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How to Use LangChain LLM-Math in Python – TheLinuxCode
How to Use LangChain LLM-Math in Python
Large language models (LLMs) like GPT-3 have demonstrated impressive capabilities in understanding and even performing mathematical reasoning when prompted appropriately. LangChain provides an easy
way to tap into these skills from Python by combining LLMs with code execution. In this comprehensive guide, we will explore using LangChain‘s LLM-Math module to solve math problems in Python.
Overview of LangChain and LLM-Math
LangChain is a Python framework that bridges large language models with code execution environments. It allows issuing text prompts to LLMs like GPT-3 and having the results executed as Python code.
LangChain handles passing data back and forth seamlessly.
The LLM-Math module within LangChain specifically equips LLMs with mathematical reasoning skills. It utilizes the LLMs‘ ability to parse complex word problems articulated in natural language and
produce mathematical expressions and solutions. LLM-Math builds upon LangChain to seamlessly integrate these mathematical capabilities into Python code.
Using LLM-Math, we can easily solve math problems ranging from basic arithmetic to complex equations, all from within Python. It harnesses the LLMs‘ latent skills at symbolic math reasoning, freeing
us from having to formally encode every step.
Installing Required Libraries
To use LLM-Math, we first need to install LangChain and a few other dependencies:
pip install langchain openai os
This will install:
• LangChain – Core library for connecting Python with LLM APIs like OpenAI
• OpenAI – Python library for accessing OpenAI‘s API
• OS – For managing environment variables
We also need an OpenAI API key which provides access to the LLM models. Go to openai.com, create an account if necessary, and grab your secret API key from the dashboard.
Next we need to set the API key as an environment variable in Python:
import os
os.environ["OPENAI_API_KEY"] = "YOUR_API_KEY"
Replace YOUR_API_KEY with your actual secret key. This allows easily authentication when we initialize the OpenAI object.
Initializing LLM and LLM-Math
With the dependencies installed, we can now initialize objects to connect with the LLM and use LLM-Math:
from langchain import OpenAI, LLMMathChain
llm = OpenAI(temperature=0)
math = LLMMathChain(llm=llm)
We initialize OpenAI with the default model (typically GPT-3 Davinci) and set temperature to 0 for deterministic results.
The LLMMathChain object wraps the LLM with math capabilities. We pass our llm instance into it.
Now math provides an interface for solving math problems!
Using LLM-Math for Basic Arithmetic
Let‘s start with some simple arithmetic using LLM-Math:
math.run("What is 123 + 456?")
# 579
math.run("If I have 350 apples and eat 115 of them, how many do I have left?")
# 235
We simply pass word problems or equations as strings to the run() method. LLM-Math handles interpreting the question and solving it.
It can handle more complex arithmetic as well:
math.run("What is the result of 12 * 62 divided by 6?")
# 124
math.run("If I walk 8.4 miles in 3 hours, what is my average speed in miles per hour?")
# 2.8
LLM-Math leverages the LLM‘s ability to parse natural language, extract the mathematical relationships, set up equations, and solve them. This makes easy work of word problems.
Comparing LLM-Math to Python Math
How does LLM-Math compare to using Python‘s built-in math module?
import math
# 16.0
llm_math.run("What is the square root of 256?")
# 16
LLM-Math produces the same results but expressed in natural language. This makes it convenient for quick math in the Python REPL without having to code formal expressions.
For more structured math code, Python‘s math module is preferable. LLM-Math shines when translating word problems into symbolic expressions.
We can even mix LLM-Math with Python math:
x = llm_math.run("What is the square root of 256?")
# 16
Solving Equations with LLM-Math
LLM-Math can also solve equations involving unknown variables:
llm_math.run("Solve the equation: 3x + 5 = 17")
# x = 4
And systems of equations:
If 3a + 2b = 14
and a - 2b = 2
What are the values of a and b?
# a = 4, b = 2
Equations with unknowns demonstrate the symbolic reasoning capabilities of LLM-Math. It manipulates the equations step-by-step symbolically to isolate the variable and solve for it.
This works for quite complex equations as well:
llm_math.run("Solve the equation: 5x^2 - 3x + 7 = 0")
# x = 1, x = -1.4
However, performance degrades for equations with many steps. Double-checking the solution is advised.
Evaluating Math Expressions
In addition to solving verbal math problems, we can also use LLM-Math to directly evaluate math expressions:
llm_math.run("What is the value of 3 * (10 + 5)^2?")
# 1215
This is helpful for quickly calculating an expression without assignment.
We can also substitute variable values:
llm_math.run("If x = 5 and y = 3, what is the value of 2x + 3y?")
# 19
Overall, LLM-Math provides an effective "math calculator" from within Python.
Tips for Best Results
Here are some tips for getting the most out of LLM-Math:
• Frame problems clearly – Use natural language and full sentences. Avoid ambiguity.
• Check results – Double-check solutions against Python‘s math module, NumPy, etc.
• Tune temperature – Lower temperature reduces randomness, higher introduces creativity.
• Retry variations – Rephrase prompts differently if stuck.
• Limit complexity – Break very complex problems into smaller steps.
• Combine tools – Use Python math where structured expressions make sense.
Limitations to Consider
While powerful, LLM-Math has some key limitations to keep in mind:
• Prone to mistakes in long mathematical reasoning chains
• Cannot match hand-coded math expressions for complex problems
• Limited manipulation of unknown variables and symbols
• Does not perform symbolic integration, differentiation, etc.
• Not optimized specifically for mathematical reasoning like a Computer Algebra System
Foranything high-precision or mission-critical, carefully hand-coded math will be superior. But LLM-Math provides a remarkably useful math assistant for everyday Python programming.
In summary, LangChain‘s LLM-Math module enables conveniently solving math problems in Python by harnessing the power of large language models. It can handle arithmetic, equations, expressions, and
more when posed in natural language. LLM-Math is an easy way to incorporate the latent mathematical reasoning skills of LLMs directly into Python code. While not a replacement for dedicated math
tools, it provides a helpful math sidekick for the Python programmer. | {"url":"https://thelinuxcode.com/use-langchain-llm-math-python/","timestamp":"2024-11-02T00:08:45Z","content_type":"text/html","content_length":"178140","record_id":"<urn:uuid:83b20ef5-cfc4-4209-84c6-4e2fa87c8b17>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00282.warc.gz"} |
Discussion: “Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review” (Warrier, G.R., and Dhir, V.K., 2006, Journal of Heat Transfer, 128, pp. 1243–1256)
Issue Section:
In their paper (1), the authors state on p. 1245 that Boyd and Meng (2) in 1996 suggested an interpolation method for calculating the heat transfer characteristics in the partial boiling region,
along with Eqs. (13)-(14). The authors of (1) further state that Kandlikar (3) in 1998 proposed a similar scheme and that the constants $a$, $b$, and $m$ were assumed to be constant.
In this discussion, we would like to point out a few errors/omissions in the above referenced paper (1).
1. In their technical note, Boyd and Meng (2) refer to the paper by Kandlikar (4), published in 1990, for the Eqs. (13)-(14) referred to in (1). Unfortunately, Boyd and Meng quoted the wrong
reference for the Kandlikar work in which the original model and equations were reported. The 1990 paper by Kandlikar (4), erroneously referred by Boyd and Meng, presents a correlation in the
saturated flow boiling region and makes no reference to subcooled flow boiling. The correct reference in Boyd and Meng’s paper should have been Kandlikar (5), which was published in 1991.
2. In his 1991 paper, Kandlikar (5) presented Eqs. (21)-(29) with an accompanying Fig. 4 explaining the construction in the subcooled partial boiling region. Immediately following Eq. (26), a note
appears regarding the exponent $m$, which matches the slope at the two ends. Kandlikar (5), however, contains typographical errors in Eqs. (25) and (26) for coefficients $a$ and $b$.
3. Those typographical errors were later corrected by Kandlikar (3) in 1998.
4. Another error that appears in (1) is the incorrect year of publication of the paper by Boyd and Meng (2), cited as Ref. [12]; the correct year of publication is 1995, not 1996.
5. Discussion related to Fig. 6 appearing in (1) is correct, except that the 1998 paper by Kandlikar (3) referred therein simply corrects the typographical error and provides a more detailed
comparison with the available experimental data.
6. Equation (21) appearing in Warrier and Dhir (
) is based on the paper by Kandlikar (
) as correctly reported by the authors of (
). However, the equation itself is incorrectly reproduced. The correct form of the equation (appearing as Eq. 14 in Ref (
)) is as follows:
is the mass flux
. The negative sign in the exponent 0.7 is missing in Eq. (21) in (
), and the wall superheat
is incorrectly replaced by
Subcooled Flow Boiling Model Description
For convenience, the correct form of the subcooled flow boiling model and coefficients are presented below.
shows a schematic representation of the subcooled flow boiling curve extending from the single-phase region at point C to the fully developed boiling at point E. In the single-phase region, to the
left of C in Fig.
, heat flux
is given by the following equation:
is the single-phase heat transfer coefficient with all flow as liquid, the local wall superheat is
and the local liquid subcooling is
, the local saturation temperature is
, the wall temperature is
and the liquid temperature is
In the fully developed boiling region, the heat transfer rate is related to the local wall superheat by the following equation:
where Bo is the boiling number
is the total mass flux
, and
is the latent heat of vaporization (J/kg), and
is the fluid-dependent parameter in the Kandlikar correlation (
). The value of
is 1 for water and all other fluids flowing in stainless steel tubes. For specific fluids in different tube materials, refer to (
) or other more recent publications.
The equation for the
plot in the partial boiling region, the main focus of the current discussion, is given by the following equation:
The constants
, and
are functions of heat flux
. The slope of the heat flux versus wall superheat in the partial boiling region is matched with the two limiting values, i.e.,
in the single-phase region at the beginning of the partial boiling region, identified by point C, and
at the beginning of the fully developed boiling region identified by point E. Thus, the values of
are obtained in terms of the heat fluxes and wall superheats at C and E, and the value of
is obtained in terms of the heat fluxes at C, E and at the desired location, where heat flux is
and wall superheat is
Note that there were typographical errors in Kandlikar (
) that erroneously omitted the exponent
in Eqs.
The value of
depends on the heat flux, and is allowed to vary linearly from
at C to
at D. Thus,
and the values of
are obtained as follows:
Note that
are constants for a system (for a given geometry and operating conditions), whereas the values of
, and
depend of the local value of
. Thus, the value of
can be obtained directly from a known value of
, while an iterative scheme is needed to calculate
for a given value of
in this region.
Further details on calculating $q̇C$ and $q̇E$ are given in Kandlikar (3).
G. R.
, and
V. K.
, 2006, “
Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review
ASME J. Heat Transfer
, pp.
R. D.
, and
, 1995, “
Boiling Curve Correlation for Subcooled Flow Boiling
Int. J. Heat Mass Transfer
, pp.
S. G.
, 1998, “
Heat Transfer Characteristics in Partial Boiling, Fully Developed Boiling, and Significant Void Flow Regions of Subcooled Flow Boiling
ASME J. Heat Transfer
, pp.
S. G.
, 1990, “
A General Correlation for Two-phase Flow Boiling Heat Transfer Inside Horizontal and Vertical Tubes
ASME J. Heat Transfer
, pp.
S. G.
, 1991, “
Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids Inside Circular Tubes
ASME J. Heat Transfer
, pp.
Copyright © 2007
by American Society of Mechanical Engineers | {"url":"https://energyresources.asmedigitalcollection.asme.org/heattransfer/article/129/9/1300/467647/Discussion-Heat-Transfer-and-Wall-Heat-Flux","timestamp":"2024-11-13T04:24:42Z","content_type":"text/html","content_length":"166668","record_id":"<urn:uuid:957effe2-0d22-42eb-b8b2-5c1facaebfaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00323.warc.gz"} |
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5 y+10=7 x-49 \\ 7 x+49-5 y-40=0 \end{array} \] is the required . 10. Find the Equation of the diamcter of the circle whose end points are and Equation of the line parsing through
Question Text and
Updated On Dec 6, 2022
Topic Coordinate Geometry
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 58
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Surface Curvature
Next: Absolute Surface Area Up: Description of Three Dimensional Previous: Relative Boundary Orientation
By the surface shape segmentation assumptions (Chapter 3), each surface region can be assumed to have constant curvature signs and approximately constant curvature magnitude. Using the orientation
information, the average orientation change per image distance is estimated and this is then used to estimate absolute curvature. This description separates surface regions into curvature classes,
which provides a first level of characterization. The absolute magnitude of the curvature then provides a second description.
Following Stevens [153] and others, the two principal curvatures, (Table 6.5). The curvature sign is arbitrary, and here convex surfaces are defined to have positive curvature.
Turner [161] classified surfaces into five different classes (planar, spherical, conical, cylindrical and catenoidal) and made further distinctions on the signs of the curvature, but here the
cylindrical and conical categories have been merged because they are locally similar. Cernuschi-Frias, Bolle and Cooper [45] classified surface regions as planar, cylindrical or spherical, based on
fitting a surface shading model for quadric surfaces to the observed image intensities. Both of these techniques use intensity data, whereas directly using the surface orientation data allows local
computation of shape. Moreover, using only intensity patterns, the methods give a qualitative evaluation of shape class, instead of absolute curvature estimates. More recently, Besl [24] used the
mean and gaussian curvature signs (calculated from range data) to produce a similar taxonomy, only with greater differentiation of the hyperboloidal surfaces.
Brady et al. [40] investigated a more detailed surface understanding including locating lines of curvature of surfaces and shape discontinuities using three dimensional surface data. This work gives
a more accurate metrical surface description, but is not as concerned with the symbolic description of surface segments. Agin and Binford [5] and Nevatia and Binford [121] segmented generalized
cylinders from light-stripe based range data, deriving cylinder axes from stripe midpoints or depth discontinuities.
To estimate the curvature magnitude, we use the difference in the orientation of two surface normals spatially separated on the object surface. The ideal case of a cross-section perpendicular to the
axis of a cylinder is shown in Figure 6.5. Two unit normals
Then, the curvature estimate at this cross-section orientation is:
To find the two principal curvatures, the curvature at all orientations must be estimated. The planar case is trivial, and all curvature estimates are
If the curvature is estimated at all orientations using the method above, then the minimum and maximum of these estimates are the principal curvatures. Part (a) of Figure 6.6 shows the path of the
intersecting plane across a cylinder. For simplicity, assume that the orientation (
Hence, the curvature estimate (from above) is:
For the ellipsoid case (part (b) of Figure 6.6), the calculation is similar. Letting
Hence, the curvature estimate (from above) is:
The special case of the cylinder can be derived from this by looking at the limit as
This analysis gives the estimated curvature versus cross-section orientation 6.7 shows a graphical presentation of the estimated curvature versus
For simplicity, this analysis used the curvature estimated by picking opposed surface normals at the extremes of the intersecting plane's path. Real intersection trajectories will usually not reach
the extrema of the surface and instead we estimate the curvature with a shorter segment using the method outlined at the beginning of this section. This produces different curvature estimates for
orientations not lying on a curvature axis. However, the major and minor axis curvature estimates are still correct, and are still the maximum and minimum curvatures estimated. Then, Euler's relation
for the local curvature estimated at orientations
One might ask why the global separated normal vector approach to curvature estimation was used, rather than using derivatives of local orientation estimates, or the fundamental forms? The basis for
this decision is that we wanted to experiment with using the larger separation to reduce the error in the orientation difference
We now determine the sign of the curvature. As Figure 6.8 shows, the angle between corresponding surface normals on similar convex and concave surfaces is the same. The two cases can be distinguished
because for convex surfaces the surface normals point away from the center of curvature, whereas for the concave case the surface normals point towards it.
Given the above geometric analysis, the implemented surface curvature computation is:
1. Let
2. Generate the curvature estimate versus
1. find cross-section length
2. find surface orientation angle difference
3. estimate curvature magnitude
3. Fit
4. Extract maximum and minimum curvature magnitudes.
5. At maximum and minimum curvature orientations, check direction of surface normal relative to surface.
1. if towards the center, then
2. if away from the center, then
The estimation of the major axis orientation for the surface region is now easy. (The major axis is that about which the greatest curvature occurs.) The plane case can be ignored, as it has no
curvature. Figures 6.9 and 6.10 illustrate the geometry for the axis orientation estimation process.
We calculate the axis direction by calculating the direction of a parallel line 6.9). Plane X contains the viewer and the nominal point
Line 6.10) so it must be perpendicular to the plane's normal
This vector is used as an estimate of the major curvature axis direction. The minor curvature axis direction is given by
The curvature and axis orientation estimation process was applied to the test scene. The curvatures of all planar surfaces were estimated correctly as being zero. The major curved surfaces are listed
in Tables 6.6 and 6.7, with the results of their curvature and axis estimates. (If the major curvature is zero in Table 6.6, then the minor curvature is not shown.) In Table 6.7, the error angle is
the angle between the measured and estimated axis vectors.
Table 6.7: Summary of Curved Surface Curvature Axis Estimates
│ IMAGE REGION │ ESTIMATED AXIS │ TRUE AXIS │ ERROR ANGLE │
│ 8 │ (0.0,0.99,-0.1) │ (0.0,1.0,0.0) │ 0.10 │
│ 16 │ (-0.99,0.0,0.0) │ (-0.99,0.0,0.1) │ 0.09 │
│ 25 │ (-0.99,0.07,0.11) │ (-0.99,0.0,0.1) │ 0.13 │
│ 31 │ (-0.86,-.21,-0.46) │ (-0.99,0.0,0.1) │ 0.53 │
│ 9 │ (-0.09,0.99,-0.07) │ (0.0,1.0,0.0) │ 0.12 │
│ 29 │ (-.14,0.99,0.0) │ (0.0,1.0,0.0) │ 0.14 │
The estimation of the surface curvature and axis directions is both simple and mainly accurate, as evidenced by the above discussion and the results. The major error is on the small, nearly
tangential surface (region 31), where the curvature estimates are acceptable, but the algorithm had difficulty estimating the orientation, as might be expected. Again, as the depth and orientation
estimates were acquired by hand, this is one source of error in the results. Another source is the inaccuracies caused by interpolating depth and orientation estimates between measured values.
The major weak point in this analysis is that the curvature can vary over a curved surface segment, whereas only a single estimate is made (though the segmentation assumption limits its variation).
Choosing the nominal point to lie roughly in the middle of the surface helps average the curvatures, and it also helps reduce noise errors by giving larger cross-sections over which to calculate the
curvature estimates.
Next: Absolute Surface Area Up: Description of Three Dimensional Previous: Relative Boundary Orientation Bob Fisher 2004-02-26 | {"url":"https://homepages.inf.ed.ac.uk/rbf/BOOKS/FSTO/node28.html","timestamp":"2024-11-06T04:43:32Z","content_type":"text/html","content_length":"31714","record_id":"<urn:uuid:5409086f-b395-4e9f-ad58-87acf4d7564a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00882.warc.gz"} |
Understanding Even and Odd Numbers: A Basic Guide
Understanding Even and Odd Numbers: A Basic Guide
Even and odd numbers are foundational concepts in mathematics, dividing integers into two distinct categories based on their divisibility by two. These simple yet profound classifications have
far-reaching implications in various areas of mathematics, computer science, and everyday life. This guide is designed to offer a comprehensive understanding of even and odd numbers, their
properties, and their applications.
What Are Even and Odd Numbers?
Even numbers are integers that can be divided evenly by two, meaning they can be split into two equal groups without any leftovers. These numbers always end in 0, 2, 4, 6, or 8. For example, numbers
such as 4, 16, and 102 are even because they can be divided by 2 with no remainder.
Odd numbers, on the other hand, cannot be evenly divided by two. They always leave a remainder of 1 when divided by two. Odd numbers always end in 1, 3, 5, 7, or 9. Examples include 3, 15, and 97,
which cannot be evenly divided by 2.
Rules and Properties of Even and Odd Numbers
Several rules and properties help define and distinguish between even and odd numbers:
• Addition and Subtraction: The sum or difference of two even numbers is always even, while the sum or difference of two odd numbers is always even as well. However, the sum or difference of an odd
and an even number is always odd.
• Multiplication: The product of two even numbers is always even. The product of two odd numbers is always odd. Moreover, the product of an even and an odd number is always even.
These properties are not just mathematical curiosities; they are pivotal in various proofs and problem-solving scenarios within mathematics.
Applications of Even and Odd Numbers
Even and odd numbers have various applications in real-world scenarios and theoretical mathematics. Here are a few examples:
• Computer Science: Even and odd numbers are used in algorithms and computing processes, such as hashing and error detection codes. Evenness or oddness, also known as parity, plays a crucial role
in binary operations and digital electronics.
• Mathematical Theorems and Proofs: Many mathematical theories and proofs rely on the properties of even and odd numbers to establish foundational arguments.
• Everyday Mathematics: The concepts of even and odd numbers are applied in tasks such as dividing groups or objects evenly, coding, and game theory.
FAQs on Even and Odd Numbers
How can I quickly determine if a large number is even or odd?
To quickly determine whether a large number is even or odd, simply examine its last digit. If the last digit is 0, 2, 4, 6, or 8, the number is even. If it ends in 1, 3, 5, 7, or 9, the number is
odd. This method works because the divisibility by two is only affected by the last digit of a number in a decimal system.
Are there any exceptions to the rules of even and odd numbers?
In general, the rules for even and odd numbers are consistent and without exceptions for integers. All whole numbers fit neatly into the category of either even or odd, including negative integers.
The properties of addition, subtraction, and multiplication involving even and odd numbers hold true across the entire set of integers.
How do even and odd numbers behave under division?
Division introduces complexity when it comes to even and odd numbers because the result of dividing two integers may not be an integer. Nevertheless, some rules can be observed. Dividing an even
number by two always results in an integer, which can be either odd or even. However, dividing an odd number by two always results in a fraction and not a whole number. Thus, in terms of
divisibility, odd numbers cannot be evenly divided by two, maintaining their odd nature.
Can the concepts of even and odd numbers be applied to non-integers?
Even and odd classifications are specific to integers. Fractions, decimals, and irrational numbers do not fit into the categories of even or odd because these concepts rely on divisibility by two, a
property that only integers possess. When dealing with non-integers, the notions of even and odd numbers don’t apply.
Are zero and negative integers considered even or odd?
Zero is considered an even number because it can be evenly divided by two, resulting in zero, which adheres to the definition of even numbers. Negative integers can also be classified as even or odd
based on the same rules that apply to positive integers. For example, -2 and -4 are even, while -1 and -3 are odd. The concept of divisibility by two applies to negative integers just as it does to
positive integers.
What are some common mistakes to avoid when working with even and odd numbers?
• Assuming that the sum of an odd and even number can be even. It’s always odd.
• Forgetting that zero is considered an even number.
• Misapplying the rules of division, expecting an odd number to result from dividing an odd number by two.
• Attempting to classify non-integers as even or odd.
Knowing these rules and common pitfalls can enhance your understanding and ability to work with the concepts of even and odd numbers effectively.
How are even and odd numbers used in cryptography?
In cryptography, even and odd numbers play a role, particularly in the realms of public key encryption and cryptographic algorithms. The RSA encryption algorithm, for example, uses properties of
prime numbers (which are odd, except for the number 2) to generate public and private keys. The security of such cryptographic methods often relies on the difficulty of factorizing large prime
numbers, a task that becomes more challenging as the numbers involved grow larger and maintain their odd characteristics, excluding 2.
What educational strategies can help students understand even and odd numbers better?
Teaching even and odd numbers can be made engaging and effective through a variety of strategies, such as:
• Using physical objects (like fruits, beads, or blocks) to visually demonstrate the divisibility and grouping of numbers.
• Implementing games and interactive activities that involve sorting numbers into even and odd categories.
• Introducing puzzles and challenges that apply the rules of even and odd numbers in different contexts.
• Applying real-life scenarios that involve grouping and dividing objects or people to illustrate the concept.
Such hands-on activities not only enhance comprehension but also make learning mathematics more enjoyable for students.
How do the concepts of even and odd numbers intersect with other mathematical concepts?
The concepts of even and odd numbers intersect with various areas within mathematics, including:
• Algebra: Understanding even and odd functions, which have symmetry properties related to the y-axis and the origin, respectively.
• Number Theory: Exploring the distribution of even and odd prime numbers (all primes are odd, except for 2) and their implications in proofs and theorems.
• Geometry: Analyzing shapes and figures based on the evenness or oddness of their sides or angles can lead to insights about their properties and relationships.
As foundational elements of arithmetic, even and odd numbers indeed weave through the fabric of mathematics, influencing various domains and enriching our understanding of the numerical world.
How can understanding even and odd numbers benefit my daily life?
Understanding even and odd numbers can significantly benefit daily life in several ways, such as:
• Simplifying tasks that involve counting, sorting, or dividing items evenly.
• Enhancing mental math skills, making it easier to calculate and make quick decisions.
• Understanding and participating in games that involve strategy based on number properties.
• Appreciating the role of mathematics in technology, such as computing and digital electronics.
Even outside of pure mathematics, the concepts of even and odd numbers enrich our interaction with the world, demonstrating the ubiquity and applicability of mathematical principles. | {"url":"https://daybook.space/even-and-odd-numbers/","timestamp":"2024-11-04T20:41:57Z","content_type":"text/html","content_length":"114164","record_id":"<urn:uuid:28860d29-0b90-478a-b686-fd8be4d59c29>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00118.warc.gz"} |
How to do sparse row vector*matrix?
I am trying to do a row vector times a matrix but it give an adjoint type instead of a row vector? should I do (A’A[1,:])'? I need the results to be a vector so I can continue other calculation.
A=sparse([1 2;3 4])
#1×2 adjoint(::SparseVector{Int64, Int64}) with eltype Int64
That is a row vector. https://www.youtube.com/watch?v=C2RO34b_oPM
Thanks. But I just did a type conversion for it like:
This solved my problem.
A little more direct (and performant) is either of
julia> (A[1,:]' * A)'
2-element SparseVector{Int64, Int64} with 2 stored entries:
[1] = 7
[2] = 10
julia> A' * A[1,:]
2-element SparseVector{Int64, Int64} with 2 stored entries:
[1] = 7
[2] = 10
For real numbers this is fine, but if your numbers are complex then you should be careful about the conjugation of the result. | {"url":"https://discourse.julialang.org/t/how-to-do-sparse-row-vector-matrix/121166","timestamp":"2024-11-08T18:02:48Z","content_type":"text/html","content_length":"23531","record_id":"<urn:uuid:8163804f-d27a-4fa3-9ac1-33322e071616>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00434.warc.gz"} |
How to Calculate Pool Volume (Gallons)
Why Is It Important to Know Your Pool's Volume?
There are some good reasons why you should know your pool volume.
It helps in order to add the right amount of chemicals when balancing and sanitizing because too much can be harmful to your pool and equipment and be uncomfortable for swimmers. Not adding enough
chemicals for your pool size can lead to increased germs and algae growth, which can be costly to fix.
Many chemical labels provide a dosage rate of 5,000 or 10,000 gallons, so you'll want to scale up or down from there as needed when adding chemicals.
You need to know how much water is required to fill the pool initially and make up for any losses over time. The volume in gallons will determine the water pool needs
Other reasons for knowing the volume of your pool have to do with properly sizing new pool equipment and determining how long your pump should run.
If it runs too long, you're wasting electricity, and if it runs too little, you won't be circulating the proper amount of water through the filter. You should be circulating about your entire pool
volume once per day or more in warmer areas, or if you're addressing water treatment issues.
How Many Gallons is My Pool?
To calculate the number of gallons in your pool, you will need to know the dimensions of your pool. Once you have that information, you can use a formula to calculate the volume of the pool in cubic
feet, and then convert that to gallons.
│ Shape │ Formula for Volume │
│ Cube │ Length x Width x Height │
│ Rectangular Prism │ Length x Width x Height │
│ Triangular Prism │ 0.5 x Base x Height x Length │
│ Cylinder │ π x Radius^2 x Height │
│ Cone │ (1/3) x π x Radius^2 x Height │
│ Sphere │ (4/3) x π x Radius^3 │
│ Pyramid │ (1/3) x Base Area x Height │
When it comes to figuring out the volume of your pool, you might be lucky enough to still have the paperwork from your pool builder or manufacturer which often lists the gallon capacity. If not,
we're going to show you how to calculate it with some simple measurements and a little math.
You'll need a tape measure, pencil, and paper, and of course, a calculator always helps to start off.
Once you have calculated the volume of your pool in cubic feet, you can convert it to gallons by multiplying by 7.5.
Gallons = Length x Width x Average Depth x 7.5
The factor of 7.5 is used to convert the volume of the pool from cubic feet to gallons.
For example, if you have a rectangular pool that is 20 feet long, 10 feet wide, and has an average depth of 5 feet, the volume would be:
• Volume = 20 x 10 x 5 = 1000 cubic feet
To convert to gallons, you would multiply by 7.48, so:
• 1000 x 7.5 = 7,500 gallons
Therefore, your pool is 7,500 gallons.
Calculate Pool Volume
Different Volume Calculations for Pool Shapes
For rectangular or square pools, you multiply the length by width by average depth by 7.5 to get your approximate volume. So, for example, if your pool is 28 feet long, 14 feet wide, and has an
average depth of five and a half feet, you'll multiply 28 * 14 * 5.5 * 7.5, which gives you an approximate pool volume of 16,000 gallons.
For circular pools and hot tubs, the diameter is your widest point with 1/2 that measurement being the radius. You can figure out the average depth of your pool by measuring the depths at the
shallowest and deepest ends, then adding them together and dividing by two. If your pool happens to be the same depth throughout, then you made out easy for this part.
For circular pools and hot tubs, you multiply π, which is 3.14 times the radius, times the radius, and again times your average depth by 7.5.
If your pool's oval, you'll multiply π, which again is 3.14 times the length by the width of your pool, times .25 times your average depth. Then you multiply that by 7.5 and that'll give you your
approximate volume and gallons.
If you have a kidney-shaped pool, you'll want to get the measurements of your two widest points. Mark one with A and the other with B. You'll add A and B together and then multiply that number by the
actual length of your pool. Once you have that, multiply it by .45, then multiply it by your average depth and lastly by 7.5 and that'll be your volume for a kidney shape pool.
For in a regular shape or free-form pool, you'll want to take the longest length of your pool times the widest width and multiply that number by your average depth. Then take that and multiply it by
5.9 to calculate your approximate volume in gallons.
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How to Quickly Lower pH in Your Pool
Keep pool clean and balanced by learning how to lower pH levels. Our easy guide will show you effective methods to maintain safe and comfortable swimming water. | {"url":"https://wikipoolcleaner.com/how-many-gallons-is-my-pool/","timestamp":"2024-11-02T01:30:59Z","content_type":"text/html","content_length":"99287","record_id":"<urn:uuid:e382857e-5f9d-4a07-a6f1-02b976b44a2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00223.warc.gz"} |
From Hoare Logic to Matching Logic - PDF Free Download
Idea Transcript
From Hoare Logic to Matching Logic Grigore Ro¸su and Andrei Stef˘ ¸ anescu Department of Computer Science, University of Illinois at Urbana-Champaign {grosu, stefane1}@illinois.edu Abstract. Matching
logic has been recently proposed as an alternative program verification approach. Unlike Hoare logic, where one defines a language-specific proof system that needs to be proved sound for each
language separately, matching logic provides a language-independent and sound proof system that directly uses the trusted operational semantics of the language as axioms. Matching logic thus has a
clear practical advantage: it eliminates the need for an additional semantics of the same language in order to reason about programs, and implicitly eliminates the need for tedious soundness proofs.
What is not clear, however, is whether matching logic is as powerful as Hoare logic. This paper introduces a technique to mechanically translate Hoare logic proof derivations into equivalent matching
logic proof derivations. The presented technique has two consequences: first, it suggests that matching logic has no theoretical limitation over Hoare logic; and second, it provides a new approach to
prove Hoare logics sound.
Operational semantics are undoubtedly one of the most accessible semantic approaches. Language designers typically do not need extensive theoretical background in order to define an operational
semantics to a language, because they can think of it as if “implementing” an interpreter for the language. For example, consider the following two rules from the (operational) reduction semantics of
a simple imperative language: while(e) s ⇒ if (e) s; while(e) s else skip proc() ⇒ body where “proc() body” is a procedure The former says that loops are unrolled and the second says that procedure
calls are inlined (for simplicity, we assumed no-argument procedures and no local variables). In addition to accessibility, operational semantics have another major advantage: they can be efficiently
executable, and thus testable. For example, one can test an operational semantics as if it was an interpreter or a compiler, by executing large test suites of programs. This way, semantic or design
flaws can be detected and confidence in the semantics can be incrementally build. We refer the interested reader to [1, 3, 6] for examples of large operational semantics (for C) and examples of how
they are tested. Because of all the above, it is quite common that operational semantics are considered trusted reference models of the programming languages they define, and thus serve as a formal
basis for language understanding, design, and implementation. With few notable exceptions, e.g. [10], operational semantics are typically considered inappropriate for program verification. That is to
a large extent due to the fact that program reasoning with an operational semantics typically reduces to reasoning within the transition system associated to the operational semantics, which can be
quite low level. Instead, semantics which are more appropriate for program reasoning are typically given to programming languages, such as axiomatic semantics under the form of Hoare
logic proof systems for deriving Hoare triples {precondition} code {postcondition}. For example, the proof rules below correspond to the operational semantics rules above: H ` {ψ ∧ e , 0} s {ψ} H `
{ψ} while(e) s {ψ ∧ e = 0} H ∪ {ψ} proc() {ψ0 } ` {ψ} body {ψ0 } where “proc() body” is a procedure H ` {ψ} proc() {ψ0 } The second rule takes into account the fact that the procedure proc might be
recursive; several instances of the rule are needed for mutually recursive procedures. Both these rules define the notion of an invariant, the former for while loops (we assume a C-like language,
where zero means false and non-zero means true) and the latter for recursive procedures. These proof rules are so compact because we are making (unrealistic) simplifying assumptions about the
language. Hoare logic proof systems for real languages are quite involved (see, e.g., [1] for C and [9] for Java), which is why, for trusted verification, one needs to prove them sound with respect
to more trusted semantics; the state-of-the-art approaches in mechanical verification do precisely that [1, 8–10, 12, 17]. Matching logic [16] is a new program verification approach, based on
operational semantics. Instead of proving properties at the low level of a transition system, matching logic provides a high-level proof system for deriving program properties, like Hoare logic. In
matching logic, program properties are specified as reduction rules ϕ ⇒ ϕ0 between patterns, abstractly capturing the idea of reachability in the corresponding transition system: program
configuration γ that matches pattern ϕ reduces in zero, one or more steps to a configuration γ0 that matches ϕ0 . Patterns are configuration terms with variables, containing both program and state
fragments like in operational semantics, but the variables can be constrained using logical formulae, like in Hoare logic. Unlike in Hoare logic, the proof rules of matching logic are all
language-independent, taking the given operational semantics as a set of axiom reduction rules. The key proof rule of matching logic is Circularity, which is meant to language-independently capture
the various circular behaviors that appear in languages, due to loops, recursion, etc. A ` ϕ ⇒+ ϕ00
A ∪ {ϕ ⇒ ϕ0 } ` ϕ00 ⇒ ϕ0 A ` ϕ ⇒ ϕ0 A initially contains the operational semantics rules. Circularity adds new reductions to A during the proof derivation process, which can be used in their own
proof! Its correctness is given by the fact that progress is required to be made (indicated by ⇒+ in A ` ϕ ⇒+ ϕ00 ) before a circular reasoning step is allowed. Everything else being equal, matching
logic has a clear pragmatic advantage over Hoare logic: it eliminates the need for an additional semantics of the same language only to reason about programs, and implicitly eliminates the need for
non-trivial and tedious correctness proofs. The soundness of matching logic has already been shown in [16]. Its practicality and usability have been demonstrated through the MatchC automatic program
verifier for a C fragment [15], which is a faithful implementation of the matching logic proof system. What is missing is a formal treatment of the completeness of matching logic. Since Hoare logic
is relatively complete [5], any semantically valid program property expressed as a Hoare triple can also be derived using the Hoare logic proof system (provided an oracle that knows all the
properties of the state model is available). 2
Of course, since Hoare logic is language-specific, its relative completeness needs to be proved for each language individually. Nevertheless, such relative completeness proofs are quite similar and
not difficult to adapt from one language to another. This paper addresses the completeness of matching logic. A technique to mechanically translate Hoare logic proof derivations into equivalent
matching logic proof derivations is presented and proved correct. The generated matching logic proof derivations are within a linear factor larger in size than the original Hoare logic proofs.
Because of the language-specific nature of Hoare logic, we define and prove our translation in the context of a specific but canonical imperative language, IMP. However, the underlying idea is
general. We also apply it to an extension with mutually recursive procedures. Although we can now regard Hoare logic as a methodological fragment of matching logic, where any Hoare logic proof
derivation can be mimicked using the matching logic proof system, experience with MatchC tells us that in general one should not want to verify programs following this route in practice. Specifying
program properties and verifying them directly using the matching logic capabilities, without going through its Hoare fragment, gives us shorter and more intuitive specifications and proofs.
Therefore, in our view, the result of this paper should be understood through its theoretical value. First, it shows that matching logic has no theoretical limitation over Hoare logic, in spite of
being language-independent and working directly with the trusted operational semantics. Second, it provides a new and abstract way to prove Hoare logics sound, where one does not need to make use of
low-level transition systems and induction, instead relying on the soundness of matching logic (proved generically, for all languages). Section 2 recalls operational semantics and Hoare logic, and
Section 3 matching logic. Section 4 illustrates the differences between Hoare logic and matching logic.Section 5 presents our translation technique and proves its correctness. Section 6 concludes.
IMP: Operational Semantics and Hoare Logic
Here we recall operational semantics, Hoare logic, and related notions, and introduce our notation and terminology for these. We do so by means of the simple IMP imperative language. Fig. 1 shows its
syntax, an operational semantics based on evaluation contexts, and a Hoare logic for it. IMP has only integer expressions, which can also be used as conditions of if and while (zero means false and
any non-zero integer means true, like in C). Expressions are built with integer constants, program variables, and conventional arithmetic constructs. For simplicity, we only show a generic binary
operation, op. IMP statements are the variable assignment, if, while and sequential composition. The IMP program configurations are pairs hcode, σi, where code is a program fragment and σ is a state
term mapping program variables into integers. As usual, appropriate definitions of the domains of integers (including arithmetic operations i1 opInt i2 , etc.) and of maps (including lookup σ(x) and
update σ[x ← i] operations) are assumed. IMP’s operational semantics has seven reduction rule schemas between program configurations, which make use of first-order variables: σ is a variable of sort
State; x is a variable of sort PVar; i, i1 , i2 are variables of sort Int; e is a variable of sort Exp; s, s1 , s2 are variables of sort Stmt. A rule mentions a context and a redex, which form a
configuration, and reduces the said configuration by rewriting the redex and possibly the context. As a notation, the context is skipped if not used. E.g., the rule op is 3
IMP language syntax PVar F program variables Exp F PVar | Int | Exp op Exp Stmt F skip | PVar := Exp | Stmt; Stmt | if(Exp) Stmt else Stmt | while(Exp) Stmt Generic IMP evaluation contexts syntax |=
ψ1 → ψ3 {ψ3 } s {ψ4 } |= ψ4 → ψ2 HL-csq Context F {ψ1 } s {ψ2 } | hContext, Statei IMP axiomatic semantics | Context op Exp | Exp op Context · | PVar := Context | Context; Stmt HL-skip {ψ} skip {ψ} |
if(Context) Stmt else Stmt · HL-asgn IMP operational semantics {ψ[e/x]} x := e {ψ} lookup op asgn seq cond1 cond2 while
{ψ1 } s1 {ψ2 } {ψ2 } s2 {ψ3 } hC, σi[x] ⇒ hC, σi[σ(x)] HL-seq {ψ1 } s1 ; s2 {ψ3 } i1 op i2 ⇒ i1 opInt i2 hC, σi[x := i] ⇒ hC, σ[x ← i]i[skip] {ψ1 ∧ e , 0} s1 {ψ2 } skip; s2 ⇒ s2 {ψ1 ∧ e = 0} s2 {ψ2 }
if(i) s1 else s2 ⇒ s1 if i , 0 HL-cond {ψ1 } if(e) s1 else s2 {ψ2 } if(0) s1 else s2 ⇒ s2 while(e) s ⇒ {ψ ∧ e , 0} s {ψ} HL-while if(e) s; while(e) s else skip {ψ} while(e) s {ψ ∧ e = 0}
Fig. 1. IMP language syntax (top), operational semantics (left) and Hoare logic (right).
in fact hC, σi[i1 op i2 ] ⇒ hC, σi[i1 opInt i2 ]. The code context meta-variable C allows us to instantiate a schema into reduction rules, one for each redex of each code fragment. We can therefore
regard the operational semantics of IMP above as a (recursively enumerable) set of reduction rules of the form “l ⇒ r if b”, where l and r are program configurations with variables constrained by
boolean condition b. There are several operational semantics styles based on such rules. Besides the popular reduction semantics with evaluation contexts [7], we also have the chemical abstract
machine [2] and K [14]. Large languages have been given semantics with only rules of the form “l ⇒ r if b”, including C [6] (defined in K with more than 1200 such rules). Matching logic works with
such rules in general (taking them as axioms), and is agnostic to the particular operational semantics or any other method used to produce them. The major role of an operational semantics is to yield
a canonical and typically trusted model of the defined language, as a transition system over program configurations. Such transition systems are important in this paper, so we formalize them here. We
also recall some mathematical notions and notations, although we generally assume the reader is familiar with basic concepts of algebraic specification and first-order logic. Given an algebraic
signature Σ, we let T Σ denote the initial Σ-algebra of ground terms (i.e., terms without variables) and let T Σ (Var) denote the free Σ-algebra of terms with variables in Var. T Σ,s (Var) is the set
of Σ-terms of sort s. Maps ρ : Var → T with T a Σ-algebra extend uniquely to morphisms of Σ-algebras ρ : T Σ (Var) → T . These notions extend to algebraic specifications. Many mathematical structures
needed for language semantics have been defined as initial Σ-algebras: boolean algebras, natural/integer/rational numbers, lists, sets, bags (or multisets), maps (e.g., IMP’s states), trees, queues,
stacks, etc. We refer the reader to the CASL [11] and Maude [4] manuals for examples. 4
Let us fix the following: (1) an algebraic signature Σ, associated to some desired configuration syntax, with distinguished sorts Cfg and Bool, (2) a sort-wise infinite set of variables Var, and (3)
a Σ-algebra T , the configuration model, which may but needs not necessarily be the initial or free Σ-algebra. As usual, TCfg denotes the elements of T of sort Cfg, which we call configurations. Let
S (from “semantics”) be a set of reduction rules “l ⇒ r if b” like above, where l, r ∈ T Σ,Cfg (Var) and b ∈ T Σ,Bool (Var). Definition 1. S yields a transition system (T , ⇒TS ), where γ ⇒TS γ0 for
γ, γ0 ∈ TCfg iff there is a “l ⇒ r if b” in S and a ρ : Var → T with ρ(l) = γ, ρ(r) = γ0 and ρ(b) holds. (T , ⇒TS ) is a conventional transition system, i.e. a set with a binary relation on it (in
fact, ⇒TS ⊆ TCfg × TCfg ), and captures the operational behaviors of the language defined by S. Hence, an operational semantics defines a set of reduction rules which can be used in some implicit way
to yield program behaviors. On the other hand, a Hoare logic defines a proof system that explicitly tells how to derive program properties formalized as Hoare triples. Operational semantics are easy
to define, test and thus build confidence in, since we can execute them against benchmarks of programs; e.g., the C semantics have been extensively tested against compiler test-suites [3, 6]. On the
other hand, Hoare logics are more involved and need to be proved sound w.r.t. another, more trusted semantics. Definition 2. (partial correctness) For the IMP language in Fig. 1, a Hoare triple {ψ}
code {ψ0 } is semantically valid, written |= {ψ} code {ψ0 }, if and only if σ0 |= ψ0 for any state σ such that σ |= ψ and hcode, σi ⇒?T hskip, σ0 i. The Hoare logic proof S system in Fig. 1 is sound
if and only if ` {ψ} code {ψ0 } implies |= {ψ} code {ψ0 }. In Definition 2, we tacitly identified the ground configurations hcode, σi and hskip, σ0 i with their (unique) interpretation in the
configuration model T . First-order logic (FOL) validity, both in Definition 2 and in the HL-csq in Fig. 1, is relative to T . Partial correctness says the postcondition holds only when the program
terminates. We do not address total correctness (i.e., the program must also terminate) in this paper.
Matching Logic
This section recalls matching logic [13, 16]. In matching logic, patterns specify configurations and reduction rules specify operational transitions or program properties. A language-independent
proof system takes a set of reduction rules (operational semantics) as axioms and derives new reduction rules (program properties). Matching logic is parametric in a model of program configurations.
For example, as seen in Section 1, IMP’s configurations are pairs hcode,σi with code a fragment of program and σ a State. Like in Section 1, let us fix an algebraic signature Σ (of configurations)
with a distinguished sort Cfg, a sort-wise infinite set of variables Var, and a (configuration) Σ-model T (which needs not be the initial model T Σ or the free model T Σ (Var)). Definition 3. [13] A
matching logic formula, or a pattern, is a first-order logic (FOL) formula which allows terms in T Σ,Cfg (Var), called basic patterns, as predicates. We define the satisfaction (γ, ρ) |= ϕ over
configurations γ ∈ TCfg , valuations ρ : Var → T and patterns ϕ as follows (among the FOL constructs, we only show ∃): (γ, ρ) |= ∃X ϕ iff (γ, ρ0 ) |= ϕ for some ρ0 : Var → T with ρ0 (y) = ρ(y) for
all y ∈ Var\X (γ, ρ) |= π iff γ = ρ(π) , where π ∈ T Σ,Cfg (Var) We write |= ϕ when (γ, ρ) |= ϕ for all γ ∈ TCfg and all ρ : Var → T . 5
A basic pattern π is satisfied by all the configurations γ that match it; the ρ in (γ, ρ) |= π can be thought of as the “witness” of the matching, and can be further constrained in a pattern. If SUM
is the IMP code “s:=0; while(n>0)(s:=s+n; n:=n-1)” e.g., then ∃s (h SUM, (s 7→ s, n 7→ n) i ∧ n ≥Int 0) is a pattern that matches the configurations with code SUM and state binding program variables
s,n to integers s,n with n ≥Int 0. Note that we use typewriter for program variables in PVar and italic for mathematical variables in Var. Pattern reasoning reduces to FOL reasoning in the
configuration model T [16]. Definition 4. A (matching logic) reduction rule is a pair ϕ ⇒ ϕ0 , where ϕ, called the left-hand side (LHS), and ϕ0 , called the right-hand side (RHS), are matching logic
patterns (which can have free variables). A reduction system is a set of reduction rules. A reduction system S induces a transition system (T , ⇒TS ) on the configuration model: γ ⇒TS γ0 for γ, γ0 ∈
TCfg iff there is a ϕ ⇒ ϕ0 in S and a ρ : Var → T with (γ, ρ) |= ϕ and (γ0, ρ) |= ϕ0 . Configuration γ ∈ TCfg terminates in (T , ⇒TS ) iff there is no infinite ⇒TS -sequence starting with γ. A rule ϕ
⇒ ϕ0 is well-defined iff for any γ ∈ TCfg and ρ : Var → T with (γ, ρ) |= ϕ, there is a γ0 ∈ TCfg with (γ0, ρ) |= ϕ0 . Reduction system S is well-defined iff each rule is well-defined, and is
deterministic iff so is (T , ⇒TS ). Operational semantics defined with rules “l ⇒ r if b”, like those in Section 2, are particular well-defined reduction systems with rules of the form l ∧ b ⇒ r (see
[16]). Matching logic reduction rules can also specify program properties. For our SUM above, ∃s (hSUM, (s 7→ s, n 7→ n)i ∧ n ≥Int 0) ⇒ hskip, (s 7→ n ∗Int (n +Int 1)/Int 2, n 7→ 0)i specifies the
property of SUM. Unlike Hoare triples, which only specify properties about the final states of programs, reduction rules can also specify properties about intermediate states. Hoare triples
correspond to reduction rules whose basic pattern in the RHS holds the code skip, like the one above. Semantic validity in matching logic captures the same intuition of partial correctness as Hoare
logic, but in more general terms of reachability: Definition 5. Let S be a reduction system and ϕ ⇒ ϕ0 a reduction rule. We define S |= ϕ ⇒ ϕ0 iff for all γ ∈ TCfg such that γ terminates in (T , ⇒TS
) and for all ρ : Var → T such that (γ, ρ) |= ϕ, there exists some γ0 ∈ TCfg such that γ ⇒?T γ0 and (γ0, ρ) |= ϕ0 . S If ϕ0 holds the empty code skip, then so does γ0 in the definition above, and, in
the case of IMP, γ0 is unique and thus we recover the Hoare validity as a special case. The reduction rule property of SUM above is valid, although the proof is tedious, involving low-level IMP
transition system details and induction. Instead, matching logic gives us an abstract proof system for deriving such reduction rules, which avoids the transition system. Fig. 2 shows the
language-independent matching logic proof system. Initially, A contains the operational semantics of the target language. Reflexivity, Axiom, Substitution, and Transitivity have an operational nature
and are needed to (symbolically) execute reduction systems. Case Analysis, Logic Framing, Consequence and Abstraction have a deductive nature. The Circularity proof rule has a coinductive nature and
captures the various circular behaviors that appear in languages, due to loops, recursion, etc. Specifically, we can derive A ` ϕ ⇒ ϕ0 whenever we can derive ϕ ⇒ ϕ0 by starting with one or more
reduction steps in A (⇒+ means derivable without Reflexivity) and continuing with steps which can involve both rules from A and the rule to be proved itself, ϕ ⇒ ϕ0 . The first step can for example
be an operational loop unrolling step in the case of loops, or a function invocation step in the case of recursive functions, etc. 6
Rules of operational nature Reflexivity : · A `ϕ⇒ϕ Axiom : ϕ ⇒ ϕ0 ∈ A A ` ϕ ⇒ ϕ0 Substitution : A ` ϕ ⇒ ϕ0 θ : Var → TΣ (Var) A ` θ(ϕ) ⇒ θ(ϕ0 ) Transitivity : A ` ϕ1 ⇒ ϕ2 A ` ϕ2 ⇒ ϕ3 A ` ϕ1 ⇒ ϕ3
Rules of deductive nature Case Analysis : A ` ϕ1 ⇒ ϕ A ` ϕ2 ⇒ ϕ A ` ϕ1 ∨ ϕ2 ⇒ ϕ Logic Framing : A ` ϕ ⇒ ϕ0 ψ is a (patternless) FOL formula A ` ϕ ∧ ψ ⇒ ϕ0 ∧ ψ Consequence : |= ϕ1 → ϕ01 A ` ϕ01 ⇒ ϕ02
|= ϕ02 → ϕ2 A ` ϕ1 ⇒ ϕ2 Abstraction : A ` ϕ ⇒ ϕ0 X ∩ FreeVars(ϕ0 ) = ∅ A ` ∃X ϕ ⇒ ϕ0
Rule for circular behavior A ` ϕ ⇒+ ϕ00 Circularity :
A ∪ {ϕ ⇒ ϕ0 } ` ϕ00 ⇒ ϕ0 A ` ϕ ⇒ ϕ0
Fig. 2. Matching logic proof system (nine language-independent proof rules)
Theorem 1. (partial correctness) [16] Let S be a well-defined and deterministic matching logic reduction system (typically corresponding to an operational semantics), and let S ` ϕ ⇒ ϕ0 be a sequent
derived with the proof system in Fig. 2. Then S |= ϕ ⇒ ϕ0 .
Hoare Logic versus Matching Logic
This section prepares the reader for our main result, by illustrating the major differences between Hoare logic and matching logic using examples. Specifically, we show how the same program property
can be specified both as a Hoare triple and as a matching logic reduction rule, and then how it can be derived using each of the two proof systems. Consider again the SUM program “s:=0; while(n>0)(s:
=s+n; n:=n-1)” in IMP. The main property of SUM can be specified as the following Hoare triple: {n = oldn ∧ n ≥ 0} SUM {s = oldn*(oldn+1)/2 ∧ n = 0} The oldn variable is needed to remember the
initial value of n. Let us derive this Hoare triple using the Hoare logic proof system in Fig. 1. Let LOOP be the actual loop of SUM, namely “while(n>0)(s:=s+n; n:=n-1)”, and let ψinv be the formula
s = (oldn-n)*(oldn+n+1)/2 ∧ n ≥ 0 We can derive our original Hoare triple by first deriving the triples {n = oldn ∧ n ≥ 0} s:=0 {ψinv } {ψinv } LOOP {s = oldn*(oldn+1)/2 ∧ n = 0} and then using the
proof rule HL-seq in Fig. 1. To keep the proof small, we skip the FOL reasoning steps (within the state model) and thus the applications of HL-csq. The first triple follows by HL-asgn. The second
follows by HL-while, after first deriving {ψinv ∧ n > 0} s:=s+n; n:=n-1 {ψinv } by using two instances of the HL-asgn rule and one instance of HL-seq. Before we discuss the matching logic proof
derivation, let us recall some important facts about Hoare logic. First, Hoare logic makes no theoretical distinction between 7
program variables, which in the case of IMP are PVar constants, and mathematical variables, which in the case of IMP are variables of sort Var. For example, in the proof above, n as a program
variable, n as an integer variable appearing in the state specifications, and oldn which appears only in state specifications but never in the program, were formally treated the same way. Second, the
same applies to language arithmetic constructs versus mathematical domain operations. For example, there is no distinction between the + construct for IMP expressions and the +Int operation that the
integer domain provides. These simplifying assumptions make proofs like above simple and compact, but come at a price: expressions cannot have side effects. Since in many languages expressions do
have side effects, programs typically suffer (possibly errorprone) transformations that extract and isolate the side effects into special statements. Also, in practice program verifiers do make a
distinction between language constructs and mathematical ones, and appropriately translate the former into the latter in specifications. Let us now show how to use the proof system in Fig. 2 to
derive the matching logic reduction rule specifying the property of SUM, already discussed in Section 3, namely ∃s (hSUM, (s 7→ s, n 7→ n)i ∧ n ≥Int 0) ⇒ hskip, (s 7→ n ∗Int (n +Int 1)/Int 2, n 7→ 0)
i The “∃s” quantifier is optional. Let us drop it and let us name the resulting rule µ1SUM ≡ (ϕLHS ⇒ ϕRHS ). The original rule follows from µ1SUM by Abstraction. Let SIMP be the operational semantics
of IMP in Fig. 1 and let ϕinv be the pattern hLOOP, (s 7→ (n −Int n0 ) ∗Int (n +Int n0 +Int 1)/Int 2, n 7→ n0 )i ∧ n0 ≥Int 0 We derive SIMP ` µ1SUM by Transitivity with µ1 ≡ (ϕLHS ⇒ ∃n0 ϕinv ) and µ2
≡ (∃n0 ϕinv ⇒ ϕRHS ). By Axiom asgn (Fig. 1, within the SUM context) followed by Substitution with θ(σ) = (s 7→ s, n 7→ n), θ(x) = s and θ(i) = 0 followed by Logic Framing with n ≥Int 0, we derive
ϕLHS ⇒ hskip; LOOP, (s 7→ 0, n 7→ n)i ∧ n ≥Int 0. This “operational” sequence of Axiom, Substitution and Logic Framing is quite common; we abbreviate it ASLF. Further, by ASLF with seq and
Transitivity, we derive ϕLHS ⇒ hLOOP, (s 7→ s, n 7→ n)i ∧ n ≥Int 0. SIMP ` µ1 now follows by Consequence. We derive SIMP ` µ2 by Circularity with SIMP ` ∃n0 ϕinv ⇒+ ϕif and SIMP ∪ {µ2 } ` ϕif ⇒ ϕRHS
, where ϕif is the formula obtained from ϕinv replacing its code with “if (n>0) (s := s+n; n := n-1; LOOP) else skip”. ASLF (while) followed by Abstraction derive SIMP ` ∃n0 ϕinv ⇒+ ϕif . For the
other, we use Case Analysis with ϕif ∧ n0 ≤Int 0 and ϕif ∧ n0 >Int 0. ASLF (lookupn , op> , cond2 ) together with some Transitivity and Consequence steps derive SIMP ∪ {µ2 } ` ϕif ∧ n0 ≤Int 0 ⇒ ϕRHS
(µ2 is not needed in this derivation). Similarly, ASLF (lookupn , op> , cond1 , lookupn , lookups , op+ , asgn, seq, lookupn , op− , asgn, seq, and µ2 ) together with Transitivity and Consequence
steps derive SIMP ∪ {µ2 } ` ϕif ∧ n0 >Int 0 ⇒ ϕRHS . This time µ2 is needed and it is interesting to note how. After applying all the steps above and the LOOP fragment of code is reached again, the
pattern characterizing the configuration is hLOOP, (s 7→ (n −Int n0 ) ∗Int (n +Int n0 +Int 1)/Int 2 +Int n0 , n 7→ n0 −Int 1)i ∧ n0 >Int 0 The circularity µ2 can now be applied, via Consequence and
Transitivity, because this formula implies ∃n0 ϕinv (indeed, pick the existentially quantified n0 to be n0 −Int 1). The matching logic proof above may seem low-level when compared to the Hoare logic
proof. However, note that it is quite mechanical, the only interesting part being to provide the invariant (ϕinv ), same like in the Hoare logic proof. The rest is automatic and 8
consists of applying the operational reduction rules whenever they match, except for the circularities which are given priority; when the redex is an if, a Case Analysis is applied. Our current
MatchC implementation can prove it automatically, as well as much more complex programs [15, 16]. Although the paper Hoare logic proofs for simple languages like IMP may look more compact, as
discussed above they make (sometimes unrealistic) assumptions which need to be addressed in implementations. Finally, note that matching logic’s reduction rules are more expressive than the Hoare
triples, since they can specify reachable configurations which are not necessarily final. For example, the rule hSUM, (s 7→ s, n 7→ n)i ∧ n >Int 0 ⇒ hLOOP, (s 7→ n, n 7→ n −Int 1)i is also derivable
and states that if the value n of n is strictly positive, then the loop is taken once and, when the loop is reached again, s is n and n is n −Int 1.
Translating Hoare Logic Proofs into Matching Logic Proofs
Here we show how proof derivations using the IMP-specific Hoare logic proof system in Fig. 1 are mechanically translated into proof derivations using the language-independent matching logic proof
system in Fig. 2 with IMP’s operational semantics in Fig. 1 as axioms. Moreover, the sizes of the two proof derivations are within a linear factor. 5.1
The Translation
Without restricting the generality, we make the following simplifying assumptions about the Hoare triples {ψ} code {ψ0 } that appear in the Hoare logic proof derivation that we translate into a
matching logic proof: (1) the variables appearing in code belong to an arbitrary but fixed finite set X ⊂ PVar; and (2) the additional variables appearing in ψ and ψ0 but not in code belong to an
arbitrary but fixed finite set Y ⊂ PVar such that X ∩ Y = ∅. In other words, we fix the finite disjoint sets X, Y ⊂ PVar, and they have the properties above for all Hoare triples that we consider in
this section. Note that we used a typewriter font to write these sets, which is consistent with our notation for variables in PVar. We need these disjointness restrictions because, as discussed in
Section 4, Hoare logic makes no theoretical distinction between program and mathematical variables, while matching logic does. These restrictions do not limit the capability of Hoare logic, since we
can always pick X to be the union of all the variables appearing in the program about which we want to reason and Y to be the union of all the remaining variables occurring in all the state
specifications in any triple anywhere in the Hoare logic proof, making sure that the names of the variables used for stating mathematical properties of the state are always chosen different from
those of the variables used in programs. Definition 6. Given a Hoare triple {ψ} code {ψ0 }, we define H2M({ψ} code {ψ0 })
∃X (hcode, σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ ψ0X,Y )
where: 1. X, Y ⊂ Var (written using italic font) are finite sets of variables corresponding to the sets X, Y ⊂ PVar fixed above, one variable x or y in Var (written using italic font) for each
variable x or y in PVar (written using typewriter font); 2. σX is the state binding each x ∈ X to its corresponding x ∈ X; and 9
3. ψX,Y and ψ0X,Y are ψ and respectively ψ0 with x ∈ X or y ∈ Y replaced by its corresponding x ∈ X or y ∈ Y, respectively, and each expression construct op replaced by its mathematical correspondent
opInt . The H2M mapping in Definition 6 is quite simple and mechanical, and can be implemented by a linear traversal of the Hoare triple. In fact, we have implemented it as part of the MatchC program
verifier, to allow users to write program specifications in a Hoare style when possible (see, e.g., the simple folder of examples on the online MatchC interface at http://fsl.cs.uiuc.edu/index.php/
Special:MatchCOnline). It is important to note that, like X, Y ⊂ PVar, the sets of variables X, Y ⊂ Var in Definition 6 are also fixed and thus the same for all Hoare triples considered in this
section. For example, suppose that X = {s, n} and Y = {oldn, z}. Then the Hoare triple {n = oldn ∧ n ≥ 0} SUM {s = oldn*(oldn+1)/2 ∧ n = 0} from Section 4 is translated into the following matching
logic reduction rule: ∃s, n (hSUM, (s 7→ s, n 7→ n)i ∧ n = oldn ∧ n ≥Int 0) ⇒ ∃s, n (hskip, (s 7→ s, n 7→ n)i ∧ s = oldn ∗Int (oldn +Int 1)/Int 2 ∧ n = 0) Not surprisingly, we can use the matching
logic proof system in Fig. 2 to prove this reduction rule equivalent to the one that we gave for SUM in Section 4. Indeed, using FOL reasoning and Consequence we can show the above equivalent to ∃s
(hSUM,(s 7→s, n 7→ oldn)i∧oldn ≥Int0) ⇒ hskip, (s 7→ oldn∗Int (oldn+Int 1)/Int 2, n 7→ 0)i which, by Substitution (n ↔ oldn), is equivalent to the reduction rule in Section 4. We also show an
(artificial) example where the original Hoare triple contains a quantifier. Consider the same X = {s, n} and Y = {oldn, z} as above. Then H2M({true} n:=4*n+3 {∃z (n = 2*z+1)}) is the reduction rule
∃s, n (hn:=4*n+3, (s 7→ s, n 7→ n)i ∧ true) ⇒ ∃s, n (hskip, (s 7→ s, n 7→ n)i ∧ ∃z (n = 2 ∗Int z +Int 1)) Using FOL reasoning and Consequence, this rule can be shown equivalent to ∃s, n hn:=4*n+3, (s
7→ s, n 7→ n)i ⇒ ∃s, z hskip, (s 7→ s, n 7→ 2 ∗Int z +Int 1)i 5.2
Helping Lemmas
The following holds for matching logic in general: Lemma 1. If S ` ϕ ⇒ ϕ0 is derivable then S ` ∃X ϕ ⇒ ∃X ϕ0 is also derivable. Proof. We have |= ϕ0 → ∃X ϕ0 . By Consequence, we derive S ` ϕ ⇒ ∃X ϕ0
. Since X ∩ FreeVars(∃X ϕ0 ) = ∅, by Abstraction we get that S ` ∃X ϕ ⇒ ∃X ϕ0 is also derivable. Symbolical evaluation of IMP expressions is actually derivable in matching logic: Lemma 2. If e ∈ Exp
is an expression, C ∈ Context an appropriate context, and σ ∈ State a state term binding each program variable in PVar of e to a term of sort Int (possibly containing variables in Var), then the
following sequent is derivable: SIMP ` hC, σi[e] ⇒ hC, σi[σ(e)] where σ(e) replaces each x ∈ PVar in e by σ(x) (i.e., a term of sort Int) and each operation symbol op by its mathematical
correspondent in the Int domain, opInt . 10
Proof. By induction on the structure of e. If e is a variable x ∈ PVar, then the result follows by Axiom with lookup in Fig. 1. If e is of the form e1 op e2 , then let C1 , C2 be the contexts
obtained from C by replacing with “ op e2 ” and respectively “σ(e1 ) op ”. Then, by the induction hypothesis, the following are derivable SIMP ` hC1 , σi[e1 ] ⇒ hC1 , σi[σ(e1 )] SIMP ` hC2 , σi[e2 ]
⇒ hC2 , σi[σ(e2 )] We also have the following pattern identities hC, σi[e] = hC1 , σi[e1 ] hC1 , σi[σ(e1 )] = hC2 , σi[e2 ] hC2 , σi[σ(e2 )] = hC, σi[σ(e1 ) op σ(e2 )] Thus, by Transitivity, we
derive SIMP ` hC, σi[e] ⇒ hC, σi[σ(e1 ) op σ(e2 )], and then the result follows by Axiom with op and by noticing that σ(e) = σ(e1 ) opInt σ(e2 ). Lemma 3. If SIMP ` ϕ ⇒ ϕ0 is derivable and s ∈ Stmt
then SIMP ` append(ϕ, s) ⇒ append(ϕ0 , s) is also derivable, where append(ϕ, s) is the pattern obtained from ϕ by replacing each basic pattern hcode, σi with the basic pattern h(code; s), σi. Proof.
(sketch) Let append(A, s) be the set of rules obtained from A by replacing each rule ϕl ⇒ ϕr ∈ A \ SIMP by the rule append(ϕl , s) ⇒ append(ϕr , s), that is append(A, s) = (A ∩ SIMP ) ∪ {append(ϕl ,
s) ⇒ append(ϕr , s) | ϕl ⇒ ϕr ∈ A \ SIMP } Let P be a proof tree deriving SIMP ` ϕ ⇒ ϕ0 . We prove the more general result that for each sequent A ` ϕl ⇒ ϕr in P, we can also derive the sequent
append(A, s) ` append(ϕl , s) ⇒ append(ϕr , s). The lemma follows as particular case. The proof goes by induction on the structure of P. If the last step is Reflexivity, the result trivially holds.
If the last step is one of Substitution, Transitivity, Case Analysis, Logic Framing, Consequence, Abstraction or Circularity, then the result holds by applying the induction hypothesis, and by
noticing that since s does not have any logical variables, then append(θ(ϕ), s) = θ(append(ϕ, s)) (Substitution), |= ϕ1 → ϕ01 iff |= append(ϕ1 , s) → append(ϕ01 , s) (Consequence) and FreeVars(append
(ϕ, s)) = FreeVars(ϕ) (Abstraction). If the last step is Axiom with a rule in A \ SIMP , again the result trivially holds. If the last step is Axiom with a rule in SIMP , then the redex always goes
to the left of “;”. Since none of the reduction rule schemas of IMP mention “;” in the LHS or in the side condition, we can conclude that ϕ ⇒ ϕ0 ∈ SIMP iff append(ϕ, s) ⇒ append(ϕ0 , s) ∈ SIMP . 5.3
The Main Result
Theorem 2. Let SIMP be the operational semantics of IMP in Fig. 1 regarded as a matching logic reduction system, and let {ψ} code {ψ0 } be derivable with the IMP-specific Hoare logic proof system in
Fig 1. Then SIMP ` H2M({ψ} code {ψ0 }) is derivable with the language-independent matching logic proof system in Fig. 2. Proof. We prove that for any Hoare logic proof of {ψ} code {ψ0 } one can
construct a matching logic proof of SIMP ` H2M({ψ} code {ψ0 })). The proof goes by structural induction on the formal proof derived using the Hoare logic proof system in Fig. 1. We consider each
proof rule in Fig. 1 and show how corresponding matching logic proofs for the hypotheses can be composed into a matching logic proof for the conclusion. · HL-skip {ψ} skip {ψ} 11
Reflexivity (Fig 2). derives SIMP ` ∃X (hskip, σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ ψX,Y ). · HL-asgn {ψ[e/x]} x := e {ψ} We have to derive SIMP ` ∃X (hx := e, σX i ∧ ψ[e/x]X,Y ) ⇒ ∃X (hskip, σX i ∧
ψX,Y ). By using Lemma 2, Logical Framing and Lemma 1, we derive SIMP ` ∃X (hx := e, σX i ∧ ψ[e/x]X,Y ) ⇒ ∃X (hx := σX (e), σX i ∧ ψ[e/x]X,Y ) Further, by using Axiom with asgn in Fig. 1,
Substitution and Logic Framing, followed by Lemma 1, we derive SIMP ` ∃X (hx := σX (e), σX i∧ψ[e/x]X,Y ) ⇒ ∃X (hskip, σX [x ← σX (e)]i∧ψ[e/x]X,Y ) Then, the result follows by Transitivity with the
rules above and by Consequence with |= ∃X (hskip, σX [x ← σX (e)]i ∧ ψ[e/x]X,Y ) → ∃X (hskip, σX i ∧ ψX,Y ), which holds because σX [x ← σX (e)] and ψ[e/x]X,Y are nothing but σX and respectively ψX,Y
with x ∈ X replaced by σX (e). HL-seq
{ψ1 } s1 {ψ2 } {ψ2 } s2 {ψ3 } {ψ1 } s1 ; s2 {ψ3 }
We have to derive SIMP ` ∃X (hs1 ; s2 , σX i ∧ ψ1 X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ3 X,Y ). By the induction hypothesis, the following sequents are derivable SIMP ` ∃X (hs1 , σX i ∧ ψ1 X,Y ) ⇒ ∃X (hskip,
σX i ∧ ψ2 X,Y ) SIMP ` ∃X (hs2 , σX i ∧ ψ2 X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ3 X,Y ) By applying Lemma 3 with the former rule, we derive SIMP ` ∃X (hs1 ; s2 , σX i ∧ ψ1 X,Y ) ⇒ ∃X (hskip; s2 , σX i ∧ ψ2 X,Y
) Further, Axiom with seq (Fig. 1), Substitution and Logic Framing, followed by Lemma 1, imply SIMP ` ∃X (hs1 ; s2 , σX i ∧ ψ1 X,Y ) ⇒ ∃X (hs2 , σX i ∧ ψ2 X,Y ). Then, the result follows by
Transitivity with the rule above and the second induction hypothesis. HL-cond
{ψ1 ∧ e , 0} s1 {ψ2 } {ψ1 ∧ e = 0} s2 {ψ2 } {ψ1 } if(e) s1 else s2 {ψ2 }
We have to derive SIMP ` ∃X (hif(e) s1 else s2 , σX i ∧ ψ1 X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ2 X,Y ) By the induction hypothesis, the following sequents are derivable SIMP ` ∃X (hs1 , σX i ∧ (ψ1 ∧ e , 0)X,Y
) ⇒ ∃X (hskip, σX i ∧ ψ2 X,Y ) SIMP ` ∃X (hs2 , σX i ∧ (ψ1 ∧ e = 0)X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ2 X,Y ) By using Lemma 2, Logical Framing, and Lemma 1, we derive SIMP ` ∃X (hif(e) s1 else s2 , σX i ∧
ψ1 X,Y ) ⇒ ∃X (hif(σX (e)) s1 else s2 , σX i ∧ ψ1 X,Y ) By using Axiom with cond1 and cond2 in Fig. 1, each followed by Substitution, Logic Framing and by Lemma 1, we also derive SIMP ` ∃X (hif(σX
(e)) s1 else s2 , σX i ∧ (ψ1 ∧ e , 0)X,Y ) ⇒ ∃X (hs1 , σX i ∧ (ψ1 ∧ e , 0)X,Y ) SIMP ` ∃X (hif(σX (e)) s1 else s2 , σX i ∧ (ψ1 ∧ e = 0)X,Y ) ⇒ ∃X (hs2 , σX i ∧ (ψ1 ∧ e = 0)X,Y ) Further, by
Transitivity with the rules above and the induction hypotheses, we derive 12
SIMP ` ∃X (hif(σX (e)) s1 else s2 , σX i ∧ (ψ1 ∧ e , 0)X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ2 X,Y ) SIMP ` ∃X (hif(σX (e)) s1 else s2 , σX i ∧ (ψ1 ∧ e = 0)X,Y ) ⇒ ∃X (hskip, σX i ∧ ψ2 X,Y ) Then the result
follows by Case Analysis, Consequence and Transitivity. {ψ ∧ e , 0} s {ψ} HL-while {ψ} while(e) s {ψ ∧ e = 0} Let µ be the matching logic rule that we have to derive, namely SIMP ` ∃X (hwhile(e) s,
σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ (ψ ∧ e = 0)X,Y ) By the induction hypothesis, we the following sequent is derivable SIMP ` ∃X (hs, σX i ∧ (ψ ∧ e , 0)X,Y ) ⇒ ∃X (hskip, σX i ∧ ψX,Y ) We derive µ by
Circularity. First, by Axiom with while (Fig. 1), Substitution, Logic Framing, and Lemma 1, we derive (note the ⇒+ , as this derivation does not use Reflexivity) SIMP ` ∃X (hwhile(e) s, σX i ∧ ψX,Y )
⇒+ ∃X (hif(e) s; while(e) s else skip, σX i ∧ ψX,Y ) Therefore, all we need to do now is to derive SIMP ∪ {µ} ` ∃X (hif(e) s; while(e) s else skip, σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ (ψ ∧ e = 0)X,Y )
Further, by Lemma 2, Logical Framing, Lemma 1 and Transitivity, we are left with SIMP ∪ {µ} ` ∃X (hif(σX (e)) s; while(e) s else skip, σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ (ψ ∧ e = 0)X,Y ) We apply Case
Analysis with σX (e) = 0 ∨ σX (e) , 0. The case σX (e) = 0 follows by Axiom with cond2 , Substitution, Logic Framing and Lemma 1. For the other case, we first use Axiom with cond1 , Substitution,
Logic Framing, Lemma 1 and Transitivity to reduce it to SIMP ∪ {µ} ` ∃X (hs; while(e) s, σX i ∧ (ψ ∧ e , 0) ) X,Y
⇒ ∃X (hskip, σX i ∧ (ψ ∧ e = 0)X,Y ) By using the induction hypothesis and Lemma 3 with s and while(e) s followed by Axiom with skip, Substitution, Logic Framing and Lemma 1 we derive SIMP ∪ {µ} ` ∃X
(hs; while(e) s, σX i ∧ (ψ ∧ e , 0)X,Y ) ⇒ ∃X (hwhile(e) s, σX i ∧ ψX,Y ) Then the result follows by using Axiom with µ and Transitivity with the rule above. 5.4
Adding Recursion
In this section we add procedures to IMP, which can be mutually recursive, and show that proof derivations done with their corresponding Hoare logic proof rule can also be done using the generic
matching logic proof system, with their straightforward operational semantics rule as axiom. We consider the following syntax for procedures: ProcedureName ::= proc | . . . Procedure ::=
ProcedureName() Stmt Stmt ::= ProcedureName()
Our procedures therefore have the syntax “proc() body”, where proc is the name of the procedure and body the body statement. Procedure invocations are statements of the form “proc()”. For simplicity,
and to capture the essence of the relationship between recursion and the Circularity rule of matching logic, we assume only no-argument procedures. The operational semantics of procedure calls is
trivial: call
proc() ⇒ body
where “proc() body” is a procedure
The Hoare logic proof rule needs to take into account that procedures may be recursive: H ∪ {ψ} proc() {ψ0 } ` {ψ} body {ψ0 } where “proc() body” is a procedure H ` {ψ} proc() {ψ0 } This rule states
that if the body of a procedure is proved to satisfy its contract while assuming that the procedure itself satisfies it, then the procedure’s contract is indeed valid. If one has more mutually
recursive procedures, then one needs to apply this rule several times until all procedure contracts are added to the hypothesis H, and then each procedure body proved. The rule above needs to be
added to the Hoare logic proof system in Fig. 1, but in order for that to make sense we need to first replace each Hoare triple {ψ} code {ψ0 } in Fig. 1 by a sequent “H ` {ψ} code {ψ0 }”. Theorem 3.
Let SIMP be the operational semantics of IMP in Fig. 1 extended with the rule call for procedure calls above, and let H ` {ψ} code {ψ0 } be a sequent derivable with the extended Hoare logic proof
system. Then SIMP ∪ H2M(H) ` H2M({ψ} code {ψ0 }) is derivable with the language-independent matching logic proof system in Fig. 2. Proof. Like in Theorem 2, we prove by structural induction that for
any Hoare logic proof of H ` {ψ} code {ψ0 } one can construct a matching logic proof of SIMP ∪ H2M(H) ` H2M({ψ} code {ψ0 })), by showing for each Hoare logic proof rule how corresponding matching
logic proofs for the hypotheses can be composed into a matching logic proof for the conclusion. The proofs for the (extended) Hoare rules in Fig. 1 are similar to those in Theorem 2, so we only
discuss the new Hoare rule for procedure calls: H ∪ {ψ} proc() {ψ0 } ` {ψ} body {ψ0 } H ` {ψ} proc() {ψ0 } Let µ be the matching logic reduction rule H2M({ψ} proc() {ψ0 }), that is, ∃X (hproc(), σX i
∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ ψ0 X,Y ). The induction hypothesis gives us that the matching logic sequent SIMP ∪ H2M(H) ∪ {µ} ` ∃X (hbody, σX i ∧ ψX,Y ) ⇒ ∃X (hskip, σX i ∧ ψ0 X,Y ) is derivable with
the generic proof system in Fig. 2. Using Axiom with call, Logic Framing with ψX,Y , and then Lemma 1, we derive (note the ⇒+ , as this derivation does not use Reflexivity): SIMP ∪ H2M(H) ` ∃X (hproc
(), σX i ∧ ψX,Y ) ⇒+ ∃X (hbody, σX i ∧ ψX,Y ) Circularity with the two rules above now derives SIMP ∪ H2M(H) ` µ. 14
Matching logic provides a sound and language-independent method of reasoning about programs, based solely on the operational semantics of the target programming language [16]. This paper addressed
the other important aspect of matching logic deduction, namely its (relative) completeness. A mechanical translation of Hoare logic proof trees into equivalent matching logic proof trees was
presented. The size of the generated proofs is linear in the size of the original proofs. The method was described and proved correct for a simple imperative language with both iterative and
recursive constructs, but the underlying principles of the translation are general and should apply to any language. The results presented in this paper have two theoretical consequences. First, they
establish the relative completeness of matching logic for a standard language, by reduction to the relative completeness of Hoare logic, and thus show that matching logic is at least as powerful as
Hoare logic. Second, they give an alternative approach to proving soundness of Hoare logics, by reduction to the generic soundness of matching logic.
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of Illinois (Feb 2012), submitted. 17. Sasse, R., Meseguer, J.: Java+ITP: A verification tool based on Hoare logic and algebraic semantics. Electr. Notes Theor. Comput. Sci. 176(4), 29–46 (2007) | {"url":"https://pdffox.com/from-hoare-logic-to-matching-logic-pdf-free.html","timestamp":"2024-11-05T12:15:35Z","content_type":"text/html","content_length":"73578","record_id":"<urn:uuid:429ed671-7037-4893-8c4a-0d19cc4d5b6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00362.warc.gz"} |
Cite as
Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three Notes on Distributed Property
Testing. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 15:1-15:30, Schloss Dagstuhl – Leibniz-Zentrum
für Informatik (2017)
Copy BibTex To Clipboard
author = {Even, Guy and Fischer, Orr and Fraigniaud, Pierre and Gonen, Tzlil and Levi, Reut and Medina, Moti and Montealegre, Pedro and Olivetti, Dennis and Oshman, Rotem and Rapaport, Ivan and Todinca, Ioan},
title = {{Three Notes on Distributed Property Testing}},
booktitle = {31st International Symposium on Distributed Computing (DISC 2017)},
pages = {15:1--15:30},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-053-8},
ISSN = {1868-8969},
year = {2017},
volume = {91},
editor = {Richa, Andr\'{e}a},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.15},
URN = {urn:nbn:de:0030-drops-79847},
doi = {10.4230/LIPIcs.DISC.2017.15},
annote = {Keywords: Property testing, Property correcting, Distributed algorithms, CONGEST model} | {"url":"https://drops.dagstuhl.de/search/documents?author=Rapaport,%20Ivan","timestamp":"2024-11-14T20:08:23Z","content_type":"text/html","content_length":"90456","record_id":"<urn:uuid:d3cd386d-61a6-46ce-8c26-e6137a29e421>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00518.warc.gz"} |
Times Interest Earned Ratio | Analysis | Formula – Get Business Strategy
Times Interest Earned Ratio | Analysis | Formula
Times Interest Earned Ratio | Analysis | Formula
The times interest earned ratio is an indicator of a corporation’s ability to meet the interest payments on its debt. The times interest earned ratio is calculated as follows: the corporation’s
income before interest expense and income tax expense divided by its interest expense.
The larger the times interest earned ratio, the more likely that the corporation can make its interest payments. The times interest earned ratio is also referred to as the interest coverage ratio.
Times Interest Earned Ratio Formula
Failing to meet these obligations could force a company into bankruptcy. TIE is also referred to as the interest coverage ratio. Generating cash flow to make principal and interest payments and
avoiding bankruptcy depends on a company’s ability to produce earnings.
A company’s capitalization refers to the amount of money it has raised by issuing stock or debt and choices about capitalization impact the TIE ratio. Businesses consider the cost of capital for
stock and debt and they use that cost to make decisions about capitalization.
The times interest earned ratio is calculated by dividing income before interest and income taxes by the interest expense.
Both of these figures can be found on the income statement. Interest expense and income taxes are often reported separately from the normal operating expenses for solvency analysis purposes. This
also makes it easier to find the earnings before interest and taxes or EBIT.
Times-Interest-Earned Ratio Example
Assume that a corporation had the following amounts for the most recent year:
• Net income after tax of $500,000
• Interest expense of $200,000
• Income tax expense of $300,000
Given these assumptions, the corporation’s income before interest and income tax expense was $1,000,000 (net income of $500,000 + interest expense of $200,000 + income tax expense of $300,000). Since
the interest expense was $200,000, the corporation’s times interest earned ratio was 5 ($1,000,000 divided by $200,000).
Times Interest Earned Ratio Analysis
The times interest ratio is stated in numbers as opposed to a percentage. The ratio indicates how many times a company could pay the interest with its before tax income, so obviously the larger
ratios are considered more favorable than smaller ratios.
In other words, a ratio of 4 means that a company makes enough income to pay for its total interest expense 4 times over. Said another way, this company’s income is 4 times higher than its interest
expense for the year.
As you can see, creditors would favor a company with a much higher times interest ratio because it shows the company can afford to pay its interest payments when they come due. Higher ratios are less
risky while lower ratios indicate credit risk.
Times Interest Earned Ratio Calculator
Assume, for example, that XYZ Company has $10 million in 4% debt outstanding and $10 million in common stock and that the firm needs to raise more capital to purchase equipment. The cost of capital
for issuing more debt is an annual interest rate of 6% and shareholders expect an annual dividend payment of 8%, plus appreciation in the stock price of XYZ.
The business decides to issue $10 million in additional debt and the firm determines that the total annual interest expense is: (4% X $10 million) + (6% X $10 million), or $1 million annually. The
company’s EBIT calculation is $3 million, which means that the TIE is 3, or three times the annual interest expense (interest payable).
What Is Difference Between The Current Ratio And The Times Interest Earned Ratio?
Current Ratio:
The current ratio is a liquidity ratio that measures a company’s ability to pay short-term obligations or those due within one year. It tells investors and analysts how a company can maximize the
current assets on its balance sheet to satisfy its current debt and other payables.
Times Earned Interest Ratio
Times interest earned (TIE) is a metric used to measure a company’s ability to meet its debt obligations. The formula is calculated by taking a company’s earnings before interest and taxes (EBIT) and
dividing it by the total interest payable on bonds and other contractual debt. TIE indicates how many times a company can cover its interest charges on a pretax earnings basis. | {"url":"https://getbusinessstrategy.com/times-interest-earned-ratio-analysis-formula/","timestamp":"2024-11-05T05:44:32Z","content_type":"text/html","content_length":"76235","record_id":"<urn:uuid:9cd38c7d-1eb6-46a0-be6a-369f904298f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00815.warc.gz"} |
In case of a simple pendulum, time period versus length is depicted by
Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP
Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc
NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in
NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS
Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi
medium and English medium for IIT JEE and NEET preparation | {"url":"https://www.doubtnut.com/qna/649437007","timestamp":"2024-11-06T23:49:37Z","content_type":"text/html","content_length":"172792","record_id":"<urn:uuid:07ee9707-95fa-4db0-b823-43204d53946a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00471.warc.gz"} |
CCOR Workshop on Information, Games and Decisions (IGD)
The Corvinus Center for Operation Research (CCOR) invites you to the workshop: Information, Games and Decisions (IGD)
Venue: Corvinus University, Building C, Room: C.714
Date: May 31 Wednesday, 10.30-14.30
Keynote speaker: Dov Samet (Tel-Aviv University)
Speakers: Péter Vida (Corvinus & CY Paris Cergy University)
Miklós Pintér (Corvinus)
10.30-11.00: Péter Vida: Simple Forward Induction in Monotonic Multi-Sender Signaling Games
11.00-11.30: Miklós Pintér: Continuous Generalized Games
11.30-12.30: Lunch break
12.30-13.30: Dov Samet: Desirability relations in Savage’s model of decision making
13.30-14.30: Discussion
Dov Samet: Desirability relations in Savage’s model of decision making (joint with David Schmeidler)
We propose a model of an agent’s probability and utility that is a compromise between Savage (1954) and Jeffrey (1965). In Savage’s model the probability-utility pair is associated with preferences
over acts which are assignments of consequences to states. The probability is defined on the state space, and the utility function on consequences. Jeffrey’s model has no consequences, and both
probability and utility are defined on the same set of propositions. The probability-utility pair is associated with a desirability relation on propositions. Like Savage we assume a set of
consequences and a state space. However, we assume that states are comprehensive, that is, each state describes a consequence, as in Aumann (1987). Like Jeffrey, we assume that the agent has a
preference relation, which we call desirability, over events, which by definition involves uncertainty about consequences.
Péter Vida: Simple Forward Induction in Monotonic Multi-Sender Signaling Games (joint with Takakazu Honryo and Helmuts Azacis)
We introduce a new solution concept called simple forward induction which is implied by strategic stability in generic finite multi-sender signaling games. We apply this notion to infinite monotonic
signaling games and show that a unique pure simple forward induction equilibrium exists and its outcome is necessarily non-distorted. Finally, we show that in this class of games the non-distorted
equilibrium outcomes are limits of stable outcomes of finite games.
Miklós Pintér: Continuous Generalized Games (joint with Imre Balog)
We consider finite stochastic games and examine the existence of equilibrium for finite stochastic games. Our goal is to use a new concept – continuous generalized game – in order to provide a
different proof of the existence for equilibrium of finite generalized discounted stochastic games. In our proof, we show that all mentioned stochastic games are continuous generalized game and then
we show that they have an equilibrium. | {"url":"http://www.ccor.hu/?p=182","timestamp":"2024-11-06T09:01:30Z","content_type":"text/html","content_length":"35583","record_id":"<urn:uuid:10d9271c-a5c6-4907-8696-2cfbec5e71d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00382.warc.gz"} |
Best ways to use MKL's Vector Math Library of function composition
03-31-2013 09:54 PM
Should I use VML to compute the sigmoid function of a vector?
If yes, is there a better way than doing 4 loops:
1. Negate the vector
2. Compute the exp of the vector
3. Add 1 to the vector
4. Compute the inverse of the vector
04-01-2013 12:37 AM
04-01-2013 01:14 AM | {"url":"https://community.intel.com/t5/Intel-oneAPI-Math-Kernel-Library/Best-ways-to-use-MKL-s-Vector-Math-Library-of-function/td-p/986258","timestamp":"2024-11-02T05:32:28Z","content_type":"text/html","content_length":"189498","record_id":"<urn:uuid:5e926c09-2d1a-4a6c-8443-37c232781893>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00293.warc.gz"} |
Graduate Student Seminar- Computing Automorphisms of Poisson Algebras | The Department of Mathematics | Columbian College of Arts & Sciences | The George Washington University
Graduate Student Seminar- Computing Automorphisms of Poisson Algebras
Fri, 27 September, 2024 12:30pm - 1:30pm
Date and Time: Friday, September 27th 12:30 - 1:30 p.m.
