channel can reduce the expected payment failure rate significantly. We assume that the potential
benefits from carefully selecting `htlc_maximum_msat` as a valve seem not to be fully exploited by routing node operators and Lightning Service Providers yet to optimize their flow control and
improve the reliability of their channels. We analyze the power of valves on the Lightning Network mathematically by using Markov Chains. These can be used to model the uncertainty about the
liquidity in a channel that arises from drain in the depleted channel model. Using these techniques we show theoretically that properly using valves may statistically lead to a more balanced channel
and expected payment failure rates drop from two digit numbers below 3%. We provide two experimental algorithm ideas and encourage Lightning node operators to investigate the potentials from
utilizing this tool that is already being shipped with the protocol.
Introduction & Motivation
As we know payments on the lightning network correspond mathematically to (minimum cost) flows and Lightning Service Providers have liquidity as a major topic to achieve a reasonably high service
level objective. Furthermore it is well known that channels frequently deplete. Depletion occurs when more sats are supposed to be sent in one direction than in the opposite direction (A phenomenon
that is usually referred to as drain). As we have discussed in this blog, depleted channels and even small drain values are a major source of expected double digit payment failure rates and reduced
reliability of the network.
Currently Lightning Node operators engage in liquidity management and mainly try to mitigate the problems arising from drain of channels via the following two strategies:
1. Refilling the depleted channel: Several techniques are known to refill the liquidity. The node operator could make a circular payment or make an off chain swap. Furthermore the channel could be
closed and reopened or an unannounced parallel shadow channel may exist. Future improvements to the protocol might also allow the possibility to splice new liquidity into the channel. Particular
to all of these mechanisms is that they are costly for the node operator.Furthermore, globally there are only limited possibilities to conduct such operations using onchain transactions as
blockspace is scarce and Mempool congestion spikes might significantly not only put a higher cost to such strategies but also put a delay to them.
2. Reduce the drain / find balanced channels: Tools like CLBOSS try to adopt the routing fees a node charges on its channel with respect to the drain. The philosophy is mainly boiling down to an
economic argument: If there is drain on a channel that one operates this means there is demand for that liquidity so increasing the routing fees would on one end increase the revenue while on the
other hand potentially lower the demand and decrease the drain. Following a similar philosophy protocol developers suggest adding negative fees and various fee rate cards to the protocol. Other
strategies that operators put effort into is to carefully select peers and predict where their liquidity may be needed or flow in a balanced way. Entire proprietary ranking and scoring systems
for nodes exist to support node operators in their decision making.
In this article we will see that there seems to be a third and very promising strategy to establish better flow control (liquidity management) that is already baked into the protocol namely the
`htlc_maximum_msat` field in the `channel_update` message. We demonstrate that `htlc_maximum_msat` can and should operate as a valve to achieve flow control. Looking at the recent channel policies on
the gossip protocol, we note that currently almost all channels have either `htlc_maximum_mset` set to their channel capacity or have the same value in both directions. We will show that both
settings are undesirable as they tend to amplify the depletion of channels with a given drain. Given these statistics we assume that as of writing almost no nodes actively make use of the
opportunities that we propose and discuss in this article.
Let us start with an openly licensed (CC-BY-SA 4.0) quote from the Wikipedia article about valves to get the right mind set and metaphors for us:
A valve is a device or natural object that regulates, directs or controls the flow of a fluid […] by opening, closing, or partially obstructing various passageways. […] In an open valve, fluid flows
in a direction from higher pressure to lower pressure. […] Modern control valves may regulate pressure or flow downstream and operate on sophisticated automation systems. […] Valves have many uses,
including controlling water for irrigation, industrial uses for controlling processes, residential uses such as on/off and pressure control to dish and clothes washers and taps in the home.
As a valve `htlc_maximum_msat` theoretically has the potential to significantly reduce expected payment failure rates in the network and produce balanced channels.
Before continuing to explain how `htlc_maximum_msat` acts as a control valve, one important side note on balanced channels: There is the widespread misbelief that a channel is balanced if 50% of the
liquidity belongs to each peer. We have shown that 50/50 channels are not desirable and practically also almost impossible to both achieve and maintain. One can think of the myth of Sisyphus where
one might rebalance a channel towards a 50/50 balance just to realize that there is drain on the channel and the channel depletes again. Thus A better definition for a balanced channel would be to
define a channel to be balanced if and only if the liquidity distribution is close to a uniform distribution. As shown the distribution of channels with drain seems to follow an exponential curve,
which is certainly far away from a uniform distribution.
`htlc_maximum_msat` as a valve to mitigate the depletion of liquidity on payment channels with (high) drain
Recall from the wikipedia quote that “Modern control valves may regulate pressure or flow“. The equivalent to pressure for a payment channel would be its drain. We define Alice’s drain on her channel
with Bob as the fraction of payments that Alice is supposed to forward to Bob in comparison to all payment requests that occur on the channel. A drain value of 0.9 would mean that if 10 payment
forwarding requests are made on the channel 9 of them ask Alice to forward sats to Bob. At the same time only one payment request asks Bob to send sats in the direction of Alice. For a short moment
we assume all payments to be of equal size. It is easy to see that because of the pressure coming from the drain most of the liquidity will be on Bob’s end of the channel very quickly. This will also
result in the fact that Alice will not be able to fulfill most of the payment routing requests. This can be measured and observed during operation as a high payment failure rate on the channel.
Since we can think of drain on the channel as pressure and knowing from the real world that the flow through a network can be regulated by a control valve we should look out for such a valve on the
Lightning Network. If one expects 9 out of 10 payments to go from Alice to Bob an obvious measure that Alice can take to reduce her failure rate on the channel is to decrease the size of the payments
that she is willing to forward and limit the liquidity that is being drained from the channel while fulfilling the payment requests. This can be done by announcing a lower `htlc_maximum_msat` value
via a `channel_update` on the gossip protocol (See the discussion section for limitations with this coming from implementations). This effectively lowers the throughput of the channel from Alice to
Bob. The exact value has to be learnt during operation or can – like we show – be theoretically estimated given some assumptions about the payment size distribution. Selected carefully the drain may
persist and the `htlc_maximum_msat` might be more stable and controlling the flow may result in a more stationary process than fiddling around with routing fees.
The following is to the best of our knowledge the first mathematically sound description and analysis that explains the effect and power of control valves on the Lightning Network. While a vast
majority of nodes currently seems to ignore the potential of using `htlc_maximum_msat` as a valve, we note that the idea to use `htlc_maximum_msat` is not new. Some operators have been playing with
the idea to signal via `htlc_maximum_msat` how much liquidity they may have available or left in the channel. For example on the gossip protocol it is notable that zerofeerouting has times where some
of the channels have a very low `htlc_maximum_msat` value. This behavior effectively deactivates routing on the channel in the direction of the drain by closing the valve and allows the channel to
undeplete before the valve is being opened again. While this seems to be a drastic measure for flow control the principle of utilizing this field as control valve is already in place.
With all this motivation and intuitive description let us dig into the mathematics of a payment channel to study the potential of using valves with the goal to achieve a better flow control:
The Markov Process for describing the uncertainty about the Liquidity in a payment channel with a given drain.
Let us assume we have a channel with capacity `c` between Alice and Bob. This means that at any given time Alice can have a liquidity of 0,1,…,c-1 or c satoshis in the channel. As usual we ignore for
simplicity of presentation the channel reserves and considerations about in-flight htlcs. We model the sats that belong to Alice as a state of a finite Markov process. The Markov process has c+1
states as Alice can have any of those values as her liquidity.
For a short moment we make the strong simplifying assumption that payments on a channel can only be of size 1 satoshi.
If Alice has `a` sats of liquidity the Markov process would be in state `a`. Additionally we take the drain of the channel as the fraction of payments that Alice is supposed to route to Bob in
relation to all payments that are requested to be routed on the channel.
This means that we have the following two transition probabilities for any value of a between 1 and c-1.
P(X=a-1| X=a)=1-d
P(X=a+1|X=a = d
We can look at this visually with the following diagram:
If the channel was of infinite capacity we could model this as a random walk. However on the Lightning Network the capacity of channels is finite and thus we have to look at what happens with the
Markov chain as soon as we hit the limits of the (depleted) channel. In such cases the payment cannot be forwarded and the state does not change. In math formulas this means:
P(X = 0 | X = 0)= 1 – d
P(X = c | X = c) = d
Which can visually for a channel of capacity 5 look like the following diagram.
Note that the only difference is in State 0 where Alice has no sats and where the probability to stay in the state equals `1-d` and state `c=5` where Alice has `5` sats and the probability to stay in
the state equals the drain `d`.
Steady State and stationary vector of the Markov Process
The main question with respect to Markov processes is how likely is it to be in a given state if the process runs long enough. In the case of the Lightning Network this translates to the question:
“Given a fixed drain and channel capacity and only 1 sats payments: How likely is it that Alice has eventually x sats of liquidity?”
From math we know that we can compute this rather easily. So let us assume we have a vector space of dimension `c+1` with each state of the Markov process (potential liquidity value of Alice) as a
base vector and a stochastic mapping that encodes as a transition matrix in the same way as described. This would result in the following transition matrix for a channel of 5 sats capacity
We note that this is a tridiagonal matrix. We will describe later how larger payments will lead to a band matrix. Without formal proof we can see that for drain values other than 0.5 the Markov
process should not oscillate but converge into a steady state. It is easy (but computationally very inefficient) to create a small program which raises the power matrix to a very high power and
evaluates any initial state vector multiplied with the high power of the matrix. This results in a stationary state which encodes the uncertainty encoded in the liquidity distribution.
In particular this Markov process results in the same liquidity distribution as the simulated constrained random walks that we have previously used to estimate the liquidity distribution of a channel
as can be seen in the following diagram:
Of course in practice we can do a bit better than the naive approach of computing a high power of the matrix. As the steady state does not change the state vector `v` (which encodes the liquidity
distribution for Alice) we can just solve the system of linear equations given by `v= vM` under the constraint that the sum of all components of `v` has to be `1`. This is equivalent to solving (MT –
I)vT = 0. The resulting vector `v` describes in each component how likely it is that the state of the Markov process is taken.
Varying the Parameters of the Markov Process
We identify 3 main parameters for the Markov process:
1. The capacity of the channel which is 1 less than the dimension of the vectors space
2. The drain of the channel which corresponds to the transition probabilities
3. The `htlc_maximum_msat` (and the distribution of payment sizes) which we have artificially set to 1 in the above example.
The main limitation of the above consideration was that we set the `htlc_maximum_msat` to one and allowed only 1-satoshi payments. This can be resolved in various ways. We present one fairly
reasonable generalization. To understand the gist let us look at the 2-satoshi case and assume that the various payment sizes (1 sat and 2 sats) are uniformly distributed and thus equally likely to
occur for a given payment routing request. This assumption can be dropped by designing the Markov chain differently. For example in our code we also looked at zipf distribution as it seems more
likely for payments to be small. In any case to show how everything works it is easiest to explain the concepts using a uniform payment size distribution.
For a given drain of `d` and for a state `a` we now have two transition probabilities:
P(X=a+1 | X = a) = d/2
P(X = a+2 | X = a) = d/2
This is because for the two satoshi case we either send 1 satoshi or 2 and both probabilities have to add up to our drain `d`
In the opposite direction we consequently get:
P(X=a-1 | X = a) = (1-d)/2
P(X = a-2 | X = a) = (1-d)/2
As in the 1 satoshi case we have to take caution when the channel depletes. Since we allow payments of size up to 2 this can happen for example in state 0 and 1. So we get:
P(X = 0 | X = 0) = 1-d # a 1 or a 2 sats payment was requested in state 0
P(X = 0 | X = 1) = (1-d)/2 # a 1 sat payment was requested in state 1
P(X = 1 | X = 1) = (1-d)/ 2 # a 2 sat payment in state 1 was requested which cannot be fulfilled
This results in the following diagram:
Setting q = (1-d) we can again look the matrix representation of the process:
Similarly we could create larger chains for channels with higher capacity or band matrices with more bands for higher `htlc_maximum_msat` values. The lnresearch repo on github contains a notebook
with code that can do this and compute the resulting stationary distributions for you.
In particular we can set a different `htlc_maximum_msat` in the direction of the drain than in the opposite direction. Recall from our motivating example with a higher drain that we want to limit the
allowed payment size in the direction of the forwadings. So please have a look at the diagram that shows the Markov process in which Alice sets her `htlc_maximum_msat` to 1. At the same time Bob
keeps his at 2 since both Alice and Bob expect more payment routing requests for Alice than for Bob, but wish to maintain a balanced channel that should not drain:
The corresponding transition Matrix looks like the following.
We can of course generalize this to arbitrary values of `htlc_maximum_msat` for Alice and for Bob. The `htlc_maximum_msat` for Alice encodes how many bands are used above the diagonal and the
`htlc_maximum_msat` for Bob encodes how many bands are used below the diagonal.
For a given drain we search for a Markov process for which the expected payment failure rate of the steady state vector on the channel is minimized.
While the payment size distribution and its change when adopting the `htlc_maximum_msat` value can only be learnt during operation of a node and with actual data, we continue our theoretical
observations. Thus, in the next section we look at how to compute expected payment failure rates.
Compute expected payment failure rates on channels with uncertainty about the liquidity (potentially arising from drain)
Once we have a probability distribution – which may or may not be derived from the depleted channel model that comes from drain – encoding the uncertainty about the liquidity, we can compute the
likelihood for a payment of size `a` to fail via the following formula:
failure_rate(a) = drain*P(X>c-a) + (1-drain)*P(X<a)
One can easily understand this by realizing that a payment of size `a` will fail in two cases.
1. If it is against the direction of the drain and the channel has less than `a` sats of liquidity available. The probability for this to happen is computed as (1-drain)*P(X<a)
2. If it is in the direction of the drain and the channel partner has more than `c-a` sats of liquidity. The probability for this to happen is computed as `drain*P(X>c-a)`
The sum of both probabilities is the expected failure rate for a payment of size `a` as described above.
The average expected failure rate is the average over all possible values of `a` which may be weighted, if we assume that the amounts are not equally distributed. In any case the payment sizes are
limited by `htlc_maximum_msat` so we can compute the expected payment failure rate via:
This assumes the same `htlc_maximum_msat` in both directions and we can see how the expected payment failure rate changes a bit by varying the setting. In particular we see that stronger drain
requires higher `htlc_maximum_msat` values to minimize the expected payment failure rate.
We can see that when increasing the `htlc_maximum_msat` for a given drain the expected error rate drops when increasing the `htlc_maximum_msat` to its minimum and then starts to increase again.
However we can note that unless we are in a scenario with hardly any drain the change of `htlc_maximum_msat` does not seem to significantly help to reduce the error rate.
However if we assume asymmetric values for `htlc_maximum_msat` where Alice sets `htlc_maximum_msat_A` for the maximum number she is willing to forward and Bob does the same via `htlc_max_size_B` we
get the following formula for the expected failure rate and a two dimensional optimization problem which Bob and Alice have a mutual interest to solve for a given drain.
The motivation for studying this is quite simple. Assume you have a channel with a drain of 0.75 from A to B. This means that statistically 3 out of 4 payments go from A to B while only one payment
goes from B to A. If node operators would now rate limit the throughput of payment sizes in direction from A to B to a fraction of what is allowed to be routed from B to A then statistically the
channel would stay balanced and thus reduce the expected payment failure rate. For drain values up to 0.95 we have computed the optimal `htlc_maximum_msat` pairs and depicted the expect payment
failure rate in the following diagram:
We can observe that for drain values up to 0.90 (meaning 9 of 10 payments flow in the same direction) the expected payment error rate drops to below 3%. This is quite some improvement. Assuming all
drain values occur equally often the median failure rate is now 2.18%. This is a stark improvement when we realize that this value was 40.88% if the channel used the same `htlc_maximum_msat` value in
both directions. As expected, failure rates increase drastically for drain values close to 1 and the improvement when looking at averages is not so strong. The average expected failure rate of
payments over all drain values is 5.43% when optimal values for `htlc_maximum_msat` are being chosen in both directions. This is still quite some improvement from the 43.1% if the `htlc_maximum_msat`
value was chosen to be equal in both directions.
We can also see how the distribution of the liquidity tends to become much closer to the uniform distribution:
While the green curve becomes more skew for larger drains we can see that the blue curve would always stay closer to the uniform distribution of liquidity and thus a balanced channel and lower
failure rates.
Of course one should be careful about the results. We don’t know the precise payment amount distributions and how they would change if node operators start to establish flow control via this
parameter. This would change the Markov model and the weighting in the optimization problem. Assuming different parameters we have however observed similar effects. In particular we have not
investigated the dynamics if this is occurring on the entire network and we have not tested this in practice yet, but invite node operators to do so and share their experience or reach out and work
together with us to test this.
It seems as if a large number of node operators are currently not aware of the importance purposefully choosing `htlc_maximum_msat` . As of writing this article 73.1% of all channels use the same
`htlc_maximum_msat` value in both directions. As discussed in this article this predicts overall higher failure rates and demonstrates that there are unused potentials by node operators to improve
flow control, increase network reliability and decrease expected payment failure rates.
An important side note: I often criticized some developers privately who used `htlc_maximum_msat` instead of capacity during payment delivery in order to compute candidate onions. I think from a min
cost flow perspective we should use `htlc_maximum_msat` as the capacity of the arcs of the min cost flow problem to plan our payment. We can still use the actual capacity (sorry for the same word but
different semantics!) of the channel to estimate our cost function. That being said, there seems to be quite some merit in actually using `htlc_maximum_msat` as the liquidity limitation of a channel
during payment delivery and computation of candidate onions. So sorry to the devs whom I criticized in the past for doing so.
Discussion: Limitations & Opportunities of the Model
The presented model has – as any model – some severe limitations and shortcomings and does not match the reality. To create the markov chain we assume to know the drain of a channel and the
distribution of payment sizes between 1 sat and the `htlc_maximum_msat`. While both may potentially be estimated during node operations, these assumptions could change dynamically. In particular a
change in the `htlc_maximum_msat` may also change these assumptions for the model. So you could ask how useful the model is? Well it helps us in a controlled setting to understand and see the effect
of valves and draw conclusions about their potential. At the end of our notebook we provide 2 experimental algorithms that node operators could use or adopt to find good values for the
`htlc_maximum_msat` value. In Particular we have one algorithm in which a node operator does not have to create a markov chain. The idea is to build a liquidity distribution in the channel of the
last 10k payments and measure how far this distribution is away from the uniform distribution. As explained before, if the distribution is far from the uniform distribution this may be an indicator
that we have a drain and a potentially a depleted channel. In such cases adopting the `htlc_maximum_msat` value can mitigate this (as motivated through the Markov Model). This is particularly useful
as a node operator can obviously know the liquidity distribution of the last k payments. The operator does not need to know the drain on the channel or the payment size distribution in order to
measure a signal to change `htlc_maximum_msat`. Thus, while the model has the typical limitations of a model it leads node operators to easy strategies which can be used to adopt their settings.
Of course node operators have to test on mainnet if changing `htlc_maximum_msat` is useful and indeed improves the reliability by a reduction of the expected payment failure rate on their channels.
From the Markov Model we can see that lower `htlc_maximum_msat` values yield lesser variance and thus a lower expected payment failure rate. Of course in reality we don’t want to select this argument
to be arbitrarily low. So again node operators will probably have to make some trade-offs here. While we suspect that the `htlc_maximum_msat` pair per channel may be rather stable it is not clear how
often it will need to be updated. The gossip protocol of course only allows on average for 4 updates per day. Also some people with whom we have discussed the ideas of this article pointed out that
there may be privacy implications. Of course node operators would implicitly signal drain on their channels with the world if they started to use the `htlc_maximum_msat` pari as a valve. This may of
course be interesting for competing routing nodes as they could now understand where liquidity may be needed. While this could overall be good for the network’s reliability it may not be of interest
for node operators to educate their competition. Assuming this is not an issue we note that operators of a channel seem to have mutual interest to find a good `htlc_maximum_msat` pair on their
channel and adapt it during operation if the drain on the channel changes. We note that the opportunity to use valves happens on a per channel base and does not need network wide adoption.
Warning: We note that according to BOLT 7 the semantics of `htlc_maximum_msat` is currently a bit vague. Bolt7 reads: A node “SHOULD consider the htlc_maximum_msat when routing“. However many
implementations currently split the amount in some way and try to deliver partial shards by solving a path finding problem on the network where all channels have a larger `htlc_maximum_msat`. This of
course may remove channels where the value is too small, which in turn would remove some of the drain. If implementations would use min cost flow solvers by default they would most likely take the
offered liquidity and not change the drain. While for the theoretical model this may lead to some inaccuracy we suspect with the presented algorithms that this will not be a problem in practice.
We have seen how for a given drain on a channel choosing an asymmetric pair of `htlc_maximum_msat` may significantly reduce the expected payment failure rate of a channel. A network wide adoption is
expected to improve the overall reliability of the network. At the same time channels that use `htlc_maximum_msat` as a valve tend to be more balanced. Even with high drain channels we see their
liquidity distribution is closer to the uniform distribution than a channel without this flow control measure. On the Internet flow control is done by the sliding window mechanism of the Transmission
Control Protocol to address some of the reliability issues of the Internet protocol. While the `htlc_maximum_msat` might make one think of the window_size in TCP it is a Lightning Network native
parameter to work towards better flow control. Other Lightning Network native parameters to improve flow control may exist and be identified in the future. While we hope that node operators can
already just selfishly choose their `htlc_maximum_msat` following the ideas from our provided algorithms, we note that an interactive communication protocol to find the pair of `htlc_maximum_msat`
values may be necessary to automate the process of flow control on a protocol level.
On a personal note: People who have been following my work more closely know that during 2022 after I have described the depleted channel model I have become rather critical and skeptical if the
Lightning Network could ever reach the desired service level objectives. With the current results I have become hopeful again that there is more potential that we have overseen and that the Lightning
Network could in fact reach the desired service level objectives. In any case I am still confident that we will need a mechanism to have redundant overpayments as well as other tools and mechanisms
to improve the reliability of the protocol.
Thank you to Stefan Richter for helpful and critical discussions about earlier ideas and versions of this text and to Andreas Antonopoulos for encouraging me to expand on the motivation section and
help me select proper analogies to explain the workings of doing flow control via `htlc_maximum_msat`. Special thanks to zerofeerouting for notifying me while I had discussed preliminary results with
him that he already uses `htlc_maximum_msat` as a valve. Christian Decker and Anoine Riard also provided very helpful feedback and critical review for which I am grateful. Finally Summer of Bitcoin
allowing research projects and the discussion with my mentees gave me quite some inspiration to derive the liquidity distribution of channels with drain from Markov chains. In that context I thank
Sebastian Alscher for stylistic proofreading. Of course I am grateful to everyone who supports my open source research! | {"url":"https://blog.bitmex.com/the-power-of-htlc_maximum_msat-as-a-control-valve-for-better-flow-control-improved-reliability-and-lower-expected-payment-failure-rates-on-the-lightning-network/","timestamp":"2024-11-06T21:09:23Z","content_type":"text/html","content_length":"204743","record_id":"<urn:uuid:6d81025e-9043-4595-9df5-df9f66c3aed1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00875.warc.gz"} |
The Probability Of Winning At Plinko
The origin of Plinko
The game of Plinko has its origin sometime in the late 1800s and was originally constructed by mathematician and statistician Francis Galton. He built what is called a "Galton box" to prove that with
a large enough sample, a binominal distribution (i.e., when there are two options with an equal chance of outcome) will result in a normal distribution (also known as the Bell curve).
If you were to collect every ball from a game of Plinko down at the bottom, you would find that the balls would form a near-perfect Bell curve.
^Your balls would pretty much end up like this
But why is that? Surely there must be some scam to all this? After all, there's a 50/50 chance for the ball to go left or right at every step to the bottom. Shouldn't that mean that every area at the
bottom has an equal chance of catching the ball? Nope, unfortunately, that's not how it works.
So how does it work?
Let's say we only have one peg; the ball would have a 50% chance of going left and a 50% chance of going right.
But if you add another row with two pegs, the results will be a bit different. We first have an equal (50%) chance for the ball to go right or left. Let's say our ball went left. Then we have a 50/50
chance of it going right or left again, but to get the probability of the ball going left and left again, we must take the whole board into account.
Because there's half of a 50% chance for the ball to go left twice, or two 25% chance of the ball going in the middle, for every row you add, it keeps dividing the chances.
The next row would be 6,25% - 25% - 37.5% - 25% - 6.25% and so on. If you add all 16 rows (16 pegs at the bottom row) the ball will have 2^16 possible paths it can go. That’s a total of 65,536
Exponential decrease in chance
Now, to calculate the probability of the ball ending up at the far left or far right, we must divide the chance above with 2—all the way down to the bottom.
Row Chance in %
2 50% / 2 = 25%
3 25% / 2 = 12.5%
4 12.5% / 2 = 6.25%
5 6.25% / 2 = 3.125%
6 3.125% / 2 = 1.5625%
7 1.5625% / 2 = 0.7812%
8 0.7812% / 2 = 0.3906%
9 0.3906% / 2 = 0.1953%
10 0.1953% / 2 = 0.0976%
11 0.0976% / 2 = 0.0488%
12 0.0488% /2 = 0.0244%
13 0.0244% / 2 = 0.0122%
14 0.0122% / 2 = 0.0061%
15 0.0061% / 2 = 0.0030%
16 0.0030% / 2 = 0.0015%
Because there’s a 0.0015% chance that the ball ends up either at the far left or the far right, there’s a total of 0.003% chance of the ball hitting the highest payout.
But what does 0.003% chance mean? Statistically, it means that for every 33,333 balls you drop, one ball should (statistically) have hit the far right or far left. But that's just how it works in
theory. In reality, you could drop two balls and have both hitting either far right or far left. That is, however, very unlikely. Playing 33,333 games without hitting far left or far right a single
time is also possible. Because the probability, or chance, is the same for every ball you drop.
We, humans, tend to make up our own logic, such as "the more times I have played without hitting the highest multiplier, the higher the chance that my next game will hit the highest multiplier".
That's not true. Let's take a coinflip as an example. There's a 50/50 chance to hit either heads or tail when flipping a coin. Let's say that we get heads on our first flip. On our second flip, we
still have a 50/50 chance to hit either heads or tail. The same goes for the third flip, the fourth and fifth and so on. The coin, physics and math don't keep track of how many times you have flipped
your coin to adjust the odds for one or the other side to end face up. Every flip is a whole new flip, with the exact same odds. You can calculate the odds of getting, i.e., five heads in a row. But
I won't cover that in this post; it will eventually get a post of its own.
Provable fair
Plinko is a provable fair game at BC.game. That means that the result is known way before the ball hits bottom. This is calculated through the client seed + nonce, and the server seed goes through an
algorithm, which will give the game's result. Since the casino provides its seed (encrypted) before you start the game, the casino can't change it without it showing.
Client seed: This is the seed you can change yourself.
Server seed: This seed is provided by BC.game.
Nonce: The number of times you have played using the client seed.
The hexadecimal numbers resulting from client seed, nonce and server seed are then converted to base10 numbers.
These will, in turn, be calculated in groups of 4 to a number between 0 and 1. If the result from the first 4 base10 numbers is less than 0.5, the ball will go left; if it's higher than 0.5, it will
go right.
The math for these four first numbers will look like this.
This will be the same as:
And equals the number: 0.6657153440173715.
So, it’s more than 0.5, which means the ball went right.
The next would be (100/256^1) + (82/256^2) + (79/256^3) + (240/256^4) = 0.39188098534941673 – This is less than 0.5, so the ball went left.
Each time you play a game of Plinko, there's a 1 in 33,333 chance that your ball will end up either at the far right or far left multiplier (as this example with 16 rows shows). This doesn't mean
that you can only hit the highest multiplier once every 33,333 games; it is just a measure of the probability of hitting the highest multiplier. You may hit the highest multiplier two times in 10
games or no times in 50,000 games.
You can, however, check to ensure that the casino isn't deceiving you by checking the provable fair algorithm. And if you don't trust the result on the provable fair link (provided for every in-house
game), you can always calculate the result yourself.
This was very interesting to read, as somebody who is new to BC but has played a lot of Plinko. Sometimes going without the highest multiplier for double the theoretical odds. But also, more than
once I've landed them in the highest multiplier on the 3rd try.
Thanks for taking the time to write this.
43 minutes ago, Vorseline said:
This was very interesting to read, as somebody who is new to BC but has played a lot of Plinko. Sometimes going without the highest multiplier for double the theoretical odds. But also, more than
once I've landed them in the highest multiplier on the 3rd try.
Thanks for taking the time to write this.
I'm glad you liked it! Probabilities don't work the way most people think it does.
This topic is now archived and is closed to further replies. | {"url":"https://forum.bcigra.com/topic/9839-the-probability-of-winning-at-plinko/","timestamp":"2024-11-06T18:28:35Z","content_type":"text/html","content_length":"116630","record_id":"<urn:uuid:19994bfb-3f39-4b77-b010-b801aac60957>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00164.warc.gz"} |
Correction to: In the Shadow of Euler’s Greatness: Adventures in the Rediscovery of an Intriguing Sum (The Mathematical Intelligencer, (2021), 43, (82–91), 10.1007/s00283-021-10106-5)
The Borweins were a recent prominent family of mathematicians. Consisting of father David (1924–2021) and two sons, the brothers Jonathan (1951–2016) and Peter (1953–2020), between them they made
important contributions to areas such as classical analysis, computational mathematics, and experimental mathematics. In my original article, when discussing the rediscovery of the quadratic harmonic
series that would later go on to become known as the series of Au-Yeung after its rediscoverer, I mentioned that on Au-Yeung’s bringing the series to the attention of Jonathan, the latter together
with David gave what amounted to a third independent evaluation for the series. I incorrectly stated that David was the brother of Jonathan, when I should have said he was his father.
ASJC Scopus subject areas
• General Mathematics
• History and Philosophy of Science
Dive into the research topics of 'Correction to: In the Shadow of Euler’s Greatness: Adventures in the Rediscovery of an Intriguing Sum (The Mathematical Intelligencer, (2021), 43, (82–91), 10.1007/
s00283-021-10106-5)'. Together they form a unique fingerprint. | {"url":"https://faculty.kaust.edu.sa/en/publications/correction-to-in-the-shadow-of-eulers-greatness-adventures-in-the","timestamp":"2024-11-07T09:35:40Z","content_type":"text/html","content_length":"56025","record_id":"<urn:uuid:ee224656-5ad9-43f8-b666-08ec2b7a6e73>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00786.warc.gz"} |
AHSEC Class 12 Accountancy Question Paper 2024
AHSEC Class 12 Accountancy Question Paper 2024 [Assam Board, HS 2nd Year]
Get AHSEC Class 12 Accountancy Question Paper 2024 [HS 2nd Year Accountancy Paper 2024], Conducted by Assam Board i.e AHSEC (Assam Higher Secondary Education Council). It is a state education
regulatory board under the jurisdiction of Ministry of Education, Government of Assam that is responsible to regulate, supervise and develop the system of Higher Secondary Education in the State of
An Overview of AHSEC Class 12 Accountancy Question Paper 2024
Name of Board AHSEC (Assam Higher Secondary Education Council)
Class 12th Standard Assam Board
Subject Finance
Mode of Exam Offline written exam.
Full Marks 80
Year 2024
Category AHSEC Preview Year question Papers
Official Website https://ahsec.assam.gov.in
AHSEC Class 12 Accountancy Question Paper 2024
For New Course Students
Full Marks: 80
Pass Marks: 24
Time: Three hours.
CODE: 34T ACOU
The figures in the margin indicate full marks for the questions.
1. (a). Fill in the blanks with appropriate word/words (any four) 1×4= 4
(i). Partners current accounts are prepared when the capital accounts are_ _ _ _ _ _ _ _ .
(ii). A company is a separate _ _ _ _ entity, It is a separate entity from its members.
(iii). Current ratio is the relationship between current assets and _ _ _ _ _ _ _ .
(iv). Equity Shareholders are _ _ _ _ _ of a company.
(v). At the time of dissolution of partnership firm, assets are transferred to Realisation Account at _ _ _ _ value.
(b). State whether the following statements are ‘True’ or ‘False’: 1×2=2
(i). Debentureholders do not have right to vote in the meetings of the company.
(ii).Premium for goodwill is shared in gaining Ratio.
(C) Choose the correct alternative: 1×2=2
(i). The portion of the authorised capital which is offered to the public for sale in the form of shares is called….?
(a) subscribed capital
(b) issued capital
(c) called-up capital
(d) paid-up capital
(ii) In the absence of partnership deed, the rate of interest allowed on partner’s capital is …….?
(a) 6%
(b) 5%
(c) 6.5%
(d) None of the above
Q. 2. What do you mean by re-issued of forfeited share ? 2
Q. 3. Write any two demerits of partnership business. 2
Q. 4. Mention two features of a debenture. 2
Q. Write the meaning of Cash flow from investing activities’. 2
Q. 5. Give two circumstances under which the fixed capitals of partners may change. 2
Q. Why is Profit and Loss Adjustment Account prepared? 2
Q. 6. What is meant by ‘calls-in-advance’ ? 2
Q. 7. Mention two limitations of financial statement Analysis. 2
Q. What is meant by the term ‘cash equivalents’? 2
Q. Write three situations when a partnership firm is compulsorily dissolved. 3
Q. 9. Give any three items that can be shown under the heading ‘Reserves and Surplus’ in a company’s Balance Sheet. 3
Name any three items of current assets. 3
10 Current liabilities of a company are 23,50,000. Its current ratio is 3:1 and liquid ratio is 1.75 :1
Mention any three objectives of preparing a comparative statement. 3
What is computerised accounting system? 3
Q. 11. A and B are partners sharing profits and losses equally. They have admitted C into the firm. A has surrendered 1/3 of his share and B has surrendered 1/6 of his share in favour of C. Ascertain
the new profit sharing ratio. 3
Explain in brief the ‘average profit method’ of goodwill valuation. 3
Write three advantages of using graphs. 3
12. Prepare a Common Size Income Statement of Maina Ltd. from the following informations:
Particulars 2022 2023
Sale 1,05,000 1,10,000
Sales Returns 5,000 10,000
Cost of Goods Sold 70,000 74,800
Office Expenses 3,000 3,200
Non-operating Incomes 5,000 6,600
Non-operating Expenses 1,000 1,100
Income Tax Rate 50% 50%
Explain in brief the tools of financial analysis. 6
Explain the concepts of ‘data validation’ and ‘data verification’. 6
13. Give Journal entries in the books of Pakhi Ltd. for issue of debentures under the following situations: 1+1+1+1+2=6
(a) Issued 5,000, 8% debentures of ₹ 100 each at par redeemable at 5% premium after 4 years.
(b) Issued 6,000, 9% debentures of ₹100 each at 5% premium, redeemable at par after 4 years.
(e) Issued 7,000, 10% debentures of 2 100 each at 5% discount, redeemable at par after 4 years.
(d) Issued 8,000, 10% debentures of 2 100 each at 5% premium, redeemable at 10% premium after 4 years.
(e) Issued 5,000, 9% debentures of ₹ 100 each to the vendors for purchasing machinery of ₹5,00,000.
Q. Give six points of distinctions between a share and a debenture.
Q. Explain the applications of Spreadsheet in Accounting. 6
14. Susanta, Ananta and Diganta were in partnership sharing profits and losses in the ratio of 3:2:1. On 1.1.2023. Susanta retires from the firm. On that date Balance Sheet of the firm was as
Balance Sheet
Liabilities ₹ Assets ₹
Creditors 50,000 Cash at Bank 6,000
Reserve Fund 60,000 Debtors 1,50,000
Susanta 80,000 1,80,000 Stock 30,000
Ananta 60,000
Diganta 40,000
Furniture 24,000
Land and Building 80,000
Total 2,90,000 Total 2,90,000
The terms of the retirement were:
(i) Goodwill of the firm were valued at 1,20,000.
(ii) Land and Building to be appreciated by 20,000.
(iii) Provision for Bad Debts to be made @ 2% on debtors.
(iv) Furniture to be depreciated by ₹4,000.
(v) Susanta’s capital is to be transferred to his Loan Account.
Give Journal entries relating to the above transactions.
Explain how the amount due to a deceased partner is ascertained? 6
15. Distinguish between dissolution of partnership and dissolution of firm.
Ravi and Vicky are partners in a firm sharing profits and losses in the ratio of 3:2. They decided to dissolve their firm on 31st December, 2022. Their Balance Sheet on that date was as under:
Assets Amounts Liability Amounts
Capitals Furniture 16,000
Ravi 17500 Investment 4,000
Vicky 10,000 Debtors 2,000
Creditors 2,000 Stock 3,000
Profit and Loss A/c 1,500 Cash at Bank 6,000
31,000 31,000
Ravi took over the investments at an agreed value of 23,800. Other assets were realised as follows:
Furniture – 218,000
Debtors 90% of Book value
Stock 22,800
Creditors of the firm agreed to accept 5% less. Expenses of realisation amounted to 2400. Close the firm’s books by preparing a Realisation Account, Partners’ Capital Accounts and Bank Account. 6
16. Anvi Ltd. has issued 10,000 equity shares of ₹10 each at a premium of 22 each payable as follows: 8
On Application – 22
On Allotment 25 (including premium)
On First and Final Call – 25
The shares have been fully subscribed, called up and paid-up except the following:
(a) Allotment and First and Final Call money on 500 shares held by Ritu, and
(b) First and Final Call money on 600 shares held by Jitu.
All these shares have been forfeited and re-issued at 10% discount as fully paid.
Give Journal entries in the books of the company.
For what purposes ‘securities premium’ can be used? 5
(b) Write three distinctions between equity share and preference share.
What are the steps involved in installation of computerised Accounting system (CAS)? 8
17. Mera and Karan are partner in a firm any profit in the ratio of 3:2 on April 1st 2022 their balance sheet was as under:
Liabilities ₹ Assets ₹
Sundry Creditors 85,000 Bank 10,000
Capitals Stock 22,000
Mihir 70,000 Plant and Machinery 40,000
Karan 60,000 1,30,000 Building 20,000
Goodwill 24,000
Debtors 24,000
Less: Provision 1000
Net Debtors 23,000
Total Liabilities 1,30,000 Total Assets 1,30,000
On the above date, they admitted Sunil as a new partner on the following terms:
(i) Sunil will bring ₹50,000 for his capital.
(ii) He would get 1/5th share in the future profits
(iii) Goodwill of the firm is valued at ₹1,20,000.
(iv) Sunil will bring necessary premium for goodwill.
Pass Journal entries to record the above transaction. Prepare Partner’s Capital Accounts and Balance Sheet of the new firm.
(i). Distinguish between Profit and Loss Account and Profit and Loss Appropriation Account. 5
(ii). Mention any three rights of a partner.3
18. Biswa and Pradip are partners in a firm. The Trial Balance of the firm as on 31st December, 2022 was as under:
Prepare Profit and Loss Account, Profit and Loss Appropriation Account and the Balance Sheet of the firm for the year ended 31st December, 2022 after considering the following information:
Debit ₹ Credit ₹
Biswas – 4000
Pradip – 3000 7,000 Capital:
Cash at Bank 45,000 Biswas – 65,000 1,05,000
Sundry Debtors 40,000 Pradip – 40,000 18,400
Insurance 19,740 Sundry Creditors 5000
Advertisement 9,000 Bank Loan 300
Closing Stock 12,500 Commission 57,200
Cash in hand 16,300 Trading A/c (Gross Profit)
Commission 5,000
Motor Car 20,860
Machinery 10,000
1,85,900 1,85,900
(a) Partners are to share profits and losses in the proportion of 3/5 and 2/5 respectively.
(b) Write off depreciation @ 10% on Machinery and 20% on Motor.
(c) Create a provision of 5% on Sundry Debtors for Doubtful Debts.
(d), Partners are entitled to Interest on Capital 5% per annum and Pradip is entitled to a salary of 2 1,800 per annum.
For Old Course: fin lieu of Project Works)
19. Answer the following questions: (any four) 5×4= 20
(a) Write distinctions between, Fixed Capital Account and Fluctuating Capital Account.
(b) What is Ratio Analysis? Mention any three limitations of ratio analysis.
(c) Explain uses of Financial Statement.
(d) What is meant by Cash Flow Statement? Mention any three objectives of preparing cash flow statement.
(e) Explain the average profit method of valuation of goodwill. What is Revaluation Account?
Click Here for : AHSEC Class 12 Accountancy Solved Question Paper 2024
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Math & CS Seminar
The Math & CS Seminar at DePaul
All talks are both in-person and over Zoom, at 1pm, on Fridays; here is a view of the public ICS file:
• Friday Nov 8: Kyle Petersen (Department of Mathematical Sciences, DePaul University) in the Loop CDM 200 or over Zoom (join the mailing list and consult the archives for the Zoom link)
Title: Zig-zag Eulerian polynomials
Abstract: For any finite partially ordered set , the P-Eulerian polynomial is the generating function for descents over the set of linear extensions of P. (The classical Eulerian polynomials
correspond to the case where P is an antichain.) In this work, we study the P-Eulerian polynomials where P is a naturally-labelled zig-zag poset; we call these the zig-zag Eulerian polynomials.
These have appeared only fleetingly in the literature, and we believe they are deserving of closer study. This talk will survey some of the known features of these polynomials by analogy with the
classical case and highlight new results, including combinatorial interpretations, gamma-positivity, and recurrence relations. Our proofs and insight come from a modification of the theory of
P-partitions. This is joint work with Yan Zhuang (Davidson College)
Announcement List
To subscribe to our announcement list, go to the list webpage or simply send an email with subject "subscribe" to math-cs-seminar-request@mailman.depaul.edu.
Important Warning: There is a strong chance that the mailing list will be marked as spam, in particular when you register to it. Please add emails from *@mailman.depaul.edu to your whitelist.
Talk archive
• Friday Week 5 (Feb 9): Emily Barnard, Department of Mathematical Sciences in the Loop CDM 228 or over Zoom (join the mailing list and consult the archives for the Zoom link)
Title: pop-stack sorting and pattern-avoiding permutations
Abstract: The pop-stack sorting method takes an ordered list or permutation and reverses each descending run without changing their relative positions. In this talk we will review recent
combinatorial results on the pop-stack sorting method, and we will extend the pop-stack sorting method to certain pattern avoiding permutations, called c-sortable. This talk will be accessible to
• Friday Week 3 (Jan 26): Umer Huzaifa, SoC in Lincoln Park Campus
Title: Walking Strategies of Humanoid Robots, with applications to assistive devices
Abstract: Human movement is characterized by precise coordination of multiple joints and muscles, resulting in complex overall body dynamics. In my presentation, I will outline a methodology I
developed to enhance the walking strategies of humanoid robots, making them more stylistic and human-like. Next I will present my extension of this work into human assistive devices and recent
results in this regard. My research also delves into the realm of soft robotics. This field presents unique challenges, as the modeling of soft robotic systems does not follow the straightforward
principles applicable to rigid robots. In one of our recent studies, we approached this challenge by conceptualizing the soft robot as a collection of interconnected rigid bodies, applying
traditional modeling techniques. Finally, I will touch upon my latest foray into biomedical engineering. Here, my focus is on identifying biomarkers that can be used to monitor the health of
active-duty personnel. This work represents a significant step towards enhancing health monitoring in demanding environments.
• Friday Week 5, Oct 6: Salman Parsa, SoC in Lincoln Park Campus
Title: Reeb Spaces and the Borsuk–Ulam Theorem
Abstract: I will talk about Reeb spaces and their discretization called mapper. These are applied tools in topological data analysis. After showing a few examples, I will use the Reeb space to
prove a partial extension of the well-known Borsuk-Ulam theorem for maps from \(2\)-sphere into \(\mathbb{R}\). This extension says that there are always two antipodal points \(S^2 \ni x, -x\)
such that \(f(x) = f(-x)\) and the two points are connected in the preimage. The proof uses the concept of the Reeb graph which is a 1-dimensional Reeb space. I also consider the relationship
between excess homology of the Reeb space of \(f: S^n \to \mathbb{R}^{n - 1}\) and the existence of the analogous extensions of the Borsuk–Ulam theorem for maps into \(\mathbb{R}^m, n > 2\).
• Friday Week 9, Nov 3: Stefan Mitsch, SoC in Lincoln Park Campus
Title: The cyber-physical systems proof workhorse: quantifier elimination in the first-order theory of real closed fields
Abstract: Cyber-physical systems are characterized by interaction between discrete computational processes and their effects in continuous differential equation models of physics, biology,
chemistry, etc. In proving universal and existential properties about such systems, many questions of interest reduce to questions in first-order real arithmetic. Conveniently, the first-order
theory of real closed fields is decidable due to a seminal result of Alfred Tarski. Algorithms for quantifier elimination, however, are highly non-trivial and not proof-producing, which makes it
difficult to construct an unbroken chain of proof arguments. This talk presents a formalization of quadratic virtual term substitution, a quantifier elimination procedure that is complete for
low-degree polynomials.
Previous years | {"url":"https://mc.cdm.depaul.edu/math-cs-seminar/","timestamp":"2024-11-09T05:44:28Z","content_type":"text/html","content_length":"17875","record_id":"<urn:uuid:70545b80-e1db-4840-bac4-45632e4acc47>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00126.warc.gz"} |
NW District Invoice Generator
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Betting Fractions To Decimals - How To Convert : Bet4bettor
How to convert Betting Fractions To Decimals (& vice versa)
Converting betting fractions to decimals can be confusing.
In this post I’ll show you how to quickly convert fractional odds to decimal odds and vice versa.
The origins of fractional and decimal odds
Betting odds in the UK were traditionally expressed as fractions. Still in common use and originating from the on-course bookmakers, it wasn’t until the introduction of betting exchanges that decimal
odds were introduced into the UK.
Decimal odds were the norm in Europe and Australia. UK sportsbooks adopted the format to appeal to the widened audience brought about by the explosion of the internet.
To those not familiar with the UK fractional odds, they could appear confusing. It was widely perceived that decimal odds were easier to understand.
Being able to quickly convert fractional odds to decimals and also in reverse, is of primary use when you are weighing up which are the best odds and across a range of bookmakers. Odds for the same
bet may be expressed in a mixture of formats.
How to convert betting fractions to decimals
To convert betting fractions to decimals, you need to apply this simple formula to the fractional odds.
Divide the first number (the numerator) by the second number (the denominator) and add 1 = decimal odds.
Numerator / denominator +1 = decimal odds
For example, to convert fractional odds of 7/2 into decimal odds
7/2+1 = 4.5
How to convert Decimal odds to Betting fractions
Converting decimal odds to fractional odds is a slightly more complex calculation, but can be done following this 4 step process.
In this example we’ll use decimal odds of 3.25. Take the decimal odds and minus 1
Multiply both the top and bottom number by multiples of 10 until the decimal point is removed from the top number and both the top and the bottom number are whole numbers.
In the case of our example
Simplify the fraction. To do this divide both the top and the bottom number by the same value until it reaches its lowest whole number.
The fractional odds of 3.25 are 9/4.
Another example using decimal odds Of 1.80
Note, you will find that not all decimal odds translate to fractional odds that are recognisable as ones regularly used traditionally in betting markets.
For example, decimal odds of 1.74 equates in its simplest form to fractional odds of 37/50.
This is down in one part to the tradition of the fractions that UK bookmakers originally used, and secondly because the expression of odds in decimals uses a fuller scale.
With the traditional UK fractional odds, bands in the odds were straddled. I.e. the next step up from 11/10 in fractional odds would traditionally be 6/5. Convert these fractional odds to decimals
2.10 and 2.20. All decimals in the range in between were not offered.
This was one-way bookmakers could build margin into their books and also to the less informed punters it was easy to disguise the books total overround and profit margin by speaking in a fractional
odds language that was quite niche.
It’s fair to say that the general introduction of decimal odds into the UK betting markets has made the odds more understandable to the masses.
To make the whole process of cross calculation easier, I’ve added the conversion table below.
Betting Fractions to Decimals - odds conversion Table (and vice Versa)
As well as the fractional and equivalent decimal odds the third and sixth column in the table titled ‘percentage chance’ shows the implied probability for these odds.
If you want to read about how a ‘book’ is formed and bookmakers margins and overrounds then you can find it here.
Calculating returns from fractional odds
Some returns from betting fractions are relatively easy to calculate.
For example, £10 at odds of 5/1 is five times your initial stake £50, plus your £10 stake added, equals £60 returns.
Similarly £4 at 11/4, for every £4 you bet, you win £11 plus your initial stake returned equals a total of £15.
These relatively simple examples are referred to as ‘money to money’.
Where it becomes a little more complicated with the money to money calculations is where you have a stake amount that isn’t the same as, or easily divisible with the denominator number in the bet.
For example a £13 bet at 11/4?
The calculation for working out returns on all fractional odds bets is…
stake x Numerator / denominator + Stake = Returns
For the example above £13 at 11/4
£13 X 11 / 4 + £13 = £48.5
Calculating returns from decimal odds
Calculating the returns from bets placed at decimal odds is far more straightforward.
For example, £13 at 3.75
£13 X 3.75 = £48.75
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Simple Definition of Business Mathematics And Advantages of Business Mathematics
Business mathematics may be define different mathematical formulas or mathematical steps which are taken for development in business . Large number of business theories which is used to solve
business problems are included in business mathematics .
Explanation of Definition of Business Mathematics
From this simple definition , we find that business mathematics is nothing more than different formulas and theories like interest rate , annuity rate , matrix theory , linear programming theory and
probability theory and many more . With these formulas and theories business can calculate many solutions of different problems .
Advantages of Business Mathematics
• With interest rate of maths , businessman can calculate the interest on debt , loan or bonds .
• With matrix , businessman can calculate salary bill of different department and branches .
• With Linear programming , business can determine the quantity of two products at which profit is maximize or cost is minimize .
• With discounting and factoring technique businessman can calculate present value of bill , rat of discounting the bill and banker's gain
• With currency translation formula of maths , businessman can cost of currency between two countries .
• With Assignment solution technique businessman can solve transportation problems.
• Large number of business estimations are done on the basis of probability theory of business mathematics .
• Market research bureau can use consistency data theory of business mathematics for calculating estimating profit , capital and sale of business . | {"url":"http://mathematics.svtuition.org/2009/12/simple-definition-of-business.html","timestamp":"2024-11-03T22:13:27Z","content_type":"application/xhtml+xml","content_length":"56131","record_id":"<urn:uuid:fff969fc-8979-4d7c-b7a8-2c546cf06f93>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00633.warc.gz"} |
(1) ∫1−x2sin−1xdx
(2) ∫2xexdx
(3) ∫1+cos2exexsinexdx
(4) ∫x3... | Filo
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Question Text (1) (2) (3) (4)
Updated On Jul 8, 2024
Topic Integration
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
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Multiple Core Usage Issues
03-13-2013 11:40 PM
I recompiled my installation of numpy and scipy agaisnt Intel MKL. I am trying to speed up a script that fits tensor for DT MRI. The script bottlenecks during the svd operation, in particular the
call to numpy.linalg.lapack_lite.dgesdd inside the svd function is very slow. Slow in the sense that I ran this calculation using the default ATLAS and now with MKL the speedup is negiigible. The
thing I noticed is like ATLAS, Intel MKL is only using one core for the bulk of the SVD calculations.
I found this topic
and he says because SVD is BLAS 2 it is usually single threaded but will be mutlithreaded on newer processors. I haev an Intel i7 2630QM(Sandy Architecture) processor and am running the latest Intel
MKL build(11). Should I be experiencing multiple thread use and if so, how can I obtain that? I'm not sure what other information would be helpful to provide, but I can provide whatever you need to
help me. Thanks in advance!
03-14-2013 02:52 PM
03-14-2013 03:29 PM
03-14-2013 03:54 PM
03-14-2013 04:34 PM
03-14-2013 05:01 PM
03-14-2013 09:47 PM
03-14-2013 10:13 PM
03-14-2013 10:22 PM
04-04-2013 07:25 PM | {"url":"https://community.intel.com/t5/Intel-oneAPI-Math-Kernel-Library/Multiple-Core-Usage-Issues/td-p/971640","timestamp":"2024-11-10T11:40:50Z","content_type":"text/html","content_length":"365869","record_id":"<urn:uuid:63f820b2-0ca6-4d12-800a-407c4d0043fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00637.warc.gz"} |
A Student Practicing For A Cross Country Meet Runs 250m In 30s...
Home>Misc>Featured>A Student Practicing For A Cross Country Meet Runs 250m In 30s. What Is Her Average Speed?
A Student Practicing For A Cross Country Meet Runs 250m In 30s. What Is Her Average Speed?
Modified: August 18, 2023
Discover the average speed of a student practicing for a cross country meet, as she runs 250m in just 30 seconds. Get insights on her impressive feat!
Participating in a cross country meet requires physical stamina, endurance, and determination. Athletes push themselves to the limit, running for long distances over various terrains. As a student
prepares for such an event, understanding their average speed becomes crucial in evaluating their performance.
In this article, we will explore the scenario of a student who takes part in a cross country meet and runs a distance of 250 meters in just 30 seconds. We will calculate the student’s average speed
during this run, providing a valuable insight into their performance.
Understanding average speed is essential because it gives us an idea of how fast an object or individual traveled over a given distance. By calculating the average speed, we can assess an athlete’s
efficiency and measure their progress.
Calculating average speed involves considering two essential factors: the distance covered and the time taken. By determining the ratio between these two elements, we can arrive at a numerical value
that represents an athlete’s average speed.
In the context of our scenario, the student runs a distance of 250 meters in a total time of 30 seconds. By dividing the distance by the time, we can calculate the average speed achieved by the
student during this specific run.
Let’s delve into the calculations to determine the student’s average speed and better understand the significance of this metric in evaluating their performance in a cross country meet.
Calculation of Average Speed
Calculating the average speed is a straightforward process that involves dividing the total distance covered by the time taken. In our scenario, we have the student running a distance of 250 meters
in a time of 30 seconds. To find the average speed, we will use the following formula:
Average Speed = Total Distance / Time Taken
This formula provides us with a clear and concise method for calculating the average speed. By plugging in the values from our scenario, we can determine the average speed achieved by the student.
As we proceed with the calculations, it’s essential to note that average speed is typically measured in a unit such as meters per second (m/s) or kilometers per hour (km/h). The unit will depend on
the unit of measurement used for the distance and time values.
The formula for calculating average speed emphasizes the importance of both distance and time. Distance represents the amount of ground covered, while time indicates the duration of the activity.
Once we have calculated the average speed, we can gain valuable insights into the student’s performance during this particular run. The average speed serves as an indicator of how efficiently the
student covered the distance and can be used to assess their overall running ability.
With an understanding of the formula and its significance in calculating average speed, let’s move on to evaluate the specific data provided in our scenario and determine the student’s average speed.
Given Data
In order to calculate the average speed of the student, we have been provided with the following data:
• Distance: 250 meters
• Time: 30 seconds
These two pieces of information are essential in determining the average speed achieved by the student during their run. The distance represents the total amount of ground covered, while the time
indicates the duration of the run.
It is crucial to have accurate and reliable data for calculating average speed. In this case, we have the specific values for the distance and time taken, providing a solid foundation for our
By utilizing this data, we can now proceed with applying the formula for average speed and obtain a numerical value that represents the student’s performance in terms of their average speed.
With the given data in hand, let’s move on to the next section, where we will apply the formula and perform the necessary calculations to determine the student’s average speed.
Formula for Average Speed
The formula for calculating average speed is a fundamental equation used to determine how fast an object or individual is moving over a given distance during a certain period of time. The formula is
simple yet powerful in providing insights into an athlete’s performance.
The formula for average speed is:
Average Speed = Distance / Time
In this formula, “Distance” represents the total distance covered by the athlete, and “Time” refers to the duration it took to cover that distance. By dividing the distance by the time, we can
calculate the average speed achieved by the athlete.
It is important to note that when using this formula, both the distance and time values must be in consistent units. For example, if the distance is given in meters, the time should be in seconds.
This ensures that the average speed is expressed in the appropriate unit of measurement.
The average speed can be measured in various units, depending on the specific context. Common units include meters per second (m/s) or kilometers per hour (km/h).
By using this formula, we can quantitatively evaluate an athlete’s performance in terms of their average speed. This measurement not only provides valuable information about their running ability but
also serves as a benchmark for improvement and comparison.
Now that we have an understanding of the formula for average speed, we can move on to the next section, where we will apply this formula to the given data and calculate the student’s average speed.
Now that we have all the necessary information, let’s apply the formula for average speed to calculate the student’s performance.
Given data:
• Distance: 250 meters
• Time: 30 seconds
Using the formula for average speed:
Average Speed = Distance / Time
Substituting the values:
Average Speed = 250 meters / 30 seconds
Performing the calculation:
• Average Speed = 8.333 meters/second
Therefore, the student’s average speed during the 250-meter run is approximately 8.333 meters per second.
This calculation shows us how quickly the student covered the distance in relation to the time it took. It provides a quantitative measure of the student’s performance in terms of their average speed
during this particular run.
Understanding the average speed helps us assess an athlete’s efficiency and progress. In the context of our scenario, the average speed of 8.333 meters per second indicates the student’s running
capability. This value can be used to evaluate their performance and potentially identify areas for improvement.
Now that we have successfully calculated the student’s average speed, let’s move on to the final section where we will present the answer and conclude our analysis.
Based on our calculations, the student’s average speed during the 250-meter run is approximately 8.333 meters per second.
This value represents the rate at which the student covered the distance, taking into account the time it took to complete the run. It provides a quantitative measure of the student’s running ability
during this particular event.
With an average speed of 8.333 meters per second, the student demonstrated a commendable level of speed and agility. This performance showcases their running prowess and highlights their ability to
cover a considerable distance in a short amount of time.
It is important to note that average speed can vary from one event to another, and different factors such as terrain, weather conditions, and the athlete’s level of fitness can influence the results.
Thus, it is crucial to consider the specific context when interpreting and comparing average speeds.
Overall, the student’s achieved average speed of 8.333 meters per second is an impressive accomplishment. It reflects their dedication and training in preparation for the cross-country meet. With
this level of performance, the student is well-positioned to excel in future competitions and continue to improve their running abilities.
Now that we have determined the student’s average speed, let’s wrap up our analysis in the next section.
Calculating the average speed of a student participating in a cross country meet provides valuable insights into their performance and running ability. In our scenario, the student ran a distance of
250 meters in a time of 30 seconds, resulting in an average speed of approximately 8.333 meters per second.
The average speed serves as a quantitative measure of the student’s efficiency and progress during this particular run. It highlights their ability to cover a significant distance in a short amount
of time, showcasing their speed and agility.
However, it is important to note that average speed can vary based on various factors, such as terrain, weather conditions, and an athlete’s level of fitness. Therefore, it is crucial to consider the
specific context when interpreting and comparing average speeds.
Calculating average speed enables us not only to evaluate an athlete’s current performance but also to set benchmarks for improvement. By striving to increase their average speed, athletes can
enhance their running abilities and achieve higher levels of success in future competitions.
In conclusion, calculating average speed provides valuable insights into an athlete’s performance, allowing for evaluation, comparison, and improvement. Understanding the significance of average
speed enables athletes to track their progress, set goals, and push their boundaries to achieve even greater results in their respective sports. | {"url":"https://runningshorts.com/misc/featured/a-student-practicing-for-a-cross-country-meet-runs-250m-in-30s-what-is-her-average-speed/","timestamp":"2024-11-11T11:46:51Z","content_type":"text/html","content_length":"348848","record_id":"<urn:uuid:945838c0-97fb-4a08-b7ad-73b92aaf3763>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00430.warc.gz"} |
Back to Papers Home Back to Papers of School of Physics
Paper IPM / P / 7737
School of Physics
Title: Semiclassical String Solutions on 1/2 BPS Geometries
Author(s): 1. H. Ebrahim
2. Amir E. Mosaffa
Status: Published
Journal: JHEP
Vol.: 01
Year: 2005
Pages: 050
Supported by: IPM
We study semiclassical string solutions on the 1/2 BPS geometry of type IIB string theory characterized by concentric rings on the boundary plane. We consider both folded rotating strings carrying
nonzero R-charge and circular pulsating strings. We find that unlike rotating strings, as far as circular pulsating strings are concerned, the dynamics remains qualitatively unchanged when the
concentric rings replace AdS_5 S^5. Using the gravity dual we have also studied the Wilson loop of the corresponding gauge theory. The result is qualitatively the same as that in AdS_5 S^5 in the
global coordinates where the corresponding gauge theory is defined on S^3 R. We show that there is a correction to 1/L leading order behavior of the potential between external objects.
Download TeX format
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Writing Whole Numbers As Fractions Worksheets 2024 - NumbersWorksheets.com
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Q. 8. The edge of the cube is increasing at the rate 3 cm/... | Filo
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Ans. Q. 8. The edge of the cube is increasing at the rate . Find the rate of increase in its volume when its edge is . (C.G. 2010 Set A,B,C)
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Question Text Ans. Q. 8. The edge of the cube is increasing at the rate . Find the rate of increase in its volume when its edge is . (C.G. 2010 Set A,B,C)
Updated On Nov 21, 2022
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 110
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I want to know more about ft-lbs, ft-lbf, in-lbs and in-lbf
As I live in sweden and thus not that familiar with the american way of measuring things, I would like to know the difference between torque units lbs and lbf. It also would be fun to know what the
abbreviation stands for, I mean it sems a bit odd to pronounce lbf/lbs - pounds....
The pound can be either a unit of mass, or a unit of force, specifically the force of standard gravity on a one pound mass (note this relationship would not hold on the moon.)
Thus, it can be hard to determine exactly what is being discussed. Not all engineers do this, but many using Imperial units insist on labelling lbf and lbm for pounds-force and pounds-mass
respectively to ensure there is no misunderstanding of what is meant.
In a torque, pound-mass makes little sense so pound-force is meant. There is really no difference between a ft-lb and a ft-lbf except clarity. I assume ft-lbs is a messy abbreviation formed from the
In the equation f = ma, using pounds for both mass and force is a major drawback of Imperial. At least three concepts are used by different groups of engineers:
1) Pound for force only, introduce the artifical unit "slug" for mass (about 32 lb.) so 1 slug accelerated at 1 ft/sec^2 requires a force of 1 lb. Invented by British, but more common in US.
2) Pound for mass only, the artifical unit "poundal" for force. 1 poundal is force to accelerate one pound mass at 1 ft/sec^2. The UK uses this more than US, or I should say, used to use.
The poundal is about 1/32th of a pound-force.
3) Pound mass, and pound force. The f=ma equation is modified to include standard gravity which should be g-sub-zero but I can't do it here. f = m*(a/g). Accelerating one pound at 1 g requires one
pound of force. Aerospace seems to like this one.
The fourth solution is to go metric and many US industries have (the auto industry for example). The three groups of engineers who use the first three solutions confuse the hell out of each other. (I
am a member of the fourth group)
Many other standard equations need to be patched for pound as force and mass. I was trained in metric and frankly don't know how to do them all. I was entirely baffled by an "English" form of
Bernoulli's equation for example.
The abbreviation lb for pound comes from the Latin word libra, a weight more or less a pound. | {"url":"https://www.convert-me.com/en/bb/viewtopic.php?f=2&t=2100","timestamp":"2024-11-13T22:45:56Z","content_type":"application/xhtml+xml","content_length":"16876","record_id":"<urn:uuid:c8894989-a80b-47bd-8830-57286adcd187>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00782.warc.gz"} |
Spreading and reachability in temporal networks
Spreading and reachability in temporal networks¶
In this example, we construct a random link-activation temporal network, then compare various reachability statistics for different types of temporal adjacency.
Let’s say you want to study the spreading of a disease or some other effect on a temporal network. There are many different models, ranging from very simple to very intricate and specialised, that
might be relavant for this use.
Making the network¶
Let’s start first by creating a random temporal network. In this case we generate a random temporal network where the nodes have an underlying random \(k\)-regular structure, using
random_regular_graph(), with each link gets activated with an exponential inter-event time distribution. In other words, in our “friendship” network, each “person” is friends with exactly \(k\) other
randomly selected people, and the time between consecutive meeting of each two friends are drawn from an exponential distribution with a given mean, i.e. meetings are a Poisson process.
import reticula as ret
max_t = 64
size = 128
k = 4
mean_iet = 1.0
gen = ret.mersenne_twister()
g = ret.random_regular_graph[ret.int64](size=size, degree=k, random_state=gen)
iet = ret.exponential_distribution[ret.double](lmbda=1/mean_iet)
residual_iet = iet
temporal_net = ret.random_link_activation_temporal_network(
The function random_link_activation_temporal_network() takes a static base network base_net, a maximum simulated time max_t, an inter-event time distribution iet_dist, a residual time distribution
res_dist and a pseudo-random number generator to generate a random link-activation network from \(t = 0\) to \(t = t_\text{max}\).
The residual time distribution generates the distribution of time to the next event, starting at a random time, i.e. if you arrive at a bus stop at a random time, how long do you have to wait for a
bus to arrive if busses arrive at intervals drawn from the inter-event time. For the case of Poisson process, this happens to be the same distribution as the waiting times.
Our temporal network is now an undirected temporal network with 128 vertices and around 16 thousand events.
>>> temporal_net
<undirected_temporal_network[int64, double] with 128 verts and 16157 edges>
Simple spreading process¶
Next, we calculate the charactristic quantities of the most simple spreading process model. In this model, once an effects, e.g., a disease or information, reaches a node, that node will forever
remain “infectious”, spreading that effect to anyone else they come in contact with. This is more or less equivalent to the Susceptible–Infected (SI) model of epidemic spreading.
In temporal network words, many of these different types of spreading processes can be understood in terms of temporal adjacency. Imagine two events \(e_1\) and \(e_2\). If the first one carries a
disease, what are the conditions for the second one to carry the disease as well as a consequence of the first event transmitting the disease? For the simple case of simple adjacency on undirected
networks, the only condition for the adjacency of these two events is that they share at lease one vertex in common, and that \(e_2\) happens after \(e_1\). This gets more and more involved for,
e.g., directed temporal networks or temporal networks where events take some time from start to finish, but here we only care about the simple definition.
Single-source simulation¶
Let’s now see what would be the result of a spreading process originating from a specific node at a certain time, using this simple adjacency rules. This can be accomplished using out_cluster():
adj = ret.temporal_adjacency.simple[temporal_net.edge_type()]()
cluster = ret.out_cluster(
vertex=0, time=0.0)
The cluster object, an instance of temporal_cluster, now contains information about the entire spreading cluster starting from vertex 0 at time 0.0, including who got infected when. We can see for
example that every node got infected at one point or another:
We can also see all the intervals when a certain vertex (for example node 5) has been infected:
>>> int_set = cluster.interval_sets()[5]
>>> int_set
<interval_set[double] from t=2.7580437394574595 to t=inf>
>>> list(int_set)
[(2.7580437394574595, inf)]
Also, it is possible to see how many events (and exactly which ones) participated in the spreading process and whether a specific vertex at a specific time.
>>> len(cluster)
>>> list(cluster)
[undirected_temporal_edge[int64, double](69, 84, time=59.1906391191876), undirected_temporal_edge[int64, double](20, 126, time=16.842526559964774), undirected_temporal_edge[int64, double](16, 107, time=4.880354855014931), ....]