Place: Rome 771
Speaker: Charlene Houchins, GWU
Title: Computing Automorphisms of Poisson Algebras
Abstract: For polynomials in one or two variables over a field of characteristic zero, all automorphisms are tame. When we encounter polynomials in three variables, some automorphisms are wild. This
makes it much harder to compute (tame?) the automorphisms. This means we don't explicitly know the generators of all automorphisms when we have three or more variables. However, if we introduce a
Poisson bracket multiplication, there is more hope to find an explicit characterization of automorphisms. We will fix a homogeneous polynomial of the form $\Omega=x^n+y^n+z^n$ to define our Poisson
bracket. We will see that for $n \geq 5$, valuation maps can be used to classify all automorphisms. Not only that, all of the automorphisms are graded. For $n=4$, we will use more elementary methods
to determine the generators of our automorphism group. For $1 \leq n \leq 3$, it is unknown what the automorphisms look like, but will certainly be explored in the future. | {"url":"https://math.columbian.gwu.edu/graduate-student-seminar-computing-automorphisms-poisson-algebras","timestamp":"2024-11-03T15:33:55Z","content_type":"text/html","content_length":"51745","record_id":"<urn:uuid:a649a149-52d6-4301-9b21-dbe68e1f96ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00622.warc.gz"} |
NCERT Solutions for Class 7 Maths Chapter 1 Integers
NCERT Solutions for Class 7 Maths Chapter 1 Integers and Class 7 Maths Chapter 1 Try These Solutions in Hindi and English Medium modified and updated for academic year 2024-25. According to new
syllabus and latest textbooks for new session 2024-25, there are only three exercises in chapter 1 of class 7th mathematics.
Class 7 Maths Chapter 1 Solutions in English Medium
Practicing math concepts like integers from Class 7 can be very beneficial for understanding and mastering the topic. Here’s a step-by-step guide on how to effectively practice Class 7 Maths Chapter
1 Try These Integers. Preparing Class 7 Maths Chapter 1 (Integers) NCERT solutions involves a structured approach to understanding the concepts, solving problems, and creating clear explanations for
students. Pay attention to definitions, examples, and the types of problems presented.
Class: 7 Mathematics
Chapter 1: Integers
Number of Exercises: 3 (Three)
Content: NCERT Exercise Solutions
Mode: Text, Images and Videos Format
Academic Session: 2024-25
Medium: English and Hindi Medium
Make sure you have a clear understanding of the basic concepts of integers, including positive and negative numbers, number line, absolute value, and addition/subtraction of integers. Integers are
the central focus of this chapter. Make sure you have a clear understanding of what integers are, how they are represented on a number line, and their properties. Break down the chapter into key
concepts and topics. These may include positive and negative integers, addition and subtraction of integers, and properties of integers. Read through the chapter in your textbook. Pay close attention
to explanations, examples, and any solved problems.
Work through the practice exercises provided in the NCERT textbook. Solve the problems step by step to ensure you understand the process. Write down the step-by-step solutions for each type of
problem. Make sure to explain each step clearly, addressing any potential difficulties students might face. Integers can be represented on a number line. Include diagrams and number lines to visually
explain concepts, especially when discussing positive and negative integers.
Integers have real-life applications, like representing temperature, bank balances, and distances. Include relatable examples to help students connect the concept to their daily experiences.
Anticipate common mistakes students might make and provide explanations to help them avoid these errors. Include a variety of problems, from basic to more complex, to challenge students’
understanding and problem-solving skills. Integers have certain rules for addition, subtraction, multiplication, and division. Note down these rules for quick reference.
NCERT Solutions for Class 7 Maths Chapter 1
7th Maths Exercise 1.1, Exercise 1.2 and Exercise 1.3 in English Medium and Prashnavali 1.1, Prashnavali 1.2 and Prashnavali 1.3 in Hindi Medium free to download in PDF format for new academic
session. CBSE NCERT (https://ncert.nic.in/) Solution are given in simplified format. Videos related to each exercises are also given for better understanding. You can use NCERT solutions 2024-25
online or download in PDF file format without any login or password. Class 7 Maths solutions apps for online as well as offline use are also given free to download. Regular practice is key to
mastering any math topic. Dedicate a specific amount of time each day for practicing integer-related problems. In videos, all the questions are described using proper properties of integers. The
language of description is kept easy so that everyone can understand easily. For any inconvenience, contact us for help.
Review your solutions for accuracy, clarity, and coherence. Ensure your explanations are in line with the language and understanding of Class 7 students. Begin with easy problems and gradually move
on to more complex ones. This will help you build confidence and progressively develop your skills. Format your solutions in a clear and organized manner. Use appropriate fonts, spacing, and headings
to make the solutions easy to read. Proofread your solutions for any grammatical or typographical errors. Well-presented solutions enhance the learning experience.
If relevant, provide additional notes or tips for teachers and parents to assist them in explaining concepts effectively to students. Test your solutions with a few students or educators to get
feedback. This helps ensure that your solutions are clear and understandable. Remember that the goal is to make Class 7 Maths Chapter 1 Integers NCERT solutions accessible, comprehensive, and
engaging for students. Work through the example problems given in your textbook. These problems are designed to illustrate the concepts discussed in the chapter. The solutions should help students
grasp the concepts, develop problem-solving skills, and build a strong foundation in mathematics.
7 Maths Chapter 1 Integers Solutions
NCERT Books for class 7 and CBSE Solutions 2024-25 for other subjects are also available for free download. We have prepared the Solutions in the simplified format so that students can understand
easily. Look for additional practice worksheets, online resources, or apps that offer integer-related problems. Many educational websites and apps provide interactive quizzes and exercises. Invent
your own integer problems based on real-life scenarios or mathematical concepts. Solving self-created problems can deepen your understanding of the subject. Class 7 Maths Chapter 1 solutions are
given below in Hindi and English Medium.
Class 7 Maths Chapter 1 Important Questions
At Srinagar temperature was C on Monday and then it dropped by C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by C. What was the temperature on this day?
On Monday, temperature at Srinagar = –5 oC
On Tuesday, temperature dropped = 2 oC
Temperature on Tuesday = –5 oC – 2 oC = –7 oC
On Wednesday, temperature rose up = 4oC
Temperature on Wednesday = –7 oC + 4 oC = –3 oC
Thus, temperature on Tuesday and Wednesday was –7 oC and –3 oC respectively.
A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?
Height of a place above the sea level = 5000 m
Floating a submarine below the sea level = 1200 m
The vertical distance between the plane and the submarine = 5000 + 1200 = 6200 m
Thus, the vertical distance between the plane and the submarine is 6200 m.
Mohan deposits ₹2,000 in his bank account and withdraws ₹1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the
amount deposited? Find the balance in Mohan’s accounts after the withdrawal?
Deposit amount = ₹2,000 and
Withdrawal amount = ₹1,642
Balance = 2,000 – 1,642 = ₹358
Thus, the balance in Mohan’s account after withdrawal is ₹ 358.
A certain freezing process requires that room temperature be lowered from 40 oC at the rate of 5 oC every hour. What will be the room temperature 10 hours after the process begins?
Present room temperature = 40 oC
Decreasing the temperature every hour = 5 oC
Room temperature after 10 hours = 40 oC + 10 x (–5 oC ) = 40 oC – 50 oC = – 10 oC
Thus, the room temperature after 10 hours is – 10 oC after the process begins.
A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹ 5 per bag of grey cement sold. The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a
month. What is its profit or loss?
Profit of 1 bag of white cement = ₹ 8
And Loss of 1 bag of grey cement = ₹ 5
Profit on selling 3000 bags of white cement = 3000 x ₹ 8 = ₹ 24,000
Loss of selling 5000 bags of grey cement = 5000 x ₹ 5 = ₹ 25,000
Since Profit < Loss Therefore, his total loss on selling the grey cement bags = Loss – Profit = ₹ 25,000 – ₹ 24,000 = ₹ 1,000 Thus, he has lost of `₹1,000 on selling the grey cement bags.
Practice problems involving addition, subtraction, multiplication, and division of integers. Try these word problems and puzzles that require application of integer concepts. Challenge yourself with
more complex problems that require critical thinking and a deep understanding of the concepts. This will help you refine your problem-solving skills.
After solving problems, review your answers. Correcting mistakes is an important part of the learning process. If you’re facing difficulties, don’t hesitate to discuss the problems with your
classmates, teacher, or parents. Sometimes, a different perspective can clarify things. Don’t overexert yourself. Take short breaks during your practice sessions to keep your mind fresh and focused.
About NCERT Solutions for Class 7 Maths Chapter 1
In 7 Mathematics Chapter 1 Integers, we will explore all the operations based on integer properties. Properties of integers on the operations like addition, subtraction, multiplication and division.
Take mock tests or create a set of problems to test your understanding. Timed tests can help you practice solving problems under pressure. Schedule regular revision sessions to reinforce your
understanding. Revise the concepts, rules, and formulas to keep them fresh in your mind. Remember, consistent practice and a positive attitude towards learning are key factors in succeeding in math.
Don’t get discouraged by challenges; instead, view them as opportunities to improve. We have to learn about all the properties like closure, commutative, associative, etc. Integers are Closure under
Addition but not under subtraction.
Class 7 Maths Chapter 1 Important Questions for Practice
1. When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient. That is, we get a negative integer.
2. The product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.
3. When we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient. We, thus, get a negative integer.
4. Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.
5. Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a.
Help and Support
We have prepared Hindi Medium NCERT Solutions for class 7 Maths on the demand of students. Now it is online on website to view online as well as for download. Time to time we modify our website on
the basis of student’s suggestions. That is why Class 7 Maths App in English or Kaksha 7 Ganit App in Hindi Medium are developed for offline use.
How many exercises, questions, and examples are there in chapter 1 of class 7th Maths?
According to new syllabus and latest textbooks for session 2024-25, there are three exercises in chapter 1 (Integers) of class 7th Maths.
In the first exercise (Ex 1.1), there are 4 questions. Question 1 has three parts, question 2 has three parts, and question 4 has five parts.
In the second exercise (Ex 1.2), there are 9 questions. Question 2 has two parts, question 3 has two parts, and question 5 has eight parts.
In the third exercise (Ex 1.3), there are seven questions. Question 1 has nine parts, and question 2 has two parts.
So, there are in all 20 questions in chapter 1 (Integers) of class 7th Maths.
There is a total of 5 examples in chapter 1 (Integers) of class 7th Maths.
What will students study in chapter 1 of class 7th Maths?
In chapter 1 of class 7th Maths, students will study:
1. Properties of integers.
2. Addition and Subtraction of integers.
3. Statement Questions- Addition.
4. Multiplication of integers.
5. Distributive law in integers.
6. Statement Questions- Multiplication.
7. Division of Integers.
8. Statement Questions- Division.
Is chapter 1 of class 7th Maths difficult?
Chapter 1 of class 7th Maths is not easy and not difficult. It lies in the mid of easy and difficult because some parts of this chapter are easy, and some parts are difficult. But, the difficulty
level of any chapter varies from child to child. So, Chapter 1 of class 7th Maths is easy or not depends on children also. Some children find it tough, some find it simple, and some find it in the
middle of simple and tough.
How much time, students need to do chapter 1 of class 7th Maths?
Students need a maximum of 7-9 days to do chapter 1 of class 7th Maths if they give at least 1-2 hours per day to this chapter. This time is an approximate time. This time can vary because no
students have the same working speed, same efficiency, same capability, etc.
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Package 'canaper'
Run a randomization analysis for one or more biodiversity metrics
The observed value of the biodiversity metric(s) will be calculated for the input community data, then compared against a set of random communities. Various statistics are calculated from the
comparison (see Value below).
n_reps = 100,
n_iterations = 10000,
thin = 1,
metrics = c("pd", "rpd", "pe", "rpe"),
site_col = "site",
tbl_out = tibble::is_tibble(comm),
quiet = FALSE
n_reps = 100,
n_iterations = 10000,
thin = 1,
metrics = c("pd", "rpd", "pe", "rpe"),
site_col = "site",
tbl_out = tibble::is_tibble(comm),
quiet = FALSE
Dataframe, tibble, or matrix; input community data with sites (communities) as rows and species as columns. Either presence-absence data (values only 0s or 1s) or abundance data (values
comm >= 0) accepted, but calculations do not use abundance-weighting, so results from abundance data will be the same as if converted to presence-absence before analysis.
phy List of class phylo; input phylogeny.
Character vector of length 1 or object of class commsim; either the name of the model to use for generating random communities (null model), or a custom null model. For full list of
null_model available predefined null models, see the help file of vegan::commsim(), or run vegan::make.commsim(). An object of class commsim can be generated with vegan::commsim() (see Examples in
n_reps Numeric vector of length 1; number of random communities to replicate.
n_iterations Numeric vector of length 1; number of iterations to use for sequential null models; ignored for non-sequential models.
thin Numeric vector of length 1; thinning parameter used by some null models in vegan (e.g., quasiswap); ignored for other models.
metrics Character vector; names of biodiversity metrics to calculate. May include one or more of: pd, rpd, pe, rpe (case-sensitive).
site_col Character vector of length 1; name of column in comm that contains the site names; only used if comm is a tibble (object of class tbl_df).
tbl_out Logical vector of length 1; should the output be returned as a tibble? If FALSE, will return a dataframe. Defaults to TRUE if comm is a tibble.
quiet Logical vector of length 1; if TRUE, suppress all warnings and messages that would be emitted by this function.
The biodiversity metrics (metrics) available for analysis include:
• pd: Phylogenetic diversity (Faith 1992)
• rpd: Relative phylogenetic diversity (Mishler et al 2014)
• pe: Phylogenetic endemism (Rosauer et al 2009)
• rpe: Relative phylogenetic endemism (Mishler et al 2014)
(pe and rpe are needed for CANAPE with cpr_classify_endem())
The choice of a randomization algorithm (null_model) is not trivial, and may strongly affect results. cpr_rand_test() uses null models provided by vegan; for a complete list, see the help file of
vegan::commsim() or run vegan::make.commsim(). One frequently used null model is swap (Gotelli & Entsminger 2003), which randomizes the community matrix while preserving column and row sums (marginal
sums). For a review of various null models, see Strona et al. (2018); swap is an "FF" model in the sense of Strona et al. (2018).
Instead of using one of the predefined null models in vegan::commsim(), it is also possible to define a custom null model; see Examples in cpr_rand_comm()
Note that the pre-defined models in vegan include binary models (designed for presence-absence data) and quantitative models (designed for abundance data). Although the binary models will accept
abundance data, they treat it as binary and always return a binary (presence-absence) matrix. The PD and PE calculations in canaper are not abundance-weighted, so they return the same result
regardless of whether the input is presence-absence or abundance. In that sense, binary null models are appropriate for cpr_rand_test(). The quantitative models could also be used for abundance data,
but the output will be treated as binary anyways when calculating PD and PE. The effects of using binary vs. quantitative null models for cpr_rand_test() have not been investigated.
A minimum of 5 species and sites are required as input; fewer than that is likely cause the some randomization algorithms (e.g., swap) to enter an infinite loop. Besides, inferences on very small
numbers of species and/or sites is not recommended generally.
The following rules apply to comm input:
• If dataframe or matrix, must include row names (site names) and column names (species names).
• If tibble, a single column (default, site) must be included with site names, and other columns must correspond to species names.
• Column names cannot start with a number and must be unique.
• Row names (site names) must be unique.
• Values (other than site names) should only include integers >= 0; non-integer input will be converted to integer.
The results are identical regardless of whether the input for comm is abundance or presence-absence data (i.e., abundance weighting is not used).
Dataframe. For each of the biodiversity metrics, the following 9 columns will be produced:
• *_obs: Observed value
• *_obs_c_lower: Count of times observed value was lower than random values
• *_obs_c_upper: Count of times observed value was higher than random values
• *_obs_p_lower: Percentage of times observed value was lower than random values
• *_obs_p_upper: Percentage of times observed value was higher than random values
• *_obs_q: Count of the non-NA random values used for comparison
• *_obs_z: Standard effect size (z-score)
• *_rand_mean: Mean of the random values
• *_rand_sd: Standard deviation of the random values
So if you included pd in metrics, the output columns would include pd_obs, pd_obs_c_lower, etc...
Faith DP (1992) Conservation evaluation and phylogenetic diversity. Biological Conservation, 61:1–10. doi:10.1016/0006-3207(92)91201-3
Gotelli, N.J. and Entsminger, N.J. (2003). Swap algorithms in null model analysis. Ecology 84, 532–535.
Mishler, B., Knerr, N., González-Orozco, C. et al. (2014) Phylogenetic measures of biodiversity and neo- and paleo-endemism in Australian Acacia. Nat Commun, 5: 4473. doi:10.1038/ncomms5473
Rosauer, D., Laffan, S.W., Crisp, M.D., Donnellan, S.C. and Cook, L.G. (2009) Phylogenetic endemism: a new approach for identifying geographical concentrations of evolutionary history. Molecular
Ecology, 18: 4061-4072. doi:10.1111/j.1365-294X.2009.04311.x
Strona, G., Ulrich, W. and Gotelli, N.J. (2018), Bi-dimensional null model analysis of presence-absence binary matrices. Ecology, 99: 103-115. doi:10.1002/ecy.2043
# Returns a dataframe by defualt
phylocom$comm, phylocom$phy,
null_model = "curveball", metrics = "pd", n_reps = 10
# Tibbles may be preferable because of the large number of columns
phylocom$comm, phylocom$phy,
null_model = "curveball", tbl_out = TRUE, n_reps = 10
# Returns a dataframe by defualt
phylocom$comm, phylocom$phy,
null_model = "curveball", metrics = "pd", n_reps = 10
# Tibbles may be preferable because of the large number of columns
phylocom$comm, phylocom$phy,
null_model = "curveball", tbl_out = TRUE, n_reps = 10 | {"url":"https://packages.ropensci.org/canaper/doc/manual.html","timestamp":"2024-11-09T03:19:04Z","content_type":"text/html","content_length":"70009","record_id":"<urn:uuid:6ac9cfc4-d764-46b6-ad40-58a312b9fa7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00894.warc.gz"} |
Revamping Algebra 2 with Rich Problems and Contexts
This is where I’m currently at for Algebra 2 (outline below). All starred items indicate an activity or rich context. All “Ch” indications are for projects I thought had some merit at the end of the
chapters. I haven’t done the work of writing up each lesson yet--just gathering the resources. Most of the “Activity” listings are from the UCSMP activities workbook. There were some real treasures
in there but none usable for me right out of the box. They gave too much away by telling the students directly what to do and making connections for them. I think 6 years ago, I would have thought
them perfect; now, I need to help make them more inquiry-based. I'll share what I have when I have it.
You’ll also see the concepts I’m cutting and those I’m adding in order to align properly with the Common Core. With the addition of more constructivist opportunities for kids, some stuff doesn’t have
time to get revisited if we’ve already covered it in Algebra 1 (goodbye linear unit?). If you've got some ideas, I'd love to hear them.
I plan to write up lessons for each of the starred sections and post them eventually both here and on
michedmath.weebly.comAlgebra 2 Course Outline (click "read more")
1. Functions
Definition of a function
Function notation
Telling whether a graph is a function
Recursive definitions of sequences
*Leaky Faucet
Ch1 none
2. Variation and Graphs
Direct Variation
Inverse Variation
Combined Variation
Graphs: names, asymptotes, symmetry, domain and range restrictions
*Modeling - This would be a good PrBL. How much weight a board can hold based on distance of supports vs Width vs Thickness. Use spaghetti noodles or straws and washers as weights.
*Mr. Potato Head Inverse Variation
*2-2 Activity 3 - Modeling the Law of the Lever
*2-7 Activity 4 - Fitting a Model to Data
Ch2 #1 Pizza Prices p128 (Pizza menus from around Cadillac)
#4 Variation and Light p129 (Need: flashlight, measuring tape, light meter)
#5 Law of the Lever p 129 (Need see-saw to check the law of the lever)
3. Linear Functions
Review of linear equations (CUT)
Slope-intercept form (CUT)
Piece-wise linear (NEED)
Standard Form (CUT)
Point-slope form (NEW)
Recursive formulas for Arithmetic Sequences (NEED)
Explicit formulas for Arithmetic Sequences (NEED)
Step-functions (NEED)
Ch3 #3 Fines for speeding pg 192
4. Matrices
Matrix addition and scalar multiplication
Matrix multiplication (NEED)
Transformations: size change, scale change, reflections, rotations. (CUT)
Perpendicular lines and bisectors. (CUT)
Ch4 #3 Overhead Projectors as Size Changers
5. Systems
Graphing (QUICK)
Substitution (QUICK)
Elimination (QUICK)
Solving systems through matrices:
Determinants of matrices
Inverses of matrices
Linear inequalities (QUICK)
Feasible regions
Linear Programming
*Gender Gap (Systems of Equations, data fitting, regression, rates, data analysis)
*5.2 Playing Catchup ACT3
* Lego furniture linear programming
Ch5 #1 Nutritious and Cheap? (Food Aid)
#5 Using Matrices to Code and Decode Messages
6. Quadratic Functions
Absolute-value functions
Translations of quadratics
Vertex Form
Standard Form
Completing the Square
Modeling based on differences
Quadratic Formula
Imaginary and Complex Numbers
Discriminant on Quadratic Formula
*6-3 Activity 11 Modeling a Quadratic with a flashlight
*6.6 Toothpicks
*6.6 Shooting hoops and fitting a curve to the data using Vernier Physics Apps
*Quadratic - Angry Birds
Ch6 #1 Projectile Motion,
#2 Sum and Products of Roots,
#4 Quadratic Models pg 408
7. Powers
Power Function
Properties of Powers
Negative Exponents
Compound interest
Geometric Sequences
Solving power functions and Equations with Rational Exponents
*7.5 Shrinking Dollar Geometric Sequences
Ch7 #3 Financing Post-High School Education #5 Local Interest Rates pg 469
8. Inverses and Radicals
Composition of Functions
Inverses of Functions (Equations, Domain, Range)
Simplifying expressions with rational exponents.
Geometric Mean
Rationalizing the denominator
Solving equations with rational exponents.
Ch8 #$ Radicals and Heights pg 522
9. Exponential and Logarithmic Functions
Exponential Growth and Decay (QUICK)
Natural number e and y = Pe^(kt)
Logarithms: converting to exponential form, base 10, non-base 10, properties of, solving equations, base e,
Solving Exponential functions using logarithms
*9.1 Exponential Growth - Domino Skyscraper
*9.? National Debt in relation to War
*9.? Moore’s Law
*9.? Motel 6 - Compound interest
*9.? One Million Dollars - Dr. Evil. Compound interest problem
*9-4 NCTM Too Hot To Handle <http://illuminations.nctm.org/LessonDetail.aspx?id=L852> along with Activity 17 Fitting Exponential Models to Data
*NCTM - Modeling Orbital Debris
Ch9 #1 CarLoans
#2 Modeling Growth of HIV
#3 Predicting Cooling Times pg 593
10. Trigonometry
Sine, Cosine, Tangent and their inverses
Angles of elevation and descent
Trigonometric properties
Exact Values of Trig functions
Unit Circle
Law of Cosines
Law of Sines
Graphs (basic) of Sine, Cosine and Tangent (NEED)
Radians (NEED)
*10-8 Activity 20 Graphs of Trigonometric Functions (Ferris Wheel)
*10.8 Tsunami of 2006 (with modifications)
Ch10 #1 Triangulation and Surveying (research into how)
#5 Sunrise and Sunset times (have students research information on their own).
11. Polynomials
Classification base on Degree and Number of terms
Multiplying and Adding Polynomials
*Polynomial Puzzler (http://illuminations.nctm.org/LessonDetail.aspx?id=L798)
Zeros in relation to factored form
Rational Zero Theorem (CUT)
Fundamental Theorem of Algebra
Modeling Polynomial Functions
*Light It Up NCTM Project (Rational Functions) This Will work for Intro of topic and project)
*Speed Dating (http://function-of-time.blogspot.com/2009/10/speed-dating.html)
*Resistors Project (Could be done with Physics)
*11.1 Paper or Plastic (with modifications)
*11-10 Activity 21 Modeling Data With Polynomials (Need pennies and dominoes)
Ch11 none
12. Quadratic Relations - (CUT)
Focus and Directrix of a Parabola (CUT)
Equations of Circles (CUT)
Ellipse and Foci (CUT)
Hyperbola and Foci (CUT)
Ch12 #2 Whispering Galleries #4 Reflection Properties of the Conics
#5 Orbits of the Planets pg 800
13. Series and Combinations - Currently don’t get to more than Sigma notation for series
Arithmetic Series (NEED)
Geometric Series (NEED)
Sigma notation for Series (NEED)
Median, Mean, Mode
Standard Deviation (NEED)
Pascal’s Triangle (NEED)
Binomial Expansion (NEED)
Series notation to explicit formulas
Sets and subsets
Independent and Mutually-exclusive events (NEED)
Trial and Experiment (NEED)
Probability (NEED)
Normal Distribution (NEED)
*13.3 Penny Pyramid - ACT 3
*13-5 Activity 26 Random Walks
*13-8 Activity 27 Probabilities and Combinations
Ch13 #5 the Kock Curve (snowflake) pg 876
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Number Systems - Edu Spot- NCERT Solution, CBSE Course, Practice Test
Number Systems
Natural numbers are - 1, 2, 3, …………….. denoted by N.
• Whole numbers are - 0, 1, 2, 3, ……………… denoted by W.• Integers - ……. -3, -2, -1, 0, 1, 2, 3, ……………… denoted by Z.
• Rational numbers - All the numbers which can be written in the form p/q,q ≠0 are called rational numbers where p and q are integers.
• Irrational numbers - A number s is called irrational, if it cannot be written in the form p/q where p and q are integers and q ≠0
• The decimal expansion of a rational number is either terminating or non terminating recurring. Thus we say that a number whose decimal expansion is either terminating or non terminating recurring
is a rational number.
• The decimal expansion of a irrational number is non terminating non recurring.
• All the rational numbers and irrational numbers taken together.
• Make a collection of real number.
• A real no is either rational or irrational.
• If r is rational and s is irrational then r+s, r-s, r.s are always irrational numbers but r/s may be rational or irrational.
• Every irrational number can be represented on a number line using Pythagoras theorem.
• Rationalization means to remove square root from the denominator.
NCERT Solutions Maths
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Gantt Chart In Excel For Multiple Projects 2024 - Multiplication Chart Printable
Gantt Chart In Excel For Multiple Projects
Gantt Chart In Excel For Multiple Projects – You can create a multiplication graph or chart in Excel by using a design. You will discover many instances of web templates and discover ways to
formatting your multiplication graph or chart utilizing them. Here are several tips and tricks to produce a multiplication chart. When you have a web template, all you want do is duplicate the
formulation and mixture it in the new mobile. You may then use this method to multiply some numbers by another establish. Gantt Chart In Excel For Multiple Projects.
Multiplication desk format
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. Initially, you have to locking mechanism row among the header line, then flourish the
quantity on row A by mobile phone B. A different way to produce a multiplication table is to use mixed references. In such a case, you might get into $A2 into line A and B$1 into row B. The end
result can be a multiplication dinner table by using a formulation that works well both for columns and rows.
If you are using an Excel program, you can use the multiplication table template to create your table. Just available the spreadsheet with the multiplication kitchen table template and change the
label for the student’s name. Also you can adjust the page to fit your individual demands. There is an choice to alter the color of the cellular material to improve the look of the multiplication
kitchen table, also. Then, you may modify all the different multiples to meet your requirements.
Building a multiplication graph in Excel
When you’re making use of multiplication table application, you can easily build a basic multiplication dinner table in Shine. Just develop a sheet with rows and columns numbered in one to 40. Where
the rows and columns intersect is the answer. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same goes for the opposite.
First, it is possible to enter in the numbers that you need to increase. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To produce the
phone numbers bigger, select the cells at A1 and A8, and after that select the correct arrow to decide on an array of tissue. You may then kind the multiplication method in the cells within the other
rows and columns.
Gallery of Gantt Chart In Excel For Multiple Projects
Genius Project Continues To Lead The Way With Release Of Their Multi
14 Excel Gantt Chart Template Excel Templates
Gantt Chart For Multiple Projects ExcelTemplate
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The KMV Sketch, First Estimator, Size = 1
For this step we are going to cheat a little so that we can learn about estimation. We are going to cheat in that we are going to predetermine that our data source only has n = 10 unique values (so
we don’t really need a sketch to estimate what we already know). We have loaded all 10 values into our ordered list. As one can see, the values are roughly evenly distributed between zero and one so
our hash transform is doing its job.
How many hash values do we have to retain to compute the estimate of n, n̂, accurately? As you might expect, the more samples we retain, the more accurate will be our estimate.
Suppose we kept only one value, so k = 1, and we chose the smallest hash value out of all the hash values in the set, which, in this case, is 0.008. We could assume that since the hash values are
random-uniform distributed that the separation between the hash values are roughly the same. Let’s label that separation between values as d. Our first estimator could be just dividing 1.0 by d.
Unfortunately, 1/0.008 is about 125, which is way larger than 10. And as one can see, there is a lot of variation in the separation of each of the hash values. If the hash values had been ordered
differently, the smallest hash value could have been the separation between the 3rd and 4th values. In this case our first estimator, 1/0.191 would be about 5, which is too small.
Clearly, our first estimator, 1/d, with a sample size of one is too noisy to be useful. However, what we do have is a simple formula for an estimate: | {"url":"https://datasketches.incubator.apache.org/docs/Theta/KMVfirstEst.html","timestamp":"2024-11-05T15:49:19Z","content_type":"text/html","content_length":"18212","record_id":"<urn:uuid:b6dbba82-b126-4674-92fc-0d7a5a263faf>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00464.warc.gz"} |
C - C to cyclinder - Mathematics Dictionary
: A symbol known as the Blackboard-bold C, commonly used to represent the set of complex numbers.
^∞ : A Blackboard-bold C with an infinity sign suffixed, commonly used to reger to the set of extended complex numbers, that is, the set of complex numbers together with a single point of infinity.
calculus: A branch of mathematics, the study of (infinitely many) infinitesimal quantities (which may be changing). It was classically made sense through the idea of limits and functions, thus they
form an important part of the study historically.
calculus of variations: The study of extremising functionals analogous to calculus and the extremising of functions. As the set of derivatives and integrals can be thought of as forming a vector
space, in addition to being analogous to, the study is naturally considered as overlapping with calculus to a certain extent.
cal: Calorie - a metric unit of energy. The equivalent SI unit of energy is joule. 1 cal (lower case c) is the energy required to raise 1 gram of water by 1 °C at roughly 4.2 joules, while 1 Cal
(capital C) is the energy required to raise 1 gram of water by 1 °C 4.2 kilojoules.
cancellation: Any method of calculation the result of which, when compared to the original form, attributes a number of components of the original form to mitigate the effects of the rest (of those
components), so that the result remains the same through their omission. A common example invloves the numerator and denominator of a fraction, and another invloves the omission of one logarithm
symbol from each side of the equation (when the each side consists of the logarithm only), even though the former is a result of the equivalence of fractions and the latter the injective nature of
the logarithmic function. Failure of understanding these can result in the misuse of such methods, and is commonplace amongst students.
canonical: Describes a representation of an object (e.g. an expression, a transformation) in a way that is preferred, perhaps unique or considered natural due to certain properties that it exhibits,
even though there may be other equivalent representations.
cantilever: A beam or such similar structures which is anchored at only one end such that it resists rotation under load.
cap: An informal class="d-title" name for symbol ∩, used to denote intersection of sets that is easier to say than the polysyllabic "intersection".
capture-recapture sampling: A method for estimating total population (usually of animals) through 2 periods of capture (observation), assuming that the total number has reamined constant and that the
probability of capture of any animal on any one visit is constant and equal. In implementing this method, a researcher captures a number of animals and mark them (or make sure that they can be
identified in the future in some way) and releases them, they come back to capture a number of animals and count the proportion of animals that were captured both times. By assuming that the
proportion of animals recaptured in an unbiased estimate of the proportion of animals marked in the first capture, combine this with the number of animals marked on the first visit, we can easily
calculate the estimated number of the total population. In practice, it is difficult to ensure that the animal population remains constant (or roughly the same) while ensuring that the chance of
recapture is truly equal amongst the population since the former requires the visits to be relatively close to each other so that the population does not change significantly and yet the latter
condition requires the visits to be sufficiently space apart so that the locations of the animals are truly randomised.
cardinal number: A number used to represent the size of a set. The abstract concept of size in this sense is called cardinality. See ordinal number.
cardinal points: North, East, South and West, the four directions equally spaced apart on the surface of the Earth. Since the idea orignated from when the Earth was generally assumed to be flat, we
need to carefully define the idea of these "directions" using the idea of geodesics, since 2 lines in the same "direction", in this sense, are not necessarily parallel.
cardioid: A curve class="d-title" named for being roughly heart-shaped, though it has a cusp, it has no point (of sudden change in direction) at the opposite point on the loop. It is the locus traced
out by a fixed point on a circle, as the circle rolls around another circle of equal radius and fixed centre.
Cartesian coordinate system: The rectilinear coordinate system used for any (finite and natural) number of dimensions. The number of axes used is commonly seen as a basic way of "defining" the number
of dimensions.
These axes must all be mutually perpendicular and the components (the coordinates) of a point are the shortest distances of a point to the hyperplane consisting the axes (except the axis related to
the component in question). Thus, the x-coordinate of a point is the shortest distance from the point to any point on the line (plane) consisting of the y-axis (both y and z-axes) in 2D (3D).
While rectilinear coordinate system is a more descriptive class="d-title" name of its function, it is more commonly called Cartesian, class="d-title" named after Cartesius, the Latin class="d-title"
name of René Descartes.
Cartesian product: The Cartesian product of two sets is the set of all ordered pairs such that the first element of the ordered pair is an element of the first set in the product and similarly, the
second element of the ordered pair is an element of the second set in the product.
casting out nines: A method of verification of integer arithmetics by checking that the answer matches the equivalent calculations under modulo arithmetic. The modulus of 9 is chosen simply because
of our decimal systems (base 10). The difference of 1 between the modulus (9) and the base (10) allows for an easy conversion of a number to modulo 9. (By adding the constituent digits of a number
represented in base 10.)
categorical data: Data that is used (or can only be used) as labels rather than quantities, as such no arithmetic structure exist and certain concepts (such as mean or median) are undefined.
categorical variable: A random variable with values which are categories. (e.g. nationality, gender etc.)
catenary: A plane curve which, if any sections of it are used to represent a string of uniform mass, has the lowest gravitational potential energy for the given length and fixed points. (Assuming
constant gravity.) Thus, it is approximately the shape of a string hanging under gravity. (It is only an approximation since gravitational strength varies at different height.)
The class="d-title" name is derived from thr latin word for chains, as it is observed to be the shape of hanging chains historically.
catenoid: A surface formed by the rotation of a catenary.
Cauchy convergence condition: 1. For sequences - a necessary and sufficient condition for the convergence of a sequence: an infinite sequqnece is convergent, if and only if, for any positive number
ε, there is always a number N(ε) (N is a function of ε), such that the difference between any two terms after the N^th term is less than ε.
where m and n are integers greater than N.
2. For series - a necessary and sufficient condition for the convergence of a series: An infinite series Σa[n] is convergent, if and only if, for any positive number ε, there is an integer N(ε) (N is
a function of ε), such that
for all integers n > N and p is a positive integer.
Cauchy distribution: A symmetric continuous probability distribution with infinite support.
Cauchy ratio test: Also known simply as the ratio test. A method of deciding the convergence of a series through the general ratio of one term of the series to the next.
Cauchy-Riemann equations: A system of partial differential equations logically equivalent to the condition of holomorphicity of a differentiable function in complex analysis. For z = x + iy and f(z)
= u + iv:
Cauchy's integral theorem: A theorem that states that, a complex function f(z) which is holomorphic and is integrated along two paths with the same endpoints are the same. Thus the integral of a
complex function f(z) holomorphic along the closed curve γ is always zero.
Cauchy's residue theorem: Mostly known as the residue theorem, it is the generalisation of Cauchy's integral Theorem and Cauchy's Integral formula.
cause variable: Also known as an independent variable, an explanatory variable, a predictor or predictor variable or regressor (see regression).
caustic: The envelope of rays refracted or reflected by a geometric object.
Cavalieri’s principle: Two geometrical figures whose cross sections are the same as each other, at the same distance away from some reference line/lines (plane/planes) have the same area (volume). As
an example, this explains why triangles whose bases have the same length, and have the same height, have also the same area, regardless of its shape.
Cayley table: The rectangular array listing all possible results of a binary operation on finitely many elements (usually of a group).
c.d.f.: Cumulative distribution function of a probability distribution.
ceiling function: A function on real numbers whose value is always rounded up, if the argument is not already an integer. The function leaves integers unchanged.
celestial mechanics: The study of motions of celestial bodies such as stars, planets and comets etc.
cell: Categories of data divided into rectangular arrays through more than one variable. It is essentially a conventional use for the analogue of groups in 1 dimensions.
Celsuis: Represented by the symbol °C. It is based on dividing the difference between the freezing point and boiling point of water into 100 equal "degrees:.
censored observations: In statistics, observations which are made incomplete systematically due to the nature of the procedure (possible) for observation or the objects under study.
census: A survey of the entire population.
centesimal measure:A metricangular measure dividing a right angle into 100 centesimal degrees.
centi-: An SI prefix for one one hundredth.
centigrade degree: Same as celsius.
central angle: An angle between two radii (of a circle or sphere).
central conic: A conic section with a centre - a hyperbola or an ellipse (which includes the case of a circle). i.e. all the conic sections except the parabola.
central difference: The absolute or directed difference between values whose argument are a fixed equal interval away from the nominated argument.
central force: A force whose direction is always toawards the centre, and whose magnitude is a function of the distance between the point of application and the centre.
central limit theorem: The theorem which states that the mean of a large number of independent random variables tends to a normal distribution as long as it satisfies certain conditions. It
essentially explains the prevalence of the normal distribution.
central polyhedral angle: The angle between a number of planes all containing the centre of the sphere.
central tendency: A common measure in summary statistics bsed around the loose idea that we can assign one location to represent the locations of a number of objects considered as one. Thus, there is
not just one but rather a number of slightly different concepts which fits the description of central tendency. It is what is commonly referred to by the similarly loose idea of an average.
centre (centre of symmetry): A point about which a geometric figure is in some way self-similar. It can either be a centre of rotational symmetry, or the intersections of multiple lines/planes of
reflective symmetry, or even the intersections of medians in a triangle.
centre of a group: The set of elements which are commutative with every elements of the group. Note that there must be at least one element in the centre, the identity. Whereas the centre can be as
large as the group, such a group is called an Abelian group.
centre of buoyancy: The centre of gravity of the body of water that an object displaces.
centre of curvature: Given a curve, the centre of curvature of a point on this curve is the centre of a circle which "locally" (for a neighbourhood of that point) describes the curvature of the
centre of gravity: A point through gavity can be considered to be acting, instead of individually on the point masses or acting on the body as a whole. It is a different concept from the centre of
mass, although due to the small differences in gravitational strength in most context, they can be considered close approximates of each other.
centre of mass: The point (not necessarily within the object) which is the weighted average of the point masses of a body (or the set of infinitely many point masses through integration), which can
be used to calculate linear motion of a rigid body as if all of the object's mass are at that point (the centre of mass) only. (Considering the object as a particle.) It is also known as the
centre of rotation: The invariant point in a rotation.
centripetal component: A component of an object's acceleration cooresponding to a centripetal force.
centripetal force: A force perpendicular to the velocity of an object which causes the object to travel on a curved (not straight) path.
centroid: The geometric centre of a plane figure.
cevian: Any line segment joining a vertex of a triangle to a point on the infinite line containing the oppositeside.
c.g.s. units: A system of units based on centimetres, grams and seconds. (Instead of metres, kilograms and hours for the SI units.) It has been largely superseded by the use of SI units nowadays.
chain: A set of elements and a total order.
chain complex: A sequence of (integer)-indexed Abelian groups together with a set of homomorphisms between adjacent groups satisfying certain conditions.
chain rule: A rule that finds the derivative of a composite function, given that the derivatives of the two components of the composite function can be found.
change of variable: A method for integrating a function with respect to a variable other than the one original given in the integral.
characteristic equation: Another class="d-title" name for the characteristic polynomial.
characteristic function: Another class="d-title" name for the indicator function. A function which returns a value of 1 if the argument is a member of the set specified and 0 otherwise.
characteristic polynomial: det(A-kI) = 0 , a polynomial derived from a swuare matrix A whose solutions are the eigenvalues of the transformation represented by the matrix.
Chebyshev: Russian mathematician known for his work in probability and statistics. Also transcribed as Tchebyshev.
Chinese postman problem: The problem of finding the least weighted circuit in a connected graph. If an Eulerian cycle exists than such a cycle is the solution to the problem, otherwise, it is
necessary to repeat at least one edge and the problem can be more complicated.
chi-squared distribution: A distribution on the value of the sum of squares of n standard normally distributed random variables.
chi-squared test: A test on the goodness of fit of an observation to the theoretical value/assumed distribution through the use of the chi-squared distribution to test its likelihood of deviation due
to natural variations.
chord: A line segment joining one point of the circumference of a circle to another. If the chord also goes through the centre than it is a diameter.
circle: The locus of all points which is equidistant from a fixed point on a plane.
circle of convergence: The set of points which is of a fixed distance from a fixed point (thus forming a circle) on an argand diagram which are the the set of arguments which converges in an infinite
circle of curvature: A circle which describes correctly the curvature of a curve for the neighbourhood of a point on the curve.
circuit: A closed path on a graph.
circular: Of or related to a circle.
circular data: The class of cyclic directional data in statistics.
circular functions: Another class="d-title" name for the trigonometric functions. Trigonometric functions are based on parametrization of a circle just as hyperbolic functions are based on
parametrization of hyperbolae. This usage highlights the analogue between the two.
circular helix: A helix with constant radius (from an axis).
circular measure: Also known as angular measure.
circular motion: The motion of an object whose path forms a circle.
circumcentre: The centre of a circle which goes through all vertices of a polygon.
circumcircle: A circle whose circumference contains all points of the polygon.
circumference: The length of the closed curve of a circle.
circumscribed: The act of enclosing a geometric figure with a (minimal) circle or sphere.
cis: A commonly defined function in the study of complex numbers. cis(θ)=cos(θ)+isin(θ) Incidentally, cis(θ)=exp(iθ). As such, cis is seldom used apart from a particular stage of learning in school
as most favour the use of the simpler exp(iθ).
Clairaut's equation: A family of differential equations of the form
which can be solved by differentiating the whole equation.
class: A collection of objects, not necessary a set. The distinction generally is that certain operations on sets are not allowed so as to allow us to refer to a (usually large) collection of objects
without causing contradictions such as Russell's paradox.
class frequency: The number of occurrence in a class.
classical mechanics: A loosely defined term generally considered to be the study of motion of objects before the drastically different quantum mechanics. Thus classical mechanics include Newtonian,
Lagrangian, Hamiltonian and arguably, relativistic mechanics.
class intervals: The intervals in which data fall into a particular class.
closed curve: A curve with no endpoints, generally used to refer to non-self-intersecting curves.
closed interval: An interval of real numbers containing its endpoints. More generally, it is a bounded set of points with a total order which is identical to its closure.
closed region: A set of points containing all limit points and is connected.
CM: An abbreviation for the centre of mass of an object.
coaxial: The property of having the same axis.
codomain: The codomain of a function is a superset of the image of the function. It is sometimes known as the range of the function. The codomain serves as a constraint as to what values a function
can take, thus there can be elements of the codomain which is not the value of the function but not vice versa.
coefficient: The quantity with which we multiply the variable in question.
coefficient matrix: A matrix formed from a system of linear equations using the coefficients of the equations only (omitting the variables). Also known as an augmented matrix
coefficient of concordance: A measure of concordance. With m orderings and n objects, let an object i be given a rank of ri,j by the ordering j, then object i's total rank can be calculated by:
which has a value between zero and 1 and a high value indicates high level of agreeement.
coefficient of correlation: A class="d-title" name for several related methods which measure the relationship between two sets of data.
coefficient of friction: A dimensionless quantity, the ratio of the friction force to the normal reaction as determined by a number of other factors. (Such the the type of surfaces in contact.)
coefficient of kurtosis: A measure of how much the data concentrates around the mean. Informally, it measured the "pointiness"/"peakedness" of the distribution curve.
coefficient of determination:A measure of how much of the variation in the data can be accounted for by the statistical model, for the purpose of inferring the likely level of determination of
coefficient of restitution: A dimensionless quantity, the ratio of the speed of separation to the speed of approach of 2 objects as determined by other factors. (e.g. the material the objects consist
of and their arrangement.)
coefficient of skewness: A measure of the asymetry of a distribution.
coefficient of variation: A measure of dispersion (standard deviation) normalised by the mean. (The absolute value of the mean is taken for a real variable to prevent the coefficient from being
cofactor: For a square matrix, augmenting the minors with a sign (positive or negative) in a "chequered" fashion forms the cofactors.
Formally, the cofactors are
where M[ij] are the minors of the matrix.
cofunction identities: A set of trigonometric identites which relates trigonometric functions to their cofunctions, e.g. sin and cos, sec and csc, tan and cot.
coincident: The property of 2 geometric figures to have all points in common.
colatitude: The angle between the vector and the polar axis (z-axis) in a Spherical polar coordinate system, whereas latitude is the value subtracting colatitude from 90 degrees, the smallest angle
between the radius vector and the plane perpendicular to the polaw axis through the origin. (the equator)
collision: The interaction of 2 objects with each other through contact transitioned from a state of non-contact prior.
column: A vertical array of elements having all but one indices in common.
column rank: The number of linearly independent columns (considered as column vectors) in a matrix. See rank.
column vector: A matrix representation of vectors using a matrix with one column. (an nx1 matrix)
combination: The way of ways that a collection of k objects can be picked from a number of n objects. (If the objects are considered to be picked the same way at the same time so that there is no way
to distinguish the ways an object is picked.)
See permutation.
combinatorics: The mathematical study of counting and countable structures.
common denominator: The common multiple of the denominators of a number of fractions. A common way to add and subtract fractions is to convert each fraction into the equivalent fraction with the
common denominator.
common difference: The constant which is the difference between each term and the next in an arithmetic sequence/series.
common factor: Also known as a common divisor. It is an integer which completely divides the two or more numbers in question.
common logarithm: Given a number, the common logarithm is the value such that if 10 is raised to such a value, gives exactly the number in question.
common multiple: An integer that is simultaneously a multiple of two or more numbers.
common ratio: The constant which is the ratio of one term to the next in a geometric sequence/series.
common tangent: A line which is the tangent of points on 2 distinct curves.
commutative: The property of a binary operation such that its operands can always be swapped around without affect its value.
compass: Also known as a pair of compasses, to avoid confusion with the instrument for telling directions. A pair of compasses is an instrument with rigid (but movable) arms for drawing circular
compatible matrices: Two matrices in a particular order so that they can be multiplied. In the usual convention, the number of columns in the first matrix and the number of rows in the second matrix
must be the same. In more abstract terms, considering the matrices as transformations, it amounts to maintaining that the number of dimensions (rank) of the codomain of the first transformation be
the same as the dimensions (rank) of the domain of the second transformation.
complement: The complement of a set is the set of elements not in the specified set. Note that this definition only makes sense with a universal set defined.
complementary angles: Two angles that sum to a right angle. In this case, each angle is called the complement of the other angle.
complementary function: Along with the particular integral, it forms the general solution of a linear differential equation. It is essentially an element in the kernel of the differential operator.
complete induction: Also known as strong induction. This method assumes that the statement is true for all values below a certain finite value in the inductive step of proving the next statement.
Logically strong induction is equivalent to weak induction and is not "stronger" in this logical sense.
complete lattice: A poset where all subsets have a supremum and an infimum.
completing the square: A method for solving quadratic equations in general by writing a quadratic expression in the vertex form. Note that the quadratic formula is derived from the method of
completing the square for the general quadratic expression.
complex analysis: The study of complex variabled functions.
complex conjugate: Given a complex number, the complex conjugate is the complex number whose real part is the same, while the imaginary part (being a real number) has opposite signs. The significance
of complex conjugates stems from the theorem that says the complex conjugates of all roots of real polynomials are also roots themselves.
complex fraction: A fraction consisting of complex numbers. Considering the division of complex numbers as complex fractions is a standard way of calculating the division. (Through algebraic methods
such as the difference of two squares.)
complex function: A function involving complex numbers. Note that, while it is true that all real variabled functions are complex functions, certain results (such as convergence) differs depending on
which kind we consider the function to be.
complex number: The set of numbers which is algebraically complete with respect to finitely many additions, multiplications, exponentiation and their inverse operations. It is a superset of the real
complex plane: The collection of complex numbers represented on a plane. It is also known as an argand diagram.
component (of a vector): Individual parts of a vector. (e.g. each element in a column/row vector represent a component of the vector.)
component analysis: The study of a set of data by an isometric transformation and subsequently dividing them into orthogonal components.
composite function: The result of composing two functions together, so that the value (output) of the first becomes the argument (input) of the second. The argument of the first function is
considered the argument of the composite function and it is the value of the second function that is considered the value of the composite function.
composite hypothesis: As opposed to a simple hypothesis. In hypothesis testing, a composite hypothesis is a hypothesis where the value of some parameters of the distribution is not specified to be a
certain value.
composite number: A positive integer that is not a prime number. Note that 1 is conventionally considered neither prime nor composite - in that sense, a composite is a positive integer with more than
2 factors, while prime has exactly 2. The number 1, having only one factor, is thus considered neither.
composition: The act of combining 2 mathematical objects in some way. (e.g. of functions, transformations or geometric figures.)
compound distribution: The compound probability distribution is the result of a probability distribution whose parameters are distributed along other probability distributions.
compound fraction: A fraction that consists of another fractions as its numerator or denominator.
compound interest: The calculation of interest payments taking into account of previous interest payments as part of the principal.
compression: A class="d-title" name for a transformation where a figure becomes proportionaly smaller.
concave: A geometric figure where it is possible to form a line between 2 points in the figure where the line consists of points not from the figure. For a plane figure, it is equivalent to a shape
having an interior angle of greater than 180 degrees.
concave function: A function whose graph is such that, for any two points of the graph, the function for arguments between the 2 points are higher than the straight line joining the 2 points. For a
differentiable function, it is equivalent to a function with a monotonically decreasingly gradient.
Note that a function can be described as concave for a certain interval only.
concave polygon: A polygon, as a plane figure, which is concave.
concentric: The property of having the same centre. Usually describes circles and spheres.
conchoid: The family of plane curves (half of) whose shape resemble thre conch shells.
concordance: A measure of agreement of pairwise bijective set of orderings on a finite number of objects.
concurrent: The property of sharing a common point.
concyclic: Having the property of being on the circumference of a circle.
conditional: A mathematical sentence that describes the implication of one from another.
conditional distribution: The distribution of a random variable given that another random variable is known to be of a certain value.
conditional equation: An equation which is only true under certain contexts. Normally known simply as equations, as opposed to identities, which is true regardless of contexts.
conditional inequality: An inequality that is only true under certain conditions.
conditionally convergent series: A series that ceases to be convergent if the modulus function is applied to all of its terms before the summation.
conditional probability: The probability of an event given the (non-)occurrence of other events.
cone: A 3-dimensional geometric figure consisting of all points on line segments that joins any points in a circle (including the interior) to another point not lying on the same plane as the circle.
confidence interval: An interval given as the estimate for a parameter, based on the theoretic value of the parameter given known information, while taking into account of the probability we require
the actual parameter to be within the given interval.
confidence level: The probability with which a parameter should fall within the given confidence interval (Since we can never be certain due to natural variations in our observations.)
confidence limits: The endpoints of a confidence interval.
confidence region: Same as confidence interval.
configuration: A particular arrangement of mathematical objects.
confocal conics: Conic sections which share the same focus (or foci).
conformable matrices: See compatible matrices.
confounding: A certain property of an extraneous variable - the property of the so-called lurking variable.
congruence: 1. A property of similarity/sameness for 2 geometric figures. See congruent.
2. Two quantities considered the same under modulo arithmetics. See congruence modulo n.
congruence modulo n: Arithmetics where 2 quantities differing by a multiple of the chosen base n are considered the same. The two quantities are said to be congruent modulo n. An example involves the
calculation of days of the week, where 5 (Friday) + 4 (2 days later) = 9 (which is congruent modulo 7 to 2 - i.e. Tuesday)
congruent: 1. One figure that would coincide with another with a combination of translation, rotation and reflection. Essentially, 2 shapes are congruent if they can be considered "the same" except
for its location and orientation.
For triangles, there are certain conditions that makes it easier to decide if two such figures are the same. (Without having to consider the transformations.)
I - SSS - The length of all three sides of a triangle are the same as the lengths of the corresponding sides of another.
II - SAS - The lengths of any two pairs of sides correspond, and the angles between those sides are also the same as each other.
III - ASA/AAS - Any pair of angles on one shape correspond with the other shape, while any side of either shape is the same as the corresponding side of the other.
IV - RHS (also known as LH) - Both triangles are right-angled, their hypotenuses are of the same length and they share another side of the same length.
2. See congruence modulo n.
congruent matrices: Two matrices A and B are congruent if A = P^TBP for some invertible matrix P.
conic: Also called a conic section, it is a plane curve which is the intersection of a plane with an infinite right double cone- i.e. a circle, an ellipse, a parabola and a hyperbola. (The
intersections can also be a single point, a line or 2 non-parallel lines, which can respectively be considered special cases of a circle/ellipse, a parabola and a hyperbola respectively.)
There exist alternative definitions of conics involving a focus and a directrix together with a ratio of distance from the two called eccentricity.
conical pendulum: A weight attached through a string to a fixed point so that the trajectory of the weight is a (horizontal) circle with the string being taut (and having constant length) at all
times. The circle drawn out by the weight together with the positions of the string forms a cone which explains the class="d-title" name.
conicoid: A surface generated by the rotation about an axis of one of the conics - ellipsoids, paraboloids, hypocycloids and a sphere. Also known as conoids.
conic section: Also known more simply as conics.
conjecture: An assertion that is not yet proven. In this sense, it is the same as an hypothesis.
conjugate arcs: Given an arc of a circle, the remaining section (also an arc) of the circle is the conjugate arc.
conjugate axis: The line of symmetry of a hyperbola which does not intersect the curve itself. Or alternatively, a segment of this line equal in length to the distance between the intersections
between the tangent at a vertex and the two asymptotes.
conjugate complex numbers: Also known as complex conjugates.
conjugate gradient method: An iterative algorithm for the numerical solutions of certain systems of linear equations.
conjugate lines: Given a line on an argand diagram, the line formed by replace each point with its complex conjugate in known as the conjugate line. It is equivalent to a reflection of a line along
the real axis of the Argand diagram.
conjugate pair theorem: A theorem which states that the complex conjugates of roots of a polynomial with real coefficients are themselves also roots of the polynomial.
conjugate points (of a conic): A pair of points on a conic which lie on the polar of the other point.
conjugate prior distribution: In Bayesian statistics, the distribution of the prior probability when the prior and the posterior distributions belong to the same family.
conjugate set: The subset of a group which are all conjugates of each other. It is also known as a conjugacy class. Note that a conjugate set is not a group since the identity must be in its own
conjugate set.
conjunct: A logical operator that returns the value true if and only if both its operands are true. It essentially captures the core meaning of the connective "and" that conbines two sentences into
one. Also known as a conjunction. The word conjunct has to it slight connotation towards natural languages or meta-language in the description of the (formal) logical conterpart.
conjunction: A logical operator that returns the value true if and only if both its operands are true. It essentially captures the core meaning of the connective "and" that conbines two sentences
into one.
connected graph: A graph is connected if there is at least one path between any two vertices of the graph.
connected set: A set is connected if it cannot be partitioned into non-empty subsets such that each subset is disjoint from the closure of every other subset.
connective: An operator that conbines 2 sentences into a new sentence such that the truth value of the new sentence is a function of the truth values of the 2 sentences.
conoid: Also known as a conicoid.
consecutive angles: A set of angles where every member can be considered to be "next" to another angle within the set. This idea of "next" or adjacency can be sharing a side or sharing a side as well
as a point depending on the context.
consecutive sides: A set of sides (edges) where every memeber can be considered to be "next" to another side within the set. This idea of "next" or adjacency can be sharing a vertex or sharing an
angle depending on the context.
consequent: In the conventional way of expressing hypothetical propositions, "If A then B", the consequent is the second part of the sentence. It is the part of the sentence whose truth value is
dependent on the other part. (Within the context of the statement itself.)
conservation laws: Any theorem or assertion that states that certain measurable quantities remains unchanged, and the condition under which this happens.
conservation of energy: The law that states that the amount of energy in a closed system remains unchanged. The exactness of this law, in our current understanding, depends on whether we use mass
-energy as the measure or simply energy in the classical sense.
conservation of momentum: The law that states that the total momentum of a closed system of objects remains unchanged.
consistent: The property of admitting possible solutions for a system of equations.
consistent estimator: A sequence of estimators that is convergent.
constant: A quantity whose value stays constant. As opposed to a variable whose value is variable.
constant of integration: The result of apply in reverse, in integration, the fact that any constants differentiate to zero. Thus, any integrals, which can also be considered as the same integral
"plus zero", integrates to an expression that always includes an arbitrary constant. Note that an arbitrary constant is not necessary in definite integration simply because whatever arbitrary value
it takes, it remains a constant and the process of subtracting the value calculated with the lower bound from the value calculated with the upper bound would ensure that the effects of the arbitrary
constant "cancel each other out".
constant of proportionality: The constant ratio between the change in 2 related variables said to be proportional (directly or partially) to each other.
constant term: The term in a polynomial which does not include any variables. (Note: that is not to say that the term cannot contain any unknowns.)
construction: The study of geometric figures which can be constructed using an idealised version of real world compasses and rulers (straight edge).
contact, point of: The point on a curve for which a tangent is considered to be parallel to and intersect the curve.
contingence, angle of: The angle between the directed tangents of 2 points on a plane curve.
contingency table: An array of frequencies (or relative frequencies and by extension, probabilities) recorded to study the relationship between 2 or more discrete variables.
continued fraction: A compound fraction where either the numerator or the denominator is also a continued fraction.
continued product: A product of more than 2 factors, usually analoguous to a sigma notation. Using this representation, it can be a product of infinitely many factors.
continuity correction: The adjustment of the argument of a discrete distribution to form a closer approximation to a continuous argument. (Usually an adjustment of 0.5 for discrete distributions
which admits integers.)
continuous distribution: A distribution where the domain of the cumulative distribution function is continuous.
continuous function A function for which a small change for every argument results in a small change in the value such that in this way, there is no limit as to how small the change in value can be.
continuous mapping A generalised analogue of a continuous function where an argument may have more than one value. (And the definition of continuous is then modified accordingly such that, a small
change CAN result in a small change in the value without limits as to how small such a change may be.)
continuous random variable: A random variable whose image is continuous.
contour integral: An integral along a contour on the complex plane/argand diagram which may be parametrized.
contour plot: A type of plane graphical representation of a function with 2 arguments where sets of points of selected values form non-intersecting closed curves, known as contours.
contractible: The property of being capable of being shrunk to a point continuously.
contrapositive: Given a statement of the form "If A then B", the contrapositive of such a statement is the one which states "If not B then not A". A statement and its contrapositive are always
equivalent, so that they share the same truth value.
convergence: The property of a mathematical object, considered as a sequence, is convergent.
convergent integral: A improper integral whose limit exists and is finite.
convergent iteration: An iterative process for which the sequence of each state (in order) is a convergent sequence
convergent product: A continued product whose sequence of partial products (analogues to partial sums) converges.
convergents: See continued fraction.
convergent sequence: Informally, a sequence is convergent if the terms gets arbitrarily close to a point (called the limit) so that given a proximity (that the sequence should get to), a sequence
eventually gets to within such proximity without leaving it.
convergent series: A series is covergent if and only if the sequence of its partial sums is covergent.
converse: Given a statement of the form "If A then B", the converse of this statement is the one which states "If B then A". Note that the converse is not equivalent to the original statement. In
fact, it is a common logical fallacy to assume that they are.
conversion period: 1. A time interval during which an investor may exchange a convertible security for another financial asset.
convex function: A function whose graph is such that, given any 2 pointst on the curve, the value of any point with argument between the 2 points lie below the straight line joining the 2 points.
Note that a function can be described as convex for a certain interval only.
convex polygon: A polygon where any line segments connecting any 2 points within the polygon is also entirely within the polygon. Alternatively, it is a polygon whose angles are all 180° or less.
convex polyhedron: A 3-dimensional geometric figure where any line segments joining any 2 points are entirely within the figure.
convex set: A set of points where, given any 2 points in the set, the line segment joining the 2 points consists of points entirely from the set itself.
coordinate: One within a set of such numbers, called coordinates, which specifies the position of a point. See abscissa and ordinate.
coordinate geometry A study of geometric objects through their representation in a coordinate system.
coplanar: The property of being contained in the same plane.
coprime: The property of 2 numbers not sharing any prime factors.
corollary: A theorem which is immediate from another proven statement. That is, a theorem that requires minimally prove, or a simple observation. As such, the difference between a corollary and a
theorem is not rigorously defined but rather subjective in its use.
correlation: The amount of interdependence in statistical relationship amongst two or more quantities.
correlation coefficient: See coefficient of correlation.
cosine rule: In trigonometry,(also known as the cosine formula oe the law of cosines ) it is a statement about a general triangle that relates the lengths of its sides to the cosine of one of its
countable: A set which can be put into a one-to-one correspondence with a subset of the natural numbers.
countably infinite: The property of a set being infinite and countable. For example, the set of natural numbers is countably infinite while the set of real numbers is not.
counting numbers: Also known as the natural numbers. Just as natural numbers, zero is variously included and excluded from the set of Counting/Natural numbers. Thus, it may be more precise to use the
terms "positive integers" (not including zero) and "non-negative integers" (including zero) instead.
Cramer’s rule: A theorem which expresses the solutions to a system of linear equations by the determinants of matrices formed from the coefficients in the equations.
critical value: Generally speaking, a value either side of which exhibits dramatically different behaviour. Sometimes the value itself exhibits one of the 2 types of behaviour, thus making an
adjacent point (if only discrete values are admitted) fitting the description also. In this case, convention usually governs which is considered to be the critical value. (As is the case in
hypothesis testing, that the "first" value which rejects H[0], rather than the "last" value which does not, is considered the critical value.)
cross product: More formally called the vector product - a way of multiplying 2 vectors and produces another vector perpendicular in direction to the "multiplicand" and the "multiplier". So called
because we use the symbol x to denote this multiplication, as in axb. (While we use a dot, such as a.b, to denote another type of multiplication of 2 vectors - scalar multiplication/dot product.
Note that unlike the dot product, the cross product is only defined for 3-dimensional vectors. In 2-dimensions, there can be no perpendicular vectors to 2 vectors in general. (Except for the trivial
case of 2 parallel vectors.) While in 4 dimensions or more, there are infinitely many vectors of a certain length perpendicular to any 2 vectors.
Thus, the vector product of 2 vectors is only defined for 2 3-dimensional vectors which are not parallel to each other.
It is called a multiplication by virtue of some of its properties being similar to multiplication in general, such as distributivity over addition and subtraction, yet, one must be careful in
checking the specific properties that are the same. For example, unlike most other operations going by the class="d-title" name of multiplication (but similar to matrix multiplication), vector
product is not commutative., That is, axb is never the same as bxa (except for the case when one of the vectors is the zero vector).
cube: A 3-dimensional geometric figure of 6 congruent faces (all squares) where the edges (all 12 of them) have the same length. It is the special case of a cuboid where all lengths are the same. A
cube is a 3-dimensional analogue of a square just as a cuboid is a 3-dimensional analogue of a rectangle.
cube root: Given a number, the cube root of this number is another number such that the former is the cube number of the latter.
cubic polynomial: A polynomial of degree 3, that is, the highest power that the variable is raised to, amongst all terms, is 3.
cuboid: A 3-dimensional analogue of a rectangle, where all faces of a cuboid must be rectangles. A cube can be considered as a special case of a cuboid.
curve: A generally loosely defined term that describes a collection of points which form a 1-dimensional object. In specific discipline or context, the word tends to be more rigorously defined. Note
that a curve can be self-intersecting, or very complicated (space-filling curve) but it must usually maintain some sense of local linearity to it to be considered a curve. (i.e. it should look like a
straight line"under a microscope".)
cusp: A sudden turn in direction in a curve.
cycle: A series of events that are regularly repeated in the same order.
cycloid: The curve traced out by a fixed point on a circle (considered as a wheel) rolling along a line (considered as the ground) without slipping.
cyclinder: A right circularprism. | {"url":"https://itseducation.asia/mathematics/c.htm","timestamp":"2024-11-09T16:17:56Z","content_type":"application/xhtml+xml","content_length":"365939","record_id":"<urn:uuid:25844665-5067-461d-abfc-355c23fdaa97>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00350.warc.gz"} |
Programme Of Study For Year 6 Mathematics
Sign In | Starter Of The Day | Tablesmaster | Fun Maths | Maths Map | Topics | More
These are the statements, each one preceeded with the words "Pupils should be taught to:"
Click on a statement above for suggested resources and activities from Transum. | {"url":"https://transum.org/Maths/National_Curriculum/Statements.asp?ID_Domain=28","timestamp":"2024-11-14T17:42:44Z","content_type":"text/html","content_length":"14342","record_id":"<urn:uuid:ba8f6b17-5354-4b1f-8528-1325fa3043f7>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00530.warc.gz"} |
About The Word "X" - Wordutopia
About The Word “X”
Everything you wanted to know about the word “x”, including spelling, parts of speech, “x” meaning and origins, anagrams, rhyming words, encodings, crossword clues and much more!
How to spell “x”
X is spelled x and has 1 letter.
How many vowels and consonants in “x”
The word “x” has 1 consonants and 0 vowels.
How many syllables in “x”?
There is 1 syllable in the word “x”.
What type of word is “x”?
The word "x" can be a conjunction.
Meaning of the word “x”
The word 'x' typically represents an unknown variable or factor in mathematical equations, symbolizing a value that needs to be discovered or solved for.
Origin of the word “x”
The word 'x' originates from the Latin letter 'X,' which was derived from the Greek letter 'Chi.' This Greek letter, in turn, was borrowed from the Phoenician letter 'Samekh,' representing the
voiceless velar fricative /x/.
Example sentences with the word “x”
1. Solve for x in the equation 2x + 3 =
1. When you find the value of x, you can determine the length of the missing side of the triangle.
1. The graph of the equation y = 3x + 2 represents a straight line with a slope of
1. In the quadratic equation x² - 5x + 6 = 0, we need to find the values of x that make the equation true.
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Words that rhyme with “x”
Box, fox, locks, mocks, rocks, socks, talks, blocks, clocks, docks
Crossword clues for “x”
Extreme unknown variable (1).
Fun facts about the word “x”
The word “x” has a Scrabble score of 8 and reads x in reverse.
Phonetic spelling of “x”
The phonetic alphabet, specifically the International Phonetic Alphabet (IPA), is a system of notation for the sounds of languages created by linguists. Unlike conventional written alphabets, which
vary across languages and can have inconsistent mappings of symbols to sounds, the IPA is designed to provide a consistent and universally understood means of transcribing the sounds of any spoken
“x” spelled in Morse code
-..- (dash dot dot dash).
Morse code is a method used in telecommunication to encode text characters as sequences of two different signal durations, called dots and dashes, or dits and dahs. It was developed in the 1830s and
1840s by Samuel Morse and Alfred Vail for their new invention, the telegraph, which required a simple way to transmit text messages across long distances.
ASCII spelling of “x”
Lowercase word: 120
Uppercase word: 88
ASCII, which stands for American Standard Code for Information Interchange, is a character encoding standard used by computers and electronic devices to understand and represent text.
Binary spelling of “x”
Lowercase word: 1111000
Uppercase word: 1011000
Binary encoding is a system that computers and digital devices use to represent and process information. It's based on binary numbers, which are composed only of zeros and ones, known as bits.
Hexadecimal value of “x”
Lowercase hexadecimal word: 0x78
Uppercase hexadecimal word: 0x58
Hexadecimal is a number system commonly used in computing as a human-friendly way of representing binary data. Unlike the decimal system, which is base 10 and uses digits from 0 to 9, the hexadecimal
system is base 16, using digits from 0 to 9 and letters from A to F to represent the values 10 to 15.
Decimal spelling of “x”
Lowercase: 120
Upprcase: 88
The decimal system, also known as base-10, is the numerical system most commonly used by people in everyday life. It's called "base-10" because it uses ten digits: 0 through 9. Each position in a
decimal number represents a power of 10.
Octal value of “x”
Lowercase: 170
Upprcase: 130
Octal is a base-8 number system used in digital computing. Unlike the decimal system which uses ten digits (0-9), and the binary system which uses two (0 and 1), the octal system uses eight digits: 0
through 7. Each position in an octal number represents a power of 8.
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We study the problem of mass transport to surfaces with heterogeneous reaction rates in the presence of shear flow over these surfaces. The reactions are first order and the heterogeneity is due to
the existence of inert regions on the surfaces. Such problems are ubiquitous in the field of heterogeneous catalysis, electrochemistry and even biological mass transport. In these problems, the
microscale reaction rate is characterized by a Damköhler number $\unicode[STIX]{x1D705}$, while the Péclet number $P$ is the dimensionless ratio of the bulk shear rate to the inverse diffusion time
scale. The area fraction of the reactive region is denoted by $\unicode[STIX]{x1D719}$. The objective is to calculate the yield of reaction, which is directly related to the mass flux to the reactive
region, denoted by the dimensionless Sherwood number $S$. Previously, we used boundary element simulations and examined the case of first-order reactive disks embedded in an inert surface (Shah &
Shaqfeh J. Fluid Mech., vol. 782, 2015, pp. 260–299). Various correlations for the Sherwood number as a function of $(\unicode[STIX]{x1D705},P,\unicode[STIX]{x1D719})$ were obtained. In particular,
we demonstrated that the ‘method of additive resistances’ provides a good approximation for the Sherwood number for a wide range of values of $(\unicode[STIX]{x1D705},P)$ for $0<\unicode[STIX]
{x1D719}<0.78$. When $\unicode[STIX]{x1D719}\approx 0.78$, the reactive disks are in a close packed configuration where the inert regions are essentially disconnected from each other. In this work,
we develop an understanding of the physics when $\unicode[STIX]{x1D719}>0.78$, by examining the inverse problem of inert disks on a reactive surface. We show that the method of resistances approach
to obtain the Sherwood number fails in the limit as $\unicode[STIX]{x1D719}\rightarrow 1$, i.e. in the dilute limit of periodic inert disks, due to the existence of a surface concentration boundary
layer around each disk that scales with ($1/\unicode[STIX]{x1D705}$). This boundary layer controls the screening length between inert disks and allows us to introduce a new theory, thus providing new
correlations for the Sherwood number that are highly accurate in the limit of $\unicode[STIX]{x1D719}\rightarrow 1$. | {"url":"https://core-cms.prod.aop.cambridge.org/core/search?filters%5BauthorTerms%5D=Eric%20S.%20G.%20Shaqfeh&eventCode=SE-AU","timestamp":"2024-11-06T16:06:53Z","content_type":"text/html","content_length":"1041155","record_id":"<urn:uuid:dee56131-24e6-442e-a91b-04aa792e12a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00785.warc.gz"} |
TR05-052 | 5th May 2005 00:00
The Computational Complexity of Nash Equilibria in Concisely Represented Games
Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional
method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows.
We study two models of concisely represented games: *circuit games*, where the payoffs are computed by a given boolean circuit, and *graph games*, where each agent's payoff is a function of only the
strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash
equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we
obtain tight results, showing that the problems are complete for various complexity classes. | {"url":"https://eccc.weizmann.ac.il/report/2005/052/","timestamp":"2024-11-14T22:12:16Z","content_type":"application/xhtml+xml","content_length":"21075","record_id":"<urn:uuid:25ee82d2-5244-451e-a9a4-31ce656f5b02>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00726.warc.gz"} |
ORMC Entry Points
UCLA Olga Radko Endowed Math Circle (ORMC) has two programs, a regular program and a Summer one. The Summer program is not open for the regular program students. It is meant to give children outside
of our regular ranks a taste of the Math Circle culture.
BNP Classes (K-1)
The first entry point to our regular program is our BNP class for students in grades K-1. We usually have two classes, BNP A and B, up to 20 students in each. The class is based on a book, called
Breaking Numbers into Parts, hence the name. The book has two parts, both available on Amazon: Part 1, Part 2.
We start BNP classes every two years because the book takes a bit more than a year to cover in the ORMC format, nine 50-min-long classes per quarter, three quarters in a school year.
Admissions are based on results of an assessment test, given at the beginning of September. The best way to prepare for the test is to solve Math Kangaroo competition problems for grades 1-2 from the
years past.
Beginners 1 Classes (grades 2-4)
The second entry point to the regular program is the Beginners 1 class for children in grades 2-4. BNP students in good standing progress to the Beginners 1 level automatically. We also expand the A
and B classes to 25 students each. This is where new students can enter the program.
Our program for grades 2-4 is based on the book Math Adventures with ORMC, Level 1, from Optical Illusions to Fighting Dragons. Both a Workbook for students and its Teacher's Companion are available
on Amazon.
If Spots Open Up
Past the Beginners 1 level, we only get an opening in the regular program if someone drops out. Please note that a student can drop out at the BNP and Beginners 1 level as well.
In the case of such an opening, preference is given to applicants placed high on national math competitions:
Summer Program
The Summer program is independent of the regular program. We try to enroll as many applicants to our Summer program as we possibly can. The limiting factor is most often the number of instructors
available in the Summer. If the number of applicants to a level exceeds the level's capacity, we give applicants an assessment test. Enrollment is then based on the test's results.
Having an opening in the regular program, we can invite a top Summer program performer to join.
Enrollment opening
Enrollment opens before the beginning of every quarter. For example, UCLA Fall quarter starts at the end of September. Therefore ORMC Fall quarter enrollment opens at the beginning of September and
closes when the Fall quarter begins. | {"url":"https://circles.math.ucla.edu/circles/entry_points.shtml?id=714&edit=true","timestamp":"2024-11-06T14:03:56Z","content_type":"text/html","content_length":"12098","record_id":"<urn:uuid:717dc365-caf0-4441-8569-62ac9179cb83>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00641.warc.gz"} |
Game semantics
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Game semantics (German: dialogische Logik, translated as dialogical logic) is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the
existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.
In the late 1950s Paul Lorenzen was the first to introduce a game semantics for logic, and it was further developed by Kuno Lorenz. At almost the same time as Lorenzen, Jaakko Hintikka developed a
model-theoretical approach known in the literature as GTS. Since then, a number of different game semantics have been studied in logic.
Shahid Rahman (Lille) and collaborators developed dialogic into a general framework for the study of logical and philosophical issues related to logical pluralism. Beginning 1994 this triggered a
kind of renaissance with lasting consequences. This new philosophical impulse experienced a parallel renewal in the fields of theoretical computer science, computational linguistics, artificial
intelligence and the formal semantics of programming languages, for instance the work of Johan van Benthem and collaborators in Amsterdam who looked thoroughly at the interface between logic and
games, and Hanno Nickau who addressed the full abstraction problem in programming languages by means of games. New results in linear logic by J-Y. Girard in the interfaces between mathematical game
theory and logic on one hand and argumentation theory and logic on the other hand resulted in the work of many others, including S. Abramsky, J. van Benthem, A. Blass, D. Gabbay, M. Hyland, W. Hodges
, R. Jagadeesan, G. Japaridze, E. Krabbe, L. Ong, H. Prakken, G. Sandu D. Walton, and J. Woods who placed game semantics at the center of a new concept in logic in which logic is understood as a
dynamic instrument of inference.
Classical logic[edit]
The simplest application of game semantics is to propositional logic. Each formula of this language is interpreted as a game between two players, known as the "Verifier" and the "Falsifier". The
Verifier is given "ownership" of all the disjunctions in the formula, and the Falsifier is likewise given ownership of all the conjunctions. Each move of the game consists of allowing the owner of
the dominant connective to pick one of its branches; play will then continue in that subformula, with whichever player controls its dominant connective making the next move. Play ends when a
primitive proposition has been so chosen by the two players; at this point the Verifier is deemed the winner if the resulting proposition is true, and the Falsifier is deemed the winner if it is
false. The original formula will be considered true precisely when the Verifier has a winning strategy, while it will be false whenever the Falsifier has the winning strategy.
If the formula contains negations or implications, other, more complicated, techniques may be used. For example, a negation should be true if the thing negated is false, so it must have the effect of
interchanging the roles of the two players.
More generally, game semantics may be applied to predicate logic; the new rules allow a dominant quantifier to be removed by its "owner" (the Verifier for existential quantifiers and the Falsifier
for universal quantifiers) and its bound variable replaced at all occurrences by an object of the owner's choosing, drawn from the domain of quantification. Note that a single counterexample
falsifies a universally quantified statement, and a single example suffices to verify an existentially quantified one. Assuming the axiom of choice, the game-theoretical semantics for classical
first-order logic agree with the usual model-based (Tarskian) semantics. For classical first-order logic the winning strategy for the verifier essentially consists of finding adequate Skolem
functions and witnesses. For example, if S denotes ${\displaystyle \forall x\exists y\,\phi (x,y)}$ then an equisatisfiable statement for S is ${\displaystyle \exists f\forall x\,\phi (x,f(x))}$. The
Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make.^[1]
The above definition was first formulated by Jaakko Hintikka as part of his GTS interpretation. The original version of game semantics for classical (and intuitionistic) logic due to Paul Lorenzen
and Kuno Lorenz was not defined in terms of models but of winning strategies over formal dialogues (P. Lorenzen, K. Lorenz 1978, S. Rahman and L. Keiff 2005). Shahid Rahman and Tero Tulenheimo
developed an algorithm to convert GTS-winning strategies for classical logic into the dialogical winning strategies and vice versa.
For most common logics, including the ones above, the games that arise from them have perfect information—that is, the two players always know the truth values of each primitive, and are aware of all
preceding moves in the game. However, with the advent of game semantics, logics, such as the independence-friendly logic of Hintikka and Sandu, with a natural semantics in terms of games of imperfect
information have been proposed.
Intuitionistic logic, denotational semantics, linear logic, logical pluralism[edit]
The primary motivation for Lorenzen and Kuno Lorenz was to find a game-theoretic (their term was "dialogical" Dialogische Logik) semantics for intuitionistic logic. Andreas Blass^[2] was the first to
point out connections between game semantics and linear logic. This line was further developed by Samson Abramsky, Radhakrishnan Jagadeesan, Pasquale Malacaria and independently Martin Hyland and
Luke Ong, who placed special emphasis on compositionality, i.e. the definition of strategies inductively on the syntax. Using game semantics, the authors mentioned above have solved the long-standing
problem of defining a fully abstract model for the programming language PCF. Consequently, game semantics has led to fully abstract semantic models for a variety of programming languages and, to new
semantic-directed methods of software verification by software model checking.
Shahid Rahman and Helge Rückert extended the dialogical approach to the study of several non-classical logics such as modal logic, relevance logic, free logic and connexive logic. Recently, Rahman
and collaborators developed the dialogical approach into a general framework aimed at the discussion of logical pluralism.^[3]
Foundational considerations of game semantics have been more emphasised by Jaakko Hintikka and Gabriel Sandu, especially for independence-friendly logic (IF logic, more recently information-friendly
logic), a logic with branching quantifiers. It was thought that the principle of compositionality fails for these logics, so that a Tarskian truth definition could not provide a suitable semantics.
To get around this problem, the quantifiers were given a game-theoretic meaning. Specifically, the approach is the same as in classical propositional logic, except that the players do not always have
perfect information about previous moves by the other player. Wilfrid Hodges has proposed a compositional semantics and proved it equivalent to game semantics for IF-logics.
Computability logic[edit]
Japaridze’s computability logic is a game-semantical approach to logic in an extreme sense, treating games as targets to be serviced by logic rather than as technical or foundational means for
studying or justifying logic. Its starting philosophical point is that logic is meant to be a universal, general-utility intellectual tool for ‘navigating the real world’ and, as such, it should be
construed semantically rather than syntactically, because it is semantics that serves as a bridge between real world and otherwise meaningless formal systems (syntax). Syntax is thus secondary,
interesting only as much as it services the underlying semantics. From this standpoint, Japaridze has repeatedly criticized the often followed practice of adjusting semantics to some already existing
target syntactic constructions, with Lorenzen’s approach to intuitionistic logic being an example. This line of thought then proceeds to argue that the semantics, in turn, should be a game semantics,
because games “offer the most comprehensive, coherent, natural, adequate and convenient mathematical models for the very essence of all ‘navigational’ activities of agents: their interactions with
the surrounding world”.^[4] Accordingly, the logic-building paradigm adopted by computability logic is to identify the most natural and basic operations on games, treat those operators as logical
operations, and then look for sound and complete axiomatizations of the sets of game-semantically valid formulas. On this path a host of familiar or unfamiliar logical operators have emerged in the
open-ended language of computability logic, with several sorts of negations, conjunctions, disjunctions, implications, quantifiers and modalities.
Games are played between two agents: a machine and its environment, where the machine is required to follow only effective strategies. This way, games are seen as interactive computational problems,
and the machine’s winning strategies for them as solutions to those problems. It has been established that computability logic is robust with respect to reasonable variations in the complexity of
allowed strategies, which can be brought down as low as logarithmic space and polynomial time (one does not imply the other in interactive computations) without affecting the logic. All this explains
the name “computability logic” and determines applicability in various areas of computer science. Classical logic, independence-friendly logic and certain extensions of linear and intuitionistic
logics turn out to be special fragments of computability logic, obtained merely by disallowing certain groups of operators or atoms.
See also[edit]
This article includes a
list of references
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its sources remain unclear
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insufficient inline citations
(May 2010) (Learn how and when to remove this template message)
• S. Abramsky and R.Jagadeesan, Games and full completeness for multiplicative linear logic. Journal of Symbolic Logic 59 (1994): 543-574.
• A. Blass, A game semantics for linear logic. Annals of Pure and Applied Logic 56 (1992): 151-166.
• D.R. Ghica, Applications of Game Semantics: From Program Analysis to Hardware Synthesis. 2009 24th Annual IEEE Symposium on Logic In Computer Science: 17-26. ISBN 978-0-7695-3746-7.
• G. Japaridze, Introduction to computability logic. Annals of Pure and Applied Logic 123 (2003): 1-99.
• G. Japaridze, In the beginning was game semantics. In Ondrej Majer, Ahti-Veikko Pietarinen and Tero Tulenheimo (editors), Games: Unifying logic, Language and Philosophy. Springer (2009).
• Krabbe, E. C. W., 2001. "Dialogue Foundations: Dialogue Logic Restituted [title has been misprinted as "...Revisited"]," Supplement to the Proceedings of the Aristotelian Society 75: 33-49.
• H. Nickau (1994). "Hereditarily Sequential Functionals". In A. Nerode; Yu.V. Matiyasevich. Proc. Symp. Logical Foundations of Computer Science: Logic at St. Petersburg. Lecture Notes in Computer
Science. 813. Springer-Verlag. pp. 253–264. doi:10.1007/3-540-58140-5_25.
• S. Rahman and L. Keiff, On how to be a dialogician. In Daniel Vanderken (ed.), Logic Thought and Action, Springer (2005), 359-408. ISBN 1-4020-2616-1.
• S. Rahman and T. Tulenheimo, From Games to Dialogues and Back: Towards a General Frame for Validity. In Ondrej Majer, Ahti-Veikko Pietarinen and Tero Tulenheimo (editors), Games: Unifying logic,
Language and Philosophy. Springer (2009).
• Johan van Benthem (2003). "Logic and Game Theory: Close Encounters of the Third Kind". In G. E. Mints; Reinhard Muskens. Games, logic, and constructive sets. CSLI Publications. ISBN
• T. Aho and A-V. Pietarinen (eds.) Truth and Games. Essays in honour of Gabriel Sandu. Societas Philosophica Fennica (2006).ISBN 951-9264-57-4.
• J. van Benthem, G. Heinzmann, M. Rebuschi and H. Visser (eds.) The Age of Alternative Logics. Springer (2006).ISBN 978-1-4020-5011-4.
• R. Inhetveen: Logik. Eine dialog-orientierte Einführung., Leipzig 2003 ISBN 3-937219-02-1
• L. Keiff Le Pluralisme Dialogique. Thesis Université de Lille 3 (2007).
• K. Lorenz, P. Lorenzen: Dialogische Logik, Darmstadt 1978
• P. Lorenzen: Lehrbuch der konstruktiven Wissenschaftstheorie, Stuttgart 2000 ISBN 3-476-01784-2
• O. Majer, A.-V. Pietarinen and T. Tulenheimo (editors). Games: Unifying Logic, Language and Philosophy. Springer (2009).
• S. Rahman, Über Dialogue protologische Kategorien und andere Seltenheiten. Frankfurt 1993 ISBN 3-631-46583-1
• S. Rahman and H. Rückert (editors), New Perspectives in Dialogical Logic. Synthese 127 (2001) ISSN 0039-7857.
• J. Redmond & M. Fontaine, How to play dialogues. An introduction to Dialogical Logic. London, College Publications (Col. Dialogues and the Games of Logic. A Philosophical Perspective N° 1). (ISBN
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Time to Abolish "Statistical Significance"?
The idea of "statistical significance" has been a basic concept in introductory statistics courses for decades. If you spend any time looking at quantitative research, you will often see in tables of
results that certain numbers are marked with an asterisk or some other symbol to show that they are "statistically significant."
For the uninitiated, "statistical significance" is a way of summarizing whether a certain statistical result is likely to have happened by chance, or not. For example, if I flip a coin 10 times and
get six heads and four tails, this could easily happen by chance even with a fair and evenly balanced coin. But if I flip a coin 10 times and get 10 heads, this is extremely unlikely to happen by
chance. Or if I flip a coin 10,000 times, with a result of 6,000 heads and 4,000 tails (essentially, repeating the 10-flip coin experiment 1,000 times), I can be quite confident that the coin is not
a fair one. A common rule of thumb has been that if the probability of an outcome occurring by chance is 5% or less--in the jargon, has a p-value of 5% or less--then the result is statistically
significant. However, it's also pretty common to see studies that report a range of other p-values like 1% or 10%.
Given the omnipresence of "statistical significance" in pedagogy and the research literature, it was interesting last year when the American Statistical Association made an official statement "ASA
Statement on Statistical Significance and P-Values" (discussed
) which includes comments like: "Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold. ... A p-value, or statistical
significance, does not measure the size of an effect or the importance of a result. ... By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis."
Now, the ASA has followed up with a
special supplemental issue of its journal The American Statistician on the theme "Statistical Inference in the 21st Century: A World Beyond p < 0.05" (January 2019).
The issue has a useful overview essay, "Moving to a World Beyond “p < 0.05.” by Ronald L. Wasserstein, Allen L. Schirm, and Nicole A. Lazar. They write:
We conclude, based on our review of the articles in this special issue and the broader literature, that it is time to stop using the term “statistically significant” entirely. Nor should variants
such as “significantly different,” “p < 0.05,” and “nonsignificant” survive, whether expressed in words, by asterisks in a table, or in some other way. Regardless of whether it was ever useful, a
declaration of “statistical significance” has today become meaningless. ... In sum, `statistically significant'—don’t say it and don’t use it.
The special issue is then packed with 43 essays from a wide array of experts and fields on the general theme of "if we eliminate the language of statistical significance, what comes next?"
To understand the arguments here, it's perhaps useful to have a brief and partial review of some main reasons why the emphasis on "statistical significance" can be so misleading: namely, it can lead
one to dismiss useful and true connections; it can lead one to draw false implications; and it can cause researchers to play around with their results. A few words on each of these.
The question of whether a result is "statistically significant" is related to the size of the sample. As noted above, 6 out of 10 heads can easily happen by chance, but 6,000 out of 10,000 heads is
extraordinarily unlikely to happen by chance. So say that you do an study which finds an effect which is fairly large in size, but where the sample size isn't large enough for it to be statistically
significant by a standard test. In practical terms, it seems foolish to ignore this large result; instead, you should presumably start trying to find ways to run the test with a much larger sample
size. But in academic terms, the study you just did with its small sample size may be unpublishable: after all, a lot of journals will tend to decide against publishing a study that doesn't find a
statistically significant effect--because it feels as if such a study isn't pointing out any new connection or insight.
Knowing that journals are looking to publish "statistically significant" results, researchers will be tempted to look for ways to jigger their results. Studies in economics, for example, aren't about
simple probability examples like flipping coins. Instead, one might be looking at Census data on households that can be divided up in roughly a jillion ways: not just the basic categories like age,
income, wealth, education, health, occupation, ethnicity, geography, urban/rural, during recession or not, and others, but also various interactions of these factors looking at two or three or more
at a time. Then, researchers make choices about whether to assume that connections between these variables should be thought of a linear relationship, curved relationships (curving up or down),
relationships are are U-shaped or inverted-U, and others. Now add in all the different time periods and events and places and before-and-after legislation that can be considered. For this fairly
basic data, one is quickly looking at thousands or tens of thousands of possible connections relationships.
Remember that the idea of statistical significance relates to whether something has a 5% probability or less of happening by chance. To put that another way, it's whether something would have
happened only one time out of 20 by chance. So if a researcher takes the same basic data and looks at thousands of possible equations, there will be dozens of equations that look like they had a 5%
probability of not happening by chance. When there are thousands of researchers acting in this way, there will be a steady stream of hundreds of result every month that appear to be "statistically
significant," but are just a result of the general situation that if you look at a very large number of equations one at a time, some of them will seem to mean something. It's a little like flipping
a coin 10,000 times, but then focusing only on the few stretches where the coin came up heads five times in a row--and drawing conclusions based on that one small portion of the overall results.