>>> cluster.covers(12, 1)
This is telling us that node 5 first got infected at time \(t = 2.7\) and remained infected forever.
Multi-source simulation¶
out_cluster() provides a powerful tool, but sometimes, you need to know the range of possible outcomes from different starting points. Of course one approach is to use out_cluster() multiple times
with different starting vertices and times. This, however, is not necessarily the most efficient approach, as it requires looping over every event in the temporal network for each call of the
We can calculate the out-cluster from all possible starting points with a single backwards scan of the network, using the method presented in Badie-Modiri et al. [17], using the out_clusters() method
(note the plural “clusters”).
clusters = ret.out_clusters(
temporal_network=temporal_net, temporal_adjacency=adj)
The variable clusters now contains a list tuples of all events on the temporal network, paired with the cluster of spreading that would result if a spreading process starts right before the time of
that event.
This on its own, however, requires sacrificing a lot of memory. Even for a small temporal network like this, you might need around to 7.7GB of RAM and it takes around 20 seconds to run on a modern
CPU core. This is due to the fact that at every stage of the algorithm we have to keep exact track of membership of every node in every cluster. Turns out, however that in many cases we don’t
actually need this information in the final results, as we mostly only care about the various statistics of a cluster such as number of unique nodes, total lifetime of the spreading process or total
“human-hours” of infection.
In cases like this, we can use out_cluster_sizes(), which throws out the actual cluster memberships when they become irrelevant to the algorithm, only keeping the afformentioned charactristic
quantities, namely mass, volume and lifetime for each possible source.
clusters = ret.out_clusters(
temporal_network=temporal_net, temporal_adjacency=adj)
This already drops the peak memory requirement by a factor of 50! The resulting charactristic quantities are exact, but there is not much detail left in each cluster, which are instances of
temporal_cluster_size. You are still able to get exact value for mass, volume and lifetime and check exactly how many events participated in each spreading cluster, but you can’t get a list of those
Multi-source estimation¶
The out_cluster_sizes() method still keeps exact track of every event involved in every spreading process, even if it throws out that list as soon as it can in the course of the traversal of the
temporal network.
We can get results much faster and a whole lot lighter (in terms of memory use) if instead of keeping accurate track every event involved in each spreading cluster, we use probabilistic counting
methods to keep track of these charactristic quantities.
out_cluster_size_estimates() method does exactly that. It uses the full force of the approach described in Badie-Modiri et al. [17] to get estimated values for mass, volume and number of events
participating in each spreading cluster.
clusters = ret.out_cluster_size_estimates(
temporal_network=temporal_net, temporal_adjacency=adj,
time_resolution=0.1, seed=0)
This results in a runtime of ~0.8 seconds, and memory usage low enough that becomes hard to measure in this scale (< 10MB). The downside is that mass, volume and number of events are now only
accurate to ±1.1% of their exact value.
The volume and event count work by directly employing a HyperLogLog++ probabilistic cardinality estimator[18]. The mass is calculated by dividing the timeline of each node to steps of size
time_resolution, then counting unique combinations of vertex name and time step that is in the spreading cluster using the probabilistic cardinality estimator. The lifetime of the cluster is still
measured exactly.
The seed parameter controls the seed used for hashing in the HyperLogLog++ cardinality estimators. By running the function multiple times with different seeds and averaging the results, you can
obtain accuracies higher than the default ±1.1%.
These sacrifices are made so that each cluster “sketch” stays within a constant memory footprint, as opposed to normal cluster objects that grow over time.
Other types of spreading processes¶
So far we only used the most simple adjacency type, corresponding to a simple Susceptible–Infected (SI) process, but this might not be the most suitable model real-world phenomena. Specifically,
processes where the effect only lingers in each node for a finite amount of time. In this section we will talk about other, slightly more realistic models of reachability and spreading processes.
Limited waiting time¶
For example, let’s say you are interested in reachability in a public transportation network. You can model each bus stop and train station as vertices in a temporal network, and each bus or train
trip between consecutive stations as a link. In this scenario, the set of all vertices and times that can be reached from a starting vertex and time for the out-cluster of that starting point and
time. To put it in other words, if vertex \(v\) at time \(t\) is in the out-cluster of \(v_0\) at \(t_0\), then there is at least one valid time-respecting path between the two.
It is, however, not very likely that a traveller would stay in a station waiting for a bus or a train to arrive for an indefinite amount of time. Let’s say we limit valid time-repsecting paths to
those with a miximum waiting time of \(\delta t\).
This limited waiting-time reachability can be expressed in terms of temporal adjacency of two events. The two events \(e_1\) and \(e_2\) are adjacent with a maximum waiting time \(\delta t\) if they
are adjacent (i.e., if \(e_2\) starts after the end of \(e_1\) and they share at least one vertex in common) and the start time of \(e_2\) is at most \(\delta t\) away from the end of \(e_1\).
lwt_adj = ret.temporal_adjacency.limited_waiting_time[temporal_net.edge_type()](dt=3.0)
cluster = ret.out_cluster(
vertex=0, time=0.0)
This creates an adjacency object with a maximum waiting time of 3.0 units of time. You can use the other cluster functions the same way as before, simply by using the new adjacency object as a
Of course, the public transportation example from above differs from the previous examples, in that the underlying temporal network is both directed and delayed, meaning that a trip between two
stations only move people in one direction and that it takes some time to move from one stop to the next and the execution of the event is not instantainious. A better temporal network type for
representing transport networks is directed_delayed_temporal_network, where each event has two distinct timestamps instead of one:
e = ret.directed_delayed_temporal_edge[ret.int64, ret.double](
tail=1, head=2, cause_time=2.0, effect_time=3.0)
The example above shows a directed_delayed_temporal_edge going from node 1 to node 2, where the “departure time” is 2.0 and the “arrival time” is 3.0.
Exponential/geometric waiting time¶
In other real-world phenomena, the idea of a limit for waiting time for valid time-respecting paths makes sense, but perhaps not a constanc cutoff described by a single value. For example, let’s say
you are modelling infromation spreading, e.g., sprading of gossip, where each person hearing a gossip has a constant probability of forgetting it at every unit of time. This means that the maximum
waiting times are drawn from an exponential distribution.
mean_cutoff_time = 2.0
exp_adj = ret.temporal_adjacency.exponential[temporal_net.edge_type()](
rate=1/mean_cutoff_time, seed=42)
cluster = ret.out_cluster(
vertex=0, time=0.0)
The seed parameter here controls the generation of waiting times. This will create a single realisation of the spreading process. To get a more accurate range of possible outcomes, re-run this
functions with different seed values.
Note that the exponential adjacency only works for networks with continues time (i.e., double time type). For discrete time types ( int64) you need to use the geometric adjacency
temporal_adjacency.geometric which has the maximum waiting times drawn from the geometric distribution.
Infectious and non-infectious events¶
Simply assuming every event will cause a transmission event might not be very realistic. Spreading processes on temporal networks sometimes assume an infection probability \(p\), similar to the role
of infectious rate \(\beta\) in SIS processes, that determines wheter a particular event can carry a disease or not. In Reticula, we can do this by creating a new temporal network where each event is
occupied (kept) independently with probability \(p\):
p = 0.8
new_net = ret.uniformly_occupy_edges(temporal_net, occupation_prob=p, random_state=gen)
We can then run the same processes on the new_net instead of the old one.
Limited waiting time processes and SIS models¶
With a little bit of work, you can see how SI processes with an infection probability 1.0 on temporal network are isomorphic to reachability. While the relationshio between SI model and simple
adjacency is easy to understand, the relationship between exponential waiting time adjacency and SIS processes are more nuanced. In exponential limited waiting-time adjacency, similar to the the
usual formulation of the SIS process, each node stays infected and contagious for a length of time drawn from an exponential distribution with a given mean. The difference arrises when thinking of
how “re-infections” affect this process. This is however, a topic very involved topic, perhaps enough for a research paper or maybe several. | {"url":"https://docs.reticula.network/examples/temporal_network.html","timestamp":"2024-11-03T05:51:25Z","content_type":"text/html","content_length":"60788","record_id":"<urn:uuid:048f1cf2-2e76-4639-ae8f-38fcde64483f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00831.warc.gz"} |
PCA (principal component analysis)
Principal Component Analysis (PCA) is a technique that reduces the dimensionality of data while retaining the variation present in the data.
Principal Component Analysis (PCA) is a technique that reduces the dimensionality of data while retaining the variation present in the data. In essence, a new coordinate system is constructed so that
data variation is strongest along the first axis, less strong along the second axis, and so on. Then, the data points are transformed into this new coordinate system. The directions of the axes are
called principal components.
If the input data is a table with p columns, there could be maximum p principal components. However, it's usually the case that the data variation along the direction of some k-th principal component
becomes almost negligible, which allows us to keep only the first k components. As a result, the new coordinate system has fewer axes. Hence, the transformed data table has only k columns instead of
p. It is important to remember that the k output columns are not simply a subset of p input columns. Instead, each of the k output columns is a combination of all p input columns.
You can use the following functions to train and apply the PCA model:
For a complete example, see Dimension reduction using PCA.
In this section | {"url":"https://docs.vertica.com/24.4.x/en/data-analysis/ml-predictive-analytics/data-preparation/pca-principal-component-analysis/","timestamp":"2024-11-10T08:45:32Z","content_type":"text/html","content_length":"41975","record_id":"<urn:uuid:e7f03228-ffc3-4294-a54e-894644c8abe4>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00317.warc.gz"} |
How to Use #chatgpt in a High School Math Class
John Spencer
13 Dec 202210:03
TLDRThe transcript explores the use of AI in a high school math class, focusing on the concept of p-values. It suggests that while AI can provide clear explanations and examples, it should not
replace teachers or direct instruction. The AI can help clarify understanding and answer specific questions, adapting its responses to the student's level. However, it also highlights the need for
students to rephrase and summarize concepts in their own words for better retention, emphasizing the importance of moving knowledge into long-term memory.
• 😲 ChatGPT can be used as a learning tool in a high school math class, particularly for complex concepts like p-values.
• 🔍 Students can use ChatGPT to get a refresher on difficult topics by asking for explanations in simpler terms.
• 📚 ChatGPT provides answers that can help build conceptual understanding, but it doesn't replace the role of a teacher or direct instruction.
• 🤔 It allows students to ask clarifying questions and get tailored responses that can aid in understanding complex mathematical concepts.
• 📉 ChatGPT can explain concepts like p-values in the context of real-world examples, such as in sports like basketball.
• 📈 The AI can guide students through different types of data visualization relevant to statistical analysis, such as scatter plots.
• 🛠️ Students can learn how to apply statistical concepts practically, for example, by finding out how to calculate p-values in Google Spreadsheets.
• 📝 ChatGPT's responses adapt to the complexity of the user's understanding, simplifying explanations when needed.
• 🚫 ChatGPT's capabilities are currently text-based and do not include visual aids like graphs or pictures.
• 💡 The tool can be particularly useful for students who are confused and need help in clarifying and summarizing mathematical concepts in their own words.
• 🌟 While ChatGPT is a powerful tool for enhancing understanding, it is not a substitute for teacher interaction or peer collaboration.
Q & A
• What is the primary concern regarding the use of AI like chat GPT in a high school math class?
-The primary concern is that AI might be used for cheating. However, the transcript explores the potential of AI as a learning tool in math, specifically for understanding complex concepts like
• What is a p-value and why is it difficult for students to understand?
-A p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming that the null hypothesis is true. It can be difficult for students
because it involves statistical concepts that require a deeper conceptual understanding.
• How can chat GPT help students understand the concept of a p-value?
-Chat GPT can provide definitions, examples, and simplified explanations of p-values. It can also answer specific questions and clarify confusions, helping students build a conceptual
• What is the significance of the Alpha level in the context of p-values?
-The Alpha level is a predetermined threshold used to determine the statistical significance of the results of a hypothesis test. If the p-value is less than the Alpha level, the null hypothesis
is rejected.
• Can chat GPT provide examples of p-values in different contexts, such as sports?
-Yes, chat GPT can provide context-specific examples, like using p-values in basketball statistics, to help students relate the concept to real-world scenarios and enhance their understanding.
• How does chat GPT adapt its responses to the user's level of understanding?
-Chat GPT can simplify its explanations and use less academic vocabulary when prompted by the user to explain concepts in simpler terms, making it more accessible to students.
• Is chat GPT capable of replacing a teacher or direct instruction in a math class?
-No, chat GPT is not intended to replace a teacher or direct instruction. It serves as a supplementary tool to help clarify concepts and provide additional help when needed.
• What limitations does chat GPT have in terms of visual aids for explaining p-values?
-Chat GPT is text-based and does not provide pictures or graphs. While it can describe what type of graph might be used to represent p-values, it cannot show a visual representation directly.
• How can students practically apply the understanding of p-values, for example, in Google Spreadsheets?
-Chat GPT can guide students on how to find the p-value using formulas in Google Spreadsheets, providing a step-by-step explanation that they can follow.
• What is the role of peer interaction compared to using AI like chat GPT for learning math concepts?
-Peer interaction has the advantage of a relational element and can be more dynamic. While AI can clarify and provide information, it lacks the personal interaction and discussion that peer
learning offers.
• How can educators ensure that students are not just memorizing answers from chat GPT but truly understanding the concepts?
-Educators can encourage students to rephrase and summarize the information in their own words, promoting deeper understanding and helping to move the knowledge into long-term memory.
📚 Exploring Chat GPT as a Math Learning Tool
This paragraph discusses the potential of Chat GPT as an educational aid, specifically in the context of teaching mathematical concepts like p-values. The speaker emphasizes that while Chat GPT is
not a substitute for traditional teaching methods, it can provide students with additional help when they need a refresher or clarification. The example of a p-value is used to illustrate how the AI
can offer explanations in simpler terms, use concrete examples, and answer follow-up questions to deepen a student's understanding. The paragraph also touches on the limitations of text-based AI in
providing visual aids but highlights the utility of Chat GPT in helping students develop a conceptual grasp of complex topics.
🤖 Adaptability and Limitations of AI in Education
The second paragraph delves into the adaptability of AI, noting how it can simplify explanations and shift perspectives to better suit the learner's needs. It contrasts the academic language with
simpler terms and acknowledges the shift in grammatical structure to aid comprehension. The speaker discusses the limitations of text-based AI, such as the inability to provide visual representations
like graphs, but also points out the value in the detailed textual explanations and step-by-step guidance that AI can offer. The paragraph concludes by emphasizing the role of AI as a supplementary
tool for clarifying concepts and enhancing understanding in mathematics, suggesting that while AI is not a replacement for teachers or peer interaction, it can be a powerful resource for students
seeking to reinforce their learning.
ChatGPT is an artificial intelligence chatbot developed by OpenAI. In the context of the video, it is presented as a learning tool that can assist students in understanding complex mathematical
concepts such as p-values. The script illustrates how students can interact with ChatGPT to receive explanations tailored to their understanding level, enhancing their learning experience.
A p-value is a statistical term that refers to the probability of obtaining results at least as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is
correct. In the video, the concept of p-values is central to the discussion on how AI can help students grasp abstract mathematical ideas, with examples provided to simplify the definition for
💡Learning Tool
In the script, a learning tool is any resource or method that aids in the educational process. ChatGPT is explored as a learning tool with the potential to clarify and deepen students' understanding
of mathematical concepts, particularly in the context of high school math classes.
💡Direct Instruction
Direct instruction is a teaching method where the teacher provides clear guidance and explanation to students. The video script contrasts ChatGPT with direct instruction, suggesting that while AI can
offer additional help, it should not replace the role of a teacher or the importance of direct instruction in learning.
💡Conceptual Understanding
Conceptual understanding refers to a deep comprehension of the underlying principles and ideas of a subject. The script emphasizes the importance of developing conceptual understanding in math, and
how ChatGPT can assist students in clarifying and building upon their understanding of complex concepts like p-values.
💡Alpha Level
The alpha level, also known as the significance level, is a threshold used in statistical hypothesis testing to determine the probability of rejecting the null hypothesis when it is true. In the
script, the alpha level is mentioned as a way to assess the reasonableness of a p-value, illustrating the practical application of statistical concepts.
💡Correlational Studies
Correlational studies are research designs used to examine the relationship between two or more variables without inferring causality. The video discusses the application of p-values in correlational
studies, indicating that p-values can be used to assess the strength of association rather than causation.
💡Cause and Effect
Cause and effect refers to a relationship where one event (the cause) influences another event (the effect). The script explores the student's curiosity about whether p-values are applicable only to
experimental designs with a cause-and-effect relationship, highlighting the importance of understanding the context in which statistical tools are used.
A graph is a visual representation of data, often used in mathematics and statistics to illustrate relationships and distributions. The video script mentions the desire for a visual representation of
a p-value, although ChatGPT's capabilities are text-based, indicating the potential limitations of AI in providing visual aids for learning.
💡Google Spreadsheets
Google Spreadsheets is a web-based application that allows users to create, edit, and share spreadsheets. In the context of the video, it is suggested as a tool for students to find and use p-values,
with ChatGPT providing instructions on how to implement this in a spreadsheet, demonstrating the practical application of AI in educational tasks.
💡Long-term Memory
Long-term memory is the ability to retain information over a long period. The script discusses the importance of moving knowledge from temporary to long-term memory, suggesting that students should
rephrase and summarize AI-provided answers in their own words to facilitate this process, emphasizing the role of active engagement in learning.
Exploring the use of AI as a learning tool in math, particularly for understanding complex concepts like p-values.
The challenge for students in grasping the concept of p-values and how AI can provide various explanations and examples.
Using AI to give a basic definition of p-values and the possibility of simplifying the explanation for better understanding.
The adaptability of AI in providing explanations at different levels of complexity, including simpler terms for younger students.
The potential of AI to answer specific questions about p-values, such as determining if a p-value is reasonable.
Comparing p-values to an Alpha level or significance threshold as a method to evaluate their reasonableness.
The importance of rephrasing questions to AI for clarity and simplicity, especially for complex topics like p-values.
Discussing the use of p-values in correlational studies and the differences from experimental studies.
The limitations of AI in providing visual aids, such as graphs, for concepts like p-values.
How AI can guide students through different types of graphs and data representations relevant to p-values.
The practical application of finding p-values using Google Spreadsheets with AI's assistance.
AI's role in enhancing conceptual understanding in math without replacing the teacher or direct instruction.
The potential of AI to serve as a tool for clarifying and developing deeper understanding in math.
The comparison between AI and other educational tools like photomath, focusing on different aspects of math learning.
The importance of students rephrasing and summarizing AI's answers in their own words for better retention.
The challenge of ensuring students use AI as a tool for clarification and not just for getting answers.
The potential of AI to be a powerful tool for students who are confused and need to clarify mathematical concepts. | {"url":"https://math.bot/blog-How-to-Use-chatgpt-in-a-High-School-Math-Class-38374","timestamp":"2024-11-07T00:26:47Z","content_type":"text/html","content_length":"125107","record_id":"<urn:uuid:50fe9ced-7fb0-43e2-a4d3-d8ddf4f1b975>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00055.warc.gz"} |
NCERT Solutions for Class 10 Maths Exercise 3.7
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.7 in Hindi and English medium updated for CBSE and State boards. Achieve academic mastery in Class 10 Maths with the up-to-date NCERT Solutions
for Chapter 3 Exercise 3.7. Optimized for both CBSE and State Boards, our guide is accessible in Hindi and English medium. Dive deep into Linear Equations in Two Variables with our expert solutions,
ideal for thorough exam prep and targeted homework help. Don’t miss this essential educational resource to attain top grades and excel in your board examinations.
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.7
10th Maths Ex. 3.7 solution Optional exercise pair of linear equations in two variables in Hindi Medium and English Medium. View Ex. 3.7 Solutions in Video Format free updated for all boards. All the
questions are described properly in simple language so that students can understand easily. UP Board students are now using the same NCERT Textbooks as CBSE Board Students. So, they also can take
help from these solutions.
Download UP Board Solution for class 10 Maths Exercise 3.7 in Hindi Medium. All chapters of Class 10 Maths solutions and Offline apps 2024-25 are prepared by experts accordance with new CBSE Syllabus
for UP Board / Uttarakhand Board / CBSE Board NCERT Books. Download (Exercise 3.7) in PDF format or see all the solutions in videos given just below the PDF solutions.
10 Maths Chapter 3 Exercise 3.7 Solutions
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.7 in English and Hindi medium for MP Board, up board / CBSE board / Gujrat and Uttarakhand board to use online or download in PDF form free.
Download solutions of All exercises of Class 10 Maths Chapter 3 from its main page. Download Offline Apps based on latest NCERT Books.
Class 10 Maths Exercise 3.7 Solution in Hindi Medium Video
Class: 10 Mathematics
Chapter 3: Exercise 3.7
Medium: English and Hindi Medium
Content: NCERT Solution
Important Questions with Answers
□ A man covers a distance of 15 km in 3 hours partly by walking and partly by running. If he walks at 3 km/h and runs at 9 km/h, find the distance he covers by running. [Answer: 9 km]
□ A and B are two points 150 km apart on a highway. Two cars start with different speeds from A and B at same time. If they move in same direction, they meet in 15 hours. If they move in
opposite direction, they meet in one hour. Find their speeds. [Answer: 80 km/h, 70 km/h]
□ A man invested ₹3500, part of it in a yearly interest rate of 4% and the rest at 5%. He receives a total annual interest of ₹153. How much did he invest at each rate? [Answer: ₹2200 at 4% and
₹1300 at 5%]
□ A boat Covers 32 km upstream and 36 km downstream, in 7 hours. Also it Covers 40 km upstream and 48 km downstream in 9 hours. Find the speed. Of boat in still water and that of the stream.
[Answer: 10 km/h, 2 km/h]
□ The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2:3. Determine the
fraction. [Answer: 5/9]
Important Questions from Board Papers
1. Raju used 2 plastic bags and 1 paper bag in a day which cost him ₹ 35. While Ramesh used 3 plastic bags and 4 paper bags per day, which cost him ₹ 65. Find the cost of each bag. [Answer: 15,
2. 8 Women and 12 men can complete a work in 10 days while 6 women and 8 men can complete the same work in 14 days. Find the time taken by one woman alone and that one man alone to finish the
work. [Answer: 1 woman in 140 days, 1 man in 280 days]
3. The ratio of incomes of two persons A and B is 3:4 and the ratio of their expenditures is 5:7. If their savings are ₹ 15,000 annually find their annual incomes. [Answer: ₹90000, ₹120000]
Feedback and Suggestions
User’s feedback is important for all online or offline intersections. So, provide your feedback and ask your doubts related to NIOS or CBSE Board and share your knowledge with your friends and other
users through Discussion Forum.
Get the NCERT Books and Offline Apps for 2024-25 based on new CBSE Syllabus for new academic session.
Do the questions ask in CBSE Board Exams from Optional Exercises 3.7 also?
Yes, there are questions in Board Papers which are asked from optional exercises.
Which question is important in Optional Exercise 3.7 of Class 10 Maths?
Class 10 Maths Exercise 3.7 Questions 8 and few parts of Questions 7 are asked in Board exams.
Is Exercise 3.7 of Class 10 Maths easy?
Class 10 Maths Exercise 3.7 is neither easy nor difficult. It needs practice to understand the concepts. Once it is practiced well, it will be an easier exercise.
Which is the most prominent exercise in Class 10 Maths Chapter 3?
All the exercises are important in 10th Maths Chapter 3 but Exercise 3.6 includes almost all type of questions for practice.
How many questions are there in exercise 3.7 of 10th Maths?
There are in all 8 questions in exercise 3.7 (Optional exercise) of class 10th mathematics chapter 3 (Pair of linear equations in two variables). All questions of this exercise are important.
Can questions come from exercise 3.7 class 10th Maths in Board exams?
Yes, questions can come from exercise 3.7 (chapter 3) of class 10th mathematics in Board exams. This exercise contains very good and logical questions.
How much time students need to complete exercise 3.7 Class 10 Maths?
Students need maximum 1 day to complete exercise 3.7 (chapter 3 Pair of linear equations in two variables) of class 10th mathematics.
Is exercise 3.7 of class 10th mathematics difficult?
Exercise 3.7 (chapter 3 Pair of linear equations in two variables) of class 10th mathematics is difficult exercise. But difficulty level of anything varies from student to student. So, Exercise 3.7
(chapter 3 Pair of linear equations in two variables) of class 10th mathematics is easy or not depends on students also.
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13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature - College Physics 2e | OpenStax
Learning Objectives
By the end of this section, you will be able to:
• Express the ideal gas law in terms of molecular mass and velocity.
• Define thermal energy.
• Calculate the kinetic energy of a gas molecule, given its temperature.
• Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
• Describe the distribution of speeds of molecules in a gas.
We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain
a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
Figure 13.21 shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide
with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus
the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Things Great and Small feature below. The
following relationship is found:
$PV = 1 3 Nm v 2 ¯ , PV = 1 3 Nm v 2 ¯ ,$
where $PP$ is the pressure (average force per unit area), $VV$ is the volume of gas in the container, $NN$ is the number of molecules in the container, $mm$ is the mass of a molecule, and $v 2 ¯ v 2
¯$ is the average of the molecular speed squared.
What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas.
Recall the previous expression of the ideal gas law:
Equating the right-hand side of this equation with the right-hand side of $PV=13Nm v 2 ¯ PV=13Nm v 2 ¯$ gives
$13Nm v 2 ¯ =NkT.13Nm v 2 ¯ =NkT.$
Making Connections: Things Great and Small—Atomic and Molecular Origin of Pressure in a Gas
Figure 13.22 shows a box filled with a gas. We know from our previous discussions that putting more gas into the box produces greater pressure, and that increasing the temperature of the gas also
produces a greater pressure. But why should increasing the temperature of the gas increase the pressure in the box? A look at the atomic and molecular scale gives us some answers, and an alternative
expression for the ideal gas law.
The figure shows an expanded view of an elastic collision of a gas molecule with the wall of a container. Calculating the average force exerted by such molecules will lead us to the ideal gas law,
and to the connection between temperature and molecular kinetic energy. We assume that a molecule is small compared with the separation of molecules in the gas, and that its interaction with other
molecules can be ignored. We also assume the wall is rigid and that the molecule’s direction changes, but that its speed remains constant (and hence its kinetic energy and the magnitude of its
momentum remain constant as well). This assumption is not always valid, but the same result is obtained with a more detailed description of the molecule’s exchange of energy and momentum with the
If the molecule’s velocity changes in the $xx$-direction, its momentum changes from $–mvx–mvx$ to $+mvx+mvx$. Thus, its change in momentum is $Δmv= +mvx––mvx=2mvxΔmv= +mvx––mvx=2mvx$. The force
exerted on the molecule is given by
$F = Δp Δt = 2 mv x Δt . F = Δp Δt = 2 mv x Δt .$
There is no force between the wall and the molecule until the molecule hits the wall. During the short time of the collision, the force between the molecule and wall is relatively large. We are
looking for an average force; we take $ΔtΔt$ to be the average time between collisions of the molecule with this wall. It is the time it would take the molecule to go across the box and back (a
distance $2l)2l)$ at a speed of $vxvx$. Thus $Δt=2l/vxΔt=2l/vx$, and the expression for the force becomes
$F = 2 mv x 2 l / v x = mv x 2 l . F = 2 mv x 2 l / v x = mv x 2 l .$
This force is due to one molecule. We multiply by the number of molecules $NN$ and use their average squared velocity to find the force
$F = N m v x 2 ¯ l , F = N m v x 2 ¯ l ,$
where the bar over a quantity means its average value. We would like to have the force in terms of the speed $vv$, rather than the $xx$-component of the velocity. We note that the total velocity
squared is the sum of the squares of its components, so that
$v 2 ¯ = v x 2 ¯ + v y 2 ¯ + v z 2 ¯ . v 2 ¯ = v x 2 ¯ + v y 2 ¯ + v z 2 ¯ .$
Because the velocities are random, their average components in all directions are the same:
$v x 2 ¯ = v y 2 ¯ = v z 2 ¯ . v x 2 ¯ = v y 2 ¯ = v z 2 ¯ .$
$v 2 ¯ = 3 v x 2 ¯ , v 2 ¯ = 3 v x 2 ¯ ,$
$v x 2 ¯ = 1 3 v 2 ¯ . v x 2 ¯ = 1 3 v 2 ¯ .$
Substituting $13 v 2 ¯ 13 v 2 ¯$ into the expression for $FF$ gives
$F = N m v 2 ¯ 3l . F = N m v 2 ¯ 3l .$
The pressure is $F/A,F/A,$ so that we obtain
$P = F A = N m v 2 ¯ 3 Al = 1 3 Nm v 2 ¯ V , P = F A = N m v 2 ¯ 3 Al = 1 3 Nm v 2 ¯ V ,$
where we used $V=AlV=Al$ for the volume. This gives the important result.
$PV = 1 3 Nm v 2 ¯ PV = 1 3 Nm v 2 ¯$
This equation is another expression of the ideal gas law.
We can get the average kinetic energy of a molecule, $12mv212mv2$, from the right-hand side of the equation by canceling $NN$ and multiplying by 3/2. This calculation produces the result that the
average kinetic energy of a molecule is directly related to absolute temperature.
$KE ¯ = 1 2 m v 2 ¯ = 3 2 kT KE ¯ = 1 2 m v 2 ¯ = 3 2 kT$
The average translational kinetic energy of a molecule, $KE¯KE¯$, is called thermal energy. The equation $KE¯=12m v 2 ¯ =32kTKE¯=12m v 2 ¯ =32kT$ is a molecular interpretation of temperature, and it
has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.
It is sometimes useful to rearrange $KE¯=12mv2¯=32kTKE¯=12mv2¯=32kT$, and solve for the average speed of molecules in a gas in terms of temperature,
$v 2 ¯ = v rms = 3 kT m , v 2 ¯ = v rms = 3 kT m ,$
where $vrmsvrms$ stands for root-mean-square (rms) speed.
Calculating Kinetic Energy and Speed of a Gas Molecule
(a) What is the average kinetic energy of a gas molecule at $20.0ºC20.0ºC$ (room temperature)? (b) Find the rms speed of a nitrogen molecule $(N2)(N2)$ at this temperature.
Strategy for (a)
The known in the equation for the average kinetic energy is the temperature.
$KE ¯ = 1 2 m v 2 ¯ = 3 2 kT KE ¯ = 1 2 m v 2 ¯ = 3 2 kT$
Before substituting values into this equation, we must convert the given temperature to kelvins. This conversion gives $T=(20.0+273)K = 293K.T=(20.0+273)K = 293K.$
Solution for (a)
The temperature alone is sufficient to find the average translational kinetic energy. Substituting the temperature into the translational kinetic energy equation gives
$KE ¯ = 3 2 kT = 3 2 1.38 × 10 − 23 J/K 293 K = 6.07 × 10 − 21 J . KE ¯ = 3 2 kT = 3 2 1.38 × 10 − 23 J/K 293 K = 6.07 × 10 − 21 J .$
Strategy for (b)
Finding the rms speed of a nitrogen molecule involves a straightforward calculation using the equation
$v 2 ¯ = v rms = 3 kT m , v 2 ¯ = v rms = 3 kT m ,$
but we must first find the mass of a nitrogen molecule. Using the molecular mass of nitrogen $N2N2$ from the periodic table,
$m = 2 14 . 0067 × 10 − 3 kg/mol 6 . 02 × 10 23 mol − 1 = 4 . 65 × 10 − 26 kg . m = 2 14 . 0067 × 10 − 3 kg/mol 6 . 02 × 10 23 mol − 1 = 4 . 65 × 10 − 26 kg .$
Solution for (b)
Substituting this mass and the value for $kk$ into the equation for $vrmsvrms$ yields
$v rms = 3 kT m = 3 1 . 38 × 10 – 23 J/K 293 K 4 . 65 × 10 –26 kg = 511 m/s . v rms = 3 kT m = 3 1 . 38 × 10 – 23 J/K 293 K 4 . 65 × 10 –26 kg = 511 m/s .$
Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature. The kinetic energy is very
small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin. The rms velocity of the nitrogen molecule is surprisingly large. These large molecular
velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood. The mean free path (the distance a molecule can move on average between
collisions) of molecules in air is very small, and so the molecules move rapidly but do not get very far in a second. The high value for rms speed is reflected in the speed of sound, however, which
is about 340 m/s at room temperature. The faster the rms speed of air molecules, the faster that sound vibrations can be transferred through the air. The speed of sound increases with temperature and
is greater in gases with small molecular masses, such as helium. (See Figure 13.23.)
Making Connections: Historical Note—Kinetic Theory of Gases
The kinetic theory of gases was developed by Daniel Bernoulli (1700–1782), who is best known in physics for his work on fluid flow (hydrodynamics). Bernoulli’s work predates the atomistic view of
matter established by Dalton.
Distribution of Molecular Speeds
The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This distribution is
called the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and has since been confirmed experimentally. (See Figure 13.24.) The distribution has a
long tail, because a few molecules may go several times the rms speed. The most probable speed $vpvp$ is less than the rms speed $vrmsvrms$. Figure 13.25 shows that the curve is shifted to higher
speeds at higher temperatures, with a broader range of speeds.
The distribution of thermal speeds depends strongly on temperature. As temperature increases, the speeds are shifted to higher values and the distribution is broadened.
What is the implication of the change in distribution with temperature shown in Figure 13.25 for humans? All other things being equal, if a person has a fever, they are likely to lose more water
molecules, particularly from linings along moist cavities such as the lungs and mouth, creating a dry sensation in the mouth.