A classic statement of this issue arises in Edward Leamer's 1983 article, "Taking the Con out of Econometrics" (
American Economic Review
, March 1983, pp. 31-43). Leamer wrote:
The econometric art as it is practiced at the computer terminal involves fitting many, perhaps thousands, of statistical models. One or several that the researcher finds pleasing are selected for
re- porting purposes. This searching for a model is often well intentioned, but there can be no doubt that such a specification search in-validates the traditional theories of inference. ... [I]n
fact, all the concepts of traditional theory, utterly lose their meaning by the time an applied researcher pulls from the bramble of computer output the one thorn of a model he likes best, the
one he chooses to portray as a rose. The consuming public is hardly fooled by this chicanery. The econometrician's shabby art is humorously and disparagingly labelled "data mining," "fishing,"
"grubbing," "number crunching." A joke evokes the Inquisition: "If you torture the data long enough, Nature will confess" ... This is a sad and decidedly unscientific state of affairs we find
ourselves in. Hardly anyone takes data analyses seriously. Or perhaps more accurately, hardly anyone takes anyone else's data analyses seriously."
Economists and other social scientists have become much more aware of these issues over the decades, but Leamer was still writing in 2010 ("Tantalus on the Road to Asymptopia," J
ournal of Economic Perspectives
, 24: 2, pp. 31-46):
Since I wrote my “con in econometrics” challenge much progress has been made in economic theory and in econometric theory and in experimental design, but there has been little progress
technically or procedurally on this subject of sensitivity analyses in econometrics. Most authors still support their conclusions with the results implied by several models, and they leave the
rest of us wondering how hard they had to work to find their favorite outcomes ... It’s like a court of law in which we hear only the experts on the plaintiff’s side, but are wise enough to know
that there are abundant for the defense.
Taken together, these issues suggest that a lot of the findings in social science research shouldn't be believed with too much firmness. The results might be true. They might be a result of a
researcher pulling out "from the bramble of computer output the one thorn of a model he likes best, the one he chooses to portray as a rose." And given the realities of real-world research, it seems
goofy to say that a result with, say, only a 4.8% probability of happening by chance is "significant," while if the result had a 5.2% probability of happening by chance it is "not significant."
Uncertainty is a continuum, not a black-and-white difference.
So let's accept the that the "statistical significance" label has some severe problems, as Wasserstein, Schirm, and Lazar write:
[A] label of statistical significance does not mean or imply that an association or effect is highly probable, real, true, or important. Nor does a label of statistical nonsignificance lead to
the association or effect being improbable, absent, false, or unimportant. Yet the dichotomization into “significant” and “not significant” is taken as an imprimatur of authority on these
characteristics. In a world without bright lines, on the other hand, it becomes untenable to assert dramatic differences in interpretation from inconsequential differences in estimates. As Gelman
and Stern (2006) famously observed, the difference between “significant” and “not significant” is not itself statistically significant.
But as they recognize, criticizing is the easy part. What is to be done instead? And here, the argument fragments substantially. Did I mention that there were 43 different responses in this issue of
American Statistician
Some of the recommendations are more a matter of temperament than of specific statistical tests. As Wasserstein, Schirm, and Lazar emphasize, many of the authors offer advice that can be summarized
in about seven words: "Accept uncertainty. Be thoughtful, open, and modest.” This is good advice! But a researcher struggling to get a paper published might be forgiven for feeling that it lacks
Other recommendations focus on the editorial process used by academic journals, which establish some of the incentives here. One interesting suggestion is that when a research journal is deciding
whether to publish a paper, the reviewer should only see a description of what the researcher did--without seeing the actual empirical findings. After all, if the study was worth doing, then it's
worthy of being published, right? Such an approach would mean that authors had no incentive to tweak their results. A method already used by some journals is
"pre-publication registration," where the researcher lays out beforehand, in a published paper, exactly what is going to be done.
Then afterwards, no one can accuse that researcher of tweaking the methods to obtain specific results.
Other authors agree with turning away from "statistical significance," but in favor of their own preferred tools for analysis: Bayesian approaches, "second-generation p-values," "false positive
"statistical decision theory," "confidence index," and many more. With many alternative examples along these lines, the researcher trying to figure out how to proceed can again be forgiven for
desiring little more definitive guidance.
Wasserstein, Schirm, and Lazar also asked some of the authors whether there might be specific situations where a p-value threshold made sense. They write:
"Authors identified four general instances. Some allowed that, while p-value thresholds should not be used for inference, they might still be useful for applications such as industrial quality
control, in which a highly automated decision rule is needed and the costs of erroneous decisions can be carefully weighed when specifying the threshold. Other authors suggested that such
dichotomized use of p-values was acceptable in model-fitting and variable selection strategies, again as automated tools, this time for sorting through large numbers of potential models or
variables. Still others pointed out that p-values with very low thresholds are used in fields such as physics, genomics, and imaging as a filter for massive numbers of tests. The fourth instance
can be described as “confirmatory setting[s] where the study design and statistical analysis plan are specified prior to data collection, and then adhered to during and after it” ... Wellek
(2017) says at present it is essential in these settings. “[B]inary decision making is indispensable in medicine and related fields,” he says. “[A] radical rejection of the classical principles
of statistical inference…is of virtually no help as long as no conclusively substantiated alternative can be offered.”
The deeper point here is that there are situation where a researcher or a policy-maker or an economic needs to make a yes-or-no decision. When doing quality control, is it meeting the standard or
not? when the Food and Drug Administration is evaluating a new drug, does it approve the drug or not? When a researcher in genetics is dealing with a database that has thousands of genes, there's a
need to focus on a subset of those genes, which means making yes-or-no decisions on which genes to include a certain analysis.
Yes, the scientific spirit should "Accept uncertainty. Be thoughtful, open, and modest.” But real life isn't a philosophy contest. Sometimes, decisions need to be made. If you don't have a
statistical rule, then the alternative decision rule becomes human judgment--which has plenty of cognitive, group-based, and political biases of its own.
My own sense is that "statistical significance" would be a very poor master, but that doesn't mean it's a useless servant. Yes, it would foolish and potentially counterproductive to give excessive
weight to "statistical significance." But the clarity of conventions and rule, when their limitations are recognized and acknowledges, can still be useful. I was struck by a comment in the essay by
Steven N. Goodman:
P-values are part of a rule-based structure that serves as a bulwark against claims of expertise untethered from empirical support. It can be changed, but we must respect the reason why the
statistical procedures are there in the first place ... So what is it that we really want? The ASA statement says it; we want good scientific practice. We want to measure not just the signal
properly but its uncertainty, the twin goals of statistics. We want to make knowledge claims that match the strength of the evidence. Will we get that by getting rid of P−values? Will eliminating
P−values improve experimental design? Would it improve measurement? Would it help align the scientific question with those analyses? Will it eliminate bright line thinking? If we were able to get
rid of P-values, are we sure that unintended consequences wouldn’t make things worse? In my idealized world, the answer is yes, and many statisticians believe that. But in the real world, I am
less sure. | {"url":"https://conversableeconomist.blogspot.com/2019/03/time-to-abolish-statistical-significance.html","timestamp":"2024-11-04T00:35:49Z","content_type":"text/html","content_length":"107090","record_id":"<urn:uuid:9dc13856-4b48-407b-8618-297e7482858d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00613.warc.gz"} |
How to compute nPr in sagemath
How to compute nPr in sagemath
How to compute nPr in sagemath? Please help
2 Answers
Sort by » oldest newest most voted
The following method avoids computing factorials explicitly, which can be computationally expensive for large values of n and r. Instead, it computes the product of a sequence of values using a
generator expression, which is more memory-efficient and faster than computing factorials. You can compute perm(n, r) using a prod function, which is a built-in SageMath function that computes the
product of a sequence of values.
See the code:
perm(n, r) = prod(n - i for i in range(r))
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Hello! I am assuming nPr means "n permutation r". Unfortunately, SageMath doesn't have a command for that purpose. However, if you remember that
$$nPr = \frac{n!}{(n-r)!} = \binom{n}{r}r!,$$
you can make something like this:
perm(n, r) = factorial(n) / factorial(n - r)
or even:
perm(n, r) = binomial(n, r) * factorial(r)
Hope this helps!
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Permutations(n,r).cardinality() ?
Max Alekseyev ( 2023-07-15 22:12:11 +0100 )edit
Or binomial(n,r)*factorial(r) ?
Emmanuel Charpentier ( 2023-07-16 20:53:45 +0100 )edit
To my surprise, the former is about 12 times faster than the latter :
sage: %timeit Permutations(5,3).cardinality()
2.31 µs ± 450 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: sage: %timeit binomial(5,3)*factorial(3)
27.3 µs ± 860 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
Go figure...
Emmanuel Charpentier ( 2023-07-16 20:59:50 +0100 )edit | {"url":"https://ask.sagemath.org/question/70185/how-to-compute-npr-in-sagemath/","timestamp":"2024-11-05T09:59:32Z","content_type":"application/xhtml+xml","content_length":"62234","record_id":"<urn:uuid:16ecf051-d333-460c-afc0-577abdc22c50>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00720.warc.gz"} |
On the Non-central Distribution of the Second Elementary Symmetric Function of the Roots of a Matrix
Ann. Math. Statist. 39(3): 833-839 (June, 1968). DOI: 10.1214/aoms/1177698314
Let $\mathbf{X}$ be a $p \times f$ matrix variate $(p \leqq f)$ whose columns are independently normally distributed with $E(\mathbf{X}) = \mathbf{M}$ and covariance matrix $\mathbf{\Sigma}$. Let
$w_1, \cdots, w_p$ be the characteristic roots of $|\mathbf{XX}' - w\mathbf{\Sigma}| = 0,$ then the distribution of $\mathbf{W} = \operatorname{diag} (w_i)$ is given by [4], [5] \begin{equation*}\tag
{1.1} e^{-\frac{1}{2}\operatorname{tr}\mathbf{\Omega}}_0F_1(\frac{1}{2}f; \frac{1}{4}\mathbf{\Omega}, \mathbf{W})\kappa (p, f) \cdot e^{-\frac{1}{2}\operatorname{tr}\mathbf{W}}|\mathbf{W}|^{\frac{1}
{2}(f-p-1)} \mathbf{\prod}_{i>j} (w_i - w_j), 0 < w_1 \leqq \cdots \leqq w_p < \infty,\end{equation*} where \begin{equation*}\tag{1.2}\kappa(p, f) = \pi^{\frac{1}{2}p^2}/\{2^{\frac{1}{2}pf}\Gamma_p(\
frac{1}{2}f)\Gamma_p (\frac{1}{2}p)\},\end{equation*} $\mathbf{\Omega} = \operatorname{diag} (\omega_i)$ where $\omega_i, i = 1, \cdots, p,$ are the characteristic roots of $|\mathbf{MM}' - \mathbf{\
omega\Sigma}| = 0$ and $_0F_1$ is the hypergeometric function of matrix argument (see Section 2) defined in [5]. The above distribution of non-central means with known covariance matrix was obtained
by James [4]. But (1.1) has also been shown, [5], to be the limiting distribution as $n \rightarrow \infty$ of $n\mathbf{R}^2 = \mathbf{W}$ such that $0 < n\mathbf{P}^2 = \mathbf{\Omega} < \infty,$
where $\mathbf{R}^2 = \operatorname{diag} (r_i^2)$ and $\mathbf{P}^2 = operatorname{diag} (\rho^2_i)$ and where the canonical correlation coefficients $r_1^2, \cdots, r_p^2$ between a $p$-set and a
$q$-set of variates $(p \leqq q)$ following a $(p + q)$ variate normal distribution, are calculated from a sample of $n + 1$ observations and $\rho_1^2, \cdots, \rho_p^2$ are population canonical
correlation coefficients. Further $q = f.$ In this paper, the first two non-central moments of $W_2^{(p)},$ the second elementary symmetric function (esf) in $\frac{1}{2}w_1, \frac{1}{2}w_2,\cdots, \
frac{1}{2}w_p$ have been obtained first by evaluating certain integrals involving zonal polynomials, and then alternately in terms of generalized Laguerre polynomials, [2], [5]. These moments were
used to suggest an approximation to the non-central distribution of $W_2^{(p)}$. The approximation is observed to be good even for small values of $f.$
Download Citation
K. C. Sreedharan Pillai. Arjun K. Gupta. "On the Non-central Distribution of the Second Elementary Symmetric Function of the Roots of a Matrix." Ann. Math. Statist. 39 (3) 833 - 839, June, 1968.
Published: June, 1968
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177698314
Rights: Copyright © 1968 Institute of Mathematical Statistics
Vol.39 • No. 3 • June, 1968 | {"url":"https://www.projecteuclid.org/journals/annals-of-mathematical-statistics/volume-39/issue-3/On-the-Non-central-Distribution-of-the-Second-Elementary-Symmetric/10.1214/aoms/1177698314.full","timestamp":"2024-11-04T11:53:25Z","content_type":"text/html","content_length":"143926","record_id":"<urn:uuid:63476462-25e6-41df-9e83-8471ac958522>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00471.warc.gz"} |
Pascal`s Identity And Triangle | BimStudies.Com
Pascal`s Identity and Triangle
Pascal’s Identity is a mathematical relationship that is closely associated with Pascal’s Triangle.
• It states that the sum of two adjacent entries in Pascal’s Triangle is equal to the entry immediately below them. Mathematically, it can be expressed as:
Pascal’s Triangle is a triangular array of numbers where the outer edges are filled with ones, and each interior number is the sum of the two numbers directly above it.
Q). Write down the expansion of (1+y)^6, using Pascal`s theorem.
Here n=6, so we use Pascal`s triangle up to 6.
When n=0
When n=1
When n=2
When n=3
When n=4
When n=5
When n=6
∴ (1+y)^6 = 1(1)^6 + 6(1)^5y + 15(1)^4y^2 + 20(1)^3y^3+ 15(1)^2y^4 + 6(1)y^5 + 1(1)^0y^6
= 1+6y+15y^2+20y^3+15y^4+6y^5+y^6 | {"url":"https://bimstudies.com/docs/discrete-structure/counting-advance-counting/pascals-identity-and-triangle/","timestamp":"2024-11-10T17:47:21Z","content_type":"text/html","content_length":"472839","record_id":"<urn:uuid:d701f406-8a5a-4429-893b-7b905f8083d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00134.warc.gz"} |
Hardware-Efficient Euler Rotations Using CORDIC
This example shows how to implement Euler rotations using a CORDIC kernel. The resulting model is deployable to FPGA or ASIC devices.
Euler Rotations
Euler rotations are a common convention for describing arbitrary three dimensional rotations. You can modify them to describe either rotation of a vector in a fixed coordinate system, or rotation of
a coordinate system with respect to a fixed vector. In this example, assume rotations of vectors with respect to fixed coordinate systems.
You can perform an Euler rotation of a vector as follows. First, rotate the vector by about the z-axis. Then, rotate by about the resultant x-axis. Finally, rotate by about the rotated z-axis to
obtain the final coordinates of the vector.
It is common to construct Euler rotations by multiplying three individual rotation matrices. For example, you can rotate a vector using the matrix transformation below.
The matrix represents a rotation about the axis . For example, the matrix is given by
While this form is easily understood, it has several inefficiencies. If , , and are variables, you must recalculate and for each angle prior to forming up the matrix. You further need to multiply and
add these results, which can result in unnecessary word length growth. Finally, the entire matrix multiplication must be properly pipelined for maximum efficiency. This can be a time consuming
Deploy Euler Rotations Using CORDIC Algorithm
Euler rotations operate by performing rotations in planes of intersecting coordinate axes. Therefore, efficient two dimensional rotations can be used to build up the full transformation. The
Coordinate Rotation DIgital Computer (CORDIC) algorithm performs these rotations using a series of shift and add operations, followed by a multiplication by a precomputed constant. It is extremely
efficient and often used as a key component of high-frequency, high-throughput systems. It also eliminates the need to explicitly calculate any trigonometric functions at runtime, thus freeing
further computational resources.
You can perform a full Euler rotation by using CORDIC to first rotate an input vector by in the XY plane, followed by a CORDIC rotation by in the resulting YZ plane. A final CORDIC rotation by in the
resulting XY plane completes the transformation. This model shows a subsystem implementing this transformation.
This subsystem shows the details of the actual Euler transformation. Note that the angles and are delayed to align all data pipelines.
This subsystem illustrates a single planar rotation. The x and y components of the vector, xIn and yIn, are input to the CORDIC Rotation block, along with the angle alpha and validIn. The z component
of the vector, zIn, is passed along in a series of registers whose latency matches the latency of the CORDIC rotation.
open_system("euler_rotations/HDL_DUT/Rotate about Z Axis")
Define Data and Simulate
Define data and simulate the model.
uIn = fi([sqrt(3)/2;0;1/2],1,16,8);
dt = numerictype(1,16,8);
nIters = 18;
out = sim("euler_rotations");
The model outputs the first 360 datapoints to the MATLAB® workspace. The model has been chosen to simulate one rotation about the z-axis.
Trajectory of Rotation
This figure shows the trajectory of the vector over the course of the simulation. The vector from the origin shows the initial vector, while the vectors on the raised line demonstrate the direction
of rotation.
outData = out.simout.Data;
quiver3(0,0,0,sqrt(3)/2, 0, 0.5);
hold on;
plot3(outData(:,1), outData(:,2), 0.5*ones(length(outData)));
0.5*[0;-1;0;1],0.5*[1;0;-1;0],[0;0;0;0], 'r-');
HDL Statistics
Generating HDL code for the system with the datatypes chosen gives excellent performance. The resource usage is shown below, and the device operates at approximately 373 MHz. All characterization was
performed using Xilinx® Vivado® using a ZC706 Evaluation Board.
Resource = ["LUT";"LUTRAM";"FF"];
Used = [3957;99;3673];
table(Resource, Used)
ans =
3x2 table
Resource Used
________ ____
"LUT" 3957
"LUTRAM" 99
"FF" 3673
Modifying and Extending Euler Rotations
You can rearrange the constituent pieces used to develop this transformation to yield countless other transformations. The only requirement is that the full transformation you build be composed of
rotations along planes where principle axes intersect. This is one way to develop efficient, FPGA-ready solutions. | {"url":"https://nl.mathworks.com/help/fixedpoint/ug/hardware-efficient-euler-rotations-using-cordic.html","timestamp":"2024-11-11T16:32:07Z","content_type":"text/html","content_length":"80400","record_id":"<urn:uuid:62b4719f-7f58-434c-ae89-025815ac44c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00605.warc.gz"} |
t S
Calculate power gain from two-port S-parameters
g = powergain(s_params,z0,zs,zl,'Gt') calculates the transducer power gain of the 2-port network by:
${G}_{t}=\frac{{P}_{L}}{{P}_{\text{avs}}}=\frac{\left(1-{|{\Gamma }_{S}|}^{2}\right){|{S}_{21}|}^{2}\left(1-{|{\Gamma }_{L}|}^{2}\right)}{{|\left(1-{S}_{11}{\Gamma }_{S}\right)\left(1-{S}_{22}{\Gamma
}_{L}\right)-{S}_{12}{S}_{21}{\Gamma }_{S}{\Gamma }_{L}|}^{2}}$
• P[L] is the output power and P[avs] is the maximum input power.
• Γ[L] and Γ[S] are the reflection coefficients defined as:
$\begin{array}{l}{\Gamma }_{S}=\frac{{Z}_{S}-{Z}_{0}}{{Z}_{S}+{Z}_{0}}\\ {\Gamma }_{L}=\frac{{Z}_{L}-{Z}_{0}}{{Z}_{L}+{Z}_{0}}\end{array}$
g = powergain(s_params,z0,zs,'Ga') calculates the available power gain of the 2-port network by:
${G}_{a}=\frac{{P}_{\text{avn}}}{{P}_{\text{avs}}}=\frac{\left(1-{|{\Gamma }_{S}|}^{2}\right){|{S}_{21}|}^{2}}{{|1-{S}_{11}{\Gamma }_{S}|}^{2}\left(1-{|{\Gamma }_{out}|}^{2}\right)}$
• P[avn] is the available output power from the network.
• Γ[out] is given by:
${\Gamma }_{\text{out}}={S}_{22}+\frac{{S}_{12}{S}_{21}{\Gamma }_{S}}{1-{S}_{11}{\Gamma }_{S}}$
g = powergain(s_params,z0,zl,'Gp') calculates the operating power gain of the 2-port network by:
${G}_{p}=\frac{{P}_{L}}{{P}_{\text{in}}}=\frac{{|{S}_{21}|}^{2}\left(1-{|{\Gamma }_{L}|}^{2}\right)}{\left(1-{|{\Gamma }_{\text{in}}|}^{2}\right){|1-{S}_{22}{\Gamma }_{L}|}^{2}}$
• P[in] is the input power.
• Γ[in] is given by:
${\Gamma }_{\text{in}}={S}_{11}+\frac{{S}_{12}{S}_{21}{\Gamma }_{L}}{1-{S}_{22}{\Gamma }_{L}}$
g = powergain(s_params,'Gmag') calculates the maximum available power gain of the 2-port network. by:
where K is the stability factor.
g = powergain(s_params,'Gmsg') calculates the maximum stable gain of the 2-port network by:
g = powergain(hs,zs,zl,'Gt') calculates the transducer power gain of the network represented by the S-parameter object hs.
g = powergain(hs,zs,'Ga') calculates the available power gain of the network represented by the S-parameter object hs.
g = powergain(hs,zl,'Gp') calculates the operating power gain of the network represented by the S-parameter object hs.
g = powergain(hs,'Gmag') calculates the maximum available power gain of the network represented by the S-parameter object hs.
g = powergain(hs,'Gmsg') calculates the maximum stable gain of the network represented by the S-parameter object hs.
Power Gain of Two-Port Network
Calculate power gains for a sample 2-port network.
s11 = 0.61*exp(1j*165/180*pi);
s21 = 3.72*exp(1j*59/180*pi);
s12 = 0.05*exp(1j*42/180*pi);
s22 = 0.45*exp(1j*(-48/180)*pi);
sparam = [s11 s12; s21 s22];
z0 = 50;
zs = 10 + 1j*20;
zl = 30 - 1j*40;
Calculate the transducer power gain of the network
Gt = powergain(sparam,z0,zs,zl,'Gt')
Calculate the available power gain of the network
Ga = powergain(sparam,z0,zs,'Ga')
Note that, as expected, the available power gain is larger than the transducer power gain, Gt. The two become identical when Gt is measured with a matched load impedance:
zl_matched = gamma2z(gammaout(sparam, z0, zs)', z0);
Gt_zl_matched = powergain(sparam, z0, zs, zl_matched, 'Gt')
Calculate the operating power gain of the network
Gp = powergain(sparam,z0,zl,'Gp')
Note that, as expected, the operating power gain is larger than the transducer power gain, Gt. The two become identical when Gt is measured with a matched source impedance:
zs_matched = gamma2z(gammain(sparam, z0, zl)', z0);
Gt_zs_matched = powergain(sparam, z0, zs_matched, zl, 'Gt')
Calculate the maximum available power gain of the network
Gmag = powergain(sparam,'Gmag')
Note that, as expected, the maximum available power gain is larger than the available power gain Ga, the transducer power gain, Gt, and the operating power gain, Gp. They all become identical when
measured with simultaneously matched source and load impedances:
zs_matched_sim = gamma2z(gammams(sparam), z0);
zl_matched_sim = gamma2z(gammaout(sparam, z0, zs_matched_sim)', z0)
zl_matched_sim =
33.6758 +91.4816i
That impedance can be also obtained directly using:
zl_matched_sim = gamma2z(gammaml(sparam), z0)
zl_matched_sim =
33.6758 +91.4816i
Ga_matched_sim = powergain(sparam, z0, zs_matched_sim, 'Ga')
Gt_matched_sim = powergain(sparam, z0, zs_matched_sim, zl_matched_sim, 'Gt')
Gp_matched_sim = powergain(sparam, z0, zl_matched_sim, 'Gp')
When the scattering parameters represent a network that is not unconditionally stable, there is no set of source and load impedances that provide simultaneous matching. In this case, the maximum
available power is infinite, but truly meaningless because the network is unstable.
To make the previously defined network conditionally stable, it is enough to increase the magnitude of the backward propagation scattering parameter, s12:
s12_cond_stable = 0.06*exp(1j*42/180*pi);
sparam_cond_stable = [s11 s12_cond_stable; s21 s22];
To verify that the network is conditionally stable, check that the stability factor, K, is smaller than 1:
K = stabilityk(sparam_cond_stable)
An attempt to calculate the maximum available gain of the network yields a NaN:
Gmag_cond_stable = powergain(sparam_cond_stable,'Gmag')
Instead, the maximum stable gain, ${\mathit{G}}_{\mathrm{msg}}$, should be used.
Calculate the maximum stable power gain of the network
Gmsg_cond_stable = powergain(sparam_cond_stable,'Gmsg')
Gmsg_cond_stable =
The maximum stable power gain is only meaningful when the network is not unconditionally stable.
Input Arguments
hs — 2-port S-parameters
S-parameter object
2-port S-parameters, specified as an RF Toolbox™ S-parameter object.
s_params — 2-port S-parameters
array of complex numbers
2-port S-parameters, specified as a complex 2-by-2-by-N array.
z0 — Reference impedance
50 (default) | positive scalar
Reference impedance in ohms, specified as a positive scalar. If the first input argument is an S-parameter object hs, the function uses hs.Impedance for the reference impedance.
zl — Load impedance
50 (default) | positive scalar
Load impedance in ohms, specified as a positive scalar.
zs — Source impedance
50 (default) | positive scalar
Source impedance in ohms, specified as a positive scalar.
Output Arguments
g — Power gain
Unit less power gain values, returned as a vector. To obtain power gain in decibels, use 10*log10(g).
If the specified type of power gain is undefined for one or more of the specified S-parameter values in s_params, the powergain function returns NaN. As a result, g is either NaN or a vector that
contains one or more NaN entries.
More About
Transducer Power Gain
G[t] = P[L]/P[avs] is the ratio of power delivered to the load to the power available from the source. This depends on both Z[S] and Z[L].
Available Power Gain
G[a] = P[avn]/P[avs] is the ratio of the power available from the two-port network to the power available from the source. Available gain is the transducer power gain when load impedance is equal to
output impedance. Thus G[a] depends only on Z[s].
Operating Power Gain
G[p] = P[L]/P[in] is the ratio of power dissipated in the load Z[L] to the power delivered to the input of the two-port network. This gain is independent of Z[S], although some active circuits are
strongly dependent on the input matching conditions.
Maximum Available Power Gain and Maximum Stable Power Gain
Maximum available power gain, G[mag] is G[a]with matched input that is Z[S] is equal to Z[in]
In the case of conditionally stable two-port networks (K<1) where the maximum available power gain result is meaningless, the maximum stable power gain, G[msg], should be used.
Version History
Introduced in R2007b | {"url":"https://it.mathworks.com/help/rf/ref/powergain.html","timestamp":"2024-11-14T21:31:36Z","content_type":"text/html","content_length":"112691","record_id":"<urn:uuid:d8a56a6e-551a-41fb-b9de-d94170b42279>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00263.warc.gz"} |
Benoit Mandelbrot
Dr. Benoit Mandelbrot
From Scholarpedia
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Yale University, New Haven, CT
20 November 1924 - 14 October 2010
Featured Author: Benoît Mandelbrot
Benoît Mandelbrot (b. 20 November 1924, d. 14 October 2010) was a Polish born French and American mathematician. He was best known as the father of fractal geometry. His Jewish family (with a strong
academic tradition) fled from Poland to France in 1936. He continued his studies in France and later he moved to US. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland then Lille,
France. Later, the couple moved to US and Mandelbrot joined IBM's Thomas J. Watson Research Center in 1958, and remained with the company ever since, eventually becoming first an IBM Fellow, and then
a Fellow Emeritus. Upon his retirement from IBM in 1987, Mandelbrot joined the Yale Department of Mathematics. At the time of his retirement in 2005, he was Sterling Professor of Mathematical
Sciences. Mandelbrot means "almond bread" in German.
His best known publications include Les objets fractals (1975) and the The Fractal Geometry of Nature (1982), both translated into several languages.
His awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003, and the Einstein Lectureship of the
American Mathematical Society in 2006. The small asteroid 27500 Mandelbrot was named in his honour. In November 1990, he was made a Knight in the French Legion of Honour. In December 2005, Mandelbrot
was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was promoted to Officer of the French Legion of Honour in January 2006. An honorary degree
from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.
Scholarpedia articles:
Fractals. Scholarpedia, Unpublished.
Mandelbrot Set. Scholarpedia, Unpublished.
(Author profile by Mortaza Doulaty)
<review>Email: benoit.mandelbrot [at] yale.edu ~~~~ </review> | {"url":"http://scholarpedia.org/article/User:Mandelbrot","timestamp":"2024-11-08T17:03:03Z","content_type":"text/html","content_length":"22772","record_id":"<urn:uuid:967b5b9b-19d5-45ba-94e6-ae39f49d890a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00393.warc.gz"} |
Limits for quantum computers: Perfect clocks are impossible, research findsLimits for quantum computers: Perfect clocks are impossible, research finds - MigZonLimits for quantum computers: Perfect clocks are impossible, research finds
The oversampling regime of an exemplary clock—a pendulum in a weakly lit environment. The two sources of entropy production for this clock are: the friction within the clockwork itself, and the
matter–light interaction necessary to track the position of the pendulum. The plot shows the elementary ticking events of this clock as a function of time, i.e., the photons reflected off the
pendulum when it is close to its maximum deflection. In the oversampling regime, the average time between two such ticks is much shorter than that of the period of the TPC (continuous line), which in
the case of this pendulum is 2 s. Due to technical limitations, one does not count photons, but rather the TPC cycles through the averaged light intensity. Credit: arXiv (2023). DOI: 10.48550/
There are different ideas about how quantum computers could be built. But they all have one thing in common: you use a quantum physical system—for example, individual atoms—and change their state by
exposing them to very specific forces for a specific time. However, this means that in order to be able to rely on the quantum computing operation delivering the correct result, you need a clock that
is as precise as possible.
But here you run into problems: perfect time measurement is impossible. Every clock has two fundamental properties: a certain precision and a certain time resolution. The time resolution indicates
how small the time intervals are that can be measured—i.e., how quickly the clock ticks. Precision tells you how much inaccuracy you have to expect with every single tick.
The research team was able to show that since no clock has an infinite amount of energy available (or generates an infinite amount of entropy), it can never have perfect resolution and perfect
precision at the same time. This sets fundamental limits to the possibilities of quantum computers.
Quantum calculation steps are like rotations
In our classical world, perfect arithmetic operations are not a problem. For example, you can use an abacus in which wooden balls are threaded onto a stick and pushed back and forth. The wooden beads
have clear states, each one is in a very specific place, if you don’t do anything the bead will stay exactly where it was.
And whether you move the bead quickly or slowly does not affect the result. But in quantum physics it is more complicated.
“Mathematically speaking, changing a quantum state in a quantum computer corresponds to a rotation in higher dimensions,” says Jake Xuereb from the Atomic Institute at the Vienna University of
Technology in the team of Marcus Huber and first author of the first paper published in Physical Review Letters. “In order to achieve the desired state in the end, the rotation must be applied for a
very specific period of time. Otherwise, you turn the state either too short or too far.”
Entropy: Time makes everything more and more messy
Marcus Huber and his team investigated in general which laws must always apply to every conceivable clock. “Time measurement always has to do with entropy,” explains Marcus Huber. In every closed
physical system, entropy increases and it becomes more and more disordered. It is precisely this development that determines the direction of time: the future is where the entropy is higher, and the
past is where the entropy is even lower.
As can be shown, every measurement of time is inevitably associated with an increase in entropy: a clock, for example, needs a battery, the energy of which is ultimately converted into frictional
heat and audible ticking via the clock’s mechanics—a process in which a fairly ordered state occurs the battery is converted into a rather disordered state of heat radiation and sound.
On this basis, the research team was able to create a mathematical model that basically every conceivable clock must obey. “For a given increase in entropy, there is a tradeoff between time
resolution and precision,” says Florian Meier, first author of the second paper, now posted to the arXiv preprint server. “That means: Either the clock works quickly or it works precisely—both are
not possible at the same time.”
Limits for quantum computers
This realization now brings with it a natural limit for quantum computers: the resolution and precision that can be achieved with clocks limits the speed and reliability that can be achieved with
quantum computers. “It’s not a problem at the moment,” says Huber.
“Currently, the accuracy of quantum computers is still limited by other factors, for example, the precision of the components used or electromagnetic fields. But our calculations also show that today
we are not far from the regime in which the fundamental limits of time measurement play the decisive role.”
Therefore, if the technology of quantum information processing is further improved, one will inevitably have to contend with the problem of non-optimal time measurement. But who knows: Maybe this is
exactly how we can learn something interesting about the quantum world.
More information:
Florian Meier et al, Fundamental accuracy-resolution trade-off for timekeeping devices, arXiv (2023). DOI: 10.48550/arxiv.2301.05173
Limits for quantum computers: Perfect clocks are impossible, research finds (2023, November 26)
retrieved 26 November 2023
from https://phys.org/news/2023-11-limits-quantum-clocks-impossible.html
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I am very happy to say that we can finally announce that Optim.jl v0.9.0 is out. This version has quite a few user facing changes. Please read about the changes below if you use Optim.jl in a
package, a script, or anything else, as you will quite likely have to make some changes to your code.
As always, I have to thank my two partners in crime: Asbjørn Nilsen Riseth (@anriseth) and Christoph Ortner (@cortner) for their help in making the changes, transitions, and tests that are included
in v0.9.0.
The last update (form v0.6.0 to v0.7.0 had) some changes that were a long time coming, and so does v0.9.0. Hopefully, these fixes to old design problems will greatly improve the user experience and
performance of Optim.jl, and pave the way for more exiting features in the future.
We’ve tried to make the transition as smooth as possible, although we do have breaking changes in this update. Please consult the documentation if you face problems, join us on gitter or ask the
community at discourse!
Okay, now to the changes.
Why not v0.8.0?
First of all, why v0.9.0? Last version was v0.7.8! The is because we are dropping support for Julia v0.4 and v0.5 simultaneously, so we are reserving v0.8.0 for backporting serious fixes to Julia
v0.5. However, v0.6 should be just around the corner. With Julia v0.7 and v1.0.0 not too far out in the horizon either, I’ve decided it’s more important to move forward than to keep v0.4 and v0.5 up
to speed. The dev time is constrained, so currently it’s one or the other. Of course, for users of Julia v0.5. they can simply continue to use Optim.jl v0.7.8. Post Julia’s proper release, backwards
compatibility and continuity will be more important, even if it comes at the expense of development speed.
Another note about the version number: The next version of Optim.jl will be v1.0.0, and we will follow SEMVER 2.0 fully.
Change order of evaluation point and storage arguments
This one is very breaking, although we have set up op a system such that all gradients and Hessians will be checked before proceeding. This check will be removed shortly in a v1.0.0 version bump, so
please correct your code now. Basically, we closed a very old issue (#156) concerning the input argument order in gradients and Hessians. In Julia, an in-place function typically has an exclamation
mark at the end of its name, and the cache as the first argument. In Optim.jl it has been the other way around for the argument order. We’ve changed that, and this means that you now have to provide
“g” or “H” as the first argument, and “x” as the second. The old version
function g!(x, g)
... do something ...
is now
function g!(g, x)
... do something ...
Since v0.7.0, we’ve moved some of the basic infrastructure of Optim.jl to NLSolversBase.jl. This is currently the Non-, Once-, and TwiceDifferentiable types and constructors. This is done to, as a
first step, share code between Optim.jl and LineSearches.jl, and but also NLsolve.jl in the future. At the same time, we’ve made the code a little smarter, such that superfluous calls to the
objective function, gradient, and Hessian are now avoided. As an example, compare the objective and gradient calls in the example in our readme. Here, we optimize the Rosenbrock “banana” function
using BFGS. Since last version of Optim we had to change the output, as it has gone from 157 calls to 53. Much of this comes from this refactoring, but some of it also comes form a better choices for
initial line search steps for BFGS and Newton introduced in #328.
As mentioned, we’ve made the *Differentiable-types a bit smarter, including moving the gradient and Hessian caches into the respective types. This also means, that a OnceDifferentiable type instance
needs to know what the return type of the gradient is. This is done by providing an x seed in the constructor
rosenbrock = Optim.UnconstrainedProblems.examples["Rosenbrock"]
f = rosenbrock.f
g! = rosenbrock.g!
x_seed = rosenbrock.initial_x
od = OnceDifferentiable(f, g!, x_seed)
If the seed also happens to be the initial x, then you do not have to provide an x when calling optimize
julia> optimize(od, BFGS(), Optim.Options(g_tol=0.1))
Results of Optimization Algorithm
* Algorithm: BFGS
* Starting Point: [1.0005999613152214,1.001138415164852]
* Minimizer: [1.0005999613152214,1.001138415164852]
* Minimum: 7.427113e-07
* Iterations: 13
* Convergence: true
* |x - x'| < 1.0e-32: false
|x - x'| = 1.08e-02
* |f(x) - f(x')| / |f(x)| < 1.0e-32: false
|f(x) - f(x')| / |f(x)| = NaN
* |g(x)| < 1.0e-01: true
|g(x)| = 2.60e-02
* stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 45
* Gradient Calls: 45
If you’ve used Optim.jl before, you’ll notice that the output carries a bit more information about the convergence criteria.
LineSearches.jl turned Julian
Line searches used to be chosen using symbols in the method constructor for line search based methods such as GradientDescent, BFGS, and Newton by use of the linesearch keyword. The new version of
LineSearches.jl uses types and dispatch exactly like Optim.jl does for solvers. This means that you now have to pass a type instance instead of a keyword, and this also means that we can open up for
easy tweaking of line search parameters through fields in the line search types.
Let us illustrate by the following example how the new syntax works. First, we construct a BFGS instance without specifying the linesearch. This defaults to HagerZhang.
julia> rosenbrock(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
rosenbrock (generic function with 1 method)
julia> result = optimize(rosenbrock, zeros(2), BFGS())
Results of Optimization Algorithm
* Algorithm: BFGS
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999926033423,0.9999999852005353]
* Minimum: 5.471433e-17
* Iterations: 16
* Convergence: true
* |x - x'| < 1.0e-32: false
|x - x'| = 3.47e-07
* |f(x) - f(x')| / |f(x)| < 1.0e-32: false
|f(x) - f(x')| / |f(x)| = NaN
* |g(x)| < 1.0e-08: true
|g(x)| = 2.33e-09
* stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 53
* Gradient Calls: 53
or we could choose a backtracking line search instead
julia> optimize(rosenbrock, zeros(2), BFGS(linesearch = LineSearches.BackTracking()))
Results of Optimization Algorithm
* Algorithm: BFGS
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999926655744,0.9999999853309254]
* Minimum: 5.379380e-17
* Iterations: 23
* Convergence: true
* |x - x'| < 1.0e-32: false
|x - x'| = 1.13e-09
* |f(x) - f(x')| / |f(x)| < 1.0e-32: false
|f(x) - f(x')| / |f(x)| = NaN
* |g(x)| < 1.0e-08: true
|g(x)| = 8.79e-11
* stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 31
* Gradient Calls: 24
this defaults to cubic backtracking, but quadratic can be chosen using the order keyword
julia> optimize(rosenbrock, zeros(2), BFGS(linesearch = LineSearches.BackTracking(order = 2)))
Results of Optimization Algorithm
* Algorithm: BFGS
* Starting Point: [0.0,0.0]
* Minimizer: [0.9999999926644578,0.9999999853284671]
* Minimum: 5.381020e-17
* Iterations: 23
* Convergence: true
* |x - x'| < 1.0e-32: false
|x - x'| = 4.73e-09
* |f(x) - f(x')| / |f(x)| < 1.0e-32: false
|f(x) - f(x')| / |f(x)| = NaN
* |g(x)| < 1.0e-08: true
|g(x)| = 1.76e-10
* stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 29
* Gradient Calls: 24
LineSearches.jl should have better documentation coming soon, but the code is quite self-explanatory for those who want to twiddle around with these parameters.
The method state is now an argument to optimize
While not always that useful to know for users, we use method states internally to hold all the pre-allocated cache variables that are needed. In the new version of Optim.jl, this can be explicitly
provided by the user such that you can retrieve various diagnostics after the optimization routine is done. One such example is the inverse Hessian estimate that BFGS spits out.
method = BFGS()
options = Optim.Options()
initial_x = rand(2)
d = OnceDifferentiable(f, g!, initial_x)
my_state = Optim.initial_state(method, options, d, initial_x)
optimize{d, method, options, my_state)
The future
We have more changes coming in the near future. There’s PR #356 for a Trust Region solver for cases where you can explicitly calculate Hessian-vector products without forming the Hessian (from
@jeff-regier from the Celeste.jl project), the interior point replacement for our current barrier function approach to box constrained optimization in PR #303, and more.
Solving a simple discrete choice model using Gaussian quadrature
In the style of some of the earlier posts, I present a simple economic problem, that uses some sort of numerical method as part of the solution method. Of course, we use Julia to do so. However, this
time we’re actually relying a bit on R, but don’t tell anyone.
Rust models
In the empirical discrete choice literature in economics, a relatively simple and popular framework is the one that matured in Rust (1987, 1988), and was later named Rust models in Aguirregabiria and
Mira (2010). Basically, we consider an agent who has to choose a (in the infinite horizon) stationary policy (sequence of actions), to solve the following problem
\(\max_{a}E\left\{\sum_{t=0}^{T} \beta^t U(a_t, s_t)|s_0\right\}\)
where \(a=(a_0, a_1, \ldots, a_T)\), and \(s_t\) denotes the states. For simplicity, we consider binary decision problems such that \(a_t\in\{1,2\}\). Assume that there an additive shock, \(\
varepsilon_t\), to utility such that
\(U(a_t,s_t)=U(a_t, x_t, \varepsilon_t) = u(a_t, x_t)+\varepsilon_t\)
where \(s_t=(x_t,\varepsilon_t)\) and \(x_t\) is usually called the observed states.
The additive and time separable nature of the problem allows us to consider a set of simpler problems instead. We reformulate the problem according to the principle of optimality, and write the
problem in its dynamic programming formulation
\( V_t(x_t, \varepsilon_t) = max_{a_t}\left[u(a_t, x_t)+\epsilon_t + \beta E_{s_{t+1}|s_t}(V_{t+1}(x_{t+1}, \varepsilon_{t+1}))\right], \forall t\in\{0,1,\ldots,T\} \)
The object \(V_t\) is called the value function, as it summarizes the optimal value we can obtain. If we assume conditional independence between the observed states and the shocks along with the
assumptions explained in the articles above, we can instead consider this simpler problem
\( W_t(x_t) = E_{\varepsilon_{t+1}|\varepsilon_t}\left\{max_{a_t}(u(a_t, x_t)+\varepsilon_t + \beta E_{x_{t+1}|x_t}((W_{t+1}(x_{t+1}))\right\}, \forall t\in\{0,1,\ldots,T\} \)
where \(W(x_t)\equiv E_{\varepsilon_{t+1}|\varepsilon}\left\{V_{t+1}(x_{t+1},\varepsilon_{t+1})\right\}\). This object is often called the ex-ante or integrated value function. Now, if we assume that
the shocks are mean 0 extreme value type I, we get the following
\( W_t(x_t) = \log\left\{\sum_{a\in\mathcal{A}} \exp\left[u(a_t,x_t)+\beta E_{x_{t+1}|x_t}\left(W_{t+1}(x_{t+1})\right)\right]\right\}, \forall t\in\{0,1,\ldots,T\} \)
At this point we’re very close to something we can calculate. Either we just recursively apply the above to find the finite horizon solution, or if we’re in the infinite horizon case, then then we
can come up with a guess for \(W\), and apply value function iterations (successive application of the right-hand side in the equation above), to find a solution. We just need to be able to handle
the evaluation of the expected value inside the \(\exp\)‘s.
Solving for a continuous function
In the application below, we’re going to have a continuous state. Rust originally solved the problem of handling a continuous function on a computer by discretizing the continuous state in 90 or 175
bins, but we’re going to approach it a bit differently. We’re going to create a type that allows us to construct piecewise linear functions. This means that we’re going to have some nodes where we
calculate the function value, and in between these, we simply use linear interpolation. Outside of the first and last nodes we simply set the value to the value at these nodes. We’re not going to
extrapolate below, so this won’t be a problem.
Let us have a look at a type that can hold this information.
type PiecewiseLinear
To construct an instance of this type from a set of nodes and a function, we’re going to use the following constructor
function PiecewiseLinear(nodes, f)
slopes = Float64[]
fn = f.(nodes)
for i = 1:length(nodes)-1
# node i and node i+1
ni, nip1 = nodes[i], nodes[i+1]
# f evaluated at the nodes
fi, fip1 = fn[i], fn[i+1]
# store slopes in each interval, so we don't have to recalculate them every time
push!(slopes, (fip1-fi)/(nip1-ni))
# Construct the type
PiecewiseLinear(nodes, fn, slopes)
Using an instance of \(PiecewiseLinear\) we can now evaluate the function at all input values between the first and last nodes. However, we’re going to have some fun with types in Julia. In Julia, we
call a function using parentheses \(f(x)\), but we generally cannot call a type instance.
julia> pwl = PiecewiseLinear(1:10, sqrt)
julia> pwl(3.5)
ERROR: MethodError: objects of type PiecewiseLinear are not callable
… but wouldn’t it be great if we could simply evaluate the interpolated function value at 3.5 that easily? We can, and it’s cool, fun, and extremely handy. The name of the concept is: call
overloading. We simply need to define the behavior of “calling” (using parentheses with some input) an instance of a type.
function (p::PiecewiseLinear)(x)
index_low = searchsortedlast(p.nodes, x)
n = length(p.nodes)
if 0 < index_low < n
return p.values[index_low+1]+(x-p.nodes[index_low+1])*p.slopes[index_low]
elseif index_low == n
return p.values[end]
elseif index_low == 0
return p.values[1]
Basically, we find out which interval we’re in, and then we interpolate appropriately in said interval. Let me say something upfront, or… almost upfront. This post is not about optimal performance.
Julia is often sold as “the fast language”, but to me Julia is much more about productivity. Sure, it’s great to be able to optimize your code, but it’s also great to simply be able to do *what you
want to do* – without too much hassle. Now, we can do what we couldn’t before.
julia> pwl(3.5)
We can even plot it together with the actual sqrt function on the interval (2,4).
julia> using Plots
julia> plot(x->pwl(x), 2, 4, label="Piecewise Linear sqrt")
julia> plot!(sqrt, 2, 4, label="sqrt")
julia> savefig("sqrtfig")
and we get
which seems to show a pretty good approximation to the square root function considering the very few nodes.
Numerical integrals using Gaussian quadrature
So we can now create continuous function approximations based on finite evaluation points (nodes). This is great, because this allow us to work with \(W\) in our code. The only remaining problem is:
we need to evaluate the expected value of \(W\). This can be expressed as
\( E_x(f) = \int_a^b f(x)w(x)dx\)
where \(w(x)\) is going to be a probability density function and \(a\) and \(b\) can be finite or infinite and represent upper and lower bounds on the values the random variable (state) can take on.
In the world of Gaussian quadrature, \(w\)‘s are called weight functions. Gaussian quadrature is basically a method for finding good evaluation points (nodes) and associated weights such that the
following approximation is good
\(\int_a^b f(x)w(x)dx\approx \sum_{i=1}^N f(x_i)w_i\)
where \(N\) is the number of nodes. We’re not going to provide a long description of the methods involved, but we will note that the package DistQuads.jl allows us to easily obtain nodes and weights
for a handful of useful distributions. To install this package write
as it is not currently tagged in METADATA.jl. This is currently calling out to R’s statmod package. The syntax is quite simple. Define a distribution instance, create nodes and weights, and calculate
the expected value of the function in three simple steps:
julia> using Distributions, DistQuads
julia> bd = Beta(1.5, 50.0)
Distributions.Beta{Float64}(α=1.5, β=50.0)
julia> dq = DistQuad(bd, N = 64)
DistQuads.DistQuad([0.000334965,0.00133945,0.00301221,0.0053512,0.00835354,0.0120155,0.0163327,0.0212996,0.0269103,0.0331578 … 0.738325,0.756581,0.774681,0.792633,0.810457,0.828194,0.845925,0.863807,0.882192,0.902105],[0.00484732,0.0184431,0.0381754,0.0603806,0.0811675,0.097227,0.106423,0.108048,0.102724,0.0920435 … 1.87035e-28,5.42631e-30,1.23487e-31,2.11992e-33,2.60541e-35,2.13019e-37,1.03855e-39,2.52575e-42,2.18831e-45,2.90458e-49],Distributions.Beta{Float64}(α=1.5, β=50.0))
julia> E(sqrt, dq)
We can then try Monte Carlo integration with many nodes to see how close they are
julia> mean(sqrt.(rand(bd, 100000000)))
and they appear to be in the same ballpark.
To replace, or not to replace
The model we’re considering here is a simple one. It’s a binary choice model very close to the model in Rust (1987). An agent is in charge of maintaining a bus fleet and has a binary choice each
month then the buses come in for maintenance: replace the engine (effectively renewing the bus) or maintain it. Replacement costs a fixed price RC, and regular maintenance has a cost that is a linear
function of the odometer reading since last replacement (or purchase if replacement has never occurred). We can use the expressions above to solve this model, but first we need to specify how the
odometer reading changes from month to month conditional on the choices made. We assume that the odometer reading (mileage) changes according to the following
\(x_{t+1}=\tilde{a}x_{t}+(1-\tilde{a}x_t)\Delta x, \quad\text{where }\Delta x \sim Beta(1.4, 50.0)\)
where \(\tilde{a}=2-a\), and as we remember \(a\in\{1,2\}\). As we see, a replacement returns the state to 0 plus whatever mileage might accumulate that month, and regular maintenance means that the
bus will end up with end of period mileage between \(x_{t+1}\) and 1. To give an idea about the state process, we see the pdf for the distribution of \(\Delta x\) below.
Solving the model
We are now ready to solve the model. Let us say that the planning horizon is 96 months and the monthly discount factor is 0.9. After the 96 months, the bus is scrapped at a value of 2 units of
currency, such that
and from here on, we use the recursion from above. First, set up the state space.
using Distributions, DistQuads, Plots
# State space
bd = Beta(1.5, 50.0)
dq = DistQuad(bd, N = 64)
Sˡ = 0
Sʰ = 1
n_nodes = 100 # arbitrary, but could be varied
nodes = linspace(Sˡ, Sʰ, n_nodes) # doesn't have to be uniformly distributed
RC = 11.5 # Replacement cost
c = 9.0 # parameter in linear maintenance cost c*x
β = 0.9 # discount factor
Then, define utility function and expectation operator
u1 = PiecewiseLinear(nodes, x->-c*x) # "continuous" cost of maintenance
u2 = PiecewiseLinear(nodes, x->-RC) # "continuous" cost of replacement (really just a number, but...)
# Expected value of f at x today given a where x′ is a possible state next period
Ex(f, x, a, dq) = E(x′->f((2-a)*x.+(Sʰ-(2-a)*x).*x′), dq)
Then, we simply
#### SOLVE
V = Array{PiecewiseLinear,1}(70)
V[70] = PiecewiseLinear(nodes, x->2)
for i = 69:-1:1
EV1 = PiecewiseLinear(nodes, x->Ex(V[i+1], x, 1, dq))
EV2 = PiecewiseLinear(nodes, x->Ex(V[i+1], x, 2, dq))
V[i] = PiecewiseLinear(nodes, x->log(exp(u1(x)+β*EV1(x))+exp(u2(x)+β*EV2(x))))
To get our solution. We can then plot either integrated value functions or policies (choice probabilities). We calculate the policies using the following function
function CCP(x, i)
EV1 = PiecewiseLinear(nodes, x->Ex(V[i], x, 1, dq))
EV2 = PiecewiseLinear(nodes, x->Ex(V[i], x, 2, dq))
We see that there are not 69/70 distinct curves in the plots. This is because we eventually approach the “infinite horizon”/stationary policy and solution.
Given the CCPs from above, it is very straight forward to simulate an agent say from period 1 to period 69.
#### SIMULATE
x0 = 0.0
x = [x0]
a0 = 0
a = [a0]
T = 69
for i = 2:T
_a = rand()<CCP(x[end], i) ? 1 : 2
push!(a, _a)
push!(x, (2-_a)*x[end]+(Sʰ-(2-_a)*x[end])*rand(bd))
plot(1:T, x, label="mileage")
Is = []
for i in eachindex(a)
if a[i] == 2
push!(Is, i)
vline!(Is, label="replacement")
This blog post had a look at simple quadrature, creating custom types with call overloading in Julia, and how this can be used to a solve a very simple discrete choice model in Julia. Interesting
extensions are of course to allow for more states, more choices, other shock distributions than extreme value type I and so on. Let me know if you try to extend the model in any of those directions,
and I would love to have a look!
Aguirregabiria, Victor, and Pedro Mira. “Dynamic discrete choice structural models: A survey.” Journal of Econometrics 156.1 (2010): 38-67.
Rust, John. “Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher.” Econometrica: Journal of the Econometric Society (1987): 999-1033.
John, Rust. “Maximum likelihood estimation of discrete control processes.” SIAM Journal on Control and Optimization 26.5 (1988): 1006-1024.
Timing in Julia
Timing code is important when you want to benchmark or profile your code. Is it the solution of a linear system or the Monte Carlo integration scheme that takes up most of the time? Is version A or
version B of a function faster? Questions like that show up all the time. Let us have a look at a few of the possible ways of timing things in Julia.
The basics
The most basic timing functionalities in Julia are the ones included in the Base language. The standard way of timing things in Julia, is by use of the @time macro.
julia> function test(n)
A = rand(n, n)
b = rand(n)
@time A\b
test (generic function with 1 method)
Do note, that the code we want to time is put in a function . This is because everything we do at the top level in the REPL is in global scope. It’s a mistake a lot of people do all the time, but
currently it is a very bad idea performance wise. Anyway, let’s see what happens for n = 1, 10, 100, and 1000.
julia> test(1);
0.000002 seconds (7 allocations: 320 bytes)
julia> test(10);
0.000057 seconds (9 allocations: 1.313 KB)
julia> test(100);
0.001425 seconds (10 allocations: 80.078 KB)
julia> test(1000);
0.033573 seconds (10 allocations: 7.645 MB, 27.81% gc time)
julia> test(1000);
0.045214 seconds (10 allocations: 7.645 MB, 47.66% gc time)
The first run is to compile both test, and then we have a look at what happens when the dimension of our problem increases. Elapsed time seems to increase, and we also see that the number of
allocations, and the amount of memory that was allocated increases. For the runs with dimension 1000 we see something else in the output. 30-50% of the time was spent in “gc”. What is this? Julia is
a garbage collected language. This means that Julia keeps track of current allocations, and frees the memory if it isn’t needed anymore. It doesn’t do this all the time, though. Running the
1000-dimensional problem once more gives us
julia> test(1000)
0.029277 seconds (10 allocations: 7.645 MB)
We see it runs slightly faster, and there is no GC time this time around. Of course, these things will look slightly different if you try to replicate them.
So now we can time. But what if we want to store this number? We could be tempted to try
t = @time 3+3
but we will realize, that what is returned is the return value of the expression, not the elapsed time. To save the time, we can either use @timed or @elapsed. Let us try to change the @time to
@timed and look at the output when we have our new test2 function return the return value.
julia> function test2(n)
A = rand(n, n)
b = rand(n)
@timed A\b
test2 (generic function with 1 method)
julia> test2(3)
We see that it returns a tuple with: the return value of A\b followed by the elapsed time, then the bytes allocated, time spent in garbage collection, and lastly some further memory counters. This is
great as we can now work with the information @time printed, but we still have access to the results of our calculations. Of course, it is a bit involved to do it this way. If we simply wanted to see
the elapsed time to act on that – then we would just use @time as we did above.
Before we move on to some simpler macros, let us consider the last “time*-family” macro: @timev. As we saw above, @timed contained more information about memory allocation than @time printed. If we
want the “verbose” version, we use @timev (v for verbose):
julia> function test3(n)
A = rand(n, n)
b = rand(n)
@timev A\b
test3 (generic function with 1 method)
Running test3 on a kinda large problem, we see that is does indeed print the contents of Base.GC_Diff
julia> test3(5000);
1.923164 seconds (12 allocations: 190.812 MB, 4.67% gc time)
elapsed time (ns): 1923164359
gc time (ns): 89733440
bytes allocated: 200080368
pool allocs: 9
malloc() calls: 3
GC pauses: 1
full collections: 1
If any of the entries are zero, the corresponding lines are omitted.
julia> test3(50);
0.001803 seconds (10 allocations: 20.828 KB)
elapsed time (ns): 1802811
bytes allocated: 21328
pool allocs: 9
malloc() calls: 1
Of the three macros, you’ll probably not use @timev a lot.
Simpler versions
If we only want the elapsed time or only want the allocations, then we used either the @elapsed or @allocated macros. However, these do not return the results of our calculations, so in many cases it
may be easier to just used @timed, so we can grab the results, the elapsed time, and the allocation information. “MATLAB”-style tic();toc()‘s are also available. toc() prints the time, while toq() is
used if we want only the returned time without the printing. It is also possible to use time_ns() to do what time.time() would do in Python, although for practically all purposes, the above macros
are recommended.
More advanced functionality
Moving on to more advanced features, we venture into the package ecosystem.
Nested timings
The first package I will present is the nifty TimerOutputs.jl by Kristoffer Carlsson. This packages essentially allows you to nest @time calls. The simplest way to show how it works, is to use the
example posted at the announcement (so credit to Kristoffer for the example).
using TimerOutputs
# Create the timer object
to = TimerOutput()
# Time something with an assigned label
@timeit to "sleep" sleep(0.3)
# Data is accumulated for multiple calls
for i in 1:100
@timeit to "loop" 1+1
# Nested sections are possible
@timeit to "nest 1" begin
@timeit to "nest 2" begin
@timeit to "nest 3.1" rand(10^3)
@timeit to "nest 3.2" rand(10^4)
@timeit to "nest 3.3" rand(10^5)
Basically we’re timing the sleep call in one time counter, all the additions in the loop in another counter, and then we do some nested generation of random numbers. Displaying the to instance gives
us something like the following
Time Allocations
────────────────────── ───────────────────────
Tot / % measured: 6.48s / 5.60% 77.4MiB / 12.0%
Section ncalls time %tot avg alloc %tot avg
sleep 1 338ms 93.2% 338ms 804KiB 8.43% 804KiB
nest 1 1 24.7ms 6.80% 24.7ms 8.52MiB 91.5% 8.52MiB
nest 2 1 9.10ms 2.51% 9.10ms 899KiB 9.43% 899KiB
nest 3.1 1 3.27ms 0.90% 3.27ms 8.67KiB 0.09% 8.67KiB
nest 3.3 1 3.05ms 0.84% 3.05ms 796KiB 8.34% 796KiB
nest 3.2 1 2.68ms 0.74% 2.68ms 92.4KiB 0.97% 92.4KiB
loop 100 6.97μs 0.00% 69.7ns 6.08KiB 0.06% 62B
which nicely summarizes absolute and relative time and memory allocation of the individual @timeit calls. A real use case could be to see what the effect is of using finite differencing to construct
the gradient for the Generalized Rosenbrock (GENROSEN) problem from CUTEst.jl using a conjugate gradient solver in Optim.jl.
using CUTEst, Optim, TimerOutputs
nlp = CUTEstModel("GENROSE")
const to = TimerOutput()
f(x ) = @timeit to "f" obj(nlp, x)
g!(g, x) = @timeit to "g!" grad!(nlp, x, g)
@timeit to "Conjugate Gradient" begin
res = optimize(f, g!, nlp.meta.x0, ConjugateGradient(), Optim.Options(iterations=5*10^10));
@timeit to "Conjugate Gradient (FiniteDiff)" begin
res = optimize(f, nlp.meta.x0, ConjugateGradient(), Optim.Options(iterations=5*10^10));
show(to; allocations = false)
the output is a table as before, this time without the allocations (notice the use of the allocations keyword in the show method)
Tot / % measured: 33.3s / 100%
Section ncalls time %tot avg
Conjugate Gradient (FiniteDiff) 1 33.2s 99.5% 33.2s
f 1.67M 32.6s 97.9% 19.5μs
Conjugate Gradient 1 166ms 0.50% 166ms
g! 1.72k 90.3ms 0.27% 52.6μs
f 2.80k 59.1ms 0.18% 21.1μs
And we conclude: finite differencing is very slow when you’re solving a 500 dimensional unconstrained optimization problem, and you really want to use the analytical gradient if possible.
Timing individual pieces of code can be very helpful, but when we’re timing small function calls, this way of measuring performance can be heavily influenced by noise. To remedy that, we use proper
benchmarking tools. The package for that, well, it’s called BenchmarkTools.jl and is mainly written by Jarrett Revels. The package is quite advanced in its feature set, but its basic functionality is
straight forward to use. Please see the manual for more details than we provide here.
Up until now, we’ve asked Julia to tell us how much time some code took to run. Unfortunately for us, the computer is doing lots of stuff besides the raw calculations we’re trying to time. From the
example earlier, this means that we have a lot of noise in our measure of the time it takes to solve A\b. Let us try to run test(1000) a few times
julia> test(1000);
0.029859 seconds (10 allocations: 7.645 MB)
julia> test(1000);
0.033381 seconds (10 allocations: 7.645 MB, 6.41% gc time)
julia> test(1000);
0.024345 seconds (10 allocations: 7.645 MB)
julia> test(1000);
0.039585 seconds (10 allocations: 7.645 MB)
julia> test(1000);
0.037154 seconds (10 allocations: 7.645 MB, 2.82% gc time)
julia> test(1000);
0.024574 seconds (10 allocations: 7.645 MB)
julia> test(1000);
0.022185 seconds (10 allocations: 7.645 MB)
There’s a lot of variance here! Let’s benchmark instead. The @benchmark macro won’t work inside a function as above. This means that we have to be a bit careful (thanks to Fengyang Wang for
clarifying this). Consider the following
julia> n = 200;
julia> A = rand(n,n);
julia> b = rand(n);
julia> @benchmark A\b
memory estimate: 316.23 KiB
allocs estimate: 10
minimum time: 531.227 μs (0.00% GC)
median time: 718.527 μs (0.00% GC)
mean time: 874.044 μs (3.12% GC)
maximum time: 95.515 ms (0.00% GC)
samples: 5602
evals/sample: 1
This is fine, but since A and b are globals (remember, if it ain’t wrapped in a function, it’s a global when you’re working from the REPL), we’re also measuring the time dynamic dispatch takes.
Dynamic dispatch happens here, because “Julia” cannot be sure what the types of A and b are when we invoke A\b since they’re globals. Instead, we should use interpolation of the non-constant
variables, or mark them as constants using const A = rand(n,n) and const b = rand(20). Let us use interpolation.
julia> @benchmark $A\$b
memory estimate: 316.23 KiB
allocs estimate: 10
minimum time: 531.746 μs (0.00% GC)
median time: 717.269 μs (0.00% GC)
mean time: 786.240 μs (3.22% GC)
maximum time: 12.463 ms (0.00% GC)
samples: 6230
evals/sample: 1
We see that the memory information is identical to the information we got from the other macros, but we now get a much more robust estimate of the time it takes to solve our A\b problem. We also see
that dynamic dispatch was negligible here, as the solution takes much longer to compute than for Julia to figure out which method to call. The @benchmark macro will do various things automatically,
to try to give as accurate results as possible. It is also possible to provide custom tuning parameters, say if you’re running these benchmarks over an extended period of time and want to track
performance regressions, but that is beyond this blog post.
Dynamic dispatch
Before we conclude, let’s have a closer look at the significance of dynamic dispatch. When using globals it has to be determined at run time which method to call. If there a few methods, this may not
be a problem, but the problem begins to show itself when a function has a lot of methods. For example, on Julia v0.5.0, identity has one method, but + has 291 methods. Can we measure the significance
of dynamic dispatch then? Sure. Just benchmark with, and without interpolation (thanks again to Fengyang Wang for cooking up this example). To keep output from being too verbose, we’ll use the @btime
macro – again from BenchmarkTools.jl (there is also an @belapsed that returns the minimum time in seconds).
julia> x = 0
julia> @btime identity(x)
1.540 ns (0 allocations: 0 bytes)
julia> @btime +x
15.837 ns (0 allocations: 0 bytes)
julia> @btime identity($x)
1.540 ns (0 allocations: 0 bytes)
julia> @btime +$x
1.548 ns (0 allocations: 0 bytes)
As we can see, calling + on the global x takes around 10 times a long as the single method function identity. To show that declaring the input a const and interpolating the variable gives the same
result, consider the example below.
julia> const y = 0
julia> @btime identity(y)
1.539 ns (0 allocations: 0 bytes)
julia> @btime +y
1.540 ns (0 allocations: 0 bytes)
julia> @btime identity($y)
1.540 ns (0 allocations: 0 bytes)
julia> @btime +$y
1.540 ns (0 allocations: 0 bytes)
We see that interpolation is not needed, as long as we remember to use constants.
There are quite a few ways of measuring performance in Julia. I’ve presented some of them here, and hopefully you’ll be able to put the tools to good use. The functionality from Base is good for many
purposes, but I really like the nested time measuring in TimerOutputs.jl a lot, and for serious benchmarking it is impossible to ignore BenchmarkTools.jl.
DynProg Class – Week 2
This post, and other posts with similar tags and headers, are mainly directed at the students who are following the Dynamic Programming course at Dept. of Economics, University of Copenhagen in the
Spring 2017. The course is taught using Matlab, but I will provide a few pointers as to how you can use Julia to solve the same problems. So if you are an outsider reading this I welcome you, but you
won’t find all the explanations and theory here. If you want that, you’ll have to come visit us at UCPH and enroll in the class!
This week we continue with a continuous choice model. This means we have to use interpolation and numerical optimization.
A (slightly less) simple model
Consider a simple continuous choice consumption-savings model:
\(V_t(M_t) = \max_{C_t}\sqrt{C_t}+\mathbb{E}[V_{t+1}(M_{t+1})|C_t, M_t]\)
subject to
\(M_{t+1} = M_t – C_t+R_t\\
C_t\leq M_t\\
where \(R_t\) is 1 with probability \(\pi\) and 0 with probability \(1-\pi\), \(\beta=0.9\), and \(\bar{M}=5\)
Last week the maximization step was merely comparing values associated with the different discrete choices. This time we have to do continuous optimization in each time period. Since the problem is
now continuous, we cannot solve for all \(M_t\). Instead, we need to solve for \(V_t\) at specific values of \(M_t\), and interpolate in between. Another difference to last time is the fact that the
transitions are stochastic, so we need to form (very simple) expectations as part of the Bellman equation evaluations.
It is of course always possible to make your own simple interpolation scheme, but we will use the functionality provided by the Interpolations.jl package. To perform interpolation, we need a grid to
be used for interpolation \(\bar{x}\), and calculate the associated function values.
f(x) = (x-3)^2
x̄ = linspace(1,5,5)
fx̄ = f.(x̄)
Like last time, we remember that the dot after the function name and before the parentheses represent a “vectorized” call, or a broadcast – that is we call f on each element of the input. We now use
the Interpolations.jl package to create an interpolant \(\hat{f}\).
using Interpolations
f̂ = interpolate((collect(x̄),), fx̄, Gridded(Linear()))
We can now index into \(\hat{f}\) as if it was an array
f̂[-3] #returns 16.0
We can also plot it
using Plots
Solving the model
Like last time, we prepare some functions, variables, and empty containers (Arrays)
# Model again
u(c) = sqrt(c)
T = 10; β = 0.9
π = 0.5 ;M₁ = 5
# Number of interpolation nodes
Nᵐ = 50 # number of grid points in M grid
Nᶜ = 50 # number of grid points in C grid
M = Array{Vector{Float64}}(T)
V = Array{Any}(T)
C = Array{Any}(T)
The V and C arrays are allocated using the type “Any”. We will later look at how this can hurt performance, but for now we will simply do the convenient thing. Then we solve the last period
M[T] = linspace(0,M₁+T,Nᵐ)
C[T] = M[T]
V[T] = interpolate((M[T],), u.(C[T]), Gridded(Linear()))
This new thing here is that we are not just saving V[T] as an Array. The last element is the interpolant, such that we can simply index into V[T] as if we had the exact solution at all values of M
(although we have to remember that it is an approximation). For all periods prior to T, we have to find the maximum as in the Bellman equation from above. To solve this reduced “two-period” problem
(sum of utility today and discounted value tomorrow), we need to form expectations over the two possible state transitions given an M and a C today, and then we need to find the value of C that
maximizes current value. We define the following function to handle this
# Create function that returns the value given a choice, state, and period
v(c, m, t, V) = u(c)+β*(π*V[t+1][m-c+1]+(1-π)*V[t+1][m-c])
Notice how convenient it is to simply index into V[t] using the values we want to evaluate tomorrow’s value function at. We perform the maximization using grid search on a predefined grid from 0 to
the particular M we’re solving form. If we abstract away from the interpolation step, this is exactly what we did last time.
for t = T-1:-1:1
M[t] = linspace(0,M₁+t,Nᵐ)
C[t] = zeros(M[t])
Vt = fill(-Inf, length(M[t]))
for (iₘ, m) = enumerate(M[t])
for c in linspace(0, m, Nᶜ)
_v = v(c, m, t, V)
if _v >= Vt[iₘ]
Vt[iₘ] = _v
C[t][iₘ] = c
V[t] = interpolate((M[t],), Vt, Gridded(Linear()))
Then we can plot the value functions to verify that they look sensible
Nicely increasing in time and in M.
Using Optim.jl for optimization
The last loop could just as well have been done using a proper optimization routine. This will in general be much more robust, as we don’t confine ourselves to a certain amount of C-values. We use
the one of the procedures in Optim.jl. In Optim.jl, constrained, univariate optimization is available as either Brent’s method or Golden section search. We will use Brent’s method. This is the
standard method, so an optimization call simply has the following syntax
using Optim
f(x) = x^2
optimize(f, -1.0, 2.0)
Unsurprisingly, this will return the global minimizer 0.0. However, if we constrain ourselves to a strictly positive interval
optimize(f, 1.0, 2.0)
we get a minimizer of 1.0. This is not the unconstrained minimizer of the square function, but it is minimizer given the constraints. Then, it should be straight forward to see how the grid search
loop can be converted to a loop using optimization instead.
for t = T-1:-1:1
update_M!(M, M₁, t, Nᵐ)
C[t] = zeros(M[t])
Vt = fill(-Inf, length(M[t]))
for (iₘ, m) = enumerate(M[t])
if m == 0.0
C[t][iₘ] = m
Vt[iₘ] = v(m, m, t, V)
res = optimize(c->-v(c, m, t, V), 0.0, m)
Vt[iₘ] = -Optim.minimum(res)
C[t][iₘ] = Optim.minimizer(res)
V[t] = interpolate((M[t],), Vt, Gridded(Linear()))
If our agent has no resources at the beginning of the period, the choice set has only one element, so we skip the optimization step. We also have to negate the minimum to get the maximum we’re
looking for. The main advantage of using a proper optimization routine is that we’re not restricting C to be in any predefined grid. This increases precision. If we look at the number of calls to “v”
(using Optim.f_calls(res)), we see that it generally takes around 10-30 v calls. With only 10-30 grid points from 0 up to M, we would generally get an approximate solution of much worse quality.
Julia bits and pieces
This time we used a few different package from the Julia ecosystem: Plots.jl, Interpolations.jl, and Optim.jl. These are based on my personal choices (full disclosure: I’ve contributed to the first
and the last), but there are lots of packages to explore. Visit the JuliaLang discourse forum or gitter channel to discuss Julia and the various packages with other users.
DynProg Class – Week 1
This post, and other posts with similar tags and headers, are mainly directed at the students who are following the Dynamic Programming course at Dept. of Economics, University of Copenhagen in the
Spring 2017. The course is taught using Matlab, but I will provide a few pointers as to how you can use Julia to solve the same problems. So if you are an outsider reading this I welcome you, but you
won’t find all the explanations and theory here. Then you’ll have to come visit us at UCPH and enroll in the class!
We start out with very simple examples and Julia code, and gradually ramp up the difficulty level.
Simple model
Consider the simplest consumption-savings model:
\(V^*(M) = \max_{C_1,C_2,\ldots,C_T}\beta^0\sqrt{C_1},\beta^1\sqrt{C2},\ldots,\beta^T\sqrt{C_T}\)
subject to
$$\bar{M} = C_1+C_2+\ldots+C_T\equiv m_1$$
$$m_{t+1} = m_t – C_t$$
Let us try to solve such models.
1) Brute force
This is not a very clever way, but let us start somewhere. We don’t have a better method – yet! Set T=2. Note, there is only one free choice to vary. Given some consumption in period t, the other is
given due to the “No cake left” boundary condition.
# Primitives
β = 0.90
α = 0.5
u(x, α) = x^α
## State
M̄ = 5.0
Cᵒᵖᵗ = u(0.0, α) + β*u(M̄, α)
Vᵒᵖᵗ = -Inf
for C1 in 1.0:1.0:M̄
V = u(C1, α) + β*u(M̄-C1, α)
if Vᵒᵖᵗ <= V
Vᵒᵖᵗ = V
Cᵒᵖᵗ = C1
println("* optimimum: ", Vᵒᵖᵗ)
println("* optimizer: ", Cᵒᵖᵗ)
This tells us that our agent should consume three units in the first period and two in the last period.
2) Vary β and M
Check that different M’s and β’s give sensible predictions.
We don’t want to rerun that code manually each time – we’re going to need something smarter. Let’s wrap it in a function. In Julia, a typical function (method) definition looks something like the
function brute_force(M̄, α, β)
Vᵒᵖᵗ = u(0.0, α) + β*u(M̄, α)
Cᵒᵖᵗ = 0
for C1 in 1.0:1.0:M̄
V = u(C1, α) + β*u(M̄-C1, α)
if Vᵒᵖᵗ <= V
Vᵒᵖᵗ = V
Cᵒᵖᵗ = C1
Vᵒᵖᵗ, Cᵒᵖᵗ
Notice that whatever is on the last line will be returned. For example, you can verify that it returns the same thing as the loop above, or check what happens if β is 0:
brute_force(M̄, α, 0.0)
The output tells us that the agent should consume everything in the first period. This is consistent with the intuition that the agent doesn’t care about the future at all. What if we have no units
at the first period?
brute_force(0, α, β)
We see that the agent should consume nothing. This might seem like a stupid thing to test for, but it is always important to sanity check your code. If it can’t predict things that are obvious, it
will probably also give wrong answers to more complex cases. It can be hard to come up with such cases, but the last two examples will always hold. Full discounting -> full consumption today. No
ressources -> no consumption today (unless you can borrow).
3) Solve a model using backwards induction
It requires little convincing to realize, that it is wasteful to check all possible combinations – especially when the number of periods increases. Instead, we use the principle of optimality to
solve the problem using dynamic programming. To solve the model using dynamic programming, we need to create a few variables to hold the value and policy functions. These will hold the solutions from
each (t,m) subproblem. We start with T=3
T = 3
M = 0.0:M̄
Vᵒᵖᵗ = [zeros(M) for t = 1:T]
Cᵒᵖᵗ = [zeros(M) for t = 1:T]
We’ve used vectors of vectors here, but could just as well have created a $5\times 3$ matrix in this case. However, in a more general class of models, the number of states might vary over time, and
there a vector of vector becomes convenient. With these variables in place, we can solve the last period by realizing that given some resource level in period T, we need to consume everything in
order fulfill the constraints of the problem (no cake left on the table!).
Vᵒᵖᵗ[end] = sqrt.(M)
Cᵒᵖᵗ[end] = M
Notice the dot after sqrt. This is Julia syntax for “broadcast the square root function over M”. Then, we solve for the period t=T-1, then t=T-2, and then we’re done. We implement a straight forward
loop over all possible m’s and feasible consumption given these m’s, by using the fact that the feasible values for consumption make up the set
for t = T-1:-1:1
for imₜ = 1:length(M)
Vᵒᵖᵗ[t][imₜ] = -Inf # some value lower than sqrt(0)
for (icₜ,cₜ) = enumerate(0:M[imₜ])
v = u(cₜ, α) + β*Vᵒᵖᵗ[t+1][1+imₜ-icₜ]
if v > Vᵒᵖᵗ[t][imₜ]
Vᵒᵖᵗ[t][imₜ] = v
Cᵒᵖᵗ[t][imₜ] = cₜ
println("* optimimum: ", Vᵒᵖᵗ)
println("* optimizer: ", Cᵒᵖᵗ)
Again, you should try to see if some of all this shouldn’t really be separate functions.
Julia bits and pieces
To close off this post, let me just briefly touch upon some of the things we used above, that might be different from Matlab. To allocate an empty array, write
V = Vector(10)
or a vector with some specific element value (here, a Float64 NaN)
W = fill(NaN, 10)
Indexing into arrays is done with square brackets. For example, let’s assign the first element of W to a variable WW
WW = W[1]
And the same goes for setting the value at some index
W[1] = 0.4;
Functions are define using either short or long form
function f(x,y)
x+y+rand() # returns the sum of the two inputs and a random number
g(x,y) = x+y+rand()
and comments are inserted using # or the #= =# blocks
# This is a comment
#= This
For-loops are written as
for a = some_iterator
# do something
or to get both the value and a counter use enumerate
for (i, a) = enumerate(some_iterator)
# do stuff with a and index something with i
Code that should be run conditional on some statement being true is written as
if condition
# do something if condition is satisfied
and that’s all for now, folks!
Julia on Azure
Microsoft recently added Julia to Azure, their cloud computing service. I got curious, and headed over to Azure to check it out. On the platform, Julia is provided in the form of the JuliaPro bundle
shipped by Julia Computing. JuliaPro consists of the latest stable Julia release, the Juno IDE (Atom based) + debugger, Jupyter notebooks, and a bundle of curated packages from different categories
such as: statistics (DataFrames.jl, Distributions.jl, …), optimization (JuMP.jl, Optim.jl), language interoperability (RCall.jl, JavaCall.jl, PyCall.jl), Deep Learning (Mocha.jl, MXNet.jl, Knet.jl),
and more. The packages come precompiled, so when JuliaPro is installed, it should “Just Work” (of course packages installed manually also works, but some packages take a long time to build).
As with many other services, you pay a price per hour you spend on the VMs. The smallest ones are quite cheap, but you can scale up to very powerful setups. Most Julia packages are quite quick to
build (precompile), but when you’re paying by the hours, precompiled packages (that JuliaPro comes with) can be quite neat. Luckily, there is a trial account where you get some “getting started”
credit. If you don’t boot up the most powerful VM configuration right away, there is plenty of credit to get started. I won’t explain the process of setting up a VM here, as it is very easy and
self-explanatory. I set up a windows/data science VM, and connected using Remmina – and wouldn’t you know it Julia is right there on the desktop: REPL, Juno, and Jupyter right next to each other:
Let’s try to open Juno, because… you’re worth it! (click to enlarge)
I’m just adding a package, loading it, creating some data, and plotting it (using a theme from PlotThemes.jl). Everything works fine, although I should note that Plots.jl took a while to load the
first time around, as I’ve chosen the smallest VM available. Since we’re using JuliaPro, we could have been using Gadfly or PyPlot instead, and it would have been precompiled and ready to go.
From here on, it’s up to you what to do. Analyse the stars, try out MXNet.jl, predict company defaults, or whatever you think is interesting. | {"url":"http://www.pkofod.com/","timestamp":"2024-11-04T15:22:41Z","content_type":"text/html","content_length":"159806","record_id":"<urn:uuid:6a3a6cbd-c218-48a5-8a7b-883f0b94d79d>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00757.warc.gz"} |
Note: This is a work in progress and will be formatting errors. Read more about the project on the home page.
the atan of number
Value returned: number equal to the trigonometric arc tangent of number expressed in radians
Note: There are 2 * pi radians in 360 degrees.
An expression that evaluates to a number.
For example:
169 / 19
(60.625 * 500)
i+1 -- (where i is a number)
line 1 of fld "Debits"
Note: Formally, HyperCard distinguishes between factors (simple values) and expressions. The difference between factors and expression matters only if you like to drop parentheses. Most functions
take factors as their parameters, which is why abs of -10 + 2 returns 12 and abs of (-10 + 2) returns 8. In short, always use parentheses to group things the way you want them to evaluate, and you
won’t have to worry about the difference between factors and expressions.
Related Topics
« annuity | HyperTalk Reference | average »
Version 0.7b1 (March 24, 2022) | {"url":"http://hypercard.center/HyperTalkReference/52019","timestamp":"2024-11-02T10:56:04Z","content_type":"text/html","content_length":"9368","record_id":"<urn:uuid:7b844519-e902-475c-8f9d-c9e581f1e004>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00826.warc.gz"} |
How Torsion as Presented by De Sabbata and Sivaram in Erice 1990 Argument as Modified May Permit Cosmological Constant, and Baseline as to Dark Energy
How Torsion as Presented by De Sabbata and Sivaram in Erice 1990 Argument as Modified May Permit Cosmological Constant, and Baseline as to Dark Energy ()
1. Introduction: Review of the Purported Role of Torsion Given by De Sabbata and Sirvaram 1990 in Its Cancelation of Vacuum Energy/Cosmological Constant. Versus a Preview of What We Will Be Doing
First of all I wish to thank the referee for his following comments which are reproduced verbatim: These are put into the introduction and they are meant as an addendum to the derivation which I
tried to put in, for the sake of readability of this document, the oversight which I did not see due to lack of experience in torsion physics. So after this synopsis is included, I will commence to
add in the derivational points adhered to in this review right after my text.
Here are some of the points raised by the referee, in his review of my document.
There are other papers in which it is pointed out that the torsion term (quoted often by Bekenstein) can indeed give rise to a residual Cosmological constant term of the observed magnitude. This has
also been used to generate inflation.
The present Reviewer has always been an advocate for a lambda term. So I agree with the Author that torsion term can give rise to a lambda term. Of course the source of the spin density in the
present paper is that of primordial BH, but essentially a similar argument. Equations 4 to 14 are just the same as in Ref. [1] .
End of quote.
To wit, what I wish to do is to adhere to the fundaments of the basic document, and then proceed to address the issues brought up by the referee, i.e. in references presented he uses different
arguments as to what generates spin density than I do, but I used, as he noted, spin density as a direct product of primordial black holes forming initially.
The other difference is in that my primordial black holes are scaled via Bose Einstein condensation, and also are linked to gravitons.
Having said that, more of the referees review is included in the end of this paper, whereas we refer to three references which the referee thought was of import and consideration at the end of my
summary of the arguments presented.
2. The Basic Argument as to Black Holes as a Source of Torsion Given for Review
To begin this look at [1] [2] [3] which purports to show a global cancellation of a vacuum energy term, which is akin, as we discuss later to cancelling the following completely [3] [4]
$\begin{array}{l}{\rho }_{\Lambda }{c}^{2}=\underset{0}{\overset{{E}_{\text{Plank}}/c}{\int }}\frac{4\pi {p}^{2}\text{d}p}{{\left(2\pi \hslash \right)}^{3}}\cdot \left(\frac{1}{2}\cdot \sqrt{{p}^{2}
{c}^{2}+{m}^{2}{c}^{4}}\right)\approx \frac{{\left(3×{10}^{19}\text{\hspace{0.17em}}\text{GeV}\right)}^{4}}{{\left(2\pi \hslash \right)}^{3}}\\ \underset{{E}_{\text{Plank}}/c\to {10}^{-30}}{\to }\
frac{{\left(2.5×{10}^{-11}\text{\hspace{0.17em}}\text{GeV}\right)}^{4}}{{\left(2\pi \hslash \right)}^{3}}\end{array}$(1)
In [1] , the first line is the vacuum energy which is completely cancelled in their formulation of application of Torsion. In our article we are arguing for the second line. In fact, in our
formulation our reduction to the second line of Equation (1) will be to confirm the following change in the Planck energy term given by [1]
$\frac{\Delta E}{c}={10}^{18}\text{\hspace{0.17em}}\text{GeV}-\frac{{n}_{\text{quantum}}}{2c}\simeq {10}^{-12}\text{\hspace{0.17em}}\text{GeV}$(2)
The term n (quantum) comes from a Corda derived expression as to energy level of relic black holes [4] .
We argue that our application of [1] [2] will be commensurate with Equation (2) which uses the value given in [2] as to the following i.e. relic black holes will contribute to the generation of a cut
off of the energy of the integral given in Equation (1) whereas what is done in Equation (1) by [1] [2] is restricted to a different venue which is reproduced below, namely cancellation of the
following by Torsion
${\rho }_{\Lambda }{c}^{2}=\underset{0}{\overset{{E}_{\text{Plank}}/c}{\int }}\frac{4\pi {p}^{2}\text{d}p}{{\left(2\pi \hslash \right)}^{3}}\cdot \left(\frac{1}{2}\cdot \sqrt{{p}^{2}{c}^{2}+{m}^{2}
{c}^{4}}\right)\approx \frac{{\left(3×{10}^{19}\text{\hspace{0.17em}}\text{GeV}\right)}^{4}}{{\left(2\pi \hslash \right)}^{3}}$(3)
Furthermore, the claim in [1] is that there is no cosmological constant, i.e. that Torsion always cancelling Equation (3) which we view is incommensurate with Table 1 as of [3] which is given below.
We claim that the influence of Torsion will aid in the decomposition of what is given in Table 1 from [3] and will furthermore lead to the influx of primordial black holes which we claim is
responsible for the behavior of Equation (2) above.
We should note in this, that we are assuming that what we refer to later as Torsion spin density is a direct consequence of primordial black holes, and we solidly link the presence of primordial
black holes as given in [2] to consequential contributions to our ideas of the cosmological constant. In doing so, we should note that there would be a causal discontinuity between the prior to
present universe, as in [2] to a modified Penrose CCC model, of which the black holes prior to Planckian space-time would be enormous whereas what is accessed at the beginning of Planckian space-time
would be Plank mass valued initial black holes. This is a breakage of earlier space-time structure, and we leave the further formation of these initially forming black holes, as further research
projects which will be necessary when we seek optimal data sets as to forming our research confirmation of this paper’s hypothesis.
As we briefly alluded to we are assuming very small initial black holes, and from [2] as using Penrose cyclic conformal cosmology as given in the document [2] .
Table 1. Pre to Post Planckian black holes, assuming Cyclic Conformal technology.
3. Now for the Statement of the Torsion Problem as Given in [1] with a Nod to [5] [6] [7] [8] , in the Massless Particle Case, Initially
The author is very much aware as to quack science as to purported torsion physics presentations and wishes to state that the torsion problem is not linked to anything other than disruption as to the
initial configuration of the expansion of the universe and cosmology, more in the spirit of [6] [7] and is nothing else. Hence, in saying this we wish to delve into what was given in [1] with a
subsequent follow up and modification: We first follow the description of [1] to remove Torsion physics from the quacks.
To do this, note that in [1] the vacuum energy density is stated to be
${\rho }_{vac}=\Lambda {}_{eff}c{}^{4}/8\pi G$(4)
whereas the application is given in terms of an antisymmetric field strength ${S}_{\alpha \beta \gamma }$ [8] .
In [1] due to the Einstein Cartan action, in terms of a SL(2, C) gauge theory, we write from [1]
$L=-R/\left(16\pi G\right)+{S}_{\alpha \beta \gamma }{S}^{\alpha \beta \gamma }/2\pi G$(5)
R here is with regards to Ricci scalar and Tensor notation and ${S}_{\alpha \beta \gamma }$ is related to a conserved current closing in on the SL(2, C) algebra as given by
${J}^{\mu }={J}^{\mu }+1/\left(16\pi G\right){\epsilon }^{\mu \alpha \beta \gamma }{S}_{\alpha \beta \gamma }$(6)
This is where we define
${S}_{\alpha \beta \gamma }={c}_{\alpha }×{f}_{\beta \gamma }$(7)
where ${c}_{\alpha }$ is the structure constant for the group SL(2, C), and
${f}_{\beta \gamma }\cdot \stackrel{¯}{g}={F}_{\beta \gamma }$(8)
Is for tangent vectors to the gauge generators of SL(2, C), and also for Gauge fields ${A}_{\gamma }$
${F}_{\beta \gamma }={\partial }_{\beta }{A}_{\gamma }-{\partial }_{\gamma }{A}_{\beta }+\left[{A}_{\beta },{A}_{\gamma }\right]$(10)
And that there is furthermore the restriction that
${\partial }_{\rho }\left({\epsilon }^{\rho \alpha \beta \gamma }{S}_{\alpha \beta \gamma }\right)=0$(11)
Finally in the case of massless particles with torsion present we have a space time metric
$\text{d}{s}^{2}=\text{d}{\tau }^{2}+{a}^{2}\left(\tau \right){\text{d}}^{2}{\Omega }_{3}$(12)
where ${\text{d}}^{2}{\Omega }_{3}$ is the metric of ${S}^{3}$ .