Calculating Temperature: Escape Velocity of Helium Atoms
In order to escape Earth’s gravity, an object near the top of the atmosphere (at an altitude of 100 km) must travel away from Earth at 11.1 km/s. This speed is called the escape velocity. At what
temperature would helium atoms have an rms speed equal to the escape velocity?
Identify the knowns and unknowns and determine which equations to use to solve the problem.
1. Identify the knowns: $vv$ is the escape velocity, 11.1 km/s.
2. Identify the unknowns: We need to solve for temperature, $TT$. We also need to solve for the mass $mm$ of the helium atom.
3. Determine which equations are needed.
• To solve for mass $mm$ of the helium atom, we can use information from the periodic table:
$m=molar massnumber of atoms per mole.m=molar massnumber of atoms per mole.$
• To solve for temperature $TT$, we can rearrange either
$KE ¯ = 1 2 m v 2 ¯ = 3 2 kT KE ¯ = 1 2 m v 2 ¯ = 3 2 kT$
$v 2 ¯ = v rms = 3 kT m v 2 ¯ = v rms = 3 kT m$
to yield
$T = m v 2 ¯ 3k , T = m v 2 ¯ 3k ,$
where $kk$ is the Boltzmann constant and $mm$ is the mass of a helium atom.
4. Plug the known values into the equations and solve for the unknowns.
$m = molar mass number of atoms per mole = 4 . 0026 × 10 − 3 kg/mol 6 . 02 × 10 23 mol = 6 . 65 × 10 − 27 kg m = molar mass number of atoms per mole = 4 . 0026 × 10 − 3 kg/mol 6 . 02 × 10 23 mol = 6
. 65 × 10 − 27 kg$
$T = 6 . 65 × 10 − 27 kg 11 . 1 × 10 3 m/s 2 3 1 . 38 × 10 − 23 J/K = 1 . 98 × 10 4 K T = 6 . 65 × 10 − 27 kg 11 . 1 × 10 3 m/s 2 3 1 . 38 × 10 − 23 J/K = 1 . 98 × 10 4 K$
This temperature is much higher than atmospheric temperature, which is approximately 250 K $(–25ºC(–25ºC$ or $–10ºF)–10ºF)$ at high altitude. Very few helium atoms are left in the atmosphere, but
there were many when the atmosphere was formed. The reason for the loss of helium atoms is that there are a small number of helium atoms with speeds higher than Earth’s escape velocity even at normal
temperatures. The speed of a helium atom changes from one instant to the next, so that at any instant, there is a small, but nonzero chance that the speed is greater than the escape speed and the
molecule escapes from Earth’s gravitational pull. Heavier molecules, such as oxygen, nitrogen, and water (very little of which reach a very high altitude), have smaller rms speeds, and so it is much
less likely that any of them will have speeds greater than the escape velocity. In fact, so few have speeds above the escape velocity that billions of years are required to lose significant amounts
of the atmosphere. Figure 13.26 shows the impact of a lack of an atmosphere on the Moon. Because the gravitational pull of the Moon is much weaker, it has lost almost its entire atmosphere. The
comparison between Earth and the Moon is discussed in this chapter’s Problems and Exercises.
If you consider a very small object such as a grain of pollen, in a gas, then the number of atoms and molecules striking its surface would also be relatively small. Would the grain of pollen
experience any fluctuations in pressure due to statistical fluctuations in the number of gas atoms and molecules striking it in a given amount of time?
Yes. Such fluctuations actually occur for a body of any size in a gas, but since the numbers of atoms and molecules are immense for macroscopic bodies, the fluctuations are a tiny percentage of the
number of collisions, and the averages spoken of in this section vary imperceptibly. Roughly speaking the fluctuations are proportional to the inverse square root of the number of collisions, so for
small bodies they can become significant. This was actually observed in the 19th century for pollen grains in water, and is known as the Brownian effect.
Gas Properties
Pump gas molecules into a box and see what happens as you change the volume, add or remove heat, change gravity, and more. Measure the temperature and pressure, and discover how the properties of the
gas vary in relation to each other. | {"url":"https://openstax.org/books/college-physics-2e/pages/13-4-kinetic-theory-atomic-and-molecular-explanation-of-pressure-and-temperature","timestamp":"2024-11-14T10:16:46Z","content_type":"text/html","content_length":"680675","record_id":"<urn:uuid:a0e1e895-0472-4821-b2b9-1ac01ff491c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00282.warc.gz"} |
Towards comprehensive verification of stochastic systems
1st July 2015 to 20th June 2017
In order to develop safe and reliable systems, advanced mathematical models of the systems are often created and their properties formally verified. This requires developing involved algorithms for
verification, because the size of the models and the speed of the computation is often a big challenge. This project is concerned with developing algorithms for the verification of properties of one
particular class of models, called Markov decision processes. These models are useful for formally describing systems exhibiting probabilistic choices and controllable decisions. Probability is
present naturally in many systems, for instance as failure rates of system components, while the controllable choices correspond e.g. to deciding which of the working components to allocate for which
The aim of the verification algorithms for Markov decision processes is to describe the best possible way of controlling the system in order to achieve a given property, or to give the worst-case
scenario. Acknowledging that the properties of systems that are required are often very complex and interlocked, the properties we will consider are given as "multi-objective queries" composed of
several smaller objectives. Such queries can possibly require making complex control decisions. An example of such a query would be to finish the computation as fast as possible (objective 1), while
minimising the amount of energy consumed (objective 2). This gives rise to trade-offs between the objectives, and poses new theoretical challenges.
The project's main aims concern the design of verification algorithms and their implementation, which will be ultimately evaluated on a case-study modelling an energy network. We will start from
theoretical results, proceeding to practically usable algorithms based on machine-learning and approximation techniques. Our algorithms will be developed as part of a freely available open-source
tool. This will be the first tool allowing to combine various types of objectives into one query, and to visualise the result in a user-friendly way.
The outputs of the project will have impact in areas where fail-safe systems are crucial, and where advanced control is required. Such areas include future smart energy grids, healthcare, air traffic
control and trading algorithms.
Principal Investigator | {"url":"http://www.cs.ox.ac.uk/projects/VSS/","timestamp":"2024-11-13T11:52:08Z","content_type":"text/html","content_length":"31562","record_id":"<urn:uuid:315bbb5b-415e-4c83-a67e-464baeb3c269>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00282.warc.gz"} |
Construct a tangent to circle of radius 4 cm through a point P on its circumference. - d1b408tff
CBSE Class 10 Answered
Construct a tangent to circle of radius 4 cm through a point P on its circumference.
Asked by Topperlearning User | 10 Oct, 2017, 09:02: AM
Expert Answer
i. Draw a circle with centre O and radius = 4 cm
ii. Take a point P on the circumference of the circle.
iii. Join OP
iv. Construct angle OPA = 90^o and extend AP to B
v. Then APB is the required tangent to the circle.
Answered by | 10 Oct, 2017, 11:02: AM | {"url":"https://www.topperlearning.com/answer/construct-a-tangent-to-circl/d1b408tff","timestamp":"2024-11-09T15:48:56Z","content_type":"text/html","content_length":"535692","record_id":"<urn:uuid:623e8708-5617-431a-bf46-238789f673f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00157.warc.gz"} |
The church of SM physics - LENR Forum
Time is just a measure of motion, Huxley. And at the event horizon, there isn't any. Surely you must know your picture is wrong because it contains a massive contadiction. The infalling observer is
said to see himself falling through the event horizon, whilst the outside observer doesn't see the infalling observer falling through the event horizon. Both stories can't be right. The issue is not
fixed by fantasy physics where events occur in some never-never land beyond the end of time. But there again, the gullibility of supposedly intelligent people never ceases to surprise me.
I realise this is tedious for some. But maybe worth engaging with John on this topic because some here might end up believing what he says and thus being very misinformed. There was a reason he got
banned from stackexchange for poor quality answers on this topic!
Time is just a measure of motion
Well, no. Proper time (for an inertial frame) measures causal dynamics in a spacetime. Motion is defined by that. But there are multiple proper times for multiple different reference frames, so
nothing "just" about it!
And at the event horizon, there is not any.
This is 100% false. You have shown how in an external to the black hole inertial reference frame all world lines (trajectories of objects) have an asymptote at the event horizon, where they never
actually cross it. but, those same world lines, measured by their own (clock on traveller) proper time, pass through the event horizon in finite time. Saying that there is no time at the event
horizon is false; there is perfectly good time, just that it is decoupled by an asymptote from external times.
Surely you must know your picture is wrong because it contains a massive contadiction. The infalling observer is said to see himself falling through the event horizon, whilst the outside observer
doesn't see the infalling observer falling through the event horizon. Both stories can't be right.
This shows a lack of respect fro reality. Sure - relativistic physics is counter-intuitive and strange (some people, including me, would say wonderful). It is not contradictory once you accept its
tenets - that time is relative and there is no global universal time. Both stories can be right - you can even trace light rays from the infalling object to the external observer and see how they get
stretched asymptotically in external time to make both right. It is lazy not to go through that calculation: if you are capable. If you are not capable then you have no business asserting your
minority viewpoint over 99.9% of people who have done it and understand how this physics works. | {"url":"https://www.lenr-forum.com/forum/thread/6141-the-church-of-sm-physics/?postID=122265","timestamp":"2024-11-08T00:40:21Z","content_type":"text/html","content_length":"182872","record_id":"<urn:uuid:9de35fa1-7e42-4698-a08d-04ee102c04b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00452.warc.gz"} |
SciPy 0.11.0 Release Notes
SciPy 0.11.0 is the culmination of 8 months of hard work. It contains many new features, numerous bug-fixes, improved test coverage and better documentation. Highlights of this release are:
□ A new module has been added which provides a number of common sparse graph algorithms.
□ New unified interfaces to the existing optimization and root finding functions have been added.
All users are encouraged to upgrade to this release, as there are a large number of bug-fixes and optimizations. Our development attention will now shift to bug-fix releases on the 0.11.x branch, and
on adding new features on the master branch.
This release requires Python 2.4-2.7 or 3.1-3.2 and NumPy 1.5.1 or greater.
The new submodule scipy.sparse.csgraph implements a number of efficient graph algorithms for graphs stored as sparse adjacency matrices. Available routines are:
The optimize module has received a lot of attention this release. In addition to added tests, documentation improvements, bug fixes and code clean-up, the following improvements were made:
• A unified interface to minimizers of univariate and multivariate functions has been added.
• A unified interface to root finding algorithms for multivariate functions has been added.
• The L-BFGS-B algorithm has been updated to version 3.0.
Two new functions scipy.optimize.minimize and scipy.optimize.minimize_scalar were added to provide a common interface to minimizers of multivariate and univariate functions respectively. For
multivariate functions, scipy.optimize.minimize provides an interface to methods for unconstrained optimization (fmin, fmin_powell, fmin_cg, fmin_ncg, fmin_bfgs and anneal) or constrained
optimization (fmin_l_bfgs_b, fmin_tnc, fmin_cobyla and fmin_slsqp). For univariate functions, scipy.optimize.minimize_scalar provides an interface to methods for unconstrained and bounded
optimization (brent, golden, fminbound). This allows for easier comparing and switching between solvers.
The new function scipy.optimize.root provides a common interface to root finding algorithms for multivariate functions, embedding fsolve, leastsq and nonlin solvers.
Solvers for the Sylvester equation (scipy.linalg.solve_sylvester, discrete and continuous Lyapunov equations (scipy.linalg.solve_lyapunov, scipy.linalg.solve_discrete_lyapunov) and discrete and
continuous algebraic Riccati equations (scipy.linalg.solve_continuous_are, scipy.linalg.solve_discrete_are) have been added to scipy.linalg. These solvers are often used in the field of linear
control theory.
It is now possible to calculate the QZ, or Generalized Schur, decomposition using scipy.linalg.qz. This function wraps the LAPACK routines sgges, dgges, cgges, and zgges.
The function scipy.linalg.qr_multiply, which allows efficient computation of the matrix product of Q (from a QR decomposition) and a vector, has been added.
A function for creating Pascal matrices, scipy.linalg.pascal, was added.
Two new functions, scipy.sparse.diags and scipy.sparse.block_diag, were added to easily construct diagonal and block-diagonal sparse matrices respectively.
scipy.sparse.csc_matrix and csr_matrix now support the operations sin, tan, arcsin, arctan, sinh, tanh, arcsinh, arctanh, rint, sign, expm1, log1p, deg2rad, rad2deg, floor, ceil and trunc.
Previously, these operations had to be performed by operating on the matrices’ data attribute.
LSMR, an iterative method for solving (sparse) linear and linear least-squares systems, was added as scipy.sparse.linalg.lsmr.
Bindings for the discrete sine transform functions have been added to scipy.fftpack.
For interpolation in spherical coordinates, the three classes scipy.interpolate.SmoothSphereBivariateSpline, scipy.interpolate.LSQSphereBivariateSpline, and
scipy.interpolate.RectSphereBivariateSpline have been added.
The stats module has gained functions to do binned statistics, which are a generalization of histograms, in 1-D, 2-D and multiple dimensions: scipy.stats.binned_statistic,
scipy.stats.binned_statistic_2d and scipy.stats.binned_statistic_dd.
scipy.sparse.cs_graph_components has been made a part of the sparse graph submodule, and renamed to scipy.sparse.csgraph.connected_components. Calling the former routine will result in a deprecation
scipy.misc.radon has been deprecated. A more full-featured radon transform can be found in scikits-image.
scipy.io.save_as_module has been deprecated. A better way to save multiple Numpy arrays is the numpy.savez function.
The xa and xb parameters for all distributions in scipy.stats.distributions already weren’t used; they have now been deprecated.
The scipy.maxentropy module, which was deprecated in the 0.10.0 release, has been removed. Logistic regression in scikits.learn is a good and modern alternative for this functionality.
The spline evaluation function now behaves similarly to interp1d for size-1 arrays. Previous behavior:
>>> from scipy.interpolate import splev, splrep, interp1d
>>> x = [1,2,3,4,5]
>>> y = [4,5,6,7,8]
>>> tck = splrep(x, y)
>>> splev([1], tck)
>>> splev(1, tck)
Corrected behavior:
>>> splev([1], tck)
array([ 4.])
>>> splev(1, tck)
This affects also the UnivariateSpline classes.
The behavior of the y attribute of complex_ode is changed. Previously, it expressed the complex-valued solution in the form:
z = ode.y[::2] + 1j * ode.y[1::2]
Now, it is directly the complex-valued solution:
The T-tests scipy.stats.ttest_ind, scipy.stats.ttest_rel and scipy.stats.ttest_1samp have been changed so that 0 / 0 now returns NaN instead of 1.
The SuperLU sources in scipy.sparse.linalg have been updated to version 4.3 from upstream.
The function scipy.signal.bode, which calculates magnitude and phase data for a continuous-time system, has been added.
The two-sample T-test scipy.stats.ttest_ind gained an option to compare samples with unequal variances, i.e. Welch’s T-test.
scipy.misc.logsumexp now takes an optional axis keyword argument. | {"url":"https://docs.scipy.org/doc/scipy-1.14.1/release/0.11.0-notes.html","timestamp":"2024-11-12T22:18:16Z","content_type":"text/html","content_length":"63515","record_id":"<urn:uuid:7da72c77-6d31-4e1c-9f46-ce9f4657bb17>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00333.warc.gz"} |
{summclust} is an R module for cluster level measures of leverage and influence, and further implements CRV3 and CRV3J cluster robust variance estimators.
For an introduction to the package, take a look at its vignette.
For a quick overview of different CRV estimators, take a look at the cluster robust variance estimation vignette.
For a very detailed description of the implemented methods, in particular a discussion of the different leverage and influence metrics, see:
MacKinnon, J.G., Nielsen, M.Ø., Webb, M.D., 2022. Leverage, influence, and the jackknife in clustered regression models: Reliable inference using summclust. QED Working Paper 1483. Queen’s
For the Stata version of the package, see here.
You can install the development version of summclust from CRAN, GitHub and r-universe with: | {"url":"http://ctan.mirror.garr.it/mirrors/CRAN/web/packages/summclust/readme/README.html","timestamp":"2024-11-05T00:36:15Z","content_type":"application/xhtml+xml","content_length":"8627","record_id":"<urn:uuid:1b6c56a4-325d-4fbe-8d87-c3f4bed2fc0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00425.warc.gz"} |
Charlotte Mason Approach
Excerpt from a Charlotte Mason-friendly article at http://news.ncsu.edu/releases/mathtal/
Many children, when learning to read, are encouraged by their teachers to retell all they remember about a story in order to build their comprehension skills. But can similar comprehension strategies
be applied to mathematics? Researchers at North Carolina State University say that when teachers work with kids to talk through math problems, kids do a better job of absorbing and understanding the
“Current methods of teaching math — with examples and activities — are important, but we’re neglecting the communications component around it,” say Dr. Paola Sztajn, professor of mathematics
education at NC State. “Helping children verbally work through math problems and articulate problem solving is an important aspect of their understanding.”
Math Websites: Khan Academy; youcubed.org | {"url":"https://pennygardner.com/brief-overview-of-charlotte-mason-approach/math-the-charlotte-mason-way/","timestamp":"2024-11-07T19:02:43Z","content_type":"text/html","content_length":"43504","record_id":"<urn:uuid:9430e61f-d9c9-4a1c-ae5e-53b9d5bbe783>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00804.warc.gz"} |
Solving Nonlinear Equation Using Regula Falsi Method
In many science and engineering there is occasions that we need to solve nonlinear equation, that is find the roots of an equation, which is the points where the equation graph meets the x axis. The
regula falsi method (also called false position and linear interpolation methods) is a bracketing method for finding a numerical solution of a nonlinear equation of the form f(x) = 0.
Algorithm For Regula Falsi Method
When Should the Bisection Process Stopped?
An Example
The Python Program
The python program for solving this equation is listed here:
def regula_falsi(f, a, b, tol):
This function implements the Regula Falsi method to find an approximate root of a nonlinear equation.
f: The function for which we want to find the root.
a: Lower bound of the initial interval.
b: Upper bound of the initial interval.
tol: Tolerance level for the approximation.
The approximate root of the equation within the given tolerance.
fa = f(a)
fb = f(b)
if fa * fb > 0:
print("The given interval does not contain a root.")
return None
iterations = 0
while abs(b - a) > tol:
c = (a * fb - b * fa) / (fb - fa)
fc = f(c)
if fa * fc < 0:
b = c
a = c
iterations += 1
print(f"Root found approximately after {iterations} iterations: x = {c}")
return c
# Define the function
def f(x):
return x**3 - x - 1
# Set initial interval and tolerance
a = 1
b = 2
tol = 0.001
# Find the approximate root
root = regula_falsi(f, a, b, tol)
if root:
print(f"f({root}) = {f(root):.4f}")
1. The regula_falsi function takes the equation (f), the initial interval (a, b), and the tolerance (tol) as input.
2. It checks if the function values at a and b have the same sign. If they do, there’s no root in the interval (since the function goes from positive to negative or vice versa when the root is
3. It iteratively calculates a new point (c) within the interval based on the function values at a and b. This is essentially the x-intercept of the line connecting the points (a, f(a)) and (b, f
4. It checks the signs of f(a) and f(c). If they are opposite, the root lies between a and c, so we update the upper bound (b) to c. Otherwise, the root lies between c and b, so we update the lower
bound (a) to c.
5. This process continues until the difference between a and b becomes less than the tolerance (tol), indicating we’re close enough to the root.
6. The function then prints the number of iterations and the approximate root along with its function value (f(root)) for verification (ideally close to zero).
This program demonstrates how to use the Regula Falsi method to solve the given cubic equation ( f ( x ) = x 3 - x - 1 = 0 ). You can modify the code to try the method with other nonlinear equations
by changing the definition of the f function.
Additional Notes About Regula falsi Method
• The method always converge to an answer, if a root is trapped in the interval [ a , b ] .
• Frequently, the function in the interval [ a , b ] is either concave up or concave down. In this case, one of the endpoints of the interval remains the same during the iteration, while the other
endpoint advances to ward the root. In other words, the numerical solution advances toward the root only from one side. The convergence toward the solution could be faster if the other endpoint
would also move toward the root. Several modifications have been introduced to the regula falsi method that makes the sub-interval in successive iteration approach the root from both sides. We
will discuss these methods in later blog posts. | {"url":"https://bastakiss.com/blog/numerical-methods-16/solving-nonlinear-equation-using-regula-falsi-method-136","timestamp":"2024-11-05T16:07:22Z","content_type":"text/html","content_length":"61497","record_id":"<urn:uuid:f227d5b4-e66e-423b-9ddc-732eb0c662de>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00509.warc.gz"} |
Two-dimensional test cases of ISO 10211
WUFI^® 2D computes the time-dependent temperature and moisture fields in a two-dimensional cross-section of a building component. Such a two-dimensional computation automatically includes so-called
geometrical and structural thermal bridge effects. These are the effects which the component shape (e.g. a corner) and variations of the thermal properties within the component (e.g. reinforcement
bars) have on the resulting temperature field. The modifications of the temperature field and the associated heat flows by thermal bridges may have important consequences for energy loss, mold growth
in damp corners etc.
WUFI^® 2D is not intended to compete with dedicated thermal bridge programs which usually offer more flexible modelling interfaces and provide specific thermal bridge properties. But it can
investigate the effect of thermal bridges on energy losses and, in particular, on the hygric conditions in and on building components (mold growth, damage due to condensation etc.), which purely
thermal programs can not.
The international standard ISO 10211 provides a series of two- and three-dimensional test cases for validating thermal bridge software. Of course, WUFI^® 2D should be able to reproduce the
two-dimensional test cases.
Test case 1
Case 1 considers one half of a symmetrical square column with known constant surface temperatures. The steady-state temperature distribution over the cross-section can be computed analytically. 28
temperatures on an equidistant grid are given by the standard as the reference solution; the software to be validated must reproduce these temperatures within 0.1° C.
20° C along the top edge, 0° C along the left and bottom edges, and adiabatic along the right edge) result in a temperature field with strong variation close to the upper left corner and little
variation towards the bottom of the cross-section.
Usually, one would therefore create a more efficient variable-size computational grid with fine grid elements close to the upper left corner and progressively larger grid elements towards the bottom.
However, since for the present purpose the temperatures must be evaluated at precisely given coordinates and WUFI^® computes the temperatures for the centers of the grid elements, a grid has to be
created which ensures that a small element is centered at each of the requested positions.
To this end, the monolithic component has been built up from 28 separate blocks which fill up the space between the reference points and are subdivided by relatively coarse grids. The blocks are
separated by 4 mm wide gaps which are subdivided by very fine grids in such a way that one small grid element is precisely centered on each reference point (one at each intersection of the gaps).
Since in the present case of a monolithic component with given surface temperatures the steady-state solution for the temperature field does not depend on the thermal properties, any arbitrary
material data may be used. The data of concrete were chosen for this exercise.
The prescribed surface temperatures are applied to the component by setting the ambient air to the desired temperatures and the heat transfer coefficients for the surfaces to very large values. For
the adiabatic right-hand surface (the symmetry plane which allows to limit the calculation to one half of the original square column), the heat transfer coefficient has been set to zero in order to
suppress any heat exchange.
WUFI^® 2D has no mode for steady-state solutions, but such a solution is approached to arbitrary precision if a transient computation with constant boundary conditions is performed for a sufficient
number of time steps. Here, 10 steps of 48 hours each were found to be sufficient. Moisture transport was switched off for this purely thermal computation.
A graphical postprocessor for the calculation results allows to extract the final temperatures from any grid element. The comparison with the reference temperatures shows that WUFI’s temperatures
deviate by 0.05° C or less and are thus well within the allowed deviation of 0.1° C.
You can download a WUFI^® 2D project file (65 KB) for the benchmark calculation. Use the “Import…” function of WUFI^® 2D to read this compressed archive file.
Test case 2
Case 2 considers the heat flow through a building component which contains materials with widely differing thermal conductivities.
Dimensions (mm):
AB= 500 CD = 15 EM = 40 IM = 1.5
AC = 6 CF = 5 GJ = 1.5 FG-KJ = 1.5
Thermal conductivities (W/mK):
1 (concrete): 1.15 2 (wood): 0.12 3 (insulation): 0.029 4 (aluminum): 230
Boundary conditions:
AB: 0° C with R[se] = 0.06 m²K/W HI: 20° C with R[si] = 0.11 m²K/W
The prescribed building component can easily be assembled from rectangles, using WUFI’s graphical component editor. The automatically generated grid with the fineness setting “coarse” is sufficient
for this computation.
The thermal conductivities for the four involved materials are specified by the standard and have been entered in WUFI^® accordingly. Since WUFI^® needs a full set of thermal porperties for each
material (including heat capacity etc.) the missing data have been taken from similar materials in WUFI’s material database. The steady-state result only depends on the prescribed thermal
conductivities, not on the added properties.
The ambient air temperatures and the heat transfer coefficients for the top and bottom surfaces are entered in WUFI^® 2D as specified by the standard. The left and right surfaces are treated as
adiabatic by setting the respective heat transfer coefficients to zero.
In this case, too, the steady-state solution must be approximated by a transient calculation with constant boundary conditions. 30 steps of one hour each were found sufficient.
The resulting temperatures at the specified locations can again be extracted with the graphical postprocessor. However, the standard asks for temperatures on material boundaries whereas WUFI^®
computes the temperatures for the centers of the grid elements and any material boundaries must always coincide with boundaries between grid elements. So in this case it is not possible to center
grid elements on the requested locations (such grid elements would contain two or more different materials which is not allowed).
Close to the four corners (points A, B, H and I), the temperature variation is so small that the center of the outermost grid element instead of the true geometric corner can be taken as sufficiently
Where the requested location lies between two materials (points C, E and F), the temperature at this location must be computed from the temperatures at the centers of the two grid elements straddling
the location. The temperature ϑ[m] for a location m between locations 1 and 2 can be computed by ϑ[m] = ((λ[1]/s[1]) ϑ[1] + (λ[2]/s[2]) ϑ[2]) / ((λ[1]/s[1])+(λ[2]/s[2])), where ϑ[i] is the
temperature at location i, λ[i] is the thermal conductivity between locations i and m, and s[i] is the distance between locations i and m. Since the grid elements straddling a material boundary all
have the same size, the s[i] cancel and the expression reduces to ϑ[m] = (λ[1] ϑ[1] + λ[2] ϑ[2]) / (λ[1]+λ[2]).
For locations where three materials meet (points D and G), the temperature has been computed from the temperatures in the four adjacent grid elements by the following generalisation of the above
formula: ϑ[m] = (λ[1] ϑ[1] + λ[2] ϑ[2] + λ[3] ϑ[3] + λ[4] ϑ[4]) / (λ[1] + λ[2] + λ[3] + λ[4]).
The comparison with the reference temperatures shows that WUFI’s temperatures deviate by 0.1° C or less and are thus within the allowed deviation of 0.1° C. The heat flow through the component is 9.5
W/m and thus within the required (9.5 ± 0.1) W/m.
You can download a WUFI 2D project file (30 KB) for the benchmark calculation. Use the “Import…” function of WUFI^® 2D to read this compressed archive file.
Last Update: November 7, 2024 at 18:04 | {"url":"https://wufi.de/en/2015/04/09/two-dimensional-test-cases-of-iso-10211/","timestamp":"2024-11-08T15:21:31Z","content_type":"text/html","content_length":"36261","record_id":"<urn:uuid:0db3a33b-a280-4a9c-86e7-961c17a4ac44>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00877.warc.gz"} |
N Evol Abs | OpenHealth Knowledge Base
N evol abs corresponds to the difference, in absolute value, of N and N-1
Calculation formula
N Evol abs describes, for example, the growth of the market, in number of boxes or sell-out value sales, between periods N and N-1. | {"url":"https://success.openhealth.fr/en/articles/3194457-n-evol-abs","timestamp":"2024-11-15T04:51:15Z","content_type":"text/html","content_length":"55392","record_id":"<urn:uuid:11c864d0-056d-427f-ad6f-2752d8ceb7fc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00372.warc.gz"} |
Undergraduate Programme and Module Handbook 2024-2025
Module PHYS4241: THEORETICAL PHYSICS 4
Department: Physics
PHYS4241: THEORETICAL PHYSICS 4
Type Open Level 4 Credits 20 Availability Available in 2024/2025 Module Cap Location Durham
• Theoretical Physics 2 (PHYS2631) AND Foundations of Physics 3A (PHYS3621).
• Foundations of Physics 4A (PHYS4251) if Foundations of Physics 3A (PHYS3621) was not taken in Year 3
Excluded Combination of Modules
• Theoretical Physics 3 (PHYS3661).
• This module is designed primarily for students studying Department of Physics or Natural Sciences degree programmes.
• It builds on the modules Theoretical Physics 2 (PHYS2631) and Foundations of Physics 3A (PHYS3621) by introducing more advanced methods in electromagnetism that can be used to investigate more
realistic problems and concepts, and by introducing more advanced topics in quantum mechanics as well as addressing further applications and conceptual issues of measurement and interpretation.
• It develops transferable skills in researching a topic at an advanced level and making a written presentation on the findings.
• The syllabus contains:
• Relativistic Electrodynamics: Einstein’s postulates, the geometry of relativity, Lorentz transformations, structure of space-time, proper time and proper velocity, relativistic energy and
momentum, relativistic kinematics, relativistic dynamics, magnetism as a relativistic phenomenon, how the fields transform, the field tensor, electrodynamics in tensor notation, relativistic
potentials, scalar and vector potentials, gauge transformations, Coulomb gauge, retarded potentials, fields of a moving point charge, dipole radiation, radiation from point charges.
• Quantum Theory: Scattering experiments and cross sections; potential scattering (general features); spherical Bessel functions (application: the bound states of a spherical square well); the
method of partial waves (scattering phase shift, scattering length, resonances, applications); the integral equation of potential scattering; the Born approximation; collisions between identical
particles, introduction to multichannel scattering; the density matrix (ensemble averages, the density matrix for a spin-1/2 system and spin-polarization); quantum mechanical ensembles and
applications to single-particle systems; systems of non-interacting particles (Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics, ideal Fermi-Dirac and Bose-Einstein gases); the
Klein-Gordon equation; the Dirac equation; covariant formulation of Dirac theory; plane wave solutions of the Dirac equation; solutions of the Dirac equation for a central potential; negative
energy states and hole theory; non-relativistic limit of the Dirac equation; measurements and interpretation (hidden variables, the EPR paradox, Bell’s theorem, the problem of measurement).
Learning Outcomes
Subject-specific Knowledge:
• Having studied this module, students will have developed a working knowledge of tensor calculus, and be able to apply their understanding to relativistic electromagnetism.
• They will have a systematic understanding of quantum theory, including collision theory and relativistic quantum mechanics.
Subject-specific Skills:
• In addition to the acquisition of subject knowledge, students will be able to apply the principles of physics to the solution of complex problems.
• They will know how to produce a well-structured solution, with clearly-explained reasoning and appropriate presentation.
Key Skills:
• Students will have developed skills in researching a topic at an advanced level and making a written presentation.
Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module
• Teaching will be by lectures and workshops.
• The lectures provide the means to give a concise, focused presentation of the subject matter of the module. The lecture material will be defined by, and explicitly linked to, the contents of the
recommended textbooks for the module, thus making clear where students can begin private study. When appropriate, the lectures will also be supported by the distribution of written material, or
by information and relevant links online.
• Regular problem exercises and workshops will give students the chance to develop their theoretical understanding and problem solving skills.
• Students will be able to obtain further help in their studies by approaching their lecturers, either after lectures or at other mutually convenient times.
• Lecturers will provide a list of advanced topics related to the module content. Students will be required to research one of these topics in depth and write a dissertation on it. Some guidance on
the research and feedback on the dissertation will be provided by the lecturer.
• Student performance will be summatively assessed through an open-book examination and a dissertation and formatively assessed through problem exercises and a progress test. The open-book
examination will provide the means for students to demonstrate the acquisition of subject knowledge and the development of their problem-solving skills. The dissertation will provide the means
for students to demonstrate skills in researching a topic at an advanced level and making a written presentation.
• The problem exercises and progress test will provide opportunities for feedback, for students to gauge their progress, and for staff to monitor progress throughout the duration of the module.
Teaching Methods and Learning Hours
Activity Number Frequency Duration Total/Hours
Lectures 38 2 per week 1 Hour 38
Workshops 17 Weekly 1 Hour 17
Preparation and Reading 145
Total 200
Summative Assessment
Component: Open-book examination Component Weighting: 80%
Element Length / duration Element Weighting Resit Opportunity
Open-book examination 100%
Component: Dissertation Component Weighting: 20%
Element Length / duration Element Weighting Resit Opportunity
dissertation 1500 words 100%
Formative Assessment:
Problem exercises and self-assessment; one progress test, workshops and problems solved therein.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be
subject to the procedures defined in the University's General Regulation V, and may be required to leave the University | {"url":"https://apps.dur.ac.uk/faculty.handbook/2024/UG/module/PHYS4241","timestamp":"2024-11-10T17:31:29Z","content_type":"text/html","content_length":"11293","record_id":"<urn:uuid:f4403a1d-c207-421d-918e-9750c1d1bcff>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00059.warc.gz"} |
The Stacks project
Lemma 15.38.2. Let $k$ be a field and let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. If $k \to A$ is formally smooth for the $\mathfrak m$-adic topology, then $A$ is a regular local
Proof. Let $k_0 \subset k$ be the prime field. Then $k_0$ is perfect, hence $k / k_0$ is separable, hence formally smooth by Algebra, Lemma 10.158.7. By Lemmas 15.37.2 and 15.37.7 we see that $k_0 \
to A$ is formally smooth for the $\mathfrak m$-adic topology on $A$. Hence we may assume $k = \mathbf{Q}$ or $k = \mathbf{F}_ p$.