Then the Einstein field equations reduce to in this torsion application, (no mass to particles) as
${\left(\text{d}a/\text{d}\tau \right)}^{2}=1-{r}_{\mathrm{min}}^{4}/{a}^{4}$(13)
With, if S is the so called spin scalar and identified as the basic $\hslash$ unit of spin
4. How to Modify Equation (13) in the Presence of Matter via Yang Mills Fields ${F}_{\mu v}^{\beta }$
First of all, this involves a change of Equation (5) to read
$L=-R/\left(16\pi G\right)+{S}_{\alpha \beta \gamma }{S}^{\alpha \beta \gamma }/2\pi G+\left(1/4{g}^{2}\right){F}_{\mu v}^{\beta }{F}_{\beta }^{\mu u }$(15)
And eventually we have a re do of Equation (13) to read as
${\left(\text{d}a/\text{d}\tau \right)}^{2}=1-{\beta }_{1}/{a}^{2}-{\beta }_{2}/{a}^{4}$(16)
If $g=\hslash c$ we have ${\beta }_{1}={r}_{\mathrm{min}}^{2},{\beta }_{2}={r}_{\mathrm{min}}^{4}$ , and the minimum radius is identified with a Planck Radius so then
${\left(\text{d}a/\text{d}\tau \right)}^{2}=1-\left({\beta }_{1}={\mathcal{l}}_{P}^{2}\right)/{a}^{2}-\left({\beta }_{2}={\mathcal{l}}_{P}^{4}\right)/{a}^{4}$(17)
Eventually in the case of an unpolarized spinning fluid in the immediate aftermath of the big bang, we would see a Roberson Walker universe given as, if $\sigma$ is a torsion spin term added due to
[1] as
${\left(\frac{\stackrel{˙}{\stackrel{˜}{R}}}{\stackrel{˜}{R}}\right)}^{2}=\frac{8\pi G}{3}\cdot \left[\rho -\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]+\frac{\Lambda {c}^{2}}{3}-\frac{\stackrel{˜}{k}
5. What [1] Does as to Equation (18) versus What We Would Do and Why
In the case of [1] we would see $\sigma$ be identified as due to torsion so that Equation (18) reduces to
${\left(\frac{\stackrel{˙}{\stackrel{˜}{R}}}{\stackrel{˜}{R}}\right)}^{2}=\frac{8\pi G}{3}\cdot \rho -\frac{\stackrel{˜}{k}{c}^{2}}{{\stackrel{˜}{R}}^{2}}$(19)
The claim is made in [1] that this is due to spinning particles which remain invariant so the cosmological vacuum energy, or cosmological constant is always cancelled.
Our approach instead will yield
${\left(\frac{\stackrel{˙}{\stackrel{˜}{R}}}{\stackrel{˜}{R}}\right)}^{2}=\frac{8\pi G}{3}\cdot \rho +\frac{{\Lambda }_{\text{0bserved}}{c}^{2}}{3}-\frac{\stackrel{˜}{k}{c}^{2}}{{\stackrel{˜}{R}}^
i.e. the observed cosmological constant ${\Lambda }_{\text{0bserved}}$ is 10^−^122 times smaller than the initial vacuum energy.
The main reason for the difference in the Equation (19) and Equation (20) is in the following observation. We will go to Table 1 and make the following assertion:
Mainly that the reason for the existence of σ^2 is due to the dynamics of spinning black holes in the precursor to the big bang, to the Planckian regime, of space time, whereas in the aftermath of
the big bang, we would have a vanishing of the torsion spin term, i.e. Table 1 dynamics in the aftermath of the Planckian regime of space time would largely eliminate the σ^2 term.
6. Filling in the Details of the Equation (19) Collapse of the Cosmological Term, versus the Situation Given in Equation (20) via Numerical Values
First look at numbers provided by [3] as to inputs, i.e. these are very revealing
${\Lambda }_{Pl}{c}^{2}\approx {10}^{87}$(21)
This is the number for the vacuum energy and this enormous value is 10^122 times larger than the observed cosmological constant. Torsion physics, as given by [3] is solely to remove this giant
In order to remove it, the reference [3] proceeds to make the following identification, namely
$\frac{8\pi G}{3}\cdot \left[-\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]+\frac{\Lambda {c}^{2}}{3}=0$(22)
What we are arguing is that instead, one is seeing, instead
$\frac{8\pi G}{3}\cdot \left[-\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]+\frac{{\Lambda }_{Pl}{c}^{2}}{3}\approx {10}^{-122}×\frac{{\Lambda }_{Pl}{c}^{2}}{3}$(23)
Our timing as to Equation (22) is to unleash a Planck time interval t about 10^−^43 seconds.
As to Equation (22) versus Equation (23) the creation of the torsion term is due to a presumed particle density of
${n}_{Pl}\approx {10}^{98}\text{\hspace{0.17em}}{\text{cm}}^{-3}$(24)
Finally, we have a spin density term of
${\sigma }_{Pl}={n}_{Pl}\hslash \approx {10}^{71}$(25)
7. Future Works to Be Commenced as to Derivational Tasks
We will assume for the moment that Equation (22) and Equation (23) share in common Equation (24) and Equation (25).
It appears to be trivial, a mere round off, but I can assure you the difference is anything but trivial. And this is where Table 1 really plays a role in terms of why there is a torsion term to begin
with, i.e. will make the following determination, i.e., the term of “spin density” in Equation (22) by Equation (25) is defined to be an ad hoc creation, as to [3] . No description as to its origins
is really offered.
We state that in the future a task will be to derive in a coherent fashion the following, i.e. the term of $\frac{8\pi G}{3}\cdot \left[-\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]$ arising as a
result of the dynamics of Table 1, as given in the manuscript.
We state that the term $\frac{8\pi G}{3}\cdot \left[-\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]$ is due to initial micro black holes, as to the creation of a Cosmological term. This would follow
from Equation (2) being utilized, i.e. what we are seeking is utilization of the following.
In the case of Pre Planckian space-time the idea is to do the following [9] , i.e. if we have an inflaton field [10]
$\begin{array}{l}|\text{d}{p}_{\alpha }\text{d}{x}^{\alpha }|\approx \frac{L}{l}\cdot \frac{h}{c}\cdot {\left[\frac{\text{d}l}{l}\right]}^{2}\\ \underset{\alpha =0}{\to }|\text{d}{p}_{0}\text{d}{x}^
{0}|\simeq |\Delta E\Delta t|\approx h/{a}_{init}^{2}\varphi \left(t\right)\\ ⇒\frac{L}{l}\cdot \frac{h}{c}\cdot {\left[\frac{\text{d}l}{l}\right]}^{2}\approx h/{a}_{init}^{2}\varphi \left({t}_{init}
Making use of all this leads to [8] to making sense of the quantum number n as given by reference to black holes, [4]
The conclusion of [3] states that Equation (22) would remain invariant for the life of the evolution of the universe. We make no such assumption. We assume that, as will be followed up later that
Equation (23) is due to relic black holes with the suppression of the initially gigantic cosmological vacuum energy.
The details of what follow after this initial period of inflation remain a task to be completed in full generality but we are still assuming as a given the following inputs [1] [9]
$\begin{array}{l}a\left(t\right)={a}_{\text{initial}}{t}^{u }\\ ⇒\varphi =\mathrm{ln}{\left(\sqrt{\frac{8\pi G{V}_{0}}{u \cdot \left(3u -1\right)}}\cdot t\right)}^{\sqrt{\frac{u }{16\pi G}}}\\ ⇒\
stackrel{˙}{\varphi }=\sqrt{\frac{u }{4\pi G}}\cdot {t}^{-1}\\ ⇒\frac{{H}^{2}}{\stackrel{˙}{\varphi }}\approx \sqrt{\frac{4\pi G}{u }}\cdot t\cdot {T}^{4}\cdot \frac{{1.66}^{2}\cdot {g}_{\ast }}{{m}_
{P}^{2}}\approx {10}^{-5}\end{array}$(28)
A possible future endeavor can also make sense of [10] as well.
8. Another Brief Reformulation of This Idea to Consider, Similar to the Above Revisiting [1]
$\begin{array}{l}\sqrt{\Lambda }=\frac{{k}_{B}E}{\hslash c{S}_{\text{entropy}}}\\ {S}_{\text{entropy}}={k}_{B}{N}_{\text{particles}}\end{array}$(29)
And then its reference to the BEC condensate given by [1] [3] as to scaling [11]
$\begin{array}{l}m\approx \frac{{M}_{P}}{\sqrt{{N}_{\text{gravitons}}}}\\ {M}_{BH}\approx \sqrt{{N}_{\text{gravitons}}}\cdot {M}_{P}\\ {R}_{BH}\approx \sqrt{{N}_{\text{gravitons}}}\cdot {l}_{P}\\ {S}
_{BH}\approx {k}_{B}\cdot {N}_{\text{gravitons}}\\ {T}_{BH}\approx \frac{{T}_{P}}{\sqrt{{N}_{\text{gravitons}}}}\end{array}$(30)
To begin this look at [1] [2] [3] which purports to show a global cancellation of a vacuum energy term, which is akin, as we discuss later to cancelling the following completely [3] [4] .
If so then we will be looking at Equation (3) to be recast as
${\left(\frac{\stackrel{˙}{\stackrel{˜}{R}}}{\stackrel{˜}{R}}\right)}^{2}=\frac{8\pi G}{3}\cdot \left[\rho -\frac{2\pi G{\sigma }^{2}}{3{c}^{4}}\right]+\frac{{k}_{B}^{2}{E}^{2}}{3{\hslash }^{2}{c}^
{2}\cdot \left[{k}_{B}^{2}{N}_{\text{particles}}^{2}\right]}-\frac{\stackrel{˜}{k}{c}^{2}}{{\stackrel{˜}{R}}^{2}}$(31)
Our analysis from here will delve into different candidate versions as to energy E put into Equation (31) as to what could be expected as to the torsion term and its implications in cosmology, i.e.
keep in mind that Equation (3) as configured in this situation is assuming in [1] that torsion completely cancels a cosmological constant.
9. What If Energy E in Equation (31) Is Thermal?
We then will be looking at
$\frac{{k}_{B}^{2}{c}_{1}^{2}{T}_{\text{Temperature}}^{2}}{12{\hslash }^{2}{c}^{2}\cdot \left[{k}_{B}^{2}{N}_{\text{particles}}^{2}\right]}-\frac{16\pi G}{9}\cdot \frac{2\pi G{\sigma }^{2}}{{c}^{4}}\
equiv \frac{{\Lambda }_{\text{observed}}{c}^{2}}{3}$(32)
Assuming that ${\Lambda }_{\text{observed}}{c}^{2}$ is of the order of 10^−^35 and, this comes up with
${N}_{\text{particles}}^{2}\approx \frac{12{\hslash }^{2}{c}^{2}}{{c}_{1}^{2}{T}_{\text{Temperature}}^{2}}/\left[\frac{16\pi G}{9}\cdot \frac{2\pi G{\sigma }^{2}}{{c}^{4}}+\frac{{\Lambda }_{\text
Becomes smaller and smaller the higher temperature we have initially, and of course this is not viable in terms of applying Equation (31), and the problem becomes well a bit, ridiculous of there is
no torsion term. i.e. we would be then be looking at N going way past 10^120, which is beyond the observed or expected entropy of the universe.
I.e. This is not going to go over well, and the only way to have a huge number of initial “particles” say of initial black holes and say gravitons from the black holes would be if we assume Table 1
is for the initial Planckian regime to have low temperature values, which is NOT what occurs.
10. By Default, We Will Be Looking Then at Changing the Energy E to Being the Corda Value of Energy for a Black Hole, So Then We Will Be Looking at the Following, Namely
$\frac{{\left(\hslash \omega \cdot {n}_{\text{quantumnumber}}\right)}^{2}}{12{\hslash }^{2}{c}^{2}\cdot \left[{k}_{B}^{2}{N}_{\text{particles}}^{2}\right]}-\frac{16\pi G}{9}\cdot \frac{2\pi G{\sigma
}^{2}}{{c}^{4}}\equiv \frac{{\Lambda }_{\text{observed}}{c}^{2}}{3}$(34)
In effect what we would be doing as to Equation (32) would be to state via Equation (2) and energy input into Equation (34).
But the term n (quantum) comes from a Corda derived expression as to energy level of relic black holes [6] after Planckian space time normalization using Equation (27) into the frequency of Equation
The term ω is here presumably Planck frequency, which is of the order of 6.62607015 × 10^−34 joule-hertz^−1 (or joule-seconds) or 3 times 10^42 Hertz.
We are presuming that in doing so that this is a GW frequency for initial relic GW, from this process.
11. Modeling Challenges Which This Presents, and Future Investigations
First what are the particle N term and the quantum n terms used in Equation (36)? This needs to be explicitly worked out.
Secondly, assume the following, namely from [1] of the following values:
Our timing as to Equation (34) is to unleash a Planck time interval t about 10^−^43 seconds.
As to Equation (34) the creation of the torsion term is due to a presumed particle density of
${n}_{Pl}\approx {10}^{98}\text{\hspace{0.17em}}{\text{cm}}^{-3}$(35)
Finally, we have a spin density term of
${\sigma }_{Pl}={n}_{Pl}\hslash \approx {10}^{71}$(36)
Would this spin density term be commensurate as to Gravitons as to a BEC condensate? This is the sort of detail which has to be worked out in future modeling of this problem.
12. Now to Include in the Overview by the Referee. FTR; While Outlining Further Research Requirements
The referee specifically delineates in [12] [13] and [14] with this quote.
There are other papers in which it is pointed out that the torsion term (quoted often by Bekenstein) can indeed give rise to a residual Cosmological constant term of the observed magnitude. This has
also been used to generate inflation. For instance see Open Astronomy Journal, 5, 7-11, 2012 and arxiv/0801.1218 (astro-ph), 2008. Here, clearly it is derived that the G × sigma^2/c^4 gives rise to
the lambda term observed. In both references the spin density is shown to be universal for all celestial bodies.
End of quote
We of course delineated spin density via primordial black holes, and it may be useful as to review further more phenomenological data set opportunities as to try to verify the existence of primordial
black holes as a contributing factor to the spin density. It is recommended that follow-ups to this document adhere to the necessity of finding equipment and data analysis protocols as to verify this
final step and to make it adhere in terms of the phenomenology as well as further intersections with instrumentation requirements which could delve into how to acquire requisite data sets which
confirm this suggestion. In addition this will by necessity go into the matter of experimental verification, via data sets of our supposition of gravitons being BEC condensates. | {"url":"https://scirp.org/journal/paperinformation?paperid=130501","timestamp":"2024-11-10T06:33:01Z","content_type":"application/xhtml+xml","content_length":"175725","record_id":"<urn:uuid:81bea85b-aef4-441e-bb48-bf0e05b8d799>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00507.warc.gz"} |
Difference Between Differential Amplifier and Operational Amplifier
Main Difference – Differential Amplifier vs. Operational Amplifier
Amplifiers are extremely vital components in electronic circuits. The main difference between differential amplifier and operational amplifier is that a differential amplifier is an amplifier that
amplifies a voltage difference between its inputs, whereas an operational amplifier is, in fact, a type of differential amplifier with a large open-loop gain, a high input impedance and a low output
What is a Differential Amplifier
A differential amplifier is an electric component which amplifies the difference between two signals applied to two input terminals, while rejecting signals which are common to both the input
terminals. A differential amplifier has two inputs: we will refer to these as $V_1$ and $V_2$. Suppose the amplifier amplifies $V_1$ by a factor $A_1$ and $V_2$ by a factor $A_2$. Then, the output
voltage $V_o$ would be:
The differential voltage ($V_d$) is defined as $V_d= V_2-V_1\qquad(2)$.
The common mode voltage ($V_c$) is defined as $V_c=\frac{ V_1+V_2}{2}\Rightarrow 2V_c=V_1+V_2\qquad(3)$.
Using these two values, we can try to derive an expression for the output voltage.
$(2)+(3)\Rightarrow 2V_c+V_d=2V_2\Rightarrow V_2=V_c+\frac{V_d}{2}$ and,
$(2)-(3)\Rightarrow V_d+2V_c=-2V_1\Rightarrow V_1=V_c-\frac{V_d}{2}$
Now we can substitute these values back into $(1)$, giving us:
$V_o=A_1\left( V_c-\frac{V_d}{2}\right)+A_2\left( V_c+\frac{V_d}{2}\right)$
Expanding this and re-factorizing, we get:
$V_o=\left( A_1+A_2\right)V_c+\left( \frac{A_2-A_1}{2}\right)V_d$
The quantity $A_1+A_2$ is called the common-mode gain ($A_c$) and the quantity $\frac{A_2-A_1}{2}$ is called the differential gain ($A_d$). In terms of these nquantitietwo quantities, we can simply
write the above equation as:
What is an Operational Amplifier
An operational amplifier is a type of differential amplifier with high gain. Operational amplifiers have large input impedances and small output impedances. For ideal operational amplifiers, the
input impedance is taken to be infinite while the output impedance is taken to be 0. Often, operational amplifiers make use of a resistor to provide itself negative feedback. By changing the value of
the resistor, it is possible to change the gain of an operational amplifier. Note that adding a resistor reduces the overall gain of an operational amplifier. When we calculated the gain in the
previous section, we did so by assuming there is no feedback. Therefore, the gain we have calculated is the maximum possible gain for an amplifier, and it is called the open-loop gain since there is
no feedback mechanism in the system.
Operational amplifiers have an extremely important place in analog computing where they are used to not only amplify signals but to invert voltages, add voltages, and to carry out differentiation and
The figure below shows the symbol for an operational amplifier:
$V_+$ and $V_-$ are are the input terminals, while $V_{S+}$ and $V_{S-}$ are the terminals to which a power supply is connected. $V_{out}$ is the output terminal.
The image below shows an operational amplifier:
Difference Between Differential Amplifier and Operational Amplifier
A differential amplifier is a type of amplifier which amplifies a voltage difference between two of its inputs.
An operational amplifier is a type of differential amplifier with a large open-loop gain, a very high input impedance and a very low output impedance.
Image Courtesy:
“Op-amp pinouts.” by User:Omegatron (Own work) [CC BY-SA 3.0], via Wikimedia Commons
“Opamp” by lambda’s (Own work) [CC BY 2.0], via flickr | {"url":"https://pediaa.com/difference-between-differential-amplifier-and-operational-amplifier/","timestamp":"2024-11-09T22:00:10Z","content_type":"text/html","content_length":"76686","record_id":"<urn:uuid:e4d7f6bd-b2e9-4080-94e5-e8de7deccd66>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00182.warc.gz"} |
Two Digits Multiplication Worksheets
Mathematics, particularly multiplication, creates the foundation of countless scholastic disciplines and real-world applications. Yet, for many students, understanding multiplication can present an
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Introduction to Two Digits Multiplication Worksheets
Two Digits Multiplication Worksheets
Two Digits Multiplication Worksheets -
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2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply 3 x 2 digits Multiply
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Two Digit Multiplication Worksheets Free Printable
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Multiplication Problems 1 X 2 Digit No Regrouping Mr R s World Of Math
Multiplication Worksheets No Regrouping PrintableMultiplication
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two Digit multiplication Worksheet 5 Stuff To Buy Pinterest Multiplication worksheets
2 Digit Multiplication Worksheet School
Multiplication 2 Digit By 2 Digit Worksheet Pdf Mundode Sophia
3 Digit By 2 Digit multiplication Games And worksheets
Multiply 2 x 2 digits worksheets K5 Learning
2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply 3 x 2 digits Multiply
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Encouraging regular practice, supplying assistance, and creating a favorable discovering environment are advantageous steps. | {"url":"https://crown-darts.com/en/two-digits-multiplication-worksheets.html","timestamp":"2024-11-06T11:01:03Z","content_type":"text/html","content_length":"28580","record_id":"<urn:uuid:ba4ba17f-17dd-43c5-a8fb-671eda655909>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00435.warc.gz"} |
OCHIAI, Hiroyuki
My research interest is algebraic analysis, especially representation theory of semisimple Lie groups and related homogeneous spaces. I also interested in special functions arising in the various
context of algebraic analysis, representation theory as well as generating functions in number theory. Such an interaction is my favorite.
Keywords Algebraic Analysis, Representation Theory, Special Function
Division Division of Fundamental Mathematics
Links Homepage | {"url":"https://www.imi.kyushu-u.ac.jp/post-department/department-4932/","timestamp":"2024-11-11T05:03:47Z","content_type":"text/html","content_length":"28055","record_id":"<urn:uuid:fd33df6e-4aa8-4aff-bce5-6c851e2be424>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00546.warc.gz"} |
VTU Engineering Mathematics 4 - June 2012 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1 (a) Using the Taylor's method, find third order approximate solution at x=0.4 of the problem dy/dx=x^2y+1, with y(0)=0. Consider terms upto fourth degree.
6 M
1 (b) Solve the differential equation dy/dx=-xy^2 under the initial condition y(0)=2, by using the modified Euler's method, at the points x=0.1 and x=0.2. Take the step size h=0.1 and carry out two
modifications at each step.
7 M
1 (c) \[ Given \ \dfrac {dy}{dx}= xy+y^2: y(0)=1, y(0,1)=1.1169, \ y(0,2)=1.2773, \ y(0, 3)=1.5049, \ find \ y(0,4) \] correct of three decimal places, using the Milne's predictor-corrector method.
Apply the corrector formula twice.
7 M
2 (a) Employing the Picard's method, obtain the second order approximate solution of the following problem at x=0.2. \[ \dfrac {dy}{dx}= x+yz; \ \dfrac {dz}{dx}=y+zx; \ y(0)=1, \ z(0)=-1 \]
6 M
2 (b) Using the Runge-Kutta method, solve the following differential equation at x=0,1 under the given condition: \[ \dfrac {d^2y}{dx^2}= x^3 \left ( y+ \dfrac {dy}{dx} \right ), \ y(0)=1, \ y'(0)=
0.5 \] Take step length h=0.1.
7 M
2 (c) Using the Milne's method, obtain an approximate solution at the point x=0.4 of the problem \[ \dfrac {d^2y}{dx^2}+3x \dfrac {dy}{dx}-6y=0, \ y(0)=1, \ y'(0)=0.1 \] Given y(0.1)=1.03995, y'(0.1)
=0.6955, y(0.2)=1.138036, y'(0.2)=1.258, y(0.3)=1.29865, y'(0.3)=1.873.
7 M
3 (a) Derive Cauchy-Riemann equations in polar form.
6 M
3 (b) If f(z) is a regular function of z, prove that \[ \left [ \dfrac {\partial^2}{\partial x^2} + \dfrac {\partial ^2} {\partial y^2} \right ]|f(z)|^2=4|f'(z)|^2 \]
7 M
3 (c) If w=?+iy represents the complex potential for an electric field and \[ y=x^2-y^2 + \dfrac {x} {x^2+y^2} \] determine the function ?. Also find the complex potential as a function of z.
7 M
4 (a) Discuss the transformation of \[ w=z+\dfrac {k^2}{z} \]
6 M
4 (b) Find the bilinear transformation that transforms the point z[1]=i, z[2]=1, z[3]=-1 on to the points w[1]=1, w[2]=0, w[3]=? respectively.
7 M
4 (c) Evaluate \[ \int_c \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)^2(z-2)} dz \] where c is the circle |z|=3, using Cauchy's integral formula.
7 M
5 (a) Obtain the solution of x^2y"+xy'+(x^2-x^2)y=0 in terms of J[n](x) and J[-n](x).
6 M
5 (b) Express f(x)=x^4+3x^3-x^2+5x-2 in terms of Legendre polynomials.
7 M
5 (c) Prove that \[ \int^{+1}_{-1}P_m(x).P_n(x)dx = \dfrac {2}{2n+1}, m=n \]
7 M
6 (a) From five positive and seven negative numbers, five numbers are chosen at random and multiplied. What is the probability that the product is a(i) negative number and (ii) positive number?
6 M
6 (b) If A and B are two events with \[ P(A)= \dfrac {1}{2}, \ P(B)= \dfrac {1}{3}, \ P(A \cap B)=\dfrac {1}{4}, \ find \ P(A/B), \ P(B/A), \ P(\bar{A}/ \bar{B}), \ P(\bar{B}/\bar{A}) \ and P(A/\bar
{B}). \]
7 M
6 (c) In a certain college 4% of boy student and 1% of girl students are taller than 1.8 m. Furthermore, 60% of the students are girls. If a student is selected at random and is found taller than 1.8
m, what is the probability that the student is a girl?
7 M
7 (a) A random variable x has the density function \[ P(x)= \left\{\begin{matrix} Kx^2, &0\le x \le 3 \\0, &elsewhere \end{matrix}\right. \] Evaluate K, and find: i) P(x?1), ii) P(1? x ? 2), iii) P
(x?2), iv) P(x >1), v) P(x>2).
6 M
7 (b) Obtain the mean and standard deviation of binomial distribution.
7 M
7 (c) If an examination 7% of students score less than 35% marks and 89% of students score less than 60% marks. Find the mean and standard deviation if the marks are normally distributed. It is given
that P(0
7 M
8 (a) A random sample of 400 items chosen from an infinite population is found to have a mean of 82 and a standard deviation of 18. Find the 95% confidence limits for the mean of the population from
which the sample is drawn.
6 M
8 (b) In the past, a machine has produced washers having a thickness of 0.50 mm. To determine whether the machine is in proper working order, a sample of 10 washers is chosen for which the mean
thickness is found as 0.53 mm with standard deviation 0.03 mm. Test the hypothesis that the machine is in proper working order, using a level of significance of (i) 0.05 and (ii) 0.01.
7 M
8 (c) Genetic theory states than children having one parent of blood type M and the other of blood type N will always be one of the three types. M, MN, N and that the proportions of these types will
on an average be 1:2:1. A report states that out of 300 children having one M parent and one N parent, 30% were found to be of type M, 45% of type MN and the remainder of type N. Test the theory by x
^2 (Chi square) test.
7 M
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Lesson 14
Nets and Surface Area
Let’s use nets to find the surface area of polyhedra.
14.1: Matching Nets
Each of the nets can be assembled into a polyhedron. Match each net with its corresponding polyhedron, and name the polyhedron. Be prepared to explain how you know the net and polyhedron go together.
14.2: Using Nets to Find Surface Area
1. Name the polyhedron that each net would form when assembled.
2. Your teacher will give you the nets of three polyhedra. Cut out the nets and assemble the three-dimensional shapes.
3. Find the surface area of each polyhedron. Explain your reasoning clearly.
1. For each net, decide if it can be assembled into a rectangular prism.
2. For each net, decide if it can be folded into a triangular prism.
A net of a pyramid has one polygon that is the base. The rest of the polygons are triangles. A pentagonal pyramid and its net are shown here.
A net of a prism has two copies of the polygon that is the base. The rest of the polygons are rectangles. A pentagonal prism and its net are shown here.
In a rectangular prism, there are three pairs of parallel and identical rectangles. Any pair of these identical rectangles can be the bases.
Because a net shows all the faces of a polyhedron, we can use it to find its surface area. For instance, the net of a rectangular prism shows three pairs of rectangles: 4 units by 2 units, 3 units by
2 units, and 4 units by 3 units.
The surface area of the rectangular prism is 52 square units because \(8+8+6+6+12+12=52\).
• base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
• face
Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.
• net
A net is a two-dimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.
• polyhedron
A polyhedron is a closed, three-dimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.
Here are some drawings of polyhedra.
• prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
• pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
• surface area
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm^2, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm^2. | {"url":"https://curriculum.illustrativemathematics.org/MS/students/1/1/14/index.html","timestamp":"2024-11-04T11:10:18Z","content_type":"text/html","content_length":"109554","record_id":"<urn:uuid:25cf0934-f0e8-4e6d-80f0-f0b265ebb21f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00751.warc.gz"} |
Finance-Magazine.com - From copulas to CDOs - pricing tranches
Finance Dublin
Finance Jobs
Thursday, 7th November 2024
From copulas to CDOs - pricing tranches Back
In FINANCE February, Finbarr Murphy and Bernard Murphy looked at how one can price Basket Default Swaps (BDSs) using Gaussian copulas. Copula functions are a relatively new tool in finance, and they
are used to construct multivariate distributions, and to investigate dependence structure between random variables. In this issue, they continue this discussion to demonstrate how one can use Copula
functions to value multi-tranche synthetic Collateralised Debt Obligations (CDOs), and they point out some of the risks inherent in these instruments.
The notional amount of outstanding Credit Default Swaps (CDSs) in June 2005 was $12.43 trillion1. This represents an annualised growth rate in excess of 100 per cent. The use of CDSs as the
underlying instrument in Synthetic Collateralised Debt Obligations (CDOs) has fuelled much of this growth. Typically, synthetic CDOs are constructed to meet client risk preferences and are sold in
different tranches. Risk adverse investors will seek tranches higher in the capital structure for lower risk and consequently lower expected returns. Conversely, those investors seeking a higher
return with associated risk will source lower (equity tranches). In tandem with the explosive growth in CDS activity, CDO issuance has grown quickly in North American with $155 billion issued in
20052 and in Europe, issuance reached €243.5 billion in 20043.
The iTraxx Credit Default Swap index in Europe and the Dow Jones CDX index in the US have allowed for greater transparency and liquidity in the Synthetic CDO market.
Standardised tranched credit products from these indices (Tracers and TriBoxx) are now quoted by banks to clients. When a bank quotes a price on standardised tranches or on a bespoke structured
tranche, by extension, it runs risk on the remaining capital structure. It is therefore critically important to measure the correlation between the constituent components in order to effectively risk
manage a disparate portfolio of credit derivatives.
The standard market model for valuing default swaps on multi-constituent credit baskets is the Gaussian copula model that uses one parameter to describe the correlations between the individual names
credit default times. This standard market approach also assumes these correlations are constant for the life of the basket swap. It is also assumed that the recovery rates and swap spreads are
constant. This simple, standard approach allows dealers to quote in terms of implied correlation rather than a swap spread.
Pricing synthetic CDOs
A CDO is a transaction that securitises a diversified pool of debt assets. A synthetic CDO is one where the underlying portfolio consists of single name Credit Default Swaps (CDSs), hence the
‘synthetic’ precursor and the reason that a synthetic CDO is classified as a credit derivative. These single name CDSs are liquid instruments with a payout based on defined credit events. The spread
of the CDS is, in part, defined by the default probability of the underlying asset or entity. Based on a time series of CDSs on a particular name, one can therefore create an implied default
probability distribution.
A CDO can be valued as a swap transaction with the premium being described by setting the default payout equal to the periodic insurance payments. We now describe how a multi-tranche CDO with n
reference assets can be valued.
Let us first assume that the jth asset has an exposure Ej and a recovery rate Rj. Nj(t) = 1 denotes the default indicator for j. The cumulative portfolio loss at t, is shown in formula 1.
A CDO tranche has attachment points at A and a detachment point at B. Default payment occurs above the A threshold and below the B threshold where 0 A B i=1Ni. When A = 0, the tranche is usually
referred as the Equity Tranche. When A > 0 and B < i=1Ni we are usually referring to the Mezzanine Tranche(s) and when B = i=1Ni we are usually dealing with the Senior, or Super Senior Tranche.
As with a CDS, a CDO tranche spread is that amount, based on the notional, paid to the investor such that the premium leg(s) equal the default payment legs. Consider first the default leg. Let M(t)
denote the cumulative loss on given tranche. We can summarise this loss in formula 2.
So, M(t) is a jump process where a payment of M(t+)-M(t) occurs at each jump. Now, we can write the price of the default payment leg of the tranche as shown in formula 3, where B(0,t) is the discount
factor to time t and EQ represents the expectation under the default probability Q.
Now turning to the premium payment, X, of a CDO tranche. Let ti denote the premium payment times where i = 1,…,T with T being the CDO maturity. Let i-1,i denote the time period, [ti-1,ti] and again,
B(0,ti) is the discount factor to maturity ti.
The tranche premium is due on the outstanding notional amount between the attachment detachment points, A and B. At time ti, if the number of names in default, given by N(ti), is less than A, then
the premium payment will be due on the full tranche notional. Similarly, if N(ti) B, the entire tranche has defaulted. Where A N(ti) < B, the premium is due on the outstanding tranche notional, B-N
We can now write the expected discounted premium payment at ti in formula 4, where Q(N(ti) = k) represents the probability of k names being in default at time ti. We can calculate the entire premium
leg by summing over the entire payment schedule. The above solution can be easily extended to include accrued premiums from the time of the last regular premium payment date to j, the default date of
entity j between ti-1 and ti.
Simulation methodology
We produce a random vector x1,…,xn, from the standard multivariate normal distribution with a mean of zero and a correlation specified by the matrix. Recall that a Copula function links the
multivariate distribution to the individual marginal distributions.
Specifically, the Gaussian copula states in formula 5, where is the correlation matrix, -1 is the generalised inverse of the univariate normal distribution function and n is the multi-variate normal
distribution function. We can transform the vector x1,…,xn, into default times through the function of the marginal distributions. E.g. i = -ln(xi).
Attributing losses to a tranche is cumulative up to the detachment point, B. Take a 100 name synthetic CDO where the recovery rate, Rj = 40% and a notional value of Hj for all names. The attachment
point, A = 0% and the detachment point, B = 5%. The cumulative losses are given as shown in formula 6.
In this case, the equity tranche takes the losses of the 1st to 8th name to default and takes 33% of the loss on the 9th name to default. Similarly, assuming a mezzanine tranche with an attachment
point at 5% and a detachment point at 15%, it will take 66% of the loss on the 9th tranche and takes all of the losses from the 9th to the 25th default. The senior tranche takes all subsequent
Simulation results
We consider a Synthetic CDO with 125 underlying reference names with equal weight.
We assume a constant recovery rate of 40% for all reference entities and a deterministic, risk-free interest rate of 5%. We further suppose the CDO to be divided into multiple tranches, the base case
assuming equity tranches with attachment points at 0, 3 and 6%.
The detachment points will be 3, 6 and 9% respectively. Finally, we assume a constant default intensity () of 0.01 for all reference names. 5000 simulations were conducted for all scenarios.
We begin by observing the loss distribution and spread for the three base tranches. Figure 1 below shows the loss distribution graph for the baseline scenario. Note the increasing probability of zero
defaults as the correlation increases. This demonstrates an increasing mark-to-market book value of a long equity tranche investor. Conversely, an increasing correlation tends to decrease the value
of senior tranches as the likelihood of all reference obligations increases.
This spread reaction, to differing correlation values, is further demonstrated in Figure 2 below. We have graphed the spreads for [0-3], [3-6] and [6-9]% tranches against increasing correlation. The
spread for the equity tranche decreases with increasing correlation but note the slightly convex shape to the higher [6-9]% tranche.
The standard Gaussian Copula model has drawbacks. It is argued that one should use stochastic models with negative correlation between recovery rates and default probabilities. The model assumes that
default rates are constant and equal which are implied from CDS spreads. Another drawback is the assumed flat correlation structure across reference names which misrepresents the complex default
relationship between entities and sectors. One of the strongest arguments against the model is the computational resource required to produce stable values, particularly when using Monte Carlo
Despite these drawbacks, the Gaussian copula model has the advantage of simplicity and tractability. The introduction of the one-factor approach has reduced the computational burden and extensions to
the model allowing clustered correlation for inter- and intrasector modelling, random recovery rates that allowed for correlation between recovery rates and default probabilities give more accurate
1 Source: International Swaps and Derivatives Association (ISDA), 2005 mid-year survey of 86 firms
2 Source: American Securitization Forum
3 Source: European Securitisation Forum
Finbarr Murphy and Dr. Bernard Murphy are lecturers in derivatives and financial engineering on the MSc in Financial Services programme in the Kemmy Business School, University of Limerick. | {"url":"https://www.finance-magazine.com/display_article.php?i=6485&pi=235","timestamp":"2024-11-07T09:05:39Z","content_type":"application/xhtml+xml","content_length":"18227","record_id":"<urn:uuid:a2c5ddd7-a5a5-4b3d-b328-16f8a7222ba5>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00777.warc.gz"} |
[Latest] Alternating Current MCQ | Assertion | CaseStudy2024
Alternating Current MCQ | Class 12 | Physics | Chapter-7 | 2024
Last updated on July 14th, 2024 at 05:29 pm
Alternating Current MCQ Chapter 7
Below are some of the very important NCERT Alternating Current MCQ Class 12 Physics Chapter 7 with answers. These Alternating Current MCQ have been prepared by expert teachers and subject experts
based on the latest syllabus and pattern of CBSE Term 1 examination.
We have given these Alternating Current MCQ Class 12 Physics questions with answers to help students understand the concept.
MCQ Questions for Class 12 Physics are very important for the latest CBSE Term 1 and Term 2 pattern. These MCQs are very important for students who want to score high in CBSE Board, NEET and JEE
We have put together these NCERT Alternating Current MCQ for Class 12 Physics Chapter 7 with answers for the practice on a regular basis to score high in exams. Refer to these MCQs questions with
answers here along with a detailed explanation.
1. Alternating current cannot be measured by DC ammeter because
1. AC is virtual
2. AC changes its direction
3. AC cannot pass through DC ammeter
4. average value of complete cycle is zero
2. An alternating current of equivalent value of I[o]/√2 is
1. RMS current
2. DC current
3. current
4. all of these
3. In an AC circuit I = 100sin200πt. The time required for current to achieve its peak value will be
1. 1/200 second
2. 1/400 second
3. 1/100 second
4. 1/300 second
4. The ratio of mean value over half cycle to RMS value of AC is
1. √2:1
2. 2:π
3. 2√2:π
4. √2:π
5. The peak value of an alternating EMF E given by E = E[o]cosωt is 10V and its frequency is 50 Hz. At that time t=1/600 s, the instantaneous EMF is
1. 5√3 V
2. 5 V
3. 10 V
4. 1 V
6. The frequency of an alternating voltage is 50 cps and its amplitude is 120 V. Then the RMS value of voltage is
1. 56.5 V
2. 70.7 V
3. 101.3 V
4. 84.8 V
7. An electric heater of 40 ohm is connected to a 200V, 50 Hz main supply. The peak value of electric current flowing in the circuit is approximately
1. 10 A
2. 5 A
3. 7 A
4. 2.5 A
8. In the case of an inductor
1. voltage leads the current by π/4
2. voltage leads the current by π/3
3. voltage leads the current by π/2
4. voltage lacks the current by π/2
9. A resistance of 20 ohm is connected to a source of an alternating potential V = 220sin(100πt). The time taken by the current to change from its peak value to RMS value is
1. 2.5 x 10^-3 s
2. 25 x 10^-3 s
3. 0.25 s
4. 0.2 s
10. The RMS value of an AC of 50 Hz is 10A. The time taken by the alternating current in reaching from zero to maximum value and the peak value of current will be
1. 1 x 10^-2 s and 7.07 A
2. 2 x 10^-2 s and 14.14 A
3. 5 x 10^-2 s and 14.14 A
4. 5 x 10^-3 s and 7.07 A
11. Determine The RMS value of the EMF given by
E (in V) = 8 sin(ωt) + 6 sin(2ωt)
1. 10√2 V
2. 10 V
3. 5√2 V
4. 7√2 V
12. An alternating current of frequency f is flowing in a circuit containing a resistor of resistance R and a choke of inductance L in series. The impedance of the circuit is
1. R + 2πfπL
2. √(R^2 + L^2)
3. √(R^2 + 2fπL)
4. √(R^2 + 4π^2f^2L^2)
13. A generator produces a voltage that is given by V = 240 sin 120t V, where t is in seconds. The frequency and RMS voltage and nearly
1. 19 Hz and 120 V
2. 19 Hz and 170 V
3. 60 Hz and 240 V
4. 754 Hz and 170 V
14. The instantaneous voltage through a device of impedance 20 ohm is e = 80sin100πt. The effective value of the current is
1. 1.732 A
2. 2.828 A
3. 3 A
4. 4 A
15. A 15μF capacitor is connected to 220V, 50Hz source. Find the capacitive reactance and the RMS current
1. 212.1 Ω; 1.037 A
2. 212.1 Ω; 2.037 A
3. 412.1 Ω; 1.037 A
4. 412.1 Ω; 2.037 A
Click Below To Learn Physics Term 1 Syllabus Chapter-Wise MCQs
Click Below To Learn Chemistry Term-1 Syllabus Chapters MCQs
16. In an AC circuit an alternating voltage V = 200√2 sin100t is connected to a capacitor of capacity 1μF. The RMS value of the current in the circuit is
1. 10mA
2. 20mA
3. 100mA
4. 200mA
17. In an LR circuit, the value of L is (0.4/π) and the value of R is 30Ω. If in the circuit, an alternating EMF of 200V at 50 cps is connected, the impedance of the circuit and current will be
1. 50 Ω, 4 A
2. 40.4 Ω, 5 A
3. 30.7 Ω, 6.5 A
4. 11.4 Ω, 17.5 A
18. In an AC circuit the voltage applied is E = E[o]sinωt. The resulting current in the circuit is I = I[o]sin(ωt – π/2). The power consumption in the circuit is given by
1. P = E[o]I[o] / 2
2. P = E[o]I[o] / √2
3. P = √E[o]I[o]
4. P = 0
19. In an LCR circuit AC circuit, the voltage across each of the components L, C and R is 50V. The voltage across the LC combination will be
1. 0 V
2. 50 V
3. 50√2 V
4. 100 V
20. Find the capacitive reactance of a 10μF capacitor, when it is part of a circuit whose frequency is 100 Hertz.
1. 159.2 Ω
2. 412.1 Ω
3. 612.1 Ω
4. 812.1 Ω
21. The resonant frequency of a circuit is f. If the capacitance is made 4 times the initial values, than the resonant frequency will become
1. f/2
2. f
3. 2f
4. f/4
22. A coil of 10Ω and 10mH is connected in parallel to a capacitor of 0.1μF. The impedance of the circuit at resonance is
1. 10^3Ω
2. 10^6Ω
3. 10^2Ω
4. 10^4Ω
23. Which of the following curves correctly represent the variation of capacitive reactance (X[c]) with frequency (f)?
1. (a)
2. (b)
3. (c)
4. (d)
24. How does the current in an RC circuit vary when the charge on the capacitor builds up?
1. it decreases linearly
2. it increases linearly
3. it decreases exponentially
4. it increases exponentially
25. The impedance in a circuit containing a resistance of 1Ω and an inductance of 0.1 H in series for AC of 50 Hz is
1. √10 Ω
2. 10√10 Ω
3. 100 Ω
4. 100√10 Ω
26. An AC circuit contains a resistance R, capacitance C and inductance L in series with the source of EMF e=e[o]sin(ωt+f). The current through the circuit is maximum when
1. ω^2 = LC
2. ωL = 1/ωC
3. R = L = C
4. ω = LCR
27. A charged 30μF capacitor is connected to a 27 mH inductor. The angular frequency of free oscillations of the circuit is
1. 1.1 x 10^3 rad s^-1
2. 2.1 x 10^3 rad s^-1
3. 3.1 x 10^3 rad s^-1
4. 4.1 x 10^3 rad s^-1
28. The frequency of the output signal becomes ________ times by doubling the value of the capacitance in the LC oscillator circuit.
1. ½
2. 2
3. √2
4. 1/√2
29. In an LCR circuit, the sharpness of resonance depends on
1. resistance
2. capacitance
3. inductance
4. all of these
30. The average power dissipation in a pure capacitor in AC circuit is
1. CV^2
2. 2CV^2
3. CV^2/2
4. zero
31. In a series resonant circuit, having L, C and R as its elements, the resonant current is ‘i’. The power dissipated in circuit at resonance is
1. Zero
2. i^2R
3. i^2ωL
4. i^2R/(ωL-(1/ωC))
32. An AC supply gives 30 V[RMS] which passes through 10Ω. The power dissipated in it is
1. 45√2 W
2. 90√2 W
3. 45 W
4. 90 W
33. In a series LCR circuit alternating EMF(e) and current(i) are given by equation v=v[o]sin(ωt), i=i[o]sin(wt+π/3). The average power dissipated in the circuit over a cycle of AC is
1. Zero
2. v[o]i[o]/2
3. v[o]i[o]/4
4. (√3/2)v[o]i[o]
34. In an AC circuit, the current flowing in inductance is I = 5sin(100t-π/2)A and the potential difference V = 200sin(100t)V. The power consumption is equal to
1. Zero
2. 20 W
3. 40 W
4. 1000 W
35. The power factor in an AC series LR circuit is
1. L/R
2. √(R^2 + L^2ω^2)
3. R√(R^2 + L^2ω^2)
4. R/(√R^2 + L^2ω^2)
36. A transformer is employed in
1. Convert DC into AC
2. Convert AC into DC
3. Obtain a suitable DC voltage
4. Obtain a suitable AC voltage
37. The loss of energy in the form of heat in the iron core of a transformer is
1. Copper loss
2. Iron loss
3. Mechanical loss
4. None of these
38. The core of any transformer is laminated so as to
1. Make it light weight
2. Make it robust and strong
3. Increase the secondary voltage
4. Reduce the energy loss due to eddy currents
39. A step up transformer has a transformation ratio 5:3. What is the voltage in secondary if voltage in primary is 60 V?
1. 60 V
2. 180 V
3. 20 V
4. 100 V
40. A transformer has 50 turns in the primary and 100 in the secondary. If the primary is connected to a 220 V DC supply, what will be the voltage across the secondary?
1. 19 V
2. 30 V
3. 62 V
4. 0 V
41. The primary of a transformer has 400 turns while the secondary has 2000 turns. If the power output from the secondary at 1000 V is 12kW, what is the primary voltage?
1. 200 V
2. 400 V
3. 300 V
4. 500 V
42. A step down transformer is used on a 1000V line to deliver 20A at 120V at the secondary coil. If the efficiency of the transformer is 80% the current drawn from the line is
1. 0.3 A
2. 3 A
3. 30 A
4. 24 A
43. If the RMS current in a 50 Hz AC circuit is 5 A, the value of the current 1/300 s after its value becomes zero is
1. 5√2 A
2. 5√(3/2) A
3. ⅚ A
4. 5√2 A
44. An alternating current generator has an internal resistance R[g] and an internal reactance X[g]. It is used to supply power to a passive load consisting of a resistance Rg and a reactance X[L].
For maximum power to be delivered from the generator to the laid, the value of XL is equal to
1. Zero
2. X[g]
3. -X[g]
4. R[g]
45. When a voltage measuring device is connected to AC mains, the meter shows the steady input voltage of 220 V. This means
1. Input voltage cannot be AC voltage, but a DC voltage
2. Maximum input voltage is 220 V
3. The meter reads not v but <v^2> and is calibrated to read √<v^2>
4. The pointer of the meter is stuck by some mechanical defect
46. To reduce the resonant frequency in an LCR series circuit with a generator
1. The generator frequency should be reduced
2. Another capacitor should be added in parallel to the first
3. The iron core of the inductor should be removed
4. Dielectric in the capacitor should be removed
47. Which of the following combinations should be selected for better tuning of an LCR circuit used for communication?
1. R = 20Ω, L = 1.5 H, C = 35 μF
2. R = 25Ω, L = 2.5 H, C = 45 μF
3. R = 15Ω, L = 3.5 H, C = 30 μF
4. R = 25Ω, L = 1.5 H, C = 45 μF
48. An inductor of reactance 1Ω and a resistor of 2Ω are connected in series to the terminals of a 6V(rms) AC source. The power dissipated in the circuit is
1. 8 W
2. 12 W
3. 14.4 W
4. 18 W
49. The selectivity of a series LCR AC circuit is large when
1. L is large, R is large
2. L is small, T is small
3. L is large , R is small
4. L = R
50. The phase difference between the current and the voltage in series LCR circuit at resonance is
1. π
2. π/2
3. π/3
4. zero
MCQ Answers
1. (4) 2. (1) 3. (2) 4. (3) 5. (1) 6. (4) 7. (3) 8. (3) 9. (1) 10. (3) 11. (3) 12. (4) 13. (2) 14. (2) 15. (1) 16. (2) 17. (1) 18. (4) 19. (1) 20. (1) 21. (1) 22. (4) 23. (2) 24. (3) 25. (2) 26. (2)
27. (1) 28. (4) 29. (4) 30. (4) 31. (2) 32. (4) 33. (3) 34. (1) 35. (4) 36. (4) 37. (2) 38. (4) 39. (4) 40. (4) 41. (1) 42. (2) 43. (2) 44. (3) 45. (3) 46. (2) 47. (3) 48. (3) 49. (3) 50. (4)
Assertion-Reasoning Based MCQ
1. Both assertion and reason are true and reason is the correct explanation of assertion.
2. Both assertion and reason are true but reason is not the correct explanation of assertion.
3. Assertion is true but reason is false.
4. Assertion is false but reason is true.
1. Assertion AC is more dangerous in use than DC
Reason It is because the peak value of AC is greater than indicated value
2. Assertion Average value of AC over a complete cycle is always zero
Reason average value of AC is always defined over half cycle
3. Assertion The alternating current lags behind the EMF by a phase angle of when AC flows through and inductor
Reason The inductive reactance increases as the frequency of AC source decreases
4. Assertion Capacitor serves as a block for DC and offers an easy path to AC
Reason Capacitive reactance is inversely proportional to frequency
5. Assertion In series LCR resonant circuit the impedance is equal to the ohmic resistance
Reason At resonance the inductive reactance exceeds the capacitive reactance
6. Assertion An alternating current shows magnetic effect
Reason Alternating current varies with time
7. Assertion In series LCR circuit resonance can take place
Reason Resonance takes place in inductance and capacitive reactance are equal and opposite
8. Assertion Power factor correction is must in heavy machinery
Reason A low power factor implies larger power loss in transmission
9. Assertion Choke coil is preferred over a registered to adjust current in an AC circuit
Reason Power factor for inductance is zero
10. Assertion When AC circuit containing resistor only its power is minimum
Reason Power of a circuit is independent of phase angle
11. Assertion A transformer cannot work on DC supply
Reason DC change is neither in magnitude nor in direction
12. Assertion A laminated core is used in transformer to increase eddy currents
Reason The efficiency of a transformer increases with increase in eddy currents
13. Assertion Soft iron is used as a core of transformer
Reason Area of hysteresis loop for soft iron is small
14. Assertion An AC generator is based on the phenomenon of electromagnetic induction
Reason In single coil we consider self induction only
Assertion-Reasoning Based MCQ Answers
1. (1)
AC is more dangerous in use than DC. It is because the peak value of AC is greater than the indicated value.
2. (2)
The mean or average value of alternating current or EMF during half cycle is given by
I[m] = 0.636 I[o]
E[m] = 0.6363 E[o]
During the next half cycle, the mean value of AC will be equal in magnitude but opposite in direction. For this reason the average value of AC over a complete cycle is always zero. So the average
value is always defined over a half cycle of AC.
3. (3)
When AC flows through an inductor current lags behind the EMF, by phase of π/2 inductive reactance
XL = ωL = 2πfL
So, when frequency increases correspondingly inductive reactance also increases.
4. (1)
The capacitive reactance of capacitor is given by
XC = 1/ ωC = 1/2πfC
So this is infinite for DC and has a very small value for AC. Hence, a capacitor blocks DC.
5. (3)
In series resonance circuit inductive reactance is equal to capacitive reactance.
ωL = 1/ωC
Z = R
6. (2)
Like direct current, an AC also produces magnetic field. But the magnitude and direction of the field goes on changing continuously with time.
7. (1)
At resonant frequency,
X[L] = X[C], Z = R (minimum)
8. (2)
A heavy machinery requires a large power.
The average power is given by,
P[av] = E[rms]I[rms]cosΦ
The required power can be supplied to the heavy machinery either by supplying larger current or by improving power factor. The first method is costly. Hence, the second one is used.
9. (1)
We can use a capacitor of suitable capacitance as a choke coil, because average power consumed per cycle in an ideal capacitor is zero. Therefore, like a choke coil a condenser can reduce AC without
power dissipation.
10. (4)
The power of an AC circuit is given by,
P = EIcosΦ
Where cosΦ is a power factor and is Φ phase angle. In case of circuit containing resistance only, phase angle is zero and power factor is equal to 1. Therefore power is maximum in case of circuit
containing resistor only.
11. (1)
Transformer works on AC only AC changes in magnitude as well as in direction and induced EMF.
12. (4)
Large eddy currents are produced in non laminated iron core of the transformer by induced EMF, as the resistance of bulk iron core is very small. By using thin iron sheets are score the resistance is
increased. Laminating the core substantially reduces the eddy currents. Eddy currents heat up the core of the transformer. More the eddy current greater the loss of energy and efficiency goes down.
13. (1)
Hysteresis loss in the core of transformer is directly proportional to the hysteresis loop area of the core material. Since soft iron has narrow hysteresis loop area, that is why soft iron core is
used in transformer.
14. (2)
According to electromagnetic induction, whenever the magnetic flux changes and EMF will be induced in the coil.
Click Below To Learn Physics Term 1 Syllabus Chapter-Wise MCQs
Case-Study Based MCQ
1. The figure shows a series LCR circuit.
For such a citcuit, the impedance Z is given by Z = √R^2 + (X[L] – X[C])^2 where X[L] and X[C] are inductive and capacitive resistances respectively. As the frequency of AC is increased, at a
particular frequency, X[L] becomes equal to X[C]. For that frequency, maximum current occurs. This is because the impedcane becomes equal to the least value which is R.
Current through the circuit is I = V/R. This circuit behaves like a pure resistive circuit and current and voltage will be in phase. This is called resonance. Frequency of AC at which resonance
occurs is called resonant frequency. If frequency is less than the resonant frequency, then the capacitive reactance will be more. The circuit will be capacitive in nature.
If frequency is more than the resonant frequency, inductive reactance will be more. Circuit is inductive in nature and the current lags behind the voltage by a phase of π/2.
An LCR circuit which has a resistance 50 ohm has a resonant angular frequency 2 x 10^3 rad/s. At resonance, the voltage across the resistance and inductance are 25 V and 20 V respectively.
(i) The value of inductance is
(a) 20mH
(b) 10mH
(c) 40mH
(d) 25mH
(ii) The value of capacitive reactance is
(a) 25μF
(b) 1μF
(c) 2μF
(d) 12.5 μF
(iii) The impedance at resonance is
(a) 50 ohm
(b) 16 ohm
(c) 64 ohm
(d) 25 ohm
(iv) Which of the following angular frequency of AC will see the circuit as inductive in nature?
(a) 1.5 x 10^3 rad/s
(b) 10^3 rad/s
(c) 2 x 10^3 rad/s
(d) 5 x 10^3 rad/s
(v) At angular frequency 10^3 rad/s, the nature of circuit is
(a) inductive
(b) capacitive
(c) resisitive
(d) none of these
2. A series LCR circuit consist of series combination of a resistance, an inductor and a capacitance. A similar series LCR circuit is shown in figure. The given series LCR circuit is connected across
a 200 V 60 Hz line consisting of capacitive reactance 30 ohm a non-inductive resistor of 44 ohm and a coil of inductive reactance 90 ohm and resistance 36 ohm.
(i) Calculate the total impedance of the circuit.
(a) 1000 ohm
(b) 100 ohm
(c) 3600 ohm
(d) 4900 ohm
(ii) Calculate the current flowing in the circuit.
(a) 1 A
(b) 5 A
(c) 2 A
(d) 10 A
(iii) What is the impedance of the coil?
(a) 97 ohm
(b) 87 ohm
(c) 100 ohm
(d) 110 ohm
(iv) what is the potential difference across the coil?
(a) 194 V
(b) 186 V
(c) 180 V
(d) 190 V
(v) Calculate the power dissipated in the coil.
(a) 100 W
(b) 122 W
(c) 130 W
(d) 144 W
Case-Study Based MCQ Answers
1. (i) (a) XL = VL/I
I = V[R]/R = 25/50 = 1/2
X[L] = 20/(1/2) = 40 ohm
X[L] = ωL ; L = 40 / (2 x 10^3) = 20 x 10^-3 = 20mH
(ii) (d) ω^2 = 1/LC ; C = 1/( ω^2L) = 1 / ((2 x 10^3)^2 x 20 x 10^-3) = 12.5 μF
(iii) (a) At resonance, the impedance equal just resistance.
(iv) (d) For inductive nature, ω > ω[r]
(v) (b) If ω < ω[r], the circuit will be capacitive in nature.
2. (i) (b) Z = √((R[1] + R[2])^2 + (X[L] – X[C])^2) = √((44 + 36)^2 + (90 – 30)^2) = 100 ohm
(ii) (c) Current, I = V/Z = 200/100 = 2A
(iii) (a) Impedance of the coil, Z[L] = √(R[2]^2 + X[L]^2) = √((36)^2 + (90)^2) = 97 ohm
(iv) (a) Potential difference across the coil, V[L] = IZ[L] = 2 x 97 = 194 V
(v) (d) Power dissipated in the inductive coil, P = I^2R[2] = (2)^2 x 36 = 144 W
Click Below To Learn Physics Term 1 Syllabus Chapter-Wise MCQs
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Final Words
From the above article, you have practiced Alternating Current MCQ of Class 12 Physics Chapter 7. We hope that the above mentioned latest Alternating Current MCQ for Term 1 of Chapter 7 will surely
help you in your exam.
If you have any doubts or queries regarding the Alternating Current MCQ with answers of CBSE Class 12 Physics, feel free to reach us and we will get back to you as early as possible.
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What exactly is calculus?
I’m an engineering student so yea I got calculus. My concern is, while I can solve calculus I don’t really understand what it is really all about. I want to understand it in like objective manner
(idk if this make sense).
In: Mathematics
9 Answers
Calculus is about slicing things (like time, distance and area) up into smaller and smaller pieces, until those pieces are infinitesimally small, and being able to use those infinitesimally small
pieces to make extremely precise calculations. For example, when we say a car travels at 100 km per hour, we can start by measuring the 100 km distance the car traveled in one hour, but that doesn’t
tell us how fast the car was travelling at any particular point in its journey unless we assume a single uniform speed (and real systems simply don’t work that way). If we take two measurements – one
at 30 min and another at 60 min – we can now understand how fast the car traveled on average during each half of it’s trip. Maybe it was 90 km per hour for the first 1/2 hour, and 110 km hour for the
second half, so overall it would be 100 km per hour. Now we have more precise information. We can take measurements at 4 intervals, or 8 or 16 or 32, etc., with each measurement getting closer and
closer to knowing the speed of the car at any particular instant. Taking this to its extreme (but logical) conclusion, we can determine the speed of the car at every instant through calculus, and it
turns out this process gives us a lot of very useful information and insights about how things work in the real world (including how the speed is changing at any particular moment).
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Registry hacks for the Windows Vista screensavers
The sad fact is Windows Vista will be shipping with screensavers without the options to be configured, however they were originally designed to be customizable. Stephen Coy, the designer of these
screensavers, tells me this feature might be available after Vista ships, perhaps in the form of a powertoy or service pack? But if you can’t wait, then you can follow the steps below to customize
the screensavers through the registry.
The registry keys that control the screensavers are located under
To apply these registry hacks, you should right click the links and “Save as…” the registry files onto your computer. Then double click on them to apply the changes.
Ribbons screensaver
This is the default look of the Ribbons screensaver. This version of Ribbons has a greater number of ribbons, which are thinner.
Download the registry file for this style.
This version of Ribbons has thicker and greater number of ribbons. This version of Ribbons has extremely thick but only few number of ribbons.
Download the registry file for this style. Download the registry file for this style.
The Ribbons screensaver has two options.
1. NumRibbons defines the maximum number of ribbons that can appear on the screen at any one time. This is a described as a simple integer count. The maximum number of ribbons is 256. An integer
higher than this will have no effect. The minimum number of ribbons is 1.
2. RibbonWidth defines the maximum width of the ribbons. This is a described as a floating point integer value.
Mystify screensaver
This is the default look of the Mystify screensaver. This version of Mystify has a greater number of lines.
Download the registry file for this style.
This version of Mystify has a few lines.
Download the registry file for this style.
The Mystify screensaver has one option.
1. NumLines defines the maximum number of lines that can appear on the screen at any one time. This is a described as a simple integer count. The minimum number of lines is 1.
Bubbles screensaver
This is the default look of the Bubbles screensaver. This version of Bubbles has a fewer number of large non-transparent bubbles. The background is the desktop.
Download the registry file for this style.
This version of Bubbles has a greater number of smaller non-transparent bubbles. The background is black. Please note: This version of Bubbles has been known to cause problems after extended period
of times. Please do not use this in a work environment as it may cause instabilities.
Download the registry file for this style.
The Bubbles screensaver has four options.
1. MaterialGlass defines if the bubbles are of a glass material or not. This is a described as a boolean value. A value of 1 will turn the bubbles into glass-like transparent bubbles. A value of 0
will turn the bubbles into metallic non-transparent bubbles.
2. Radius defines the radius size of the bubbles. This is described as a floating point integer. The larger the radius, the less number of bubbles will appear on the screen, and vice versa.
3. ShowShadows defines if the bubbles have a shadow or not. This is described as a boolean value. A value of 1 will enable shadows below the bubbles. A value of 0 will disable shadows under the
4. ShowBubbles defines if the bubbles are displayed on the desktop or not. This is described as a boolean value. A value of 1 will render the bubbles on the desktop. A value of 0 will render the
bubbles against a solid black background.
Restore to default
To return the screensavers to their original state, you would simply delete the registry values under the key related to the screensaver.
For example, if you were to restore the Bubbles screensaver to the default options, you would navigate to ...\CurrentVersion\Screensavers\Bubbles\ and delete all the associated values (Material
Glass, Radius, ShowBubbles, ShowShadows).
49 insightful thoughts
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3. My first thought is that ribbons have infected Windows too, not just Office.
4. Thanks for the files. Very cool.
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8. after i add them to the registry, how do i apply them? i go to my display properties and i dont see it in the list…
9. @kody: Are you running XP? These are for Windows Vista.
11. I just got vista business today (part of msdnaa) and I was testing out the screen savers… I saw the bubbles screensaver, watched it, and just before I was going to move the mouse, I thought to
myself, wouldnt it becool if they were all to pop and then go back to the original screen. So I moved my mouse hoping they would pop, but no š they just disapear. But hey, that would of been
12. Do you know how to get the Screensavers to work at all in Vista?
This is what I have tries so far:
13. Hi, I can`t even get my screensavers to work regardless. They just don`t, I have tried everything but alas , no screensaver. I have Vista home basic. Any suggestions? Also the email. It sucks, I
wnt to use my Yahoo email but even though I disable mail. it still goes to windows mail when I click send email. I hate that. I have made Yahoo mail my default but Windows ignore it.
14. And…….any news on updates for vista re d-link another suck feature, I have to keep exiting no can find link in library messages, Yet I have internet
15. @barbara: RE: Screensavers. Do they load at all? If you go to the control panel and find the screensavers, do they give you a preview? If not, then there is something wrong with your graphics
drivers or graphics card that is preventing these screensavers from being drawn.
Any version of Windows Vista should be able to load screensavers.
RE: Email. I’m not sure how Yahoo Mail handles the default action for clicking on email links, but I know by default it uses Windows Mail, but can be disabled.
16. Yes they load and you can preview but they don`t come on when the com is quiet at all.. I have tried to disable Micro mail and add Yahoo as my default but when I click on send email Micromail
loads regardless. As for the D-Link internet key, well internet does come on but the com throws up messages saying not located in dynamic library etc.
17. Well I finally got my screensavers. I downloaded driver detective, did the scan and it told me to update Nvidia, I did so and screensaver works!!!!!Thanks
18. The first 3 keys worked well.
But the 4th key ShowBubble didn’t work in my laptop (Win XP Egyptian).
If I set 0, the background is solid black.
If I set 1, it turn to solid white. I cannot see the wallpaper.
Any tips ? Tks a lot.
19. hey,
I want the bubbles screeny on my XP system, is there a way to download them or wont they work on XP?
21. Hey there is an update that fixes the issue with microsoft usb keyboards and mice that cause the screensavers not to start. It’s on windows update something like HID input filter
22. i loved the bubble hack, love the bight colors that the bubbles are now
23. Dude
When I add ShowBubbles DWORD (1) in the reg for this blasted screensaver I still get the Black background. I am using build 5308.17? What do I have to do to sort this out?
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25. I cannot get the screensaver to work at all on my Windows Vista computer, it previews fine, but nothing nada….. any and all suggestions would be greatly appreciated….
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28. hey barbara these bubble screen savers unfortunately dont work on vista basic
29. Is there any way to get the default Vista screensaver (Windows Logo) to work on XP?
30. I’ve been looking on my computer, but can’t find the registry. Can anyone help?
31. ROFLMAO! Never mind, I found it. I’ve never played around witht he registry before. So all I had to do was lauch the app, cool….
Sorry for the posting twice.
32. Is it just me or are the mystify screen savers kinda slow? Or is it simply because I tried the modified one with dual monitors?
33. hope someone can help me with this:
1. no matter how i change the wallpaper on my Vista, only the background color can be change , the pix somehow “stays behind” .I can see it for a bried 2 or 3 seconds when shutting down.
2. i created another user, and the problem disappear which means some setting is wrong with my original user/administrator But I find where to activate/deactivate the Active Desktop in Vista that
was suggested in some other forum
3.this is a different problem: i am using SonyEricsson EDGE PC Card GC85 and a GSM GPRS for net access when i am on the road. Problem is my card able to locate service provider signal but the
access not possible as the card seems forever stuck at ” Updating Setting”
p.s. i wish i had stayed with XP and not bought a new Vista laptop
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37. Pingback: Connected to Vista Bookmarks
39. Legendary…..
Not only did it all work…now i have something to do…lol
40. My screen when i turn on the computer is all white now. it used to be blue. I can’t get the color change to work ont he background when I try to change it? what do I do?
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A Lattice-Based Method for Recovering the Unknown Parameters of Truncated Multiple Recursive Generators with Constant
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Journal of ACM
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, vol. 28, no. 3, pp. 193–203, 2017.
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, vol. 830–831, pp. 20–42, 2020.
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[6] J. Gutierrez, “Attacking the linear congruential generator on elliptic curves via lattice techniques,”
Cryptography and Communications
, vol. 14, no. 3, pp. 505–525, 2022.
doi: 10.1007/s12095-021-00535-6
[8] L. Y. Deng and H. Q. Xu, “A system of high-dimensional, efficient, long-cycle and portable uniform random number generators,”
ACM Transactions on Modeling and Computer Simulation
, vol. 13, no. 4, pp. 299–309, 2003.
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[9] L. Y. Deng, “Efficient and portable multiple recursive generators of large order,”
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, vol. 15, no. 1, pp. 1–13, 2005.
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[10] L. Y. Deng, J. J. H. Shiau, H. H. S. Lu,
et al
., “Secure and fast encryption (SAFE) with classical random number generators,”
ACM Transactions on Mathematical Software
, vol. 44, no. 4, article no. 45, 2018.
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[12] A. M. Frieze, J. Hastad, R. Kannan,
et al
., “Reconstructing truncated integer variables satisfying linear congruences,”
SIAM Journal of Computing
, vol. 17, no. 2, pp. 262–280, 1988.
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[13] J. L. Massey, “Shift-register synthesis and BCH decoding,”
IEEE Transactions on Information Theory
, vol. 15, no. 1, pp. 122–127, 1969.
doi: 10.1109/TIT.1969.1054260
[14] E. R. Berlekamp, Algebraic Coding Theory. McGraw-Hill, New York, NY, USA, pp. 178−188, 1968.
[15] J. A. Reeds and N. J. A. Sloane, “Shift register synthesis (modulo
SIAM Journal on Computing
, vol. 14, no. 3, pp. 505–513, 1985.
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[16] M. Q. Huang, “Analysis and cryptologic evaluation of primitive sequences over an integer residue ring,” Ph.D. Thesis, Graduate School of USTC, Academia Sinica, Beijing, China, pp. 20–23, 1988.
(in Chinese)
[17] A. S. Kuzmin and A. A. Nechaev, “Linear recurring sequences over Galois rings,”
Algebra and Logic
, vol. 34, no. 2, pp. 87–100, 1995.
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[18] J. B. Yang, “Reconstructing truncated sequences derived from primitive sequences over inter residue rings,” Master Thesis, PLA Information Engineering University, Zhengzhou, pp. 9−13, 2017. (in
[19] D. Coppersmith, “Finding a small root of a univariate modular equation,” in
International Conference on the Theory and Application of Cryptographic Techniques, Advances in Cryptology-EUROCRYPT’96
, U. Maurer, Ed. Springer, Saragossa, Spain, vol. 1070, pp. 155–165, 1996.
[20] H. Y. Sun, X. Y. Zhu, and Q. X. Zheng, “Predicting truncated multiple recursive generators with unknown parameters,”
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, vol. 88, no. 6, pp. 1083–1102, 2020.
doi: 10.1007/s10623-020-00729-8
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et al
., “An improved method for predicting truncated multiple recursive generators with unknown parameters,”
Designs, Codes and Cryptography
, vol. 91, no. 5, pp. 1713–1736, 2023.
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Mathematics of Operations Research
, vol. 12, no. 3, pp. 415–440, 1987.
doi: 10.1287/moor.12.3.415
[23] H. Y. Sun, X. Y. Zhu, and Q. X. Zheng, “Reconstructing truncated sequences derived from second-order linear congruential generator with unknown coefficients,”
Journal of Cryptologic Research
, vol. 6, no. 4, pp. 496–511, 2019. (in Chinese)
doi: 10.13868/j.cnki.jcr.000318
[24] M. Ward, “The arithmetical theory of linear recurring series,”
Transactions of the American Mathematical Society
, vol. 35, no. 3, pp. 600–628, 1933.
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[25] H. J. Chen and W. F. Qi, “On the distinctness of maximal length sequences over
) modulo 2,”
Finite Fields and Their Applications
, vol. 15, no. 1, pp. 23–39, 2009.
doi: 10.1016/j.ffa.2008.07.005
[27] M. Ajtai, “Generating random lattices according to the invariant distribution,” Draft of March, 2006.
[29] P. Q. Nguyen and D. Stehlé, “LLL on the average,” in
7th International Symposium on Algorithmic Number Theory, ANTS-VII
, F. Hess, S. Pauli, and M. Pohst, Eds. Springer, Berlin, Germany, vol. 4076, pp. 238–256, 2006.
[30] A. K. Lenstra, H. W. Jr. Lenstra, and L. Lovász, “Factoring polynomials with rational coefficients,”
Mathematische Annalen
, vol. 261, no. 4, pp. 515–534, 1982.
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[31] C. P. Schnorr and M. Euchner, “Lattice basis reduction: Improved practical algorithms and solving subset sum problems,”
Mathematical Programming
, vol. 66, no. 1, pp. 181–199, 1994.
doi: 10.1007/BF01581144
[33] M. Ajtai, R. Kumar, and D. Sivakumar, “A sieve algorithm for the shortest lattice vector problem,” in
Proceedings on 33rd Annual ACM Symposium on Theory of Computing
, Heraklion, Crete, Greece, pp. 601–610, 2001.
[34] N. M. Korobov, Exponential Sums and Their Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 56–58, 1992.
[35] Q. X. Zheng, D. D. Lin, and W. F. Qi, “Distribution properties of binary sequences derived from primitive sequences modulo square-free odd integers,” in
14th International Conference on Information Security and Cryptology, Inscrypt 2018
, F. C. Guo, X. Y. Huang, and M. Yung, Eds. Springer, Fuzhou, China, vol. 11449, pp. 568–585, 2018. | {"url":"https://cje.ejournal.org.cn/article/doi/10.23919/cje.2022.00.387","timestamp":"2024-11-14T15:35:58Z","content_type":"text/html","content_length":"85718","record_id":"<urn:uuid:4db45f95-c833-4151-b6da-8b78cabab63f>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00510.warc.gz"} |
How to count several non-blank rows based on specific criteria across two sheets
Hi Community!
I have two different grid sheets within smarthseet with the following information:
• In sheet 1 I have several products that need a certain number of photography shots and being delivered in different channels. Each row has a due date.
• In sheet 2 I want to find a formula that would give me the numbers in blue. So is there a formula that says, go to the shotlist sheet, and calculate how many handbags shots that are being
delivered in a print chanel and due on the 17th (here there are 5 nonblank cells for example). And so on and so forth for each product and due date.
Thanks in advance for your help!
• Hi @emka,
I don't recommend using your second sheet to solve this problem, because the formulas will be very complicated and hard to manage.
Instead, I recommend adjusting your first sheet a little bit, and then using a Report to track the number of shots for each date. Here's how I would do it.
In your first sheet, create 3 new columns named Print Count, Digital Count, and Email Count. These columns will calculate the number of shots for each type.
You can right click these columns and hide them, after you set the formulas. After you enter the formula, you can right click the cell and hit Convert to Column Formula to apply the formula to
the entire column.
Print Count's formula is:
=Print@row * (IF(ISBLANK([Shot 1]@row), 0, 1) + IF(ISBLANK([Shot 2]@row), 0, 1) + IF(ISBLANK([Shot 3]@row), 0, 1))
Digital Count's formula is:
=Digital@row * (IF(ISBLANK([Shot 1]@row), 0, 1) + IF(ISBLANK([Shot 2]@row), 0, 1) + IF(ISBLANK([Shot 3]@row), 0, 1))
Email Count's formula is:
=Email@row * (IF(ISBLANK([Shot 1]@row), 0, 1) + IF(ISBLANK([Shot 2]@row), 0, 1) + IF(ISBLANK([Shot 3]@row), 0, 1))
Then in your report you will use the Group, Summary, and Sort buttons to make the data look exactly how you want.
In the screenshot below, I Group by Date and Type, and I have a Summary for Print Count, Digital Count, and Email Count. My summaries all use SUM.
The report will also let you do pretty cool things like filtering, which might be useful in the future. For example, if you wanted to find all of the dates where you need to print at least 10
images of shoes. Or if you want to find dates where you needed to deliver prints, digitals, and emails for handbags all on the same day.
I hope this helps, let me know if you have any questions!
SSFeatures - The browser extension that adds more features into SmartSheet.
□ Automatic sorting, sorting with filters, saving sort settings
□ Spell checking
□ Report PDF generator that supports grouped and summarized reports
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/131919/how-to-count-several-non-blank-rows-based-on-specific-criteria-across-two-sheets","timestamp":"2024-11-03T02:55:30Z","content_type":"text/html","content_length":"439567","record_id":"<urn:uuid:74a5ceb2-9df5-48a8-b396-2542764298ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00014.warc.gz"} |
Vec3 Transform(Vec3 point)
Transforms a point through the Matrix! This is basically just multiplying a vector (x,y,z,1) with the Matrix.
Vec3 point The point to transform.
RETURNS: Vec3 The point transformed by the Matrix.
Shorthand to transform a ray though the Matrix! This properly transforms the position with the point transform method, and the direction with the direction transform method. Does not normalize, nor
does it preserve a normalized direction if the Matrix contains scale data.
Ray ray A ray you wish to transform from one space to another.
RETURNS: Ray The transformed ray!
Pose Transform(Pose pose)
Shorthand for transforming a Pose! This will transform the position of the Pose with the matrix, extract a rotation Quat from the matrix and apply that to the Pose’s orientation. Note that extracting
a rotation Quat is an expensive operation, so if you’re doing it more than once, you should cache the rotation Quat and do this transform manually.
Pose pose The original pose.
RETURNS: Pose The transformed pose.
Found an issue with these docs, or have some additional questions? Create an Issue on Github! | {"url":"https://stereokit.net/Pages/StereoKit/Matrix/Transform.html","timestamp":"2024-11-13T06:00:59Z","content_type":"text/html","content_length":"7088","record_id":"<urn:uuid:bb70551b-b527-452b-a254-7b16d0c40112>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00101.warc.gz"} |
cm2feet.com | Converters and Calculaters for all your needs
Linear Current Density Converter
Abampere Per Centimeter To Abampere Per Meter
1. Abampere Per Centimeter To Abampere Per Meter
Abampere per centimeter to Abampere per meter Conversion Formula:
abampere/meter (abA/m) = abampere/centimeter (abA/cm) × 100
How to Convert abampere/centimeter (abA/cm) to abampere/meter (abA/m)?
To get Abampere per meter linear current density, simply multiply Abampere per centimeter by 100. With the help of this linear current density converter, we can easily convert Abampere per centimeter
to Abampere per meter. Here you are provided with the converter, proper definitions,relations in detail along with the online tool to convert abampere/centimeter (abA/cm) to abampere/meter (abA/m).
How many Abampere per meter in one Abampere per centimeter?
1 abampere/centimeter (abA/cm) is 100 abampere/meter (abA/m).
abampere/centimeter (abA/cm) to abampere/meter (abA/m) converter is the linear current density converter from one unit to another. It is required to convert the unit of linear current density from
Abampere per centimeter to Abampere per meter, in linear current density. This is the very basic unit conversion, which you will learn in primary classes. It is one of the most widely used operations
in a variety of mathematical applications. In this article, let us discuss how to convert abampere/centimeter (abA/cm) to abampere/meter (abA/m), and the usage of a tool that will help to convert one
unit from another unit, and the relation between Abampere per centimeter and Abampere per meter with detailed explanation.
Abampere per centimeter Definition
An abampere per centimeter (abA/cm) is a unit of the linear current density in the cgs-emu system of units.
Abampere per meter Definition
An abampere per meter (abA/m) is a unit of the linear current density in the cgs-emu system of units.
abampere/centimeter (abA/cm) to abampere/meter (abA/m) Conversion table: | {"url":"https://cm2feet.com/converter/electricity/linear-current-density/abampere-per-centimeter-to-abampere-per-meter/","timestamp":"2024-11-14T04:59:35Z","content_type":"text/html","content_length":"21489","record_id":"<urn:uuid:db16c46c-1025-496f-a2c0-167ead7cf040>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00821.warc.gz"} |
Developing an Impairment Loss Given Default Model Using Weighted Logistic Regression Illustrated on a Secured Retail Bank Portfolio
Centre for Business Mathematics and Informatics, North-West University, Potchefstroom 2531, South Africa
SAS Institute Canada, Toronto, ON M5A 1K7, Canada
Author to whom correspondence should be addressed.
Submission received: 11 October 2019 / Revised: 13 November 2019 / Accepted: 6 December 2019 / Published: 13 December 2019
This paper proposes a new method to model loss given default (LGD) for IFRS 9 purposes. We develop two models for the purposes of this paper—LGD1 and LGD2. The LGD1 model is applied to the
non-default (performing) accounts and its empirical value based on a specified reference period using a lookup table. We also segment this across the most important variables to obtain a more
granular estimate. The LGD2 model is applied to defaulted accounts and we estimate the model by means of an exposure weighted logistic regression. This newly developed LGD model is tested on a
secured retail portfolio from a bank. We compare this weighted logistic regression (WLR) (under the assumption of independence) with generalised estimating equations (GEEs) to test the effects of
disregarding the dependence among the repeated observations per account. When disregarding this dependence in the application of WLR, the standard errors of the parameter estimates are
underestimated. However, the practical effect of this implementation in terms of model accuracy is found to be negligible. The main advantage of the newly developed methodology is the simplicity of
this well-known approach, namely logistic regression of binned variables, resulting in a scorecard format.
1. Introduction
The International Accounting Standard Board published the IFRS 9 standard in 2014, (
IFRS 2014
), which replaced most of International Accounting Standard (IAS) 39. Amongst others, it contains impairment requirements that allow for earlier recognition of credit losses. The financial statements
of banks are expected to reflect the IFRS 9 accounting standards as of 1 January 2018 (
European Banking Authority (EBA) 2016
). Banks found that IFRS 9 had a significant impact on systems and processes (
Beerbaum 2015
). While the IAS 39 standard made use of provisions on incurred losses, the financial crisis showed that expected losses, instead of incurred losses, are better used to calculate provisioning for
banks (
Global Public Policy Committee (GPPC) 2016
). In addition, under IFRS 9, the expected credit losses (ECL) should be equivalent to the lifetime ECL, if the credit risk has increased significantly. When the converse is true, a financial entity
may allow for credit losses equal to a 12-month ECL. The ECL model is a forward-looking model and should result in the early detection of credit losses, which is anticipated to contribute to overall
financial stability (
IFRS 2014
). The ECL is a function of the probability of default (PD), the loss given default (LGD) and the exposure at default (EAD).
In this paper, we focus on the LGD component within the impairment calculation under IFRS 9. There are many methodologies to model LGD, see e.g.,
Joubert et al.
) and the references therein. These methodologies include the run-off triangle method, beta regression, survival analysis, fractional response regression, inverse beta transformation, and Box–Cox
transformation. Most of these techniques are quite complex and very difficult to understand, including the monitoring and validation thereof. This is confirmed by
Bijak and Thomas
), who indicate that more than 15 different performance measures can be found in the literature concerning LGD models, possibly due to the difficulty of modelling the distribution shape of LGD. The
LGD can be modelled through either the direct or the indirect approach. When using the direct approach, the LGD is equal to one minus the recovery rate (
De Jongh et al. 2017
). The indirect approach uses two components that are modelled separately, namely the probability component and the loss severity component. Independent of the methodology, the LGD is always assessed
over the life of the lending exposure (
BCBS 2015a
Different modelling approaches are usually followed for accounts in different stages. An account can reside in one of three stages. Stage 1 accounts are performing accounts, Stage 2 have significant
deterioration in credit risk, but are not in default, while defaulted accounts are in Stage 3 (
Aptivaa 2016
This paper describes the proposed new methodology to model the LGD for IFRS 9 purposes. We estimated both the LGD1 and LGD2 values, where the LGD1 was applied to non-defaulted accounts and the LGD2
to defaulted accounts. For the non-defaulted accounts (accounts in Stages 1 and 2, according to the IFRS 9 definition) we used the historically observed the LGD value (LGD1) and segmented this value
according to variables with business importance using a lookup table. The weighted logistic regression was applied on the defaulted accounts (accounts in Stage 3, according to the IFRS 9 definition)
to obtain the LGD2. This therefore resulted in two models: one for the LGD1 and one for the LGD2. The LGD1 was applied for Stage 1 (12 months) and Stage 2 (lifetime) because, while the PD component
differentiates between 12 months and lifetime, the LGD is the loss expected for the remaining life of the account. Since logistic regression is well known and regularly used in banks, established
monitoring metrics and governance practices have been embedded in the industry. These metrics, as well as the methodology, are thoroughly understood by stakeholders, which leads to a high degree of
confidence in the results. Logistic regression using the scorecard format provides an even more transparent and user-friendly technique that is easy to understand and communicate to stakeholders. For
this reason, we propose this new methodology to model the LGD for IFRS 9 purposes.
The paper consists of five sections. The modelling approach is described in
Section 2
Section 3
follows with a case study where the proposed methodology is applied to a secured retail portfolio. The effect of the dependency of the observations used in the logistic regression is tested by
comparing the results from the logistic regression with that of a generalised estimating equation (that takes dependency into account). We also investigate whether a decision tree could outperform
the weighted logistic regression.
Section 4
discusses the strengths and weaknesses of our new methodology and
Section 5
2. LGD Methodology
This section describes the newly proposed LGD methodology. First, the methodology used to estimate the LGD of the non-defaulted accounts is provided (LGD1) under
Section 2.1
, followed by
Section 2.2
that discusses the methodology employed to model the LGD of the defaulted accounts (LGD2).
2.1. LGD1 Methodology
The LGD1 was obtained by calculating the loss as the exposure at default minus the net present value (NPV) of recoveries, divided by the EAD. The LGD1 is typically modelled on a smaller data sample
than for the LGD2, since the loss is only calculated on accounts in Stages 1 and 2 that eventually transition into default. The probability of transitioning from Stages 1 and 2 directly into default
is typically very low. The data sample for LGD2 is typically much larger as it considers all accounts in default and not only those that transition into it in a specific cohort. In this specific case
study, the discounted write-off amount served as a proxy for the NPV of recovery cash flows. A more granular estimate was obtained by using the most important LGD drivers to segment the LGD1 values.
The historical LGD values were calculated (and averaged) per segment and the estimated LGD1 values were derived using a lookup table (see e.g.,
)). The results of the case study are shown in
Section 3.1
. Note that the number of variables available to use for the LGD2 modelling is much larger than that for the LGD1. The reason is that some of the default related variables are not available at the
LGD1 model stage, e.g., months since default.
2.2. LGD2 Methodology
We use a weighted logistic regression to model the LGD for the defaulted account including all available data. The actual loss experienced is transformed to a binary format related to the “fuzzy
augmentation” technique commonly used to introduce “rejects” in scorecard development (
Siddiqi 2006
). This means that each observation has both a target of 1 (Y = 1) as well as a target of 0 (Y = 0). Furthermore, a weight variable is created, where the sum of the weights of these two events adds
up to the full exposure of the account at observation. This is related to
Van Berkel and Siddiqi
) who used a scorecard format for modelling LGD. This newly proposed methodology for LGD2 only considers worked-out accounts. A worked-out account can either cure or be written off. Note that the
point of write-off is taken as that specific point where the institution (e.g., bank) no longer expects any recovery. This is specifically prescribed by the reporting standard: “IFRS 7 (35F) (e): The
Group writes off financial assets, in whole or in part, when it has exhausted all practical recovery efforts and has concluded there is no reasonable expectation of recovery. Indicators that there is
no reasonable expectation of recovery include (i) ceasing enforcement activity and (ii) where the Group’s effort to dispose of repossessed collateral is such that there is no reasonable expectation
of recovering in full” (
PWC 2017
). In effect, with our methodology, all write-offs and cures are included regardless of the time spent in default and no filter is applied on default cohort.
We calculated the LGD for accounts that cured and for accounts that were written off. The modelling approach can be subdivided into five steps: (1) sample creation; (2) target and weight variables
created; (3) input variables; (4) weighted logistic regression; and (5) test for independence.
Note that if any accounts in the considered dataset originated as credit impaired accounts (i.e., accounts starting in default), their loss behaviour will most likely be different from other Stage 3
accounts and should therefore be modelled separately (i.e., segment the portfolio based on this characteristic). In this specific case study here presented, no such accounts existed.
2.2.1. Step 1: Sample Created
This approach would first need to identify all worked-out accounts (i.e., write-off, cure) over an appropriate reference period. Note that one account can appear multiple times, but an account will
only appear once per month. This violates the logistic regression assumption of independent observations. The effect of this dependence (
Sheu 2000
) is tested at the end of the paper, by comparing a generalised estimating equation (GEE) model with the logistic regression model. In statistics, a GEE is used to estimate the parameters of a
generalised linear model with a possible unknown correlation between outcomes (
Kuchibhatla and Fillenbaum 2003
The sample of observations was split into two datasets for out of sample testing. Evaluating the performance of a classifier on the same data used to train the classifier usually leads to an
optimistically biased assessment (
SAS Institute 2010
). The simplest strategy for correcting the optimism bias is to hold out a portion of the development data for assessment (
Baesens et al. 2016
), i.e., data splitting. We therefore split the data into a training and a validation dataset. The validation data is used only for assessment and not for model development.
2.2.2. Step 2: Target and Weight Variables Created
Two rows (
$Y = 1$
$Y = 0$
) are created for each observation (i.e., per account per month). Each row is weighted. Cured and written-off accounts are weighted differently. Mathematically, the weight for observation
is defined as
$w i = { E x p o s u r e i × L G D i i f Y i = 1 E x p o s u r e i × ( 1 − L G D i ) i f Y i = 0 ,$
where the loss given default of observation
$L G D i$
) is defined as
$L G D i = { P ( C u r e ) × P ( r e d e f a u l t ) × L G D 1 . U n a d j if observation i is a cured W O i / E x p o s u r e i if observation i is written off ,$
• $i$ is the number of observations from $1$ to $N$;
• $E x p o s u r e i$
is the exposure of observation
; and therefore,
$E A D i = ∑ ∀ Y i E x p o s u r e i = E x p o s u r e i IND ( Y i = 1 ) + E x p o s u r e i IND ( Y i = 0 ) ,$
$I N D ( Y i = 1 ) := { 1 i f Y i = 1 0 i f Y i = 0$
$I N D ( Y i = 0 ) := { 1 i f Y i = 0 0 i f Y i = 1$
• $P ( C u r e )$ is the proportion of cured observations over the total number of worked-out accounts (over the reference period);
• $P ( r e d e f a u l t )$ is the proportion of observations that re-default over the reference period;
• $L G D 1 . U n a d j$
is the exposure at default (EAD) minus the net present value (NPV) of recoveries from first point of default for all observations in the reference period divided by the EAD—see e.g.,
) and
Volarević and Varović
• $W O i$ is the discounted write-off amount for observation $i$; and
• $P ( C u r e )$
$P ( r e d e f a u l t )$
$L G D 1 . U n a d j$
are therefore empirical calculated values. This should be regularly updated to ensure the final LGD estimate remains a point in time estimate as required by IFRS (
IFRS 2014
Note that the write-off amount is used in Equation (2) to calculate the actual LGD. An alternative method employs the recovery cash flows over the work out period. A bank is required to use its “best
estimate” (a regulatory term, e.g.,
Basel Committe on Banking Supervision
) and
European Central Bank
)) to determine actual the LGD. In this case, this decision was based on the data available. Only the write-off amount was available for our case study, not the recovered cash flows. In Equation (2),
the write-off amount needs to be discounted using the effective interest rate (
PWC 2017
), to incorporate time value of money. When recoveries are used, each recovery cash flow needs to be similarly discounted. In the case study, the length of the recovery time period exists in the data
and differs for each account. The length of this recovery time period will have an influence on the calculation of LGD: the longer the recovery process, the higher the effective discount rate. In the
case study, we used the client interest rate as the effective interest rate when discounting.
Note that, in special circumstances, accounts may be partially written off, leading to an overestimation of provision. This should be taken into account during the modelling process. However, in our
case study no such accounts existed.
Illustrative Example
Consider one observation with an exposure of
50,000. Assume it is a written-off account, for a specific month, with an
$L G D i$
= 27% (based on the written-off amount divided by the exposure, i.e.,
$W O i / E x p o s u r e i$
). The weight variable for
$Y = 1$
will be
$27 % × 50,000 = 13,500$
$Y = 0$
will be
$( 1 − 27 % ) × 50,000 = 36,500$
Table 1
2.2.3. Step 3: Input Variables (i.e., Variable Selection)
All input variables were first screened according to the following three requirements: percentage of missing values, the Gini statistic and business input. If too many values of a specific variable
were missing that variable was excluded. Similarly, if a variable had a too low value for the Gini statistic, then that variable was also excluded. Note that business analysts should investigate
whether there are any data issues with variables that have low Gini statistics. For example, traditionally strong variables may appear weak if the data has significant sample bias. This forms part of
data preparation that is always essential before predictive modelling should take place.
The Gini statistic (
Siddiqi 2006
) quantifies a model’s ability to discriminate between two possible values of a binary target variable (
Tevet 2013
). Cases are ranked according to the predictions and the Gini then provides a measure of correctness. It is one of the most popular measures used in retail credit scoring (
Baesens et al. 2016
Siddiqi 2006
Anderson 2007
) and has the added advantage that it is a single value (
Tevet 2013
• Sort the data by descending order of the proportion of events in each attribute. Suppose a characteristic has $m$ attributes. Then, the sorted attributes are placed in groups $1 , 2 , … , m$.
Each group corresponds to an attribute.
• For each of these sorted groups, compute the number of events
$( ( # ( Y = 1 ) j )$
and the number of nonevents (#(Y=0)_j)in group
. Then compute the Gini statistic:
$( 1 − 2 ∑ j = 2 m ( ( # ( Y = 1 ) j × ∑ j = 1 j − 1 # ( Y = 0 ) j ) + ∑ j = 1 m ( ( # ( Y = 1 ) j × # ( Y = 0 ) j ) # ( Y = 1 ) × # ( Y = 0 ) × 100 ) ,$
$# ( Y = 1 )$
$# ( Y = 0 )$
are the total number of events and nonevents in the data, respectively.
Only variables of sufficient Gini and which were considered important from a business perspective were included in the modelling process. All the remaining variables after the initial screening were
then binned. The concept of binning is known by different names such as discretisation, classing, categorisation, grouping and quantification (
Verster 2018
). For simplicity we use the term binning throughout this paper. Binning is the mapping of continuous or categorical data into discrete bins (
Nguyen et al. 2014
). It is a frequently used pre-processing step in predictive modelling and considered a basic data preparation step in building a credit scorecard (
Thomas 2009
). Credit scorecards are convenient points-based models that predict binary events and are broadly used due to their simplicity and ease of use; see e.g.,
) and
). Among the practical advantages of binning are the removal of the effects of outliers and a convenient way to handle missing values (
Anderson 2007
). The binning was iteratively done by first generating equal-width bins, followed by business input-based adjustments to obtain the final set. Note that if binned variables are used in logistic
regression, the final model can easily be transformed into a scorecard.
All bins were quantified by means of the average LGD value per bin. The motivation behind this was to propose an alternative to using dummy variables. Logistic regression cannot use categorical
variables coded in its original format (
Neter et al. 1996
). As such, some other measure is needed for each bin to make it usable—the default technique of logistic regression is a dummy variable for each class less one. However, expanding categorical inputs
into dummy variables can greatly increase the dimension of the input space (
SAS Institute 2010
). One alternative to this is to quantify (e.g., using weights of evidence (WOE)—see
)) each bin using the target value (in our case the LGD value), which will reduce the number of estimates. An example of this is using the natural logarithm (ln) of the good/bad odds (i.e., the WOE)
—see for example
Lund and Raimi
). We used the standardised average LGD value in each bin.
Some of the advantages of binning and quantifying the bins are as follows:
• The average LGD value can be calculated for missing values, which will allow ”Missing” to be used in model fit (otherwise these rows would not have been used in modelling). Note that not all
missing values are equal and there are cases where they need to be treated separately based on reason for missing, e.g., “No hit” at the bureau vs. no trades present. It is therefore essential
that business analysts investigate the reason for missing values and treat them appropriately. This again forms part of data preparation that is always a key prerequisite to predictive modelling.
• Sparse outliers will not have an effect on the fit of the model. These outliers will become incorporated into the nearest bin and their contributions diminished through the usage of bin WOE or
average LGD.
• Binning can capture some of the generalisation (required in predictive modelling). Generalisation refers to the ability to predict the target of new cases and binning improves the balance between
being too vague or too specific.
• The binning can capture possible non-linear trends (as long as they can be assigned logical causality).
• Using the standardised average LGD value for each bin ensures that all variables are of the same scale (i.e., average LGD value).
• Using the average LGD value ensures that all types of variables (categorical, numerical, nominal, ordinal) will be transformed into the same measurement type.
• Quantifying the bins (rather than using dummy variables) results in each variable being seen as one group (and not each level as a different variable). This aids in reducing the number of
parameter estimates.
Next, each of these average LGD values was standardised using the weight variable by calculating the average LGD per bin. An alternative approach could have been to calculate the WOE for each bin.
The WOE is regularly used in credit scorecard development (
Siddiqi 2006
) and is calculated using only the number of 1’s and the number of 0’s for each bin. Note that our underlying variable of interest (LGD) is continuous. However, since our modelled target variable was
dichotomous, we wanted the quantification of the bin to reflect our underlying true target, e.g., the LGD value, which ranges from 0 to 1. This average LGD value per bin was then standardised by
means of the weight variable. The weighted mean LGD,
$L G D ¯ w$
is defined as
$L G D ¯ w = ∑ i w i L G D i ∑ i w i ,$
$L G D i$
is the LGD value of observation
$w i$
is the weight of observation
. The weighted standard deviation LGD is defined as
$s w = ∑ i w i ( L G D i − L G D ¯ w ) 2 N − 1 ,$
is the number of observations. The weighted standardised value for LGD,
$L G D * i ,$
for observation
will then be
$L G D * i = L G D i − L G D ¯ w s w .$
The standardisation of all input variables implies that the estimates from the logistic regression will be standardised estimates. The benefit is that the absolute value of the standardised estimates
can serve to provide an approximate ranking of the relative importance of the input variables on the fitted logistic model (
SAS Institute 2010
). If this was not done, the scale of each variable could also have had an influence on the estimate. Note that the logistic regression fitted was a weighted logistic regression with the exposure as
weight (split for
$Y = 1$
$Y = 0$
) and therefore to ensure consistency, we also weighted the LGD with the same weight variable as used in the logistic regression.
Furthermore, pertaining to the month since default as input variable: The model that is developed does not require the length of default for incomplete accounts in order to estimate LGD. It assumes
that the length of default for these accounts will be comparable to similar accounts that have been resolved. This is an assumption that can be easily monitored after implementation.
2.2.4. Step 4: Weighted Logistic Regression
A weighted logistic regression was then fitted using the available data. The log of the odds in a weighted logistic regression is given as:
$l o g i t ( p i ) = ln ( p i 1 − p i ) = β 0 + β w i X i T ,$
• $p i = E ( Y i = 1 | X i , β )$ is the probability of loss for observation $i$;
• $β 0 , β$ are regression coefficients with $β$ = {$β 1 , … , β K }$;
• $X i$ is the vector of the predictor variables $X i 1 , , … , X i K$ for observation $i$; and
• $w i$ is the weight of each observation $i$, calculated by the actual loss amount ($s) and given in Equation (1).
Note that in this weighted logistic regression, we estimated the regression coefficients where the repeated observation within the same individual was assumed to be independent (i.e., disregarding
the dependence among repeated observation of the same account).
2.2.5. Step 5: Test the Effect of the Dependence Assumption
A single account would appear multiple times in our dataset (depending on the number of months the account is present), which violates the assumption of independent observations in logistic
regression. We therefore tested the effect of this violation using a GEE that can handle the statistical dependence of repeated data by assuming some correlation structure (
Kuchibhatla and Fillenbaum 2003
) among observations. This approach estimates regression coefficients without completely specifying the joint distribution of the multivariate responses, but the parameters of the within-subjects
correlation are explicitly accounted for in the estimation process (
Sheu 2000
). It is also shown in
) that the GEE approach is not sensitive to the choice of correlated structure.
Kuchibhatla and Fillenbaum
) also found that when comparing the model fit using the GEE with that using the logistic regression, the logistic regression overestimated the standard errors of the dependent variables.
3. Case Study: Secured Retail Portfolio from a South African Bank
This section illustrates the newly proposed LGD methodology on a secured retail portfolio from one of the major banks in South Africa.
Section 3.1
shows the results for the non-defaulted accounts (i.e., LGD1). Then,
Section 3.2
shows results for the defaulted accounts (LGD2). Note that the data was split into LGD1 and LGD2, resulting in 95% of the data in the LGD2 dataset and 5% of the data in the LGD1 dataset. The reason
for the much smaller LGD1 dataset is that very few of the “non-defaulted” sub-set of total accounts actually defaulted. In reality, the LGD1 model is applied to the non-defaulted portfolio (which is
typically the bigger dataset), whereas the LGD2 model is applied to the defaulted portfolio (which is typically a much smaller dataset). The datasets used for modelling therefore appear
counterintuitive to real world conditions.
3.1. LGD1 Results
The empirical observed LGD described in
Section 2.1
was applied to the pre-default book, i.e., accounts not in default. This number was further enhanced by using segmentation variables. As it is important that the variables used for segmentation do
not change over time, the final set of variables was selected on the basis of stability and business sense. These variables were then binned. The final variables selected for segmentation were loan
to value (LTV) at origination, channel/manufacturer and new/old/used indicator (NOU). The channel/manufacturer variable was derived the channel and manufacturer code. The empirical LGDs at the point
of default were subsequently calculated by these variables in a matrix type approach (lookup table). The final lookup table for accounts not in default (Stage 1 and 2) is in
Table 2
. Note that the standardised LGD values are shown to protect the confidential information surrounding this portfolio’s observed values. The final segmentation is consistent with business sense (note
the LGD separation from very negative to very positive). This lookup table approach—separating risks into different bins (slots)—is closely related to the concept of slotting (
BCBS 2019a
3.2. LGD2 Results
The results are described according to the five steps discussed in
Section 2.2
3.2.1. Step 1: Sample Created
A 24-month reference period, based on business input, was used. Only worked-out accounts were selected in this reference period. The LGD2 dataset was then split into a 70% training (946,285
observations, 38,352 unique accounts) and 30% validation dataset (405,630 observations, 37,720 unique accounts).
3.2.2. Step 2: Target and Weight Variables Created
Two rows (
$Y = 1$
$Y = 0$
) were created for each observation (i.e., per account per month). Each row was weighted, as described in
Section 2.2
3.2.3. Step 3: Input Variables
All input variables were screened using the following three requirements: percentage of missing values (more than 50% missing was used as a cut-off), the Gini statistic (variables with low Gini
statistic values were excluded) and business input. All bins were quantified using the average LGD value per bin, which was then standardised with the weight variable.
Table 3
lists the binned and quantified variables used in the weighted logistic regression. The final decision on binning was a combination of bucket stability (CSI), logical trends as well as consistency of
logical trends over time. Some of the observations on these variables, with respect to LGD values, include (variable names indicated in brackets):
• Higher LTV values are associated with higher LGD values (LTV).
• The higher the month on book (MOB) value for a customer, the lower the expected LGD value (MOB).
• The more months a customer has been in default, the higher the LGD value (Default).
• Customers buying old vehicles are associated with higher LGD values (New/Old).
• Certain channels and certain manufacturers are associated with higher LGD values (Channel Manufacturer).
This binning approach (separating risks in different bins or slots) is related to the underlying principle used in slotting (
BCBS 2019a
3.2.4. Step 4: Weighted Logistic Regression
A stepwise weighted logistic regression was fitted on the dataset, with a 5% significance level. The analyses were performed using SAS. The SAS code is provided as
Supplementary Material
for reference. While the authors of this paper used SAS, users can implement these techniques using any available analytic tool including Python and R. The final variables for accounts in default
(Stage 3) are given in
Table 4
. The Gini statistic on the training dataset was 45.49% and on the validation dataset 36.04%. The difference between the training and validation Gini is quite large and could be an indication of the
model not generalising well. However, it should be acknowledged that the Gini describes how well the model distinguishes between the two groups
$Y i = 1$
$Y i = 0$
Breed and Verster 2017
), while our underlying target is the LGD value which is a continuous value between 0 and 1. Therefore, a better measure for both model performance and comparing training and validation samples is
the mean squared error (MSE), although several other measures could have been used (see
Bijak and Thomas
) for an extensive list of performance measures applicable to LGD).
The mean squared error was therefore calculated as follows:
$M S E i = ( L G D ^ i − L G D i ) 2 N ,$
$L G D ^ i$
is the predicted LGD value of observation
from the model,
$L G D i$
the best estimate of the actual LGD (as defined in Equation (1)) and where
is the number of observations from
The MSE for the training and validation datasets were 0.0473 and 0.0427 respectively, thus showing a small mean squared error.
The R-square value (
Bijak and Thomas 2018
) was calculated as follows:
$R s q u a r e d = 1 − ∑ i ( L G D ^ i − L G D i ) 2 ∑ i ( L G D ^ i − LGD ¯ ) 2 ,$
$L G D ¯$
is the expected value of the actual LGD values. The R-squared value for the training and validation datasets were 0.3202 and 0.2727, respectively.
Furthermore, it showed that the model generalises well, as it also predicted well on the validation dataset (small difference between train and validation datasets). Here, the MSE on the training and
validation dataset are very close. Note that the MSE and the R-squared value were calculated on the LGD values and not the standardised LGD values.
3.2.5. Step 5: Test the Effect of the Dependence Assumption
Next, we estimated the regression coefficients using the GEE, first assuming an independent correlation structure and then with an autoregressive correlation structure of the order one. In
Table 5
, the results of the GEE using an independent correlation structure are shown. The code is provided in the
Supplementary Material
for reference, although any other analytics tool could be used to fit a GEE model.
The Gini on the training and validation datasets came to 45.49% and 36.04%, respectively, while the MSE on the training and validation datasets were 0.0473 and 0.0427. We used a significance level of
5% throughout, and all six variables were statistically significant. We note that the results (parameter estimates, Gini and MSE) are almost identical to that of the weighted logistic regression.
Next, an autoregressive correlation structure to the order of one was assumed. The results are shown in
Table 6
The Gini on the training data was 45.48% and on the validation dataset 36.04%, with the MSE values being 0.0522 and 0.0406, respectively, for training and validation. Note that all six variables were
again statistically significant.
Next, the three models were compared in terms of parameter estimates, standard errors of the estimates and on model performance.
Table 7
provides the parameter estimates comparisons, which indicate similar numbers for the weighted logistic regression (LR) and the GEE (independent correlation). This is similar to the results found by
). The parameter estimates were quite different, however, when using an autoregressive correlation matrix.
Table 8
the most significant difference between using a weighted logistic regression (disregarding the dependence among repeated observations of the same account) and using a GEE (addressing the dependence)
can be seen. The weighted logistic regression underestimates the standard error of the parameter estimates. This is also confirmed by
) and
Kuchibhatla and Fillenbaum
). Disregarding the dependence leads to the incorrect estimation of the standard errors. Although this is a problem from a statistical standpoint, resulting in incorrect inferences of the parameters,
the practical effect is negligible, as evident from the goodness-of-fit statistics (MSE) of the different models.
Table 9
summarises the model performance of all three models. It is interesting to note that all three models have almost identical performance. From a practical point of view, there was no difference in
using any of these three techniques. When the model is productionalised, the bank will use the model to predict a specific LGD value and the accuracy of this predicted LGD was almost identical with
either technique. If we suppose that the standard errors are not used by the bank, then there is no reason to refrain from using logistic regression.
One additional issue that bears mentioning is the low number of variables in the model itself. The banking industry prefers to see models that are consistent with how they would make
decisions—meaning models must have variables that not only make business sense, but also cover as many of the different information types that should be considered. Typically, between eight to
fifteen variables are considered normal in the industry (
Siddiqi 2017
). In a business setting, it is also common to add weaker variables, albeit those that display satisfactory correlations with the target, into the model itself.
3.3. Additional Investigation: Decision Tree
An additional modelling technique, namely the decision tree (
Breiman et al. 1984
), was considered to determine whether it could improve on the results above. First, the distribution of the actual LGD was analysed, as shown in
Figure 1
(training dataset). Note that the LGD values were standardised, by subtracting the average and then dividing by the standard deviation. It can be seen that the LGD has a huge spike to the left and a
much smaller spike closer to the right. This bimodal type of distribution is typical of an LGD distribution (
Joubert et al. 2018a
The decision tree (i.e., classification tree), was developed with three different settings (see the
Supplementary Material
for the specific code used). First, the default settings were used. Second, the decision tree was pruned by means of the average squared error (ASE), and lastly, a setting of “no pruning” was used.
For each of the decision trees, we used the same target variable (binary variable), the same weight variable and the same six explanatory variables as with the other models developed in this section.
The MSE (
Table 10
) for the decision tree was worse than the weighted logistic regression and the GEE models that were developed (
Table 9
). Note that we only show the MSE values here and not the Gini values, because, as noted before, the MSE is a better measure to indicate model performance of the “true” target value, i.e., the LGD
4. Strengths and Weaknesses of the Methodology
The method discussed in this paper presents several advantages. The first is that it is a relatively simplistic approach. Logistic regression is a well-known technique that has a long history in the
financial services industry. In contrast, for secured products, indirect more complex methodologies are often used. One example is using a haircut model for the loss severity component and survival
analysis for the probability component (
Joubert et al. 2018b
). Because logistic regression is well known and regularly used in banks, established monitoring metrics and governance practices have been embedded in the industry. These metrics, as well as the
methodology, are thoroughly understood by stakeholders, which leads to a high degree of confidence in the results. Logistic regression using the scorecard format provides an even more transparent and
user-friendly technique that is easy to understand and communicate to stakeholders.
A second advantage is that all variables are first binned and then quantified using the standardised LGD rate in each bin. Some of the specific advantages of this type of data transformation, as
noted earlier in the paper, are:
• Better handling of missing values, and their usage in the model.
• Better way to deal with outliers by minimising their influence.
• Improved generalisation of data.
• Easier way to capture non-linear trends.
• Easier comparison across variables through the usage of standardised average LGD value for each bin and standardised estimates.
• A reduction in the degrees of freedom introduces stability into the model.
A weakness of the weighted regression is that it disregards the assumption of independence and this results in the statistical inference of the parameter estimates being incorrect. In particular, the
standard errors of the parameter estimates are underestimated. Yet, there is no apparent difference in model accuracy.
5. Conclusions and Recommendation
This paper presented a new methodology to model LGD for IFRS 9 purposes, consisting of two components. First, the LGD1 model was applied to the non-default accounts and is an empirical value obtained
through a lookup table, based on a specified reference period. This LGD1 was further segmented across the most important variables to obtain a more granular estimate. Second, the LGD2 was applied to
defaulted accounts and is estimated using an exposure weighted logistic regression. This new methodology was tested by applying it on a real dataset, using a secured retail bank portfolio.
A comparison of this weighted logistic regression was done with GEE models to test the effect of the dependence among repeated observation of the same account. We discovered that when disregarding
the repeated accounts, the standard errors of the parameter estimates were underestimated. However, the practical effects of such disregard were found to be negligible.
In conclusion, we propose this new methodology to model LGD for IFRS 9 purposes based on the following reasons mentioned in the paper:
• This methodology presents a relatively simple approach using logistic regression, which is a well-known and accepted technique in the banking industry.
• The results are easy to interpret and understand, and when converted to the scorecard format, provide a transparent user-friendly output.
• The method also uses transformations that offer better alternatives for dealing with issues such as missing data and outliers.
• Most banks have well-established processes for monitoring and implementing logistic regression models and they are well understood by stakeholders.
• From a practical perspective, there was no discernible difference in model accuracy when comparing the logistic regression model to the GEE model or the decision tree.
From a purely theoretical point of view, we recommend using the GEE approach. However, as some banks do not use the parameter estimates or the associated standard errors for any decisions (e.g.,
variable selection), the weighted logistic regression approach may be preferable in such situations.
We suggest future research ideas to include comparing this new methodology to other LGD modelling techniques. We could also explore alternative data transformations from the current binning and
quantification using standardised LGD rates. We also did not include any direct costs in the calculation of the LGD, and determining how to split costs into direct and indirect components could be a
further research idea. According to IFRS 9, the LGD should include forward-looking macro-economic scenarios (
Miu and Ozdemir 2017
). This has also not been considered in this paper and could be researched in future.
Author Contributions
Conceptualization, D.G.B., T.V. and W.D.S.; formal analysis, D.G.B., T.V. and W.D.S.; investigation, D.G.B., T.V. and W.D.S.; methodology, D.G.B., T.V., W.D.S. and N.S.; software, D.G.B., T.V. and
W.D.S.; validation, D.G.B., T.V., W.D.S. and N.S.; visualization, D.G.B., T.V. and W.D.S.; writing—original draft, D.G.B., T.V. and W.D.S.; writing—review and editing, D.G.B., T.V., W.D.S. and N.S.
This research received no external funding.
This work is based on research supported in part by the Department of Science and Technology (DST) of South Africa. The grant holders at the Centre for Business Mathematics and Informatics
acknowledges that opinions, findings and conclusions or recommendations expressed in any publication generated by DST-supported research are those of the author(s) and that the DST accepts no
liability whatsoever in this regard.
Conflicts of Interest
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the
decision to publish the results.
$Binary Outcome ( Y )$ Exposure Weight Variable
0 $50,000 $13,500
1 $50,000 $36,500
LTV Channel & Manufacturer New/Old Standardised LGD
<=1 Group 1 New −1.0553
<=1 Group 1 Old −1.00075
<=1 Group 2 New −0.87389
<=1 Group 2 Old −0.18252
<=1 Group 1 New −0.2155
<=1 Group 1 Old −0.10513
<=1 Group 3 New −0.67346
<=1 Group 3 Old 0.050902
>1 Group 1 New −0.22311
>1 Group 1 Old 0.519007
>1 Group 2 New −0.24721
>1 Group 2 Old 0.532962
>1 Group 1 New 0.365509
>1 Group 1 Old 0.957936
>1 Group 3 New 0.647134
>1 Group 3 Old 1.503425
LTV LTV Range # Standardised LGD
Bin 1 LTV <=1 18188 −0.00566
Bin 2 LTV <=1.2 10461 −0.00268
Bin 3 LTV > 1.2 9703 0.004802
MOB MOB Range # Standardised LGD
Bin 1 MOB <=24 17593 0.005193244
Bin 2 MOB <=42 10431 −0.000342394
Bin 3 MOB > 42 10328 −0.006198457
Default Default Range # Standardised LGD
Bin 1 0 1043 −0.005747327
Bin 2 1 7706 −0.004411893
Bin 3 2+ 16150 −0.000289465
Bin 4 Other/Missing 13453 0.006032881
New/Old New/Old Range # Standardised LGD
Bin 1 New 15249 −0.004677389
Bin 2 Old 23103 0.004428005
Channel Manufacturer Channel Manufacturer Range # Standardised LGD
Bin 1 Group 1 3870 −0.008325
Bin 2 Group 2 5984 −0.004694
Bin 3 Group 3 26422 0.001172
Bin 4 Group 4 2076 0.011212
Analysis of Maximum Likelihood Estimates
$Parameter ( X )$ DF $Estimate ( β )$ Standard Error Wald Chi-Square Pr > ChiSq
Intercept 1 −1.0977 0.000012 8528907254 <0.0001
$LTV ( X 1 )$ 1 32.6329 0.00256 161977546 <0.0001
$Months on books ( X 2 )$ 1 10.3046 0.00261 15622966.5 <0.0001
$Default event ( X 3 )$ 1 173.9 0.00253 4709270394 <0.0001
$New / Old ( X 4 )$ 1 18.5934 0.00252 54593987.2 <0.0001
$Channel / Manufacturer ( X 5 )$ 1 17.3602 0.00248 48935118.5 <0.0001
Analysis of GEE Parameter Estimates
Empirical Standard Error Estimates
$Parameter ( X )$ $Estimate ( β )$ Standard Error 95% Confidence Limits Z Pr > |Z|
Intercept −1.0978 0.0116 −1.1205 −1.0750 −94.44 <0.0001
$LTV ( X 1 )$ 32.6348 2.4257 27.8805 37.3891 13.45 <0.0001
$Months on books ( X 2 )$ 10.3055 2.3708 5.6587 14.9522 4.35 <0.0001
$Default event ( X 3 )$ 173.8758 1.8297 170.2897 177.4619 95.03 <0.0001
$New / Old ( X 4 )$ 18.5943 2.4984 13.6976 23.4910 7.44 <0.0001
$Channel Manufacturer ( X 5 )$ 17.3607 2.5861 12.2921 22.4293 6.71 <0.0001
Analysis of GEE Parameter Estimates
Empirical Standard Error Estimates
$Parameter ( X )$ $Estimate ( β )$ Standard Error 95% Confidence Limits Z Pr > |Z|
Intercept −0.7973 0.0080 −0.8131 −0.7816 −99.15 <0.0001
$LTV ( X 1 )$ 24.8404 1.8335 21.2468 28.4339 13.55 <0.0001
$Months on books ( X 2 )$ 6.8528 1.7314 3.4592 10.2463 3.96 <0.0001
$Default event ( X 3 )$ 129.6377 1.3393 127.0126 132.2627 96.79 <0.0001
$New / Old ( X 4 )$ 12.5228 1.8139 8.9677 16.0779 6.90 <0.0001
$Channel Manufacturer ( X 5 )$ 11.7312 1.8959 8.0154 15.4470 6.19 <0.0001
Weighted LR GEE (Ind Corr) GEE (Ar1 Corr)
$β 0$ −1.0977 −1.0978 −0.7973
$β 1$ 32.6329 32.6348 24.8404
$β 2$ 10.3046 10.3055 6.8528
$β 3$ 173.9 173.8758 129.6377
$β 4$ 18.5934 18.5943 12.5228
$β 5$ 17.3602 17.3607 11.7312
Weighted LR GEE (Ind Corr) GEE (Ar1 Corr)
$β 0$ 0.000012 0.0116 0.0080
$β 1$ 0.00256 2.4257 1.8335
$β 2$ 0.00261 2.3708 1.7314
$β 3$ 0.00253 1.8297 1.3393
$β 4$ 0.00252 2.4984 1.8139
$β 5$ 0.00248 2.5861 1.8959
Technique Train MSE Valid MSE Train Gini Valid Gini
Weighted logistic regression 0.04727719 0.04274367 0.45492145910 0.36039085030
GEE (independent correlation) 0.04727703 0.04274417 0.45492145910 0.36039085030
GEE (AR 1 correlation) 0.05222953 0.04062386 0.45482289180 0.36037450660
Technique Valid MSE
Decision tree (default settings) 0.1012884759
Decision tree (prune on ASE) 0.1002412789
Decision tree (no pruning) 0.1041756997
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Breed, D.G.; Verster, T.; Schutte, W.D.; Siddiqi, N. Developing an Impairment Loss Given Default Model Using Weighted Logistic Regression Illustrated on a Secured Retail Bank Portfolio. Risks 2019, 7
, 123. https://doi.org/10.3390/risks7040123
AMA Style
Breed DG, Verster T, Schutte WD, Siddiqi N. Developing an Impairment Loss Given Default Model Using Weighted Logistic Regression Illustrated on a Secured Retail Bank Portfolio. Risks. 2019; 7(4):123.
Chicago/Turabian Style
Breed, Douw Gerbrand, Tanja Verster, Willem D. Schutte, and Naeem Siddiqi. 2019. "Developing an Impairment Loss Given Default Model Using Weighted Logistic Regression Illustrated on a Secured Retail
Bank Portfolio" Risks 7, no. 4: 123. https://doi.org/10.3390/risks7040123
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Tony Vazzana
Professor of Mathematics
Violette Hall 2262
(660) 785-4284
Current Semester Schedule
Teaching and Office Hour Schedule for the current semester.
Careers in Mathematics
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internship, research experience, or other math related summer opportunities.
MCTM Elementary and Middle School Math Contest
The Truman State Math Department will be hosting a an MCTM Qualifying Round competition this winter.
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Getting sub-optimal solution after time limit for large LP problems
I have a large LP problem that takes several hours to find the optimal solution. The solution is needed in a time sensitive environment, so I would like to be able to stop the optimization at point
in time and retrieve a feasible sub-optimal solution from Gurobi. Currently I am using the TimeLimit parameter but when Gurobi terminates due to the time limit no solution is found (model.SolCount==
0). Is there any way get some sub-optimal solution?
PS: The only constraints of my model are a set of mutually exclusive norm constraints, so finding a feasible solution should be trivial.
number of variables = 10390740, number of constraints = 266730
Set parameter Threads to value 6
Gurobi Optimizer version 11.0.0 build v11.0.0rc2 (win64 - Windows 10.0 (19045.2))
CPU model: 13th Gen Intel(R) Core(TM) i7-1365U, instruction set [SSE2|AVX|AVX2]
Thread count: 10 physical cores, 12 logical processors, using up to 12 threads
Academic license 2432445 - for non-commercial use only - registered to da___@ugent.be
Optimize a model with 273870 rows, 10390741 columns and 50528006 nonzeros
Model fingerprint: 0xe079cdae
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+00]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
Presolve removed 0 rows and 0 columns (presolve time = 5s) ...
Presolve removed 0 rows and 0 columns (presolve time = 14s) ...
Presolve removed 0 rows and 0 columns (presolve time = 15s) ...
Presolve removed 7140 rows and 7140 columns (presolve time = 33s) ...
Presolve removed 7140 rows and 7140 columns (presolve time = 38s) ...
Presolve removed 7140 rows and 7140 columns (presolve time = 40s) ...
Presolve removed 7140 rows and 7140 columns (presolve time = 50s) ...
Presolve removed 7140 rows and 7140 columns (presolve time = 56s) ...
Presolve removed 7140 rows and 7140 columns
Presolve time: 62.42s
Presolved: 266730 rows, 10383601 columns, 50513726 nonzeros
Concurrent LP optimizer: primal simplex, dual simplex, and barrier
Showing barrier log only...
• Hi Dante,
You should be able to access the best solution found when the optimization ends, the same way you would retrieve the solution when it finds the optimal solution. It is not clear if this is the
case, but if presolve is taking a long time and you have very limited time, you may consider reducing the value of the Presolve parameter.
• Hi Michel,
Thanks for you response.
In most case the presolve is not the problem, see e.g. this smaller problem below. I set the time limit to 30 seconds. After the timeout I check if there is a solution using
print(f"Solutions found? {model.SolCount}") -> log shows "Solutions found? 0"
so I would expect no solution was found. To be sure I tried to retrieve the solution like I would normally do with
but this throws:
AttributeError: Unable to retrieve attribute 'x'
So I am not able to retrieve any sub-optimal solution.
Full log is shown below:
Set parameter Username
Set parameter WLSAccessID
Set parameter WLSSecret
Set parameter LicenseID
number of variables = 1391544, number of constraints = 72336
Set parameter Threads to value 6
Set parameter TimeLimit to value 30
Gurobi Optimizer version 11.0.0 build v11.0.0rc2 (win64 - Windows 10.0 (19045.2))
CPU model: 13th Gen Intel(R) Core(TM) i7-1365U, instruction set [SSE2|AVX|AVX2]
Thread count: 10 physical cores, 12 logical processors, using up to 12 threads
Optimize a model with 75240 rows, 1391545 columns and 6595704 nonzeros
Model fingerprint: 0x89f40056
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+00]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
Presolve removed 2904 rows and 2904 columns
Presolve time: 4.98s
Presolved: 72336 rows, 1388641 columns, 6589896 nonzeros
Concurrent LP optimizer: primal simplex, dual simplex, and barrier
Showing barrier log only...
Elapsed ordering time = 5s
Ordering time: 10.12s
Barrier statistics:
Dense cols : 1
AA' NZ : 4.434e+06
Factor NZ : 7.560e+06 (roughly 700 MB of memory)
Factor Ops : 8.337e+09 (less than 1 second per iteration)
Threads : 8
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 8.64118532e+01 0.00000000e+00 1.76e+03 0.00e+00 1.79e-02 23s
1 8.94762291e+01 -2.71494429e+03 1.17e+01 3.48e-03 1.19e-03 24s
2 8.93820253e+01 -1.69906375e+01 2.00e-15 1.80e-05 3.89e-05 25s
3 8.55828510e+01 3.48615786e+01 2.00e-15 9.78e-06 1.85e-05 27s
4 7.67951497e+01 5.12615496e+01 3.62e-14 4.25e-06 9.32e-06 28s
5 7.30615169e+01 5.62575457e+01 2.29e-14 2.49e-06 6.12e-06 29s
6 6.93874338e+01 5.94058357e+01 1.18e-14 4.32e-07 3.62e-06 30s
7 6.73249316e+01 6.06981889e+01 6.00e-15 2.08e-07 2.39e-06 30s
Barrier performed 7 iterations in 30.32 seconds (29.88 work units)
Barrier solve interrupted - model solved by another algorithm
Stopped in 94976 iterations and 30.53 seconds (36.23 work units)
Time limit reached
Solutions found? 0
Traceback (most recent call last):
AttributeError: Unable to retrieve attribute 'x'
• Ok, let me explain a few things that can help and the challenges in having a result for this model in 30 seconds.
1. If the presolve takes the full 30 seconds, you will not have a solution within your time limit. For this last log it does not seem to be the case, but it may happen for other models. If that
is a problem, you would have to change the Presolve parameter.
2. You may want to try different values for the parameter Method, so Gurobi will focus all computing power in a single algorithm, which may help you finding a solution in time.
3. There are a lot of other parameters you can experiment with to try to have a solution faster, but for this model size and time limit, it may be impossible. Try using 1 minute or more and see
if you are able to retrieve a solution.
• Hi Dante,
Michel has provided some good advice.
Ultimately though, this:
Getting sub-optimal solution after time limit for large LP problems
is not possible.
Following on from Michel's recommendation, first try setting Method=2. Barrier will typically work better on problems of this size. If you do not need a basic solution then setting
SolutionTarget=1 or (alternatively Crossover=0) could potentially save a lot of time if the simplex algorithms are struggling on your LP.
- Riley
• Thanks for your help, Michel and Riley!
Setting the parameters did help to reduce computation time. Unfortunately, the speedup was not high enough. I will have to resort to heuristicst to solve this problem.
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prime number
prime number
An integer
greater than one
is called a
prime number
if its only positive
) are one and itself. For example, the prime
of 10 are 2 and 5, and the first six primes are 2, 3, 5, 7, 11, and 13. By the
fundamental theorem of arithmetic
we know that all integers greater than one factor uniquely into a product of primes.
Technical comment on the definition: In the integers we can easily prove the following
1. A positive integer p, not one, is prime if whenever it divides the product of integers ab, then it divides a or b (perhaps both).
2. A positive integer p, not one, is prime if it can not be decomposed into factors p=ab, neither of which is 1 or -1.
When we study other number systems, these properties may not hold. So in these systems of integers (often called rings) we often make the following definitions:
1. Any element which divides one is a unit.
2. An element p, not a unit, is prime if whenever it divides the product of integers ab, then it divides a or b (perhaps both).
3. An element p, nonzero and not a unit, is called irreducible if it can not be decomposed into factors p=ab, neither of which is a unit.
See Also: PrimeNumberThm, PrimeGaps
Related pages (outside of this work)
Printed from the PrimePages <t5k.org> © Reginald McLean. | {"url":"https://t5k.org/glossary/page.php?sort=Prime","timestamp":"2024-11-02T23:29:16Z","content_type":"text/html","content_length":"10953","record_id":"<urn:uuid:7242f36e-1b79-4300-9aa2-d806c07ba245>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00160.warc.gz"} |
secret_key formats a secret_key in this fashion: (Sign.secret_key AAAAAA==).
public_key formats a public_key in this fashion: (Sign.public_key AAAAAA==).
keypair formats a keypair in this fashion: ((Sign.secret_key AAAAAA==) (Sign.public_key AAAAAA==)).
signature formats a signature in this fashion: (Sign.signature AAAAAA==).
seed formats a seed in this fashion: (Sign.seed AAAAAA==).
module type S = sig ... end
Bytes offers versions of each of the above formatters, which get their value from a bytes value.
Bigbytes offers versions of each of the above formatters, which get their value from a Sodium.bigbytes value. | {"url":"https://schube.srht.site/sodium-fmt/Sodium_fmt/Sign/index.html","timestamp":"2024-11-09T18:59:05Z","content_type":"application/xhtml+xml","content_length":"5717","record_id":"<urn:uuid:bf68c918-d5e8-41c8-ab15-cbde2277af84>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00308.warc.gz"} |
Fast Rank-1 NMF for Missing Data with KL Divergence
Fast Rank-1 NMF for Missing Data with KL Divergence
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:2927-2940, 2022.
We propose a fast non-gradient-based method of rank-1 non-negative matrix factorization (NMF) for missing data, called A1GM, that minimizes the KL divergence from an input matrix to the reconstructed
rank-1 matrix. Our method is based on our new finding of an analytical closed-formula of the best rank-1 non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is known to exactly
solve NMF for missing data if positions of missing values satisfy a certain condition, and A1GM transforms a given matrix so that the analytical solution to NMMF can be applied. We empirically show
that A1GM is more efficient than a gradient method with competitive reconstruction errors.
Cite this Paper
Related Material | {"url":"https://proceedings.mlr.press/v151/ghalamkari22a.html","timestamp":"2024-11-09T17:26:29Z","content_type":"text/html","content_length":"14833","record_id":"<urn:uuid:23f17691-89e3-4bf4-81a1-07990556c0d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00079.warc.gz"} |
Fields Medalist Martin Hairer Visited Department of Applied Mathematics
Fields Medalist Martin Hairer Visited Department of Applied Mathematics
November 1, 2016Posted in: University News
Martin Hairer, who won the Fields Medal in 2014, visited Illinois Tech’s Department of Applied Mathematics on October 27 to meet with faculty and students and deliver a lecture, “Ergodic theory of
singular stochastic PDEs.” The Fields Medal is regarded as the highest honor a mathematician can receive–the “Nobel Prize in math,” as there is no Nobel Prize in math.
Hairer developed a mathematical theory for singular stochastic partial differential equations, so that solutions to these equations make sense. This allows for stochastic modeling of various
mechanisms in science and engineering.
Hairer is currently the Regius Professor of Mathematics, Mathematics Department, University of Warwick. He is a Fellow of the Royal Society, a Fellow of the American Mathematical Society, and has
received many other honors and awards. He also created the sound-editing program Amadeus.
Jinqiao (Jeffrey) Duan, professor of applied mathematics, hosted the visit. | {"url":"https://today.iit.edu/fields-medalist-martin-hairer-visited-applied-mathematics-department/","timestamp":"2024-11-01T22:48:47Z","content_type":"text/html","content_length":"34559","record_id":"<urn:uuid:5b7bbaaf-72a7-46b5-b6ff-45bf160b06e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00713.warc.gz"} |
Statistical Tests for comparing data
Statistical Tests for comparing data by Kelvin Ho
1. Categorical
1.1. 1-Sample
1.1.1. 1-Sample proportion (Z test)
1.1.1.1. **Purpose:** To test the proportion of a sample to a global value
1.1.1.2. Assumptions
1.1.1.2.1. -Sample size ~ 20 -Mutually exclusive groups -Constant
1.1.1.3. Data model
1.1.2. Sign test (non-parametric)
1.1.2.1. Purpose: Test a claim regarding the proportion of a population meeting a particular criterion using a small sample of data. Because the sample is small, the normal approximation to the
binomial does not apply, and the Z test should not be used.
1.1.2.2. Assumptions
1.1.2.2.1. -Sampling is binominally distributed -the probability of success in a given trial equal to the assumed proportion specified in the null
1.1.2.3. Data model
1.2. 2-Sample
1.2.1. 2-Sample proportion (Z test)
1.2.1.1. **Purpose:** Test a claim regarding how the proportions of two populations compare
1.2.1.2. Data model
1.3. 2-Sample (Paired)
1.3.1. 2-Sample proportion (Z test)
1.4. Categorical with observations
1.4.1. Goodness of Fit Tests (Frequency Dist)
1.4.1.1. Purpose: To test the population data if they are normally/uniformly dist
1.4.1.2. Assumptions
1.4.1.2.1. -Independent data -The expected frequencies should be theoretically determined based on a specified distribution or null hypothesis.
1.4.1.3. Data model
1.4.2. Goodness of Fit Tests (Contingency table)
1.4.2.1. Purpose: Evaluate whether there is an association or relationship between two categorical variables based on their observed frequencies, compared to expected frequencies
1.4.2.2. Data model
2. Quantitative
2.1. 1-Sample
2.1.1. 1-Sample T
2.1.1.1. Purpose: Test a claim regarding how the mean of a population relates to a given number.
2.1.1.2. Assumptions
2.1.1.2.1. -Normal distribution -Interval level data -Sample size ~30
2.1.1.3. Data model
2.1.2. χ2 Test of Variance (Chisquare)
2.1.2.1. Purpose: Test a claim that the variance (or standard deviation) of a population equates to a given number.
2.1.2.2. Assumptions
2.1.2.2.1. -Normally distribution -Sample size 20+
2.2. 2-Sample
2.2.1. 2-Sample T
2.2.1.1. Equal Var
2.2.1.1.1. Purpose: Test a claim regarding how the means of two populations relate to each other
2.2.1.1.2. Assumptions
2.2.1.1.3. Data model
2.2.1.2. Unequal Var (Welch's t-test)
2.2.2. F test
2.2.2.1. Purpose: to test the variance of two pop
2.2.2.2. Assumptions
2.2.2.2.1. -Both population should be normal dist -Sample size 20+
2.2.3. Wilcoxon Rank Sum test (non-parametric)
2.2.3.1. Purpose: Test a claim regarding how the means of two populations relate to each other when the samples are independently selected but small and it is believed that the populations of data
are non-normal.
2.2.3.2. Assumptions
2.2.3.2.1. -Ordinal & independent data -Does not assume equal variances between the two groups.
2.2.3.3. Data model
2.3. 2-Sample (Paired)
2.3.1. 2-Sample T paired test
2.3.1.1. Purpose: Before-and-After Comparisons
2.3.1.2. T paired test Assumptions
2.3.1.2.1. -The differences between paired observations should follow an approximately normal distribution. -The paired observations should be independent of each other -Random sampling -Interval or
Ratio Data (meaningful magnitudes)
2.3.1.3. Data model
2.3.2. Wilcoxon Signed rank test (Non-parametric)
2.3.2.1. Purpose: Test a claim regarding how the means of two populations relate to each other when the samples are dependent and small, and it is believed that the population of differences is
2.3.2.2. Assumptions: The sample data must be at least ordinal level so that they can be ranked
2.3.2.3. Data model
2.4. 3 or more samples
2.4.1. ANOVA
2.4.1.1. 1-Factor
2.4.1.1.1. Data model
2.4.1.2. 2-Factor (Without rep)
2.4.1.2.1. Data model
2.4.2. KW (Kruskal-Wallis) (non-parametric)
2.4.2.1. Purpose: Test a claim that 2+ populations have the same central location. Use when the populations are non-normal and sample sizes are small (since ANOVA cannot be used under these
2.4.2.2. Assumptions
2.4.2.2.1. -Ordinal data -Populations are assumed to have similar shapes (not identical variances). If the populations are symmetric, then this is a test comparing medians.
2.4.2.3. Data model | {"url":"https://www.mindmeister.com/3258472421/statistical-tests-for-comparing-data","timestamp":"2024-11-06T11:40:25Z","content_type":"application/xhtml+xml","content_length":"53807","record_id":"<urn:uuid:933f0581-5b2a-4d20-8101-e9035edb2ee1>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00554.warc.gz"} |
Hyperparameter - Data Science Wiki
Hyperparameter :
Hyperparameters are a type of
machine learning
algorithms that cannot be directly learned from the data. They are set prior to training the
and are used to control the behavior of the learning algorithm.
One example of a hyperparameter is the learning rate in a neural
. The learning rate determines the step size at which the model updates its weights during training. A higher learning rate can lead to faster convergence, but also has the potential to overshoot the
optimal solution. On the other hand, a lower learning rate may lead to slower convergence but can help the model avoid getting stuck in a local minimum.
Another example of a hyperparameter is the
term in a linear
model. Regularization is a method used to prevent
, which occurs when a model fits the training data too closely but fails to generalize to new data. The regularization term controls the strength of the regularization, with a higher value indicating
stronger regularization. This can help the model avoid overfitting by penalizing large weights, but can also lead to underfitting if the regularization is too strong.
Hyperparameters play a critical role in the performance of a machine learning model. They can have a significant impact on the accuracy and generalizability of the model, and can therefore be
considered a key part of the model selection process.
When selecting hyperparameters for a model, it is important to consider the characteristics of the data and the goals of the model. For example, if the data is highly imbalanced, a different set of
hyperparameters may be needed compared to a
with balanced classes. Additionally, the specific performance metrics that are important for the model should be considered when choosing hyperparameters.
For instance, if the goal is to maximize
, a different set of hyperparameters may be needed compared to a model that is focused on maximizing
One common approach to selecting hyperparameters is through trial and error. This involves training the model with different combinations of hyperparameters and evaluating their performance using a
validation set. The hyperparameters that produce the best performance on the validation set are then selected for the final model.
Another approach is to use a grid search, where a grid of hyperparameter combinations is defined and the model is trained and evaluated for each combination. The combination with the best performance
is selected as the final set of hyperparameters.
Alternatively, some machine learning algorithms include methods for automatically setting hyperparameters, such as the Bayesian optimization algorithm. This algorithm uses a probabilistic model to
explore the space of hyperparameters and select the values that are most likely to produce the best performance.
Overall, hyperparameters are an important part of machine learning algorithms, as they can significantly impact the performance of the model. Careful selection of hyperparameters can help improve the
accuracy and generalizability of the model, and can therefore be a crucial step in the machine learning process. | {"url":"https://datasciencewiki.net/hyperparameter/","timestamp":"2024-11-13T16:36:00Z","content_type":"text/html","content_length":"42710","record_id":"<urn:uuid:691139ec-c42c-4dea-bfeb-6288181a65ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00232.warc.gz"} |
Multiply Rational Numbers Worksheet
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Check more of Multiply Rational Numbers Worksheet below
Multiplying Rational Numbers Worksheet
Rational Numbers Worksheet For 8th 9th Grade Lesson Planet
Multiplying Rational Numbers Worksheet
Multiply Rational Expressions Worksheet
Multiplying Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet
Learn And Practice Rational Numbers Grade 6 Printable Worksheets
To multiply two rational numbers we multiply the numerators and the denominators For example 2 3 x 4 5 2 x 4 3 x 5 8 15 For example 2 3 x 4 5 2 x 4 3 x 5 8 15 To divide two rational numbers we flip
the second number and multiply it
Multiplying Rational Expressions Kuta Software
Multiplying Rational Expressions Date Period Simplify each expression 1 59 n 99 80 33 n 2 53 43 46 n2 31 3 93 21 n 34 n 51 n 4 79 n 25 85 27 n2 5 96 38 n Create your own worksheets like this one with
Infinite Algebra 1 Free trial available at KutaSoftware Title Multiplying Rational Expressions
To multiply two rational numbers we multiply the numerators and the denominators For example 2 3 x 4 5 2 x 4 3 x 5 8 15 For example 2 3 x 4 5 2 x 4 3 x 5 8 15 To divide two rational numbers we flip
the second number and multiply it
Multiplying Rational Expressions Date Period Simplify each expression 1 59 n 99 80 33 n 2 53 43 46 n2 31 3 93 21 n 34 n 51 n 4 79 n 25 85 27 n2 5 96 38 n Create your own worksheets like this one with
Infinite Algebra 1 Free trial available at KutaSoftware Title Multiplying Rational Expressions
Multiply Rational Expressions Worksheet
Rational Numbers Worksheet For 8th 9th Grade Lesson Planet
Multiplying Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet
Multiplying Rational Expressions Worksheet
Multiplying Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet Pdf Thekidsworksheet | {"url":"https://szukarka.net/multiply-rational-numbers-worksheet","timestamp":"2024-11-08T11:03:18Z","content_type":"text/html","content_length":"25782","record_id":"<urn:uuid:7e09966a-1160-422a-86f0-d4f0f95ef7a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00117.warc.gz"} |
SMRI Seminar: Wiesel -- SMRI Seminar: Johannes Wiesel
SMS scnews item created by Catherine Meister at Wed 18 Sep 2024 0928
Type: Seminar
Modified: Thu 3 Oct 2024 1511; Thu 3 Oct 2024 1518
Distribution: World
Expiry: 16 Oct 2024
Calendar1: 10 Oct 2024 1300-1400
CalLoc1: SMRI Seminar Room (A12 Room 301)
CalTitle1: SMRI Seminar: Johannes Wiesel TBA
Auth: cmeister@w95s10l3.staff.sydney.edu.au (cmei0631) in SMS-SAML
SMRI Seminar: Wiesel -- SMRI Seminar: Johannes Wiesel
Speaker: Johannes Wiesel Title: Estimating processes using optimal transport Abstract:
The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal
transport cost. Its numerous mathematical properties have been well studied in recent
years. The adapted Wasserstein distance $\mathcal{AW}_p$ extends this theory to laws of
discrete time stochastic processes in their natural filtrations, making it particularly
well suited for analyzing time-dependent stochastic optimization problems. Recently,
$\mathcal{W}_p$ has found a lot of interesting applications in statistics and machine
learning. In this talk I will explain how some of these results can be extended to
stochastic processes and $\mathcal{AW}_p$. In particular I will focus a new measure of
dependence between two random objects, which I call the Wasserstein correlation. | {"url":"https://www.maths.usyd.edu.au/s/scnitm/cmeister-SMRISeminar-Wiesel-SMRISe?Clean=1","timestamp":"2024-11-10T14:24:51Z","content_type":"text/html","content_length":"2183","record_id":"<urn:uuid:06998e2d-0653-40c6-900a-a62b1f9f6949>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00824.warc.gz"} |
torch.distributed.tensor is currently in alpha state and under development, we are committing backward compatibility for the most APIs listed in the doc, but there might be API changes if necessary.
PyTorch DTensor (Distributed Tensor)¶
PyTorch DTensor offers simple and flexible tensor sharding primitives that transparently handles distributed logic, including sharded storage, operator computation and collective communications
across devices/hosts. DTensor could be used to build different paralleism solutions and support sharded state_dict representation when working with multi-dimensional sharding.
Please see examples from the PyTorch native parallelism solutions that are built on top of DTensor:
DTensor follows the SPMD (single program, multiple data) programming model to empower users to write distributed program as if it’s a single-device program with the same convergence property. It
provides a uniform tensor sharding layout (DTensor Layout) through specifying the DeviceMesh and Placement:
• DeviceMesh represents the device topology and the communicators of the cluster using an n-dimensional array.
• Placement describes the sharding layout of the logical tensor on the DeviceMesh. DTensor supports three types of placements: Shard, Replicate and Partial.
DTensor Class APIs¶
DTensor is a torch.Tensor subclass. This means once a DTensor is created, it could be used in very similar way to torch.Tensor, including running different types of PyTorch operators as if running
them in a single device, allowing proper distributed computation for PyTorch operators.
In addition to existing torch.Tensor methods, it also offers a set of additional methods to interact with torch.Tensor, redistribute the DTensor Layout to a new DTensor, get the full tensor content
on all devices, etc.
class torch.distributed.tensor.DTensor(local_tensor, spec, *, requires_grad)¶
DTensor (Distributed Tensor) is a subclass of torch.Tensor that provides single-device like abstraction to program with multi-device torch.Tensor. It describes the distributed tensor sharding
layout (DTensor Layout) through the DeviceMesh and following types of Placement:
□ Shard: Tensor sharded on the tensor dimension dim on the devices of the DeviceMesh dimension
□ Replicate: Tensor replicated on the devices of the DeviceMesh dimension
□ Partial: Tensor is pending reduction on the devices of the DeviceMesh dimension
When calling PyTorch operators, DTensor overrides the PyTorch operators to perform sharded computation and issue communications whenever necessary. Along with the operator computation, DTensor
will transform or propagate the placements (DTensor Layout) properly (based on the operator semantic itself) and generate new DTensor outputs.
To ensure numerical correctness of the DTensor sharded computation when calling PyTorch operators, DTensor requires every Tensor argument of the operator be DTensor.
Return type
property device_mesh: DeviceMesh¶
The DeviceMesh attribute that associates with this DTensor object.
device_mesh is a read-only property, it can not be set.
static from_local(local_tensor, device_mesh=None, placements=None, *, run_check=False, shape=None, stride=None)[source]¶
Create a DTensor from a local torch.Tensor on each rank according to the device_mesh and placements specified.
Keyword Arguments
Return type
When run_check=False, it is the user’s responsibility to ensure the local tensor passed in is correct across ranks (i.e. the tensor is sharded for the Shard(dim) placement or replicated for
the Replicate() placement). If not, the behavior of the created DTensor is undefined.
from_local is differentiable, the requires_grad of the created DTensor object will depend on if local_tensor requires_grad or not.
full_tensor(*, grad_placements=None)[source]¶
Return the full tensor of this DTensor. It will perform necessary collectives to gather the local tensors from other ranks in its DeviceMesh and concatenate them together. It’s a syntatic
sugar of the following code:
dtensor.redistribute(placements=[Replicate()] * mesh.ndim).to_local()
Keyword Arguments
grad_placements (List[Placement], optional) – the placements describes the future layout of any gradient layout of the full Tensor returned from this function. full_tensor converts
DTensor to a full torch.Tensor and the returned torch.tensor might not be used as the original replicated DTensor layout later in the code. This argument is the hint that user can give to
autograd in case the gradient layout of the returned tensor does not match the original replicated DTensor layout. If not specified, we will assume the gradient layout of the full tensor
be replicated.
A torch.Tensor object that represents the full tensor of this DTensor.
Return type
full_tensor is differentiable.
property placements: Tuple[Placement, ...]¶
The placements attribute of this DTensor that describes the layout of this DTensor on the its DeviceMesh.
placements is a read-only property, it can not be set.
redistribute(device_mesh=None, placements=None, *, async_op=False)[source]¶
redistribute performs necessary collective operations that redistribute the current DTensor from its current placements to a new placements, or from is current DeviceMesh to a new DeviceMesh.
i.e. we can turn a Sharded DTensor to a Replicated DTensor by specifying a Replicate placement for each dimension of the DeviceMesh.
When redistributing from current to the new placements on one device mesh dimension, we will perform the following operations including communication collective or local operation:
1. Shard(dim) -> Replicate(): all_gather
2. Shard(src_dim) -> Shard(dst_dim): all_to_all
3. Replicate() -> Shard(dim): local chunking (i.e. torch.chunk)
4. Partial() -> Replicate(): all_reduce
5. Partial() -> Shard(dim): reduce_scatter
redistribute would correctly figure out the necessary redistribute steps for DTensors that are created either on 1-D or N-D DeviceMesh.
○ device_mesh (DeviceMesh, optional) – DeviceMesh to place the DTensor. If not specified, it would use the current DTensor’s DeviceMesh. default: None
○ placements (List[Placement], optional) – the new placements that describes how to place the DTensor into the DeviceMesh, must have the same number of elements as device_mesh.ndim.
default: replicate on all mesh dimensions
Keyword Arguments
async_op (bool, optional) – whether to perform the DTensor redistribute operation asynchronously or not. Default: False
A DTensor object
Return type
redistribute is differentiable, which means user do not need to worry about the backward formula of the redistribute operation.
redistribute currently only supports redistributing DTensor on the same DeviceMesh, Please file an issue if you need to redistribute DTensor to different DeviceMesh.
to_local(*, grad_placements=None)[source]¶
Get the local tensor of this DTensor on its current rank. For sharding it returns a local shard of the logical tensor view, for replication it returns the replica on its current rank.
Keyword Arguments
grad_placements (List[Placement], optional) – the placements describes the future layout of any gradient layout of the Tensor returned from this function. to_local converts DTensor to
local tensor and the returned local tensor might not be used as the original DTensor layout later in the code. This argument is the hint that user can give to autograd in case the
gradient layout of the returned tensor does not match the original DTensor layout. If not specified, we will assume the gradient layout remains the same as the original DTensor and use
that for gradient computation.
A torch.Tensor or AsyncCollectiveTensor object. it represents the local tensor on its current rank. When an AsyncCollectiveTensor object is returned, it means the local tensor is not
ready yet (i.e. communication is not finished). In this case, user needs to call wait to wait the local tensor to be ready.
Return type
to_local is differentiable, the requires_grad of the local tensor returned will depend on if the DTensor requires_grad or not.
DeviceMesh as the distributed communicator¶
DeviceMesh was built from DTensor as the abstraction to describe cluster’s device topology and represent multi-dimensional communicators (on top of ProcessGroup). To see the details of how to create/
use a DeviceMesh, please refer to the DeviceMesh recipe.
DTensor Placement Types¶
DTensor supports the following types of Placement on each DeviceMesh dimension:
Different ways to create a DTensor¶
There’re three ways to construct a DTensor:
□ distribute_tensor() creates a DTensor from a logical or “global” torch.Tensor on each rank. This could be used to shard the leaf torch.Tensor s (i.e. model parameters/buffers and inputs).
□ DTensor.from_local() creates a DTensor from a local torch.Tensor on each rank, which can be used to create DTensor from a non-leaf torch.Tensor s (i.e. intermediate activation tensors during
□ DTensor provides dedicated tensor factory functions (e.g. empty(), ones(), randn(), etc.) to allow different DTensor creations by directly specifying the DeviceMesh and Placement. Compare to
distribute_tensor(), this could directly materializing the sharded memory on device, instead of performing sharding after initializing the logical Tensor memory.
Create DTensor from a logical torch.Tensor¶
The SPMD (single program, multiple data) programming model in torch.distributed launches multiple processes (i.e. via torchrun) to execute the same program, this means that the model inside the
program would be initialized on different processes first (i.e. the model might be initialized on CPU, or meta device, or directly on GPU if enough memory).
DTensor offers a distribute_tensor() API that could shard the model weights or Tensors to DTensor s, where it would create a DTensor from the “logical” Tensor on each process. This would empower the
created DTensor s to comply with the single device semantic, which is critical for numerical correctness.
torch.distributed.tensor.distribute_tensor(tensor, device_mesh=None, placements=None)¶
Distribute a leaf torch.Tensor (i.e. nn.Parameter/buffers) to the device_mesh according to the placements specified. The rank of device_mesh and placements must be the same. The tensor to
distribute is the logical or “global” tensor, and the API would use the tensor from first rank of the DeviceMesh dimension as the source of truth to perserve the single-device semantic. If you
want to construct a DTensor in the middle of the Autograd computation, please use DTensor.from_local() instead.
☆ tensor (torch.Tensor) – torch.Tensor to be distributed. Note that if you want to shard a tensor on a dimension that is not evenly divisible by the number of devices in that mesh
dimension, we use torch.chunk semantic to shard the tensor and scatter the shards. The uneven sharding behavior is experimental and subject to change.
☆ device_mesh (DeviceMesh, optional) – DeviceMesh to distribute the tensor, if not specified, must be called under a DeviceMesh context manager, default: None
☆ placements (List[Placement], optional) – the placements that describes how to place the tensor on DeviceMesh, must have the same number of elements as device_mesh.ndim. If not specified,
we will by default replicate the tensor across the device_mesh from the first rank of each dimension of the device_mesh.
A DTensor or XLAShardedTensor object.
Return type
When initialize the DeviceMesh with the xla device_type, distribute_tensor return XLAShardedTensor instead. see this issue for more details. The XLA integration is experimental and subject to
Along with distribute_tensor(), DTensor also offers a distribute_module() API to allow easier sharding on the nn.Module level
torch.distributed.tensor.distribute_module(module, device_mesh=None, partition_fn=None, input_fn=None, output_fn=None)¶
This function expose three functions to control the parameters/inputs/outputs of the module:
1. To perform sharding on the module before runtime execution by specifying the partition_fn (i.e. allow user to convert Module parameters to DTensor parameters according to the partition_fn
specified). 2. To control the inputs or outputs of the module during runtime execution by specifying the input_fn and output_fn. (i.e. convert the input to DTensor, convert the output back to
☆ module (nn.Module) – user module to be partitioned.
☆ device_mesh (DeviceMesh) – the device mesh to place the module.
☆ partition_fn (Callable) – the function to partition parameters (i.e. shard certain parameters across the device_mesh). If partition_fn is not specified, by default we replicate all module
parameters of module across the mesh.
☆ input_fn (Callable) – specify the input distribution, i.e. could control how the input of the module is sharded. input_fn will be installed as a module forward_pre_hook (pre forward
☆ output_fn (Callable) – specify the output distribution, i.e. could control how the output is sharded, or convert it back to torch.Tensor. output_fn will be installed as a module
forward_hook (post forward hook).
A module that contains parameters/buffers that are all DTensor s.
Return type
When initialize the DeviceMesh with the xla device_type, distribute_module return nn.Module with PyTorch/XLA SPMD annotated parameters. See this issue for more details. The XLA integration is
experimental and subject to change.
DTensor Factory Functions¶
DTensor also provides dedicated tensor factory functions to allow creating DTensor directly using torch.Tensor like factory function APIs (i.e. torch.ones, torch.empty, etc), by additionally
specifying the DeviceMesh and Placement for the DTensor created:
When launching the program, you can turn on additional logging using the TORCH_LOGS environment variable from torch._logging :
• TORCH_LOGS=+dtensor will display logging.DEBUG messages and all levels above it.
• TORCH_LOGS=dtensor will display logging.INFO messages and above.
• TORCH_LOGS=-dtensor will display logging.WARNING messages and above.
Debugging Tools¶
To debug the program that applied DTensor, and understand more details about what collectives happened under the hood, DTensor provides a CommDebugMode:
class torch.distributed.tensor.debug.CommDebugMode¶
CommDebugMode is a context manager that counts the number of functional collectives within its context. It does this using a TorchDispatchMode.
Example usage
mod = ...
comm_mode = CommDebugMode()
with comm_mode:
Generates detailed table displaying operations and collective tracing information on a module level. Amount of information is dependent on noise_level
0. prints module-level collective counts
1. prints dTensor operations not included in trivial operations, module information
2. prints operations not included in trivial operations
3. prints all operations
generate_json_dump(file_name='comm_mode_log.json', noise_level=3)[source]¶
Creates json file used to build browser visual 0. prints module-level collective counts 1. prints dTensor operations not included in trivial operations 2. prints operations not included in
trivial operations 3. prints all operations
Returns the communication counts as a dictionary.
The communication counts as a dictionary.
Return type
Dict[Any, int]
Return type
Dict[str, Dict[str, Any]]
Return type
Dict[str, Dict[str, Any]]
Return type
log_comm_debug_tracing_table_to_file(file_name='comm_mode_log.txt', noise_level=3)[source]¶
Alternative to console CommDebugMode output, writes to file specified by the user
To visualize the sharding of a DTensor that have less than 3 dimensions, DTensor provides visualize_sharding():
torch.distributed.tensor.debug.visualize_sharding(dtensor, header='')¶
Visualizes sharding in the terminal for DTensor that are 1D or 2D.
This requires the tabulate package. No sharding info will be printed for empty tensors
Experimental Features¶
DTensor also provides a set of experimental features. These features are either in prototyping stage, or the basic functionality is done and but looking for user feedbacks. Please submit a issue to
PyTorch if you have feedbacks to these features.
torch.distributed.tensor.experimental.local_map(func, out_placements, in_placements=None, device_mesh=None, *, redistribute_inputs=False)¶
local_map() is an experimental API that allows users to pass DTensor s to a function that is written to be applied on torch.Tensor s. It is done by extracting the local components of DTensor,
call the function, and wrap the outputs to DTensor according to the out_placements.
☆ func (Callable) – the function to be applied on each local shard of DTensor s.
☆ out_placements (Union[PlacementType, Tuple[PlacementType, …]]) – the desired placements of the DTensor s in func’s flattened output. If the flattened output is a single value, the
out_placements should be of type PlacementType. Otherwise if the flattened output has multiple values, the out_placements should be a tuple of PlacementType values 1:1 mapping to the
flattened output. Besides, for Tensor output, we use PlacementType as its placements (a Tuple[Placement] value). For non-Tensor output, the PlacementType should be None. Note that the
only exception is when no DTensor argument is passed in. In this case, even if out_placements is not None, the result function should ignore the desired placements because the function is
not running with DTensor s.
☆ in_placements (Tuple[PlacementType, …], optional) – the required placements of the DTensor s in the flattened inputs of func. If in_placements is specified, local_map() would examine
whether the placements of each DTensor argument is the same as the required placements or not. If the placements are not the same and redistribute_inputs is False, an exception will be
raised. Otherwise if redistribute_inputs is True, the argument will be first redistributed to the required sharding placements before passing its local tensor to func. The only exception
is when required placements are not None and the argument is a torch.Tensor. In this case, the placements examination will be skipped and the argument will be directly passed to func. If
in_placements is None, no placements examination will be performed. Default: None
☆ device_mesh (DeviceMesh, optional) – the device mesh that all the DTensor s are placed on. If not specified, this will be inferred from the input DTensor s’ device mesh. local_map
requires every DTensor s to be placed on the same device mesh. Default: None.
☆ redistribute_inputs (bool, optional) – the bool value indicating whether to reshard the input DTensor s when their placements are different from the required input placements. If this
value is False and some DTensor input has a different placement, an exception will be raised. Default: False.
A Callable that applies func to each local shard of the input DTensor and returns a DTensor constructed from the return value of func.
☆ AssertionError – If the input DTensor is not placed on the same device mesh, or if they are placed on a different device mesh than the device_mesh argument passed in.
☆ AssertionError – For any non-DTensor output, we require its corresponding output placement in out_placements be None. An AssertionError will be raised if this is not the case.
☆ ValueError – If redistribute_inputs=False but the input DTensor needs a redistribution according to in_placements.
>>> def mm_allreduce_forward(device_mesh, W, X):
>>> partial_sum_tensor = torch.mm(W, X)
>>> reduced_tensor = funcol.all_reduce(partial_sum_tensor, "sum", device_mesh)
>>> return reduced_tensor
>>> W = torch.randn(12, 8, requires_grad=False)
>>> X = torch.randn(8, 16, requires_grad=False)
>>> Y = torch.mm(W, X)
>>> row_wise = [Shard(0)] # row-wise sharding placements on 1-d mesh
>>> col_wise = [Shard(1)] # col-wise sharding placements on 1-d mesh
>>> # local_mm_allreduce_forward is the function wrapped with DTensor/Tensor convertion
>>> local_mm_allreduce_forward = local_map(
>>> mm_allreduce_forward,
>>> out_placements=[Replicate()],
>>> in_placements=[col_wise, row_wise],
>>> device_mesh=device_mesh,
>>> )
>>> W_dt = distribute_tensor(W, device_mesh, (col_wise)) # col-wisely sharded W tensor
>>> X_dt = distribute_tensor(X, device_mesh, (row_wise)) # row-wisely sharded X tensor
>>> Y_dt = local_mm_allreduce_forward(device_mesh, W_dt, X_dt) # apply local_mm_allreduce_forward to DTensors
This API is currently experimental and subject to change
register_sharding() is an experimental API that allows users to register sharding strategies for an operator when the tensor inputs and outputs are DTensor. It can be useful when: (1) there
doesn’t exist a default sharding strategy for op, e.g. when op is a custom operator that is not supported by DTensor; (2) when users would like to overwrite default sharding strategies of
existing operators.
op (Union[OpOverload, List[OpOverload]]) – An op or a list of ops to register the customized sharding function.
A function decorator which can be used to wrap a function that defines the sharding strategy for the operator specified in op. The defined sharding strategy will be registered to DTensor and
will override the default sharding strategy if DTensor has already implemented the operator. The customized sharding function takes the same inputs as the original op (except that if an arg
is a torch.Tensor, it will be replaced by a tensor-like object that DTensor uses internally). The function should return a sequence of 2-tuples, each specifying acceptable output placements
and its corresponding intput placements.
>>> @register_sharding(aten._softmax.default)
>>> def custom_softmax_sharding(x, dim, half_to_float):
>>> softmax_dim = dim if dim >= 0 else dim + x.ndim
>>> acceptable_shardings = []
>>> all_replicate = ([Replicate()], [Replicate(), None, None])
>>> acceptable_shardings.append(all_replicate)
>>> for sharding_dim in range(x.ndim):
>>> if sharding_dim != softmax_dim:
>>> all_sharded = (
>>> [Shard(sharding_dim)],
>>> [Shard(sharding_dim), None, None],
>>> )
>>> acceptable_shardings.append(all_sharded)
>>> return acceptable_shardings
This API is currently experimental and subject to change | {"url":"http://pytorch.org/docs/stable/distributed.tensor.html","timestamp":"2024-11-04T04:14:21Z","content_type":"text/html","content_length":"183255","record_id":"<urn:uuid:3d226c6d-780f-467c-a851-f9965f182c85>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00570.warc.gz"} |
Bounding the locality of distributed routing algorithms
We examine bounds on the locality of routing. A local routing algorithm makes a sequence of distributed forwarding decisions, each of which is made using only local information. Specifically, in
addition to knowing the node for which a message is destined, an intermediate node might also know (1) its local neighbourhood (the subgraph corresponding to all network nodes within k hops of
itself, for some fixed k), (2) the node from which the message originated, and (3) the incoming port (which of its neighbours last forwarded the message). Our objective is to determine, as k varies,
which of these parameters are necessary and/or sufficient to permit local routing on a network modelled by a connected undirected graph. In particular, we establish tight bounds on k for the
feasibility of deterministic k-local routing for various combinations of these parameters, as well as corresponding bounds on dilation (the worst-case ratio of actual route length to shortest path
• Dilation
• Distributed algorithms
• Local routing
ASJC Scopus subject areas
• Theoretical Computer Science
• Hardware and Architecture
• Computer Networks and Communications
• Computational Theory and Mathematics
Dive into the research topics of 'Bounding the locality of distributed routing algorithms'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/bounding-the-locality-of-distributed-routing-algorithms-6","timestamp":"2024-11-11T07:22:36Z","content_type":"text/html","content_length":"57489","record_id":"<urn:uuid:d7bd7ef8-516d-439c-88b6-3a94bc7fda9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00054.warc.gz"} |
The aim of this Laboratory in the first two years will be to study and develop expertise in the theory and implementation aspects of symmetric and asymmetric (public key) encryption methods and the
mathematics underlying them. Topics of interest include:
• Symmetric Protocols: DES, 3DES, AES
• Arithmetic of Finite Fields
• Arithmetic of Abelian varieties and Elliptic Curves over finite fields
• RSA and ECC
• Computational algebraic number theory, the number field sieve, in particular calculation of Hecke polynomials, their factorization
• Modular forms
• Index calculus and the discrete log problem
• Arithmetic on Jacobians of Hyperelliptic curves
foto by:Jane M Sawyer | {"url":"https://www.math.toronto.edu/ganita/","timestamp":"2024-11-04T17:11:46Z","content_type":"application/xhtml+xml","content_length":"3229","record_id":"<urn:uuid:02d98b89-0db3-4941-a28a-2500306799a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00036.warc.gz"} |
Summing a List of Lists in Python
Summing a List of Lists in Python
In Python, a list of lists is a common data structure used to represent a collection of elements, where each element is also a list. Sometimes, we may need to sum up all the elements of all the lists
in this structure. In this article, we will discuss different ways to do this in Python.
Using a For Loop
The most basic way to sum up the elements of a list of lists is to use a for loop to iterate through each list and add up the elements. Here is an example:
list_of_lists = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
total = 0
for sublist in list_of_lists:
for element in sublist:
total += element
print(total)Code language: PHP (php)
This will output the sum of all the elements in the list of lists, which is 45.
The time complexity of this method is O(n^2) where n is the total number of elements in the list of lists. This is because we are iterating through each list and then iterating through each element
in that list.
Using a List Comprehension
We can also use a list comprehension to accomplish the same task. Here is an example:
list_of_lists = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
total = sum([element for sublist in list_of_lists for element in sublist])
print(total)Code language: PHP (php)
This will also output the sum of all the elements in the list of lists, which is 45.
The time complexity of this method is also O(n^2). This is because we are iterating through each list and then iterating through each element in that list, similar to the for loop method.
Using the Built-in sum() Function
Python has a built-in function called sum() that can be used to add up the elements of a list. We can use this function to sum up the elements of a list of lists by using it in combination with the *
operator. Here is an example:
list_of_lists = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
total = sum(*list_of_lists)
print(total)Code language: PHP (php)
This will also output the sum of all the elements in the list of lists, which is 45.
The time complexity of this method is O(n) where n is the total number of elements in the list of lists. This is because the built-in sum() function has a time complexity of O(n).
Using numpy
Another way to sum up the elements of a list of lists is to use the numpy library. Here is an example:
import numpy as np
list_of_lists = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
total = np.sum(list_of_lists)
print(total)Code language: JavaScript (javascript)
This will also output the sum of all the elements in the list of lists, which is 45.
The time complexity of this method is O(n) where n is the total number of elements in the list of lists. This is because NumPy uses a more efficient algorithm to sum up the elements of a list.
In this article, we have discussed different ways to sum up the elements of a list of lists in Python. We have seen that we can use a for loop, a list comprehension, the built-in sum() function and
numpy to accomplish this task.
The time complexity of each method varies depending on the specific implementation. The for loop and list comprehension methods have a time complexity of O(n^2), while the built-in sum() function and
numpy have a time complexity of O(n).
When deciding which method to use, it is important to consider the specific requirements of your use case. If performance is a concern and the list of lists is large, using the built-in sum()
function or numpy may be more efficient. If you are working with small lists or if readability is more important, using a for loop or list comprehension may be more suitable.
Overall, Python provides many ways to sum up the elements of a list of lists, and the best approach will depend on your specific use case. | {"url":"https://www.4dev.pro/blog/summing-a-list-of-lists-in-python/","timestamp":"2024-11-11T12:37:21Z","content_type":"text/html","content_length":"54870","record_id":"<urn:uuid:5f23028e-41cd-43bd-b9a2-a74028256880>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00604.warc.gz"} |
Prediction Markets 101 - All Best Posts Ever - Midas Oracle.ORG - Predictions & Innovation
Prediction Markets 101
Prediction markets produce dynamic, objective probabilistic predictions on the outcomes of future events by aggregating disparate pieces of information that traders bring when they agree on prices.
Prediction markets are meta forecasting tools that feed on advanced indicators (like polls and surveys). Garbage in, garbage out…- Intelligence in, intelligence out…-
A prediction market is a market for a contract that yields payments based on the outcome of a partially uncertain future event, such as an election. A contract pays $100 only if candidate X wins the
election, and $0 otherwise. When the market price of an X contract is $60, the prediction market believes that candidate X has a 60% chance of winning the election. The price of this event derivative
can be interpreted as the objective probability of the future outcome (i.e., its most statistically accurate forecast). A 60% probability means that, in a series of events each with a 60%
probability, then 60 times out of 100, the favored outcome will occur- and 40 times out of 100, the unfavored outcome will occur.
Each prediction exchange organizes its own set of real-money and/or play-money markets, using either a CDA or a MSR mechanism.
Any comment, Michael Giberson?
Credits given to:
– Chris Masse-.
– Justin Wolfers.
– Robin Hanson.
– Jason Ruspini.
– Caveat Bettor.
– John Tierney.
– Jonathan Kennedy.
– Mike Giberson.
– Eric Zitzewitz.
– Cass Sunstein.
– Steve Roman,
– Nigel Eccles.
– The Everyday Economist.
– Adam Siegel.
– George Tziralis.
– Leighton Vaughan-Williams.
– Emile Servan-Schreiber.
– “-Thrutch“-.
– Panos Ipeirotis. | {"url":"https://www.midasoracle.org/2008/01/12/prediction-markets-101/","timestamp":"2024-11-06T17:23:37Z","content_type":"text/html","content_length":"35217","record_id":"<urn:uuid:7877555c-9fbe-4195-ba29-eed75e0a926a>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00472.warc.gz"} |
Gdansk Logic Colloquium
IMPAN and University of Gdansk
1. December 6, 2023
Speaker: Juliette Kenedy, University of Helsinki (Part of Simon's Semester Program)
Time: 16:00-17:00
Place: University of Gdansk, Department of Mathematics, Room D003
Title: On the mathematical sublime
Speaker: Jouko Vaananen, University of Helsinki (Part of Simon's Semester Program)
Time: 17:00-18:00
Place: University of Gdansk, Department of Mathematics, Room D003
Title: Inner models from extended logics
2. November 23, 2023
Matteo Viale, University of Torino (Part of Simon's Semester Program)
Title: Strong forcing axioms and the continuum problem
ABSTRACT: A topological approach to forcing axioms considers them as strong forms of the Baire category theorem; an algebraic approach describes certain properties of "algebraic closure" for the
universe of sets that can be derived from them. The goal of the talk is to outline the link betwen the geometric and algebraic points of view.
The talk is meant for a general mathematical audience. In particular familiarity with logic or set theory is not assumed.
3. November 2, 2023
Ralf Schindler, University of Muenster (Part of Simon's Semester Program)
Title:The *-version of Martin's Maximum
Time: 16:45-17:45
Place: University of Gdansk, Department of Mathematics, Room D003
4. Date: November 2nd, 2023
Boban Velickovic, Institut de Mathématiques Jussieu - Paris Rive Gauche (IMJ-PRG)
Université Paris Cité
Title: Higher forcing axioms
Time: 15:30-16:30
Place: University of Gdansk, Department of Mathematics, Room D003
5. October 26, 2023
John Steel, UC Berkeley (Part of Simon's Semester Program)
Title: Mouse Pairs and Soulsin Cardinals
Place: University of Gdansk, Room D003
6. October 14, 2022
Maciej Malicki, IMPAN
Title: Continuous logic and equivalence relations
7. July 25-August 6, 2022
Gabriel Goldberg, University of California, Berkeley
Title: The Ultrafilter Axiom (4 Lectures)
8. February 9, 2022
Ralf Schindler, University of Muenster
Title: Set theory and the Continuum Hypothesis
Abstract: In a 2021 Annals paper, D. Aspero and the speaker showed that two prominent axioms of set theory which were introduced independently from one another in the late 80's early 90's and
which both decide the size of the continuum are compatible, in fact one implies the other. Both axioms are so-called forcing axioms which are also exploited in other areas of mathematics. I am
going to provide an accessible introduction to our result. | {"url":"https://grigorsarg.github.io/seminars/LC/","timestamp":"2024-11-07T03:11:42Z","content_type":"text/html","content_length":"14715","record_id":"<urn:uuid:e25d321d-c58c-4cc9-a73c-0b155230a128>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00788.warc.gz"} |
What Happens If We Connect A Ac Supply To A Capacitor? | Yasir Arafin
Connect a Ac Supply to a Capacitor causes the capacitor to charge and discharge continuously due to the alternating voltage. This process allows the AC current to flow through the capacitor.
When an AC source is connected to a capacitor, the alternating voltage continuously charges and discharges the capacitor, allowing the AC current to flow through it. This process leads to the
accumulation and subsequent release of charge across the plates of the capacitor in sync with the alternating voltage.
It’s crucial to understand how capacitors behave in an AC circuit, as the frequency of the supply voltage directly affects their performance. Let’s delve into the effects of connecting AC supply to a
capacitor and how it influences the behavior of the circuit.
What Happens When Connect a Ac Supply to a Capacitor
1. Initial Charge:
• As the AC voltage first connects, a large current surge momentarily flows. This is because the capacitor needs to initially charge up to the applied voltage.
• This initial charging current is limited by the impedance of the capacitor, which depends on its capacitance and the AC frequency. Higher capacitance or lower frequency means slower charging and
lower initial current.
2. Charge Accumulation:
• As the AC voltage rises in the positive direction, positive charge builds up on one plate of the capacitor, and negative charge accumulates on the other.
• This charge accumulation creates an electric field across the dielectric material separating the plates, opposing the applied voltage.
3. Voltage Reversal:
• When the AC voltage reaches its peak and starts to reverse, the charges on the capacitor plates also switch sides.
• The previously positive plate becomes negative, and vice versa. This process allows the capacitor to store and release energy in sync with the alternating voltage.
4. Continuous Charge/Discharge Cycle:
• This cycle of charging and discharging repeats continuously with the AC frequency.
• At any given moment, the voltage stored across the capacitor lags behind the applied voltage by 90 degrees, meaning the peak voltage across the capacitor occurs slightly later than the peak
voltage of the AC source.
• This phase shift is a fundamental property of capacitors in AC circuits and has various practical applications, like tuning circuits and power factor correction.
5. Energy Storage and Release:
• While the capacitor doesn’t technically “conduct” current in the same way as a wire, it stores and releases electrical energy by accumulating and discharging opposite charges on its plates.
• This stored energy can be used in various ways depending on the circuit configuration, such as smoothing voltage fluctuations, filtering out unwanted frequencies, or providing short bursts of
high current.
6. Additional Points:
• The actual current flowing through the capacitor depends on the AC voltage, frequency, and capacitance. Higher voltage, higher frequency, or lower capacitance result in higher current flow.
• Real capacitors have internal resistance and leakage currents, which can impact their performance and energy storage capabilities.
• The type of capacitor (electrolytic, ceramic, film, etc.) also influences its behavior in AC circuits, with different characteristics regarding voltage rating, frequency response, and temperature
Can AC flow through a capacitor?
Yes, but not like current flows through a wire. A capacitor rapidly builds up and releases charge on its plates as the AC voltage changes back and forth. It’s like an electrical seesaw, storing and
releasing energy but not letting actual current flow directly through the insulator (dielectric) between the plates.
When an AC source is connected to a capacitor?
The capacitor charges and discharges in sync with the AC voltage changes. At one voltage peak, one plate builds up positive charge while the other gathers negative charge. As the voltage reverses,
the charges swap sides. This charging/discharging cycle happens continuously with the AC frequency.
Can capacitors hold AC current?
No, not really. Capacitors don’t hold AC current directly. They store electrical energy by accumulating opposite charges on their plates. This stored energy can be released later (discharged), but
it’s not the same as holding current flow.
Can AC run without capacitor?
Yes, many AC circuits function without capacitors. However, capacitors play crucial roles in:
• Smoothing AC waveforms: They filter out unwanted fluctuations and stabilize voltage levels.
• Blocking DC: They prevent DC leakage current from flowing through AC circuits.
• Tuning circuits: In conjunction with inductors, they create resonant circuits for filtering specific frequencies.
• Power factor correction: They improve the efficiency of AC power transmission by compensating for lagging current
Is capacitor connected to AC or DC?
A capacitor can be connected to either AC or DC circuits, but its behavior differs in each:
• AC: It charges and discharges continuously, allowing for the aforementioned roles.
• DC: It initially charges to the DC voltage level and then blocks any further current flow (acting like an open circuit).
Can I use a DC capacitor for AC?
No, using a DC-rated capacitor for AC can be dangerous. DC capacitors aren’t designed for the rapid charge/discharge cycles of AC and may overheat or rupture. Always use capacitors rated for the
intended AC voltage and frequency.
How to convert AC to DC capacitor?
A capacitor alone cannot directly convert AC to DC. You need a rectifier circuit with diodes and other components to convert the AC to pulsating DC, and then a filter circuit with capacitors and
inductors to smooth out the pulsations and get pure DC.
Why capacitor block DC but allows AC?
A capacitor blocks DC because the initial charging creates an opposing electric field within the dielectric that prevents further current flow. In AC, the constant polarity switching keeps the
charging/discharging cycle going, effectively allowing AC “through” the capacitor (though not in the same way as through a wire).
Can we use diode in AC?
Yes, diodes are essential components in many AC circuits. They rectify AC to DC, isolate parts of the circuit, and perform other functions based on their non-linear voltage-current characteristic.
What happens when DC supply is given to capacitor?
With DC, the capacitor initially charges to the DC voltage level and then acts as an open circuit, blocking further current flow.
Why add capacitor to DC power supply?
Capacitors in DC power supplies help:
• Smoothen the DC voltage: They filter out any remaining voltage ripples after rectification.
• Store and release energy: They provide short bursts of current during peak loads, stabilizing the voltage supply.
• Protect against transients: They absorb voltage spikes to protect sensitive electronics.
How does a capacitor work in AC and DC?
• AC: The changing AC voltage creates an electric field that alternately attracts and repels charges on the capacitor plates, causing them to swing in opposite directions. This continuous charging/
discharging cycle allows the capacitor to influence the AC current flow.
• DC: The DC voltage initially charges the capacitor plates to a specific level, creating an opposing electric field within the dielectric that prevents further current flow. The capacitor then
behaves like an open circuit for DC. | {"url":"https://yasirarafin.com/connect-a-ac-supply-to-a-capacitor/","timestamp":"2024-11-06T01:53:54Z","content_type":"text/html","content_length":"102169","record_id":"<urn:uuid:08d5bc76-32a7-4ae6-9fe2-c7dcb16566cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00744.warc.gz"} |
Hadrons and Their Interactions
Current and Field Algebra, Soft Pions, Supermultiplets, and Related Topics
• 1st Edition - November 14, 2012
• Paperback ISBN:
9 7 8 - 0 - 1 2 - 4 1 2 3 7 4 - 8
• eBook ISBN:
9 7 8 - 0 - 3 2 3 - 1 4 2 8 9 - 2
Hadrons and Their Interactions: Current and Field Algebra, Soft Pions, Supermultiplets, and Related Topics focuses on formulas, principles, and interactions involved in the study… Read more
Save 50% on book bundles
Immediately download your ebook while waiting for your print delivery. No promo code needed.
Hadrons and Their Interactions: Current and Field Algebra, Soft Pions, Supermultiplets, and Related Topics focuses on formulas, principles, and interactions involved in the study of physics. The
compilation contains the papers presented at the ""Ettore Majorana,"" held in Erice on July 1-14, 1967. Divided into three parts with 22 chapters, the compilation focuses first on lectures on soft
pions; the method of phenomenological lagrangians and algebra of fields; and radiative corrections to beta decay and the structure of hadrons. The second part focuses on seminars. The areas covered
include a review of coherent production in strong interactions; spontaneous breakdown and the weak interaction angle; and the symmetries of the S-matrix. The concluding part also focuses on lectures,
including lectures on the present status of the fundamental interactions; a pedagogical exercise in binning and resolution; and the pomeranchuk affair and twisting trajectories. The compilation is a
valuable source of data for readers and physicists wanting to explore the interactions of hadrons.