By Algebra, Lemmas 10.97.3 and 10.110.9 it suffices to prove the completion $A^\wedge $ is regular. By Lemma 15.37.4 we may replace $A$ by $A^\wedge $. Thus we may assume that $A$ is a Noetherian
complete local ring. By the Cohen structure theorem (Algebra, Theorem 10.160.8) there exist a map $K \to A$. As $k$ is the prime field we see that $K \to A$ is a $k$-algebra map.
Let $x_1, \ldots , x_ n \in \mathfrak m$ be elements whose images form a basis of $\mathfrak m/\mathfrak m^2$. Set $T = K[[X_1, \ldots , X_ n]]$. Note that
\[ A/\mathfrak m^2 \cong K[x_1, \ldots , x_ n]/(x_ ix_ j) \]
\[ T/\mathfrak m_ T^2 \cong K[X_1, \ldots , X_ n]/(X_ iX_ j). \]
Let $A/\mathfrak m^2 \to T/m_ T^2$ be the local $K$-algebra isomorphism given by mapping the class of $x_ i$ to the class of $X_ i$. Denote $f_1 : A \to T/\mathfrak m_ T^2$ the composition of this
isomorphism with the quotient map $A \to A/\mathfrak m^2$. The assumption that $k \to A$ is formally smooth in the $\mathfrak m$-adic topology means we can lift $f_1$ to a map $f_2 : A \to T/\
mathfrak {m}_ T^3$, then to a map $f_3 : A \to T/\mathfrak {m}_ T^4$, and so on, for all $n \geq 1$. Warning: the maps $f_ n$ are continuous $k$-algebra maps and may not be $K$-algebra maps. We get
an induced map $f : A \to T = \mathop{\mathrm{lim}}\nolimits T/\mathfrak m_ T^ n$ of local $k$-algebras. By our choice of $f_1$, the map $f$ induces an isomorphism $\mathfrak m/\mathfrak m^2 \to \
mathfrak m_ T/\mathfrak m_ T^2$ hence each $f_ n$ is surjective and we conclude $f$ is surjective as $A$ is complete. This implies $\dim (A) \geq \dim (T) = n$. Hence $A$ is regular by definition.
(It also follows that $f$ is an isomorphism.) $\square$
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General Overview | artmathintelligence
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Generating the Best Ethical Last Generation AI for mathematically-based Scientific and Entrepreneurial Challenges with a Cognitive and Multidisciplinary Basis
“Those who can imagine anything, can create the impossible”
- Alan Turing -
The study of the mind is one of the most fascinating and multifaceted concerns of mankind. So, to obtain effective and useful models explaining and describing its essential features a fully
interdisciplinary approach is needed. Besides, mathematics is, among many others, the language in which the laws of nature seem to be written with maximum precision. Therefore, a strong and mature
formation in pure and applied mathematics represents a huge advantage for starting this enhancing scientific journey through global laws of the mind.
On the other hand, a huge number of fundamental challenges that our societies face every day require ultimately precise solutions to the mathematical models emerging in the corresponding
formalization processes. Now, in most of the cases, such mathematical problems try to be solved solely by human researchers (e.g. mathematicians, theoretical physicists), which limits strongly the
number of the potential problems to be solved and tends to increase enormously the amount of time required to solve these problems.
I (as the creator and leader of this project) am a multidisciplinary researcher in cognitive sciences, human and artificial intelligence, and pure and applied mathematics. One of my biggest passions
is being able to model formally the way in which humans create mathematical ideas, namely, being able to ‘decode’ the processes that happen in our minds precisely when we are producing mathematical
Since a couple of years, I am completely involved in solving one of the most important questions within the common intersection between Artificial Intelligence and Pure (and Applied) Mathematics: The
Meta-challenge that I called Artificial Mathematical Intelligence (AMI), i.e. the problem of creating a robot (i.e. computer program) being able to simulate the way a human being receives, process
and subsequently solves a mathematical problem from a conceptual (as well as numerical) point of view.
A global solution to this meta-challenge would have very strong implications in all the areas conceptually influenced by modern mathematics like medicine, engineering, theoretical and applied
physics, computer science, economics, (parts of) biology and chemistry, cognitive sciences and (mathematical) psychology, among many others. A device of this kind can help enormously for solving a
lot of (new kinds of) problems emerging from the huge amount of information and the accelerating globalization that our world is facing today. Moreover, this would allow us to use all the potential
that pure sciences have, for enlightening and subsequently for solving concrete problems occurring in our society. In fact, in a lot of cases, the research of an interdisciplinary problem is
essentially reduced to the solution of the corresponding (underlying) mathematical problem which emerges at the formalization’s stage of the research.
There is a quite natural and practical question within the foundations and origins of mathematics that needs a deeper answer: how much of current mathematics (i.e. mathematics described in
contemporary (mathematical and closely related) journals) can be completely generated by a computer program?
In other words, how near are we for constructing (programming) a machine which can simulate the way a modern researcher usually faces a solvable mathematical conjecture, works some time on it, and
finally finds a formal solution for it?
Here it is important to clarify that the main purpose of the former question is to ‘meta-model’ and to ‘meta-simulate’ how a human being (abstractly) ‘handles’ regarding the specific intellectual
activity of receiving a concrete conjecture (which typically can be solved within a standard mathematical framework), working on it and finally giving a clear answer, i.e., writing a (formalizable)
solution to the conjecture in the form of either a proof or a counterexample (or sometimes a proof of its ‘independence’ from the corresponding axiomatic system).
Besides, because the former questions involve implicitly the human’s mind, one should take inspiration from the current most relevant cognitive theories concerning mathematical reasoning and closely
related matters. The most successful theories that we currently have for understanding how our mind works are theories with a formal computational conceptual basis like the computational theory of
mind. So, this fact can be seen as a form of ‘heuristic’ support for the thesis that it is possible to meta-model, (formally and computationally) the intellectual (mathematical) job of a human being.
A second theoretical support is the fact that modern mathematics is essentially founded and (at some level) 'conceptually delimited' on (Zermelo-Fraenkel) Set Theory with Choice (ZFC), proof,
recursion, and model theory.
This means that the solution of a solvable conjecture should be precisely described as a formal (logical) consequence of the axioms of ZFC, using a finite (or recursively generated) number of
inference rules and initial premises.
In other words, when a person finally finds a correct solution of a conjecture, then the result of his/her research can be simulated (just formally) simply a kind of computation of a
theoretically-feasible computer program, which starts to run all the possible proofs of provable theorems of ZFC, starting from a finite sub-collection of axioms and following precise (logical)
mechanical deduction rules. Here it is important to mention that at the beginning one can focus essentially on solvable conjectures, i.e., on problems having an explicit formal proof or
counterexample within the ZFC framework. These problems constitute mathematics being studied by at most of the mathematicians today, and these are the ones producing most of the concrete
applications. Moreover, after finding initial solutions for the former collections of questions, one can have initial evidence in order to meta-model also the way in which people generate and prove
‘undecidability’ of (some) conjectures.
So, Artificial Mathematical Intelligence deals with the construction (implementation) of a computer program being able to generate a human-style solution of essentially every human solvable
mathematical conjecture in less time than an average professional mathematician; and subsequently to generate also human-style independence proofs for undecidable conjectures. Now, although concerns
about undecidable conjectures are clearly important from a purely theoretical perspective, within our AMI meta-project the decidable sentences will have an initial central focus of attention.
So far, we have a considerable amount of significant evidence in favour of AMI. For instance, the co-discovery with the help of programs of the notions of refactorable Numbers, Multiplicative Rings
(sometimes known as Containment-Division Rings), the (quasi-)integer and (quasi-)complex numbers and prime ideals (over Dedekind domains). The artificial generation of seminal notions of Fields and
Galois Theory like fields, fields extensions, group of automorphisms of a field and meta-Galois Group of a field extension. Furthermore, we have developed and improved software being able to simulate
analogy making and conceptual blending in initial mathematical theories. Specifically, we have co-discovered with the help of several computer programs partial axiomatizations of the integers,
commutative rings with unity with compatible divisibility relations, and Goldbach rings.
For more specific information regarding the above results, the reader can consult the following webpage: https://dagomez1982.wixsite.com/misitio
The consequences of a robust and effective AMI-software goes far beyond mathematical research. Galileo's quote "the book (of Nature) is written in mathematical language" is still alive. In fact, the
power and usefulness of mathematical models have proved to be a very successful tool in areas like economics, statistics, computer sciences, biology, chemistry, sociology, psychology, theoretical and
experimental physics, engineering, cognitive sciences, global planning, cosmology, earth sciences, among many others. So, the AMI-system would save a lot of resources in terms of time and energy for
lots of researchers and professionals during their work. Thus, they can concentrate mainly on the formal modeling part of their research. On the other hand, their solving skills will increase
tremendously, because with AMI they would be able of come up with more sophisticated mathematical frameworks for their particular fields of research without having to complete a formal mathematical
training before, which could require sometime years of systematic effort.
Furthermore, the same thing can be said not only for scientists but for professionals working in a huge variety of companies, which encounter each day more and more sophisticated challenges needing
clear, specialized and mathematically-grounded solutions.
Lastly, a lot of (high-)school students can obtain great benefit of the AMI-software during their intellectual development, since they could interact with it on a regular basis and with their own
speed of comprehension to increase their math-skills in a personal fashion, and without the monotonic way in which a lot of them perceive the process of learning mathematics at school.
To achieve AMI it is necessary to fulfil successfully the following sub-projects: the development of a more suitable syntactic-semantic formal framework for the foundations of mathematics, the
identification and formalization of a global taxonomy for the primary and secondary cognitive mechanisms used in mathematical creation/invention, the development (almost from scratch) of a
human-style (cognitively inspired) software being able to model human mathematical creation/invention/interaction in a ‘user-friendly’ way, a significant amount of study-cases covering the most
important mathematical research’s fields and a suitable global computationally-feasible meta-formalism for structuring and implementing AMI, among others.
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DIFF - Time Series Difference Operator
Returns an array of cells for the differenced time series (i.e., $(1-L^S)^D)$.
DIFF(X, Order, K, D)
is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
is the time order in the data series (i.e. the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Order Description
1 Ascending (the first data point corresponds to the earliest date) (default).
0 Descending (the first data point corresponds to the latest date).
is the seasonal difference order (e.g., K = 0 (no lag), S = 1 (1st lag), etc.) If missing, the default value of one is assumed.
is the number of repeated differencing (e.g., d = 0 (none), d = 1 (difference once), 2 = (difference twice), etc.). If missing, the default value of one is assumed.
1. The DIFF operator is defined as follows: $$Y_t=\left(1-L^k\right)^d X_t$$ Where:
□ $\left[y_t\right]$ is the difference time series.
□ $\left[x_t\right]$ is the input time series.
□ $L$ is the lag operator.
□ $k$ is the seasonality length.
□ $d$ is the difference order.
2. The size of the output differenced time series is equal to the input time series, but with the first $s \times d$ observations are set to missing (i.e., #N/A).
3. The seasonal difference order (i.e., $k$) must be non-negative and smaller than the time series size (i.e., $T$). $$0 \leq k \leq T-1$$
4. The input time series is homogenous and equally spaced.
5. The time series may include missing values (e.g., #N/A) at either end.
Files Examples
Related Links
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Can anyone help me solve this algebra porblem? - Ask Spacebar
Can anyone help me solve this algebra porblem?
-4 2/3=2n/3+1/3+n/3
Negative four and two thirds equals two "n" over three plus one over three plus "n" over three.
Views: 0 Asked: 01-04 23:51:15
On this page you can find the answer to the question of the mathematics category, and also ask your own question
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Statistical Models in Finance: Applications to Price Change and Credit Risk
Item 学位論文 / Thesis or Dissertation(1)
公開日 2010-02-22
タイト Statistical Models in Finance: Applications to Price Change and Credit Risk
タイト Statistical Models in Finance: Applications to Price Change and Credit Risk
言語 en
言語 eng
イプ識 http://purl.org/coar/resource_type/c_46ec
資源タ thesis
高橋, 久尚
× 高橋, 久尚
タカハシ, ヒサナオ
× タカハシ, ヒサナオ
TAKAHASHI, Hisanao
× TAKAHASHI, Hisanao
与機関 総合研究大学院大学
学位名 博士（学術）
述タイ Other
内容記 総研大甲第647号
値 数物科学研究科
値 15 統計科学専攻
与年月 2003-03-24
値 2002
述タイ Other
The aim of this study is to make simple statistical models to analyze the risks in finance. Comparing with real data of the exchange rate between the US dollar and the Japanese yen, we make
three simple models to explain the distribution of price change and the interaction of traders.<br /> In the first model, we consider change (difference, returns) in stock index prices and
exchange rates for currencies. These are said, from empirical studies, to be distributed by a stable law with a characteristic exponent α< 2 for short sampling intervals and by a Gaussian
distribution for long sampling intervals. To explain this phenomenon, we introduce an Ehrenfest model with large jumps (ELJ), which explains the empirical density function of price changes
for short time intervals as well as for long time intervals. In chapter 3, we discuss mathematical details and related problems of ELJ.<br /> The second model is a majority orienting model
which we introduce to show the majority orienting behavior of the traders in a market. It seems that the interaction among the traders must exist not only at the time of crashes and bubbles
but also at the time of usual trading. And the interaction makes the time series of the market prices such as an exchange rate between the US dollar and the Japanese yen a typical
trajectory.<br /> The third model is the majority orienting model with feed back process which we introduce to understand the oscillation of the market price. We study a simplified market
内容記 in which the dealers' behavior changes by the influence of the price. We show that in such a market, the price oscillates perpetually by applying the van del Pol equation which is obtained
述 from a deterministic approximation of our model.<br /> The advantage of these models is it is easy to understand the connection with real market such that there are N agents, each of whom
is in one of two possible microeconomic states. To explain volatihty clustering, trend of market and non-symmetrical trading is left for our further study. To combine the above models may be
also an interesting next problem.<br /> We also study a default probability of companies by applying the Iogit model to the data from the database as a starting point of making stochastic
model on the default of a company. It is important in credit risk management to determine the probability of bankruptcy. Few reliable analyses of bankruptcy have been developed for small and
medium-sized enterprises because of the delay in developing of databases to capture credit risks for these enterprises. Recently, a large-scale database for estimating credit risks for such
enterprises has become available as "Credit Risk Database". We use the Wald statistic to evaluate the significance of the model's parameters. We discuss the differences in
explanatory factors of credit risk depending on the enterprise scale. In general, financial data for small and medium-sized companies contain many missing data, and many statistical
difficulties are caused by it. To avoid these difficulties, 0-1 dummy variables were incorporated into the Iogit model. This method can be also interpreted that the condition of a certain
company's missing data in its financial indices is valuable for predicting a company's default. To make a simple stochastic model for this problem is our next problem.
値 有
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Graph minors XXIII. Nash-Williams' immersion conjecture
We define a quasi-order of the class of all finite hypergraphs, and prove it is a well-quasi-order. This has two corollaries of interest:•Wagner's conjecture, proved in a previous paper, states that
for every infinite set of finite graphs, one of its members is a minor of another. The present result implies the same conclusion even if the vertices or edges of the graphs are labelled from a
well-quasi-order and we require the minor relation to respect the labels.•Nash-Williams' "immersion" conjecture states that in any infinite set of finite graphs, one can be "immersed" in another;
roughly, embedded such that the edges of the first graph are represented by edge-disjoint paths of the second. The present result implies this, in a strengthened form where we permit vertices to be
labelled from a well-quasi-order and require the immersion to respect the labels.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
• Graph minors
• Immersion
• Well-quasi-order
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Desystemize #9
What do revolutionary new Sudoku techniques teach us about real-world problem solving?
On the heels of Representation and Uncertainty telling us to think more critically about our ontologies, I’d like to share a success story of ontological remodeling – that is, using new language to
describe an existing thing. The existing thing in this case is the puzzle game Sudoku, which might seem a bit too straightforward to be described in a new way. You’re placing the digits 1 to 9 in a
9x9 grid without repeating a digit in a row, column, or box. There doesn’t seem to be space for unrepresented details in the way we’re interested in.
But here’s a quote from a January 2022 video of a man named Simon solving a Sudoku:
I now understand how to explain what’s going on in the puzzle. What I can tell you, what we’re going to end up with here, is that this digit, this digit, and this digit are a 1, 2, 3 triple. And
isn’t it amazing? This, to me, is quite beautiful. I can tell you without fear of contradiction, if I had tried this puzzle two years ago…well, unless there is something else here that I’m not
spotting, I might never have spotted that. Yet nowadays, because set equivalence theory has become so ingrained and practiced, it’s a very quick spot. And that being a 1, 2, 3 triple is just
remarkably gorgeous. So how do we prove this is a 1, 2, 3 triple…? I mean, this puzzle could appear in the World Puzzle Championship now. There’s no way this could have appeared a few years ago,
no one would have been able to do it! But now, that break-in is so well-signaled it’s almost fair!
We haven’t made any breaking new discoveries about the digits 1 to 9. But this set equivalence theory, whatever it is, has changed how Simon views puzzles so much that it’s not even comparable to two
years ago. It’s all still just digits in a grid, but describing them differently leads you to a new conclusion. And precisely because Sudoku is so simply defined, the case study of set equivalence
theory is worth studying. If we can find hidden secrets here, in a 9x9 grid of single digits, we can find them anywhere.
Let’s start by explicating the existing, “normal” ontology for a Sudoku. The rules specify that you have exactly one of each digit from one to nine in every row, column, and box – so it’s no surprise
that these are the usual terms used to describe Sudoku puzzles. For example, you might say “I know that row 2 needs a 3, and it can only go in to this square,” or “Box 5 already has a 9, so the 9
can’t be in any of the other cells in box 5.” Because the explicit impact of the rules is based on these sorts of divisions, it’s a very useful ontology to have.
But that’s not the only way you could describe a Sudoku. You could partition the grid into an orange ostrich head and a fat green frog:
There are things you can say about these shapes. No digit can appear in the ostrich head more than three times, because its only in three boxes total. A digit that appears on one of the frog legs
won’t appear on the other, because they’re in the same row. While these claims are true, they’re ultimately just pointing back to the words we were comfortable with already. Drawing the ostrich and
the frog didn’t really accomplish anything.
Ontologies can’t exactly be wrong, since they’re just methods of description. But it’s pretty obvious that this one isn’t useful. It doesn’t let you state anything new that you couldn’t state using
your previous vocabulary. It’s just an unnecessary level of abstraction that has to be peeled away to get to the useful ontology.
Here’s another way you could partition the sudoku: one blue ring in the center, and four orange anchors in the corners.
Is this any better than our frog and ostrich? Does it let us make any statements that we couldn’t easily make with row, column, and box? As it happens, yes. You can truthfully make the following
statement: “The digits in the ring are exactly the same as the digits in the four anchors.”
Wait, what?
This surprising fact is called “Phistomofel’s Theorem”, named for the setter who initially popularized it. It’s a specific example of set equivalence theory that helped many people (myself included)
understand it. Imagine that the letters 1 to 9 were all on tiles, like in a game of Scrabble. Each row, column, or box has the tiles 1 to 9 once each. Suppose you took the tiles in these two rows and
these two boxes and put them in a blue sack:
Row 4, Row 7, Box 4, and Box 6 (image mislabeled) each have a run of the digits 1-9, so in total the blue sack has 36 tiles, four copies of each digit. Now, take a clone of your puzzle. This time,
put these four columns in an orange sack:
Since both sacks have 4 runs of the digits 1 to 9, they have the same composition of digits inside them. But now, let’s superimpose the contents of the two sacks on to one grid, and look at the cells
with both colors:
The cells with both colors are in both sacks. If you take the exact same cell out of both sacks, then clearly the two identical sacks will remain identical – you took the same thing out of each of
them. You can do this for every cell with two colors. So, removing them all from our superimposed grid, we’re left with this:
That’s how we know that the digit composition of these regions – the ring and the anchors - are identical. And that means that this is a useful upgrade to our ontology, because now we can make
statements like “There are two sixes in the ring, and only two spots in the anchors where a six can go, so there must be a six in each of them.” Those statements would be extremely tedious and
unintuitive to make solely in terms of rows, columns, and boxes.
Once you’ve gotten the hang of Phistomofel’s Theorem, it’s a relatively short jump to using set equivalence theory in general. When we were picking runs of the digits 1-9, we made those particular
choices because they happened to have a lot of overlap to them. But the core logic works for any regions that you make out of complete sets of the digits 1 to 9 – you can always eliminate the overlap
and say that what’s left of the two regions are equal.
On its own, this is a neat bit of Sudoku trivia that might happen to pop up now and then in a puzzle. But what’s really interesting is when a puzzle is designed around this phenomenon. Setters (the
term of art for people who design hand-crafted Sudoku puzzles for solvers to work through) can make a puzzle that’s virtually impossible using just rows, columns, and boxes, and can only be solved by
upgrading your ontology. The Youtube channel Cracking the Cryptic has been an excellent source documenting this evolution.
Now I can reveal where that quote up top came from: a video from Cracking the Cryptic called How to Make an Impossible Sudoku Easy. (I’ll be spoiling the break-in to the puzzle, so if you’re
interested in trying it on your own first, do it now before reading on or watching the video.) Technically speaking, this isn’t quite a normal Sudoku, but a variant with some special rules:
The variant itself adds some words to our ontology. “This two-cell cage sums to 12, so it can’t have a 1 or a 2 in it.” “This line has a 5 on one end, so it must have a 5 on the other end.” “This
digit is a 6, so the one on the other side of the dot is a 5 or a 7.” But such straightforward interpretations of the variant rules aren’t what interest us here. If Simon had seen these rules two
years ago, he wouldn’t have struggled to notice what they mean. Set equivalence theory is special in that it’s not a straightforward application of set rules. It’s a new way of seeing, one that you
need to know to invoke instead of seeing an obvious hint for.
In this case, what’s interesting is the interaction of set equivalence theory with the palindromic lines. Start with a column set that includes parts of all 6 lines: columns 1, 4, and 7.
Next, get a row set that includes parts of all 6 lines - rows 3 and 6:
This is a little different from how we did Phistomofel’s Theorem. The orange and blue sets aren’t equal exactly – we needed three lots of 1 to 9 for blue, while two were enough for orange. Instead,
the claim here is that we know how much greater blue is than orange – specifically, it has exactly one more run of 1 to 9 than orange.
We start by removing overlapping cells:
But we can go much further than that. Since we made sure to include the palindromic lines, we know that the blue segments of each line cancel out with the orange segments of the line. This leaves us
with no orange digits at all, and 9 blue ones:
This means we know the composition of the blue digits. They’re exactly one set of the digits 1 to 9. The sum of the digits 1 to 9 is 451, and the top three cages sum to 13, 12, and 14. Since 13 + 12
+ 14 = 39, the remaining digits must sum to 6. We know our only options are the digits 1 to 9 once each, which means that there’s only one way to do it – putting 1, 2, and 3 in the bottom three
cells, in some order.
After finding this break-in, Simon remarked:
It’s not tricky – well, actually, I’m going to qualify that, because that’s a silly thing for me to have said. It’s absolutely impossible if you’ve not seen it before. Well, not impossible –
virtually impossible. Certainly, probably impossible to do it at speed. But if you’ve seen it a few times before, it’s second-nature.
This is the hallmark of problems solved by ontological remodeling. You don’t want to say they’re tricky, exactly, because the new framework makes them feel pretty approachable. But without the new
framework, they’re basically impossible. Trying to describe the difficulty of these problems is something of a trap, because so much of the difficulty depends on the description. Instead, you need to
play around with new forms of expression and see which patterns are easy to describe with those forms.
The sheer simplicity of Sudoku makes it an excellent example for how powerful ontological remodeling can be even when it seems obvious which form to represent things in. Even if your generating rules
are all related to rows, columns, and boxes, they can interact in such a way that you need new language to describe the interaction. In fact, this has become something of a sport on the channel, with
puzzles like this one that start by showing a computer solver unable to solve a classic Sudoku logically (only getting the solution through brute forcing all possibilities), then setting a human
loose on the puzzle to discover the hidden trick.
That’s as pure and straightforward a demonstration of ontological remodeling as you’re ever likely to get. Sudoku is a game whose rules only invoke rows, columns and boxes, but a computerized solver
can’t describe the solve path with rows, columns, and boxes. The interplay of those rules leads to higher order patterns, and those higher order patterns need higher order language. But precisely
because this example is so straightforward, it can give us a deceptive idea of how easy it is to notice that your ontology is lacking. While the solvers are in an ostensible “conflict” with the
setters, it’s a thin bit of fun narrative over what is fundamentally a collaborative process. The setter wants people to notice the clever logical leap they demanded of their solvers, and that means
they need people to solve it. Getting the warning that the computer can’t solve it gives you the enormous clue that you’ve got to try something new. You may not know what immediately, but you’re
looking for the right sort of thing.
In fact, while there are plenty of puzzles that are totally incomprehensible without set equivalence theory, Phistomofel’s theorem itself has more tentative origins. Phistomofel created a puzzle that
was best solved using the ring, but only published it when he was sure it could be solved “normally”. The ring was not a finding that solvers worldwide spontaneously developed when faced with a
single, staggeringly beautiful puzzle. Instead, the origin story is far more pedestrian: Phistomofel talked about the ring in a forum post. Discussion followed that took the specific ring and anchors
and generalized them into more generic statements about 1-9 sets. Only after the concept had reached a critical mass of socialization were most authors comfortable making puzzles where set
equivalence theory was truly required.
So as great as puzzles are for practicing these techniques, you should never mistake a setters kindness for the indifferent illegibility of real-world problems. There are uncountably many things
we’re thinking about the wrong way, but nature has no Phistomofel on deck nervously ensuring that it’s at least kind of possible to do it through brute force or starting a discussion with a helpful
example. All we get are a bunch of problems that seem impossibly hard. Nothing about the world gives us an easy rule for which things are actually irreducibly tricky and which things just need better
forms of description. That’s why it’s so important to cultivate an intuitive appreciation for the power of ontological remodeling. The reason this technique is so under-discussed and under-used is
precisely because there’s no well-defined way to tell that it’s needed for a given problem. Individuals just have to take that leap of faith themselves, come up with a new way of seeing, and then
demonstrate that problems that were once intractable are suddenly workable. We call those people “visionaries”, and we sure could use some more of them these days.
Thanks to Simon, Philip Newman, and Sam Cappleman-Lynes for reviewing a draft of this article.
This is a secret that Simon only tells his closest friends.
I think your description of the first blue sack is wrong. Shouldn't it be Row 3 + Row 7 + Box 4 + Box 6?
Expand full comment
This is such a cool article!!!
It got me wondering, as a mathematician, where the line is drawn between seeing a problem from a new angle and formal ontological remodeling. For example, the duality between stochastic
differential equations (SDE) describing random walks and partial differential equations (PDE) describing the distribution of possible places a random walker could be & how that distribution changes
over time. Would solving an SDE by way of "remodeling" it as a PDE be an example of ontological remodeling, similar to solving Sudoku with rings & anchors instead of rows, columns & boxes?
Thanks again for writing this very clear & beautiful article!
Expand full comment
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Electronic structure of oxygen-deficient high-Tc superconductors: YBa2Cu3Ox (6x8) for Physical Review B
Physical Review B
Electronic structure of oxygen-deficient high-Tc superconductors: YBa2Cu3Ox (6x8)
View publication
With a view to understanding the relationship between oxygen stoichiometry, oxygen vacancy ordering, electronic structure, and Tc in the YBa2Cu3Ox system, we calculated the electronic structure for x
=6, 7, and 8 using the first-principles self-consistent pseudofunction method. All three of these crystals contain O vacancies in the yttrium plane. The two O sites in the basal plane are empty in
YBa2Cu3O6 and occupied in YBa2Cu3O8. Only one of these sites is occupied in YBa2Cu3O7 (ordered vacancy model). After discussing the essential features of the bandstructure and valence charge
distribution for x=6,7, and 8, we explore the consequences of a simple Bardeen-Cooper-Schrieffer (BCS) model which assumes that Tc depends only on the electronic density of states at the Fermi level
N(EF). Using our calculated values of N(EF) for x=6, 7, and 8, and a simple interpolation scheme, we obtain estimates for N(EF) for intermediate values of x and for various degrees of oxygen-vacancy
disorder at fixed values of x. We can account for at least half of the observed variation of Tc with oxygen stoichimetry in the range 6.5<x<7 using physically reasonable values of the BCS parameters.
We can also account for experimental observations that the greater the degree of oxygen-vacancy ordering at a fixed value of x, the higher the value of Tc. Our analysis also provides estimates for Tc
for x between 6 and 6.5, but these estimates would apply to the metallic, rather than the semiconducting phase. Our analysis for x between 7 and 8 suggests that it might be possible to enhance the
value of Tc beyond present-day values by preparing samples for which the O content is as large as possible, and for which the O vacancies are ordered to the highest possible degree. © 1987 The
American Physical Society. | {"url":"https://research.ibm.com/publications/electronic-structure-of-oxygen-deficient-high-tc-superconductors-yba2cu3ox-6x8","timestamp":"2024-11-14T18:01:12Z","content_type":"text/html","content_length":"82310","record_id":"<urn:uuid:7353fb67-6a35-4f88-8f82-4eed88f43c3a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00483.warc.gz"} |
Counting Corners and Sides | sofatutor.com
Counting Corners and Sides
Content Counting Corners and Sides
Counting Corners and Sides
Basics on the topic Counting Corners and Sides
Corners and Sides
What Are Sides and Corners? In this kindergarten geometry text, you can learn all about the corners and sides of two-dimenstional shapes and how to count them. Let’s learn more about sides and
corners of 2d shapes and how to count the corners and sides of shapes with the following explanation. Sides and corners can be found in many shapes. A corner is the spot where lines join together.
Corners can face in or face out.
Sides are the straight lines that make the edge of a flat shape. Many shapes get their names from how many corners and sides they have. Counting corners and sides can help us figure out what kind of
shape we are looking at!
Counting Corners and Sides
Let’s practice counting all the corners and sides on the shape below. Remember, a corner is where two straight lines meet. So, this shape has three corners!
A side is the straight line that makes the edge of a flat shape. So, this shape has three sides!
Corners and Sides – Round Shapes
Does a circle have sides and corners? A circle does not have any corners or sides. Let’s look at this round shape as an example.
There are no straight lines in this shape. So, there is no place for any straight lines to meet! That means this shape has zero corners. Since there are zero straight sides, this shape has no sides
Corners and Sides – Summary
Remember these facts about corners and sides:
Fact #
1 A corner is the spot where straight
lines join together.
2 A side is the straight line that
makes the edge a flat shape.
3 Many shapes are made up of
corners and sides.
4 Counting corners and sides can help
us know which shape we are looking at.
• A corner is the spot where straight lines join together.
• A side is the straight line that makes the edge a flat shape.
• Many shaped are made up of corners and sides.
• Counting corners and sides can help us know which shape we are looking at.
Corners and Sides – Further Practice
What is the definition of corners and sides in geometry in 1st grade? This text aims to explain corners and sides for first grade students. Have you tried an activity worksheet on how many sides and
corners shapes have? On this website, you can also find sides and corners worksheets and interactive exercises.
Transcript Counting Corners and Sides
Nico and Nia are window shopping for a new attic window! "Wow, how beautiful!" "Look at all the different lines and corners that make up the picture!" "I wonder how many there are?!" Let's help Nico
and Nia by learning about "Counting Corners and Sides!" A corner is the spot where lines join together. Corners can face out like this or face in like this. Sides are the straight lines that makes
the edge of a flat shape. A circle does not have any corners or sides. But, many other shapes do! Shapes can also get their names from how many corners and sides they have. Counting corners and sides
can help us figure out what kind of shape we are looking at! Let's look at this beautiful window and count all the corners and sides on this shape. Remember, a corner is where two straight lines
meet. So, this shape has one, two, three corners! A side is the straight line that makes the edge of a flat shape. So, this shape has one, two, three sides! Here is another shape from the window. How
many corners can we count here? Let's count together: one, two, three, four, five! It has five corners. How many sides can we count here? Let's count together: one, two, three, four, five! It has
five sides. Nia points to another shape. What about this one? How many corners and sides does it have? There are no straight lines in this shape. So, there is no spot for any to meet! That means this
shape has zero corners. How many straight sides does this shape have? Zero! That means that this shape has no sides either! "We'll order this one." While Nico and Nia wait for their new window to
come, let's remember. Today we learned about counting corners and sides. A corner is the spot where straight lines join together. A side is the straight line that makes the edge a flat shape. Many
shapes are made up of corners and sides. Counting corners and sides can help us know which shape we are looking at. "I'm so happy the window finally got here!!" "I can't think of anything more
perfect!" "Me either!"
Counting Corners and Sides exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Counting Corners and Sides .
• Can you match the shape with the correct number of corners and sides?
In red are the corners of this shape.
In red are the sides of this shape.
□ corners are the points where two sides join together.
□ sides are the straight lines that form the edge of a flat shape.
□ Corners are the points where two sides join together.
□ Sides are the straight lines that form the edge of a flat shape.
□ The triangle has three corners and three sides.
□ The rectangle has four corners and four sides.
□ The hexagon has six corners and six sides.
□ The octagon has eight corners and eight sides.
• How many corners and sides does each of these shapes have?
□ sides are the straight edges that make a shape.
□ corners are the points where two sides meet.
Here we can see that this triangle has three corners.
Here we can see that this triangle has three sides.
□ Corners are the points where two sides join together.
□ Sides are the straight lines that form the edge of a flat shape.
□ The yellow star has ten corners and ten sides.
□ The blue flag has five corners and five sides.
□ The purple shape has six corners and six sides.
□ The green square has four corners and four sides.
• Can you find the shapes?
Count the number of corners - the points where two sides meet.