Opening Ceremony
Soft Pions
1. The Reduction Formula
2. The Weak Interactions: First Principles
3. The Goldberger-Treiman Relation and a First Glance at PCAC
4. A Hard Look at PCAC
5. The Gradient-Coupling Model
6. Adler's Rule for the Emission of One Soft Pion
7. Current Commutators
8. The Weinberg-Tomozawa Formula and the Adler-Weisberger Relation
9. Pion-Pion Scattering Ala Weinberg
10. Kaon Decays
Appendix 1. Notational Conventions
Appendix 2. No-renormalization Theorem
Appendix 3. Threshold S-Matrix and Threshold Scattering Lengths
Discussion 1
Discussion 2
Discussion 3
Discussion 4
The Method of Phenomenological Lagrangians and that of the Algebra of Fields
1. Introduction
2. The Method of Phenomenological Lagrangians
3. The Algebra of Fields
Discussion 1
Discussion 2
Discussion 3
Discussion 4
Breaking Chiral SU(3) X SU(3)
I. Introduction
II. Chiral SU(3) X SU(3)
III. The Mass Formula and its Interpretation
IV. The Next Approximation
V. A Sum Rule for Scalar and Pseudoscalar Mesons
VI. Spectral Functions for Vector and Axial Mesons
Discussion 1
Discussion 2
Discussion 3
Discussion 4
Radiative Corrections to Beta Decay and the Structure of Hadrons
1. Introduction
2. Radiative Corrections to the Beta Decay of Point-like Particles
3. Ultraviolet Divergence in the Correlations to Beta Decays of Real Hadrons
4. A Sum Rule
5. The Sub Model
Appendix A — The Fierz Transformation
Discussion 1
Discussion 2
Recent Work on Representation of Current Algebra
I. Introduction—Formulation of the Program
II. Mathematical Preliminaries
III. The Single Charge-Bearing-Quark Model
IV. The (l/µ) Expansion
V. Further Possibilities—Conclusions
Discussion 1
Discussion 2
Discussion 3
Discussion 4
Meson Resonances
1. Introduction
2. Experimental Problems
3. Classification
4. Pseudoscalar Nonet
5. Vectornonet
6. The 2+ Nonet
7. The Quark Model and Candidates for other Nonets
8. High Mass Pion Resonances
9. Other Particles
10. Conclusion
Discussion 1
Discussion 2
Discussion 3
Discussion 4
The Infrared Radiative Corrections for Colliding Beam (Electrons and Positrons) Experiments
I. The Administration of Infrared Radiative Corrections
II. Details of the Classical Model and other Applications
Discussion 1
Discussion 2
A Review of Coherent Production in Strong Interactions
Spontaneous Breakdown and the Weak Interaction Angle
All Possible Symmetries of the S Matrix
Neutrino Physics
Empirical Mass Formula for Mesons and Baryons
Recent Experimental Investigations of Electromagnetic Interactions at DESY
Low Mass Structure in the (Kππ) System
A Pedagogical Exercise in Binning and Resolution
The Pomeranchuk Affair and Twisting Trajectories
A Brief Review of the Nimrod Experimental Program
Photoproduction of Pairs at High Energies
Classification of Particle Multiplets
Closing Lecture
Present Status of the Fundamental Interactions
Closing Ceremony
• Published: November 14, 2012
• Paperback ISBN: 9780124123748
• eBook ISBN: 9780323142892 | {"url":"https://shop.elsevier.com/books/hadrons-and-their-interactions/zichichi/978-0-12-395588-3","timestamp":"2024-11-06T12:29:35Z","content_type":"text/html","content_length":"184299","record_id":"<urn:uuid:69126e98-38ca-484a-925e-a8c42efe02f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00297.warc.gz"} |
What is OpenBTS?
OpenBTS is a Unix application that uses a software radio to present a GSM air interface to standard 2G GSM handset and uses a SIP softswitch or PBX to connect calls. (You might even say that OpenBTS
is a simplified form of IMS that works with 2G feature-phone handsets.) The combination of the global-standard GSM air interface with low-cost VoIP backhaul forms the basis of a new type of cellular
network that can be deployed and operated at substantially lower cost than existing technologies in many applications, including rural cellular deployments and private cellular networks in remote
Where can I get the latest code?
The best way to get OpenBTS is by pulling the code directly from there source code repository as an anonymous read-only user.
You do this by entering the following command into a new terminal window
git clone https://github.com/RangeNetworks/dev.git
like seen here
when u have entered that command
u can press Enter
and next u will see this
now that this i done we are going to change that Directory name into OpenBTS so u can remember it easily
to do that u need to enter the following command:
mv dev OpenBTS
like seen here
when u typed this command u can press Enter
So now we renamed the folder dev to OpenBTS
Next we need to enter that directory, you do this by entering the following command:
cd OpenBTS
like seen here
after thats done press Enter
and you should now be here
Now, to download all of the components simply run the clone.sh script.
u do this entering the following command
like seen here
after that u can press Enter
Now if u pressed enter it’s going to pull all the needed software from the github
Component master status
it’s a whole list, so i am not going to post the whole list in screenshots, but it should end with this
now that this is done we need to build it
The build.sh script will automatically install any build dependencies (building them manually when required).
After dependencies are taken care of, each component is compiled into an installable package.
To do this we need to enter the following command:
./build.sh SDR1
like seen here
after u entered that command press Enter
This is a very long process, so go grab yourself a drink and lay back
After a while when it’s done
you should see this
Now the building is done !
Next we need to configure all the things
Configuring OpenBTS
With OpenBTS built, you now need to configure it to run correctly. There are a two key files that must be created for this to happen.
OpenBTS.db is the database store for all OpenBTS configuration. It must be installed at /etc/OpenBTS, which likely does not exist. So, to create this file, we first need to enter the OpenBTS
you do this by entering the following command
cd openbts
like seen here
now that this in entered u can press Enter
and should be here
now that u are in this location
we are going to make a directory elsewhere, it’s the one we need for putting the database
so enter the following command:
mkdir /etc/OpenBTS
like seen here
now that u typed this command press Enter
You will see nothing happening but should come out back here again
next we need to run the following command:
sqlite3 -init ./apps/OpenBTS.example.sql /etc/OpenBTS/OpenBTS.db ".quit"
like seen here
when u typed that command u can press Enter
and should see this
To make sure that it worked we are going to test it
Test this by running the following command :
sqlite3 /etc/OpenBTS/OpenBTS.db .dump
like seen here
after u pasted that command press Enter
If you see a lot of configuration variables, the DB has been installed correctly.
so like this
this means that the database is configured correctly
At this point, we should be able to perform a basic sanity check of OpenBTS.
so we are going to do this
the openbts executable is in the apps folder
so we need to enter that by typing the following command here:
cd apps
like seen here
after that press Enter
and you should now be here
next we need to create a script with the filename transceiver with the following content and make it executable:
exec <your path to osmocom-bb>/src/host/layer23/src/transceiver/transceiver 1
so to do that we need to type the following command:
nano transceiver
like seen here
when u typed out this u can press Enter
and should then see this
now that u have this open
we should paste the following lines here:
exec /root/osmocom-bb/src/host/layer23/src/transceiver/transceiver 1
like seen here
If you followed all my guides, it should be correct..But if u installed osmocom-bb in another location u should change that here
Whereas 1 needs to be replaced with the ARFCN of the reference cell you want to use for synchronization
(find a strong one with the rssi-app for example).
But when that is done we can continue
Press CTRL+X to close
and you should then see this
now that u see this u can press Y on your keyboard
and should then see this
now when you see this u can just press Enter
and it should be saved now
And you should now come back out here again
now that u are here again, we need to make the file we just created executable
we do this by running the following command:
chmod +x transceiver
like seen here
now that u typed this command, u can press Enter
You will see nothing happening but the file should be executable
you can check this by running the following command:
and should then see this
the file transceiver in green (which means it’s executable!)
Now that this is done we need to make some changes to the OpenBTS database.
We do this by entering the following command:
sqlitebrowser /etc/OpenBTS/OpenBTS.db
like seen here
now that u entered that command press Enter
and should now see this
now that u see this we need to click on the Tab Browse Data
like seen here
when you clicked that, u should see this
now we need to make some changes here
find the line
like seen here
now that u see this double click on the 0 (zero) next to it
and you should then see this
when u have this edit window open
change the 0 (zero) to 1 (one)
like seen here
and then click OK
and should then come back out here again
now that this is done find the following line
like seen here
and make sure the VALUESTRING is set to 900
next look for the following line
and change that VALUESTRING 1 to 3
like seen here
and then press OK
and you should then have this
now that this is done
we need to look for the following line
like seen here
now that u found this change the VALUESTRING from 14 to 8
like seen here
now that this is done click on OK
and you should then have this
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Sensor m
Sensor measurement from states
Since R2022a
z = measurement(sensor,filter) returns the measurement z from the state maintained in the filter object. You must implement this method when you define a sensor object based on the
positioning.INSSensorModel abstract class.
Customize Sensor Model Used with insEKF
Customize a sensor model used with the insEKF object. The sensor measures the velocity state, including a bias affected by random noise.
Customize the sensor model by inheriting from the positioning.INSSensorModel interface class and implementing its methods. Note that only the measurement method is required for implementation in the
positioning.INSSensorModel interface class. These sections provide an overview of how the BiasSensor class implements the positioning.INSSensorModel methods, but for details on their implementation,
see the details of the implementation are in the attached BiasSensor.m file.
Implement sensorStates method
To model bias, the sensorStates method needs to return a state, Bias, as a structure. When you add a BiasSensor object to an insEKF filter object, the filter adds the bias component to the state
vector of the filter.
Implement measurement method
The measurement is the velocity component of the filter state, including the bias. Therefore, return the summation of the velocity component from the filter and the bias.
Implement measurementJacobian method
The measurementJacobian method returns the partial derivative of the measurement method with respect to the state vector of the filter as a structure. All the partial derivatives are 0, except the
partial derivatives of the measurement with respect to the velocity and bias state components.
Implement stateTransition method
The stateTransiton method returns the derivative of the sensor state defined in the sensorStates method. Assume the derivative of the bias is affected by a white noise with a standard deviation of
0.01. Return the derivative as a structure. Note that this only showcases how to set up the method, and does not correspond to any practical application.
Implement stateTransitionJacobian method
Since the stateTransiton function does not depend on the state of the filter, the Jacobian matrix is 0.
Create and add inherited object
Create a BiasSensor object.
biSensor =
BiasSensor with no properties.
Create an insEKF object with the biSensor object.
filter = insEKF(biSensor,insMotionPose)
filter =
insEKF with properties:
State: [17x1 double]
StateCovariance: [17x17 double]
AdditiveProcessNoise: [17x17 double]
MotionModel: [1x1 insMotionPose]
Sensors: {[1x1 BiasSensor]}
SensorNames: {'BiasSensor'}
ReferenceFrame: 'NED'
The filter state contains the bias component.
ans = struct with fields:
Orientation: [1 2 3 4]
AngularVelocity: [5 6 7]
Position: [8 9 10]
Velocity: [11 12 13]
Acceleration: [14 15 16]
BiasSensor_Bias: 17
Show customized BiasSensor class
classdef BiasSensor < positioning.INSSensorModel
%BIASSENSOR Sensor measuring velocity with bias
% Copyright 2021 The MathWorks, Inc.
function s = sensorstates(~,~)
% Assume the sensor has a bias. Define a Bias state to enable
% the filter to estimate the bias.
s = struct('Bias',0);
function z = measurement(sensor,filter)
% Measurement is the summation of the velocity measurement and
% the bias.
velocity = stateparts(filter,'Velocity');
bias = stateparts(filter,sensor,'Bias');
z = velocity + bias;
function dzdx = measurementJacobian(sensor,filter)
% Compute the Jacobian, which is the partial derivative of the
% measurement (velocity plus bias) with respect to the filter
% state vector.
% Obtain the dimension of the filter state.
N = numel(filter.State);
% The partial derviative of the Bias with respect to all the
% states is zero, except the Bias state itself.
dzdx = zeros(1,N);
% Obtain the index for the Bias state component in the filter.
bidx = stateinfo(filter,sensor,'Bias');
dzdx(:,bidx) = 1;
% The partial derivative of the Velocity with respect to all the
% states is zero, except the Velocity state itself.
vidx = stateinfo(filter,'Velocity');
dzdx(:,vidx) = 1;
function dBias = stateTransition(~,~,dt,~)
% Assume the derivative of the bias is affected by a zero-mean
% white noise with a standard deviation of 0.01.
noise = 0.01*randn*dt;
dBias = struct('Bias',noise);
function dBiasdx = stateTransitonJacobian(~,filter,~,~)
% Since the stateTransiton function does not depend on the
% state of the filter, the Jacobian is all zero.
N = numel(filter.State);
dBiasdx = zeros(1,N);
Input Arguments
sensor — Sensor model used with INS filter
object inherited from positioning.INSSensorModel class
Sensor model used with an INS filter, specified as an object inherited from the positioning.INSSensorModel abstract class.
Output Arguments
z — Measurement
M-by-1 real-valued vector
Measurement, returned as an M-by-1 real-valued vector.
Version History
Introduced in R2022a | {"url":"https://ww2.mathworks.cn/help/fusion/ref/positioning.inssensormodel.measurement.html","timestamp":"2024-11-03T16:23:13Z","content_type":"text/html","content_length":"88171","record_id":"<urn:uuid:8ea25b71-3267-4f44-b562-e977ddc75ee8>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00380.warc.gz"} |
Conglomerate Blog: Business, Law, Economics & Society
September 26, 2008
What is the NPV of a JD?
Posted by Karl Okamoto
Is law school a good value? That’s the question I ask my students to figure out, hoping to teach them a bit about finance. Using crude numbers, the answer looks like a resounding “yes.” As they say
in the investment business, it looks like a “three bagger.” Even if you have to put $230,000 in, you get over $700,00 back! Sweet! Hey Deans, maybe we should raise tuitions (and law professor
What crude numbers, you may ask? Well the Wall Street Journal reported recently on salary statistics. While the median salary for persons holding just a BA has slipped to $47,240, those of us with
professional degrees have gone up to $89,602. Even better, recent Labor Department numbers show the median salary for lawyers at $106,120. So, as I say to my students, think of your law degree as an
annuity. It represents a payment stream that lasts for a career (say 40 years) that equals the spread between what you would have earned without your law degree versus what you can with it. Using the
median salary numbers, that spread is almost $60,000. Discounted at 8%, the annuity has a present value of over $700,000. The present value of three years of tuition (at $40,000 a year), books and
foregone salary (at the median) is about $230,000. So, as your stockbroker used to say about Lehman bonds, a “no brainer!”
So what’s all the brouhaha about misleading students about the value of a law school education? Well, here is where class gets fun.
According to the Department of Labor, the distribution of annual lawyer compensation in 2001 was: first 10%, less than $45,000; next 15%, $45-60,000; next 50%, $61-137,000; next 15%, 137-145,000; and
top 10%, over $145,000. 2007 numbers are not much higher, but I can’t find similar granularity for my model. Now if you are standing in the ex ante position, i.e., where law school applicants stand,
how do you decide what salary statistic to use in your present value model? Might it not be rational to use the “expected value” that comes from this distribution? You might immediately object that
we need more data. True, while we can ballpark the first four quintiles, how do we come up with a value for the top 10% contingency? And if we already know (which I do not tell my students) that the
“mean” salary was $92,000, isn’t that our answer? Mathematically, yes, but the exercise is revealing.
So much of one’s answer to the question of law school value depends on how you envision the top quintile. So the bi-modal distribution in starting salaries that Bill Henderson has identified has
serious implications. The more skewed the distribution, the less helpful means and medians become. I find particularly interesting one result from my class discussions I did not expect. I assumed
that students would use a very high average for the top quintile of lawyer salaries (based on AmLaw 100 profits for partner, for example), and that that would skew their present values even higher
than median salaries would support. In other words, I assumed like many concerned law faculty that some students were being bedazzled by the headlines. In my in-class surveys, numbers were skewed
higher than the labor statistics support, but not because of multi-million dollar outliers. They were high because so many students assumed they would earn just about twice the median income (not 50
times like partners at Wachtell).
In other words, none of my students assumed they would climb atop the Olympus of the AmLaw 100 (funny, I assume a few will). But they revealed a commonly-held notion of “average” incomes that far
exceed what the numbers show us is realistic. It appears that my students, as a group, have a very similar notion of what lawyers make on average. Unfortunately, it happens to be more than what at
least 90 percent of lawyer actually do earn.
I have a theory. While student are indeed savvy enough to discount starting salary stories that focus only on the big Wall Street firms and the like, these stories serve as anchors in making their
judgments. And like most humans, they assume the world is evenly distributed and linear. So if multi-million dollar draws are common among big firm partners, isn’t it reasonable to assume an average
law grad can do, say 10%. After all, we’re only asking to go just a few yards down the field! Nope.
At least I can take comfort in knowing that on average I'm still offering them a good deal (even if it isn't the one they thought they were getting)!
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Strassen algorithm not fast enough for RRFRNDS
Reading the editorial for RRFRNDS problem from COOK48 it describes a method to solve the problem using Strassen algorithm, but mine solution using is TLE, may someone help me?
If someone have ideas to optimize this in python it would be great!
python implementation of Strassen algorithm
Also I saw some posts asking for references of the Strassen algorithm, this implementation + wikipedia is enough to understand it.
Apparently it isn’t fast enough in c++ either c++ implementation of Strassen algorithm
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Quasi-morphism and the poisson bracket
For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties.
On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a measure of non-commutativity of functions in the sense of the Poisson bracket, the operation which
involves first derivatives of the functions. Furthermore, the same functional gives rise to a non-trivial lower bound for the error of the simultaneous measurement of a pair of non-commuting
Hamiltonians. These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold. The above-mentioned
functional comes from a special homogeneous quasi-morphism on the universal cover of the group, which is rooted in the Floer theory.
ASJC Scopus subject areas
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An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations
An intuitive formulation for the reproductive number for the
spread of diseases in heterogeneous populations
James M. Hyman
, Jia Li
a[Theoretical Division, MS-B284, Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos,] NM 87545, USA
b[Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA] Received 1 February 1999; received in revised form 14 May 1999; accepted 18 May 1999
The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide signi®cant insight into the transmission
dynamics of a disease and can guide strategies to control its spread. We de®ne the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for
certain epidemiological models, and construct a simpli®ed formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the
heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate
several models with di erent heterogeneous structures and demonstrate how to de®ne the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it
seems necessary to de®ne the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the
reproductive number for the total population. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords:Epidemiology; Mathematical model; Threshold conductors; Reproductive number; Transmission dynamics; Endemic equilibrium; Dynamical systems; Stability
1. Introduction
One of the fundamental questions of mathematical epidemiology is to ®nd threshold conditions that determine whether an infectious disease will spread in a susceptible population when the
*[Corresponding author. Tel.: +1-256 890 6470; fax: +1-256 895 6173.] E-mail address:[email protected] (J. Li).
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S0025-5564(00)00025-0
disease is introduced into the population. The threshold conditions are characterized by the so-called reproductive number, the reproduction number, the reproductive ratio, basic reproductive
value, basic reproductive rate, or contact number, commonly denoted by R0 in mathematical
epidemiology [5,10,17,19±21,23,29,35,39,43]. The concept ofR0, introduced by Ross in 1909 [39],
is de®ned in epidemiological modeling such that if R0<1, the modeled disease dies out, and if
R0 >1, the disease spreads in the population.
There have been intensive studies in the literature to calculateR0 for a wide class of
epidemi-ological models of infectious diseases [6,8,9,12,17,18,25,26,28,30,32±34,41]. In mathematical models, the reproductive number is determined by the spectral radius of the next-generation
operator in continuous models and, in particular, is determined by the dominant eigenvalue of the Jacobian matrix at the infection-free equilibrium for models in a ®nite-dimensional space
[8,9,24,27]. It can also be obtained, in certain models, by suitable Lyapunov functions [28,41].
The biological meaning of the reproductive number is the average number of secondary cases produced by one infected individual during the infected individual's entire infectious period when
the disease is ®rst introduced. Let r be the average number of contacts per unit of time per
in-dividual,bbe the probability of transmitting the infection per contact, andsbe the mean duration
of the infectious period. Then the reproductive number can be estimated by the following intuitive formula:
R0 rbs: 1:1
This formula can give insight into the transmission dynamics of infectious diseases for various relatively simple epidemiological models [3±5,7,40].
For simple homogeneous models, it is easy to de®ner,b, ands. For example, consider a simple
homogeneous AIDS model governed by the following system of ordinary di erential equations: dS dt l S0 ÿ ÿS t ÿk t S t ; dI dt k t S t ÿ l m I t ; dA dt mI t ÿdA t ;
whereS,I, andAdenote the individuals susceptible to infection, the infected individuals, and the
AIDS cases, respectively;lS0[is the input ¯ow into the susceptible group;][l][the removal rate;][m][the]
rate of contracting AIDS;dthe removal rate due to the death from AIDS or other reasons; andk
is the rate of infection given by k t br I t
S t I t :
Here, bis the transmission probability per contact and r is the average number of contacts per
individual per unit of time. We assume here that transmission by the AIDS cases is neglected. To focus our attention on the issues we will address, we assume, for simplicity, that the mixing is
proportional for this model and other models in this paper.
The system has the infection-free equilibrium S0[;][0][ ][. The stability of] [ ][S]0[;][0][ ] [determines the]
R0 [l]rb[ ][m]:
Formula (1.1) then holds when we de®ne the duration of infection assdef 1= l m . (We will use
the symboldef to indicate that the equation is the de®nition of a quantity.)
However, as more heterogeneous structures or subgroups for the infected population are
in-cluded in an epidemiological model, the calculation of R0 becomes more complicated, and it is
di cult to ®nd an explicit formula forR0. Even when an explicit formula can be obtained, it is not
always clear whether it is appropriate to de®ne a mean contact rate, a mean duration of infection, and a mean transmission probability so that the reproductive number can still be estimated by
formula (1.1). Furthermore, even if it can be claimed that such an estimate is adequate, a deep understanding of the model is absolutely necessary so that those means can be well de®ned.
Moreover, for models of the diseases for which di erentiation of the contact rates or the partner acquisition rates must be addressed, such as sexually transmitted disease (STD) models, not only the
mean but also the second moment or the variance about the mean must be taken into account. Then, formula (1.1) can no longer be applied. For certain simple models, a more accurate formula for the
reproductive number is
R0 r r2 r bs; 1:2
whereris the mean number of contacts per individual andris the variance or standard deviation
of the mean number of contacts [1±3,5,8,28,36].
Formula (1.2) is an e ective formulation for providing insight into the transmission dynamics of diseases. Unfortunately, as more heterogeneities are considered, it becomes impracticable to de®ne the
variance or the standard deviation, and expression (1.2) becomes inadequate.
For risk-group models, Hethcote and Yorke [23] ®rst introduced and Jacquez et al. [28,29] speci®ed the idea of de®ning a mean reproductive number as the average number of infected individuals
generated per infected individual over the duration of the infected state. They de®ned a reproductive number for each subgroup and then express the mean reproductive number as a weighted mean of
those group reproductive numbers.
In this paper, we use the models in [26] as a basis and formulate new heterogeneous models to demonstrate how di erent cases can be treated so that an appropriate reproductive number can be
estimated. We show that for models with no risk structure, that is, the models with a homoge-neous susceptible population in the contact rates, it is still possible to de®ne the mean quantities and
to apply formula (1.1). We show, however, for susceptible populations with heterogeneous structure such as risk structure and age structure, it is more appropriate to de®ne a reproductive number for
each subgroup or each cohort and then express the reproductive number for the whole population as the weighted average of those reproductive numbers for the subgroups or cohorts. 2. Models without
risk structure
We ®rst consider the models in which the risk level is assumed to be uniform for all the sus-ceptible individuals. The sussus-ceptible population may still be divided into subgroups, but they are not
based on the risk level, that is, the number of partners, or the number of contacts.
2.1. Di erential infectivity models
There is evidence that HIV serum and plasma levels or individual variations a ect trans-mission, that infected individuals have di erent levels of virus after the acute phase, and that those with
high levels progress to AIDS more rapidly than those with low levels in clinical studies. As a result, a hypothesis that some individuals are highly infectious over long periods of time and a new
model that accounts only for di erences between infected individuals, re-ferred to as a di erential infectivity (DI) HIV model, were proposed in [26]. In that DI model, it is assumed that individuals
enter a speci®c group when they become infected and stay in that group until they are no longer involved in transmitting the disease. Their infectivity and progression rates to AIDS are assumed to
depend upon which group they are in; the sus-ceptible population is assumed to be homogeneous; and variations in susceptibility, risk be-havior, and many other factors associated with the dynamics of
the spread of HIV are neglected. In this section, we use the simple DI model formulated in [26] and generalize it to a model in which the risk level of infected individuals depends on the group to
which they be-long. We demonstrate how a mean number of contacts, a mean transmission probability, and a mean duration of infection can be de®ned so that formula (1.1) can be used to obtain the
reproductive number.
The following DI model with homogeneous contact rate was formulated for HIV transmission in [26] dS dt l S0ÿS ÿkS; dIi dt pikSÿ l mi Ii; i 1;. . .;n; 2:1 dA dt Xn j 1 mjIjÿdA;
where the infected population is subdivided into n subgroups, I1;I2;. . .;In. Upon infection, an
individual enters subgroupiwith probabilitypiand stays in this group until becoming inactive in
transmission, where Pn[i][ ][1]pi 1. The variable Adenotes the group of individuals removed from
the population due to end stage disease or behavioral changes. Individuals inAare assumed to die
at a rate dPl. The ratemi of leaving the infected population because of behavioral changes
in-duced by either HIV-related illnesses or testing positive for HIV (presumably changing behavior so as not to transmit infection) depends on the subgroups.
The rate of infection k depends on the transmission probability per contact of individuals in
subgroupi,b[i], the proportion of individuals in the subgroup,Ii=N, and the number of contacts of
an individual per unit of time r, so that
k t rXn
i 1
bi[N]Ii [ ]t[t] [ ];
The reproductive number for this model is R0 r Xn i 1 pibi l mi:
By de®ning the mean duration of infectiousness for infected individuals and the mean proba-bility of transmission as sdef Xn i 1 pi l mi; bdef 1[] s Xn i 1 pibi l mi;
respectively, the reproductive number can be expressed as the product of the number of contacts and these two means
R0 rbs:
Now we generalize the DI model (2.1) by assuming that the number of contacts per individual per unit of time depends on the subgroups, because people may change their behavior according
to how ill they are. Let ri be the average number of contacts per individual per unit of time in
subgroupi. Then the rate of infection is generalized to
k t rXn
i 1
bi[rS][ ][t][ ]riPIi nt
j 1rjIj t ;
where r is the contact rate of the susceptible individuals and ri is the contact rate of infected
individuals in subgroup. We refer to model (2.1) with the generalized infection rate as a general DI model.
Similarly as in [26], a simple stability analysis for the infection-free equilibrium gives the re-productive number for the general DI model as
i 1
l mi:
The mean duration of infectiousness for infected individuals in this general DI model is the same as for model (2.1).
Since ri= l mi is the average number of contacts per individual in group i made during the
whole infection period, the total average number of contacts per infected individual during the whole infection period is
rtotaldef[ ] Xn i 1
l mi;
and hence the mean number of contacts per infected individual per unit time for the general DI
model, denoted byr, is rdef rtotal s 1 s Xn i 1 piri l mi:
The total transmission probability through all contacts with infected individuals in subgroupi
i def [l]b[ ]iri[m] i:
Hence, the mean probability per contact per unit of time for the general DI model, denoted byb,
is bdef Xn i 1 pibtotal[i] rs 1 rs Xn i 1 piribi l mi: Therefore, R0 rbs:
2.2. The staged progression models
A common mathematical model for the spread of AIDS assumes that infected individuals pass through several stages, being highly infectious in the ®rst few weeks after becoming infected, then having
low infectivity for many years, and ®nally becoming gradually more infectious as their immune systems break down, and they progress to AIDS, with the rates of progression to AIDS being also very low
in the ®rst few years after infection. Based on this hypothesis, epidemiological models that we refer to as staged progression (SP) models have been studied by many researchers (see the references in
2.2.1. The general discrete SP model
The following SP model with a homogeneous contact rate is studied in [26]. It assumes that the susceptible population is homogeneous and is maintained by the same type of in¯ow. It assumes that the
population of infected individuals is subdivided into subgroupsI1;I2;. . .;Inwith di erent
infection stages such that infected susceptible individuals enter the ®rst subgroup I1 and then
gradually progress from this subgroup to subgroupIn. Letci be the average rate of progression
from subgroup i to subgroupi 1, for i 1;. . .;nÿ1, and let cn be the rate at which infected
individuals in subgroup In become sexually inactive or no longer infectious due to end-stage
disease or behavioral changes. The dynamics of the transmission are governed by the following system: dS dt l S0ÿS ÿkS; dI1 dt kSÿ c1 l I1; dIi dt ciÿ1Iiÿ1ÿ ci l Ii; 26i6n; 2:2 dA dt
k rXn
i 1
bi[N]Ii: 2:3
Here, r is the average number of contacts per individual per unit time, b[i] the transmission
probability per contact with an individual in subgroupi,dPlthe removal rate of individuals in
groupA, andN S Pn
i 1I. Notice again that the transmission by theAgroup is neglected just
as it was in model (2.1).
The reproductive number for model (2.2) with (2.3) is de®ned by R0 r Xn i 1 biqi l c[i]; where qidef Yiÿ1 j 1 cj l cj:
The mean duration of infection is de®ned by
sdef Xn
i 1
l c[i];
and the mean probability of transmission per partner from an infected individual during the course of infection is de®ned by
bdef 1 s Xn i 1 biqi ci l:
Then, the reproductive number is again expressed as R0 rbs:
(See [26] for details.)
Instead of assuming that the all infected individuals have the same contact rate, we now assume that infected individuals with di erent stages may have di erent rates of contacts because of possible
changes in behavior. Then the infection rate is given by
k rXn
i 1
b[i] riIi
rS Pn[j][ ][1]rjIj: 2:4
Here,ris the average number of contacts per susceptible individual per unit of time,rithe average
number of contacts per infected individual with stage i per unit of time, bi the transmission
probability per contact with an infected individual in subgroup i, and d is the removal rate of
individuals in groupA.
The reproductive number for the general SP model given by (2.2) and (2.4) can be de®ned as R0
i 1
l ci:
The termriqi= l ci is the number of contacts per infected individual during the individual's
infection period in subgroup i. Then the total number of contacts that an infected individual
makes during all of the individual's infection period is
i 1
l ci def[r][total][:]
Hence, the average number of contacts per infected individual per unit of time is
rdef rtotal[][s] ;
so that rtotaldef[ ][][r][][s.]
The transmission probability through all contacts with an infected individual in subgroupi is
biriqi= l ci . Then the total transmission probability from all contacts with an infected
indi-vidual during the indiindi-vidual's entire infection period, denoted by btotal[, can be de®ned as]
btotaldef[ ] Xn i 1
l ci:
Hence, the average transmission probability per contact is
bdef b[r][total]total [][r]1[][s] Xn
i 1
l ci;
and the reproductive number can be expressed as R0 rbs:
2.2.2. The continuous SP model
In this section, we consider a simple SIR (susceptible±infected±removed) STD model with
continuous infection stages (see [25,42] for further references). Let u be the infection age, and
denote the distribution functions of susceptible, infected, and removed individuals byS t ,I t;u ,
and R t , respectively. We again neglect the transmission by the group of removed individuals,
assuming that they are a small portion of the infected population and that they are less active in transmitting the disease. We also neglect migration between populations and assume that the only
recruitment into the population is a constant in¯ow of susceptible individuals and that all infected individuals are infectious and will eventually be removed.
Under these assumptions, the dynamics of the population are governed by the following system of equations and associated boundary conditions:
dS dt l S0ÿS ÿk t S; oI ot oI ou ÿ l c u I; I t;0 k t S;
I 0;u W u ; 2:5
dt ÿdR
Z [1]
0 c s I t;u du;
where l is the attrition rate caused by natural death or movement out of the sexually active
population, kthe infection rate, lS0 [the rate at which individuals migrate into the population,] [c]
the removal rate of infected individuals,dthe death rate of individuals in the removed group, and
Wis the initial distribution of the infected population.
We consider the infection rate that can be represented as k t r 0
Z 1
0 b u r u
I t;u
r 0 S t R[0]1r v I t;v dvdu: 2:6 Here we assume that the individuals at di erent infection ages have di erent activity levels such
that r u is the number of contacts that an infected individual with infection ageu has,r 0 the
number of contacts that a susceptible individual has, andb u is the probability that an infected
partner with infection ageu will infect a susceptible partner.
By linearizing S t and I t;u about S0[;][0][ ] [and assuming the solutions initially change ]
expo-nentially, a characteristic equation can be obtained. Analyzing the characteristic equation to locate the eigenvalues of the equation in the left-half complex plane yields the following formula
for the reproductive number for the model governed by (2.5) and (2.6):
R0 Z 1 0 r u b u exp ÿ lu Z u 0 c w dw du: 2:7
(Details of the derivation of formula (2.7) can be found in Appendix A.) It is similar to the discrete SP models that the mean duration of infection is
sdef Z 1 0 exp ÿ lu Z u 0 c w dw du;
the total contact rate is rtotaldef[ ] Z 1 0 r u exp ÿ lu Z u 0 c w dw du;
the mean number of contacts is rdef rtotal[] s 1 s Z 1 0 r u exp ÿ lu Z u 0 c w dw du;
the total transmission probability is btotaldef[ ]Z 1 0 r u b u exp ÿ lu Z [u] 0 c w dw du;
and then the mean probability of transmission is
bdef b[r][total]total [][r]1[][s]
Z [1] 0 r u b u exp ÿ lu Z [u] 0 c w dw du:
Therefore, the reproductive number can be again expressed as R0 rbs:
2.3. The di erential susceptibility model
We have shown in Section 2.1 that for the DI model and the SP models, in so far as we assume a homogeneous susceptible population such that there is one group of susceptible individuals, the mean
number of contacts, the mean transmission probability, and the mean duration of infection can be de®ned so that the reproductive number can be always given as the product of these three means. In
this section, we consider a simple di erential susceptibility (DS) model in which the
infected population is homogeneous, but the susceptible population is divided into n groups
according to their susceptibilities. The model equations are given by dSi dt l Si0ÿSi ÿkiSi; dI dt Xn k 1 kkSkÿ l c I; 2:8 dA dt cIÿdA:
The rate of infection is
ki rbI[N] ai; i 1;. . .;n; 2:9
whereai is the susceptibility of susceptible individuals in subgroupi, b the infectious rate of
in-fected individuals, r the average number of contacts per sexually active individual, and
N Pn[i][ ][1]Si I.
By the local stability analysis of the infection-free equilibrium, the reproductive number for model (2.8) with (2.9) can be de®ned as
R0 rb
i 1aiSi0
l c Pn[i][ ][1]S0 i :
Since there is only one group of infected individuals, the mean duration issdef 1= l c . The
biological de®nition of the reproductive number is the number of secondary cases produced when a primary case is introduced into a totally susceptible population. Hence, the mean susceptibility
of susceptible individuals in all the groups, denoted by a, should be weighted by all susceptible
groups at the infection-free equilibrium. That is, adef Pn i 1aiSi0 P[n] i 1Si0 ;
and hence the total mean infectivity isbdef ba. With these notations, the reproductive number can
be rewritten as R0 rbs :
2.4. The combined DS and DI models
We showed in Section 2.3 that even if the susceptible population is divided into subgroups, whereas there is a homogeneous infected population, we can still de®ne the mean infectivity, and the
reproductive number can be given by the compact and intuitive formula. Now if the infected population is divided into subgroups based on their di erent infectivities and the average numbers of
contacts of infected individuals in the infected subgroups are distinct, can those means be well de®ned and the reproductive number still be the product of those means? We combine the DS and DI
models and derive the formula of the reproductive number as follows.
Divide the susceptible population into n groups according to their susceptibilities and the
in-fected population intomgroups based on their infectivities and how ill they are. Then we have the
following system of equations: dSi dt l S0i ÿSi ÿkiSi; i 1;. . .;n; dIj dt Xn k 1 pkjkkSkÿ l mj Ij; j 1;. . .;m; 2:10 dA dt Xm k 1 mkIkÿdA;
where the fractions satisfyPm
j 1pkj 1,k 1;. . .;n.
The rate of infection is ki rai
j 1
bj[r]Pn rjIj
l 1Sl Pmk 1rkIk; 2:11
where r is the mean number of contacts per susceptible individual, rj the average number of
contacts per infected individual in subgroupj,aithe susceptibility of the susceptible individuals in
subgroupi, and bj is the infectiousness of the infected individuals in subgroupj.
By investigating the stability of the infection-free equilibrium, the reproductive number for the model given by (2.10) and (2.11) can be de®ned by
R0 Xm j 1 Xn k 1 pkjakSk0rjbj l mj Pn[l][ ][1]Sl0: 2:12
(The detailed proof of formula (2.12) is given in Appendix B.)
The infected individuals in each subgroup are infected from all susceptible subgroups. Their mean duration of infection needs to be weighed by the fractions and the sizes of susceptibles at the
infection-free equilibrium. Denote the mean duration of infection by s. Then
sdef Xn j 1 1 l mj P[m] k 1pkjSk0 Pn l 1Sl0 :
The term rj= l mj is the number of contacts per infected individual in group j during the
Xn k 1 rj l mj pkjSk0 Pn l 1S0l
is the average number of contacts per infected individual in subgroupjduring the whole infection
period with susceptible individuals that induce the transmission of infection. Summing them over and dividing by the mean duration of infection gives the mean number of contacts per infected
individual over all infected subgroups per unit of time r. That is,
rdef 1[][s] Xm j 1 Xn k 1 rj l mj pkjSk0 P[n] l 1S0l:
The termrjbj= l mj is the total infectivity per infected individual in subgroupj through all
contacts during the whole infection period. Since transmission of a disease results from the in-fectivity of infected individuals and the susceptibility of susceptible individuals, the probability of
transmission per contact from an infected individual in subgroupjwith all susceptible individuals
during the whole infection period is rjbj l mj Xn k 1 pkjakS0[k] Pn l 1Sl0:
Again, summing over all subgroups of infected individuals and dividing by the mean number of contacts and the mean duration of infection yields the mean probability of transmission
bdef [][s]1[][r] Xm j 1 rjbj l mj Xn k 1 pkjakS0k Pn l 1Sl0 1 srR0:
Therefore, the reproductive number can be rewritten as R0 rbs:
3. The segregated risk DI model
In this section, we consider a segregated risk DI model. We divide the susceptible population
into n groups based on their risk behavior. Then, each risk-based infected population group is
further subdivided intomsubgroups. Upon infection, a susceptible individual in the risk-groupSi
enters infected subgroupIi
jwith probabilitypjand stay in this subgroup until becoming inactive in
transmission, where Pm
j 1pj 1. The rate at which infected individuals are removed from
jto the group of removed individuals,R, ismij. Again, we assume that individuals in group
Rare no longer actively transmitting disease. The model is then de®ned by the following system:
dSi dt l Si0ÿSi ÿkiSi; i 1;. . .;n; dIi j dt pjkiSiÿ l mij Iji; j 1;. . .;m; 3:1 dR dt Xn i 1 Xm j 1 mi jIjiÿdR;
wherelis the removal rate, including the natural death rate and other rates at which people leave
the investigated population,lS0
i the recruitment of new susceptible individuals into the population
with risk i, and d is the death rate of individuals in groupR.
The rate of infection for the individuals with risk i,ki, for proportional mixing, is de®ned by
ki ri Pm j 1bjPni 1riIji P[n] k 1rk Sk P[m] l 1Ilk ÿ ; 3:2
where bj is the infectivity of the individuals in the infected subgroup Iji and is assumed to be
independent of their risk level.
The reproductive number for model (3.1) with (3.2) can be de®ned by R0 Xn i 1 r2 iSi0 Pn l 1rlSl0 Xm j 1 pjbj l mi j: 3:3
(We give a complete derivation of formula (3.3) in Appendix C.)
Note that if the removal rates mi
j mj are independent of risk level, then the reproductive
number becomes R0 Xm j 1 pjbj l mj Xn i 1 r2 iSi0 P[n] l 1rlS0l: 3:4
The term Pm[j][ ][1]pjbj= l mj is the product of the mean duration of infection and the mean
transmission probability. ThisR0 in (3.4) involves the second mean of the risk levelPn[i][ ][1]r[i]2S[i]0. As
Diekmann et al. pointed out in [8],
Pn i 1r2iSi0 P[n] l 1rlSl0 mean variance mean :
Hence, (3.4) is consistent with the results in [8,11,37].
However, if the removal rates are not risk-level independent, it is unclear how to de®ne the mean duration of infection and the mean transmission probability. Here we provide an alternative way to
make the formula of the reproductive number more intuitive.
De®ne the mean duration of infectivity for infected individuals with risk level i by
sidef Xm j 1 pj l mi j;
the mean probability of transmission per partner from those infected individuals by b[i]def 1 si Xm j 1 pjbj l mi j;
and the reproductive number for the subgroup with risk level i by
0def ribisi:
R0 Xm i 1 riSi0 Pm l 1rlSl0R i 0:
Here,riSi0 is the total number of contacts of an individual in the group with risk leveli. Then the
reproductive number for the whole population is equal to the mean reproductive numbers of the risk groups weighted by their risks.
4. A simple age-structured model
We consider a simple SIR model with age structure in this section (see [33]). Denote the dis-tribution functions of susceptible, infected, and removed individuals byS t;a ,I t;a , andR t;a ,
respectively, wheretis the time andais the age. We neglect transmission of the virus by groupR.
We also neglect migration between populations and assume that the only recruitment into the population is a constant in¯ow of susceptible individuals.
Under these assumptions, the dynamics of the population are governed by the following system of equations and associated boundary conditions:
oS ot oS oa K a ÿ l a k t;a S; S t;a0 B; S 0;a U a ; oI ot oI oa ÿ l a c a I k t;a S; I t;a0 0; 4:1 I 0;a W a ; oR ot oR oa ÿd a R c a I; R t;a0 0; R 0;a 0;
wherel is the attrition rate due to natural death or movement out of the sexually active
popu-lation,kthe infection rate,Bthe number of individuals in the susceptible class at agea0,Kthe rate
at which individuals ¯ow into the population at ages greater thana0, cthe removal rate,dis the
death rate in group R, andU and W are the initial distributions of the susceptible and infected
We consider the infection rate that can be represented as k t;a
Z [1]
b a;a0[ ][p][ ][t][;][a][;][a]0[ ] I t;a0
with the total sexually active population given by N t;a S t;a I t;a :
Hereb a;a0[ ][is the probability that an infected partner of age][a]0[will infect a susceptible partner of]
agea during their partnership,p t;a;a0[ ][the rate of pair formation between individuals of age] [a]
and individuals of agea0[, and][I][=][N] [is the probability that a randomly selected partner is infected.]
We assume that the transmission probability is the product of the susceptibility of the sus-ceptible individual and the infectiousness of the infected individual. They can also both depend on age.
However, in order to keep the analysis of the model tractable, we allow susceptibility to be age-dependent, but make the somewhat restricting assumption that infectiousness is age-inde-pendent.
Hence,b a;a0[ ][b][ ][a][ ][.]
In order to simplify the analysis, we assume that there are no strong biases at work and partners are chosen at random, according to their availability. The random partner selection process leads
to a proportionate mixing ratepwith the form
p t;a;a0[ ][R]r a r a0 N t;a0 1
a0 r a N t;a da
wherer a is the partner acquisition rate of individuals of agea, or the number of contacts per
individual of ageaper unit of time.
Under these assumptions, the infection rate is k t;a b a r a Z 1 a0 r a0[ ][I][ ][t][;][a]0[ ] R1 a0 r a N t;a da da0[:] [ ][4][:][2][ ]
Using the same technique for showing (2.7), we can de®ne the reproductive number for model (4.1) with (4.2) as R0 R1 a0 r a Ra a0b g r g expfÿ Ra g l a c a dagS0 g dgda R1 a0 r a S 0[ ][a][ ]
[da] ; where S0[ ][a][ ][Be]ÿM a [ ][e]ÿM a Z a a0 eM x [K][ ][x][ ][dx][;] and M a Z a a0 l s ds:
By interchanging the order of the integration, R0 can also be expressed as
R0 R[1] a0 r g S 0[ ][g][ ][b][ ][g][ ]R1 g r a expfÿ R[a] g l a c a dagdadg R[1] a0 r a S 0[ ][a][ ][da] :
Note that expfÿR[g]a l s c s dsgis the probability that an individual who is infected at agegis still in the infected population at age a. Then, r a expfÿR[g]a l s c s dsg is the number of
contacts from partners who are infected at age g and survive to age a, and the total contacts
inducing transmission from all surviving infected individuals of all agesaPg is
Z 1 g r u exp ÿ Z u g l s c s ds du:
Again, since expfÿR[g]u l s c s dsgis the probability of infected individuals who are infected at
agegand survive to ageu, the mean duration of infections of the cohort of agegcan be expressed as
s g def Z [1] g exp ÿ Z [u] g l s c s ds du:
Then the mean contact rate of the cohort of ageg can be de®ned by
r g def [][s][ ]1[g][ ] Z 1 g r u exp ÿ Z u g l s c s ds du:
De®ne the reproductive number of the cohort of age gby
R0 g defr g b g s g :
Then the reproductive number for the total population is the in®nite sum of the reproductive numbers of all cohorts weighted by the fractions of the total contacts of the cohorts at the
infection-free equilibrium, where the reproductive number or the initial transmission is determined; that is,
R0 Z [1] a0 r g S0[ ][g][ ] R[1] a0 r a S 0[ ][a][ ][da]R0 g dg: 5. Discussion
The reproductive numberR0is one of the most important concepts in epidemiological theory. It
characterizes the threshold behavior such that ifR0 <1, the modeled disease will die out if a small
number of infected individuals are introduced into a susceptible population, and if R0 >1, the
disease will spread in the population. A good estimate of the reproductive number can provide signi®cant insight into the transmission dynamics of the disease and can lead to e ective strategies to
control and eventually eradicate the disease.
Formulas (1.1) and (1.2) are useful estimates. They have been applied to various models for di erent purposes and, in particular, have been widely used in biology and the medical com-munity. Their
contributions are signi®cant. For example, sensitive studies of those estimates on di erent parameters have been used to investigate the e ects of changes in sexual behavior on the transmission
dynamics of STDs such as HIV [7,31,38]. It was shown in [31] that, in a preferred mixing, single-sex model, reductions in the frequency of partner change by low-activity people can increase the
long-term prevalence of HIV/AIDS in populations that would have low steady-state prevalence given current activity levels. Such ®ndings can be used to plan educational campaigns.
Formulas forR0 can also be used to establish e ective vaccination programs [2,13±16,22]. E ects
For simple homogeneous models, it is easy to estimate the mean duration, mean number of contacts, and transmission probability so that formulas (1.1) and (1.2) can be applied. As shown in Section 2,
it seems that if there is no risk structure involved in the model, no moments higher than the ®rst will be needed in the formula for the reproductive number. More speci®cally, if the susceptible
population is not divided into risk groups, the reproductive number is always based on the ®rst moments. Hence, it should be possible to de®ne the mean number of contacts, the mean
duration of infection, and the mean transmission probability in appropriate ways and thenR0 can
be estimated with an intuitive formula. Even if there are subgroups in the infected population with di erent contact rates, this still seems true. That is, the heterogeneity of the infected
population may not be as crucial as that of the susceptible population. This observation can be explained as follows. From the biological point of view, the reproductive number characterizes the
where a small number of infected individuals are ®rst introduced into an entirely susceptible
population. Hence, the heterogeneous structure of the infected population will not play a critical role in the transmission dynamics, at least at the early stage of the transmission. From the
mathematical modeling perspective, R0 is determined by the stability of the infection-free
equi-librium for whichthe components of infected individuals are 0. Therefore, the heterogeneity of the
infected individuals is negligible.
On the other hand, if there is heterogeneous structure in the susceptible population concerned,
this heterogeneity cannot be neglected. If an explicit formula ofR0 can be obtained, higher
mo-ments will be naturally involved, and it may be necessary to include the variance or deviation. That is, for di erent models, although those means may be de®ned in the same way, their
het-erogeneous di erence may cause signi®cant deviation about the means and then may lead to very di erent transmission dynamics.
Ideally, appropriate de®nitions of the means and their variances for the total population can be de®ned. However, as shown in Sections 3 and 4, if more heterogeneous structures are included in the
model, it may not be possible to de®ne those means in practice. More importantly, the devi-ations from the means have to be taken into consideration. Then it will be more reasonable and practical to
de®ne the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the
total population. Heathcote and Yorke [23] and Jacquez et al. [28,29] introduced this idea for risk-group models. Our studies in Sections 3 and 4 support and generalize this idea.
Finally, it is worthwhile to point out from the study of the DS model in Section 2.3 that, al-though there seems to be heterogeneous structure in the susceptible population for the DS model, since
the infection transmission has to be through contacts with infected individuals, and there is only one infected group, higher moments do not appear in this particular situation.
The authors would like to thank Ann Stanley for leading them in seeking this general for-mulation of the reproductive number and for her insightful comments and contributions in de-®ning the
reproductive number for the DI and SP models. They are also very grateful for Herbert Hethcote for his careful reading throughout the manuscript and valuable comments. This research was supported by
the Department of Energy under contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program KC-07-01-01.
Appendix A. Derivation of formula (2.7)
Letx SÿS0 [and] [y][ ][I][. Then]
k r 0 Z 1 0 b u r u y t;u r 0 x S0[ ]R1 0 r v y t;v dv du[S]1[0] Z 1 0 b u r u y t;u du;
and the linearization of (2.5) about the infection-free equilibrium is given by dx dt ÿlxÿ Z [1] 0 b u r u y t;u du; oy ot oy ou ÿ l c u y; A:1 y t;0 Z [1] 0 b u r u y t;u du:
Substituting x x 0 eqt [and][y] [ ][k][ ][u][ ][e]qt [into (A.1) yields the following system of equations:]
q l x 0 ÿ Z [1] 0 b u r u k u du; A:2 dk du ÿ q l c u k u ; A:3 k 0 Z 1 0 b u r u k u du: A:4
Solving Eq. (A.3) for k u and employing the initial condition (A.4) leads to the following
characteristic equation: Z 1 0 b u r u exp ÿ Z u 0 q l c v dv du 1: A:5
Then, it is easy to see if R0<1, all rootsqof (A.5) have negative real part, and if R0 >1, there
exists at least one positive root qof (A.5).
Appendix B. Derivation of formula (2.12) The Jacobian at an equilibrium has the form
0 D
D ÿr1 Pnk 1pk1Skok[oI]k 1 Pn k 1pk1Skok[oI]k 2 Pn k 1pk1Skok[oI]k m P[n] k 1pk2Sk okk oI1 ÿr2 P[n] k 1pk2Sk okk oI2 P[n] k 1pk2Sk okk oIm ... ... .. . ... Pn k 1pkmSkok[oI]k 1 Pn k 1pkmSkok[oI]k 2
ÿrm Pn k 1pkmSkok[oI]k m 0 B B B B B B B B @ 1 C C C C C C C C A
evaluated at the infection-free equilibrium withridef l mi.
SetQi 1=N0P[k]n[ ][1]pkiS[k]0ak, i 1;. . .;m, withN0 Pn[i][ ][1]S[i]0. Then Dhas the form of
D ÿr1 r1Q1b1 r2Q1b2 rmQ1bm r1Q2b1 ÿr2 r2Q2b2 rmQ2bm ... ... .. . ... r1Qmb1 r2Qmb2 ÿrm rmQmbm 0 B B B @ 1 C C C A:
ConsiderÿDand let V def 1=r1;. . .;1=rn . Then
ÿDV 1ÿXm i 1 riQibi ri ! E;
where E is the vector each of whose elements is one. Let R0def Pm[i][ ][1]riQibi=ri. Then, from
M-matrix theory, all eigenvalues ofD have negative real part ifR0 <1 which leads to the local
stability of the infection-free equilibrium. On the other hand, by mathematical induction, it can be
shown that the determinant ofDis given by
detD ÿ1 m 1Ym
i 1
ri R0ÿ1 :
Hence, ifR0 >1,Dhas at least one positive eigenvalue. Therefore, the reproductive number for
the model (2.10) can be de®ned as R0 [N]1[0] Xm i 1 Xn k 1 pkiakS0kribi l mi :
Appendix C. Derivation of formula (3.3)
The Jacobian matrix at the infection-free equilibrium S1 S01;. . .;Sn Sn0;I11 0;
2 0;. . .;Im1 0;. . .;I1n 0;. . .;Imn 0 has the following form:
0 B
Bdef B11 B12 B1n B21 B22 B2n ... ... ... ... Bn1 Bn2 Bnn 0 B B B @ 1 C C C A; with Biidef ai 1 rib1ÿni1 a1irib2 ai1ribm ai 2rib1 ai2 rib2ÿni2 ai2ribm ... ... ... ... ai mrib1 aimrib2 aim ribmÿnim
0 B B B B @ 1 C C C C A; and Bijdef ai 1rjb1 ai1rjbm ... .. . ... ai mrjb1 aimrjbm 0 B @ 1 C A; i6 j: Here, we write ai jdef pjriS 0 i P[n] k 1rkS0k and n i jdef l mi j ai j :
The stability of the Jacobian matrix at the infection-free equilibrium is completely determined
by the stability of B. Note that all o -diagonal elements of Bare positive. We consider ÿBand
take V def 1 n1 1 ;. . .; 1 n1 m ;. . .;[n]1n 1;. . .; 1 nn m !T : Since ÿBV 1ÿXn k 1 Xm l 1 rkbl nk l ! E; where Edef a1 1;. . .;a1m;. . .;an1;. . .;anm ÿ T[;] if we de®ne R0def Xn k 1 Xm l 1
rkbl nk l Xn i 1 r2 iSi0 Pn l 1rlSl0 Xn j 1 pjbj l mi j;
On the other hand, by mathematical induction, it can be shown that the determinant of Bis detB ÿ1 nm Yn i 1 Ym j 1 ai jribj ! Yn k 1 Ym l 1 nk l rkbl ! 1ÿXn k 1 Xm l 1 rkbl nk l ! ÿ1 nmYn i 1 Ym
j 1 l mi j 1 ÿR0 :
Hence, if R0 >1, matrix B has at least one positive eigenvalue. Therefore, the infection-free
equilibrium is unstable. References
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[38] M. Morris, L. Dean, E ects of sexual behavior change on long-term human immunode®ciency virus prevalence among homosexual men, Am. J. Epidemiol. 140 (1994) 217.
[39] R. Ross, The Prevention of Malaria, Murray, London, 1909.
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[43] P. Waltman, Deterministic Threshold Models in the Theory of Epidemics, Lecture Notes in Biomathematics, vol. 1, Springer, Berlin, 1974. | {"url":"https://1library.net/document/ynn698jy-intuitive-formulation-reproductive-number-spread-diseases-heterogeneous-populations.html","timestamp":"2024-11-14T22:10:53Z","content_type":"text/html","content_length":"191831","record_id":"<urn:uuid:0fc1fba0-16b6-48eb-b06a-e88bc50aee6d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00619.warc.gz"} |
Peirce’s Deductive Logic
First published Fri Dec 15, 1995; substantive revision Fri May 20, 2022
Charles Sanders Peirce was a philosopher, but it is not easy to classify him in philosophy because of the breadth of his work. (Please refer to the table of contents of the entry Charles Sanders
Peirce.) Logic was one of the main topics on which Peirce wrote. If we focus on logic, however, it becomes apparent that both Peirce’s concept of logic and his work on logic were much broader than
his predecessors’, his contemporaries’, and ours. First, Peirce located logic in his large architectonic framework of philosophy, which is why some strongly believe that Peirce’s logic cannot be
properly understood without understanding his pragmatism and his semiotics, to mention but two of his other contributions. Even within the traditional boundaries of logic, Peirce made too many
contributions to outline in a single article.
Acknowledging the nature of this next-to-impossible task, we single out the common theme of Peirce’s various contributions to modern logic—to extend logic, as characterized by the three different
i. the scope of formalism (from monadic to relations),
ii. the kinds of systems (from symbolic to diagrammatic systems), and
iii. semantic values (from bivalence to three values).
The main goal of this entry is not only to present Peirce’s accomplishments in each of these three extensions, but also to explore the relations, if any, among these novel developments. The three
sections of the entry will be devoted, respectively, to each of these three ways how the horizon of deductive logic is expanded by Peirce.
Peirce’s journey on formal deductive logic started with Boolean calculus and De Morgan’s logic of relatives. Boolean algebra created a path to generalize Aristotelian syllogism and De Morgan’s
ambition to formalize relations opened a new territory to conquer. However, Peirce’s predicate logic is neither a mechanical expansion from the existing logics nor a simple combination of these two.
A leap made by Peirce from his contemporary logic is qualitatively substantial enough to call Peirce a founder of modern deductive logic, as the entry explains. The first section explores Peirce’s
development of predicate logic presented in his several well-known papers, by locating the root of Peirce’s introduction of quantifiers and bound variables. While formal details and notations of
Peirce’s first-order logic can be overwhelming, one should not lose the sight of the bigger picture by paying attention to the main motivation behind Peirce’s enterprise for a new logic. Conquering
new territory—relations—with new formal notation, Peirce’s adventure was launched into another dimension—a new mode of representation, that is, diagrammatic representation. This is the topic of the
second section. While two systems of Peirce’s Existential Graphs (“EG” henceforth) are presented, the following perspective is at the background: Peirce’s EG were not invented just as random
alternatives, logically equivalent to his own predicate logical notation, but a reflection of Peirce’s new approach to logic and formalization. As Peirce’s tireless attempts for predicate logic
brought to us more powerful formal notation, Peirce’s search for better representation of relational states of affairs was pursued beyond his own symbolic system. Spatial, as opposed to linear,
notation is still not familiar to some of us, and the second section introduces the basic notational aspects of Peirce’s EG, and discusses the fundamental differences between EG and symbolic systems.
The third section about three-valued logic examines another new enterprise of Peirce, not in syntactic notation, but in semantic values. Peirce scholars propose various motivations behind Peirce’s
three value semantics and those different views will be briefly discussed.
While the first contribution, that is, an extension from monadic to predicate logic, has positioned Peirce as a founder of modern logic along with Frege, it took much longer for Peirce’s other
achievements to receive proper attention from logicians or philosophers. The entry aims to draw a road map for Peirce’s journey in deductive logic so that one may realize his accomplishments are very
much connected with each other. More specifically, Peirce’s achievements in deductive logic were accumulative. After being able to formalize polyadic relations with new symbolic notation, Peirce
devised a totally new form of representation—diagrammatic systems. What we can formalize is extended and how we can formalize what we can formalize is extended. And Peirce ventured into what our
formalization represents and suggested a more fine-grained or a bigger territory of semantic values than binary True or False values.
1. From Monadic to Polyadic Logic
Peirce and Frege, independently of each other, took us from the traditional Aristotelian logic to modern logic—a large leap. Nobody could deny the power of formalization which has led early twentieth
century mathematicians to surprising achievements and results.^[1] What is the essence of the leap made by Peirce and Frege? Is it just a matter of introducing new formal notations, i.e., quantifiers
and variables, so that we may easily formalize our reasoning? If so, modern logic would be just dressing up Aristotelian logic with quantifiers/variables. This would equate one of Peirce’s main
contributions in logic to the increase in formal vocabulary.
The enormous impact of the adoption of quantifiers/bound variables on the world of logic and mathematics cannot be denied. However, that should not overshadow Peirce’s insight behind the new extended
formalism. This section will explore how Peirce’s conviction about the novelty of the logic of relations led him to the introduction of quantifiers/variables. Hence, quantification theory, according
to Peirce, is not a matter of a linear extension of formal vocabulary, but an expansion into territory that is qualitatively different from what Aristotelian logic covers. At the same time, we should
not forget that Peirce extended the territory of logic in the spirit of Boole’s algebra of logic.
In “An Improvement in Boole’s Calculus of Logic” (1867) Peirce hints at the need for an improvement of Boolean logic, not in the context of predicate logic, but in its inability to express
existential statements in the context of term logic. His 1870 paper “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus”
(DNLR) reveals his ambition to marry Boole’s algebraic notation with De Morgan’s effort at relational representation. Many agree that this paper introduces essential vocabulary of first-order
predicate logic for the first time in history. Subsequently, in “On the Algebra of Logic” (1880) Peirce investigates two kinds of operations over relations—relative sum and relative product—and “The
Logic of Relatives” (known as “Note B”) published in his edited book Studies in Logic by Members of the Johns Hopkins University (1883) shows a major progress in quantification, influenced by the
work of O. H. Mitchell (who was his student). Finally, Peirce’s 1885 paper “On the Algebra of Logic: A Contribution to the Philosophy of Notations” has been considered to be the place where Peirce
fully presented his quantification theory.
Starting with DNLR, the first subsection examines Pierce’s subsequent steps until he presented the final form of his first-order logic in his 1885 paper “On the Algebra of Logic: a Contribution to
the Philosophy of Notations”. (For a number of manuscripts written between these two papers, refer to Beatty 1969; Dipert 2004: 297–299; and Merrill 1978.) The second subsection locates Peirce’s
first-order predicate logic work in a larger context.
1.1 Relations and quantification formalized
Peirce’s quantification theory is presented in a comprehensive way together with axiomlike “icons” in his 1885 paper “On the Algebra of Logic: A contribution to the Philosophy of Notations”. Peirce’s
not-short journey to modern logic started with his attempt to extend the territory of formalization. In this, Peirce was inspired by De Morgan’s struggle for the representation of relations, and at
the same time Peirce was empowered with Boolean calculus which formalizes Aristotelian term logic. That is, Peirce took De Morgan’s ambition as a road map for the direction while being equipped with
Boole’s method and notation for getting there. This subsection will follow Peirce’s journey to see how he reached the destination, by checking in at his main stops.
The title of Peirce’s 1870 paper “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus” (DNLR) is spelled out at the beginning
of the paper in the following way:
[I]t is interesting to inquire whether it [Boole’s logical algebra] cannot be extended over to the whole realm of formal logic, instead of being restricted to that simplest and least useful part
of the subject, the logic of absolute terms,…The object of this paper is to show that an affirmative answer can be given to this question. (DNLR [CP 3.45])
Boolean logic needs to be “extended” if we want to cover the entire realm of formal logic, Peirce states. What does Peirce mean by “the whole realm of formal logic”? Peirce answers: “Deductive logic
can really not be understood without the study of the logic of relatives” (1911a [CP 3.641]).^[2]
Being encouraged by Boole’s algebra of logic, but at the same time taking his father, Professor Benjamin Peirce’s negative view of logic seriously,^[3] Peirce explored a way to apply Boole’s method
to a larger domain of our reasoning so that relations may be formalized.
What are relations and why are they so special? Let’s compare three sentences: “John is an American”, “John is taller than Tom”, and “John is between Tom and Mary”. The first sentence has a unary
predicate “is an American”, the second sentence a binary predicate “is taller than”, and the third sentence a ternary predicate “is between…and…”. A unary predicate stands for a property or quality,
while a binary or ternary predicate stands for a relation. If a first-order logical system has only unary predicates, then we say it is monadic. Otherwise, predicate logic is assumed to have binary
or other higher predicates.
When we move from monadic to polyadic logic, substantial changes take place. The following three changes are at the top of the list. First of all, a move from property to relation is a territory
expansion. Noting that Aristotelian syllogisms are limited to unary predicates, one expects polyadic logic to represent more than the reasoning involved in Aristotelian syllogism, that is, term
logic. Second, monadic logic is decidable while polyadic logic is not decidable, as Church’s theorem proved. In some sense, as the territory is expanded, we are losing its grip. Third, a change in
notation is inevitable, which necessitates modern quantification theory. How are these three important aspects are handled by Peirce?
The realm of relations was a frontier where De Morgan did much of his creative and novel work on logic.^[4] However, his inquiry on the topic resides within the patterns of traditional syllogisms.^[5
] Even more importantly, De Morgan did not have enough tools to formalize this newly extended realm.^[6] Hence, not surprisingly De Morgan’s relations are rather limited to a certain group which fit
in syllogistic reasoning. As Merrill points out,
De Morgan develops the general logic of relations only to the point where it can be used for his familiar syllogistic purposes. This means that he is especially interested in relations which are
convertible and/or transitive,…. (Merrill 1990: 113)
It is somewhat unclear and controversial whether Peirce’s interest in the logic of relations started independently of De Morgan’s work on the subject.^[7] Regardless of the origin of Peirce’s inquiry
into relations, many have come to agree that it is Peirce (not De Morgan) who successfully formalized the logic of relations. Merrill, a De Morgan scholar, puts the matter in the following way:
The most obvious problem with this view of the proposition [De Morgan’s way of handling relational arguments] is that it does not seem general enough. If we can unite two terms into a proposition
by relating them, why not three or four or ten terms? De Morgan’s concern with the relational syllogism seems to have precluded this generalization; but there is no reason in principle why it
could not be made. For this, though, we must wait for Frege and Peirce. (Merrill 1990: 110)
Interestingly enough, Peirce’s writings before 1870 DNLR show Peirce also attempted to solve relational arguments by traditional syllogistic reasoning rules,^[8] but the approach taken in DNLR is
totally different—not within a syllogistic frame but by introducing Boolean algebra notation. Peirce must have realized the power of generalization that Boolean notation could provide. Boole’s
algebra formalized Aristotelian categorical syllogisms and opened a way for generalization of term logic.^[9] Peirce, who was impressed with Boole’s mathematical treatment of Aristotelian syllogisms,
not surprisingly aimed to apply this method to relations. In that sense, modern predicate logic started in Peirce’s 1870 pioneering work. Hence, the goal of Peirce’s project—that is, to broaden the
scope of formalization in logic—was a main motivation for the introduction of new vocabulary for quantifiers and bound variables. If so, Peirce’s insight early on as to the importance of reasoning
involving relations is a key element in understanding a difference between Peirce’s and Frege’s developments of first-order logic.^[10] Furthermore, the next section will show how Peirce’s obsession
with the logic of relations led him to the invention of Existential Graphs.
The logic of relations formalizes a larger territory than monadic logic, but there is a price to pay for obtaining the additional expressive power: While monadic logic is decidable, polyadic logic is
not. Even though we need to wait until Church’s theorem to see the undecidability of first-order predicate logic, Peirce intuited a fundamental difference between the logic of non-relatives versus
the logic of relations. Here are Peirce’s suggestive ideas over the comparisons between monadic and relational logic:
The logic of relatives is highly multiform; it is characterized by innumerable immediate inferences, and by various distinct conclusions from the same set of premises. (1883a [CP 3.342])
[T]he old syllogistic inference can be worked by machinery, but characteristic relative inferences cannot be performed by any mere mechanical rule whatever. (1896: 330)
As Dipert correctly points out, Peirce’s remarks reveal his “understanding of the richness and difficulty which relations introduce into logic” (Dipert 1984a: 63).^[11]
In order to increase expressive power, Peirce left the traditional syllogistic pattern and brought in Boolean algebra of logic. The following comment emphasizes that Peirce’s choice of notation marks
a clear departure from De Morgan’s pursuit of relational logic:
De Morgan’s methodology is governed by the logic of syllogism while Peirce’s methodology is entirely algebraic. This algebraic model taken over from Boole is foreign to De Morgan’s methods. This
difference in methodology reflects a significant difference at the level of definition. (Brunning 1991: 36)
And after realizing the complicated nature of the logic of relations, Peirce explored new notation beyond Boolean calculus. That move is predicted in the following passage:
The effect of these peculiarities [the non-mechanic nature of relative logic] cannot be subjected to hard and fast rules like those of the Boolian calculus. (1883a [CP 3.342])
Here is the third aspect of a transition from monadic to polyadic logic: The complication that relations bring in our reasoning, obviously, pushed Peirce to develop a new notational system. As the
rest of this subsection shows, the process made over the course of 15 years—from DNLR to “On the algebra of logic”—is rather complicated. Importantly, Peirce’s introduction of quantifiers and bound
variables could be seen as an inevitable outcome of his ambitious goal to expand the scope of formalization to cover relations, as Merrill says “The quantification complexities of many relational
statements cried out for quantifiers” (1997: 158).
The third section of DNLR, as the title “Application of the Algebraic Signs to Logic” says, is one of the first places where Boolean algebraic notation and relational logic joined each other. In the
first subsection Peirce makes it clear that the territory he aims to cover is relational, by including polyadic predicates in the following way:
(DNLR [CP 3.63–64]; the entry adopts our modern terminology instead of Peirce’s.)
Predicates Peirce’s terminology Letters Examples
unary absolute terms \(\unary{a}, \unary{b}, \unary{c},\)… (Roman alphabet) Frenchman \((\unary{f})\), violinist \((\unary{u})\),…
binary simple relative terms \(\binary{a}, \binary{b}, \binary{c}\),… (italics) wife \((\binary{w})\)), lover \((\binary{l})\), owner \((\binary{o})\),…
ternary conjugative terms \(\ternary{a}, \ternary{b}, \ternary{c}\)… (Kennerly [Kennerley]) giver to — of — \((\ternary{g})\)
For the rest of the third section, four kinds of algebraic signs are introduced to be applied on these predicate letters: the inclusion sign (\(\inclusion\)), the addition sign (\(\cunion\)), the
multiplication sign (juxtaposition or “,”), and the involution sign (exponentiation).
First, he combines the sign equality “=” and the sign \(<\) for “less than” to come up with the sign “\(\inclusion;\)” to represent inclusion:
\[\unary{f} \inclusion \unary{m}\]
means “every Frenchman is a man”, without saying whether there are any other men or not. So,
\[\binary{m} \inclusion \binary{l}\]
will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. (DNLR [CP 3.66])
Note that “\(\unary{f} \inclusion \binary{m}\)” (unlike “\(\unary{f} \inclusion \unary{m}\)”) would be ungrammatical since \(\binary{m}\), being a binary predicate, cannot have an inclusion relation
with a unary predicate \(\unary{f}\).
For the sign of addition, Peirce brings in Boolean sign \(+\), but with slight variation:
The sign of addition is taken by Boole, so that
\[x + y\]
denotes everything denoted by \(x\), and besides, everything denoted by \(y\).…But if there is anything which is denoted by both the terms of the sum, the latter no longer stands for any logical
term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example,
\[\unary{f} + \unary{u}\]
means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. For this reason alone…I preferred to take as
the regular addition of logic a non-invertible process, such that
\[\unary{m} \cunion \unary{b}\]
stands for all men and black things, without implication that the black things are to be taken besides the men. (DNLR [CP 3.67])
Hence, Peirce’s slightly modified addition sign, \(\cunion\) , denotes inclusive disjunction. “\(\unary{f} \cunion \unary{u}\)” denotes all those who are either a Frenchman or a violinist. The
notation does not imply that no Frenchman is a violinist or no violinist is a Frenchman. Even though Peirce’s example is limited to unary predicates, we can extend the idea to binary. Using modern
notation, \[\binary{l} \cunion \binary{s} = \{\langle x, y\rangle \mid lover(x, y) \lor servant(x,y)\}.\] That is, it corresponds to a union of relations.
When the multiplication sign enters the picture, the logic of relations becomes powerful, and here is Peirce’s hallmark for the interpretation of multiplication:
I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, \(\binary{l}\unary{w}\) shall denote whatever is a lover of a woman.…\(\binary
{s}(\unary{m} \cunion \unary{w})\) will, then, denote whatever is a servant of anything of the class composed of men and women taken together. (DNLR [CP 3.68])
When polyadic predicates are in the picture, how to form a new relation becomes more interesting and complicated. This is why the multiplication operation of relative product is extremely important
for further work on relational logic. A product between two predicates is much more interesting than an addition between two, depending on what kinds of predicates are involved:
i. The product between two properties is another property, being the intersection between two properties,
ii. the product of a relation and a property is another new property, and
iii. the product between relations produces a new relation.
Let’s try to understand Peirce’s concept of relative product in terms of modern terminology:
• “\(\unary{w}\)” be a unary predicate, being a woman,
• “\(\unary{u}\)” be a unary predicate, being a violinist,
• “\(\binary{l}\)” a binary predicate, being a lover of, and
• “\(\binary{s}\)” be a binary one, being a servant of.
• \(\unary{w}\bcomma\unary{u} = \{x \mid \textit{woman}(x) \land \textit{violinist}(x)\}\).^[12]
• \(\binary{l}\unary{w} = \{ x \mid \exists y (\textit{lover}(x, y) \land \textit{woman}(y))\}\).
• \(\binary{l}\binary{s} = \{ \langle x, z\rangle \mid \exists y (\textit{lover}(x, y) \land \textit{servant}(y, z))\}\).
In this modern translation, the existence of an existential quantifier is noticeable, even though Peirce himself did not mention it at all in DNLR.
Hidden quantifier implication becomes even more obvious in the operation of involution below.
I shall take involution in such a sense that \(x^y\) will denote everything which is an \(x\) for every individual of \(y\). Thus \(\binary{l}^{\unary{w}}\) will be a lover of every woman. (DNLR
[CP 3.77])
That is, \(\binary{l}^{\unary{w}} = \{ x \mid \forall y (\textit{woman} (y) \rightarrow \textit{lover} (x,y))\}\). Here, a universal quantifier is present!
Before we go into the details of Peirce’s quantifiers, let’s summarize algebraic signs Peirce adopted to handle polyadic predicates:
Algebraic Meanings/ Examples
Signs Operations
\(\unary{w} \inclusion \unary{u}\)
\(\inclusion\) inclusion \(\quad\forall x (\textit{woman} (x) \rightarrow \textit{violinist}(x))\)
\(\binary{l} \inclusion \binary{s}\)
\(\quad\forall x \forall y (\textit{lover}(x, y) \rightarrow \textit{servant}(x, y))\)
\(\unary{w} \cunion \binary{u}\)
\(\cunion\) union \(\quad\{ x\mid \textit{woman}(x) \lor \textit{violinist}(x)\}\)
\(\binary{l} \cunion \binary{s}\)
\(\quad\{ \langle x, y \rangle \mid \textit{lover}(x,y) \lor \textit{servant} (x, y)\}\)
\(\bcomma\) intersection \(\unary{w}\bcomma \unary{u}\)
\(\quad\{ x\mid \textit{woman}(x) \land \textit{violinist}(x)\}\)
\(\binary{l}\unary{w}\) (a lover of some woman)
(no comma) relative product \(\quad\{ x \mid \exists y (\textit{lover}(x, y) \land \textit{woman}(y))\}\)
\(\binary{l}\binary{s} = \{ \langle x, z \rangle \mid \exists y (\textit{lover}(x, y) \land \textit{servant}(y, z)\}\)
\(x^y\) \(x\) of every \(y\) \(\binary{l}^{\unary{w}}\) (a lover of every woman)
\(\{ x \mid \forall y (\textit{woman} (y) \rightarrow \textit{lover} (x,y)\}\)
Let’s focus on hidden but assumed presence of quantifiers in the case of multiplication and exponentiation: \(\binary{l}\unary{w}\) is interpreted as “a lover of some woman” and \(l^{\unary{w}}\) as
“a lover of every woman”. Interestingly, in the process of introducing polyadic predicates, Peirce ends up bringing in quantifiers, some and every. On the other hand, considering that Aristotle’s
syllogisms have two quantifiers, Peirce’s algebraic notation for quantifiers—some and every—should not surprise us. However, a crucial aspect of this development is that Boole’s unsatisfactory
representation of existential propositions (as opposed to universal propositions) pushed Peirce and his student O. H. Mitchell to go beyond Boole’s logic.^[13] The way Peirce interprets a relative
product—“\(\binary{l}\unary{w}\)” meaning “lover of some woman”—allows an existential quantifier to be expressed implicitly in terms of multiplication.
Let us see several different ways Peirce pursued to represent existential statements in a more explicit way. In the above for “\(\binary{l}\unary{w}\)”, the existential quantifier is carried out in
the way relation \(\binary{l}\) is applied to a unary predicate “\(\unary{w}\)”, but not explicitly. There are several different explicit ways Peirce represents existential statements. One method:
borrowing exponentiation (in which a binary predicate is applied to a unary predicate, e.g., \(\binary{l}^{\unary{w}}\)), Peirce expresses an existential statement as a contradiction of a universal
Particular [existential] propositions are expressed by the consideration that they are contradictory of universal propositions. Thus, as \(\unary{h}\bcomma(1-\unary{b})=0\) means every horse is
black, so \(0^{\unary{h}, (1-\unary{b})} = 0\) means that some horse is not black; and as \(\unary{h}\bcomma \unary{b}= 0\) means that no horse is black, so \(0^{\unary{h}, \unary{b}} = 0\) means
that some horse is black. (DNLR [CP 3.141])
The number 1 represents the universe class and 0 the null class. However, Peirce’s notation of exponentiation whose base is 0 has some slightly different nuance. Let’s recall Peirce’s exponentiation
\[ \binary{l}^{\unary{w}} = \{ x \mid \forall y (\textit{woman} (y) \rightarrow \textit{lover}(x,y))\}. \]
\[ 0^{0} = \{ x \mid \forall y (\textit{null-class}(y) \rightarrow \textit{null-relation}(x,y))\}. \]
[Note: The base 0 denotes a relation—the null relation, while the exponent 0 a class—the null class.]
There is no \(y\) such that \(\textit{null-class}(y)\), since noting could be in the null-class. Hence, vacuously, every object in the domain gets in \(0^{0}\). That is, the class of things which has
no relation to the null class is the universe class which is represented by 1. Hence, \(0^{0} = 1\).
Suppose \(\unary{m}\not = 0\).
\[ 0^{\unary{m}} = \{ x \mid \forall y (\textit{non-null-class}(y) \rightarrow \textit{null-relation}(x,y))\}. \]
Since nothing can bear the null-relation to every member of any non-null class, \(0^{\unary{m}} = 0\). Hence, we get the following exponentiation notation:
\[\tag{*} 0^x = 0 &\quad \textrm{if } x \not =0\\ 0^x = 1 &\quad \textrm{if } x =0\\ \]
Using this result, let’s unpack Peirce’s above quotation:
1. 1 represents the universe class and 0 the null class. (Boolean symbols)
2. “\(\unary{h}\bcomma(1-\unary{b})\)” denotes the class of non-black horses. (Multiplication operation for intersection)
3. “\(\unary{h}\bcomma(1-\unary{b})=0\)” says that there is nothing that is non-black horse. I.e., every horse is black. (by 1 and 2)
4. “\(\unary{h}\bcomma(1-\unary{b}) \not = 0\)” means it is not the case that every horse is black. I.e., some horse is not black. (by 3)
5. Since \(\unary{h}\bcomma (1-\unary{b}) \not = 0, 0^{\unary{h},(1-\unary{b})} = 0\) (by (*) above)
Similarly, “\(\unary{h}\bcomma\unary{b}=0\)” means no horse is black. Hence, “\(\unary{h}\bcomma \unary{b} \not = 0\)” says some horse is black. Therefore, “\(0^{\unary{h}\bcomma\unary{b}} = 0\)”
(being the exponent being not-zero) means that some horse is black.
The method presented in DNLR [CP 3.141] is interesting for several reasons. First of all, Peirce maintains Boole’s theme that all propositions are represented as equations. Another is that Peirce
utilizes the contradictory relation between universal and existential propositions. Even more interestingly and importantly, Peirce brings in his exponentiation notation, exponentiation between
relation to property (that is, between binary to unary predicates), to express existential statements.
After the rather complicated presentation of existential statements, focusing on the exponent part, Peirce suggests another, much simpler, way to express existential statements by using the
inequality sign:
Particular [Existential] propositions may also be expressed by means of the signs of inequality. Thus, some animals are horses, may be written \(\unary{a}\bcomma\unary{h} > 0\). (DNLR [CP 3.143])
Another method Peirce adopts is to utilize the sign \(\inclusion\) for inclusion and the sign \(\bar{ \ \ }\) for complement. That is,
All \(a\) is \(b\). \(a \inclusion b\)
No \(a\) is \(b\). \(a \inclusion \bar{b}\)
Some \(a\) is \(b\). \(\overline{[a \inclusion \bar{b}]}\)
Some \(a\) is not \(b\). \(\overline{[a \inclusion b]}\)
However, these explicit ways to handle existential statements are limited to unary predicates or Aristotelian syllogisms, and cannot go beyond them. Instead, we would like to examine Peirce’s
multiplication and exponential expressions between relations and properties more carefully. Let’s recall \(\binary{l}\unary{w}\) means “a lover of some woman” while \(\binary{l}^{\unary{w}}\) “a
lover of every woman”. Here Peirce’s proposal for individual terms gets in the picture so that existential and universal quantifiers may become more explicitly represented. Peirce suggests
individuals be denoted by capitals (DNLR [CP 3.96]). For example, for unary predicate \(\unary{w}\), if \(\unary{w} > 0\), then \(\unary{w} = \unary{W}' \cunion \unary{W}'' \cunion \unary{W}''' \
cunion \cdots\), where each of \(\unary{W}'\), \(\unary{W}''\), \(\unary{W}'''\),…denotes an individual woman. Hence,
\[ \binary{l}\unary{w} & = \binary{l}(\unary{W}' \cunion \unary{W}'' \cunion \unary{W}''' \cunion \cdots)\\ &= \binary{l}\unary{W}' \cunion \binary{l}\unary{W}'' \cunion \binary{l}\unary{W}''' \
cunion \cdots\\ \binary{l}^{\unary{w}} &= \binary{l}^{(\unary{W} ' \cunion \unary{W}'' \cunion \unary{W}''' \cunion \cdots)}\\ & = \binary{l}^{\unary{W}'}, \binary{l}^{\unary{W}''}, \binary{l}^{\
unary{W}'''},\ldots\\ & = \binary{l}\unary{W}', \binary{l}\unary{W}'', \binary{l}\unary{W}''',\ldots (\binary{l}^{\unary{W}}=\binary{l}\unary{W}, \unary{W} \textrm{ being an individual term.}) \]
At this point, Peirce makes a connection (i) between existential statements and the sign \(\Sigma\) (as logical addition) and (ii) between universal statements and \(\Pi\) (as logical
multiplication), and the following passage is a precursor for the notations which appear in subsequent papers:
\[\Pi' \inclusion \Sigma',\]
where \(\Pi'\) and \(\Sigma'\) signify that the addition and the multiplication with commas are to be used. From this it follows that
\[s^{\unary{w}} \inclusion s\unary{w}. \qquad \textrm{(DNLR [CP 3.97])}\]
We have now come quite close to a modern notation of quantifiers. Dipert emphasizes the significance of Peirce’s notations found in DNLR:
C. S. Peirce was the first person in the history of logic to use quantifier-like variable binding operators (briefly in 1870, W2, 392f, predating Frege’s Begriffsschrift (1879)). (Dipert 2004:
Ten years later, in a more comprehensive and more survey-style paper, “On the algebra of logic” (1880), we find Peirce’s ideas of DNLR emerge more tightly and more systematically. Below we will now
summarize the developments of quantification presented in “Brief description of the algebra of relatives” (1882a), “The logic of relatives” (1883a), and “On the algebra of logic: A contribution to
the philosophy of notations” (1885a plus 1885b).
First, he modifies the previous idea of representing a property in terms of individual terms and extends it to relations. In DNLR (1870), “\(\unary{w}\)”, which stands for the property “being a
woman”, is expressed as
\[\unary{w} = \unary{W}' + \unary{W}'' + \unary{W}''' + \cdots\]
(where \(\unary{W}'\),… denotes each individual woman). However, in 1882a, Peirce expresses a unary predicate using coefficients: For each unary predicate \(x\) and for each object \(a\) in the
domain, Peirce defines the coefficient \((x)_a\) in the following way.
Continuing the example for the unary predicate \(\unary{w},\) suppose the domain has objects \(\unary{A},\) \(\unary{B},\) \(\unary{C},\)….^[14] Some object is a woman and some is not. A coefficient
\((\unary{w})_a\) is defined as follows:
\[ (\unary{w})_{a} A & = 1 \textrm{ if } A \textrm{ is woman.}\\ & = 0 \textrm{ if } A \textrm{ is not a woman.}\\ \]
\[ \unary{w} & = (w)_{a} A + (w)_{b} B+ (w)_{c} C + \cdots\\ & = \Sigma_{i} (w)_{i} \binary{I}.\\ \]
A unary predicate is successfully represented as a sum of individuals. Moving on to a binary predicate, a relation is modeled by a pair of objects:^[15]
A dual relative term [binary predicate], such as “lover”, “benefactor”, “servant”, is a common name signifying a pair of objects. (1883a [CP 3.328])
And he expresses a pair of objects as “\(A:B\)”, where \(A\) and \(B\) are individual objects. Let \(\binary{l}\) stand for “a lover”. Peirce defines a coefficient for each ordered pair of objects in
the following way:
\[ (\binary{l})_{i, j} (A_i: A_j) & = 1 \textrm{ if } A_i \textrm{ is a lover of } A_j.\\ & = 0 \textrm{ if } A_i \textrm{ is not a lover of } A_j.\\ \]
\[ \binary{l} & = (l)_{1,1}(A_{1}: A_{1}) + (l)_{1,2}(A_{1}: A_{2}) + (l)_{2,1}(A_{2}: A_{1}) + (l)_{2,2}(A_{2}: A_{2}) + \cdots\\ & = \Sigma_{i} \Sigma_{j}(l)_{i,j}(A_i: A_j).\\ \]
Peirce’s generalization for a unary predicate \(\unary{x}\) and a binary predicate \(\binary{l}\) goes like this (1883a [CP 3.329]):
\[ \unary{x} & = \Sigma_{i} (\unary{x})_i \binary{I} & \textrm{ (1882a [CP 3.306])}\\ \binary{l} & = \Sigma_{i} \Sigma_{j}(\binary{l})_{i,j}(I: J) &\textrm{ (1883a [CP 3.329])}\\ \]
This is what Peirce had in mind when he wrote “Every term [predicate] may be conceived as a limitless logical sum of individuals” (1880 [CP 3.217]). We take only objects whose coefficients are 1.
Suppose \(A\), \(B\), and \(D\) are women. Applying Peirce’s sign “+” as inclusive-or, \(\unary{w} = \{ A, B, D\}\). Suppose \(A\) is a lover of \(C\), and \(B\) is a lover of \(D\). Then, \(l = \{(
A: C), (B: D)\}\).^[16]
The next task is how to utilize this tool to express an existential proposition. If there is at least one woman, say \(K\), in the domain, there is at least one coefficient \(\unary{w}_k\) such that
it is 1. Hence, the sum of coefficients of the individuals in the domain is greater than 0. That is, \(\Sigma_{i} w_i > 0\). If nobody is a woman, we will get \(\Sigma_{i} w_i = 0\). In the case of
the universal proposition that everybody is a woman, the product of coefficients is 0 as long as one coefficient is 0, that is, there is one person who is not a woman. That is, \(\Pi_{i} w_i = 0\).
When everybody is a woman, \(\Pi _{i} w_i = 1\).
While the Boolean approach for quantifiers is limited to term logic, this way of handling quantifiers is so general that we can apply it directly to relations as well, as the following passage
Any proposition whatever is equivalent to saying that some complexus of aggregates [sums] and products of such numerical coefficients is greater than zero. Thus,
\[\Sigma_{i}\Sigma_{j} \binary{l}_{ij} > 0\]
means that something is a lover of something; and
\[\Pi_{i}\Sigma_{j} \binary{l}_{ij} > 0.\]
means that everything is a lover of something. (1883a [CP 3.351])
And Peirce proposes to drop the \(“>0”\) part:
We shall, however, naturally omit, in writing the inequality, the \(“>0”\) which terminates them all; and the above two propositions will appear as
\(\Sigma_{i}\Sigma_{j} \binary{l}_{ij}\) and \(\Pi_{i}\Sigma_{j} \binary{l}_{ij}.\qquad\) (1883a [CP 3.351])
Getting to Peirce’s 1883 paper, we witness two significant steps taken: One is the use of index-subscripts in a crucial way, and the other is the alternation of \(\Sigma\) and \(\Pi\). After dropping
“\(>0\)” Peirce draws our attention to the role of subscripts:
The following are other examples:
\[\Pi_i \Sigma_j (l)_{ij} (b)_{ij}\]
means that everything is at once a lover and a benefactor of something.
\[\Pi_i \Sigma_j (l)_{ij} (b)_{ji}\]
means that everything is a lover of a benefactor of itself. (1883a [CP 3.352])
The order of index-subscripts is crucial to make a distinction between these two propositions. On the other hand, a difference between “Everybody loves some woman” and “There is a woman everybody
loves” relies on the order of \(\Pi\) and \(\Sigma\):
\[\Pi_i \Sigma_j (\binary{l})_{ij} ({\unary{w}})_j \quad \textrm{ vs.}\quad \Sigma_j \Pi_i (\binary{l})_{ij }({\unary{w}})_j.\]
Finally in “On the Algebra of Logic: A contribution to the philosophy of notation” (1885a) all of these developments—(i) \(\Sigma\) (sum) for some and \(\Pi\) (product) for all, (ii) utilizing
coefficients and their subscripts to drop denotation of individuals (e.g., from \((l)_{ij} (A_{i}: A_{j})\) to \((l)_{ij}\)), (iii) mixing \(\Sigma\) and \(\Pi\), and (iv) omitting the “\(>0\)”
part—have become official:
In general, according to 1885a [CP 3.393]:
\(\Sigma_i x_i\) means that \(x\) is true of some one of the individuals noted by \(i\) or
\[\Sigma_{i}x_i = x_i + x_j + x_k + \textit{etc.}\]
In the same way, \(\Pi_i x_i\) means that \(x\) is true of all these individuals, or
\[\Pi_{i}x_i = x_i x_j x_k, \textit{etc.}\]
And if \(x\) is a simple relation [binary predicate],
□ \(\Pi_i\Pi_j x_{i,j}\) means that every \(i\) is in this relation to every \(j\),
□ \(\Pi_j\Sigma_i x_{i,j}\) that every \(j\) some \(i\) or other is in this relation,
□ \(\Sigma_i\Sigma_j x_{i,j}\) that some \(i\) is in this relation to some \(j\).