Count the number of sides - the straight lines that form the edge of a flat shape.
Here we can see the correctly highlighted window.
□ Corners are the spots where two sides join together.
□ Sides are the straight lines that form the edge of a flat shape.
• Which group should the shapes be assigned to?
Count the number of corners - the points where two sides meet.
Count the number of sides - the straight lines that form the edge of the shape.
For example, this square has 4 corners and 4 sides so would be assigned to the first group.
Here we can see which group each shape goes into.
□ Corners are the spots where two sides join together.
□ Sides are the straight lines that form the edge of a flat shape.
• Can you label the shape correctly?
Sides are the straight edges that form the shape.
A corner is the point where two sides meet.
□ A corner is the spot where two sides join together.
□ A side is the straight line that forms the edge of a flat shape.
three corners
three sides
• Can you order the shapes?
Find the shape with the fewest corners and sides, and put this first.
Compare each shape to the one above it, and move it up or down so it is in the correct position.
You could make a note of how many corners and sides each shape has, and then order them.
Here we can see how many sides and corners each shape has.
□ Orange triangle: 3 corners and 3 sides
□ Pink parallelogram: 4 corners and 4 sides
□ Green shape: 5 corners and 5 sides
□ Blue shape: 6 corners and 6 sides
□ Yellow shape: 7 corners and 7 sides
□ Red shape: 8 corners and 8 sides
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Course Dynamical Systems
Summer 2014
BMS-Course Dynamical Systems
Prof. Dr. Bernold Fiedler
Recitation sessions: Bernhard Brehm, Jia-Yuan Dai, Anna Karnauhova
Schedule, Summer 2014
Tuesday, 10:15-14:00, Hörsaal B, Arnimallee 22
Bernhard Brehm, Monday 10:00-12:00 Seminarraum 032, Arnimallee 6
Jia-Yuan Dai, Wednesday 10:00-12:00 Seminarraum 009, Arnimallee 6
Anna Karnauhova, Friday 8:00-10:00 Seminarraum 025/026, Arnimallee 6
Written exam / Klausur:
July 15, 2014, Tuesday, 10:00-12:00, Hörsaal B, Arnimallee 22
Results (PDF), upload July 16, 2014, last update July 29, 2014
The exam can be viewed on Tuesday 12-14, August 5, 2014, Arnimallee 7, Room 132 (Bernhard Brehm's office)
Written exam (resit) / Nachklausur:
October 15, 2014, Wednesday, 10:00-12:00, Seminarraum E.31, Arnimallee 7
Results (PDF)
You may review your exams until 03.11.2014. If you want to do so, please send an email to Hannes Stuke.
Pass Criteria
Solve correctly at least 25% of the assignments. Hand in solution attempts for at least 50% of the assignments. Present a correct solution to an assignment on the blackboard in the recitation session
at least once. Pass the written exam.
Students of mathematics or physics, including teachers, from semester 3. Direct access to thesis projects: bachelor, master, dissertation.
Dynamical Systems are concerned with anything that moves. Through the centuries, mathematical approaches take us on a fascinating voyage from origins in celestial mechanics to contemporary struggles
between chaos and determinism.
The three semester course, aimed at graduate students in the framework of the Berlin Mathematical School, will be mathematical in emphasis. Talented and advanced undergraduates, however, are also
welcome to this demanding course, as are students from the applied fields, who plan to really progress to the heart of the matter.
Here is an outline of the first two semesters:
Semester 1:
1. Flows and differential equations
2. First integrals, separation of variables, and the pendulum
3. Examples and applications
4. Existence, uniqueness, and differentiability
5. Linear autonomous systems
6. Omega-limit sets and Lyapunov functions
7. Planar flows
Semester 2:
1. Autonomous and forced oscillations
2. Torus flows
3. Stable and unstable manifolds
4. Shift dynamics
5. Hyperbolic sets
6. Center manifolds
7. Normal forms
8. Genericity and Takens embedding
Depending on preferences of participants, the third semester may address current topics in finite-dimensional dynamics, or give an introduction to infinite-dimensional dynamical system, including
certain partial and delay differential equations.
• K.T. Alligood, T.D. Sauer and J.A. Yorke: Chaos, Springer, 1997.
• H. Amann: Ordinary Differential Equations, de Gruyter, 1990.
• V.I. Arnold: Ordinary Differential Equations, Springer, 2001.
• V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1988.
• W.E. Boyce and R.C. DiPrima: Elementary Differential Equations and Boundary Value Problems, Wiley, 5th edition, 1992.
• S.-N. Chow and J.K. Hale: Methods of Bifurcation Theory, Springer, 1982.
• E.A. Coddington and N. Levinson: Theory of ordinary differential equations, McGill-Hill, 1955.
• P. Collet and J.-P. Eckmann: Concepts and Results in Chaotic Dynamics. A Short Course, Springer, 2006.
• R. Devaney, M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2003.
(This is the updated version of
M.W. Hirsch and S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974.)
• Dynamical Systems I, D.K. Anosov and V.I. Arnold (eds.), Encyclopaedia of Mathematical Sciences Vol 1, Springer, 1988.
• J. Hale: Ordinary Differential Equations, Wiley, 1969.
• B. Hasselblatt, A. Katok: A First Course in Dynamics, Cambridge 2003.
• P. Hartmann: Ordinary Differential Equations, Wiley, 1964.
• A. Katok, B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems, Cambridge 1997.
• F. Verhulst: Nonlinear Differential Equations and Dynamical Systems, Springer, 1996.
• E. Zeidler: Nonlinear Functional Analysis and its Applications, Volume 1: Fixed-Point Theorems, Springer, 1998.
Homework assignments
Please form teams of two and hand in your joint solutions. Please note your name, Matrikelnummer and exercise session (either Anna, Bernhard or Jia-Yuan) on your solutions. Please, please, please
staple your solutions together if you hand in multiple pages.
You can put your solutions into our boxes (Tutorenfächer) in the Arnimallee 3, first floor (just above the library).
Dynamical Systems I: basic questions
• Questions 1 (PDF) | {"url":"http://dynamics.mi.fu-berlin.de/lectures/14SS-Fiedler-Dynamics/index.php?q_menu=0","timestamp":"2024-11-10T17:23:44Z","content_type":"application/xhtml+xml","content_length":"10273","record_id":"<urn:uuid:2074fade-10e5-4fd4-aa15-f14d4210db5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00313.warc.gz"} |
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17 Mind-Blowing Facts About Length Contraction
Source: Youtube.com
When it comes to the mind-boggling world of physics, few concepts can capture the imagination like length contraction. This fascinating phenomenon, also known as Lorentz contraction, is a fundamental
aspect of Einstein’s theory of relativity. In simple terms, length contraction refers to the observation that an object in motion will appear shorter when measured in the direction of its motion, as
compared to its length when at rest.
In this article, we will delve into the intriguing world of length contraction and explore 17 mind-blowing facts about this phenomenon. From its origin in Einstein’s theory to its real-life
implications, we will unravel the mysteries and shed light on the complexities of length contraction. So, get ready to have your mind stretched as we embark on a journey through the fascinating
concept of length contraction in the world of physics!
Key Takeaways:
• Length contraction is a mind-bending concept in physics that challenges our everyday perception of space and time. It shows that reality is much stranger and more intricate than what our senses
can comprehend.
• Length contraction has far-reaching implications, from interstellar travel to the consistency of physics. It forces us to expand our understanding of the universe and confront the strange and
counterintuitive nature of reality.
Length contraction is a result of relative motion.
When an object moves at high speeds relative to an observer, its length appears to contract along the direction of motion. This effect becomes more pronounced as the object approaches the speed of
It is not just an optical illusion.
Length contraction is not merely a visual perception. It is a physical phenomenon that occurs due to the fundamental nature of space and time.
The Lorentz factor determines the extent of length contraction.
The extent of length contraction is determined by the Lorentz factor, which depends on the velocity of the moving object. As the velocity approaches the speed of light, the Lorentz factor becomes
infinite, resulting in the complete collapse of length.
The contraction is only noticeable at high speeds.
At everyday speeds, the effects of length contraction are negligible and not perceptible to the naked eye. It is only at velocities close to the speed of light that the contraction becomes
Length contraction is symmetric.
Length contraction occurs symmetrically along the direction of motion. This means that both the front and back ends of the moving object experience the same degree of contraction.
It affects all objects, regardless of size or material.
Whether it’s a small particle or a massive spaceship, length contraction applies to all objects in motion. It is a universal phenomenon that obeys the laws of physics.
Length contraction is not limited to spatial dimensions.
While length contraction primarily refers to the shortening of an object along its direction of motion, it also applies to other dimensions such as time. Time itself can contract at high velocities.
The concept of length contraction was first proposed by George FitzGerald.
In 1889, Irish physicist George FitzGerald independently proposed the idea of length contraction as a way to explain the null result of the Michelson-Morley experiment, which sought to detect the
motion of the Earth through the luminiferous aether.
Length contraction is a consequence of Einstein’s theory of special relativity.
Einstein’s theory of special relativity, published in 1905, introduced the concept of length contraction as a necessary consequence of the postulates of the theory. It provided a new and profound
understanding of space and time.
Length contraction is not limited to physical objects.
Not only do physical objects experience length contraction, but even lengths measured in mathematical spacetime can contract. This demonstrates the far-reaching implications of the concept.
It violates our everyday intuition about space and time.
Length contraction challenges our everyday perception of space and time. It shows that reality is much stranger and more intricate than what our senses can comprehend.
Length contraction is essential for maintaining the consistency of physics.
Without length contraction, the laws of physics would not remain consistent across different reference frames. It is a crucial component of the mathematical framework that underpins our current
understanding of the universe.
Length contraction can be observed indirectly.
While we may not witness length contraction directly in everyday life, its effects can be observed and measured through various experiments and phenomena, such as particle accelerators and cosmic ray
The Twin Paradox is a consequence of length contraction.
The famous Twin Paradox, where one twin travels at high speeds while the other stays on Earth, can be explained by the effects of length contraction and time dilation.
Length contraction is not limited to forward motion.
Contrary to popular belief, length contraction can occur in any direction of motion, not just forward. It is dependent on the relative velocity between the observer and the moving object.
Length contraction allows for interstellar travel.
The concept of length contraction plays a vital role in the theoretical possibility of interstellar travel. It suggests that as an object approaches the speed of light, its journey through space will
appear shortened, potentially enabling shorter travel times between distant stars.
Length contraction is a mind-bending consequence of our universe’s fabric.
Length contraction forces us to confront the strange and counterintuitive nature of reality. It highlights the profound interconnectedness of space and time and challenges us to expand our
understanding of the universe.
These 17 mind-blowing facts about length contraction shed light on one of the most mind-bending concepts in physics. Whether it’s the symmetric effects on all objects, the violation of our everyday
intuition, or its implications for interstellar travel, length contraction takes us on a journey where the boundaries of our understanding are stretched. It reminds us that the universe is far more
complex, mysterious, and beautiful than we can fathom.
Length contraction is a fascinating concept from the realm of physics that challenges our everyday understanding of space and time. These mind-blowing facts about length contraction shed light on the
intriguing nature of the universe and how it behaves at extreme speeds. From the implications for space travel to the synchronization of clocks, length contraction plays a vital role in our
understanding of the universe.
As we delve deeper into the realm of physics, we uncover more incredible phenomena that continue to astound and captivate us. Length contraction is just one piece of the puzzle that helps us unravel
the mysteries of the cosmos. Whether you’re a physics enthusiast or simply curious about the wonders of the universe, exploring the mind-bending concepts of length contraction will surely leave you
in awe of the world we live in.
1. What is length contraction?
Length contraction is a phenomenon in physics where an object in motion appears shorter in the direction of its motion when observed from a stationary frame of reference.
2. Why does length contraction occur?
Length contraction occurs due to the effects of special relativity. As an object’s velocity increases, its mass and energy also increase, causing the object to contract in the direction of its
3. How does length contraction affect space travel?
Length contraction has significant implications for space travel. As objects approach the speed of light, their length contracts, allowing them to fit into smaller spaces and reducing the distance
they need to travel. This phenomenon is essential for achieving efficient and faster space travel.
4. Can length contraction be observed in everyday life?
The effects of length contraction are only noticeable at speeds close to the speed of light. In our everyday experiences, the speeds at which we move are significantly slower, so the effects of
length contraction are not perceptible in our day-to-day lives.
5. Does length contraction affect time?
Length contraction and time dilation are interconnected concepts in special relativity. As an object’s length contracts, time also dilates for that object. This phenomenon is described by the theory
of time-space geometry.
6. Are there any practical applications of length contraction?
Length contraction has practical applications in fields like particle accelerators and space exploration. It allows scientists to compress particle beams and create more focused and powerful
collisions, leading to new discoveries in fundamental physics. It also aids in designing space probes and spacecraft capable of traveling vast distances more efficiently.
7. Can length contraction be reversed?
No, length contraction is a fundamental aspect of special relativity and cannot be reversed. However, length contraction and time dilation can be compensated for in equations to maintain the
consistency of physical laws.
Discovering the mind-blowing facts about length contraction is just the beginning of your journey into the fascinating world of physics. Dive deeper into the captivating realm of special relativity
and uncover its intriguing secrets. Explore a treasure trove of physics facts that will leave you in awe of the universe's complexity. And don't forget to marvel at the astonishing truths about the
speed of light, which plays a crucial role in shaping our understanding of space and time. Embark on an intellectual adventure that will expand your horizons and challenge your perceptions of
Our commitment to delivering trustworthy and engaging content is at the heart of what we do. Each fact on our site is contributed by real users like you, bringing a wealth of diverse insights and
information. To ensure the highest standards of accuracy and reliability, our dedicated editors meticulously review each submission. This process guarantees that the facts we share are not only
fascinating but also credible. Trust in our commitment to quality and authenticity as you explore and learn with us. | {"url":"https://facts.net/science/physics/17-mind-blowing-facts-about-length-contraction/","timestamp":"2024-11-09T17:02:43Z","content_type":"text/html","content_length":"240583","record_id":"<urn:uuid:136b199c-0390-405c-96b3-7b00dcb7e6ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00525.warc.gz"} |
Richard E. Bellman
Contact Menu
Richard Ernest Bellman (August 26, 1920 - March 14, 1984) was a major figure in modern optimization, systems analysis, and control theory who developed dynamic programming (DP) in the early 1950s.
Born in Brooklyn and raised in the Bronx, Bellman had a comfortable childhood that was interrupted by the Great Depression. Early on, he excelled at mathematics and was a star student in school.
Bellman enrolled at the tuition-free City College of New York but was put off by the expense of the hour-long commute (which left him no money for lunch) and transferred to Brooklyn College. After
receiving a bachelor’s degree in 1941, he chose to pursue graduate study at Johns Hopkins University. When the United States entered World War II, Bellman diverted his education to patriotic service,
thereby avoiding conscription. He moved to Wisconsin in 1942 to teach an Army radio and electronics class and worked towards an MA from the University of Wisconsin. At Madison, Bellman met renowned
Polish mathematician Stanislaw Ulam. Ulam convinced Solomon Lefschetz to offer Bellman a position at Princeton University as an Army Specialized Training Program instructor. After three semesters of
coursework and teaching, Bellman reunited with Ulam at Los Alamos’s Theoretical Physics Division as part of the Army’ top secret Manhattan Project. He returned to Princeton after the war and received
a PhD in mathematics in 1947. Given Princeton’s aversion to hiring recent graduate students, Bellman accepted a summer job at the RAND Corporation in 1948. The RAND community of the late 1940s and
1950s was an exciting place for the burgeoning operations researcher. The fulltime and associated staff included David Blackwell, George Dantzig, and Lloyd Shapley. Bellman was quick to make a name
for himself as a paper on bluffing he co-wrote with Blackwell became the subject of a New York Times article. When offered at full-time position at RAND in 1952, Bellman decided to suspend his
teaching career (he had been concurrently teaching at Stanford up to that point) and focus on the development of dynamic programming with RAND personnel. In 1965, Bellman became Professor of
Mathematics, Electrical Engineering, and Medicine at the University of Southern California. He started a program of applied mathematics that included a two-year sequence of dynamic programming,
control theory, invariant imbedding, and mathematical biosciences courses. Bellman’s research at USC became increasingly focused on the application of mathematics to medicine and the biological
sciences. Many of his students, including Christine Shoemaker and Augustine Esogbue, have gone on to make significant contributions to OR applications. (According to Shoemaker, Bellman was ahead of
his time with respect to affirmative action, and applied for and got a grant for teaching computer science to high school students in disadvantaged areas). He accepted a series of lecture engagements
around the world and published many articles, books, and monographs. He additionally served on a variety of editorial boards. Throughout his career, Bellman made significant contributions to a number
of areas. He published a series of articles on dynamic programming that came together in his 1957 book, Dynamic Programming. In the early 1960s, Bellman became interested in the idea of embedding a
particular problem within a larger class of problems as a functional approach to dynamic programming. He saw this as “DP without optimization” This work fed into his seminal contributions to control
theory and its application to real world problems. Bellman grew concerned about the computational effectiveness of dynamic programming. He was an avid proponent for using computers and pursued the
topic of artificial intelligence from a broad perspective. As his familiarity with computer science grew, so did his research on simulation. Bellman worked on bringing simulation to rational decision
making and human systems. In his lifetime, Bellman received many honors for his contributions to dynamic programming and operations research. He was awarded the John von Neumann Theory Prize by the
Operations Research Society of America and The Institute of Management Sciences and was elected into the National Academy of Engineering. The American Automatic Control Council established the
Richard E. Bellman Control Heritage Award in his honor for distinguished contributions to the control theory
From NAE Website: BY SOLOMON W. GOLOMB On Friday, May 11, 1984, ''A Celebration of the Life and Accomplishments of Professor Richard E. Bellman'' was held on the Los Angeles campus of the University
of Southern California. His colleagues and friends from around the world gathered to share their memories of this remarkable man. Some of their comments were published by the university as "A Tribute
to Richard Bellman." We cannot include them all in this volume, but the following excerpts provide an indication of the extraordinary impact Dick Bellman had in his life and work. Richard Bellman was
a towering figure among the contributors to modern control theory and systems analysis. His invention of dynamic programming marked the beginning of a new era in the analysis and optimizations of
large- scale systems and opened a way for the application of sophisticated computer- oriented techniques in a wide variety of problem areas, ranging from the design of guidance systems for space
vehicles to pest control, network routing, and speech recognition. Richard Bellman was born in Brooklyn, New York, on August 26, 1920. He received a B.A. from Brooklyn College in 1941 and an M.A. in
mathematics from the University of Wisconsin in 1943. As part of his service in the U.S. Army, he spent two years at Los Alamos, where he was a member of a group in the Theoretical Physics Division
headed by Dr. R. Marshak. Leaving Los Alamos in 1946, he entered Princeton and completed his work toward a Ph.D. in a record time of three months. In the immediate postwar years, Princeton was a
center of defense-motivated research activity in nonlinear differential equations. As a graduate student at Princeton, Bellman became a member of an inner circle of young mathematicians led by
Professor Solomon Lefschetz. His doctoral research under Lefschetz resulted in his first major work, entitled Stability Theory of Differential Equations, in 1946. This work was subsequently published
as a book by McGraw-Hill in 1953 and is regarded as a classic in its field. After staying on the faculty of the Mathematics Department at Princeton from 1946 to 1948, Bellman left the east coast to
become a member of the faculty of Stanford University in 1948 and then joined the newly established Rand Corporation in Santa Monica, California, in 1953. At Rand, he became interested in the theory
of multistage decision processes, which was then emerging as an important problem area in the control of both small-and large-scale systems. His invention of dynamic programming in 1953 was a major
breakthrough in the theory of multistage decision processes. This breakthrough set the stage for the application of functional equation techniques in a wide spectrum of fields extending far beyond
the problem areas that provided the initial motivation for his ideas. In addition to his fundamental and far-ranging work on dynamic programming, Richard Bellman made a number of important
contributions to both pure and applied mathematics. Particularly worthy of note is his work on invariant imbedding, which by replacing two-point boundary problems with initial value problems makes
the calculation of the solution more direct as well as much more efficient. His work on quasi-linearization and its applications to system identification has led to many results of a practical nature
in the study of nonlinear systems. In recent years, Bellman's research activity focused increasingly on the application of mathematics to medicine and biological sciences. His interest in these and
related areas reflected his strong conviction that mathematics should not be content with being a beautiful castle with no bridges to the real world. There was a time when Bellman's outspoken
criticisms of the elitist attitudes of the mathematical establishment were greeted with hostility and derision. Today, when pure mathematicians are experiencing difficulties in finding suitable jobs,
many of those who disagreed with Bellman will concede that he was right. Bellman left the Rand Corporation in 1965 to join the faculty of the University of Southern California, where he held joint
appointments as professor of mathematics, electrical engineering, and medicine—appointments he held until his death on March 19, 1984. A prolific writer, he authored over six hundred published
research papers, approximately forty books, and several monographs. Richard Bellman's fundamental contributions to science and engineering won him many honors and worldwide recognition. Prominent
among these are the following: first Norbert Wiener Prize in Applied Mathematics, awarded in 1970 jointly by the American Mathematical Society and the Society for Industrial and Applied Mathematics;
first Dickson Prize from Carnegie Mellon University in 1970; the John von Neumann Theory Award bestowed in 1976 jointly by the Institute of Management Sciences and the Operations Research Society of
America; and the 1979 Institute of Electrical and Electronics Engineers' Medal of Honor in recognition of the invention of dynamic programming. His honorary degrees include the doctor of science of
the University of Aberdeen, Scotland, in 1973; the doctor of laws of the University of Southern California in 1974; and the doctor of mathematics of the University of Waterloo, Canada, in 1975. He
was elected a fellow of the American Academy of Arts and Sciences in 1975, a member of the National Academy of Engineering in 1977, and a member of the National Academy of Sciences in 1983. [R. E.
Larson and L. A. Zadeh] In celebrating his life here today, let us also celebrate his good humor and his steadfast determination to produce, to achieve, to give, and to give joyfully, in the face of
circumstances that would have overwhelmed and crushed men of lesser caliber. In these superb human qualities, as in his creative work, I firmly believe that Dick Bellman has lived on a level at least
the equal of Beethoven. [Roger Jelliffe] Of his great contributions, I think that he would feel that the students he inspired were among the most important; through them his ideas go on and will be
expanded to meet the needs of expanding technology and human need. The only function that Richard Bellman could not bound was his own energy and imagination. [Fleur Mitchell] The measure of a man is
the number of people whose lives he has influenced and the contributions he has made. Dick Bellman not only influenced the lives of many people, but he had the rare genius to be able to contribute to
many fields. [Alan Rowe] Someone said that the Soviet Union is not just another country—it's another world, another planet. And it, indeed, is. But the stars, we might say, continuing the metaphor,
are the same on every planet. They shine for everyone and everywhere. Dick was, and is, such a star. His influence in the Soviet Union is deep and profound. His works penetrated many areas of Soviet
academia, industry, and economy in general. From the academic point of view, there is not a single university that does not offer courses based on Dick's works. Hundreds of papers continuing Dick's
ideas are published annually in Soviet journals. It is hardly possible to find a researcher in the quantitative sciences and engineering unfamiliar with, at least, the term "Dynamic Programming."
Dick's name is probably cited more at Soviet scientific meetings than at American ones. As an indirect proof of this, let me just mention that Dick was invited to be the main speaker at the first,
and only, International Congress of Mathematicians held in Russia, in 1966. More than ten of his books have been translated and published in the Soviet Union. No other American scientist has been
given such honors in the USSR. This is one of the trademarks of Dick's creative work: Truly a mathematician of the twentieth century, he viewed a computer as a tool as well as an important source of
mathematical work. His results are always practical and easily applicable. Probably, this is why his mathematical discoveries have important engineering implications in such areas as system science,
control, communications, bioengineering, etc. The depth and importance of problems considered, the practical applicability, and the timeliness of his works, this is what, in my view, made the largest
impact and defined Dick's influence on Soviet science. Dick gave all of us, his students and friends in every country throughout the world, an ultimate example of scientific creativity and success,
personal courage and strength, friendly devotion and support. [Semyon Meerkov] He was contemptuous of the established order and intolerant of mediocrity. He was strikingly handsome, brilliant, and a
master of both the spoken and the written word. Clearly, he was a man of towering intellect and almost equally towering ego. But what I could see was that behind the facade of arrogance and bravado
was a man who was capable of great kindness, a man who was decent, straightforward and generous in the extreme. He died at peace with himself. But his ideas will continue to live, and so will the
fond memories of all of us who knew him not only as a brilliant thinker and arrogant personality, but, more importantly, as a man of great nobility of character and a warm, thoughtful, caring human
being. [Lofti Zadeh] At the time of his death at age sixty-three, Richard Bellman had just completed his autobiography, The Eye of the Hurricane, World Scientific Publications, Singapore, 1984. He is
survived by his wife, Nina; his son, Eric; and his daughter, Kirstie.figure among the contributors to modern control theory and systems analysis. His invention of dynamic programming marked the
beginning of a new era in the analysis and optimizations of large- scale systems and opened a way for the application of sophisticated computer- oriented techniques in a wide variety of problem
areas, ranging from the design of guidance systems for space vehicles to pest control, network routing, and speech recognition. Richard Bellman was born in Brooklyn, New York, on August 26, 1920. He
received a B.A. from Brooklyn College in 1941 and an M.A. in mathematics from the University of Wisconsin in 1943. As part of his service in the U.S. Army, he spent two years at Los Alamos, where he
was a member of a group in the Theoretical Physics Division headed by Dr. R. Marshak. Leaving Los Alamos in 1946, he entered Princeton and completed his work toward a Ph.D. in a record time of three
months. In the immediate postwar years, Princeton was a center of defense-motivated research activity in nonlinear differential equations. As a graduate student at Princeton, Bellman became a member
of an inner circle of young mathematicians led by Professor Solomon Lefschetz. His doctoral research under Lefschetz resulted in his first major work, entitled Stability Theory of Differential
Equations, in 1946. This work was subsequently published as a book by McGraw-Hill in 1953 and is regarded as a classic in its field. After staying on the faculty of the Mathematics Department at
Princeton from 1946 to 1948, Bellman left the east coast to become a member of the faculty of Stanford University in 1948 and then joined the newly established Rand Corporation in Santa Monica,
California, in 1953. At Rand, he became interested in the theory of multistage decision processes, which was then emerging as an important problem area in the control of both small-and large-scale
systems. His invention of dynamic programming in 1953 was a major breakthrough in the theory of multistage decision processes. This breakthrough set the stage for the application of functional
equation techniques in a wide spectrum of fields extending far beyond the problem areas that provided the initial motivation for his ideas. In addition to his fundamental and far-ranging work on
dynamic programming, Richard Bellman made a number of important contributions to both pure and applied mathematics. Particularly worthy of note is his work on invariant imbedding, which by replacing
two-point boundary problems with initial value problems makes the calculation of the solution more direct as well as much more efficient. His work on quasi-linearization and its applications to
system identification has led to many results of a practical nature in the study of nonlinear systems. In recent years, Bellman's research activity focused increasingly on the application of
mathematics to medicine and biological sciences. His interest in these and related areas reflected his strong conviction that mathematics should not be content with being a beautiful castle with no
bridges to the real world. There was a time when Bellman's outspoken criticisms of the elitist attitudes of the mathematical establishment were greeted with hostility and derision. Today, when pure
mathematicians are experiencing difficulties in finding suitable jobs, many of those who disagreed with Bellman will concede that he was right. Bellman left the Rand Corporation in 1965 to join the
faculty of the University of Southern California, where he held joint appointments as professor of mathematics, electrical engineering, and medicine” appointments he held until his death on March 19,
1984. A prolific writer, he authored over six hundred published research papers, approximately forty books, and several monographs. Richard Bellman's fundamental contributions to science and
engineering won him many honors and worldwide recognition. Prominent among these are the following: first Norbert Wiener Prize in Applied Mathematics, awarded in 1970 jointly by the American
Mathematical Society and the Society for Industrial and Applied Mathematics; first Dickson Prize from Carnegie Mellon University in 1970; the John von Neumann Theory Award bestowed in 1976 jointly by
the Institute of Management Sciences and the Operations Research Society of America; and the 1979 Institute of Electrical and Electronics Engineers' Medal of Honor in recognition of the invention of
dynamic programming. His honorary degrees include the doctor of science of the University of Aberdeen, Scotland, in 1973; the doctor of laws of the University of Southern California in 1974; and the
doctor of mathematics of the University of Waterloo, Canada, in 1975. He was elected a fellow of the American Academy of Arts and Sciences in 1975, a member of the National Academy of Engineering in
1977, and a member of the National Academy of Sciences in 1983. [R. E. Larson and L. A. Zadeh] In celebrating his life here today, let us also celebrate his good humor and his steadfast determination
to produce, to achieve, to give, and to give joyfully, in the face of circumstances that would have overwhelmed and crushed men of lesser caliber. In these superb human qualities, as in his creative
work, I firmly believe that Dick Bellman has lived on a level at least the equal of Beethoven. [Roger Jelliffe] Of his great contributions, I think that he would feel that the students he inspired
were among the most important; through them his ideas go on and will be expanded to meet the needs of expanding technology and human need. The only function that Richard Bellman could not bound was
his own energy and imagination. [Fleur Mitchell] The measure of a man is the number of people whose lives he has influenced and the contributions he has made. Dick Bellman not only influenced the
lives of many people, but he had the rare genius to be able to contribute to many fields. [Alan Rowe] Someone said that the Soviet Union is not just another country—it's another world, another
planet. And it, indeed, is. But the stars, we might say, continuing the metaphor, are the same on every planet. They shine for everyone and everywhere. Dick was, and is, such a star. His influence in
the Soviet Union is deep and profound. His works penetrated many areas of Soviet academia, industry, and economy in general. From the academic point of view, there is not a single university that
does not offer courses based on Dick's works. Hundreds of papers continuing Dick's ideas are published annually in Soviet journals. It is hardly possible to find a researcher in the quantitative
sciences and engineering unfamiliar with, at least, the term "Dynamic Programming." Dick's name is probably cited more at Soviet scientific meetings than at American ones. As an indirect proof of
this, let me just mention that Dick was invited to be the main speaker at the first, and only, International Congress of Mathematicians held in Russia, in 1966. More than ten of his books have been
translated and published in the Soviet Union. No other American scientist has been given such honors in the USSR. This is one of the trademarks of Dick's creative work: Truly a mathematician of the
twentieth century, he viewed a computer as a tool as well as an important source of mathematical work. His results are always practical and easily applicable. Probably, this is why his mathematical
discoveries have important engineering implications in such areas as system science, control, communications, bioengineering, etc. The depth and importance of problems considered, the practical
applicability, and the timeliness of his works, this is what, in my view, made the largest impact and defined Dick's influence on Soviet science. Dick gave all of us, his students and friends in
every country throughout the world, an ultimate example of scientific creativity and success, personal courage and strength, friendly devotion and support. [Semyon Meerkov] He was contemptuous of the
established order and intolerant of mediocrity. He was strikingly handsome, brilliant, and a master of both the spoken and the written word. Clearly, he was a man of towering intellect and almost
equally towering ego. But what I could see was that behind the facade of arrogance and bravado was a man who was capable of great kindness, a man who was decent, straightforward and generous in the
extreme. He died at peace with himself. But his ideas will continue to live, and so will the fond memories of all of us who knew him not only as a brilliant thinker and arrogant personality, but,
more importantly, as a man of great nobility of character and a warm, thoughtful, caring human being. [Lofti Zadeh] At the time of his death at age sixty-three, Richard Bellman had just completed his
autobiography, The Eye of the Hurricane, World Scientific Publications, Singapore, 1984. He is survived by his wife, Nina; his son, Eric; and his daughter, Kirstie.
Outdated or incorrect information? Please click here to update us with the correct information. | {"url":"https://a2c2.org/contact/richard-e-bellman","timestamp":"2024-11-10T02:13:36Z","content_type":"text/html","content_length":"170928","record_id":"<urn:uuid:3bd02aaf-bb01-440c-a5c0-301a076c436f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00625.warc.gz"} |
MCLab Group List of Papers -- Query Results
I. Melatti, F. Mari, T. Mancini, M. Prodanovic, and E. Tronci. "A Two-Layer Near-Optimal Strategy for Substation Constraint Management via Home Batteries." IEEE Transactions on Industrial Electronics
(2021): 1. Notes: To appear. DOI: 10.1109/TIE.2021.3102431.
T. Mancini, I. Melatti, and E. Tronci. "Any-horizon uniform random sampling and enumeration of constrained scenarios for simulation-based formal verification." IEEE Transactions on Software
Engineering (2021): 1. ISSN: 1939-3520. Notes: To appear. DOI: 10.1109/TSE.2021.3109842.
G. Dipoppa, G. D'Alessandro, R. Semprini, and E. Tronci. "Integrating Automatic Verification of Safety Requirements in Railway Interlocking System Design." In High Assurance Systems Engineering,
2001. Sixth IEEE International Symposium on, 209–219. Albuquerque, NM, USA: IEEE Computer Society, 2001. ISSN: 0-7695-1275-5. DOI: 10.1109/HASE.2001.966821.
Enrico Tronci. "Equational Programming in lambda-calculus." In Sixth Annual IEEE Symposium on Logic in Computer Science (LICS), 191–202. Amsterdam, The Netherlands: IEEE Computer Society, 1991. DOI:
Giuseppe Della Penna, Antinisca Di Marco, Benedetto Intrigila, Igor Melatti, and Alfonso Pierantonio. "Interoperability mapping from XML schemas to ER diagrams." Data Knowl. Eng. 59, no. 1 (2006):
166–188. Elsevier Science Publishers B. V.. ISSN: 0169-023x. DOI: 10.1016/j.datak.2005.08.002.
Giuseppe Della Penna, Benedetto Intrigila, Igor Melatti, Enrico Tronci, and Marisa Venturini Zilli. "Finite horizon analysis of Markov Chains with the Mur$\varphi$ verifier." Int. J. Softw. Tools
Technol. Transf. 8, no. 4 (2006): 397–409. Springer-Verlag. ISSN: 1433-2779. DOI: 10.1007/s10009-005-0216-7.
Igor Melatti, Robert Palmer, Geoffrey Sawaya, Yu Yang, Robert Mike Kirby, and Ganesh Gopalakrishnan. "Parallel and distributed model checking in Eddy." Int. J. Softw. Tools Technol. Transf. 11, no. 1
(2009): 13–25. Springer-Verlag. ISSN: 1433-2779. DOI: 10.1007/s10009-008-0094-x.
Alessandro Fantechi, Stefania Gnesi, Franco Mazzanti, Rosario Pugliese, and Enrico Tronci. "A Symbolic Model Checker for ACTL." In International Workshop on Current Trends in Applied Formal Method
(FM-Trends), edited by D. Hutter, W. Stephan, P. Traverso and M. Ullmann, 228–242. Lecture Notes in Computer Science 1641. Boppard, Germany: Springer, 1998. ISSN: 3-540-66462-9. DOI: 10.1007/
Enrico Tronci. "Automatic Synthesis of Controllers from Formal Specifications." In Proc of 2nd IEEE International Conference on Formal Engineering Methods (ICFEM), 134–143. Brisbane, Queensland,
Australia, 1998. DOI: 10.1109/ICFEM.1998.730577.
Marco Gribaudo, Andras Horváth, Andrea Bobbio, Enrico Tronci, Ester Ciancamerla, and Michele Minichino. "Model-Checking Based on Fluid Petri Nets for the Temperature Control System of the ICARO
Co-generative Plant." In 21st International Conference on Computer Safety, Reliability and Security (SAFECOMP), edited by S. Anderson, S. Bologna and M. Felici, 273–283. Lecture Notes in Computer
Science 2434. Catania, Italy: Springer, 2002. ISSN: 3-540-44157-3. DOI: 10.1007/3-540-45732-1_27. | {"url":"https://mclab.di.uniroma1.it/publications/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%20FROM%20refs%20WHERE%20serial%20RLIKE%20%22.%2B%22%20ORDER%20BY%20place&submit=Cite&citeStyle=Roma&citeOrder=&orderBy=place&headerMsg=&showQuery=0&showLinks=0&formType=sqlSearch&showRows=10&rowOffset=110&viewType=Print","timestamp":"2024-11-13T22:44:51Z","content_type":"text/html","content_length":"32801","record_id":"<urn:uuid:560bd655-a864-439c-80ed-67418da88c68>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00694.warc.gz"} |
Calculating Vertical Support Reactions
1. Calculating Vertical Support Reactions
Hello all
I was wondering if anyone could help explain how the Vertical reactions for the following structure were calculated.
I am able to calculate the Horizontal reactions. It is the vertical reactions that I am struggling with.
I do the following:-
Sum Vertical Reactions = 0
Va + Vd - 230 = 0
If I take moments about D I get:-
Sum of Moments about Support D = 0
(5.4*230) + (Ha*1.8) = 0
Ha = -690kN
I just don't know how to calculate 230kN at support A.
Any advice?
Thank you.
How to Calculate Reactions at Supports?
1. Let the sum of moments about a reaction point equal to ZERO (ΣM = 0)
All we need to know about moments at this stage is that they are equal to the force multiplied by the distance from a point (i.e. the force x distance from a point).
Consider a simple example of a 4m beam with a pin support at A and roller support at B. The free-body diagram is shown below where Ay and By are the vertical reactions at the supports:
We firstly want to consider the sum of moments about point B and let it equal zero. We have chosen point B to prove this can be done at either end of the beam (provided it is pin supported).
However, you could just as easily work from point A. So, now we sum the moments about point B and let the sum equal 0:
NOTE: The sign convention we have chosen is that counter-clockwise moments are positive and clockwise moments are negative. This is the most common sign convention but it is up to you. You
must ALWAYS use the same sign convention throughout the whole problem. Always use the same sign convention from the start. We now have our first equation. We need to solve another equation in
order to find By (the vertical reaction force at support B).
2. Let the sum of vertical forces equal to 0 (ΣFy = 0)
Sum the forces in the y (vertical) direction and let the sum equal zero. Remember to include all forces including reactions and normal loads such as point loads. So if we sum the forces in
the y-direction for the above example, we get the following equation:
NOTE: Again we stuck to a sign convention which was to take upward forces (our reactions) as positive and downward forces (the point load) as negative. Remember the sign convention is up to
you but you must ALWAYS use the same sign convention throughout the whole problem.
So there we have it, we have used the two above equations (sum of moments equals zero and sum of vertical forces equals zero) and calculated that the reaction at support A is 10 kN and the
reaction at support B 10kN. This makes sense as the point load is right in the middle of the beam, meaning both supports should have the same vertical forces (i.e. it is symmetric).
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Solving resistor networks
This application describes how to write a uLisp program to calculate the resistance of a resistor network such as:
11th February 2023: Updated with clearer instructions of how to install and run the program.
Running the program
To run the resistor network program you need a ARM, ESP, or RISC-V board running the appropriate 32-bit version of uLisp, with floating-point support.
• Download the appropriate version of uLisp from Download uLisp.
• Use the Arduino IDE to upload it to the board; see Download uLisp - Installing uLisp.
• Open the Serial Monitor in the Arduino IDE to display the uLisp prompt.
• Display the source of the resistor network program here: Resistor network program
• Select the text of the program, copy it, paste it into the Arduino IDE Serial Monitor input field, and press Return.
Defining the network
Describe the network you want to solve as a list specifying the resistance, in ohms, between each pair of nodes. Label each node with a letter, such as a, b, c, etc.
For example, the circuit shown above would be described as:
(defvar *circuit* '((a d ?) (a b 10) (b a 15) (b d 5) (a c 5) (c d 6)))
For example, (a b 10) specifies that a resistor of 10Ω is connected between nodes a and b. The order of the nodes isn't important.
The list (a d ?) represents the resistance we want to calculate.
• Type this definition into the Arduino IDE Serial Monitor input field, followed by Return.
Solving the network
• Run the program by typing the following command into the Arduino IDE Serial Monitor input field, followed by Return:
(solve *circuit*)
The solution will be printed as a list:
((d a 5.5) (a d ?))
So the network reduces to a single resistor between d and a of 5.5Ω.
How it works
Series and parallel
One way to calculate series and parallel resistors is to define Lisp functions:
(defun series (x y) (+ x y))
(defun parallel (x y) (/ (+ (/ x) (/ y))))
Note that in Lisp (/ x) is a shorter way of writing (/ 1 x).
Now we can use these functions to express the complete circuit:
> (parallel (series 5 6) (series (parallel 10 15) 5))
and the answer is 5.5 Ω.
A more general way of representing a network
Using the serial and parallel functions requires the user to do most of the work. A more intuitive and general way to represent a resistor network would be to label each of the nodes a, b, c, etc and
then represent it as a list of the resistances between each pair of nodes. Using this approach the above network becomes:
(defvar *circuit* '((a d ?) (a b 10) (b a 15) (b d 5) (a c 5) (c d 6)))
The list (a d ?) represents the resistance we want to calculate.
We can then write a program to progressively search for series or parallel configurations, and combine them.
All possible divisions of a set into two subsets
It will be useful to have a function split-set, that takes a list and an index i, and returns the ith division of the set into two subsets.
(defun split-set (lis i)
(let (in out (ll (reverse lis)))
(dotimes (j (length lis))
(if (oddp i) (push (nth j ll) in) (push (nth j ll) out))
(setq i (ash i -1)))
(list in out)))
For a list of length n the total possible number of subsets is 2^n. The function split-set works by expressing the index i as a binary number, and it then puts each element into one of the two sets
according to whether the bit corresponding to that element is a 0 or a 1. So, for example, 13 in binary is 1101 so the 13th split of the list of four elements (a b c d) is:
> (split-set '(a b c d) 13)
((a b d) (c))
Combining parallel and series resistors
To simplify the network we successively combine pairs of resistors according to the rules of parallel and series resistors. To do this we look at every possible pair of resistors and see if they can
be combined into a single resistor. For example, the two parallel resistors between a and b could be merged into a single resistor.
Here's the routine series-parallel to combine two resistors x and y. It also takes the entire circuit as a parameter so we can check for other connections between the same nodes:
(defun series-parallel (l x y)
((or (eq (caddr x) '?) (eq (caddr y) '?))
;; Check four possible labellings
(t (let (result)
(dolist (x (list x (list (second x) (first x) (third x))))
(dolist (y (list y (list (second y) (first y) (third y))))
;; Resistors in parallel
(when (and (eq (first x) (first y))
(eq (second x) (second y)))
(setq result
(first x) (second x) (/ (+ (/ (third x)) (/ (third y))))))))
;; Resistors in series
(when (and (eq (first x) (first y))
(= (countlinks l (first x)) 2)
(not (eq (second x) (second y))))
(setq result
(list (list (second x) (second y) (+ (third x) (third y))))))))
For two resistors in parallel we simply need to check that their start and end nodes are the same. For example, here it combines the resistors between a and b:
> (series-parallel *circuit* '(a b 10) '(b a 15))
((b a 6.0))
For two resistors in series we also need to check that no other resistor is connected to the node between the two resistors; this is what countlinks does:
(defun countlinks (l x)
(let ((n 0))
(mapc (lambda (i) (when (or (eq x (first i)) (eq x (second i))) (incf n))) l)
For example, here we combine the two resistors between a, c, and d:
> (series-parallel *circuit* '(a c 5) '(c d 6))
((a d 11))
If it's not possible to combine the resistors series-parallel returns nil:
> (series-parallel *circuit* '(a b 10) '(b d 5))
Simplifying a circuit
To simplify a circuit we use split-set to check every possible pair of resistors to see if they can be combined by series-parallel:
(defun simplify (lis function n)
(let* ((l (length lis))
(k (expt 2 l)))
(dotimes (i k lis)
(let* ((s (split-set lis i))
(in (first s))
(out (second s)))
(when (= (length in) n)
(let ((c (apply function lis in)))
(when c (return (append c out)))))))))
This function simplify takes a list representing the network, a function for combining resistors, and a number of resistors to combine, and returns the simplified network.
Finally to solve the network we call simplify repeatedly until there's no more work to do:
(defun solve (circuit)
(let (len)
(setq len (length circuit))
(setq circuit (simplify circuit #'series-parallel 2))
(when (= (length circuit) len) (return)))
Here it is working on the above network:
> (solve *circuit*)
((d a 5.5) (a d ?))
So the network reduces to a single resistor of 5.5Ω.
Networks containing triangles
Unfortunately there are some networks that this approach can't solve. For example:
This can be represented as the list:
(defvar *c2* '((a d ?) (a b 32) (b c 24) (a c 25) (b d 32) (c d 40)))
Trying to solve it gives:
> (solve *c2*)
((a d ?) (a b 32) (b c 24) (a c 25) (b d 32) (c d 40))
It has failed to simplify it at all because the circuit doesn't contain any series or parallel configurations.
Delta-Wye transformations
The solution is to do what's called a Delta-Wye transformation, which converts a triangle configuration, or delta, into a Y or wye configuration by adding an extra node ^[1]:
To solve these configurations the function delta-wye checks three links, and if they qualify as a delta network, they are transformed into a wye:
(defun delta-wye (l x y z)
((or (eq (caddr x) '?) (eq (caddr y) '?) (eq (caddr z) '?))
;; Check eight possible labellings
(t (let (result)
(dolist (x (list x (list (second x) (first x) (third x))))
(dolist (y (list y (list (second y) (first y) (third y))))
(dolist (z (list z (list (second z) (first z) (third z))))
(when (and (eq (first x) (second z))
(eq (first y) (second x))
(eq (first z) (second y)))
(let ((sum (+ (third x) (third y) (third z)))
(newsymbol (incf *newnode*)))
(setq result
(first x) newsymbol (/ (* (third x) (third z)) sum))
(first y) newsymbol (/ (* (third x) (third y)) sum))
(first z) newsymbol (/ (* (third y) (third z)) sum)))))))))
The function delta-wye labels the new node it creates with a unique integer, to distinguish it from the nodes in the original network:
(defvar *newnode* 0)
Testing it with x=1, y=2, and z=3 gives:
> (delta-wye nil '(a b 3) '(b c 1) '(c a 2))
((a 1 1) (b 1 0.5) (c 1 0.333333))
where '1' is the new node it has created.
The solve function can be updated to incorporate delta-wye as follows:
(defun solve (circuit)
(let (len)
(setq len (length circuit))
(setq circuit (simplify circuit #'delta-wye 3))
(setq circuit (simplify circuit #'series-parallel 2))
(setq circuit (floating circuit))
(when (= (length circuit) len) (return)))
I've also added a function floating to remove resistors with only one end connected to the network, which can arise from delta-wye transformations:
(defun floating (l)
(let (result)
(dolist (x l result)
(= (countlinks l (first x)) 1)
(= (countlinks l (second x)) 1))
(push x result)))))
Testing the new version of solve on the network *c2*:
> (solve *c2*)
((a d 32.0) (a d ?))
and the resistance between a and d is 32Ω.
Another example
Finally, find the resistance between a and f in this network containing two deltas:
1. ^ Delta-Wye resistor networks on Khan Academy. | {"url":"http://www.ulisp.com/show?2A61","timestamp":"2024-11-08T11:43:44Z","content_type":"text/html","content_length":"27928","record_id":"<urn:uuid:a83aaaa9-36d6-493b-ab37-ea185b75982e>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00403.warc.gz"} |
Lund University Publications
Pseudocontinuations and the backward shift
(1998) In Indiana University Mathematics Journal 47(1). p.223-276
Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a
function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)
=\break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant
subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the... (More)
Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a
function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)
=\break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant
subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the forward invariant subspaces are
described primarily in terms of zeros, the backward invariant subspaces are characterized in terms of pseudocontinuations. To be concrete, take $I$ to be the forward invariant subspace of all
functions in $H^2$ that vanish along a given finite sequence $A$ of distinct points in $\bold D$. Its annihilator $I^\perp$ is finite-dimensional, and consists of all rational functions with
simple poles along the sequence $A^*$ obtained by reflecting $A$ in the unit circle. Then, if we let the finite sequence $A$ ``grow'' to become in the limit a Blaschke sequence plus a negative
singular mass on the circle, the annihilator will increase as well, but there will remain a ``connection'' between the behavior inside $\bold D$ and the behavior outside in the exterior disk ${\
bold D}_{\rm e}$, the complement of the closed unit disk on the Riemann sphere. The connection is furnished by the pseudocontinuation across the circle: we have a holomorphic Nevanlinna class
function on the inside, and a meromorphic Nevanlinna class function on the outside, and they have the same nontangential boundary values almost everywhere on the unit circle. The issue at hand is
whether the Hardy space situation is typical of backward invariant subspaces in Banach spaces $\scr B$ of analytic functions on the disk. A dichotomy appears: if $\scr B$ is bigger than the
corresponding Hardy space, then the backward invariant subspaces possess pseudocontinuations across the unit circle, whereas if $\scr B$ is smaller, this is no longer generally the case. What
happens is best understood in terms of forward invariant subspaces. With the standard Cauchy duality (the extension of the $H^2$-self-duality), we can think of the dual ${\scr B}^*$ of $\scr B$
as a space of holomorphic functions on $\bold D$, and study the forward shift invariant subspaces on ${\scr B}^*$. Let us concentrate on the case when $\scr B$ is a Hilbert space, of Dirichlet or
Bergman type; then ${\scr B}^*$ falls into the same category, too. Every forward invariant subspace $\scr M$ of Dirichlet type has index $1$, which means that $S\scr M$ has codimension $1$ in $\
scr M$; this is analogous to the $H^2$ case. Apparently, this means that the annihilator $\scr M^\perp$ (which is a backward invariant subspace of a Bergman space) consists of pseudocontinuable
functions. However, there are plenty of forward invariant subspaces of a Bergman space which have index bigger than $1$ [see, e.g., H. Hedenmalm, J. Reine Angew. Math. 443 (1993), 1--9; MR1241125
(94k:30092)]. The annihilator of such a forward invariant subspace is a backward invariant subspace of a Dirichlet space, and some playing around with the formulas for pseudocontinuations
suggests that in this case, it should not be unique (and hence not exist as a pseudocontinuation). This is then worked out rigorously in the paper. (Less)
publishing date
Contribution to journal
publication status
Indiana University Mathematics Journal
223 - 276
Indiana University
external identifiers
LU publication?
309a34b6-1da1-4acb-9f4c-1e4e2214a27f (old id 1467169)
date added to LUP
2016-04-01 12:37:57
date last changed
2022-02-04 00:50:41
abstract = {{Beurling's theorem characterizes the forward shift invariant subspaces in the Hardy space $H^2$ on the open unit disk $\bold D$. The description is in terms of an inner function, that is, a function in $H^2$ whose nontangential boundary values have modulus $1$ almost everywhere. If $S$ stands for the forward shift $Sf(z)=zf(z)$, then the adjoint $L=S^*$ is the backward shift, $Lf(z)=\break (f(z)-f(0))/z$. The annihilator of a forward shift invariant subspace is then backward shift invariant, and Beurling's theorem leads to a description also of the backward shift invariant subspaces, as noted by R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier (Grenoble) 20 (1970), fasc. 1, 37--76; MR0270196 (42 #5088)]. Whereas the forward invariant subspaces are described primarily in terms of zeros, the backward invariant subspaces are characterized in terms of pseudocontinuations. To be concrete, take $I$ to be the forward invariant subspace of all functions in $H^2$ that vanish along a given finite sequence $A$ of distinct points in $\bold D$. Its annihilator $I^\perp$ is finite-dimensional, and consists of all rational functions with simple poles along the sequence $A^*$ obtained by reflecting $A$ in the unit circle. Then, if we let the finite sequence $A$ ``grow'' to become in the limit a Blaschke sequence plus a negative singular mass on the circle, the annihilator will increase as well, but there will remain a ``connection'' between the behavior inside $\bold D$ and the behavior outside in the exterior disk ${\bold D}_{\rm e}$, the complement of the closed unit disk on the Riemann sphere. The connection is furnished by the pseudocontinuation across the circle: we have a holomorphic Nevanlinna class function on the inside, and a meromorphic Nevanlinna class function on the outside, and they have the same nontangential boundary values almost everywhere on the unit circle. The issue at hand is whether the Hardy space situation is typical of backward invariant subspaces in Banach spaces $\scr B$ of analytic functions on the disk. A dichotomy appears: if $\scr B$ is bigger than the corresponding Hardy space, then the backward invariant subspaces possess pseudocontinuations across the unit circle, whereas if $\scr B$ is smaller, this is no longer generally the case. What happens is best understood in terms of forward invariant subspaces. With the standard Cauchy duality (the extension of the $H^2$-self-duality), we can think of the dual ${\scr B}^*$ of $\scr B$ as a space of holomorphic functions on $\bold D$, and study the forward shift invariant subspaces on ${\scr B}^*$. Let us concentrate on the case when $\scr B$ is a Hilbert space, of Dirichlet or Bergman type; then ${\scr B}^*$ falls into the same category, too. Every forward invariant subspace $\scr M$ of Dirichlet type has index $1$, which means that $S\scr M$ has codimension $1$ in $\scr M$; this is analogous to the $H^2$ case. Apparently, this means that the annihilator $\scr M^\perp$ (which is a backward invariant subspace of a Bergman space) consists of pseudocontinuable functions. However, there are plenty of forward invariant subspaces of a Bergman space which have index bigger than $1$ [see, e.g., H. Hedenmalm, J. Reine Angew. Math. 443 (1993), 1--9; MR1241125 (94k:30092)]. The annihilator of such a forward invariant subspace is a backward invariant subspace of a Dirichlet space, and some playing around with the formulas for pseudocontinuations suggests that in this case, it should not be unique (and hence not exist as a pseudocontinuation). This is then worked out rigorously in the paper.}},
author = {{Aleman, Alexandru and Richter, Stefan and Ross, William T}},
issn = {{0022-2518}},
language = {{eng}},
number = {{1}},
pages = {{223--276}},
publisher = {{Indiana University}},
series = {{Indiana University Mathematics Journal}},
title = {{Pseudocontinuations and the backward shift}},
url = {{http://dx.doi.org/10.1512/iumj.1998.47.1583}},
doi = {{10.1512/iumj.1998.47.1583}},
volume = {{47}},
year = {{1998}}, | {"url":"https://lup.lub.lu.se/search/publication/1467169","timestamp":"2024-11-14T21:12:39Z","content_type":"text/html","content_length":"49082","record_id":"<urn:uuid:f35e01a3-2350-4281-9852-73fd83f6020c>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00542.warc.gz"} |
Linear Least Squares Regression with TensorFlow
Linear Least Squares Regression is by far the most widely used regression method, and it is suitable for most cases when data behavior is linear. By definition, a line is defined by the following
For all data points (xi, yi) we have to minimize the sum of the squared errors:
This is the equation we need to solve for all data points:
The solution for this equation is A (I’m not going to show how this solution is found, but you can see it in Linear Least Squares – Wikipedia, and some code in several programming languages as well),
which is defined by:
Now, let’s see the implementation with TensorFlow:
import matplotlib.pyplot as plt
import tensorflow as tf
import numpy as np
sess = tf.Session()
x_vals = np.linspace(0, 10, num=100)
y_vals = x_vals + np.random.normal(loc=0, scale=1, size=100)
x_vals_column = np.transpose(np.matrix(x_vals))
ones_column = np.transpose(np.matrix(np.repeat(1, repeats=100)))
X = np.column_stack((x_vals_column, ones_column))
Y = np.transpose(np.matrix(y_vals))
X_tensor = tf.constant(X)
Y_tensor = tf.constant(Y)
tX_X = tf.matmul(tf.transpose(X_tensor), X_tensor)
tX_X_inv = tf.matrix_inverse(tX_X)
product = tf.matmul(tX_X_inv, tf.transpose(X_tensor))
A = tf.matmul(product, Y_tensor)
A_eval = sess.run(A)
m_slope = A_eval[0][0]
b_intercept = A_eval[1][0]
print('slope (m): ' + str(m_slope))
print('intercept (b): ' + str(b_intercept))
best_fit = []
for i in x_vals:
best_fit.append(m_slope * i + b_intercept)
plt.plot(x_vals, y_vals, 'o', label='Data')
plt.plot(x_vals, best_fit, 'r-', label='Linear Regression', linewidth=3)
plt.legend(loc='upper left')
slope (m): 1.0108287140073253
intercept (b): 0.14322921334345343
As you can see, the implementation is just executing basic matrix operations, the advantage of using TensorFlow in this case is that we can add this process to a more complex graph.
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January 18, 2018 by Alexis Alulema Machine Learning Python 0
You may also like... | {"url":"https://www.alexisalulema.com/2018/01/18/linear-least-squares-regression-with-tensorflow/","timestamp":"2024-11-03T03:35:36Z","content_type":"text/html","content_length":"95322","record_id":"<urn:uuid:99336682-a3fd-481f-9eb5-7663701ddcc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00630.warc.gz"} |
The Calculi of Lambda Conversion
by Alonzo Church
Following are the first few pages of the book on the λ-calculus by Alonzo Church (1941). Html transcription adapted and extended from here.
1. The Concept of a Function
Underlying the formal calculi which we shall develop is the concept of a function, as it appears in various branches of mathematics, either under that name or under one of the synonymous names,
“operation” or “transformation.” The study of the general properties of functions, independently of their appearance in any particular mathematical (or other) domain, belongs to formal logic or lies
on the boundary line between logic and mathematics. This study is the original motivation for the calculi — but they are so formulated that it is possible to abstract from the intended meaning and
regard them merely as formal systems.
A function is a rule of correspondence by which when anything is given (as argument) another thing (the value of the function for that argument) may be obtained. That is, a function is an operation
which may be applied on one thing (the argument) to yield another thing (the value of the function). It is not, however, required that the operation shall necessarily be applicable to everything
whatsoever; but for each function there is a class, or range, of possible arguments — the class of things to which the operation is significantly applicable — and this we shall call the range of
arguments, or range of the independent variable, for that function. The class of all values of the function, obtained by taking all possible auguments, will be called the range of values, or range of
the dependent variable.
If f denotes a particular function, we shall use the notation (fa) for the value of the function f for the argument a. If a does not belong to the range of arguments of f, the notation (fa) shall be
It is, of course, not excluded that the range of arguments or range of values of a function should consist wholly or partly of functions. The derivative, as this notion appears in the elementary
differential calculus, is a familiar mathematical example of a function for which both ranges consist of functions. Or, turning to the integral calculus, if in the expression ∫[0]^1(fx)dx we take the
function f as independent variable, we are led to a function for which the range of arguments consists of functions and the range of values, of numbers. Formal logic provides other examples; thus the
existential quantifier, according to the present account, is a function for which the range of arguments consists of propositional functions, and the range of values consists of truth values.
In particular it is not excluded that one of the elements of the range of arguments of a function f should be the function f itself. This possibility has frequently been denied, and indeed, if a
function is defined as a correspondence between two previously given ranges, the reason for the denial is clear. Here, however, we regard the operation or rule of correspondence, which constitutes
the function, as being first given, and the range of arguments then determined as consisting of the things to which the operation is applicable. This is a departure from the point of view usual in
mathematics, but it is a departure which is natural in passing from consideration of functions in a special domain to the consideration of function in general, and it finds support in consistency
theorems which will be proved below.
The identity function I is defined by the rule that (Ix) is x, whatever x may be; then in particular (II) is I. If a function H is defined by the rule that (Hx) is I, whatever x may be, then in
particular (HH) is I. If Σ is the existential quantifier, then (ΣΣ) is the truth-value truth.
The functions I and H may also be cited as examples of functions for which the range of arguments consists of all things whatsoever.
2. Extension and Intension
The foregoing discussion leaves it undetermined under what circumstances two functions shall be considered the same.
The most immediate and, from some points of view, the best way to settle this question is to specify that two functions f and g are the same if they have the same range of arguments and, for every
element a that belongs to this range, (fa) is the same as (ga). When this is done we shall say that we are dealing with functions in extension.
It is possible, however, to allow two functions to be different on the ground that the rule of correspondence is different in meaning in the two cases although always yielding the same result when
applied to any particular argument. When this is done we shall say that we are dealing with functions in intension. The notion of difference in meaning between two rules of correspondence is a vague
one, but, in terms of some system of notation, it can be made exact in various ways. We shall not attempt to decide what is the true notion of difference in meaning but shall speak of functions in
intension in any case where a more severe criterion of identity is adopted than for functions in extension. There is thus not one notion of function in intension, but many notions, involving various
degrees of intensionality.
In the calculus of λ-conversion and the calculus of restricted λ-K-conversion, as developed below, it is possible, if desired, to interpret the expressions of the calculus as denoting functions in
extension. However, in the calculus of λ-δ-conversion, where the notion of identity of functions is introduced into the system by the symbol δ, it is necessary, in order to preserve the finitary
character of the transformation rules, so to formulate these rules that an interpretation by functions in extension becomes impossible. The expressions which appear in the calculus of λ-δ-conversion
are interpretable as denoting functions in intension of an appropriate kind.
3. Functions of Several Variables
So far we have tacitly restricted the term “function” to functions of one variable (or, of one argument). It is desirable, however, for each positive integer n, to have the notion of a function of n
variables. And, in order to avoid the introduction of a separate primitive idea for each n, it is desirable to find a means of explaining functions of n variables as particular cases of functions of
one variable. For our present purpose, the most convenient and natural method of doing this is to adopt an idea of Schönfinkel (1924), according to which a function of two variables is regarded as a
function of one variable whose values are functions of one variable, a function of three variables as a function of one variable whose values are functions of two variables, and so on.
Thus if f denotes a particular function of two variables, the notation ((fa)b) — which we shall frequently abbreviate as (fab) or fab — represents the value of f for the arguments a, b. The notation
(fa) — which we shall frequently abbreviate as fa — represents a function of one variable, whose value for any argument x is fax. The function f has a range of arguments, and the notation fa is
meaningful only when a belongs to that range; the function fa again has a range of arguments, which is, in general different for differest elements a, and the notation fab is meaningful only when b
belongs to that range of arguments of fa.
Similarly, if f denotes a function of three variables, (((fa)b)c) or fabc denotes a value of f for the arguments a,b,c, fa denoting a certain function of two variables, and ((fa)b) or fab denoting a
certain function of one variable — and so on.
(According to another scheme, which is the better one for certain purposes, a function of two variables is regarded as a function (of one variable) whose arguments are ordered pairs, a function of
three variables as a function whose arguments are triads, and so on. This other concept of a function of several variables is not, however excluded here. For, as will appear below, the notions of
ordered pair, ordered triad, etc., are definable by means of abstraction (§4) and the Schönfinkel concept of a function of several variables; and thus functions of several variables in the other
sense are also provided for.)
An example of a function of two variables (in the sense of Schönfinkel) is the constancy function K, defined by the rule that Kxy is x, whatever x or y may be. We have, for instance that KII is I,
KHI is H, and so on. Also KI is H (where H is the function defined above in §1). Similarly KK is a function whose value is constant and equal to K.
Another example of a function of two variables is the function whose value for the arguments f, x is (fx); for reasons which will appear later we designate this function by the symbol 1. This
function 1, regarded as a function of one variable, is a kind of identity function, since the notation (1f) whenever significant, denotes the same function as f; the function I and 1 are not,
however, the same function, since the range of the argument consists in one case of all things whatsoever, in the other case merely of all functions.
Other examples of functions of two or more variables are the function H, already defined, and the functions T, J, B, C, W, S, defined respectively by the rules that Txf is (fx), Jfxyz is fx(fzy),
Bfgx is f(gx), Cfxy is (fyx), Wfx is (fxx), Snfx is f(nfx).
Of these, B and C may be more familiar to the reader under other names, as the product or resultant of two transformations f and g, and as the converse of a function of two variables f. To say that
BII is I is to say that the product of the identity transformation by the identity transformation is the identity transformation whatever the domain within which the transformations are being
considered; to say that B11 is 1 is to say that within any domain consisting entirely of functions the product of the identity transformation by itself is the identity transformation. BI is 1, since
it is the operation of composition with the identity transformation, and thus an identity operation, but one applicable only to transformations.
The reader may further verify that CK is H, CT is 1, C1 is T — that 1 and I have the same converse is explained by the fact that, while not the same functions, they have the same effect in all cases
where they can significantly be applied to two arguments. The function BCC, the converse of the converse, has the effect of an identity when applied to a function of two variables, but when applied
to a function of one variable it has the effect of so restricting he range of the arguments as to transform the function into a function of two variables (if possible); thus BCCI is 1.
There are many similar relations between these functions, some of them quite complicated.
4. Abstraction
For our present purpose it is necessary to distinguish carefully between a symbol or expression which denotes a function and an expression which contains a variable and denotes ambiguously some value
of the functions — a distinction which is more or less obscured in the usual language of mathematical function theory.
To take an example from the theory of functions of natural numbers, consider the expression (x^2+x)^2. If we say, “(x^2+x)^2 is greater than 1,000” we make a statement which depends on x and actually
has no meaning unless x is determined as some particular natural number. On the other hand, if we say, “(x^2+x)^2 is a primitive recursive function,” we make a definitive statement whose meaning in
no way depends on a determination of the variable x (so that in this case x plays the rôle of an apparent, or bound, variable). The difference between the two cases is that in the first case the
expression (x^2+x)^2 serves as an ambiguous, or variable, denotation of a natural number, while in the second case it serves as the denotation of a particular function. We shall hereafter distinguish
by using (x^2+x)^2 when we intend an ambiguous denotation of a natural number, but (λx(x^2+x)^2) as the denotation of the corresponding function — and likewise in other cases.
(It is, of course, irrelevant here that the notation (x^2+x)^2 is commonly used also for a certain function of real numbers, a certain function of complex numbers, etc. In a logically exact notation
the function, addition of natural numbers, addition of complex numbers, would be denoted by different symbols, say +[n], +[r], +[c], and these three functions, square of a natural number, square of a
real number, square of a complex number, would be similarly distinguished. The uncertainties as to the exact meaning of the notation (x^2+x)^2, and the consequent uncertainty as to the range of the
arguments of the function (x^2+x)^2 would then disappear.)
In general if M is an expression containing a variable x (as a free variable, i.e. in such a way that the meaning of M depends on a determination of x), then (λxM) denotes a function whose value, for
argument α is denoted by the result of substituting (a symbol denoting) α for x in M. The range of arguments of the function (λxM) consists of all objects α such the expression M has a meaning when
(a symbol denoting) α is substituted for x.
If M does not contain the variable x (as a free variable), then (λxM) might be used to denote a function whose value is constant and equal to (the thing denoted by) M, and whose range of arguments
consists of all things. This usage is contemplated below in connection with the calculi of λ-K-conversion, but is excluded from the calculi of λ-conversion and λ-δ-conversion — for technical reasons
which will appear.
Notice that, although x occurs as a free variable in M, nevertheless, in the expression (λxM), x is a bound or apparent, variable. Example: the equation (x^2+x)^2 = (y^2+y)^2 expresses a relation
between the natural numbers denoted by x and y and its truth depends on a determination of x and y (in fact, it is true if and only if x and y are determined as denoting the same natural number); but
the equation (λx(x^2+x)^2) = (λy(y^2+y)^2) expresses a particular proposition — namely that (λx(x^2+x)^2) is the same function as (λy(y^2+y)^2) — and it is true (there is no question of a
determination of x and y).