For example, according to 1885a [CP 3.394]:
Let \(l_{ij}\) mean that \(i\) is a lover of \(j\) and \(b_{ij}\) that \(i\) is a benefactor of \(j\).
Let \(g_i\) mean that \(i\) is a griffin, and \(c_i\) that \(i\) is a chimera.
Then \(\Pi_i\Sigma_j (l)_{ij }(b)_{ij}\) means that everything is at once a lover and a benefactor of something; and
□ [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land \textit{Benefactor} (x, y)]\), meaning that everybody is a lover and a benefactor of someone.]
And \(\Pi_i\Sigma_j (l)_{ij}(b)_{ji}\) means that everything is a lover of a benefactor of itself.
□ [I.e., \(\forall x \exists y [\textit{Lover} (x, y) \land \textit{Benefactor} (y, x)]\), meaning that everybody is a lover of a benefactor of himself/herself.]
And we find ourselves arriving at the land of modern logic. At the same time, we realize that the key concepts and vocabulary of first-order logic were already formed in his previous work discussed
above. We also note that the goal outlined in the first section of the 1885 paper is more or less the same as the proposal made in his 1870 paper: “The first is the extension of the power of logical
algebra over the whole of its proper realm” (1885a [CP 3.364]). Also, he almost reiterates the limit of the project which he “regrets” in 1870: “I shall not be able to perfect the algebra
sufficiently to give facile methods of reaching logical conclusions” (1885a [CP 3.364]). That is, we should not expect a full-blown deductive system in this paper, but “I can only give a method by
which any legitimate conclusion may be reached and any fallacious one avoided” (1885a [CP 3.364]). He carries out his promise in section 3 of the paper titled as “§3. First-intentional logic of
relatives”,^[17] by suggesting a list of methods of transformation. He did not mean to claim this list is exhaustive, but “the one which seems to me the most useful tool on the whole” (1885a [CP
3.396]). The following are some of the rules involving quantifiers:^[18]
\[ \forall x \phi(x) \land \forall y \phi(y) & = \forall x\forall y (\phi(x) \land \phi(y))\\ \exists x \phi(x) \land \forall y \phi(y) & = \exists x\forall y (\phi(x) \land \phi(y))\\ \exists x \phi
(x) \land \exists y \phi(y) & = \exists x\exists y (\phi(x) \land \phi(y))\\ \] \[ \forall x \forall y \chi(x,y) & = \forall y\forall x \chi(x,y)\\ \exists x \exists y \chi(x,y) & = \exists y\exists
x \chi(x,y)\\ \forall x \exists y (\phi(x) \land \psi(y)) & = \exists y\forall x (\phi(x) \land \psi(y))\\ \forall x \exists y \chi(x,y) & \not = \exists y\forall x \chi(x,y), \textrm{ but}\\ \exists
x\forall y \chi(x,y) & \Rightarrow \forall y \exists x \chi(x,y)\\ \] \[\exists x \forall y \chi(x,y) = \exists x \forall y (\chi(x,y)\land \chi(x,x))\]
In spite of the six-year interval between Frege’s Begriffsschrift (1879) and Peirce’s quantification theory in 1885, credit has been given to both logicians. We call them both the founders of modern
logic, since Peirce was not aware of Frege’s work on the topic. Also, it should be noted that Frege presented a logical system equipped with axioms and rules, which was not pursued in Peirce’s work.
1.2 Boolean tradition—algebraic and model-theoretic
Boole’s aspiration to capture logic in terms of an algebraic system has inspired many mathematicians and logicians who are interested in connecting the two disciplines, logic and mathematics. As seen
in the previous section, Peirce is clearly one of them. He pushed the Boolean idea of an algebraic system further in two ways: One is to improve Boole’s representation of particular proposition
(i.e., “Some A is B”) so that traditional Aristotelian syllogisms may fit in an algebraic system. The other is to represent not only qualities but also relations so that a new algebraic system may
reach beyond traditional syllogisms. In that process, new notations for quantifiers and variables were invented.
Peirce’s two-decade work made a major contribution to the algebra of logic tradition in two important ways. First, Peirce’s introduction of quantifiers and variables itself is a significant advance
in formal logic, close to the predicate logic we know. Second, subsequent momentous work on mathematical logic was built on the new notation and extended logic by Peirce and his students, O. H.
Mitchell and C. Ladd (later Ladd-Franklin). A decade after Peirce’s “On the Algebra of Logic”, (1885) Ernst Schröder published three volumes of mathematical logic Vorlesungen über die Algebra der
Logik (1890–1905).^[19] His work was carried out squarely in the Boolean algebraic tradition, and two important aspects of the book reflect Peirce’s influence: He adopted Peirce’s notation (over
Frege’s), and the third volume is devoted to the logic of relations. Goldfarb’s insightful paper expresses these two aspects of Peirce’s influence in the following way:
Building on earlier work of Peirce, in the third volume of his Lectures on the algebra of logic [1895] Schröder develops the calculus of relatives (that is, relations). Quantifiers are defined as
certain possibly infinite sums and products, over individuals or other relations. (Goldfarb 1979: 354)
As seen in the previous subsection, including relations (not just qualities) in algebraic expressions and representing the universal quantifier as products (i.e., \(\Pi\)) and the existential
quantifier as sums (i.e., \(\Sigma\)) were the main output of Peirce’s two decade tireless work. Considering Schröder’s book was the most popular logic textbook for mathematical logic students during
that era, we can easily say Peirce’s legacy has lived on. Peckhaus, working on a delicate relation between Frege’s and Schröder’s quantification theory, locates where Schröder’s modern quantification
theory originates from:
This [Schröder’s Vorlesungen über die Algebra der Logik was the result of his learning modern quantification from Frege’s Begriffsschrift] is a simple and plausible answer, but it is false.
Schröder never claimed any priority for his quantification theory, but he did not take it from Frege. Schröder himself gives the credit for his use of \(\Sigma\) and \(\Pi\) to Charles S. Peirce
and Peirce’s student Oscar Howard Mitchell (Schröder 1891, 120–121). (Peckhaus 2004: 12)
Afterwards well-known mathematicians and logicians, Löwenheim, Skolem and Zermelo, all used Peirce-Schröder notation. Peano was also very much familiar with Peirce-Schröder algebraic logic. Putnam
includes Whitehead in this tradition as well:
This [Whitehead’s Universal Algebra] is a work squarely in the tradition to which Boole, Schröder, and Peirce belonged, the tradition that treated general algebra and logic as virtually one
subject. (Putnam 1982: 298)
Interestingly enough, Putnam points out that this portion of Whitehead’s work was prior to his collaboration with Russell, and during this early period Whitehead’s work, especially on quantifiers,
mentions Peirce and his students, but not Frege. Clearly we needed to wait until Russell drew our attention to Frege, but “it was Peirce who seems to have been known to the entire world logical
community” (Putnam 1982: 297). Putnam’s label of the Peirce group as “effective” discoverers of the quantifier and Frege as a discoverer could be a resolution to the Frege-first versus Peirce-first
While many have focused on the development of quantifiers, it is quite noteworthy that Tarski drew our attention to the importance of the algebra of relations in his 1941 paper “On the Calculus of
Relations”.^[20] Just as Peirce did in his DNLR (1870) paper, Tarski acknowledges De Morgan’s contribution: It is De Morgan who first realized the necessity of representing relations as well as
qualities and struggled over the limits of traditional logic. And Tarski gives full credit to Peirce in terms of solid advance over the calculus of relations.
The title of creator of the theory of relations was reserved for C. S. Peirce. In several papers published between 1870 and 1882a, he introduced and made precise all the fundamental concepts of
the theory of relations and formulated and established its fundamental laws.…In particular, his investigations made it clear that a large part of the theory of relations can be presented as a
calculus which is formally much like the calculus of classes developed by G. Boole and W. S. Jevons, but which greatly exceeds it in richness of expression and is therefore incomparably more
interesting from the deductive point of view. (Tarski 1941: 73)
This passage not only situates Peirce’s logical achievement in the context of the algebra of logic tradition but also characterizes Peirce’s work as an extension of Boole’s and Jevons’ monadic logic.
(For more details about Peirce’s position in Boole’s tradition, see the entry the algebra of logic tradition.)
Some Peirce scholars have also claimed that Peirce’s invention of quantifiers is a product of Peirce’s own philosophy of logic, which is different from Frege’s (Brady 1997; Burch 1997; Iliff 1997;
Merrill 1997). Hintikka’s proposal (1997) to explain the main difference between Frege’s and Peirce’s contributions to modern logic is quite intriguing. Tracing back to Frege’s own distinction
between calculus ratiocinator versus a lingua characterica, van Heijenoort adds a new dimension to these two opposing views of logic beyond what Frege alluded to (van Heijenoort 1967: footnote 1, p.
329).^[21] While Frege emphasized a difference between propositional and quantification logic, van Heijenoort located a difference in what is taken as a totality. Boole’s tradition does not make any
ontological commitment about a totality, but it “can be changed at will” (1967: 325). On the other hand, Frege’s language is about the universe. Borrowing van Heijenoort’s distinction between Boole’s
logic as a calculus and Frege’s universality of logic, Hintikka locates Peirce in Boole’s camp, calling it the model-theoretic tradition. Unlike Frege’s view of the universe, the model-theoretic
tradition allows us to reinterpret a language and thus assign different universes to quantifiers. According to Hintikka, Peirce’s development of modal logic is a good piece of evidence to show how
fruitful Peirce’s way of understanding quantifiers could be (Hintikka 1997). In the next section where Peirce’s graphical systems are introduced, we will revisit this issue.
2. From Symbolic to Iconic Representation
So far, we have argued that Peirce’s insight on relations pushed him to extend the territory of logic from monadic, non-relational, propositional logic to polyadic, relational, quantification logic.
This is the beginning of modern logic as we know it. In this section, taking up a different angle of Peirce’s adventure—to extend forms of representation from symbolic systems to diagrammatic
systems, we present a story where his two different kinds of extension—one from non-relations to relations and the other from symbolic to diagrammatic—are connected with each other.
Peirce presented propositional logic, quantification logic, and modal logic in a graphical way, and invented three systems of Existential Graphs (EG)—Alpha, Beta, and Gamma, respectively. In spite of
Peirce’s own evaluation of Existential Graphs as “my chef d’oeuvre”, EG had to wait to be understood for a half century until two philosophers—Don Roberts and Jay Zeman—produced their impressive
work. In the 1980s, EG was receiving attention from new disciplines—computer science and artificial intelligence—thanks to John Sowa’s novel application of EG to knowledge representation in
Conceptual Structure (1984). More recently, toward the end of the twentieth century interdisciplinary research on multi-modal reasoning has drawn our attention to non-symbolic systems (see, e.g.,
Barwise & Allwein [eds] 1996 and Barwise & Etchemendy 1991) and EG, not surprisingly, occupied the top of their list. In that context, Shin (2002) focused on differences between symbolic versus
diagrammatic systems and suggested a new way of understanding the EG system, though this was criticized in Pietarinen 2006.
While Peirce mainly presented linear expressions in his official writings from 1870 to 1885,^[22] the notation adopted in Frege’s 1879 Begriffsschrift is more iconic; it is at least not as linear as
Peirce’s in the above period. However, it is Peirce, not Frege, who invented a full-blown non-symbolic system for first-order logic—Existential Graphs. It is Peirce’s EG, not his linear first-order
notation, which is presented as a deductive system with inference rules. As the EG system has been investigated more rigorously, philosophical questions involving Peirce’s invention of the system
have been raised as well. The discovery of EG’s power and novelty has naturally led us to other parts of Peirce’s philosophy. Why and how did the invention of EG come about? What does EG reveal about
Peirce’s view of logic and representation?
Many have pointed to Peirce’s theory of signs, which classifies signs as being of three kinds—symbols, indices, and icons—as the foremost theoretical background for Peirce’s EG.^[23] For example, as
will be shown below, ovals and lines, along with letters, are the basic vocabulary of Peirce’s EG. It is natural to connect Peirce’s interest in icons with his invention of graphical systems, and the
connection is real (Shin 2002: 22–35). However, to pinpoint the features of icons and the iconic nature of Peirce’s graphical systems requires much more work than our intuition provides. Moreover,
there is a big gap between Peirce’s discussions of icons^[24] and his invention of full-blown graphical systems; something else has to be brought into the picture to explain how Peirce got from his
initial ideas about icons all the way to his EG.
In a slightly different and bigger picture, van Heijenoort’s distinction between Boole’s calculus ratiocinator versus Frege’s lingua characteristica could be related to the topic. Agreeing with both
Hintikka’s and Goldfarb’s evaluation that Peirce belongs to Boole’s tradition, Shin finds a connection between the model-theoretic view of logic (where Boole and Peirce are placed) and EG’s birth
(see Shin 2002: 14–16 and Pietarinen 2006). However, Peirce’s awareness of the re-interpretation of language is necessary, but not sufficient, for his pursuit of a different form of representation.
While the acknowledgment of the possibility of different models of a given system was presupposed by Peirce’s project for various kinds of systems, not every Boolean has presented multiple systems.
Boole himself
was quite conscious of the idea of disinterpretaion, of the idea of using a mathematical system as an algorithm, transforming the signs purely mechanically without any reliance on meanings.
(Putnam 1982: 294)
On the other hand, Burris and Legris’s entry shows us how Boole’s algebra of logic tradition has led us to the development of model theory (see the entry on the algebra of logic tradition).
2.1 Pragmatic maxim applied to the logic of relations
Without challenging these existing explanations involving Peirce’s EG, in this entry we would like to bring in one overlooked but crucial aspect of Peirce’s journey to EG so that our story may fill
in part of the puzzle of Peirce’s overall philosophy. Peirce’s mission for a new logic started with how to represent relations, which led him to invent quantifiers and bound variables, as we
discussed in the previous section. The same commitment, that is, to represent relations in a logical system, we claim, was a main motivation behind Peirce’s search for a new kind of sign
systems—iconic representation of relations. Peirce’s work on Euler/Venn diagrams provides us with another piece of evidence to support our claim that the main motivation behind EG was to represent
relations. While improving Venn systems, Peirce realizes that the following defect cannot be eliminated:
[T]he system [Venn’s] affords no means of exhibiting reasoning, the gist of which is of a relational or abstractional kind. It does not extend to the logic of relatives. (Peirce 1911b [CP 4.356])
Again, we do not think this is the crucial ingredient for the creation of EG, but one key element which works nicely together with his theory of signs and his model-theoretic view of logic.
Peirce’s graphical representation first appears in his 1897 paper “The Logic of Relatives”. After his own new linear notation came out in 1885 as seen above, why did Peirce revisit the logic of
relations? The first paragraph of the paper provides a direct answer:
I desire to convey some idea of what the new logic is, how two “algebras”, that is, systems of diagrammatical representation by means of letters and other characters, more or less analogous to
those of the algebra of arithmetic, have been invented for the study of the logic of relatives, and…. (1897a [CP 3.456])
Two things should be noted. One is that diagrammatic systems are also called “algebra” by Peirce. That is, according to Peirce, algebra is not limited to symbolic systems. The other is that Peirce
makes it clear that two different forms of algebra carry out the new logic, not new logics.
In thinking about the scope of the logic of relations, the question arises: Why did Pierce feel the need for another form of representation different from the 1885 notation? “I must clearly show what
a relation is” (1897a [CP 3.456]). The clear understanding of “relations”, Peirce believes, is a guide for his excursion into different forms of logical systems. Here we would like to draw reader’s
attention to Peirce’s well-known paper “How To Make Our Ideas Clear ” (1878), where three sections are devoted to the three grades of meaning (see the entry on Peirce’s theory of signs).
The first grade of understanding the word “relation” comes from our ordinary experience, and the second grade is to have a more abstract and general definition-like understanding. According to
Peirce, that is not enough to achieve a full understanding of the word “relation”. Finally, Peirce’s hallmark of the pragmatic maxim leads us to the third grade of clarity:
It appears, then, the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, which might conceivably have practical bearings, we conceive the object
of our conception to have. Then the whole of our conception of those effects is the whole of our conception of the object. (1878 [CP 5.402])
In order to understand what a relation is, we need to know what follows from it. Then, the question is how we know what its consequences are. Here is one answer given by Peirce in the 1897a paper, as
far as the term “relation” goes:
The third grade of clearness consists in such a representation of the idea that fruitful reasoning can be made to turn upon it, and that it can be applied to the resolution of difficult practical
problems. (1897a [CP 3.457])
Therefore, how a relation is represented is crucial in figuring out what follows from a relational state of affairs. Better representations will yield more “fruitful reasoning” and hence, will be
more helpful for solving practical problems. It is obvious that in the paper Peirce intends to search for more desirable representations. Importantly, in section 4 when the third grade of clearness
of the meaning “relation” is discussed, diagrammatic representation of relations makes its first appearance.
Influenced by A. B. Kempe’s graphic representation,^[25] Peirce finds an analogy between relations and chemical compounds:
A chemical atom is quite like a relative in having a definite number of loose ends or “unsaturated bonds”, corresponding to the blanks of the relative. (1897a [CP 3.469])
A chemical molecule consists of chemical atoms, and how atoms are connected with one another is based on the number of loose ends of each atom. For example, chemical atom H has one loose end and
chemical atom O has two. So, the following combination is possible, and it is a representation of the water molecule, H[2]O:
An analogy to the logic of relations runs like this: A sentence consists of names (proper names or indices) and predicates, and each predicate has a fixed arity. For example, the predicate “love”
needs two names and “give” three. Hence, the following diagrammatic representation is grammatical and it is a representation of the proposition “John loves Mary”.
Peirce created a novel and productive analogy in representation between chemistry and the logic of relation by adopting the doctrine of valency as the key element for the analogy, as shown in the
above two diagrams. Believing that this graphic style of representation would help us conceive the consequences or effects of a given relation in a more efficient way,^[26] Peirce presents Entitative
Graphs, which is a predecessor of EG.^[27]
EG keeps the representation of a relation developed here, and remains as Peirce’s final and the most cherished notation for the logic of relations (1903a). EG consists of three parts, Alpha, Beta,
and Gamma, which correspond to propositional, first-order, and modal logic, respectively. After presenting the Alpha system in a formal way, we discuss the Beta system of EG focusing on Peirce’s
novel ideas in expanding a propositional graphic system to a quantificational graphic system. For more details, we recommend works on EG by Roberts, Zeman, Sowa, and Shin.
2.2 Alpha system
Peirce’s Alpha graphs may be drawn on a blackboard, on a whiteboard, or on a sheet of paper. The basic unit is a simple sentence without any sentential connectives, that is, negation, conjunction,
disjunction or conditional, etc. The following is an example of a basic Alpha graph, asserting that it is sunny.
When we would like to assert that it is sunny and windy, we juxtapose two basic Alpha graphs in the following way:
In order to make the Alpha Graph Boolean-functionally complete, all we need is to represent negation. The following Alpha graph says that it is not the case it is sunny, by enclosing the above graph
with a cut:
When we have negation and conjunction, it is important to keep the order in the right way. “It is not sunny and it is not windy” is different from “It is not the case that it is sunny and windy”.
Hence, the sentence “It is not the case that it is sunny and it is windy” is ambiguous, depending on the scope of “it is not the case”. In the case of sentential logic, parentheses get in to prevent
this ambiguity: \(\neg (S \land W)\) versus \(\neg S \land W\). Peirce’s warning follows:
The interpretation of existential graphs is endoporeutic, that is proceeds inwardly; so that a nest sucks the meaning from without inwards unto its centre, as a sponge absorbs water. (Peirce
1910a: 18, Ms 650)
Hence, the following Alpha graph should be read not as “\(\neg P \land \neg Q\)”, but as “\(\neg(P \land \neg Q)\)”:
This way of understanding Alpha Graphs is not incorrect, but has given a wrong impression that the Alpha system is equivalent to a sentential system with two connective symbols, negation and
conjunction. We all prefer having more connectives than these two, especially when we use the language. The section explores an alternative reading of Alpha diagrams, beyond negation and conjunction
only, without introducing any new syntactic device.
Below we introduce Alpha Graphs as a formal system equipped with its syntax and semantics. These tools not being available to Peirce, the presentation aims to show Peirce’s EG is not intrinsically
different from other formal systems. At the same time, in order to place Peirce’s graphical systems in the traditional well-developed discourse of logic, there will be an intermediate stage, that is,
to read off Peirce’s graphs into symbolic language. This will make Peirce’s graphs more accessible, and at the same time support our claim that Peirce extended forms of representations with the same
scope of logic as symbolic representation.
1. Sentence symbols: \(A_{1},\) \(A_{2},\)…
2. Cut
Well-formed diagrams
1. An empty space is a well-formed diagram.
2. A sentence symbol is a well-formed diagram.
3. If \(D\) is a well-formed diagram, then so is a single cut of \(D\) (we write “\([D]\)]”).
4. If \(D_{1}\) and \(D_{2}\) are well-formed diagrams, then so is the juxtaposition of \(D_{1}\) and \(D_{2}\) (write “\(D_{1}\ D_{2}\)”).
5. Nothing else is a well-formed diagram.
Here we present two equivalent reading methods for the system. The Endoporeutic reading algorithm, formalized based on Peirce’s own suggestion (as quoted above), is a traditional way to understand
EG. An alternative reading method, the Multiple reading algorithm, was more recently presented to approach EG in a more efficient way.^[28]
Endoporeutic Reading Algorithm
1. If \(D\) is an empty space, then it is translated into \(\top\).
2. If \(D\) is a sentence letter, say \(A_{i}\), then it is translated into \(A_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then, \([D]\) is translated into \((\neg \alpha)\).
4. Suppose the translation of \(D_{1}\) is \(\alpha_{1}\) and the translation of \(D_{2}\) is \(\alpha_{2}\).
Then, the translation of \(D_{1}\ D_{2}\) is \((\alpha_{1} \land \alpha_{2})\).
Multiple Readings Algorithm
1. If \(D\) is an empty space, then it is translated into \(\top\).
2. If \(D\) is a sentence letter, say \(A_{i}\), then it is translated into \(A_{i}\).
3. Suppose the translation of \(D\) is \(\alpha\). Then, \([D]\) is translated into \((\neg \alpha)\).
4. Suppose the translation of \(D_{1}\) is \(\alpha_{1}\) and the translation of \(D_{2}\) is \(\alpha_{2}\).
a. the translation of \(D_{1} D_{2}\) is \((\alpha_{1} \land \alpha_{2})\),
b. the translation of \([D_{1} D_{2}]\) is \((\neg\alpha_{1} \lor \neg \alpha_{2})\),
c. the translation of \([D_{1}\ [D_{2}]]\) is \((\alpha_{1} \rightarrow \alpha_{2})\), and
d. the translation of \([[D_{1}]\ [D_{2}]]\) is \((\alpha_{1} \lor \alpha_{2})\).
Each of these two readings has its own strength.^[29] The Endopreutic reading assures us that the Alpha system is truth-functionally complete, since it has power to express conjunction and negation.
However, this traditional method has been partly responsible for the following two incorrect judgments about Alpha graphs:
i. There is not much difference between the Alpha system and a propositional language with only two connectives, \(\land\) and \(\neg\), except that Alpha graphs have cuts instead of symbolic
ii. When it comes down to practical use, just as we do not want to use only two connectives in a language, we have no reason to adopt the Alpha system over propositional languages with more
Challenging these misconceptions, the Multiple readings algorithm shows that Alpha diagrams do not have to be read off as a sentence with “\(\land\)” and “\(\neg\)” only, but can be directly read off
in terms of other connectives as well. Two questions may be raised:
i. Is there a redundancy in the Multiple readings method? For example, is clause 4(b) above dispensable in terms of clause 3 and clause 4(a)?
ii. Does this new reading show that the Alpha system is just like a propositional language with various connectives?
Let us answer these questions through the following example.
The following graph is translated into the following four formulas:
The Endoporeutic reading allows us to get the first reading only, but we may obtain different sentences by the Multiple Readings. Of course, all of these sentences are logically equivalent. Here is
an interesting point: In the case of symbolic systems, we need to prove the equivalence among the above sentences by using inference rules. But, derivation processes are dispensable in the case of
the Alpha system when the Multiple readings are adopted.^[30] Hence, having the clause 4(b) above in addition to clause 3 and clause 4(a) is not redundant, but instead highlights a fundamental
difference between the Alpha system and a symbolic language with various connectives (see Shin 2002: §§4.3.2, 4.4.4, and 4.5.3).
Since we have the semantics for propositional logic and our reading methods translate Alpha diagrams into a propositional language, we can live without the direct semantics. However, if one insists
on the direct semantics:
Let \(v\) be a truth function such that it assigns t or f to each sentence letter and t to an empty space. Now, we extend this function to \(\overline{v}\) as follows:
1. \(\overline{v}(D) = v(D)\) if \(D\) is a sentence symbol or an empty space.
2. \(\overline{v} ( {\bf [}D{\bf ]})\) = t iff \(\overline{v}(D)\) = f.
3. \(\overline{v}(D_{1} D_{2})\) = t iff \(\overline{v}(D_{1}) =\)t and \(\overline{v}(D_{2})\) = t.
We also would like to emphasize that this is not the only way to approach Peirce’s EG. For example, some claim that game-theoretic semantics were foreshadowed by Peirce, and thus argue for a more
dynamic understanding of EG from the game-theoretic point of view (Burch 1994; Hilpinen 1982; Hintikka 1997; Pietarinen 2006).
Peirce makes it clear his EG is a deductive system equipped with inference rules:
The System of Existential Graphs is a certain class of diagrams upon which it is permitted to operate certain transformations. (1903a [CP 4.414])
The inference rules for the Alpha system are presented as follows: (1903a [CP 4.415])^[31]
Code of Permissions
• Permission No. 1. In each special problem such graphs may be scribed on the sheet of assertion as the conditions of the special problem may warrant.
• Permission No. 2. Any graph on the sheet of assertion may be erased, except an enclosure with its area entirely blank.
• Permission No. 3. Whatever graph it is permitted to scribe on the sheet of assertion, it is permitted to scribe on any unoccupied part of the sheet of assertion, regardless of what is already on
the sheet of assertion.
• Permission No. 4. Any graph which is scribed on the inner area of a double cut on the sheet of assertion may be scribed on the sheet of assertion.
• Permission No. 5. A double cut may be drawn on the sheet of assertion; and any graph that is scribed on the sheet of assertion may be scribed on the inner area of any double cut on the sheet of
• Permission No. 6. The reverse of any transformation that would be permissible on the sheet of assertion is permissible on the area of any cut that is upon the sheet of assertion.
• Permission No. 7. Whenever we are permitted to scribe any graph we like upon the sheet of assertion, we are authorized to declare that the conditions of the special problem are absurd.
Emphasizing the symmetry both in erasure versus insertion and in even versus odd number of cuts, Shin rewrote the rules (Shin 2002: 84–85):
Reformulated Transformation Rules
1. RR1: In an E-area,^[32] say, area \(a\),
a. we may erase any graph, and
b. we may draw graph \(X\), if there is a token of \(X\)
i. in the same area, i.e., area \(a\), or
ii. in the next-outer area from area \(a\).
2. RR2: In an O-area,^[33] say, area \(a\),
a. we may erase graph \(X\), if there is another token of \(X\)
i. in the same area, i.e., area \(a\), or
ii. in the next-outer area from area \(a\), and
b. we may draw any graph.
3. RR3: A double cut may be erased or drawn around any part of a graph.
For examples of deduction sequences, refer to Roberts (1973: 45–46) and Shin (2002: 91).
2.3 Beta system
In §1.1, we showed that formalizing relations was a key motivation behind Peirce’s new logic—first-order logic. In §2.1, we established a connection between Peirce’s own pragmatic maxim and his
graphic representation of relations. Peirce did not aim to present a new logic by inventing a graphic system, but rather to present another new notation for the logic carried out by quantifiers and
bound variables. He almost took it for granted that a graphic representation of relations helps us observe their consequences in a more efficient way. Hence, the Beta system may be considered to be
the final stop of Peirce’s long journey to search for better notation for the logic of relations, which started in 1870 at the latest.^[34]
We will not go into the formal details of the Beta system in this entry but will instead refer to Chapter 5 of Shin, where three slightly different approaches to Beta graphs—Zeman’s, Roberts’, and
Shin’s—are discussed at a full length. While Zeman’s reading is comprehensive and formal, Roberts’ method seems to appeal to a more intuitive understanding of the system. Taking advantage of the
merits of these two existing works, Shin developed a new reading method of Beta graphs and reformulated the transformation rules of the system.^[35] Her approach focuses on visual features of Beta
graphs and highlights fundamental differences between symbolic versus diagrammatic systems. In the remaining part of the entry, we would like to examine how the essence of the logic of relations is
graphically represented in the Beta system so that the reader may place EG in the larger context of Peirce’s enterprise.
The introduction of quantifiers and bound variables is believed to be one of the key steps of first-order logic in symbolic systems. This is why some logicians take Peirce’s 1885 paper “On the
Algebra of Logic: A contribution to the Philosophy of Notations” to be the birthplace of modern logic. If this is the case, then how does Peirce represent quantifiers and bound variables in Beta
Interestingly enough, when Peirce considered a graphic system his first concern was representation of relations, not representation of quantifiers. As we said in §3.1, Peirce presented diagrammatic
representation based on an analogy to chemical molecules for a full understanding of relations. Hence, the arity of a predicate is represented by the number of lines radiating from the predicate
term. Next, Peirce extends the use of a line to connect predicates:
In many reasonings it becomes necessary to write a copulative proposition in which two members relate to the same individual so as to distinguish these members.… [I]t is necessary that the signs
of them should be connected in fact. No way of doing this can be more perfectly iconic than that exemplified in [the following graph]:
(1903b [CP 4.442])
The line connecting two predicates, representing one and the same object, is called a line of identity by Peirce. That is, the sameness is represented visually in Beta diagrams.^[36] In the case of a
symbolic language, we may adopt one and the same quantified variable-type to represent the identity. For example, the above diagram says \(\exists x(x <A \ \land \ B < x)\), and hence, the
variable-type \(x\) (roughly) corresponds to the identity line. However, the same variable-type is not sufficient for expressing the sameness in other cases, e.g., \(\exists x(x <A \ \land \ B < x) \
rightarrow \exists x (x < C)\).
The way universal and existential statements are represented in the Beta system highlights a difference between graphic and symbolic systems. Rather than adopting one more syntactic device for
quantification, Peirce relies on the following visual features:
[A]ny line of identity whose outermost part is evenly enclosed refers to something, and any one whose outermost part is oddly enclosed refers to anything there may be. (1903b [CP 4.458]^[37])
Let us borrow the two following graphs from Roberts (1973: 51):^[38]
The first graph (where the outermost part of the line is evenly, zero, enclosed) says that something good is ugly, and the second graph (where the outermost part is enclosed once) says that
everything good is ugly.^[39]
How about the scope problem which arises when multiple quantifiers are used? In the case of a symbolic system, the linear order takes care of the problem. Peirce’s solution for EG is to read off
another kind of visuality: The less enclosed the outermost part of a line is, the larger the scope that the line gets.
Roberts’ following example illustrates the scope matter nicely (1973: 52):
The first graph says
\[\forall x (\textit{Catholic}(x) \rightarrow \exists y [\textit{Adores} (x,y) \land \textit{Woman}(y)])\]
and the second
\[\exists y (\textit{Woman}(y) \land \forall x [\textit{Catholic}(x) \rightarrow \textit{Adores} (x,y)]).\]
In the first graph, the line whose outermost part is oddly enclosed is less enclosed than the line whose outermost part is evenly enclosed. Therefore, the universal quantifier has larger scope than
the existential quantifier. In the second graph, it is the other way around.
Let us summarize three interesting features of the Beta system:
i. Relations are represented graphically, not symbolically, in the Beta system, in terms of a line. We argued that ultimately Peirce’s pragmatic maxim was behind this alternative way of
ii. A distinction between universal versus existential statements is represented by the visual fact about whether the outermost part of a line lies in an area enclosed either by an odd number or by
even number of cuts.
iii. The order of quantification is represented by the following visuality: The less enclosed a line is, the more extensive scope it has.
3. From Bivalent to Triadic Logic
Fisch and Turquette (1966) discovered three crucial pages out of Peirce’s Logic Notebook (1865–1909, Ms 339).^[40] This shows that Peirce’s invention of three-valued sentential logic predates by at
least a decade Jan Lukasiewicz’s and Emil Post’s achievements on the same topic. The three pages contain the essential elements of triadic logic and an intriguing passage about Peirce’s motivation
behind triadic logic. If we put Peirce’s development of triadic logic in contemporary terms, Peirce seemed to be branching out to non-standard logic. If so, this adventure would be qualitatively
different from the other two we have just discussed in the previous sections.
When Peirce developed relational logic, the territory of formalization was vastly expanded. New vocabulary, hence, new syntactic rules and semantic rules, were added. Naturally we welcome the
territory of formalization and, hence, theoretical justification is not needed. From sentential to relational logic—this is an extension in a literal sense: We do not discard the previous
results—hence they are preserved—but all we do is to expand them.
On the other hand, in the case of extension to non-symbolic languages, the logic itself stays the same, without addition or subtraction, but a new form of representation is introduced. That is, what
to represent is not extended, but how to represent is. Some might not see the need for various forms of representation, and might not be convinced of the necessity of graphical systems. Nonetheless,
at a theoretical level, Peirce’s EG does not demand lengthy theoretical justification. In some sense, the proof is in the pudding: Can this new graphical system carry out the same task as existing
symbolic systems do? If so, which system is easier to use? Which system is more efficient? We might not arrive at a clean consensus, but the discussions are more or less predictable.
However, when one more semantic value is added to T (true) and F (false), logic is not preserved any more. When semantics is extended or changed, the new logic is neither a monotonic expansion of the
territory of logic nor an alternative syntactic form of representation to existing symbolic systems. Triadic logic, by introducing one more semantic value, departs from the standard logic which is
based on bivalence. Here, the status of the principle of excluded middle (“Q or not-Q”) is shaken. So is the law of contradiction. It is the burden of any non-standard logic to justify its being
non-standard: Why the third value? What is the third value? Unknown? If so, is it an epistemological issue? Indeterminate? If so, does this require a metaphysical explanation?
The first subsection summarizes Peirce’s calculus of triadic logic and the second briefly discusses Peirce’s own motive for triadic logic.
3.1 Truth table of a three-valued system
Three values, V, L, and F, are introduced, where V is true, L indeterminate, and F false. The traditional semantic domain for sentential logic, true and false, is extended to include “indeterminate”.
Based on this extended semantic territory, Peirce presents the semantics of several sentential operators, one unary and the others binary. Modifying Peirce’s presentation slightly to make it more
similar to our conventional truth table style without changing content, we present the three operators’ truth tables.
The semantics of a unary operator, which corresponds to negation:
\(x\) \(\bar{x}\)
V F
L L
F V
The semantics for six binary connectives is presented:
\(x\) \(y\) \(\Phi(x, y)\) \(\Theta(x, y)\) \(\Psi(x, y)\) \(Z(x, y)\) \(\Omega(x, y)\) \(\Gamma(x, y)\)
V V V V V V V V
L V V V V L L L
F V V V F F F V
V L V V V L L L
L L L L L L L L
F L F L F F L L
V F V V F F F V
L F F L F F L L
F F F F F F F F
Why six? What is the rationale behind the semantics of these connectives? One way to understand them is to figure out a dominance hierarchy among the three values.
In the case of \(\Phi\),
i. if at least one is V, then \(\Phi(x, y)\) is V,
ii. else if at least one is F, then \(\Phi(x, y)\) is F, and
iii. else \(\Phi(x, y)\) is L.
That is, V is the most dominant, F next, and L is the least.
Six patterns of hierarchy emerges:
\(\Phi\) V \(>\) F \(>\) L
\(\Theta\) V \(>\) L \(>\) F
\(\Psi\) F \(>\) V \(>\) L
\(Z\) F \(>\) L \(>\) V
\(\Omega\) L \(>\) F \(>\) V
\(\Gamma\) L \(>\) V \(>\) F
Peirce’s \(\Theta\) is our familiar disjunction and Peirce’s Z conjunction.
3.2 Why the third value?
In Peirce’s own words:
Triadic Logic is that logic, which though not rejecting entirely the Principle of Excluded Middle, nevertheless recognizes that every proposition, S is P, is either true or false, or else has a
lower mode of being such that it can neither be determinately P, nor determinately not-P, but is at the limit between P and not P. (Ms 339, copied from Fisch & Turquette 1966: 75)
When do we have value L (indeterminate) for the proposition “S is P”? Sometimes S, Peirce says, has a lower mode of being P and is at the limit between P and not P. The crux of the matter is how to
interpret the two phrases—“lower mode of being P” and “being at the limit between P and not P”. Existing literature offers two different explanations—modality versus continuity.
Fisch and Turquette, upon the discovery of Peirce’s notes on triadic logic, locate the root of indeterminacy in potentiality. That is, indeterminacy is the semantic value assigned to an unrealized
situation; hence, we can say neither “S is P” nor “S is not P” at this point. Potentiality, according to this view, cannot be captured by dyadic logic. If so, Peirce’s triadic logic is directly
related to modality talk, which Fisch and Turquette conclude:
Essentially, Peirce seems to be saying that triadic logic may be interpreted as a modal logic which is designed to deal with the indeterminacies resulting from that mode of being which Peirce has
called “Potentiality” and “Real Possibility”. Under such an interpretation, dyadic logic becomes a limiting case of triadic modal logic resulting from removing indeterminacy and being determined
entirely by “Actuality”. (Fisch & Turquette 1966: 79)
According to the modality interpretation, Peirce’s “lower mode of being P” means P not being actual, and Peirce’s third value L, being potential, is “at the limit between P (i.e., T) and not P (i.e.,
F)”. Later in the paper, suggesting a possible relation between Peirce’s triadic logic and MacColl’s implication (as opposed to material implication), the authors make an interesting remark:
Considering MacColl’s rejection of Russell’s material implication, it is interesting to notice also that MacColl’s “Def. 13” gives what is now called “C. I. Lewis’s strict implication”. (Fisch &
Turquette 1966: 83)
Even though the connection was not pursued further in their paper, one cannot help realizing that their modality interpretation is boosted by a relation to MacColl’s implication since C. I. Lewis’
strict implication is a beginning of modal logic. However, equating Peirce’s triadic logic with modal logic, the modality view needs to explain the relation between Peirce’s Gamma graph lecture in
1903c (which is about modality) and the triadic logic notes written in 1909 ([Ms 339] 340v, 341v, 344r). Modal logic explored in Gamma graphs is the extension of classical logic, which required new
vocabulary, e.g., broken cuts and tincture. Modal logic does not have to be non-standard. On the other hand, triadic logic does not add any vocabulary, but brings in different interpretations, and
becomes non-standard logic. On a slightly different note, Fisch and Turquette’s suggests Peirce’s tychism (the view that indeterminacy is part of the reality) as a motivation for Peirce’s invention
of triadic logic. If so, Peirce’s triadic logic is a reflection of his own metaphysics.
Challenging the modality view, Robert Lane proposes the continuity interpretation for Peirce’s triadic logic. According to Lane, Peirce’s indeterminate value L has nothing to do with modality, hence,
Peirce’s development of triadic logic is not another mechanism for modal logic, but with Peirce’s synechism—the doctrine “that all that exists is continuous” (c. 1897b [CP 1.172])! How does Peirce’s
philosophy of continuity justify the third value?
First, Lane makes a distinction between the principle of excluded middle (PEM, henceforth) being false with regard to a proposition and PEM not being applied to a proposition. If PEM is true or
false, it means the principle is applied to it. And, Lane claims that PEM is applied to only non-general and non-modal propositions, citing the following passages from Peirce:
anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it. (1905: 488 [CP 5.448])
an assertion is said to be made in “the mode of necessity” if, and only if, the affirmation and the denial that [sic] which is so asserted could conceivably be both alike false. Thus if a person
says “It will certainly rain tomorrow”, it may be alike false that it is certain to rain and that it is certain not to rain. (1910b: 26–28, Ms 678)
If a proposition is either general or expresses necessity, PEM is not false, but is not applied. Hence, focusing on individual and non-modal propositions, Lane draws our attention to a special nature
of a predicate in L-propositions. Lane calls the kind of the property which results in L-propositions a “boundary-property”. Here is Peirce’s own example of a boundary-property:
Thus, a blot is made on the sheet. Then every point of the sheet is unblackened or blackened. But there are points on the boundary line, and those points are insusceptible of being unblackened or
of being blackened, since these predicates refer to the area about S and a line has no area about any point of it. (Ms 339: 344r, quoted in Lane 1999: 294)
Those points on the boundary line are neither black nor non-black. Consider the propositions “Point O is black” and “Point O is not black” (where point O is on the boundary line of a black blot).
Neither of them is true, but false, either. These are prime examples of Peirce’s L-proposition. Using Peirce’s own truth-tables in the previous subsection, let’s compute the truth value of “Point O
is black or point O is not black”.
Let \(\alpha\) be the value of “Point O is black”, which is L.
\(\alpha\) \(\bar{\alpha}\) \(\Theta(\alpha, \bar{\alpha})\)
L L L
Note that PEM is applied, but not true, period.
Lane’s following conclusion would be welcomed by many Peirce scholars:
[B]oundary-propositions were important to Peirce because continuity was important to him;…this [the thought that an actual breach of continuity possesses neither of the properties that are the
boundary properties relative to that breach] led him to think that boundary-propositions are neither true nor false. To accommodate such propositions, and thus the phenomenon of continuity,
within the bounds of formal reasoning, was, I contend, the motivation behind Peirce’s experiments in triadic logic. (Lane 1999: 304)
Regardless of endorsement of the continuity talk, some might not welcome metaphysics getting into logic. Moreover, if Peirce’s synechism is not embraced, Peirce’s triadic logic, which Lane argues is
an attempt to formalize the continuity phenomenon, might lose its force.
The previous two sections showed that Peirce’s relational logic and graphical systems push us further both in what we do with logic and how we do logic so that we may formalize more and we may
formalize in more diverse ways. Triadic logic, as explained at the beginning of this section, is neither just a monotonic extension to get further nor an alternative to get to the same place. By
extending semantic entities, we have a different logic, for example, PEM being not true. That is why we call triadic logic a non-standard logic. However, Peirce’s way of introducing the third value
gives us some pause. First of all, unlike with contemporary triadic logic, Peirce does not discard PEM completely:
Triadic Logic…not rejecting entirely the Principle of Excluded Middle,… (Ms 339: 344r, copied from Fisch & Turquette 1966: 75)
I do not say that the Principle of Excluded Middle is downright false; (1909: 21–22 [NEM 3/2: 851], quoted in Fisch & Turquette 1966: 81)
For certain (not all) properties, because of the way things are, we find ourselves caught at limits between clearly P and clearly non-P. If we want to formalize those cases as well, the third value,
L, is needed to express the indeterminacy of boundary cases. Hence, Peirce himself does not think triadic logic is a new logic, but an addition or extension of the existing dyadic logic:
The recognition [that there is an intermediate ground between positive assertion and positive negation which is just as Real as they are] does not involve any denial of existing logic, but it
involves a great addition to it. (1909: 21–22 [NEM 3/2: 851], quoted in Fisch & Turquette 1966: 81)
If we accept Peirce’s suggestion literally, his triadic logic is not a typical form of non-standard logic but Peirce’s another way to extend the territory of logic, along with his relational logic.
A. Primary Sources: Works by C. S. Peirce cited in this entry
• [CP], Collected Papers of Charles Sanders Peirce, Charles Hartshorne and Paul Weiss (eds), Cambridge, MA: Harvard University Press, 1960, Volumes 1—5.
• [W], Writings of Charles S. Peirce: A Chronological Edition, 6 volumes, The Peirce Edition Project, M. Fisch, C. Kloesel, and N. Houser (eds), Bloomington, IN: Indiana University Press.
• [Ms 339] 1865–1909, “Logic Notebook”, unpublished manuscript, Ms 339. Pages 340v, 341v, 344r replicated in Fisch and Turquette 1966: 73–75.
• 1866, “Lowell Lectures on the Logic of Science 1866, Lecture II”, Ms 353. Parts quoted in Merrill 1978.
• 1867, “An Improvement in Boole’s Calculus of Logic”, Proceedings of the American Academy of Arts and Sciences, 7: 249–261. Reprinted in CP 3.1–19. doi:10.2307/20179565
• [DNLR] 1870, “Description of a Notation for the Logic of Relatives, Resulting from An Amplification of the Conceptions of Boole’s Calculus”, Memoirs of the American Academy of Arts and Sciences,
9(2): 317–378. Reprinted in CP 3.45–148. doi:10.2307/25058006
• 1878, “How To Make Our Ideas Clear”, Popular Science Monthly, 12(January): 286–302. Reprinted in CP 5.388–410.
• 1880, “On the Algebra of Logic”, American Journal of Mathematics, 3(1): 15–57. Reprinted in CP 3.154–251. doi:10.2307/2369442
• 1882a, “Brief description of the algebra of relatives”, manuscript. Reprinted in CP 3.306–322.
• 1882b, Letter to Oscar H. Mitchell, MS L 294 (21 December 1882), quoted in Roberts 1973, p. 18.
• 1883a, “The Logic of Relatives” also known as “NoteB”, in Peirce (ed.) 1883b: 187–203. Reprinted in CP 3.328–358.
• 1883b, editor, Studies in Logic by Members of the Johns Hopkins University, Boston: Little, Brown, and Company.
• 1885a, “On the Algebra of Logic: A Contribution to the Philosophy of Notation”, The American Journal of Mathematics, 7(2): 180–202. Reprinted in CP 3.359–403. doi:10.2307/2369451
• 1885b, “Note”, undated but written for the issue of The American Journal of Mathematics just after the previous article. Reprinted in CP 4.403A–403M.
• 1896, “Review of Schröder’s Algebra und Logik der Relative”, The Nation, 62: 330–331.
• 1897a, “The Logic of Relatives”, The Monist, 7(2): 161–217. Reprinted in CP 3.456–552. doi:10.5840/monist18977231
• c. 1897b, “Fallibilism, Continuity, and Evolution” (originally untitled), unpublished manuscript. Printed CP 1.141–175.
• 1903a, “Existential Graphs”, in his A Syllabus of Certain Topics of Logic, Boston: Alfred Mudge & son, pp. 15–23. Reprinted in CP 4.394–417.
• 1903b, “On Existential Graphs, Euler’s Diagrams, and Logical Algebra”, from “Logical Tracts, No. 2”. Reprinted in CP 4.418–529.
• 1903c, “The Gamma Part of Existential Graphs”, Lowell Lectures of 1903, Lecture IV, unpublished. Printed in CP 4.510–529.
• c. 1903d, “Nomenclature and Divisions of Triadic Relations, as Far as They Are Determined”, manuscript, Printed in CP: 2.233–272.
• 1905, “The Issues of Pragmaticism”:, The Monist, 15(4): 481–499. Reprinted in CP 5.438–463. doi:10.5840/monist19051544/
• 1906, “Prolegomena to an Apology for Pragmaticism”, The Monist, 16(4): 492–546. Reprinted in CP 4.530–572. doi:10.5840/monist190616436
• 1909, Letter to William James, 26 February 1909, 40 pages. Reprinted in Peirce’s New Elements of Mathematics, volume 3/2: Mathematical Miscellanea 2, Carolyn Eisele (ed.), Boston: De Gruyter,
1976, NEM 3/2: 836–866.
• 1910a, “Diversions of Definitions”, unpublished, Ms 650.
• 1910b, “The Art of Reasoning Elucidated”, unpublished, Ms 678.
• 1911a, “Notes on Symbolic Logic and Mathematics”, with H. B. Fine, in Dictionary of Philosophy and Psychology, second edition, J. M. Baldwin (ed.), New York: Macmillan, volume 1, p. 518;.
Reprinted in CP 3.609—3.645.
• 1911b, “Euler’s Diagrams”, in Dictionary of Philosophy and Psychology, second edition, J. M. Baldwin (ed.), New York: Macmillan, vol. 2, p. 28. Reprinted in CP 4.347–371.
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• Barwise, Jon and Gerard Allwein (eds), 1996, Logical Reasoning with Diagrams, New York: Oxford University Press.
• Barwise, Jon and John Etchemendy, 1991, “Visual Information and Valid Reasoning”, Visualization in Teaching and Learning Mathematics, Walter Zimmerman and Steve Cunningham (eds), Washington, DC:
Mathematical Association of America, 9–24.
• Beatty, Richard, 1969, “Peirce’s Development of Quantifiers and of Predicate Logic.”, Notre Dame Journal of Formal Logic, 10(1): 64–76. doi:10.1305/ndjfl/1093893587
• Boole, George, 1847, The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Cambridge: Macmillan, Barclay, & Macmillan.
• Brady, Geraldine, 1997, “From the Algebra of Relations to the Logic of Quantifiers”, in Houser, Roberts, and Van Evra 1997: 173–192 (ch. 10).
• –––, 2000, From Peirce to Skolem: A Neglected Chapter in the History of Logic, Amsterdam: Elsevier.
• Brunning, Jacqueline, 1991, “C. S. Peirce’s Relative Product”, Modern Logic, 2(1): 33–49. [Brunning 1991 available online]
• Burch, Robert W., 1994, “Game-Theoretical Semantics for Peirce’s Existential Graphs”, Synthese, 99(3): 361–375. doi:10.1007/BF01063994
• –––, 1997, “Peirce on the Application of Relations to Relations”, in Houser, Roberts, and Van Evra 1997: 206—233 (ch. 12).
• Dau, Frithjof, 2006, “Fixing Shin’s Reading Algorithm for Peirce’s Existential Graphs”, in Diagrammatic Representation and Inference, Dave Barker-Plummer, Richard Cox, and Nik Swoboda (eds.),
(Lecture Notes in Computer Science 4045), Berlin: Springer Berlin Heidelberg, 88–92. doi:10.1007/11783183_10
• De Morgan, Augustus, 1847, Formal Logic or, The Calculus of Inference, Necessary and Probable, London: Taylor and Walton.
• –––, 1864, “On the Syllogism, No. IV, and on the Logic of Relations”, Cambridge Philosophical Transactions, 10: 331—358. Originally written in 1859.
• Dipert, Randall R., 1984a, “Peirce, Frege, the Logic of Relations, and Church’s Theorem”, History and Philosophy of Logic, 5(1): 49–66. doi:10.1080/01445348408837062
• –––, 1984b, “Essay Review: Studies in Logic by Members of the Johns Hopkins University, 1883, Edited by Charles S. Peirce”, History and Philosophy of Logic, 5(2): 227–232. doi:10.1080/
• –––, 1984c, “Review of Studies in Logic by Members of Johns Hopkins University”, Transactions of the Charles S. Peirce Society, 20(4): 469–472.
• –––, 1995, “Peirce’s Underestimated Place in the History of Logic: A Response to Quine”, in Ketner 1995: 32–58.
• –––, 1996, “Reflections on Iconicity, Representation, and Resemblance: Peirce’s Theory of Signs, Goodman on Resemblance, and Modern Philosophies of Language and Mind”, Synthese, 106(3): 373–397.
• –––, 2004, “Peirce’s Deductive Logic: Its Development, Influence, and Philosophical Significance”, in The Cambridge Companion to Peirce, Cheryl Misak (ed.), Cambridge: Cambridge University Press,
287–324. doi:10.1017/CCOL0521570069.012
• Frege, Gottlob , 1879 [1972], Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a. S.: Louis Nebert. Translated as Conceptual Notation, and Related
Articles, Terrell Ward Bynum (trans.), Oxford: Oxford University Press, 1972.
• Fisch, Max and Atwell Turquette, 1966, “Peirce’s Triadic Logic”, Transactions of the Charles S. Peirce Society, 2(2): 71—85.
• Goldfarb, Warren D., 1979, “Logic in the Twenties: The Nature of the Quantifier”, Journal of Symbolic Logic, 44(3): 351–368. doi:10.2307/2273128
• Grattan-Guinness, I., 2002, “Re-interpreting ‘\(\lambda\)’: Kempe on multisets and Peirce on graphs, 1886–1905”, Transactions of the Charles S. Peirce Society, 38, 327–350.
• Hawkins, Benjamin S., Jr, 1995, “De Morgan, Victorian Syllogistic and Relational Logic”, in Modern Logic, 5(2) : 131–166. [Hawkins 1995 available online]
• Herzberger, Hans G., 1981, “Peirce’s Remarkable Theorem”, in Pragmatism and Purpose: Essays Presented to Thomas A. Goudge, Leonard Wayne Sumner, John G. Slater, and Fred Wilson (eds), Toronto:
University of Toronto Press, 41–58 and 297–301.
• Hilpinen, Risto, 1982, “On C. S. Peirce’s Theory of the Proposition: Peirce as a Precursor of Game-Theoretical Semantics”, The Monist, 65(2): 182–188. doi:10.5840/monist198265213
• –––, 2004, “Peirce’s Logic”, Handbook of the History of Logic: Volume 3, The Rise of Modern Logic: From Leibniz to Frege, Dov M. Gabbay and John Woods (eds), Amsterdam: Elsevier, 611—658.
• Hintikka, Jaakko, 1980, “C. S. Peirce’s ‘First Real Discovery’ and Its Contemporary Relevance”, The Monist, 63(3): 304–315. doi:10.5840/monist198063316
• –––, 1988, “On the Development of the Model-Theoretic Viewpoint in Logical Theory”, Synthese, 77(1): 1–36. doi:10.1007/BF00869545
• –––, 1990, “Quine as a Member of the Tradition of the Universality of Language”, in Perspectives on Quine, Robert B. Barret and Roger F. Gibson (eds), Cambridge, MA: B. Blackwell, 59–175.
• –––, 1997, “The Place of C.S. Peirce in the History of Logical Theory”, in The Rule of Reason: The Philosophy of Charles Sanders Peirce, Jacqueline Brunning and Paul Forster (eds), Toronto:
University of Toronto Press, 13–33.
• Houser, Nathan, 1997, “Introduction: Peirce as Logician”, in House, Roberts, and Van Evra 1997: 1–22.
• Houser, Nathan, Don D. Roberts, and James Van Evra (eds.), 1997, Studies in the Logic of Charles Sanders Peirce, Bloomington, IN: Indiana University Press.
• Ketner, Kenneth Laine (ed.), 1995, Peirce and Contemporary Thought: Philosophical Inquiries, New York: Fordham University Press.
• Lane, Robert, 1999, “Peirce’s Triadic Logic Revisited”, Transactions of the Charles S. Peirce Society, 35(2): 284–311.
• Ladd [Ladd-Franklin], Christine, 1883, “On a New Algebra of Logic”, in Peirce (ed.) 1883b: 17–71.
• Iliff, Alan, 1997, “The Role of the Matrix Representation in Peirce’s Development of the Quantifiers”, in Houser, Roberts, and Van Evra 1997: 193–205 (ch. 11).
• Merrill, Daniel D., 1978, “DeMorgan, Peirce and the Logic of Relations”, Transactions of the Charles S. Peirce Society, 14(4): 247–284.
• –––, 1990, Augustus De Morgan and the Logic of Relations, Dordrecht: Kluwer. doi:10.1007/978-94-009-2047-7
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• Michael, Emily, 1974, “Peirce’s Early Study of the Logic of Relations, 1865–1867”, Transactions of the Charles S. Peirce Society, 10(2): 63–75.
• Mitchell, O. H., 1883, “On a New Algebra of Logic”, in Peirce (ed.) 1883b: 72–106.
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Economics - UvoCorpEssays
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Please complete the following two applied problems. Show all your calculations and explain your results. Problem 1:
A generous university benefactor has agreed to donate a large amount of money for student scholarships. The money can be provided in one lump sum of $12 million in Year 0 (the current year), or in
parts, in which $7 million can be provided at the end of Year 1, and another $7 million can be provided at the end of Year 2.
Describe your answer for each item below in complete sentences, whenever it is necessary. Show all of your calculations and processes for the following points:
a. Assuming the opportunity interest rate is 8%, what is the present value of the second alternative mentioned above? Which of the two alternatives should be chosen and why?
b. How would your decision change if the opportunity interest rate is 12%?
c. Provide a description of a scenario where this kind of decision between two types of payment streams applies in the “real-world” business setting.
Problem 2:
The San Diego LLC is considering a three-year project, Project A, involving an initial investment of $80 million and the following cash inflows and probabilities:
Describe your answer for each question in complete sentences, whenever it is necessary. Show all of your calculations and processes for the following points:
a. Describe and calculate Project A’s expected net present value (ENPV) and standard deviation (SD), assuming the discount rate (or risk-free interest rate) to be 8%. What is the decision rule in
terms of ENPV? What will be San Diego LLC’s decision regarding this project? Describe your answer.
b. The company is also considering another three-year project, Project B, which has an ENPV of $32 million and standard deviation of $10.5 million. Project A and B are mutually exclusive. Which of
the two projects would you prefer if you do not consider the risk factor? Explain.
c. Describe the coefficient of variation (CV) and the standard deviation (SD) in connection with risk attitudes and decision making. If you now also consider your risk-aversion attitude, as the CEO
of the San Diego LLC will you make a different decision between Project A and Project B? Why or why not?
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Born 22 Feb 1936.
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Sur l'homme et le developpement de ses facultés, essai d'une physique sociale
. Whereas the normal curve had previously been applied to error correction, Quetelet used it to illustrate a distribution of measured human traits about the central value, giving the concept of the
average man at the peak. In this way, for example, he applied a statistical view to the nature of criminal behaviour in society.«
Image: from a 1974 Belgian postage stamp.
Born 22 Feb 1785; died
27 Oct 1845
at age 60.
French physicist who discovered the Peltier
(1834), that at the junction of two dissimilar metals an electric current will produce heat or cold, depending on the direction of current flow. In 1812, Peltier received an inheritance sufficient to
retire from clockmaking and pursue a diverse interest in phrenology, anatomy, microscopy and
. Peltier made a thermoelectric thermoscope to measure temperature distribution along a series of thermocouple circuits, from which he discovered the Peltier effect. Lenz succeeded in freezing water
by this method. Its importance was not fully recognized until the later thermodynamic work of Kelvin. The effect is now used in devices for measuring temperature and non-compressor
[Image: Peltier's atmospheric electricity gauge.]
Born 22 Feb 1778; died
3 Oct 1860
at age 82.
American artist and naturalist, son of
Charles Willson Peale
, who followed his father as a portrait painter with an interest in natural history. Many of Rembrandt Peale's portraits are of scientists. He also took an interest in the technology of his era,
including the pioneering steam navigation of
John Fitch
Robert Fulton
, gas lighting of Baltimore's streets, and the chemistry of pigments. In 1801, he assisted his father's excavation of mastodon bones from the peat bogs of Orange County, New York, from which they
assembled two skeletons. One was mounted in his father's Philadelphia Museum. Rembrandt displayed another as a travelling exhibit around New York (1802) and London (1802, 1803). From the details of
its fossil teeth, he believed the mastodon was a carnivor, though
Georges Cuvier
shortly established it was a herbivor.«
Born 22 Feb 1732; died
14 Dec 1799
at age 67.
American surveyor, military leader and president who was an eager student of mathematics in his youth, teaching himself geometry and trigonometry. This led to his early career as a surveyor,
proficient at drafting, mapmaking, and designing tables of data. Surveying let him explore regions of Virgina, and earn income to be a landowner by age 19. Mathematics courses in early American
education included applications in surveying. For example, it was part of state law in Massachusetts (1827) that any locality with 500 families should have a master capable of instructing “geometry,
surveying and algebra.” Thus, long before that law, in a less-known aspect of his life, Washington was equipped with a technical education of service to his community—although he is most famous for
fighting in the Revolutionary war and becoming the the first President of the U.S.A.«
Died 22 Feb 2002 at age 100 (born
25 Mar 1901
New Zealander social anthropologist whose major
was with the Maori and other peoples of Oceania and Southeast Asia. Firth conducted his first
in the British Solomon Islands 1928-29. The economic organization of primitive societies became one of Firth's primary interests as indicated by his works on the Kauri gum industry and the fishing
industry of Malaysia. Among his other chief interests were social structure and religion, especially of the Tikopia of the Solomon Islands, and the anthropological treatment of symbols. Firth was
also well know for his work concerning sacrifices. In 1963, Raymond began his work on the influence of economics on the ideology of sacrifice.
Died 22 Feb 1984 at age 12 (born
21 Sep 1971
American patient who lived his twelve years of
in a sterile plastic “bubble” to protect him from any chance of infection, because he was born with the a genetic disease, severe combined deficiency syndrome (
). He was publicly identified only as “David,” or “the
Bubble Boy
.” He died after an unsuccessful bone marrow stem cells transplant that had been hoped could save him. The confinment and loneliness caused psychological turmoil that was kept from the media, which
wrote of a more benign experience. A made-for-TV movie was inaccurate. The bone marrow from his sister, despite screening to avoid such a problem, contained Epstein-Barr virus. He
of Burkitt's lymphoma (from which it was learned, for the first time, that a virus can cause cancer.) He lived his last 15 days outside of the bubble.«
Died 22 Feb 1949 at age 75 (born
25 Apr 1873
Canadian-French bacteriologist who is generally known as the discoverer of the bacteriophage, a virus that infects bacteria. (The earlier identification of the bacteriophage by the British
Frederick W. Twort
in about 1915 became obscured by Twort's disinclination to take credit for or to pursue his initial findings.)
Died 22 Feb 1945 at age 71 (born
15 Nov 1873
American physician who was a
in public health and child welfare in the United States. She was
assistant to the Commissioner for Public Health of New York City, later heading the city's Department of Health in 'Hell's Kitchen' for 25 years. Convinced of the value of well-baby care and the
prevention of disease, in 1908 she founded the Bureau of Child Hygiene after visiting mothers on the lower east side, thus helping to decrease the death rate by 1200 from the previous year. Her work
made the New York City infant mortality rate the lowest in the USA or Europe at the time. She set up free milk clinics, licensed midwives, and taught the use of silver nitrate to prevent blindness in
Died 22 Feb 1944 at age 86 (born
21 Jun 1857
Hugh Frank Newall was an English astronomer and physicist who held the first chair of astrophysics at Cambridge University (1909-1928). After teaching at Wellington College, he went to Cambridge to
be an assistant to
J. J. Thomson
. He changed his interests from being senior demonstrator in experimental physics to astronomy when he facilitated the university's acquisition of the 25-inch Newall Telescope after the death of his
father, Robert Stirling Newall, in 1889. His father, an engineer in manufacturing wire ropes and submarine telegraph cables, had the telescope built for private use at his Gateshead home. Hugh paid
the moving expenses. When built, it was the largest in the world, and remained so for many years. He designed spectrographs and studied the solar corona, became director of the Solar Physics
Observatory (1913) and led many eclipse expeditions.«
Died 22 Feb 1941 at age 74 (born
13 Mar 1866
American physicist.
The Science of Musical Sounds
(1916). Miller's
of nearly 1,650 flutes and other instruments, and other materials mostly related to the flute, is now at the Library of Congress. To provide a mechanical means of recording sound waves
photographically, he invented the phonodeik (1908). He became expert in architectural ecoustics. During WW I, he was consulted concerning using his photodeik to help locate enemy guns. Miller spent
considerable research effort on repeating the
experiment, proposed by
, to detect a stationary aether. He spent some time working with Morley (1902-4), then more time at Mt. Wilson, recording results favoring the presence of the aether.
Died 22 Feb 1925 at age 88 (born
20 Jul 1836
English physician who invented the short
clinical thermometer
(1866) to meet the need for a convenient method to follow the progress of a fever by temperature measurements at the bedside. Allbutt invented the pocket-size, six-inch clinical thermometer, which
could take a temperature in five minutes. Before his improvement, the instruments used were a foot long, and required 20 minutes to measure a patient’s temperature. Allbutt also demonstrated that
angina was caused by a narrowing of the coronary artery. This understanding was an important contribution to improving procedures to treat arterial diseases.«
Died 22 Feb 1913 at age 55 (born
26 Nov 1857
, born in Geneva, whose ideas on structure in language laid the foundation for much of the approach to and progress of the linguistic sciences in the 20th century. The work by which he is best known,
Cours de linguistique générale
Course in General Linguistics
) was compiled from the lecture notes of his students after his death. His focus on language as an 'underlying system' inspired a great deal of later semiology and structuralism, and he is often
described as the founder of modern linguistics.
Died 22 Feb 1875 at age 77 (born
14 Nov 1797
Charles Lyell (Baronet) was a Scottish geologist who promoted a theory of gradualism, by which all features of the Earth's surface are produced by physical, chemical, and biological processes through
long periods of geological time. This extended the ideas of uniformitarianism as earlier stated by
James Hutton
. Lyell rejected the idea of
Abraham Werner
that a single great historic deluge had been responsible for producing the Earth's present surface topology. The concept of uniformitarianism also opposed the theory of catastrophism whereby
zoologists such as
Georges Cuvier
believed it was by dramatic changes that flora and fauna appeared in their present form. Instead, Lyell maintained that changes were gradual, shaped by forces over unlimited time, in a similar way
throughout, and were still operating in the present time.«
Died 22 Feb 1827 at age 85 (born
15 Apr 1741
American artist and naturalist who opened the first U.S. popular Museum of Natural Science and Art. Alongside fame as a portraitist, Peale maintained a diverse interest in science. He used a
physiognotrace machine used to record profiles and make silhouettes. He patented a fireplace, porcelain false teeth, and a new kind of wooden bridge. He invented a technique to put motion with
pictures and wrote papers on engineering and hygiene. He perfected a kind of portable writing desk, named the polygraph, which reproduced several copies of a manuscript at once. In 1786, he
established the first U.S. scientific museum with both living and stuffed specimens, and later a complete mastodon skeleton he helped excavate (1801).«
Died 22 Feb 1815 at age 53 (born
30 Nov 1761
English chemist.
Died 22 Feb 1794 at age 60 (born
18 Jan 1734
, known as the "founder of modern embryology." In
Theoria Generationis
(1759) he first wrote an epigenetic
of development: that the organs of living things take shape gradually from non-specific tissue. Wolff applied the microscope to the study of animal embryology and remarked that "the particles which
constitute all animal organs in their earliest inception are little globules, which may be distinguished under a microscope." The book was ignored for half a century, as the prevailing idea was held
that life begins preformed in a small body that grows larger in the same form. His name is found describing parts of the kidneys of embryos: the Wolffian body and the Wolffian ducts.
Died 22 Feb 1512 at age 60 (born
9 Mar 1451
Italian-Spanish navigator, explorer and cartographer whose name was given to the New World - America - because it was he and not Columbus, who realized and announced that Columbus had discovered a
. He was well educated, including studies of physics, geometry and astronomy. On 10 May 1497, he began his first
of discovery. Three ships were provided by by King Ferdinand of Spain. After four voyages of exploration, on 14 Apr 1505, he naturalized as a Spaniard. He made two more voyages. In 1507, German
mapmaker Martin Waldseemuller, printed the first map applying the name
for the New World. The title
piloto mayor de España
(chief of navigation) for Spain was bestowed on Vespucci, by royal decree, on 6 Aug 1508.«
[a.k.a. Americus Vespucius. EB gives years 1454?-1512.]
In 1995, Steve Fossett completed the first hot air balloon flight over Pacific Ocean (9600 km). On
3 Mar 2005
, he completed the first solo non-stop and fastest
around the world without refueling, in
The Global Flyer
, a single-engine, single-use experimental jet plane.«
In 1984, a 12-year-old Houston boy, known publicly only as “David,” died. He had spent nearly all his
in a sterile plastic bubble because he had no immunity to disease. He was born at Texas Children's Hospital, Houston, with a rare disorder called severe combined immune deficiency (
David Vetter
lacked T-cells. It was thought that transplanted marrow stem cells—precursors to blood cells—could evolve and become the patient's own T-cells. So, on
21 Oct 1983
, he received a bone marrow transplant from his older sister. It was intended to stimulate his immune system. By New Year's Day, he was becoming ill. Sadly, despite cleansing treatment, an undetected
virus was in the transferred cells. When his death was imminent, on
7 Feb 1984
, he left his bubble, and had 15 days of freedom.«
In 1946, Dr
Selman Abraham Waksman
announced his
of the antibiotic streptomycin, the first specific antibiotic effective against tuberculosis. In 1943, he had isolated streptomycin from a mold he had known and studied early in his life. For this
work, he was awarded the 1952 Nobel Prize.«
The antibiotic era
: A history of the antibiotics..., by Selman Abraham Waksman. - book suggestion.
In 1918, the documented world record tallest human, Robert Pershing Wadlow, was
in Alton, Illinois. Shortly after graduating from high school, he was 8-ft 4-in (2.54m) tall. He grew to a final height, shortly before his death, of 8-ft 11.1-in (2.72m). This unique size was
attributed to an over active pituary gland, which produced much higher than normal levels of growth hormone. Today's medical science can compensate for such problems, but in the 1920s there was no
therapy available. He died on 15 Jul 1940, at age 22, from a fatal infection which set in due to a blister on his foot, despite emergency surgery and blood transfusions. At the time of his death he
weighed 490 pounds. The 1,000-pound casket required a dozen pallbearers, assisted by eight other men.
[Image: Wadlow with his brother, 1936]
In 1896, the first
of clinical radiology in England was reported in
The Lancet*
by surgeon
Sir Robert Jones
and the head of Liverpool University's physics department,
Oliver Lodge
. A 12-year-old boy, who had shot himself in his wrist the previous month, was
in Lodge's laboratory on
7 Feb 1896
at Jones' request. Merely probing could not locate the bullet. Jones had heard of
's discovery of X-rays a few months earlier. Using X-rays, the pellet was identified embedded in the third carpo-metacarpal joint. Jones subsequently financed an X-ray apparatus for his senior
assistant who had been with him then,
Charles Thurstan Holland
, to pioneer radiology at Royal Southern Hospital, Liverpool.«
[Image: X-ray from 1896 of hand with buckshot made at Columbia University, USA]
In 1828, German biochemist
Friedrich Wöhler
Jakob Berzelius
that he had synthesized the organic chemical, urea. This was a landmark event, for it was the first time a material previously only associated with the body function of a living thing, was made from
inorganic chemicals of non-living origin. In this case, urea had formerly been known only from the urine of animals.
In 1630, popcorn was introduced to the English colonists by an Indian named Quadequina who brought it in deerskin bags as his contribution at their first Thanksgiving dinner. Popcorn is a type of
corn with smaller kernels than regular corn, and when heated over a flame, it "pops" into the snack we know it as today. Native Americans were growing it for more than a thousand years before the
arrival of European explorers. In 1964, scientists digging in southern Mexico found a small cob of popcorn discovered to be 7,000 years old. Today, the United States grows nearly all of the world's | {"url":"https://todayinsci.com/2/2_22.htm","timestamp":"2024-11-04T17:17:33Z","content_type":"text/html","content_length":"115078","record_id":"<urn:uuid:8bf35e59-4bd9-4efd-963a-052f66d58e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00586.warc.gz"} |
bug in numpy.mean() ?
24 Jan 2012 24 Jan '12
7:33 p.m.
I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be: 2-D data file here: http://dl.dropbox.com/u/139035/data.npy Then: In [3]: data.mean() Out
[3]: 3067.0243839999998 In [4]: data.max() Out[4]: 3052.4343 In [5]: data.shape Out[5]: (1000, 1000) In [6]: data.min() Out[6]: 3040.498 In [7]: data.dtype Out[7]: dtype('float32') A mean value
calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend version with the same
results. Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1
and my whole setup is based on Enthought's EPD. Best regards, Michael
I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be:
2-D data file here: http://dl.dropbox.com/u/139035/data.npy
Then: In [3]: data.mean() Out[3]: 3067.0243839999998
In [4]: data.max() Out[4]: 3052.4343
In [5]: data.shape Out[5]: (1000, 1000)
In [6]: data.min() Out[6]: 3040.498
In [7]: data.dtype Out[7]: dtype('float32')
A mean value calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend
version with the same results.
Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1 and my
whole setup is based on Enthought's EPD.
Best regards, Michael
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion You have a million 32-bit floating
point numbers that are in the
On 01/24/2012 12:33 PM, K.-Michael Aye wrote: thousands. Thus you are exceeding the 32-bitfloat precision and, if you can, you need to increase precision of the accumulator in np.mean() or change the
input dtype:
a.mean(dtype=np.float32) # default and lacks precision 3067.0243839999998 a.mean(dtype=np.float64) 3045.747251076416 a.mean(dtype=np.float128) 3045.7472510764160156 b=a.astype
(np.float128) b.mean() 3045.7472510764160156
Otherwise you are left to using some alternative approach to calculate the mean. Bruce
You have a million 32-bit floating point numbers that are in the thousands. Thus you are exceeding the 32-bitfloat precision and, if you can, you need to increase precision of the accumulator in
np.mean() or change the input dtype:
a.mean(dtype=np.float32) # default and lacks precision 3067.0243839999998 a.mean(dtype=np.float64) 3045.747251076416 a.mean(dtype=np.float128) 3045.7472510764160156 b=a.astype
(np.float128) b.mean() 3045.7472510764160156
Otherwise you are left to using some alternative approach to calculate the mean.
Interesting -- I knew that float64 accumulators were used with integer arrays, and I had just assumed that 64-bit or higher accumulators would be used with floating-point arrays too, instead of the
array's dtype. This is actually quite a bit of a gotcha for floating-point imaging-type tasks -- good to know! Zach
I have confirmed this on a 64-bit linux machine running python 2.7.2 with the development version of numpy. It seems to be related to using float32 instead of float64. If the array is first converted
to a 64-bit float (via astype), mean gives an answer that agrees with your looped-calculation value: 3045.7472500000002. With the original 32-bit array, averaging successively on one axis and then on
the other gives answers that agree with the 64-bit float answer to the second decimal place. In [125]: d = np.load('data.npy') In [126]: d.mean() Out[126]: 3067.0243839999998 In [127]: d64 = d.astype
('float64') In [128]: d64.mean() Out[128]: 3045.747251076416 In [129]: d.mean(axis=0).mean() Out[129]: 3045.7487500000002 In [130]: d.mean(axis=1).mean() Out[130]: 3045.7444999999998 In [131]:
np.version.full_version Out[131]: '2.0.0.dev-55472ca' -- On Tue, 2012-01-24 at 12:33 -0600, K.-MichaelA wrote:
I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be:
2-D data file here: http://dl.dropbox.com/u/139035/data.npy
Then: In [3]: data.mean() Out[3]: 3067.0243839999998
In [4]: data.max() Out[4]: 3052.4343
In [5]: data.shape Out[5]: (1000, 1000)
In [6]: data.min() Out[6]: 3040.498
In [7]: data.dtype Out[7]: dtype('float32')
A mean value calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend
version with the same results.
Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1 and my
whole setup is based on Enthought's EPD.
Best regards, Michael
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
-- -------------------------------------------------- Kathleen M. Tacina NASA Glenn Research Center MS 5-10 21000 Brookpark Road Cleveland, OH 44135 Telephone: (216) 433-6660 Fax: (216) 433-5802
On Jan 24, 2012, at 1:33 PM, K.-Michael Aye wrote:
I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be:
2-D data file here: http://dl.dropbox.com/u/139035/data.npy
Then: In [3]: data.mean() Out[3]: 3067.0243839999998
In [4]: data.max() Out[4]: 3052.4343
In [5]: data.shape Out[5]: (1000, 1000)
In [6]: data.min() Out[6]: 3040.498
In [7]: data.dtype Out[7]: dtype('float32')
A mean value calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend
version with the same results.
Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1 and my
whole setup is based on Enthought's EPD.
I get the same result: In [1]: import numpy In [2]: data = numpy.load('data.npy') In [3]: data.mean() Out[3]: 3067.0243839999998 In [4]: data.max() Out[4]: 3052.4343 In [5]: data.min() Out[5]:
3040.498 In [6]: numpy.version.version Out[6]: '2.0.0.dev-433b02a' This on OS X 10.7.2 with Python 2.7.1, on an intel Core i7. Running python as a 32 vs. 64-bit process doesn't make a difference. The
data matrix doesn't look too strange when I view it as an image -- all pretty smooth variation around the (min, max) range. But maybe it's still somehow floating-point pathological? This is fun too:
In [12]: data.mean() Out[12]: 3067.0243839999998 In [13]: (data/3000).mean()*3000 Out[13]: 3020.8074375000001 In [15]: (data/2).mean()*2 Out[15]: 3067.0243839999998 In [16]: (data/200).mean()*200 Out
[16]: 3013.6754000000001 Zach
Just what Bruce said. You can run the following to confirm: np.mean(data - data.mean()) If for some reason you do not want to convert to float64 you can add the result of the previous line to the
"bad" mean: bad_mean = data.mean() good_mean = bad_mean + np.mean(data - bad_mean) Val On Tue, Jan 24, 2012 at 12:33 PM, K.-Michael Aye <kmichael.aye@gmail.com>wrote:
I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be:
2-D data file here: http://dl.dropbox.com/u/139035/data.npy
Then: In [3]: data.mean() Out[3]: 3067.0243839999998
In [4]: data.max() Out[4]: 3052.4343
In [5]: data.shape Out[5]: (1000, 1000)
In [6]: data.min() Out[6]: 3040.498
In [7]: data.dtype Out[7]: dtype('float32')
A mean value calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend
version with the same results.
Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1 and my
whole setup is based on Enthought's EPD.
Best regards, Michael
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Thank you Bruce and all, I knew I was doing something wrong (should have read the mean method doc more closely). Am of course glad that's so easy understandable. But: If the error can get so big,
wouldn't it be a better idea for the accumulator to always be of type 'float64' and then convert later to the type of the original array? As one can see in this case, the result would be much closer
to the true value. Michael On 2012-01-24 19:01:40 +0000, Val Kalatsky said:
Just what Bruce said.
You can run the following to confirm: np.mean(data - data.mean())
If for some reason you do not want to convert to float64 you can add the result of the previous line to the "bad" mean: bad_mean = data.mean() good_mean = bad_mean + np.mean(data - bad_mean)
On Tue, Jan 24, 2012 at 12:33 PM, K.-Michael Aye <kmichael.aye@gmail.com> wrote: I know I know, that's pretty outrageous to even suggest, but please bear with me, I am stumped as you may be:
2-D data file here: http://dl.dropbox.com/u/139035/data.npy
Then: In [3]: data.mean() Out[3]: 3067.0243839999998
In [4]: data.max() Out[4]: 3052.4343
In [5]: data.shape Out[5]: (1000, 1000)
In [6]: data.min() Out[6]: 3040.498
In [7]: data.dtype Out[7]: dtype('float32')
A mean value calculated per loop over the data gives me 3045.747251076416 I first thought I still misunderstand how data.mean() works, per axis and so on, but did the same with a flattenend
version with the same results.
Am I really soo tired that I can't see what I am doing wrong here? For completion, the data was read by a osgeo.gdal dataset method called ReadAsArray() My numpy.__version__ gives me 1.6.1 and my
whole setup is based on Enthought's EPD.
Best regards, Michael
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@scipy.org http://mail.scipy.org/mailman/listinfo/numpy-discussion
Hi, Oddly, but numpy 1.6 seems to behave more consistent manner: In []: sys.version Out[]: '2.7.2 (default, Jun 12 2011, 15:08:59) [MSC v.1500 32 bit (Intel)]' In []: np.version.version Out[]:
'1.6.0' In []: d= np.load('data.npy') In []: d.dtype Out[]: dtype('float32') In []: d.mean() Out[]: 3045.7471999999998 In []: d.mean(dtype= np.float32) Out[]: 3045.7471999999998 In []: d.mean(dtype=
np.float64) Out[]: 3045.747251076416 In []: (d- d.min()).mean()+ d.min() Out[]: 3045.7472508750002 In []: d.mean(axis= 0).mean() Out[]: 3045.7472499999999 In []: d.mean(axis= 1).mean() Out[]:
3045.7472499999999 Or does the results of calculations depend more on the platform? My 2 cents, eat
On Wed, Jan 25, 2012 at 01:12:06AM +0200, eat wrote:
Or does the results of calculations depend more on the platform?
Floating point operations often do, sadly (not saying that this is the case here, but you'd need to try both versions on the same machine [or at least architecture/bit-width]/same platform to be
certain). David
Last active (days ago)
8 comments
7 participants
participants (7)
• Bruce Southey
• David Warde-Farley
• eat
• K.-Michael Aye
• Kathleen M Tacina
• Val Kalatsky
• Zachary Pincus | {"url":"https://mail.python.org/archives/list/numpy-discussion@python.org/thread/2LEXC5V55FR7QJY5KEDOAGDVRDXIOBN4/?sort=thread","timestamp":"2024-11-09T18:55:20Z","content_type":"text/html","content_length":"64598","record_id":"<urn:uuid:1bca7c7e-d821-4bc2-9b24-e0524a25b674>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00538.warc.gz"} |
Math in Practice
A Grade-by-Grade Guide for Teachers
Math in Practice, the new resource from Susan O’Connell and colleagues, is a comprehensive, grade-by-grade resource designed to fit with any math program or resource you are using. It is not a
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professional development.
Math in Practice bridges this gap with support for these key questions:
• How do we promote deeper, more thoughtful learning in math?
• Why should we approach math instruction differently?
• What resources are needed to do all of this effectively?
Each Grade Level text comes packaged with A Guide for Teachers.
The Guide for Teachers is the linchpin of the entire series. It lays out key instructional ideas and approaches, providing a foundation for the accompanying grade-level books. Throughout the Guide
for Teachers, you'll find what standards and research say about these topics, extensive support for effectively incorporating these strategies into your everyday instruction, and opportunities to
reflect on your teaching. Explore instructional strategies such as.
• Asking questions that stimulate student thinking
• Exploring math concepts through modeling
• Using formative assessment to guide instruction.
Grade level topic coverage may vary across Provinces in Canada.
Items marked with can only be purchased by schools and/or school districts. Please call 1-800-361-6128 for more information or to place an order. | {"url":"https://estore.pearsoncanadaschool.com/math-in-practice/","timestamp":"2024-11-08T20:49:32Z","content_type":"text/html","content_length":"82320","record_id":"<urn:uuid:29fa2403-abf5-481d-9dcb-e1ecf5702a2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00073.warc.gz"} |
Angular Velocity - Definition, Unit, Formula, Example, Videos
Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. It is also known as rotational velocity. Angular
velocity is measured in angle per unit time or radians per second (rad/s). The rate of change of angular velocity is angular acceleration. Let us learn in more detail about the relation between
angular velocity and linear velocity, angular displacement and angular acceleration.
Suggested Videos
Angular Velocity
Angular velocity plays an eminent role in the rotational motion of an object. We already know that in an object showing rotational motion all the particles move in a circle. The linear velocity of
every participating particle is directly related to the angular velocity of the whole object.
These two end up as vector products relative to each other. Basically, the angular velocity is a vector quantity and is the rotational speed of an object. The angular displacement of in a given
period of time gives the angular velocity of that object.
Browse more Topics Under System Of Particles And Rotational Dynamics
Relation Between Angular Velocity and Linear Velocity
For understanding the relation between the two, we need to consider the following figure:
The figure above shows a particle with its center of the axis at C moving at a distance perpendicular to the axis with radius r. v is the linear velocity of the particle at point P. The point P lies
on the tangent of the circular motion of the particle. Now, after some time(Δt) the particle from P displaces to point P1. Δθ or ∠PCP1 is the angular displacement of the particle after the
time interval Δt. The average angular velocity of the particle from point P to P1 = Angular displacement / Time Interval = Δθ/Δt
At smallest time interval of displacement, for example, when Δt=0 the rotational velocity can be called an instantaneous angular (ω) velocity, denoted as dt/dθ for the particle at position P.
Hence, we have ω = dt/dθ
Linear velocity (v) here is related to the rotational velocity (ω ) with the simple relation, v= ωr, r here is the radius of the circle in which the particle is moving.
Angular Velocity of a Rigid Body
This relation of linear velocity and angular velocity apply on the whole system of particles in a rigid body. Therefore for any number of particles; linear velocity v[i] = ωr[i]
‘i’ applies for any number of particles from 1 to n. For particles away from the axis linear velocity is ωr while as we analyze the velocity of particles near the axis, we notice that the value of
linear velocity decreases. At the axis since r=0 linear velocity also becomes a zero. This shows that the particles at the axis are stationary.
A point worth noting in case of rotational velocity is that the direction of vector ω does not change with time in case of rotation about a fixed axis. Its magnitude may increase or decrease. But
in case of a general rotational motion, both the direction and the magnitude of angular velocity (ω) might change with every passing second.
Browse more Topics under System Of Particles And Rotational Dynamics
Rotational Velocity of Revolutions
When a rigid body rotates around an axis, after the lapse of some time it completes a revolution. The time taken by that rigid body to complete a revolution is called the frequency of that body.
Rotational velocity and frequency hence have a relation between each other. Here one revolution is equal to 2π, hence ω= 2π/ T
The time taken to complete one revolution is T and ω= 2Ï€ f. ‘f’ is the frequency of one revolution and is measured in Hertz.
Angular Acceleration
When an object follows a rotational path, it is said to move in an angular motion or the commonly known rotational motion. In the course of such motion, the velocity of the object is always
changing. Velocity being a vector involves a movement of an object with speed that has direction. Now, since in a rotational motion, the particles tend to follow a circular path their direction at
every point changes constantly. This change results in a change in velocity. This change in velocity with time gives us the acceleration of that object.
Angular acceleration is a non-constant velocity and is similar to linear acceleration of translational motion. Understanding linear displacement, velocity, and acceleration are easy and this is why
when we intend to study rotational motion, we compare its vectors with translational motion. Like linear acceleration, angular acceleration (α) is the rate of change of angular velocity with time.
Therefore, α = dω/ dt
Now since for rotation about a fixed axis the direction of angular velocity is fixed therefore the direction of angular momentum α is also fixed. For such cases, the vector equation transforms
into a scalar equation.
Solved Question For You
 1. The angular velocity of a scooter tire of diameter 6 inches rotates 6 times a second is:
a. 16π     b.2π        c.12π            d.none
Solution: c) 12 π. We have: ω= 2π f. The frequency of the tire is 6 revolution per second;
Therefore we can write, ω = 2π × 6 = 12π | {"url":"https://www.toppr.com/guides/physics/system-of-particles-and-rotational-dynamics/angular-velocity-and-angular-acceleration/","timestamp":"2024-11-04T23:13:35Z","content_type":"text/html","content_length":"230237","record_id":"<urn:uuid:801e4f17-9254-41a4-a418-0f8b0aa4794a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00246.warc.gz"} |
Let us consider the impact of reversing the chain rule for differentiation when it comes to finding antiderivatives and indefinite integrals.
Recall that the chain rule states for differentiable functions $f(x)$ and $g(x)$, we have
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$
Expressing this in terms of integration, yields
$$\int f'(g(x)) \cdot g'(x) \, dx = f(g(x)) + C$$
Rephrased slightly, if $F(x)$ is an antiderivative of $f(x)$, then
$$\int f(g(x)) \cdot g'(x) \, dx = F(g(x)) + C$$
Of course to use this rule, we need to see to the right of the integral sign a composition with some expression $g(x)$ on the inside of that composition, along with the derivative of that $g(x)$ on
the outside of that composition, as a factor of the entire expression to be integrated.
Suppose we denoted by $u$ the inside of the composition. That is to say, $u = g(x)$.
Differentiating both sides and then multiplying by the differential $dx$ then tells us that $du = g'(x) \, dx$. This let's us write the above integral in a different way, as
$$\int f(u) \, du$$
which may be easier to integrate.
Let us consider a few examples of this process, called $u$-substitution, to make the technique more clear:
Find $\int \sqrt{3x+4} \, dx$
Noting that we seek the integral of a composition, let $u = 3x+4$ so that it equals the inside of that composition.
Then note that $du = 3 \, dx$, which immediately tells us that $dx = \frac{1}{3} \, du$.
Replacing the expressions in terms of $x$ (including $dx$) with their corresponding expressions in terms of $u$ then gives
$$\int \sqrt{3x+4} \, dx = \int \sqrt{u} \cdot \frac{1}{3} \, du$$
Pulling out the factor of $\frac{1}{3}$ and rewriting the $\sqrt{u}$ as a rational power reveals an expression easy to integrate.
$$\begin{array}{rcl} \int \sqrt{u} \cdot \frac{1}{3} \, du &=& \frac{1}{3} \int \sqrt{u} \, du\\ &=& \frac{1}{3} \int u^{1/2} \, du\\ &=& \frac{1}{3} \cdot \frac{2}{3} u^{3/2} + C\\ &=& \frac{2}{9} u
^{3/2} + C \end{array}$$
Finally, we would of course prefer to have our answer in terms of the original variable $x$, so let us appeal to $u = 3x+4$ one more time, as we conclude
$$\int \sqrt{3x+4} \, dx = \frac{2}{9} (3x+4)^{3/2} + C$$
Find $\int t (5+3t^2)^8 \, dt$
Again, we see a composition in the expression to be integrated. Let $u = 5 + 3t^2$, so it equals the inside of that composition.