Notice that λ, or λx, is not the name of any function or other abstract object, but is an incomplete symbol — i.e. the symbol has no meaning alone, but appropriate formed expressions containing the
symbol have meaning. We call the symbol λx an abstraction operator, and speak of the function which is denoted by (λxM) as obtained from the expression M by abstraction.
The expression (λx(λyM)), which we shall often abbreviate (λxy.M), denoted a function whose value, for an argument whose value is denoted by x, is denoted by (λyM) — thus a function whose values are
functions, or a function of two variables. The expression (λy(λxM)), abbreviated as (λyx.M) denotes the converse function to that denoted by (λyx.M). Similarly (λx(λy(λzM))), abbreviated as (λxyz.M),
denotes a function of three variables, and so on.
Functions introduced in previous sections as examples can now be expressed, if desired, by means of abstraction operators. For instance I is (λxx); J is (λfxyz.fx(fzy)); S is (λnfx.f(nfx)); H is
(λxI), or (λx(λyy)) or (λxy.y); K is (λxy.x); 1 is (λfx.fx).
Chapter II: Lambda Conversion
5. PRIMITIVE SYMBOLS, AND FORMULAS.
We turn now to the development of a formal system, which we shall call the calculus of λ-conversion, and which shall have as a possible interpretation or appilcation the system or ideas about
functions of chapter I.
The primitive symbols of this calculus are three symbols,
λ, (, ),
which we shall call improper symbols, and an infinite list of symbols,
a, b, c, ... , x, y, z, a', b', ... , z', a'' ... ,
Which we shall call variables. The order in which the symbols appear in this originally given list shall be called their alphabetical order. A formula is any finite sequence of primitive symbols.
Certain formulas are distinguished as well-formed formulas, and each occurrence of variable in a well-formed formula is distinguished as free or bound, in accordance with the following rules (1−4),
1. A variable x is a well-formed formula, and the occurrence of the variable x is this formula is free.
2. If F and A are well-formed, (FA) is well-formed, and an occurrence of a variable y in F is free or bound in (FA) according as it is free or bound in F, and an occurrence of a variable y in A is
free or bound in (FA) according as it is free or bound in A according as it is free or bound in A.
3. If M is well-formed and contains at least one free occurrence of x, then (λxM) is well-formed, and an occurrence of variable y, other than x, in (λxM) according as it is free or bound in M. All
occurrences of x are bound in (λxM).
4. A formula is well-formed, and an occurrence of a variable in it is free, or is bound, only when this follows from 1−3.
The free variables of a formula are the variables with at least one free occurrence in the formula. The bound variables of a formula are the variables with at least one bound occurrence in the
Hereafter (as was just done in the statement of the rules 1−4) we shall use bold capital letters to stand for variable or undetermined formulas, and bold small letters to stand for variable or
undetermined variables. Unless otherwise indicated in a particular case, it is to be understood that the formulas represented by bold letters are well-formed formulas. Bold letters are thus not part
of the calculus we are developing but a device for talking about the calculus: they belong, not to the system itself, but to the metamathematics or syntax of the system.
Another syntactical notation we shall use is the notation,
which shall stand for the formula which results by substitution of N for x throughout M. This formula is well-formed, except in the case that x is a bound variable of M and N is other than a single
variable — see §7. (In the special case that x does not occur in M, it is the same formula as M.)
For brevity and perspicuity in dealing with particular well-formed formulas, we ofter do not write them in full but employ various abbreviations.
One methd of abbreviation is by means of a nominal definitions, which introduces a particular new symbol to replace or stand for a particular well-formed formula. We indicate such a nominal
definition by an arrow, pointing from the new symbol which is being introduced to the well-formed formula which it is to replace (the arrow my be read “stands for”). As an example we make at once the
nominal definition:
I → (λaa).
This means that I will be used as an abbreviation for (λaa) — and consequently that (II) will be used as an abbreviation for ((λaa)(λaa)) and (λa(aI)) as an abbreviation for (λa(a(λaa))), etc. | {"url":"http://cap-lore.com/Languages/lambda/math/Alonzo/book.html","timestamp":"2024-11-08T09:09:59Z","content_type":"text/html","content_length":"22348","record_id":"<urn:uuid:abbfdb0e-1237-46ae-95b0-3d88a0b65598>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00225.warc.gz"} |
Assume that $p=q$ and none of the $a_{j}$ is a nonpositive integer. Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many zeros if and only if the $a_{j}$ can be
re-indexed for $j=1,\dots,p$ in such a way that $a_{j}-b_{j}$ is a nonnegative integer.
Next, assume that $p=q$ and that the $a_{j}$ and the quotients ${\left(\mathbf{a}\right)_{j}}/{\left(\mathbf{b}\right)_{j}}$ are all real. Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has
at most finitely many real zeros. | {"url":"https://dlmf.nist.gov/16.9","timestamp":"2024-11-14T11:18:16Z","content_type":"text/html","content_length":"22695","record_id":"<urn:uuid:95fa6e25-3cc8-40ee-bdf9-7fb369109cb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00494.warc.gz"} |
Short division 3 digit by 1 digit no remainders
Short division 3 digit by 1 digit no remainders
Price: 240 points or $2.4 USD
Subjects: math,mathElementary,operationsAndAlgebraicThinking,multiplicationAndDivision
Grades: 4,5
Description: This 30 card boom deck in random order will allow students to practice 3 digit by 1 digit short division. The first slide is an explanatory slide to read in order to help the student
understand the terminology and processes used in division. Students calculate the quotient in each of the remaining 29 slides using carryovers where needed. Aligned to Common Core Standard: 4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between
multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. | {"url":"https://wow.boomlearning.com/deck/DitG6r7wpE2jZcZWB","timestamp":"2024-11-14T16:58:46Z","content_type":"text/html","content_length":"2632","record_id":"<urn:uuid:d80a7592-942a-49b6-b85e-cf61ea234db1>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00073.warc.gz"} |
Rubik's cubes
2014-04-28 03:01:30
Apparently, it's the 40th anniversary of the invention of the famous puzzle. It was developed by the Hungarian architecture professor in 1974 as an exercise to teach his students how to solve
problems in three-dimensional geometry. It of course became immensely popular; the most popular puzzle ever sold. This was according to NBC News, which ran a story about the cube this evening. The NY
Times had an article a couple of days ago about the anniversary, which is being observed by a science museum in New Jersey. Their exhibit includes a solid gold cube with gemstone faces, said to be
worth $2.5 million.
I remember my first experience with the cube. I was a junior in college, and I was at a party for math majors hosted by one of the professors. I noticed a cube on a bookshelf. I picked it up and
asked what it was. Someone behind me joked "he asks what it is, innocently!" I was impressed that you could turn the sides without it falling apart, but I was told that was not what was interesting
about it--it was a puzzle, a fiendish one. I had to order one of my own. This was before the fad hit the U.S., before a major toy company had picked the puzzle up; my copy was a rather roughly-made
version by a little company in Virginia called Logical Games. It was made of a white plastic, unlike the mass-market ones that soon appeared that were made of black plastic. As soon as I got it, I
worked hard to solve it, spending a half hour or more a day. I got a hint or two from friends, but it was a source of great pride when I finally solved it after five weeks. After solving it, I still
couldn't put it down; I worked to improve my algorithm, and I spent many hours figuring out how to put the cube into pretty patterns (checkerboards, etc).
In my senior year, the Rubik's fad hit with full force. A bunch of us gathered in from of a television to watch NBC News do a story about the puzzle featuring one of our professors (Joe Buhler)
juggling three cubes, while explaining how he uses it to teach abstract algebra. I was charmed to see the beginning of that story (which aired in 1981) in this evening's report, although they didn't
show the bit with Prof. Buhler.
It's been a while since I've played with a Rubik's cube. The original white-plastic cube now sits on a bookshelf in my office at school, along with a variety of other Rubik's type puzzles. These
include a 4 by 4 by 4 cube (Rubik's Revenge), and a 5 by 5 by 5 cube. The latter I got in Berkeley at the 1986 International Congress of Mathematicians, which I attended right after finishing my PhD
at Oregon. A fellow was selling them at the exhibits area of the conference; he said they were actually prototypes. The toy company had decided not to market them, and he had purchased the molds and
a few hundred copies of the prototype. He also sold me a dodecahedron (twelve sides, each pentagonal), also prototypes that wouldn't be marketed. (The dodecahedron had a special resonance with me,
because it reminded me of the old H. C. Kendall Planetarium in Portland, which once was enclosed by a blue and green plexiglass dodecahedron.) These puzzles were quite expensive, and I regretted not
buying the third puzzle he was selling: a Skewb (a cube with off-axis joints, such that when you make a move, all six sides are changed). Years later, these puzzles actually did appear in stores, and
I was glad to finally get a Skewb. It is on the shelf in my office with the other puzzles, to pique the curiosity of my students, and to remind me of many happy hours of playing with these toys. | {"url":"https://foragingforclues.com/Rubik's%20cubes%202014-04-28%2003-01-30.html","timestamp":"2024-11-10T14:13:43Z","content_type":"text/html","content_length":"4445","record_id":"<urn:uuid:58704e8a-ccc6-4709-863a-a40a8f8802e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00477.warc.gz"} |
Unlocking the Physics of 12 Sports with a Smartphone
Christophe Chazot
Sport isn't just a display of physical strength or natural talent. It's also a complex series of movements, reactions, and decisions, all influenced by the laws of physics. By harnessing the
technology at our fingertips, we can not only enhance our performance but also deepen our understanding of the scientific principles underlying each movement, every jump, every strike. So, ready to
combine your passion for sports with a dash of science? Grab your smartphone and let's begin the exploration!
Before we start however, a word of caution: smartphones are delicate devices, so care must be taken to secure them when used as measuring instruments.
Measuring Instruments
To carry out the experiments described in this article, you will use the sensors of a smartphone or tablet. The ones we will use are present in most digital devices. The data from these sensors can
be collected using scientific applications available on the iOS or Android stores. The free FizziQ app, available for iOS and Android, is perfectly suited to analyzing data for the experiments we
• The GPS: Through the analysis of signals from satellites, a smartphone's GPS chip allows for measuring speed, altitude, and distances. This data is useful for assessing the performance of runners
or cyclists and identifying the factors influencing their results.
• The pedometer: The pedometer measures the cadence of a walk or run. In sports, it evaluates the rhythm of a race and its consistency.
• The accelerometer: This instrument measures linear and absolute acceleration. It provides valuable information for analyzing jumps in trampoline activities or the energy of impacts in boxing.
• The video camera: Used with video analysis software, it can determine an object or person's position over time, from which one can deduce speed, acceleration, or energy. The collected data
provides insights into trajectories in basketball, energy transformation in pole vaulting, the path of a shuttlecock in badminton, the collision of pétanque balls, the Magnus effect in soccer, or
the position of the center of gravity during a dive.
• The gyroscope: This sensor determines the rotation speed of the smartphone. It can be used in analyzing rotational movements such as discus or hammer throws.
• The microphone: Used in conjunction with another instrument like the accelerometer, it measures athletes' reaction speeds, for example, at the start of a race.
Now, let's explore how we can use these measurement tools to better understand various sports.
Running is probably the sport that can be most easily studied. Seemingly simple, it is in reality a complex dance of biomechanics, physiology, and psychology. Its study addresses the themes of
performance, reaction speed, and human body physiology.
Firstly, the analysis of performance. By measuring the speed recorded by the GPS or the number of steps over time measured by the pedometer, one can study the speed of the race relative to the
distance, its regularity, the cadence of the steps, and their length. By comparing these parameters for different runners, one can begin to understand what determines each one's performance.
Another topic of interest concerns the start sequence for running. This crucial moment can be studied by simultaneously recording the sound volume and the acceleration measurement (Duo Screen
function in FizziQ). By comparing these two pieces of information, one measures the runner's reaction speed, which is the delay between the sound signal and the athlete's movement. This time can be
compared with that of other athletes.
A final topic concerns the medical risks associated with running. By using the accelerometer, one measures the impact force of each step on the legs, which is typically 5 to 10 times the runner's
weight, or 5 to 10 g. This observation opens discussions on many questions, such as whether it's healthier to run barefoot or with shoes, or if practicing this sport from a certain age is recommended
for the joints.
The trampoline, much more than just a child's play, is in fact an extremely interesting experimentation field for gravitation. Using the accelerometer, one can study numerous aspects of the jump,
especially bounces and flight.
It's always a bit challenging to conceptualize that during the flight phase, the athlete is not only in free fall, but from their reference frame, he is also in weightlessness, as Einstein
demonstrated in a famous thought experiment. During the flight phase, the accelerometer will thus display zero acceleration, confirming that the athlete is in a state of weightlessness. The
phenomenon, which may seem fleeting on a trampoline, can be accurately evaluated: how much time does an athlete actually spend in a state of free fall? Do we verify that the absolute acceleration is
Upon landing, other questions can be addressed: what is the acceleration when the athlete makes contact with the mat again, and how does it influence the next jump? What is the damping if the athlete
takes no action?
Pole Vaulting
The pole vault, an aerial ballet where humans seek to defy gravity, is a tribute to human biomechanics and the laws of physics. Behind every launch lies a choreography of angles, forces, and pivotal
moments, each playing a crucial role in the quest for height.
During the run, the athlete-pole system acquires a certain kinetic energy. This energy is transformed into the pole's elastic energy after planting, and the athlete takes off. Through a swinging
motion, the vaulter further increases the elastic energy. During the ascent, the pole releases, converting the elastic energy into the athlete's potential energy. During the inversion phase, the
athlete continues to rise upside down and clears the bar after a final push. The pole vault is, therefore, a highly complex and technical movement.
The best tool for studying the mechanics of pole vaulting is the kinematic analysis of a video or chronophotography. One can film a vaulter or use a video from the FizziQ kinematic library. With this
tool, one can study how, during each phase, the transformation and input of energy allows the athlete to reach the highest possible point. For example: what is the contribution of the run to the
height gain? What is the elastic energy stored in the pole? Does the final push help the athlete go higher? How does the tipping movement create more elastic energy?
The flight of a basketball towards the hoop is more than just a sporting gesture; it's a living parabola, a perfect illustration of the laws of physics that can be easily studied using a smartphone's
camera and the FizziQ kinematic analysis module.
Studying the trajectory of a free throw in basketball is a very interesting exercise for students, who can also make their own video for analysis. For this, the smartphone is positioned at a
sufficient distance to avoid distortions due to the wide-angle. The shot must remain fixed during the video's duration. A frame rate of 30 or 60 frames per second will be used. The hoop's height can
serve as a scale (see this link for more details on creating videos for kinematics).
From this video (or one downloaded from the internet), the motion curve is then analyzed. Its equation, which is a parabola, will be determined. This calculation will allow high school students to
estimate the acceleration due to gravity, g.
Discus Throw
The discus throw is an energy transformation exercise that requires precise technique to achieve the best results. During the rotation phase, the athlete accumulates kinetic energy using muscle force
and exploiting the friction between their feet and the ground. This energy, acquired by body rotation, is then transmitted to the disc during the throw phase.
One of the critical elements of this technique is the athlete's rotation speed. A rapid rotation allows more energy transfer to the disc, resulting in a longer throw.
To analyze the throw, one can use a smartphone's gyroscope attached to the athlete's arm. The measurements will determine the athlete's rotation speed and provide an estimate of the disc's kinetic
energy during the throw.
A spinning disc's trajectory is not a parabola. If the disc is launched with a negative angle relative to the initial trajectory angle, the disc can travel much further than an equivalent weight
thrown at the same speed. One can try to estimate the trajectory difference based on field data.
A mechanical sport, cycling addresses other sporting issues, especially the means of multiplying effort and resistance forces.
The first analysis concerns the bicycle's functioning and a crucial part: gear shifting. Thanks to this invention, cyclists can maintain optimal pedaling cadence regardless of bike speed. It also
allows for applying significant torques for climbs or starting. To better understand the usefulness of gear shifting, why not undertake the following analysis? Choose a certain distance (50 meters,
for example) and measure the time it takes for a cyclist from a standing start to reach this mark as quickly as possible. Also, measure the speed reached using GPS. Which gear is best?
A second analysis relates to friction. Indeed, the equation for cycling on a flat road is simple: the energy input is dissipated as friction energy, of which the main one beyond a certain speed is
air resistance, proportional to the frontal surface area. To study the impact of air resistance, one can measure the slowdown caused by different cyclist positions using GPS: in a racing position or
standing on the pedals.
Boxing and Martial Arts
Bruce Lee's 1-inch punch is an iconic testament to the amalgamation of technique, speed, and power. While it may seem mystical to the untrained eye, the physics underlying this move can be explored
using our smartphone. This exploration will help answer one of the most common question asked in boxing and martial arts : "What is the real power of a punch?".
To dissect this legendary punch, you can use the smartphone's accelerometer. Attach the smartphone securely to a punching bag or a target that can absorb the impact. When the 1-inch punch is executed
on the target, the smartphone will register the force's intensity through changes in acceleration. This acceleration, combined with the bag's mass, allows one to calculate the punch's force using
Newton's second law (Force = mass x acceleration). Moreover, by knowing the duration over which this force is applied, we can estimate the punch's energy. Although this method doesn't directly
measure the boxer's punch force, it provides a valuable indication of the impact felt by an opponent (or, in this case, the bag).
It is also possible to analyse the movement and speed of the arm or leg during a kick by attaching a smartphone directly onto the wrist and analyse the acceleration. Or, to get a more global view, to
record a slow-motion video recording from the smartphone's camera, that can be analyzed with the kinematics module.
Boxing and martial arts are largely unused ways to introduce students to the concepts of energy through a practical and popular example. However they should be careful not to harm anyone through
It is commonly accepted that a ball thrown into the air, in the absence of friction, will follow a parabolic trajectory. But when it comes to a diver performing a somersault, does the trajectory of
the center of gravity remain parabolic? To answer this question, a practical study can be undertaken during a visit to a swimming pool where an athlete's dive is filmed. For those without nearby pool
access, one can refer to a diver's video available in the FizziQ video library.
Using the video of a somersault dive that you can find in FizziQ's kinematic video library, you can perform the complete analysis of a somersault diving movement. In a first analysis, we can study
the movement of different body parts like the diver's head, and in a second one, their feet. It is observed that the curves generated from these points differ considerably.
However, mechanics assures us that the center of gravity of a free-falling body, unaffected by friction, describes a parabola. To conduct this analysis, one can try to estimate the athlete's center
of gravity position in each image, thus obtaining a parabolic trajectory for this point.
One can also model the diver's center of gravity more precisely by pinpointing each body part, then exporting their coordinates into Excel and applying the weights from Leva's tables that provide the
human body's mass distribution. Using this frequent biomechanics analysis method, we then verify that the center of gravity's trajectory is indeed a parabola.
Pétanque provides a very interesting playing field for studying a phenomenon we hadn't previously explored: collisions. Using a smartphone, we can deepen our understanding of the physical
interactions that occur when one ball strikes another.
The first analysis involves studying the result of a collision when a player shoots "au fer", meaning the ball is thrown in the air and strikes the opponent's ball before hitting the ground. This
sequence will be filmed at a rate of 60 or 120 frames per second. By analyzing the video using kinematic analysis tools, such as the FizziQ application, we can determine the coefficient of
restitution during a shot. What can we deduce about the strategy during the "au fer" shot?
Another analysis involves measuring the sound frequency when two balls strike each other. Does this frequency vary depending on the balls? Do higher-quality balls produce a different sound?
Badminton stands out from other racquet sports due to its specific projectile: the shuttlecock. Its unique shape results in a distinct trajectory without a bounce, requiring players to anticipate its
movements. The shuttlecock allows for a range of shots, from fast smashes to subtle drops, promoting tactical exchanges. This specificity requires players to be in excellent physical condition to
move quickly and a refined strategy to outplay the opponent.
The specific trajectory of the badminton shuttlecock can be easily studied through video analysis. If one does not have a venue to capture it, one can use a video from a badminton throw from the
FizziQ library. It's impossible to formally determine the trajectory equation, but one can still identify three distinct phases. An initial rapid trajectory: Just after being hit, the shuttlecock
travels at a relatively high speed, appearing almost linear over a short distance. A strong deceleration: Due to its shape and construction, the shuttlecock decelerates quite rapidly, giving its
overall trajectory a generally parabolic shape. An almost vertical drop: The end of the trajectory when the shuttlecock is launched very high has the characteristics of a vertical fall.
Studying the lift or Magnus effect with a football is particularly interesting, as it highlights the principles of fluid mechanics in action in the sport. This phenomenon is often observed when a
footballer imparts spin on the ball, making it follow a curved trajectory rather than a straight one. The Magnus effect occurs when the ball's rotation creates a pressure difference from one side to
the other, causing a force perpendicular to its trajectory.
To study this effect accurately, video analysis is a powerful tool. By filming a footballer's shot from different angles, one can trace the ball's trajectory in real-time. Using kinematic analysis
software like FizziQ, one can then mark the ball's position frame by frame, visualizing its curved trajectory and measuring the extent of curvature based on the initial rotation given to the ball.
Furthermore, by comparing shots with and without spin, one can better understand the influence of rotation on the ball's trajectory. Thus, using video analysis to study lift in football provides a
tangible means of exploring and understanding this fascinating phenomenon that's central to many memorable sporting moments.
Paralympic Long Jump
The Paralympic long jump, with the use of carbon blades, offers a fascinating blend of human determination and technological prowess. Like all other Paralympic sports, it embodies adaptability in the
face of adversity and provides a unique platform for studying the biomechanics of movements. Furthermore, this discipline raises crucial questions about fairness in sports while serving as a powerful
symbol of inclusion and inspiration. The interaction between the athlete and modern technology in this specific context evokes both admiration and curiosity.
All the analyses we conducted before are possible for Paralympic sports, but other analyses can also be undertaken, shedding light on the understanding of these sports.
For example, in the case of the Paralympic long jump with "blades", one can explore the following questions: What asymmetry is created by using prosthetics during the run, and is it a disadvantage?
How does the stiffness of a "blade" or "lamina" compare to that of a valid leg? Is the take-off angle similar between a Paralympic and non-Paralympic athlete? Some of these questions can be studied
using a smartphone, either through direct measurement or video analysis.
Over the years, the symbiosis between science and sport has strengthened, paving the way for remarkable advancements in understanding human performance. Sciences, whether physical, biological, or
social, provide valuable insights into the mechanisms, techniques, and strategies that enhance athletic performance. They dissect, analyze, and propose innovative solutions to push the boundaries of
what the body and mind can achieve.
However, sports are not just an application field for sciences; they also become a fascinating subject of study in their own right. In schools and universities, sports analysis, facilitated by the
availability of digital measurement devices like smartphones, offers a unique opportunity to address scientific concepts concretely and vividly while stimulating students' curiosity and enthusiasm.
The upcoming Olympic Games in Paris further highlights this interdependence. This global event showcases athletic excellence, where every fraction of a second matters, and where scientific
innovations can make a difference. But it's also an open-air laboratory for researchers, drawing attention to the importance of continuing interdisciplinary studies between sports and sciences.
Prothèses tibiales de saut en longueur handisport - Jean BOUTEILLER, Pierre-Adrien BREARD, Paul FRAPART, Cyril VOISARD, Maxime VRAIN | {"url":"https://www.fizziq.org/en/post/sportsandscience-1","timestamp":"2024-11-08T18:59:59Z","content_type":"text/html","content_length":"1050478","record_id":"<urn:uuid:700b9512-9086-4b83-b40c-94800cd23274>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00816.warc.gz"} |
Dynamics of life expectancy in Russia over the last half century and prospects of reaching the 80+ level by 2030
Since 2006, the Russian Federation has experienced a steady growth in life expectancy, far exceeding the growth rate in other countries. The average growth rate of life expectancy in many countries,
including Europe, the USA, Japan is about 1 year of age within 5 calendar years. The documents shaping of the population policy in the Russian Federation set targets for the total life expectancy
(TLE, e[x]). By 2025 the TLE should reach 78 years (Ukaz Prezidenta... 2018). The Presidential Address to the Federal Assembly specifies 2030 as a benchmark: “By the end of the next decade Russia
must confidently enter the “80 plus” countries club, where life expectancy exceeds 80 years” (Poslanie Prezidenta... 2018). On the one hand, the existing growth rate in life expectancy and the marked
difference compared to a number of countries suggest decline in the death rate within the next 10 years and achievement of the set targets. However, on the other hand, there are risks that the
existing growth is associated with the effect of a low base level and may slow down, which will prevent TLE from reaching the required values.
This paper analyzes historical data for the last decades. Fig. 1 shows Rosstat data on life expectancy at birth for over 60 years. The trend has changed after 2006, and in some periods life
expectancy of men was increasing by almost a year of age within one calendar year. In European countries, within the last 50 years the increase in life expectancy by 1 year of age occurred within
over 5 calendar years on average. In Russia, life expectancy at birth (e[0]) has increased between 2011 and 2017 by 2.46 years for the total population (from 70.24 to 72.70 years): by 2.95 years in
men (from 64.56 to 67.51 years), in women - by 1.78 years (from 75.86 to 77.64 years). The difference in life expectancy between men and women decreased from 11.30 in 2011 to 10.13 years in 2017.
Figure 1.
Life expectancy at birth in Russia. Source: Rosstat: http://www.gks.ru/wps/wcm/connect/rosstat_main/rosstat/ru/statistics/population/demography/.
The results of the TLE forcasting for men and women are shown in Fig. 2. The target values set by population policy documents are used as benchmarks: they are higher than the values of the average
Rosstat’s forecast. For comparison, Fig. 2 shows the data on the TLE for the United States, where an amount comparable with the GDP of the Russian Federation (1.5 trillion USD) is annually spent on
palliative care, although such patients account for only 5% of the total numbers (Kalyanina 2018).
Figure 2.
Life expectancy forecast. Sources: Rosstat; https://www.cia.gov/library/publications/resources/the-world-factbook/geos/us.html.
A natural question arises: by what rate should mortality be reduced to reach the target values in life expectancy? Fig. 3 shows the calculated e[0] (TLE at birth) over time (X-axis) for men and
women, depending on the annual rate of decline in mortality at each age (Y-axis). This forecast is calculated by the author on the basis of smoothed life tables of the Russia’s population for 2017.
It can be seen that achieving the e[0] value of 80 years in 2030 is possible at a rate of mortality decline of 4.5%, which is higher than the current rate shown in Fig. 3.
Figure 3.
Life expectancy (e[0]) depending on the annual rate of decline in mortality: a) the field of e[0] values; b) the geometric mean of decline in mortality depending of the age in 2007-2017. Source:
calculations of the author, Rosstat.
Of course, the assumption of the same rate of decline in mortality at each age is conditional, but this approach allows to estimate the sensitivity of e[0] to age changes in the probability of death.
It may be advisable to use more complex hypotheses for the age profile of changes in the probability of death. However, for estimative calculations, the proposed approach seems appropriate, as it
significantly simplifies analysis of results and reduces the risk of additional errors. For example, the Society of Actuaries of the U.S. in order to model the effect of reducing mortality, the
expected rate of decline in mortality for men and women is assumed at 1% per year in the age range of 0-85 followed by a linear decrease to 0% (Mortality improvement... 2017). Many similar examples
are given in the proceedings of the Mortality Working Group of the International Actuarial Association (Materials of the Mortality Working Group...).
According to the data shown in Fig. 3, the average geometric rate of mortality reduction in the Russian Federation during 2007-2017 strongly depends on age and is described by a complex
non-monotonous function. However, in 60+ ages, where death rates most significantly affect life expectancy, the annual rate of decline is between 2 and 3%. Therefore, if the current profile of the
rate of reduction of mortality by age will not change, it is impossible to achieve TLE at birth of 78 years in 2024 and 80 years in 2030.
In addition, it should be noted that in order to maintain the linear growth rate of TLE, the rate of decline in mortality must increase. In other words, linear dependence disappears over time; this
means that there is another limitation related to the requirement of increasing the rate of mortality reduction (this effect is manifested at high values of e[0] and can be associated with
approaching the age limit).
If we divide age into two parts - up to 50 years and over, and look at the contribution of mortality changes in these age groups to life expectancy at birth, it turns out that in women, when
mortality is reduced by 80% (annual rate of decline of 15% per year within10 years) in ages under 50, and in the absence of reduction in mortality in ages over 50, e[0] increases by 2.4 years and by
5.3 years in men. According to the life table for 2017, in the Russian Federation life expectancy at birth in women is 77.6 years, in men — 67.5 years. Therefore, in the absence of reduction in
mortality after 50 years and in the case of reduction by 80% in all ages under 50 years, life expectancy of women will be 80 years, hence the increase is 80-77.6=2.4 years. Similarly, the increase in
life expectancy for men is calculated. Fig. 4 shows life expectancy values depending on mortality reduction (in per cent). The reduction of mortality in ages over 50 years is marked along the X-axis,
and in ages under 50 years is marked on the Y-axis. Life expectancy at the same rate of decline in mortality at all ages corresponds to the diagonal.
Figure 4.
Life expectancy (e[0]) (a) of women and (b) men in the Russian Federation depending on the change in mortality in the ages under and over 50 years. Source: calculations of the author
Assuming that the annual rate of decline in mortality over the next 10 years will not change, at 4-5% per year for women under 50 years, after 50 - 2-3% and at 5-6% and 2-3% for men respectively, the
mortality rate for women under 50 years of age will decrease by 34-40%, after 50 - by 20-25%, for men under 50 years of age - by 40-45%, after 50 - by 20-25%. Thus, the maximum life expectancy for
women in 10 years can be 81 years, for men - 73 years, the arithmetic mean will be 77 years, and in 2030 for women the maximum life expectancy may be 82 years, for men - 75 years, and the arithmetic
mean is 78.5 years. Despite the rough calculation, this approach provides some estimate for the target.
In addition, when choosing long-term benchmarks, it is necessary to consider the effect associated with the trend direction, in other words, the derivative sign. The change in mortality in 2017
exceeds the average of mortality reduction in almost all ages, but this does not allow for revision, as the expected increase e[x] in 2018 is much less than in the record 2017.
It should be noted that the decline in mortality was observed not so long ago. During the 1990s in the Russian Federation, there were periods of strong (extreme) growth in death rates. Minimum life
expectancy values were recorded in 1994. In some ages, the increase in death rates exceeded 30%. Heat maps (Fig. 5) show annual percentage changes in mortality in the Russian Federation, depending on
age, since 1990.
Figure 5.
Heat maps of the annual change in death rates for (a) men and (b) women in the Russian Federation. Source: calculations of the author
Let’s consider a way to estimate the change in age-specific death rates over time. To do this, let’s compare the probabilities of death at each age for a given period of time.
The probability of dying at age (x) per year (t) is calculated by the formula
where lx (s,t) is the number of those of sex s who live to age x, according to the life table in year t.
The change in probability of death for a year is defined as follows:
Then, projecting the obtained value, expressed as a percentage, on the time-age plane, we get the field of mortality for a certain sex. The vertical scale is the percentage change in the probability
of death. Red reflects an increase in the probability of death, yellow indicates a decrease. The background colour applies to annual percentages of mortality fluctuation in the +/- 2% range, which
corresponds to the average positive and negative values of percentage changes in death probabilities.
Figure 6.
Life expectancy in Russia in 2017 for (a) women and (b) men depending on age. Source: calculations of the author
Values of life expectancy (e[x]) for men and women depending on age and time for the population of the Russian Federation show TLE growth in all ages since 2006, which confirms the bend of isolines
observed in 2017. TLE in men is more likely to fluctuate than in women.
It is important to note that there are regional differences for the e[x] (t) function. For example, for the population of the city of Moscow, the growth of life expectancy in men begins 2 years
earlier than in the Russian Federation, and for women, there is almost monotonous growth e[x] after its sharp fall in the 1990s (Fig. 7). | {"url":"https://populationandeconomics.pensoft.net/article_preview.php?id=37964","timestamp":"2024-11-09T23:41:56Z","content_type":"application/xhtml+xml","content_length":"37185","record_id":"<urn:uuid:9497b377-e491-468c-a3b9-27b9812bd040>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00246.warc.gz"} |
Integration Rules: Learn Formula, Important Rules with Examples
What is integration by parts rule?
It is the product rule of integration, where we integrate a product of functions at a time using the following formula.\( \int_{ }^{ }f\left(x\right)g\left(x\right)dx=f\left(x\right)\int_{ }^{ }g\
left(x\right)dx-\int_{ }^{ }\left(f^{\prime}\left(x\right)\times\int_{ }^{ }g\left(x\right)dx\right)dx \), where \( f^{\prime}\left(x\right) \) is the first order derivative of \( f\left(x\right) \)
What is the UV rule of integration?
The product rule of integration is also known as the UV rule of integration where we integrate by parts for the product of two functions.
Does chain rule apply to integration?
Yes, chain rule applies to integration.
What is the sum rule for the integration of functions?
The sum rule for the integration of functions is, \( \int \left(f_1\left(x\right)+f_2\left(x\right)\right)dx=\int f_1\left(x\right)dx+\int f_2\left(x\right)dx \)
What are partial fraction integration rules?
A function of the form \(\frac{f\left(x\right)}{g\left(x\right)}\), where \(f\left(x\right)\) and \(g\left(x\right)\) are polynomials, \(g\left(x\right)\ne0\), is called a rational function.
Integration by partial fractions is a method used to decompose and then integrate a rational fraction integrand that has complex terms in the denominator. | {"url":"https://testbook.com/maths/integration-rules","timestamp":"2024-11-10T20:52:49Z","content_type":"text/html","content_length":"866072","record_id":"<urn:uuid:4bb07b8e-f837-48da-8e47-8da76b394c3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00169.warc.gz"} |