Then note that $du = 6t \, dt$, so $t \, dt = \frac{1}{6} \, du$.
$$\begin{array}{rcl} \int t (5 + 3t^2)^8 \, dt &=& \int (5+32^2)^8 \cdot (t \, dt)\\ &=& \int u^8 \cdot (\frac{1}{6} \, du)\\ &=& \frac{1}{6} \int u^8 \, du\\ &=& \frac{1}{6} \cdot \frac{1}{9} u^9 +
C\\ &=& \frac{1}{54} u^9 + C \end{array}$$
Finally, to get our integral back in terms of the original variable $t$, we appeal to $u = 5 + 3t^2$ one more time to obtain:
$$\int t (5+3t^2)^8 \, dt = \frac{1}{54} (5 + 3t^2)^9 + C$$
Find $\int x^2 \sqrt{1+x} \, dx$
This time we may worry slightly at not seeing the derivative of the inside of the composition (or a simple multiple of it) sitting on the outside of the composition as a factor of the overall
expression to be integrated.
Sometimes such a worry is well-founded. However, here there is an easy way to resolve the situation.
Again pick $u$ to be the inside of the composition seen. So $u = 1+x$. This can be used not only to give us a way to "substitute out" the $dx$, as $du = dx$ -- but also to replace other $x$
expressions as well. Note, if $u = 1+x$, then $x = u-1$.
Hence, we can rewrite the integral in the following way:
$$\begin{array}{rcl} \int x^2 \sqrt{1+x} \, dx &=& \int (u-1)^2 u^{1/2} \, du\\ &=& \int (u^2 - 2u + 1) u^{1/2} \, du\\ &=& \int u^{5/2} \, du -2 \int u^{3/2} \, du + \int u^{1/2} \, du\\ &=& \frac
{2}{7} u^{7/2} - \frac{4}{5} u^{5/2} + \frac{2}{3} u^{3/2} + C \end{array}$$
Finally, using $u = 1+x$ one last time, we write the expression sought in terms of the original variable $x$:
$$\int x^2 \sqrt{1+x} \, dx = \frac{2}{7} (1+x)^{7/2} - \frac{4}{5} (1+x)^{5/2} + \frac{2}{3} (1+x)^{3/2} + C$$ | {"url":"https://mathcenter.oxford.emory.edu/site/math111/uSubstitution/","timestamp":"2024-11-10T17:25:25Z","content_type":"text/html","content_length":"8184","record_id":"<urn:uuid:11f1f49d-b090-4dca-9c15-d6e772fa1311>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00729.warc.gz"} |
Capital Flows and Growth: Some New Results | RDP 2002-03: International Financial Liberalisation and Economic Growth
RDP 2002-03: International Financial Liberalisation and Economic Growth 5. Capital Flows and Growth: Some New Results
5.1 Data and Methodology
Our empirical analysis employs annual data for a set of 40 countries, consisting of 20 developed and 20 emerging and developing countries in Asia, Latin America and Africa.^[9] The sample period
spans 1976–1995. The choice of countries and the sample period are dictated by data availability. The estimations use a panel regression framework, in which data for each country are averaged over
five non-overlapping years. Averaging the data for a number of years helps abstract from short-term business cycle effects and capture the longer-run effects of capital mobility on growth.
The base specification of our model is as follows:
The dependent variable growth is the annual growth rate in real GDP per capita for country i averaged over each 5-year interval t. The first set of explanatory variables includes our state variables
– the stock of human capital (proxied by the average years of secondary education in the adult population) and the level of real per capita GDP – both measured at the beginning of each 5-year
interval. In the neoclassical framework, the coefficient on the initial per capita GDP captures the rate of convergence (i.e., the rate at which poor countries catch up with rich countries) and is
expected to be negative.^[10]
The second set of explanatory variables includes a number of control variables that have been found to be important determinants of growth by previous studies. The coefficient on the openness to
trade variable (proxied by ratio of the sum of total exports and imports to GDP) is expected to be positive. Government consumption is expected to have a negative effect on growth. Similarly the
black-market exchange rate premium, which we use to proxy financial market distortions, is also expected to negatively affect growth.
The third set of variables includes the different capital flow measures we use. The broadest measure we use is total capital inflows. We also consider the three main components of capital inflows –
foreign direct investment, portfolio inflows and bank inflows. The different measures are entered sequentially into the regressions.
An important consideration in these regressions is the possible endogeneity of financial liberalisation and capital flows. As noted by Kraay (1998) there are two main sources of endogeneity. The
first is that capital flows themselves may be influenced by economic performance. If a country relaxes controls in ‘good’ times and imposes them in ‘bad’ times, we would find a spuriously large
positive effect of liberalisation on growth. Another source of endogeneity is that the extent of capital mobility may be correlated with other fundamental determinants of growth and investment. For
example, Grilli and Milesi-Ferretti (1995) observe that countries with small public sectors and relatively independent central banks are less likely to impose capital controls. If having a small
public sector and an independent central bank were good for growth, then the benefits of capital account liberalisation would be overstated. In principle, this problem can be addressed by using
instrumental variables that are correlated with financial openness, but uncorrelated with the disturbance term. Finding good instruments, however, is difficult.
In selecting the instruments for our estimations we draw on the literature on the determinants of capital flows. Following the work of Calvo et al (1993) a number of studies have sought to explain
the movements in capital flows by looking at the relative importance of the external (‘push’) factors and internal (‘pull’) factors. Their findings suggest that US interest rates have played a
dominant role in driving capital flows to developing countries. We also use total flows to developing countries to reflect broader supply-side factors. Other instruments include lagged capital flows,
lagged GDP growth, and change in the terms of trade.
5.2 Main Results and Discussion
Table 2 presents the main results from our regressions. Regression 2.1 is the base regression without the capital flow variables. The results are consistent with theory and previous empirical
findings. The coefficient on initial GDP per capita is negative and statistically significant suggesting strong convergence. Education has a positive effect on growth, but the coefficient is not
statistically significant. Openness to foreign trade has a positive and significant effect on growth. The coefficients on black market premium and government spending are both negative and
Table 2: Effect of Capital Flows on Economic Growth in Developed and Developing Countries
Dependent variable: growth rate of real per capita GDP
2.1 2.2 2.3 2.4 2.5
Total flows FDI Portfolio Bank loans
Initial GDP −0.056^*** −0.057^*** −0.046^*** −0.063^*** −0.063^***
(0.014) (0.013) (0.010) (0.011) (0.019)
Human capital 0.027 0.038^* 0.014 0.036^** 0.077^**
(0.021) (0.022) (0.013) (0.018) (0.034)
Government spending −0.259^*** −0.223^*** −0.146^** −0.280^*** −0.265^***
(0.075) (0.069)^*** (0.070) (0.072) (0.015)
International trade 0.041^*** 0.033^*** 0.035^*** 0.047^*** 0.037^**
(0.015) (0.015) (0.014) (0.014) (0.017)
Black market premium −0.034^*** −0.031^** −0.026^*** −0.037^*** −0.006
(0.011) (0.016) (0.010) (0.010) (0.017)
Capital flows 0.086^* 0.406^** 0.239^*** −0.271
(0.050) (0.176) (0.067) (0.319)
Adjusted R^2 0.53 0.63 0.59 0.59 0.56
No of observations 155 126 146 145 131
Notes: Two-stage least squares panel regressions for 1976–1995 using 5-year averages. Numbers in parenthesis are White heteroscedasticity robust standard errors. Instruments include US interest
rate, total capital flows to all countries in sample, current and lagged terms of trade, lagged capital flows and lagged GDP. Significance at 10%, 5% and 1% denoted by *, ** and *** respectively.
Regressions 2.2–2.5 augment the base regression with the different measures of capital flows. Total flows have a positive effect on growth, with the coefficient significant at the 10 per cent level.
Regressions 2.3–2.5 look at FDI, portfolio, and bank flows individually. Foreign direct investment and portfolio flows have a statistically significant positive effect on growth. Bank flows have a
negative but statistically insignificant effect.
Given our focus on the effect of capital flows on developing countries, we next consider the results for the developing countries in our sample (Table 3). The results for the base regression do not
differ markedly from those of the full sample. Capital flows, however, are found to have a negative effect on growth, though the coefficient is not statistically significant. As in the full sample
case, foreign direct investment and portfolio flows both have a statistically significant positive effect on growth. Bank flows are found to have a statistically significant negative effect on
growth. These results are also economically significant. For example, an increase in FDI of 1 percentage point would result in a 0.40 percentage point higher real per capita growth rate per year. A 1
percentage point increase in portfolio flows is associated with a 0.35 percentage point increase, whereas a 1 percentage point increase in bank inflows results in a 0.33 percentage point decline in
the real per capita GDP growth rate.
Table 3: Effect of Capital Flows on Economic Growth in Developing Countries
Dependent variable: growth rate of real per capita GDP
3.1 3.2 3.3 3.4 3.5
Total flows FDI Portfolio Bank loans
Initial GDP −0.044^*** −0.031^** −0.036^*** −0.052^*** −0.039^**
(0.016) (0.017) (0.014) (0.013) (0.026)
Human capital 0.021 0.026 0.020^* 0.029 0.063^**
(0.025) (0.025) (0.017) (0.021) (0.033)
Government spending −0.268^** −0.095^** −0.187^* −0.276 −0.122
(0.136) (0.109) (0.124) (0.130) (0.123)
International trade 0.041^** 0.035^** 0.034^** 0.047^** 0.025
(0.017) (0.018) (0.019) (0.021) (0.018)
Black market premium −0.033^*** −0.031^** −0.031^*** −0.036^*** −0.030^**
(0.015) (0.018) (0.011) (0.011) (0.014)
Capital flows −0.045 0.412^* 0.348^** −0.329^*
(0.104) (0.254) (0.194) (0.176)
Adjusted R^2 0.55 0.66 0.63 0.61 0.60
No of observations 75 58 67 67 58
Notes: As for Table 2.
Note that we do not include the investment rate in these regressions, even though investment is an important determinant of economic growth. This has implications for the interpretation of the effect
of capital flows on growth. The coefficient on the capital flow variables without investment captures the effect of capital flows on growth through all possible channels, including through
investment. The coefficient on capital flow variables with investment on the other hand, captures the effect of capital flows on growth above and beyond its effect on total investment. When
investment is included in the regression, the effect of FDI on growth is positive but no longer statistically significant at conventional levels. While the coefficient on portfolio flows becomes
marginally smaller, it is statistically significant at the 5 per cent level. This suggests that portfolio flows affect economic growth above and beyond their effect on domestic investment. The
coefficient on bank flows remains negative and statistically significant at the 5 per cent level.
In order to check the robustness of these results, we introduce a variety of changes to our specification. These include replacing the black market premium with the measure of the size of the banking
sector (bank assets/GDP), adding a measure of institutional strength (proxied by an index of law and contract enforcement), using a currency crisis dummy, and dummies for the 1980s to represent the
period of the debt crisis and the ‘lost decade’ for the Latin American countries. Our findings for FDI and portfolio flows remain fairly robust to these changes. While the coefficients on bank flows
remain negative, they are not always statistically significant.
Our findings are consistent with the conventional wisdom on the composition of capital flows. Foreign direct investment has historically played a larger role in developing countries than have other
forms of capital flows. Though some countries have experienced periods of large bank inflows, they haven't been sustained over time. For the countries in our sample, FDI constituted the largest
component of capital flows followed by portfolio flows and bank flows. During 1976–1998 average annual foreign direct investment represented 1.4 per cent of GDP, and portfolio flows and bank flows
were approximately 1.1 and 0.5 per cent of GDP. Similarly, simple measures of volatility indicate that FDI was the most stable form of capital flows, while bank flows were the most volatile. For
instance, the coefficient of variation of annual FDI, portfolio and banks flows to our sample countries during 1976–1998 was 1.2, 2.8 and 4.8 per cent respectively. Given that bank flows have been
small and volatile, it is likely that they have not made a meaningful contribution to investment. Our results also suggest that portfolio flows affect growth above and beyond their effect on
investment. While the identification of the exact channels is beyond the scope of this paper, the most likely channel (besides investment) through which foreign investment in the domestic equity and
debt markets could contribute to growth is through the development and deepening of these markets.
The hypothesis that the quality of domestic financial and regulatory institutions determines the effect of liberalisation on growth is not firmly supported by the data. Our attempts to test this
hypothesis by using alternative measures of institutional strength generally produce results that are either statistically insignificant or contradict the hypothesis. The measures we considered
included the ratio of liquid liabilities to GDP, the index of law and contract enforcement, and the index of the quality of countries' accounting and reporting standards. Our guess is that this is a
consequence of the incomplete and imprecise nature of these measures, and not because institutions do not play a role in this process.
Details are provided in Appendix A. [9]
This property derives from the assumption of diminishing returns to capital – economies that have less capital per worker (relative to their long-run ratio) tend to have higher rates of return and
higher growth rates. [10] | {"url":"https://www.rba.gov.au/publications/rdp/2002/2002-03/capital-flows-and-growth-some-new-results.html","timestamp":"2024-11-05T09:30:44Z","content_type":"application/xhtml+xml","content_length":"44419","record_id":"<urn:uuid:32bae098-f625-4f20-87c1-27e96f42def8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00378.warc.gz"} |
The Elongated Square Pyramid
The elongated square pyramid is the 8th Johnson solid (J8). It has 9 vertices, 16 edges, and 9 faces (4 equilateral triangles and 5 squares).
The elongated square pyramid can be constructed by attaching a square pyramid to a cube.
Attaching another square pyramid to the opposite end of the cube produces the elongated square bipyramid.
Here are some views of the elongated square pyramid from various angles:
Projection Envelope Description
Square Top view.
Pentagon Side view.
Pentagon Diagonal side view.
The Cartesian coordinates of the elongated square pyramid with edge length 2 are:
• (0, 0, 1+√2)
• (±1, ±1, ±1) | {"url":"http://www.qfbox.info/4d/J8","timestamp":"2024-11-04T11:00:53Z","content_type":"text/html","content_length":"9920","record_id":"<urn:uuid:c09e4930-8e5b-4b1a-a7f4-a2306117e8a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00600.warc.gz"} |
Higher Order Derivatives:
Dive deep into higher order derivatives, from basic concepts to real-world applications. Master calculation techniques, understand patterns, and apply your knowledge to solve complex problems in
various fields.
Higher order derivatives are a fundamental concept in calculus, extending our understanding of rate of change beyond the first derivative. Our introduction video serves as a crucial starting point,
offering a visual and intuitive grasp of this complex topic. This article delves deeper into higher order derivatives, exploring their definition, notation, and real-world applications. We'll unpack
how these derivatives allow us to analyze acceleration, jerk, and even more nuanced changes in various systems. From physics to economics, higher order derivatives play a vital role in modeling
complex phenomena. By mastering this concept, you'll gain powerful tools for advanced mathematical analysis. Whether you're a student grappling with calculus or a professional seeking to enhance your
analytical skills, understanding higher order derivatives is essential. Join us as we unravel the intricacies of this fascinating mathematical concept, building upon the foundation laid in our
introductory video.
A higher order derivative is the result of differentiating a function multiple times. For example, the second derivative is the derivative of the first derivative, the third derivative is the
derivative of the second derivative, and so on. These derivatives provide deeper insights into the behavior of functions, such as acceleration, jerk, and more complex rates of change.
Higher order derivatives can be notated using prime notation or superscript notation. In prime notation, f'(x) represents the first derivative, f''(x) the second, and f'''(x) the third. For higher
orders, superscript notation is often used, where f^(n)(x) represents the nth derivative of f(x).
Higher order derivatives have numerous applications in various fields. In physics, they're used to analyze motion, with the second derivative representing acceleration. In economics, they help in
cost analysis and optimization. In engineering, they're crucial for structural analysis and signal processing. They're also used in computer graphics for creating smooth curves and realistic
Yes, certain functions exhibit patterns in their higher order derivatives. For example, the exponential function e^x has all derivatives equal to itself. Trigonometric functions like sin(x) and cos
(x) show a repeating pattern every four derivatives. Polynomial functions have derivatives that decrease in degree until they become constant.
Higher order derivatives are fundamental in constructing Taylor series, which are used to approximate functions as polynomial expressions. Each term in a Taylor series involves a higher order
derivative of the function at a specific point, divided by a factorial. This allows for increasingly accurate approximations of complex functions using simpler polynomial expressions.
Understanding higher order derivatives is a crucial concept in calculus, but to truly grasp this topic, it's essential to have a solid foundation in several prerequisite areas. One of the fundamental
concepts you need to master is the rate of change. This concept forms the basis for understanding derivatives and how they represent the changing nature of functions.
As you delve deeper into derivatives, you'll encounter the power of a product rule, which is vital for differentiating more complex functions. This rule, along with others like the chain rule,
becomes increasingly important when dealing with higher order derivatives, as they allow you to break down complicated expressions into manageable parts.
When working with higher order derivatives, you'll often encounter various types of functions. Understanding how to work with polynomial function derivatives is crucial, as these form the basis for
many mathematical models. Similarly, being comfortable with trigonometric function derivatives opens up a whole new world of applications, particularly in physics and engineering.
Don't overlook the importance of exponential function derivatives either. These functions and their derivatives play a significant role in modeling growth and decay processes, which are fundamental
in many scientific fields.
As you progress to higher order derivatives, you'll find that these prerequisite topics intertwine in increasingly complex ways. For instance, you might need to apply the chain rule multiple times
when finding the second or third derivative of a composite function. Or, you might encounter a situation where you need to differentiate a product of polynomial and trigonometric functions, requiring
a combination of product rule and trigonometric differentiation skills.
Moreover, understanding these prerequisite topics doesn't just help with the mechanics of calculating higher order derivatives. They also provide crucial intuition about what these derivatives
represent. For example, while the first derivative gives you information about the rate of change of a function, the second derivative tells you about the rate of change of that rate of change. This
concept becomes much clearer when you have a solid grasp of basic rate of change principles.
In conclusion, mastering these prerequisite topics is not just about ticking boxes on a curriculum. It's about building a robust foundation that will enable you to tackle higher order derivatives
with confidence and understanding. Each of these topics contributes to your overall comprehension, making the journey into advanced calculus concepts smoother and more intuitive. So, take the time to
reinforce these fundamental concepts your future self will thank you when you're effortlessly calculating nth order derivatives! | {"url":"https://www.studypug.com/calculus-help/higher-order-derivatives","timestamp":"2024-11-07T07:44:46Z","content_type":"text/html","content_length":"389482","record_id":"<urn:uuid:f2b6cd9b-bdcc-46f5-b4ca-631e541749b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00026.warc.gz"} |
The Idea
Picture Book Edition - The Idea
You can find a lot of attempts to represent Pi digits in an artistic way for the last few years. One of these attempts was our MAGIC PIWORLD project (HAEL YGGS&JVS 1998) where we built a computer
tool for executing digit-colour and digit-geometrical transformations. Using this tool HAEL YGGS created several artistic objects (see HAEL YGGS site for details).
The basic representation of MAGIC PIWORLD was the so-called SQUARE WORLD where we have a M x N - Matrix and the shade of every square stands for 1 digit. Scaling the square down to 1 pixel we can see
more than 1.000.000 digits on the screen at a glance.
But alas! All this is virtual!
We want to hold real PI things in our hands.
In spring 2000, I persuaded my mother CHRISTA SCHMIDT to crochet a PI-carpet from a SQUARE WORLD pattern. And one year later this marvellous carpet was shown at the ZIRKUMFERENZ exhibition. The
carpet (3.14m x 3.14m) represent the first 2.500 decimal dgits of PI.
CHRISTA SCHMIDT - Pi Carpet
(Photo: Amira Fritz)
The FRIENDS OF PI took an active part in the exhibition. HAEL YGGS showed MAGIC PIWORLD generated Scotch prints on glass that represented some hundred thousands of PI digits. And Albert McWasi
printed THE PIBEL containing 10.000.000 PI digits.
After the ZIRKUMFERENZ my studies reached the outstanding book "PI - COMPUTER, ALGORITHMEN, ARITHMETIK" by Joerg Arndt and Christoph Haenel. The book includes a CD with 400.000.000 PI digits from the
YASUMASA-KANADA-LABS. And this was the moment I decided to create an object that should be based on this data: THE PICTURE BOOK. | {"url":"https://piworld.de/index.php/pi-presetations/pi-art-projects/picture-book-project/19-the-idea","timestamp":"2024-11-03T13:13:03Z","content_type":"text/html","content_length":"13668","record_id":"<urn:uuid:972fa966-602e-4ab6-b4ea-7c604adc434f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00118.warc.gz"} |
Debugging nonsymmetric square warning
One of the most common mistakes found when users face infeasibility or unexpected solutions is unintended definition of linear inequalities instead of semidefinite constraints. This is so common that
YALMP now tries to detect this mistake to warn about it.
Incorrect definition of variables
Although it is much more common to define intended full matrices as symmetric by mistake, sometimes users misunderstand the meaning of full vs symmetric in sdpvar. A typical mistake arise when trying
to prove stability of a linear system by finding a solution to the semidefinite programming problem \(A^TP + PA\preceq 0,~P\succeq I,~P = P^T\).
We purposely create this model incorrectly, and note when displaying the constraint (always do that in the initial phase of your model development!) that we do not have 2 semidefinite constraints but
two sets of elementwise constraints of dimension 4. A warning will be issued for both constraints. In the end, the solver will fail as this linear program is infeasible.
A = [-4 2;3 -4];
P = sdpvar(2,2,'full') %WRONG
Model = [A'*P + P*A <= 0, P >= eye(2), P == P']
| ID| Constraint| Coefficient range|
| #1| Element-wise inequality 4x1| 2 to 8|
| #2| Element-wise inequality 4x1| 1 to 1|
| #3| Equality constraint 2x2| 1 to 1|
The correct approach is to define a structurally symmetric matrix \(P\). Since we now have semidefinite constraints a semidefinite programming solver will be called and easily solve the problem.
P = sdpvar(2,2)
Model = [A'*P + P*A <= 0, P >= eye(2)]
| ID| Constraint| Coefficient range|
| #1| Matrix inequality 2x2| 2 to 8|
| #2| Matrix inequality 2x2| 1 to 1|
Mistakes in definition of constraints
The most common reason is some minor mistake with a misplaced transpose or similiar in the construction of a constraint. Consider once again a control problem where we want to find a positive
definite \(P \succeq I\) with
\[ \begin{pmatrix} A^T P + PA & PB \\ B^T P & 1-\gamma \end{pmatrix} \preceq 0\].
Since we are prone to make mistakes, we display the constraint object which correctly gives us a warning when we define it. The code below contains a mistake which turns the intended semidefinite
constraint into 9 elementwise constraints. Trying to solve this model leads to infeasibility.
A = [-4 2;3 -4];B = [1;1];
P = sdpvar(2,2);
sdpvar gamma
M = [A*P + P*A P*B;B'*P 1-gamma];
Model = [P >= 0, M <= 0]
| ID| Constraint| Coefficient range|
| #1| Matrix inequality 2x2| 1 to 1|
| #2| Element-wise inequality 9x1| 1 to 8|
Although it is easy to spot the mistake here (right!) we need a strategy in the general case. The warning will be issued when we create the Model object, so it is not obvious which of the two
constraints YALMIP deems suspcious. Define them separately to see this, or look at the generated constraint which lists the second constraint as an elementwise constraint instead of the intended
semidefinite constraints.
We thus know the matrix \(M\) accidentally has beome non-symmetric. To find the mistake in this matrix, it is convenient to use spy which shows non-zero elements in a matrix. Without any output, it
gives a graphical view. Alternatively catch the output and display it. \(M\) is supposed to be symmetric, so check this
s = full(spy(M - M'))
s =
3×3 logical array
The upper left block is not symmetric, and we can hone in on \( A^TP + PA\) to find the missing tranpose on \( A\).
Bad data
Sometimes the model is correctly constructed, the variables are correctly defined, but YALMIP still thinks the obviously symmetric matrix is non-symmetric and issues a warning about a full matrix
being used in a square constraint.
The typical cause then is numerical issues where floating-point limitations are causing small errors in computations causing a theoretically symmetric matrix to become non-symmetric in practice. For
this to happen, you model has to involve very bad data, and this is an issue you should adress first
We might have something like this
Z = A'*P+P*A
Linear matrix variable 20x20 (full, real, 210 variables)
Coeffiecient range: 1.7462e-10 to 2763983.0299
If we try to use this matrix in a constraint \(Z\succeq 0\) warnings about a full matrix in a square constraint will rightfully appear. To see the small terms causing the non-symmetry, we can look at
the distance to symmetry
Linear matrix variable 20x20 (symmetric, real, 210 variables)
Coeffiecient range: 1.7462e-10 to 8.6512e-10
Just as above, we can look at the pattern of \(Z-Z^T\) which in theory should be 0
To circumvent this, you should treat the root-cause with bad data which most likely will cause issues in the solver too, but a quick fix for the noise terms is to symmetrize the matrix which
hopefully will cancel the small terms
It is not a mistake!
Sometimes you want to add elementwise constraints on a square full matrix. To avoid this warning you have to make it into a non-square constraint such as
Z = A-B;
Model = [Z(:) >= 0]
Alternatively, if you think YALMIP is too clever, and you want to keep your code as it is, you can turn off the warning (not recommended) | {"url":"https://yalmip.github.io/inside/debuggingnonsymmetricsquare/","timestamp":"2024-11-09T17:26:27Z","content_type":"text/html","content_length":"32957","record_id":"<urn:uuid:971b7cb3-5fae-4919-a0df-a69caa56d471>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00313.warc.gz"} |
[Solved] (paragraph form each question) how do you | SolutionInn
Answered step by step
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(paragraph form each question) how do you formulate a strategy?, how do you implement a strategy? how do you evaluate a strategy?
(paragraph form each question) how do you formulate a strategy?,
how do you implement a strategy?
how do you evaluate a strategy?
There are 3 Steps involved in it
Step: 1
Formulating a Strategy To formulate a strategy I would follow a structured approach that involves several steps First I would identify the organizations goals and objectives as well as the external
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Seminars 2014
Date: Friday, November 14, 2014
Time: 4:00pm
Location: CTR Conference Room
Speaker: Dr. Elif Karatay, FPCE Postdoctoral Scholar, ME Department
Title: Mass and Momentum Transfer near Interfaces: From bubble surfaces to charged materials
Abstract: The transport phenomena at interfaces often determine the bulk transport rates and thereby determine the performance of both micro- or macro- scale systems. Therefore, a better
understanding of the fluid motion and the associated transport processes at these interfaces is essential for further optimization of various micro- and macro- scale technologies. Microfluidics offer
an ideal platform allowing for the integration of surfaces with precise and controllable interfaces and direct measurements of transport phenomena driven at these interfaces. Within this context, I
will present aspects of my past and ongoing research relevant to interfacial mass and momentum transport. The first part of the talk covers experimental and numerical investigations on the momentum
and mass transfer near gas-liquid interfaces established in hydrophobic silicon micro-grooves. The results reveal the impact of bubble geometry on hydrodynamic slippage and mass transfer rates of
solutes at curved bubble interfaces. In the second part of the talk, I will present our recent study on chaotic electrokinetic transport near ion selective boundaries with the objective of assessing
efficiency of commercial software for prediction of such phenomena. I will present comparisons of detailed statistics against a reference custom-made code that is tailored to the specific physics of
electrokinetic transport. Our results indicate that while accuracy can be guaranteed with proper mesh resolution and avoiding numerical dissipation, commercial solvers are generally at least an
order of magnitude slower than custom-made DNS codes. Finally I will conclude with remarks on our ongoing projects aiming at (i) understanding the role of buoyancy forces driven by the variations in
salt concentration in electrochemical systems and (ii) direct imaging of electrokinetic chaos in close proximity (~nm) of ion-selective interfaces.
Date: Friday, October 31, 2014
Time: 4:00pm
Location: CTR Conference Room
Speaker: Dr. Yongle Du, Post Doctoral Candidate, Aerospace Engineering, Penn State University
Title: A New CFD/CAA Methodology for Installed Jet Noise Simulations
Abstract: Over a half century of research has yielded a significant reduction of aircraft engine jet noise with innovative noise reduction devices, such as chevrons and fluidic injections. However,
experiments have shown that for installed jet engines, their noise reduction effects may be compromised, and the spectral directivity of the radiated noise is altered due to the complex flap-jet
interaction, airframe reflection/diffraction, etc. Therefore, understanding these installation effects is crucially important for the noise reduction design at the system level.
At present, estimate of the installed jet noise relies primarily on semi-empirical models. Although successful for simple isolated jets, the efficient approach using RANS solutions and noise source
models showed relatively large errors for installed jets. The more accurate hybrid method combining high-fidelity LES and computational aero-acoustic (CAA) techniques is still very expensive for the
complex airframe-jet configurations.
This presentation introduces a new CFD/CAA methodology under development for accurate yet efficient installed jet noise simulations. The CFD/CAA is closely coupled based on the recently developed
compact disturbance equations (CDE). The CDE solve the small disturbances about the given base flow. Various reduced governing equations, such as the linearized Navier-Stokes (LNS) and linearized
Euler equations (LEE) are included in the same formulation. A seamless switch of the equations can be made between the zones with different flow physics.
An installed jet noise simulation using this methodology is performed in two steps. The steady base flow is obtained first to determine the noise source region and the noise propagation region.
Existing methodologies can provide reasonably accurate solutions for complex configurations. A third-party solver and unstructured grids can be used to significantly reduce the computational cost.
Second, the coupled CFD/CAA solves the unsteady disturbances in the greatly reduced, simpler sub-domain of interest with an optimal grid design. The simulation restricts the computationally
expensive, high-fidelity LES in the confined source region (for example, near the jet shear layer and jet/flap interaction region), and applies the less expensive and more accurate LEE in the vast
noise propagation region.
Benchmark tests show that this is an accurate and affordable approach for complex installed jet noise simulations. With good base flow solutions, more accurate boundary treatments and less numerical
errors can be achieved in a greatly reduced, simpler CFD/CAA domain in terms of the small disturbances. Furthermore, the reduced equations bring an additional ~35% reduction of the computational
Date: Friday, October 24, 2014
Time: 4:00pm
Location: CTR Conference Room
Speaker: Dr. Manav Vohra, Post-Doctoral Candidate, Mechanical Engineering and Materials Science, Duke University
Title: Development of Reaction Models for Novel Energetic Materials
Abstract: Metallic multilayered systems referred to as energetic materials have attracted immense interest owing to their characteristic reaction properties. Rapid intermixing in the multilayers due
to steep concentration gradients and atomic diffusion at length scales of the order of tens of nanometers leads to a large amount of localized heat; thus making such energetic materials suitable for
joining applications such as welding and soldering. Moreover, the reaction self-propagates once initiated by means of a high energy source such as an electric spark. Numerous computational studies
have focused on capturing the transient reaction phenomena and understanding its dependence on the microstructure and composition of the multilayered systems. During the talk, I plan to discuss my
work on developing new models as well as refining the existing reaction models. In particular, the focus would be on calibrating intermixing rates in the multilayers using regression analysis and
Bayesian statistical approaches. Recent experimental investigations by Joress et al. [Appl. Phys. Lett. 101.11:111908, 2012] revealed that the oxidation of equimolar Zr-Al multilayers would help
extend the duration of heat release by three orders of magnitude as compared to conventional multilayered systems. A simplified computational model to reproduce experimental observations as well as
understand the kinetics of oxide layer growth in the equimolar Zr-Al system will also be presented.
Date: Friday, October 17, 2014
Time: 4:00pm
Location: CTR Conference Room
Speaker: Dr. Kwitae Chong, Post-Doctoral Candidate, Mechanical and Aerospace Engineering Department, University of California, Los Angeles
Title: Particle manipulation in viscous streaming
Abstract: A probe of circular cross section, undergoing rectilinear oscillation, creates large-scale steady circulatory cells by viscous streaming. We have shown that inertial particles can be
effectively trapped inside these streaming cells, regardless of particle size and density and Reynolds number. we extend this study to various arrangements of oscillating probes. High _delity
computations (Viscous Vortex Particle Method) are used to simulate the ow _eld. It is shown that, by controlling the sequence of starting and stopping the oscillation of individual probes, inertial
particles can be transported in a predictable manner between trapping points. In order to reduce the considerable expense of generating the ow _eld, we also explore the use of steady Stokes ow to
serve as an approximate surrogate for the ow between probes. The boundary conditions for this ow are obtained by matching with the inner Stokes layer solution.
Date: Friday, May 23, 2014
Time: 4:00pm
Location: CTR Conference Room
Speaker: Farid Karimpour, PhD Candidate, Civil and Environmental Engineering Department, Colorado State University
Title: Mixing in stably stratified wall-bounded turbulent flows: Insights and Modeling
Abstract: Stably stratified wall-bounded flows are ubiquitous in nature such as in estuaries, lakes, oceans and atmospheric boundary layer. In such flows, the simultaneous existence of density
stratification and solid wall results in anomalous mixing of momentum and active scalar (density) compared to other turbulent flows. Hence, there is no surprise that stratified wall-bounded flows are
usually considered as one of the most complex flows. The focus of this study is to understand and model stratified wall-bounded turbulent flows. The equilibrium assumption between the production rate
of the turbulent kinetic energy the dissipation rate of the turbulent kinetic energy and the dissipation rate of the turbulent potential energy is invoked to discuss a number of pertinent issues that
have direct implications for prediction of the turbulent mixing in stably stratified wall-bounded turbulence. Simple formulations for the flux Richardson number which is commonly considered as a
measure of the turbulent mixing in stratified flows and the turbulent viscosity are proposed. Further, RANS simulations of stratified wall-bounded flows are performed. The mixing of density in
stratified flows is usually modeled by employing a turbulent Prandtl number which is the linking bridge between the turbulent viscosity and diffusivity. Most parameterizations for Prandtl number are
developed based on data obtained from homogeneous shear flows. A one-dimensional stratified turbulent channel flow is modeled and the efficacy of a number of homogeneous turbulent Prandtl number
formulations are evaluated. The numerical simulation results highlight the inadequacy of such formulations. We introduce a modified parameterization for Prandtl number that takes into account the
inhomogeneity caused by the wall coupled with the effects of density stratification and evaluate its performance. Comparisons with data of direct numerical simulation of stably stratified channel
flow show remarkably good agreement.
Date: Friday, May 16, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: High-fidelity numerical simulations of compressible turbulence and mixing generated by hydrodynamic instabilities
Speaker: Pooya Movahed, PhD Candidate, Mechanical Engineering Department, University of Michigan, Ann Arbor
Abstract: The Rayleigh-Taylor (RT) instability occurs in a variety of applications at different scales, ranging from inertial confinement fusion to supernova explosion. In RT unstable configurations,
initial perturbations at a heavy-light interface may evolve to a turbulent mixing region, in a process in which the initial potential energy feeds the instability growth. Due to the gravitational
field and the density gradient, the resulting turbulence is anisotropic. However, these two effects are generally coupled, such that it is difficult to assess each one individually. The goal of this
study is to better understand anisotropy in turbulence through the Rayleigh-Taylor instability using direct numerical simulation (DNS). We use a novel set-up to study the temporal evolution of the
mixing region starting from an unperturbed material interface in an existing isotropic field in the presence and absence of gravity. First, we ignore gravity and focus on the temporal evolution of
the mixing region due to turbulence diffusion. This set-up allows us to study the role of density gradient across the mixing region individually. At large scales, the mixing region grows
self-similarly after an initial transient period; a one-dimensional turbulence-diffusion model in conjunction with Prandtl's mixing length theory is applied to describe the growth of the mixing
region. The observed growth exponent tends to 2/7, as expected for Batchelor turbulence based on energy budget arguments for large Reynolds numbers. At small scales, flow isotropy and intermittency
are measured. Results suggest that a large density ratio between the two fluids is required to make the velocity field anisotropic at the Taylor microscale, while the flow remains isotropic at the
Kolmogorov microscale. Second, we assess the role of gravity in a RT unstable configuration, in comparison to our first set of runs. Now, the baroclinic vorticity due to the gravitational field
provides energy driving the initial decaying isotropic field. A comparison of relevant physical quantities regarding isotropy and mixing is made between both cases. The role of different initial most
energetic wave numbers of the initial decaying field and Reynolds number are investigated. Current DNS are performed using a high-order accurate minimally dissipative kinetic-energy preserving and
interface capturing scheme.
Date: Friday, May 9, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: A fast pressure-correction method for simulating two-fluid flows and DNS of droplet-laden isotropic turbulence
Speaker: Antonino Ferrante, Assistant Professor in Aeronautics and Astronautics, University of Washington
Abstract: Direct numerical simulation (DNS) studies of droplet-laden turbulent flows have mostly been limited to sub-Kolmogorov (d < η) size droplets using the point-particle approach. DNS of
finite-size droplets (d > η), characteristic of the size of fuel droplets during secondary atomization, requires fully-resolving the flow inside and around the droplets while accounting for the
effects of surface tension. The main goal of the present study is to investigate via DNS the effects of finite-size deformable droplets on decaying isotropic turbulence.
In order to achieve this objective, first, we have developed a three-dimensional volume of fluid (VoF) method for tracking droplets accurately and efficiently in incompressible velocity fields. The
novelty of the developed approach is that besides conserving mass globally, a condition not always satisfied by VoF methods, mass conservation is also ensured locally while requiring half the number
of advection and reconstruction steps of conventional methods. Then, we have developed and coupled a new pressure-correction method with the VoF method for simulating incompressible two-fluid flows.
The method's main advantage is that the variable coefficient Poisson equation that arises in solving the incompressible Navier-Stokes equations for two-fluid flows is reduced to a constant
coefficient equation. This equation can then be solved directly using, e.g., the FFT-based parallel Poisson solver that we have developed for petascale supercomputers. For a 1024 mesh, our new
pressure-correction method using the FFT-based parallel Poisson solver is ten to forty times faster than the standard pressure-correction method using multigrid. In general, the new
pressure-correction method could be coupled with other interface advection methods such as level-set, phase-field, or front-tracking.
Our new pressure-correction/VoF flow solver has been verified up to density and viscosity ratios of 10,000 against theoretical results, validated against experimental results, and shown to conserve
mass, momentum, and kinetic energy in the inviscid limit. Finally, I will present results from DNS of non-evaporating droplet-laden isotropic turbulence and the effects of varying the droplet Weber
number and the density ratio on the time development of the turbulence kinetic energy budget.
Date: Friday, May 2, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: Linear solver and multiscale modeling methods for enhanced performance in cardiovascular simulations
Speaker: Mahdi Esmaily-Moghadam, PhD Student, Mechanical and Aerospace Engineering, University of California, San Diego
Abstract: Numerical simulations in cardiovascular disease present a number of important challenges that necessitate specialized algorithm development for enhanced performance in a high performance
computing environment. First, multiscale modeling methods to incorporate dynamic coupling between local hemodynamics and circulatory physiology result in ill-conditioned systems dominated by
eigenvalues coming from the coupled boundaries. Second, computational expense becomes increasingly important when coupling simulations to optimization and uncertainty quantification, and in problems
with fluid structure interaction. Third, high aspect ratio complex geometries present a challenge for conventional linear solvers. In this talk, I will present our recent developments in numerical
algorithms to reduce the cost of solving the linear system of equations arising from incompressible flow and fluid structure interaction problems, and efficiently and implicitly couple flow
simulations to reduced order circulatory system models. First, I will present a novel bi-partitioned algorithm for CFD and FSI problems that is shown to reduce computational cost compared to
standard GMRES methods. Second, I will discuss methods for implicitly coupling flow simulations with reduced order cardiovascular models, and introduce a specialized preconditioner for problems
involving coupled boundary conditions. Third, I will present an efficient data structure for handling iterative solver operations in parallel. Finally, I will demonstrate the effectiveness of the
presented algorithms through several relevant examples. I will then discuss ongoing research on a new multi-partitioned algorithm and my future plans.
Date: Friday, April 11, 2014
Time: 3:00pm
Location: CTR Conference Room
Title: Thermonuclear Ignition of ICF Capsules: Challenges and opportunities
Speaker: Dr. Baolian Cheng, Los Alamos National Lab
Abstract: Ignition is required to make fusion energy a viable alternative energy source. Recently National Ignition Facility (NIF) has achieved record high neutron yield (9e+15) in the high foot
inertial confinement fusion (ICF) capsule experiments. However, it is still far away from achieving the gain threshold and ignition. In this talk, I will present the ignition citron, scaling laws,
NIF successes, areas to improve and challenges as well as opportunities in both designs and simulations.
Date: Friday, April 4, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: The space-time correlation models for turbulent flows
Speaker: Prof. Guowei He, LNM, Institute of Mechanics, Chinese Academy of Sciences
Abstract: Space-time correlations are simple but fundamental measures of the relationships between turbulent fluctuations in space and time. The space-time correlation models are used to provide the
indispensable time scales for two-point closure approach and develop temporally accurate turbulence models. The most well-known model for space-time correlations is Taylor’s frozen flow model.
However, it has many limitations such as a weak shear rate and low turbulence intensity. In this talk, I will introduce our recent work on space-time correlations: (1) I will first introduce the EA
model for the space-time correlations in turbulent shear flows. The EA model accounts for both propagation and sweeping effects by a second approximation to iso-correleation contours, while Taylor’s
model is the first approximation. This model is verified by the DNS of turbulent channel flows and used to transform the temporally varying signals into spatially varying ones in the Rayleigh-Benard
experiments. (2) I will further introduce the swept-wave model for compressible turbulence. This model is the extension of the linear-wave propagation model originally developed by Lee, Lele and
Moin. It is shown that the temporal decorrelations in dilatational fluctuations are dominated by both random sweeping and wave propagation. The swept-wave model is validated by DNS of compressible
and isotropic turbulence and used in the EDQNM approach to derive the scaling of energy spectra.
Date: Friday, March 7, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: Separation of scales in spray combustion
Speaker: Dr. Javier Urzay, Research Associate, Flow Physics and Computational Engineering, Stanford University
Abstract: This talk addresses the analytical description of multiscale processes governing spray vaporization and combustion downstream from the injector in liquid-fueled burners. The focus of the
presentation is placed on the phenomena involved in the collective combustion of fuel sprays and their aerodynamic interactions with the surrounding flow rather than on the processes occurring at
droplet scale. Relevant spray-combustion scales and related dimensionless parameters are presented. Laminar problems are identified that can shed light on modeling different aspects of
spray-combustion phenomena. Besides consideration of spherical spray clouds, specific attention is given to group ignition in mixing layers and counterflow spray flames, including inertial effects of
droplets with order-unity Stokes numbers. The presentation ends with a brief account of some open problems and modeling challenges in need of additional work.
Date: Friday, February 14, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: Scale Coupling in Richtmyer-Meshkov Flows
Speaker: Prof. Snezhana I. Abarzhi, Carnegie Mellon University
Abstract: We systematically study the Richtmyer-Meshkov instability (RMI) induced by strong shocks for fluids with contrasting densities and with small and large amplitude initial perturbations
imposed at the fluid interface. The Smoothed particle hydrodynamics code (SPHC) is employed to ensure accurate shock capturing, interface tracking, and accounting for the dissipation processes.
Simulations results achieve good agreement with existing experiments and with the theoretical analyses including zero-order theory describing the post-shock background motion of the fluids, linear
theory providing RMI growth-rate in a broad range of the Mach and Atwood numbers, weakly nonlinear theory accounting for the effect of the initial perturbation amplitude on RMI growth-rate, and
highly nonlinear theory describing evolution of RM bubble front. We find that for strong-shock-driven RMI the background motion is supersonic, and the interfacial mixing can be sub-sonic or
supersonic. Significant part of the shock energy goes
into compression and background motion of the fluids, and only a small portion remains for interfacial mixing. The initial perturbation amplitude appears a key factor of RMI evolution. It strongly
influences the dynamics of the interface, in the fluid bulk, and the transmitted shock. In case of large amplitudes, the vector and scalar fields in the fluid bulk are non-uniform. The flow
heterogeneities include cumulative reverse jets, checkerboards velocity pattern, shock-focusing effects, and local hot spots with temperature substantially higher than that in the ambient. The
dynamics of the nonlinear flow is shown to have a multi-scale character.
Date: Friday, January 10, 2014
Time: 4:00pm
Location: CTR Conference Room
Title: Jet-noise reduction as an inverse problem
Speaker: Dr. Jeonglae Kim, Postdoctoral Research Associate, Mechanical and Aerospace Engineering, Cornell University
Abstract: Noise radiation from high temperature jet-engine exhaust is a prime interest to aviation industry and engine manufacturers. However, jet-noise reduction has not been sufficiently guided in
general based upon the true physics of turbulent jet; instead, empirical models or extensive parametric studies have been employed. In this study, adjoint-based optimization is utilized so that
controls which reduce the sound radiation of a Mach 1.3 turbulent jet are directly explored in conjunction with high-fidelity solutions of Navier–Stokes equations. Jet-noise reduction is formulated
as an optimization problem and adjoint of the linearized Navier–Stokes equations is used to provide the control sensitivity. The space–time resolved solutions of high-fidelity simulations are
formulated to determine noise-reducing controls, which model heat release effects near the nozzle exit. Controls found by the optimization procedure suppress the jet-noise radiation and make the jet
quieter. The simulation generates an unique set of acoustically loud and quiet states of the turbulent jet. Studies demonstrate that radiation of particularly loud acoustic waves is triggered by
large-scale interactions of axisymmetric, slowly-propagating disturbances.
Date: Tuesday January 7, 2014
Time: 2:00pm
Location: CTR Conference Room
Title: Environmental fluid mechanics: from density stratified to multi-phase flows
Speaker: Dr. Mona Rahmani, Postdoctoral Student, IFPEN Energies Nouvelles, France
Abstract: Large-scale phenomena in geophysical flows provide energy for stirring, mixing, and transport of mass and momentum. The input energy cascades down from macro to meso, and then to micro
scales, through chaotic processes. The future challenge for environmental fluid dynamists is to understand the interaction of these multi-scale processes. Environmental flows commonly exhibit these
features: density stratification caused by variations in temperature, salinity, or concentration of suspended particles, and addition of a secondary phase(s) such as solid particles, vegetation,
liquid droplets, or air bubbles to the primary phase of liquid or gas. This talk addresses some topics in environmental fluid mechanics of density stratified and multi-phase flows, that involve
studying natural processes at different scales. The first part of the talk focuses on direct numerical simulations of mixing caused by shear instabilities in density stratified flows. As the Reynolds
number increases a transition in the overall amount of mixing is found, which is in agreement with previous experimental studies. The effect of Prandtl number on mixing is studied to understand the
characteristics of high Prandtl number mixing events in the ocean; these cases have usually been approximated by low Prandtl number simulations. The increase in the Prandtl number has some
significant implications for the evolution of the flow, the time variation of mixing properties, and the overall mixing. The second part of the talk focuses on fully resolved simulations of
particulate flows. Path instabilities of the free motion of spherical and angular particles are examined for increasing Reynolds numbers. The results reveal mechanisms of path instabilities for
angular particles that are different from those for spherical ones. Some results of ongoing research on suspension of spherical particles is also discussed. The goal is to understand how the
micro-scale interactions of the particles influences the properties of the suspension as a whole. | {"url":"https://swap.stanford.edu/was/20160311035101mp_/https://ctr.stanford.edu/seminars-2014","timestamp":"2024-11-03T22:11:41Z","content_type":"text/html","content_length":"62313","record_id":"<urn:uuid:e28d92f2-efb7-4add-99fd-1d1196dc5999>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00100.warc.gz"} |
Introduction to Writing Ratios and Calculating Rates
What you’ll learn to do: Write ratios as fractions and calculate unit rates
When you go grocery shopping, do you sometimes find it difficult to compare prices when you’re choosing between packages of different sizes or quantities of an item? For example, imagine you want to
buy a block of cheese. One brand isĀ $4.99 for a 16-ounce brick. Another brand is on sale for $6.99 for a 24-ounce brick. Which one is the better deal? To make sure you get the most for your money,
you’ll need to figure out the price of cheese per ounce so that you can compare equal quantities. In this section, we’ll explore ratios and rates, which will help you calculate unit rates and unit
Before you get started, take this readiness quiz.
readiness quiz
If you missed this problem, review this video.
Divide: [latex]2.76\div 11.5[/latex]
Solution: [latex]0.24[/latex]
If you missed this problem, review the video below. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/ratios-and-rate/","timestamp":"2024-11-09T15:59:52Z","content_type":"text/html","content_length":"50653","record_id":"<urn:uuid:1d0eae4d-d50e-4000-af04-ef1d862c3658>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00705.warc.gz"} |
1,879 research outputs found
We discuss a maximally localized Wannier function approach for constructing lattice models from first-principles electronic structure calculations, where the effective Coulomb interactions are
calculated in the constrained random-phase-approximation. The method is applied to the 3d transition metals and a perovskite (SrVO_3). We also optimize the Wannier functions by unitary transformation
so that U is maximized. Such Wannier functions unexpectedly turned out to be very close to the maximally localized ones.Comment: 22 pages, 6 figure
We show a method to solve the problem of the brachistochrone as well as other variational problems with the help of the soap films that are formed between two suitable surfaces. We also show the
interesting connection between some variational problems of dynamics, statics, optics, and elasticity.Comment: 16 pages, 11 figures. This article, except for a small correction, has been submitted to
the American Journal of Physic
The Hartree-Fock equations are modified to directly yield Wannier functions following a proposal of Shukla et al. [Chem. Phys. Lett. 262, 213-218 (1996)]. This approach circumvents the a posteriori
application of the Wannier transformation to Bloch functions. I give a novel and rigorous derivation of the relevant equations by introducing an orthogonalizing potential to ensure the orthogonality
among the resulting functions. The properties of these, so-called a priori Wannier functions, are analyzed and the relation of the modified Hartree-Fock equations to the conventional,
Bloch-function-based equations is elucidated. It is pointed out that the modified equations offer a different route to maximally localized Wannier functions. Their computational solution is found to
involve an effort that is comparable to the effort for the solution of the conventional equations. Above all, I show how a priori Wannier functions can be obtained by a modification of the Kohn-Sham
equations of density-functional theory.Comment: 7 pages, RevTeX4, revise
A new formulation is presented for a variational calculation of $N$-body systems on a correlated Gaussian basis with arbitrary angular momenta. The rotational motion of the system is described with a
single spherical harmonic of the total angular momentum $L$, and thereby needs no explicit coupling of partial waves between particles. A simple generating function for the correlated Gaussian is
exploited to derive the matrix elements. The formulation is applied to various Coulomb three-body systems such as $e^-e^-e^+, tt\mu, td\mu$, and $\alpha e^-e^-$ up to $L=4$ in order to show its
usefulness and versatility. A stochastic selection of the basis functions gives good results for various angular momentum states.Comment: Revte
This study is focused on describing the molecular mechanism beyond the molecular picture provided by the evolution of molecular orbitals, valence bond structures along the reaction progress, or
conceptual density functional theory. Using bonding evolution theory (BET) analysis, we have deciphered the mechanism of the 1,3-dipolar rearrangement between acetonitrile oxide and (1S,2R,4S)
-2-cyano-7-oxabicyclo[2.2.1]hept-5-en-2-yl acetate derivatives. The BET study revealed that the formation of the C−C bond takes place via a usual sharing model before the O−C one that is also
formed in the halogenated species through a not very usual sharing model. The mechanism includes depopulation of the electron density at the N−C triple bond and creation of the V(N) and V(C)
monosynaptic basins, depopulation of the former C−C double bond with the creation of V(C,C) basins, and final formation of the V(O,C) basin associated with the O−C bond. The topological changes
along the reaction pathway take place in a highly synchronous way. BET provides a convenient quantitative method for deriving curly arrows and electron flow representation to unravel molecular
We report a first-principles theoretical study of hyperfine interactions, zero-point effects and defect energetics of muonium and hydrogen impurities in silicon and germanium. The spin-polarized
density functional method is used, with the crystalline orbitals expanded in all-electron Gaussian basis sets. The behaviour of hydrogen and muonium impurities at both the tetrahedral and
bond-centred sites is investigated within a supercell approximation. To describe the zero-point motion of the impurities, a double adiabatic approximation is employed in which the electron, muon/
proton and host lattice degrees of freedom are decoupled. Within this approximation the relaxation of the atoms of the host lattice may differ for the muon and proton, although in practice the
difference is found to be slight. With the inclusion of zero-point motion the tetrahedral site is energetically preferred over the bond-centred site in both silicon and germanium. The hyperfine and
superhyperfine parameters, calculated as averages over the motion of the muon, agree reasonably well with the available data from muon spin resonance experiments.Comment: 20 pages, including 9
figures. To appear in Phys. Rev.
We present a method that uses the one-particle density matrix to generate directly localized orbitals dedicated to multireference wave functions. On one hand, it is shown that the definition of local
orbitals making possible physically justified truncations of the CAS ~complete active space! is particularly adequate for the treatment of multireference problems. On the other hand, as it will be
shown in the case of bond breaking, the control of the spatial location of the active orbitals may permit description of the desired physics with a smaller number of active orbitals than when
starting from canonical molecular orbitals. The subsequent calculation of the dynamical correlation energy can be achieved with a lower computational effort either due to this reduction of the active
space, or by truncation of the CAS to a shorter set of references. The ground- and excited-state energies are very close to the current complete active space self-consistent field ones and several
examples of multireference singles and doubles calculations illustrate the interest of the procedur
We study a single particle which obeys non-relativistic quantum mechanics in R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2, then q > -2, and if N = 1, then q > -1.
The discrete eigenvalues E_{n\ell} may be represented exactly by the semiclassical expression E_{n\ell}(q) = min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) = ln(r). By
writing one power as a smooth transformation of another, and using envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are monotone increasing. Recent refinements to the
comparison theorem of QM in which comparison potentials can cross over, allow us to prove for n = 1 that Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q} is monotone
decreasing. Thus P(q) cannot increase too slowly. This result yields some sharper estimates for power-potential eigenvlaues at the bottom of each angular-momentum subspace.Comment: 20 pages, 5 figure | {"url":"https://core.ac.uk/search/?q=author%3A(S.%20F.%20Boys)","timestamp":"2024-11-09T16:50:32Z","content_type":"text/html","content_length":"162107","record_id":"<urn:uuid:73e02ef9-b2f8-49ed-addb-0827196104ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00507.warc.gz"} |
17.4 Bayesian \(p\)-values & model checking | An Introduction to Data Analysis
17.4 Bayesian \(p\)-values & model checking
The previous section showed how to approximate a \(p\)-value with Monte Carlo sampling. Notice that nothing in this sampling-based approach hinges on the model having no free parameters. Indeed, we
can similarly approximate so-called Bayesian predictive \(p\)-values. Bayesian predictive \(p\)-values have a good role to play in Bayesian data analysis: they are one possible tool for model
checking a.k.a. model criticism.
Suppose we have a Bayesian model for the binomial 24/7 data. The model consists of the usual likelihood function, but also has a prior (maybe from previous research, or maybe obtained from training
the model on a training data set):
\[ \theta_c \sim \text{Beta}(11,2) \]
Notice that this is a biased prior, placing more weight on the idea that the coin is biased towards heads. In model checking we ask whether the given model could be a plausible model for some data at
hand. We are not comparing models, we just “check” or “test” (!) the model as such. Acing the test doesn’t mean that there could not be much better models. Failing the test doesn’t mean that we know
of a better model (we may just have to do more thinking).
Let’s approximate a Bayesian predictive \(p\)-value for this Bayesian model and the 24/7 data. The calculations are analogous to those in the previous section.
# 24/7 data
k_obs <- 7
n_obs <- 24
# specify how many Monte Carlo samples to take
x_reps <- 500000
# build a vector of likelihoods (= the relevant test statistic)
# for hypothetical data observations, which are
# sampled based on the assumption that the
# Bayesian model to be tested is true
lhs <- map_dbl(1:x_reps, function(i) {
# hypothetical data assuming the model is true
# first sample from the prior
# then sample from the likelihood
theta_hyp <- rbeta(1, 11, 2)
k_hyp <- rbinom(1, size = n_obs, prob = theta_hyp)
# likelihood of that hypothetical observation
dbinom(k_hyp, size = n_obs, prob = theta_hyp)
# likelihood (= test statistic) of the observed data
# determined using MC sampling
lh_obs = map_dbl(1:x_reps, function(i){
theta_hyp <- rbeta(1, 11, 2)
dbinom(k_obs, size = n_obs, prob = theta_hyp)
}) %>% mean()
# proportion of samples with a lower or equal likelihood than
# the observed data
mean(lhs <= lh_obs) %>% show()
## [1] 0.000176
This Bayesian predictive \(p\)-value is rather low, suggesting that this model (prior & likelihood) is NOT a good model for the 24/7 data set.
We can use Bayesian \(p\)-values for any Bayesian model, whether built on a prior or posterior distribution. A common application of Bayesian \(p\)-values in model checking are so-called posterior
predictive checks. We compute a Bayesian posterior for observed data \(D_\text{obs}\) and then test, via a Bayesian posterior predictive \(p\)-value, whether the trained model is actually a good
model for \(D_\text{obs}\) itself. If the \(p\)-value is high, that’s no cause for hysterical glee. It just means that there is no cause for alarm. If the Bayesian posterior predictive \(p\)-value is
very low, the posterior predictive test has failed, and that means that the model, even when trained on the data \(D_\text{obs}\), is NOT a good model of that very data. The model must miss something
crucial about the data \(D_\text{obs}\). Better start researching what that is and build a better model if possible.
Most importantly, these considerations of Bayesian \(p\)-values show that frequentist testing has a clear analog in the Bayesian realm, namely as model checking. | {"url":"https://michael-franke.github.io/intro-data-analysis/bayesian-p-values-model-checking.html","timestamp":"2024-11-05T00:13:44Z","content_type":"text/html","content_length":"89131","record_id":"<urn:uuid:082e0fee-cba9-4e7b-9062-aaa89f460ee6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00556.warc.gz"} |
Installation Guide · JuMP
This guide explains how to install Julia and JuMP. If you have installation troubles, read the Common installation issues section below.
JuMP is a package for Julia. To use JuMP, first download and install Julia.
You can install the "Current stable release" or the "Long-term support (LTS) release."
• The "Current stable release" is the latest release of Julia. It has access to newer features, and is likely faster.
• The "Long-term support release" is an older version of Julia that has continued to receive bug and security fixes. However, it may not have the latest features or performance improvements.
For most users, you should install the "Current stable release," and whenever Julia releases a new version of the current stable release, you should update your version of Julia. Note that any code
you write on one version of the current stable release will continue to work on all subsequent releases.
For users in restricted software environments (for example, your enterprise IT controls what software you can install), you may be better off installing the long-term support release because you will
not have to update Julia as frequently.
JuMP is installed using the built-in Julia package manager. Launch Julia, and then enter the following at the julia> prompt:
julia> import Pkg
julia> Pkg.add("JuMP")
We recommend you create a Pkg environment for each project you use JuMP for, instead of adding lots of packages to the global environment. The Pkg manager documentation has more information on this
When we release a new version of JuMP, you can update with:
julia> import Pkg
julia> Pkg.update("JuMP")
JuMP depends on solvers to solve optimization problems. Therefore, you will need to install one before you can solve problems with JuMP.
Install a solver using the Julia package manager, replacing "HiGHS" by the Julia package name as appropriate.
julia> import Pkg
julia> Pkg.add("HiGHS")
Once installed, you can use HiGHS as a solver with JuMP as follows, using set_attribute to set solver-specific options:
julia> using JuMP
julia> using HiGHS
julia> model = Model(HiGHS.Optimizer);
julia> set_attribute(model, "output_flag" => false)
julia> set_attribute(model, "primal_feasibility_tolerance" => 1e-8)
Most packages follow the ModuleName.Optimizer naming convention, but exceptions may exist. See the README of the Julia package's GitHub repository for more details on how to use a particular solver,
including any solver-specific options.
Most solvers are not written in Julia, and some require commercial licenses to use, so installation is often more complex.
• If a solver has Manual in the Installation column, the solver requires a manual installation step, such as downloading and installing a binary, or obtaining a commercial license. Consult the
README of the relevant Julia package for more information.
• If the solver has Manualᴹ in the Installation column, the solver requires an installation of MATLAB.
• If the Installation column is missing an entry, installing the Julia package will download and install any relevant solver binaries automatically, and you shouldn't need to do anything other than
Solvers with a missing entry in the Julia Package column are written in Julia. The link in the Solver column is the corresponding Julia package.
Solver Julia Package Installation License Supports
Alpine.jl Triad NS (MI)NLP
Artelys Knitro KNITRO.jl Manual Comm. (MI)LP, (MI)SOCP, (MI)NLP
BARON BARON.jl Manual Comm. (MI)NLP
Bonmin AmplNLWriter.jl EPL (MI)NLP
Cbc Cbc.jl EPL (MI)LP
CDCS CDCS.jl Manualᴹ GPL LP, SOCP, SDP
CDD CDDLib.jl GPL LP
Clarabel.jl Apache LP, QP, SOCP, SDP
Clp Clp.jl EPL LP
COPT COPT.jl Comm. (MI)LP, SOCP, SDP
COSMO.jl Apache LP, QP, SOCP, SDP
Couenne AmplNLWriter.jl EPL (MI)NLP
CPLEX CPLEX.jl Manual Comm. (MI)LP, (MI)SOCP
CSDP CSDP.jl EPL LP, SDP
DAQP DAQP.jl MIT (Mixed-binary) QP
DSDP DSDP.jl DSDP LP, SDP
EAGO.jl MIT (MI)NLP
ECOS ECOS.jl GPL LP, SOCP
FICO Xpress Xpress.jl Manual Comm. (MI)LP, (MI)SOCP
GLPK GLPK.jl GPL (MI)LP
Gurobi Gurobi.jl Manual Comm. (MI)LP, (MI)SOCP
HiGHS HiGHS.jl MIT (MI)LP, QP
Hypatia.jl MIT LP, SOCP, SDP
Ipopt Ipopt.jl EPL LP, QP, NLP
Juniper.jl MIT (MI)SOCP, (MI)NLP
Loraine.jl MIT LP, SDP
MadNLP.jl MIT LP, QP, NLP
MAiNGO MAiNGO.jl EPL 2.0 (MI)NLP
Manopt.jl MIT NLP
MiniZinc MiniZinc.jl Manual MPL-2 CP-SAT
Minotaur AmplNLWriter.jl Manual BSD-like (MI)NLP
MOSEK MosekTools.jl Manual Comm. (MI)LP, (MI)SOCP, SDP
NLopt NLopt.jl GPL LP, QP, NLP
Octeract AmplNLWriter.jl Comm. (MI)NLP
Optim.jl MIT NLP
OSQP OSQP.jl Apache LP, QP
PATH PATHSolver.jl MIT MCP
Pajarito.jl MPL-2 (MI)NLP, (MI)SOCP, (MI)SDP
Pavito.jl MPL-2 (MI)NLP
Penbmi Penopt.jl Comm. Bilinear SDP
Percival.jl MPL-2 NLP
PolyJuMP.KKT PolyJuMP.jl MIT NLP
PolyJuMP.QCQP PolyJuMP.jl MIT NLP
ProxSDP.jl MIT LP, SOCP, SDP
RAPOSa AmplNLWriter.jl Manual RAPOSa (MI)NLP
SCIP SCIP.jl Apache (MI)LP, (MI)NLP
SCS SCS.jl MIT LP, QP, SOCP, SDP
SDPA SDPA.jl, SDPAFamily.jl GPL LP, SDP
SDPLR SDPLR.jl GPL LP, SDP
SDPNAL SDPNAL.jl Manualᴹ CC BY-SA LP, SDP
SDPT3 SDPT3.jl Manualᴹ GPL LP, SOCP, SDP
SeDuMi SeDuMi.jl Manualᴹ GPL LP, SOCP, SDP
SHOT AmplNLWriter.jl EPL (MI)NLP
StatusSwitchingQP.jl MIT LP, QP
Tulip.jl MPL-2 LP
• LP = Linear programming
• QP = Quadratic programming
• SOCP = Second-order conic programming (including problems with convex quadratic constraints or objective)
• MCP = Mixed-complementarity programming
• NLP = Nonlinear programming
• SDP = Semidefinite programming
• (MI)XXX = Mixed-integer equivalent of problem type XXX
• CP-SAT = Constraint programming and Boolean satisfiability
Developed a solver or solver wrapper? This table is open for new contributions. Edit the installation.md file, and use the checklist Adding a new solver to the documentation when opening the pull
Developing a solver or solver wrapper? See Models and the MathOptInterface docs for more details on how JuMP interacts with solvers. Please get in touch via the Developer Chatroom with any questions
about connecting new solvers with JuMP.
Use AmplNLWriter to access solvers that support the NL format.
Some solvers, such as Bonmin, Couenne and SHOT can be installed via the Julia package manager. Others need to be manually installed.
Consult the AMPL documentation for a complete list of supported solvers.
Use GAMS.jl to access solvers available through GAMS. Such solvers include: AlphaECP, Antigone, BARON, CONOPT, Couenne, LocalSolver, PATHNLP, SHOT, SNOPT, SoPlex. See a complete list here.
Use NEOSServer.jl to access solvers available through the NEOS Server.
When in doubt, run import Pkg; Pkg.update() to see if updating your packages fixes the issue. Remember you will need to exit Julia and start a new session for the changes to take effect.
Each package is versioned with a three-part number of the form vX.Y.Z. You can check which versions you have installed with import Pkg; Pkg.status().
This should almost always be the most-recent release. You can check the releases of a package by going to the relevant GitHub page, and navigating to the "releases" page. For example, the list of
JuMP releases is available at: https://github.com/jump-dev/JuMP.jl/releases.
If you post on the community forum, please include the output of Pkg.status().
Did you get an error like Unsatisfiable requirements detected for package JuMP? The Pkg documentation has a section on how to understand and manage these conflicts.
Another common complaint is that after adding a new package, code that previously worked no longer works.
This usually happens because the new package is not compatible with the latest version of JuMP. Therefore, the package manager rolls-back JuMP to an earlier version. Here's an example.
First, we add JuMP:
(jump_example) pkg> add JuMP
Resolving package versions...
Updating `~/jump_example/Project.toml`
[4076af6c] + JuMP v0.21.5
Updating `~/jump_example/Manifest.toml`
... lines omitted ...
The + JuMP v0.21.5 line indicates that JuMP has been added at version 0.21.5. However, watch what happens when we add JuMPeR:
(jump_example) pkg> add JuMPeR
Resolving package versions...
Updating `~/jump_example/Project.toml`
[4076af6c] ↓ JuMP v0.21.5 ⇒ v0.18.6
[707a9f91] + JuMPeR v0.6.0
Updating `~/jump_example/Manifest.toml`
... lines omitted ...
JuMPeR gets added at version 0.6.0 (+ JuMPeR v0.6.0), but JuMP gets downgraded from 0.21.5 to 0.18.6 (↓ JuMP v0.21.5 ⇒ v0.18.6)! The reason for this is that JuMPeR doesn't support a version of JuMP
newer than 0.18.6.
Pay careful attention to the output of the package manager when adding new packages, especially when you see a package being downgraded. | {"url":"https://jump.dev/JuMP.jl/stable/installation/","timestamp":"2024-11-09T15:51:08Z","content_type":"text/html","content_length":"63829","record_id":"<urn:uuid:81fa3ce3-a5a6-4d85-bbda-2f235de05855>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00384.warc.gz"} |
Rounding Up the Research Results
So it appears velocity routing was handled 1-deep in the
Can paper
(but you have to look carefully at 3.3 to notice it) although this is for a hypercube and explored with synthetic numbers only. There implementation closely matches (if not identically) the method I
proposed which boils down to "maximum ratio of progress to rtt". Then there is discussion of optimizing shortcuts by using latency and even the convergence of the
. Though his metrics really don't make much sense, I quote "To select the candidate set, the distance to the destination is expressed in binary notation, and neighbor i is chosen to the set if there
is a 1 in the ith position." I have NO idea what that has to do with proximity routing and perhaps it is revealed in their results where they suggest that proximity routing has no advantage when used
with proximity-based shortcuts. Finally, all the results are based on very synthetic benchmarks and I can easily lits many cases where PR definitely help PS, think of the case where I am surrounded
by all but 1 low latency node, if a packet is incoming, it has a 1 / N chance of coming in on a low latency branch and (N-1)/N chance of not. Considering that routes in a stable system are static,
that could lead to very poor performance! So while PS helps traversing the ring faster, PR helps when we're getting closer. Perhaps there is a way to reconcile this.
So what is left undone: we have a system where there are 2*K near neighbors, when we are routing nearby, most likely the packet will be routed through one of them. If we make the assumption that each
node also has at miniumum 2*K neighbors we can approximate how many hops it would take for this packet to arrive at the remote node. I propose that we may see a reduction in hops and latency if we
come up with some heuristic to evaluate our latency to our neighbors and the potential for increased hop count and make a routing decision based upon that. If we know our neighbors, neighbors and
their latency to them, we could further develop this routing mechanism to bring us even closer.
What makes this relevant? Well I guess that will be another post.
1 comment:
1. Golden Goose Outlet, Canada Goose Outlet, Golden Goose Shoes, Golden Goose Sale, Golden Goose Sneakers, Moncler Outlet, Valentino Shoes, Golden Goose Sneakers, Golden Goose | {"url":"http://blog.davidwolinsky.com/2008/12/rounding-up-research-results.html","timestamp":"2024-11-07T09:47:06Z","content_type":"application/xhtml+xml","content_length":"44922","record_id":"<urn:uuid:3ae2f58f-7542-4110-9fd0-e63ff1733ca6>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00224.warc.gz"} |
Impedance Matching Network Design for Class C Power Amplifier
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Impedance Matching Network Design for Class C Power
Amenah I. Kanaan
M.Sc In Electronic And Communication Engineering
Assistant Lecturer
Electronics Engineering College
Electronic Department
Mosul University / Iraq
Power amplifiers (PA) are typically the most power-consuming building blocks of RF transceivers. Therefore, the design of a high-efficiency radio frequency power amplifier is the most obvious
solution to overcoming the battery lifetime limitation in the portable communication systems. In order to obtain the maximum output power, the reference impedance (usually 50 Ohm) must be transformed
to the optimum input and output impedance of the selected transistor. Matching networks are therefore necessary at the input and at the output of a power amplifier circuit. In this research we
designed a class C power amplifier operates in frequency range (200MHz -500MHz) with input power
0.63watt and output power 10watt. At high radio frequencies, the spurious elements (like wire inductances, interlayer capacitances, and conductor resistances) have a significant yet unpredictable
impact on the matching network. In our design the matching network for input and output is implementing for frequency range (300MHz-350MHz) because of the wide band frequency range for transistor
used. Two ways to implement the matching network: Theoretical calculations method (smith chart) and simulations using computer programs method are often presented.
Keywords: amplifier design, High frequencies, microwave amplifiers, Class C power amplifier, smith chart, voltage standing wave reflection (VSWR).
نﺎ ﻓ ﻚﻟﺬ ﻟ .
يﻮ ﯾداﺮﻟا ﻞ ﺳﺮﻤﻟا ﻞﺒﻘﺘ ﺴﻤﻟا ﻲ ﻓ ةرﺪ ﻘﻠﻟ ﺎﻛﻼﮭﺘﺳا ﺮﺜﻛ ﻷا ﺔﯿﺟذﻮﻤﻨﻟا ءﺎﻨﺒﻟا ﺐﻟاﻮﻗ ﻞﺜﻤﺗ ةرﺪﻘﻟا تاﺮﺒﻜﻣ
ﺔﯾرﺎﻄﺒﻟا ﻞﻤﻋ ﺪﯿﯿﻘﺗ ﻰﻠﻋ ﺐﻠﻐﺘﻠﻟ ﺎﺣﻮﺿو ﺮﺜﻛﻷا ﻞﺤﻟا ﻞﺜﻤﯾ ﺔﯿﻟﺎﻌﻟا تاددﺮﺘﻟا ﻦﻤﺿ ﻞﻤﻌﯾ ﺔﯿﻟﺎﻋ ةءﺎﻔﻛ وذ ةرﺪﻗ ﺮﺒﻜﻣ ﻢﯿﻤﺼﺗ
جاﺮﺧﻹاو لﺎﺧدﻹا ﺔﻌﻧﺎﻤ ﻣ ﻰﻟإ لﻮﺤﺗ نإ ﺐﺠﯾ ( موا 50 نﻮﻜﺗﺎﻣ ةدﺎﻋ ) رﺪﺼﻤﻟا ﺔﻌﻧﺎﻤﻣ نﺎﻓ ﻚﻟﺬﻟ ،ﺔﻟﺎﻘﻨﻟا لﺎﺼﺗﻻا ﺔﻤﻈﻧأ ﻲﻓ
ﻲ ﻓ .
ةرﺪ ﻘﻟا ﺮ ﺒﻜﻤﻟ جاﺮ ﺧﻹاو لﺎ ﺧدﻹا ﺮ ﺋاود ﻲﻓ ﺔﯾروﺮﺿ نﻮﻜﺗ ﺔﻤﺋاﻮﻤﻟا تﺎﻜﺒﺷ نﺎﻓ ﻚﻟﺬﻟ .
رﺎﺘﺨﻤﻟا رﻮﺘﺳﺰﻧاﺮﺘﻠﻟ ﺔﯿﻟﺎﺜﻤﻟا
C ىﺪ ﻤﻟا ﻦﻤ ﺿ ﻞ ﻤﻌﯾ (200MHz -500MHz) . ( طاو 10 ) جاﺮﺧا ةرﺪ ﻗو ( طاو 0.63
) لﺎ ﺧدإ ةرﺪ ﻘﺑ ﺎﻨﻤﻗ ﺚﺤﺒﻟا اﺬھ
تﺎﻌ ﺴﺘﻣو ﺔ ﯿﺋﺎﺑﺮﮭﻜﻟا ﻚﻠ ﺴﻟا ﺔ ﺛﺎﺤﻣ ﻞ ﺜﻣ ﺔ ﯿﻘﯿﻘﺤﻟا ﺮ ﯿﻏ ﺮ ﺻﺎﻨﻌﻟا ، ﺔ ﯿﻟﺎﻌﻟا ﺔ ﯾﻮﯾداﺮﻟا ت اددﺮ ﺘﻟا ﻲ ﻓ ةرﺪ ﻗ ﺮ ﺒﻜﻣ ﻢﯿﻤ ﺼﺘﺑ
ﻢﺗ ﻢﯿﻤ ﺼﺘﻟا اﺬ ھ ﻲ ﻓ .
ﺔ ﻤﺋاﻮﻤﻟا تﺎﻜﺒ ﺷ ﻰ ﻠﻋ نﻵا ﺪ ﺤﻟ ﺎﮭﺑ ﺆﺒﻨﺘﻟا ﻦﻜﻤﯾﻻ تاﺮﯿﺛﺄﺗ ﺎﮭﻟ ﻞﺻﻮﻤﻟا ﺔﻣوﺎﻘﻣو ﺔﯿﻠﺧاﺪﻟا تﺎﻘﺒﻄﻟا
(300MHz -350MHz) .
مﺪﺨﺘ ﺴﻤﻟا رﻮﺘ ﺳﺰﻧاﺮﺘﻠﻟ تاددﺮ ﺘﻟا ﺔ ﻣﺰﺣ ضﺮ ﻋ ﺐﺒ ﺴﺑ لﺎ ﺧدﻺﻟ ﺔ ﻤﺋاﻮﻤﻟا تﺎﻜﺒ ﺷ ﺬ ﯿﻔﻨﺗ
ﺚﻤﺳ ﻂﻄﺨﻣ ) ﺔﯿﻄﯿﻄﺨﺘﻟا ﺔﻘﯾﺮﻄﻟا : ﺔﻤﺋاﻮﻤﻟا تﺎﻜﺒﺷ ﺬﯿﻔﻨﺘﻟ نﺎﺘﻘﯾﺮﻃ ﺚﺤﺒﻟا اﺬھ ﻲﻓ ﺖﻣﺪﻗ ﻦﻣ حواﺮﺘﺗ تاددﺮﺘﻟ جاﺮﺧﻹاو
ﺔﺟﻮﻣ سﺎﻜﻌﻧا ،ﺚﻤﺳ ﻂﻄﺨﻣ ، C
بﻮﺳﺎﺤﻟا ﺞﻣاﺮﺑ ماﺪﺨﺘﺳﺎﺑ ةﺎﻛﺎﺤﻤﻟا ﺔﻘﯾﺮﻃو
عﻮﻧ ةرﺪﻗ ﺮﺒﻜﻣ ،ﻒﯾووﺮﻜﯾﺎﻤﻟا تاﺮﺒﻜﻣ ،ﺔﯿﻟﺎﻌﻟا تاددﺮﺘﻟا ،ﺮﺒﻜﻣ ﻢﯿﻤﺼﺗ : ﺔﻟاﺪﻟا تﺎﻤﻠﻜﻟا
ﺔﻔﻗاﻮﻟا ﺔﯿﺘﻟﻮﻔﻟا
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
1. Introduction
Solid-state microwave amplifiers play an important role in communication. Usually, signals provided by the transducers are weak; typically, it is in the order of microvolt (µV) or millivolt (mV). It
is not easy, and sometimes not possible, to have reliable processing for signals with low levels. For this reason, the need for a signal amplifier arises. In a transceiver circuit, a signal amplifier
has different applications, including low noise, high gain, and high power amplifiers [1].
The intent of the research reported in this thesis is three-fold: to survey an amplifier classifications and definitions, to give an overview of some basic principles used in the analysis and design
of the microwave transistor amplifier, and to design high efficiency power amplifiers for possible use in portable cellular telephone units.
Most RF power amplifiers fit into one of six common classes: A,B,C,D,E, or F. The distinctions between these classes lie primarily in the biasing conditions of the transistor and the design of the
output network that couples the drain to the load. Each class has its own strengths and weaknesses, and choosing a class amounts to compromising between various power amplifier figures of merit,
which include gain, linearity, and efficiency. For example,
Class A and B power amplifiers offer high gain and a wide linear range, but are inefficient.
On the other hand, class E and F power amplifiers can achieve high efficiency but do not provide linear amplification.
The applications of our proposed device include many products in the field of microwave communications. One of the important applications of a Microwave power amplifier is in the output stage of a
transmitter where a signal needs amplification before it is transmitted. A high power amplifier is needed for transmitting a signal through an antenna and a medium.
The Microwave power amplifier amplifies the input signal after the signal has been modulated in the transmitter. The High power amplification step is necessary for every application of antenna
transmission [2].
2. Class C Power Amplifier
The operating point of a class C power amplifier is located between zero and the pinchoff point in the transfer characteristic of an enhancement FET device. The conduction angle of a class C power
amplifier is between 0 and π. The output waveform of a class C power amplifier using FET devices is shown in Fig. 1. Clearly, the drain-source voltage can also swing over its maximum range of zero to
2Udd [3, 4].
On the other hand, the entire negative part and a fraction of the positive part of the drain current are cut off; the current waveform is reduced to a train of short pulses, which have lower DC
component compared to the other classes of power amplifiers mentioned above, but also a lower fundamental RF component. Consequently, very high efficiencies can be obtained, but at the expense of
lower RF output power and heavy input drive requirements.
The maximum drain efficiency of a class C power amplifier can even reach 100 % [4], if the operating points close to the zero point are selected.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822 wt wt
Fig .1. Waveforms of a class C power
For this design a class C power amplifier was chosen for various reasons. The class C amplifier is biased below its turn-on voltage and the input drives the device on for a small portion, which is
less than half of the input cycle. This results into a pulsed current in the device. This current is filtered to extract the fundamental frequency component, which is then passed to the resistive
load. The output waveform is thus at the fundamental frequency. It was noted that power consumption is a high concern for RF applications. Choosing a class D, E, or F design would then result in
higher efficiency. This, in turn will lower power consumption. Despite the obvious benefit an agreement was reached to move forward with a class C design and perform optimization to maximize the
achievable efficiency.
We were encouraging of our decision because of the several reasons why class C can be considered an optimal choice for design. Compared to the A-B classes, there is significantly increased efficiency
for a relative light degradation of linearity. As well, the class C is preferred over the switch mode PA’s for the following reasons:
The output amplitude of the class C varies with a varied input level, whereas the output amplitude of the switch mode PA’s is fixed, relative to the input amplitude.
Once the input is large enough, the class C PA switches on and stays on, whereas the switch mode PA’s need voltage regulators for effective switching. This adds more complexity to the block.
The class C PA is able to transmit different power levels at different times, whereas the power levels of switch mode PA’s are fixed [5].
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
4. Matching Circuits and High Power Components
Matching circuits are an important part of the design of high-power RF amplifiers. A number of different types are discussed in the following sections along with components capable of withstanding
the voltage and current stresses encountered.
4.1 Transmission Line Matching
Figures (2) through (4) show a number of matching methods classified as transmission line transformers.
Figure 2(a) is a quarter-wave transformer whose characteristic impedance (Z0) equals the square root of the product of Zin × ZL.
Figure 2(b) is a quarter-wave transmission line used as a balanced to unbalanced transformer or balun.
Figure 3(a) is a transmission line used as a balun but loaded with ferrite cores to reduce the length. The choking reactance should be at least 4 × Z0 in order to present a high impedance to common
mode currents and thereby preserve the balanced to unbalanced properties.
Figure 3(b) is a ferrite loaded to unbalanced 4:1 transformer known as an unun. Z0 equals ZL/2 and the choking reactance should be at least 4 × Z0.
Figure 4(a) is a high-power transformer balun is used to provide the balanced to unbalanced function. Z0 of the two coaxial cables is ZL/2.
Figure 4(b) is not actually of the transmission line class but is included for discussion.
Fig. 2 · (A) A Quarter-Wave
Line Transformer, And (B) A
Quarter-Wave Line Used As A
Fig. 3 · (A) Ferrite-Loaded
Line Used As A Balun, And (B) A
Ferrite-Loaded 4:1 Transmission Line
Transformer [6] .
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
A single turn primary uses brass tubes that are loaded with ferrite toroids. A secondary is passed through the tubes and the impedance transformation varies as N2 where N is the number of turns. This
is a popular selection because of its simplicity, but is limited in bandwidth and power. The bandwidth can be extended if the secondary is made to be an union as shown in Figure 3(b) using a
semi-rigid coaxial cable. The outer conductor is insulated and placed inside of the brass tubes. The brass tubes and semi-rigid outer conductor then become a 1:1 transformer with close coupling and
also provide the isolation requirements. This arrangement has been called a triaxial transformer and can extend the frequency response substantially.
Fig. 4 · (A) An Improved High Power 4:1 Transformer With A
Separate Balun, And (B) A Conventional Transformer With
Tightly-Coupled Windings[6].
4.2 LC Matching Circuits
Figures (5) and (6) show various forms of LC matching circuits. The equations for calculating the matching components are included.
The circuits in Figure 5(a) and 5(b) transform RL to a higher input Rin using either a series L and a shunt C, or series C and shunt L.
The circuits in Figure 5(c) and 5(d) transform RL to a lower input Rin using either a shunt C and series L, or a shunt L and series C.
Figure 6(a) is a pi network that can match either a higher or lower input resistance.
The pi can either be a high pass or low pass version. The low pass is a lum ped constant version of a quarter wave transmission line. The inductance and capacitance values are equal to the square
root of the product of Rin × RL.
Figure 6(b) is the design of a high power pi match with a 10:1 ratio between RL and
The component values, current and voltages have been calculated and the performance using two coaxial cables wound on a common ferrite core that has a 4:1 impedance transformation. The cores are
connected as shown. The power capability is 1 kW and its frequency response is (5-100) MHz A separate ohm, and a power dissipation of 50 watts.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
The 220 pF chip capacitor has a breakdown rating of 3600 volts and a current rating of
10 amps. A FET with a breakdown voltage of 900 volts is used to switch the pin diode on and off. A 450 volt DC voltage is used to back bias the diode in the off position. Switching speed is on the
order of 10μsec. An optoisolator connects the input to the switching circuit
Fig.5· LC Matching Networks: (A) And (B) Provide
Transformation To A Load Resistance Lower Than
Rin, While (C) And (D) Provide Transformation To
Higher Resistances[6].
Fi g.6. High power pi network matching circuits. (a) shows the highpass and low pass configurations;
(b) is a high power pi network for a 10:1
Transformation and 90-degree phase shift[6].
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
5. Design Environment: Microwave Office 2000
It was obvious from the start that the propose amplifier would need to be designed in the software environment if we actually wanted to build it. There are several software packages in the industry
that are used for the design and simulation of RF circuits. The one that we chose to use was Applied Wave Research’s Microwave Office. The primary reason for this choice was that we could obtain our
own trial copy which gave us much more flexibility in the design process.
Microwave Office is one of the top three industry standard RF design and simulation packages which also made it very attractive. Learning the use and capabilities of the software through the design
process turned out to be very time consuming but the experience gained with the software will no doubt be invaluable in an RF career.
6. Obtaining nonlinear model of transistor
The first step in the whole design process was to choose a transistor. We chose the RF
Line NPN silicon RF power transistor MRF321. The MRF321 is a packaged Aluminum
Gallium Arsenide / Indium Gallium Arsenide (AlGaAs/InGaAs) pseudomorphic High
Electron Mobility Transistor (pHEMT). This transistor was chosen because it met all of the requirements for our target specifications.
The most unexpected problems that we encountered when we started our design was that there are no perfect non-linear models for microwave transistors. We found the solution to this problem by
consulting a professional in the design field. We were advised to optimize the non-linear device model for our design frequency by adjusting various parameter values in the non-linear model. It
should be noticed that optimization would not necessarily be needed if the amplifier circuit were only to designed and tested in the software environment. Fig.7.
shows some of the proposed model transistor show it successful in particular field.
Figure.7. Input And Output Impedance Model For
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Theoretical smith chart is used to design matching network for class C power amplifier by using Chebyshev low-pass ladder circuit with number of sections depends upon the bandwidth and required
impedance transfer ratio, each section have quality factor Q increase by increasing transfer ratio given in expression[7]:-
Q = f0 / fH –fL = fL+ fH/2(fH-fL) ...(1)
Where: fo: middle frequency of frequency band.
fH: high frequency of frequency band.
fL: low frequency of frequency band.
The input and output impedance of transistor as given in the data sheet for band from
200MHz-400MHz given in the table (1), which is used as initial point, than from this point one can move on the smith chart circuit and Table (2) gives input and output impedance values calculated for
MRF321 through frequency band (300MHz- 350MHz).
Table (1): Input And Output Impedance Of MRF Band321 (200-500) Mhz Transistor
Frequency (MHz)
0.68- j 0.75
0.89+ j 2.7
1.3 + j 4.3
14.2 – j 22
9.8 – j 14.4
9.3 – j 13
Table (2): Input And Output Impedance For MRF 321 Through Frequency Band (300mhz-
350mhz) Without Matching.
0.812 + 1.03
0.774 + j 1.419
0.875 +j 1.816
11.267 – j17.486
10.843 – j 16.786
10.271 – j 15.943
We draw the Q circuit with radius equal to
1 + 1 / Q 2
and center at point ±1/Q on the
Imaginary axis. Figures (8) and (9) show the theoretical input matching circuit for bandwidth
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
At point G
Z in
Fig.8. Input Matching Network Design Using Smith
Fig.9. Input Matching Circuit Result From Using Smith Chart
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
We note from figure (8) that at point G the value of Zin equal to 55+j3.158 at 325 MHz.
figure (10) show output impedance design using smith chart method and fig.11. state the output matching circuit result.
At point F
Fig.10. Output Impedance Network Design Using Smith Chart.
Fig.11. Output Impedance Circuit Result From Using Smith
From figure (10) we note that Zo equal to 52.476+j0.195 at point f at frequency equal to
325MHz. after that we used microwave office 2000 to build the impedance matching network for the transistor.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
We use the propose model shows in fig.7. to represent the transistor in the program, give initial value for component then make optimizations to get a suitable value that cover the values of the
input and output impedance for transistor as shown in fig.12. The optimization procedure of the non-linear model involved changing arbitrary values one at a time using these model we get the
nonlinear model to represent transistor in the microwave office 2000 and we get the exact value of input and output impedance for transistor as shown in figures
(13) and(14).
Fig .12. Input And Output Impedance Model Propose For MRF321
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Fig.13. Input Impedance For Transistor For Frequency
Band(200mhz-500mhz) Using A) Smith Chart. B) Rectangular
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Fig.14. Output Impedance For Transistor For Frequency Band
(200mhz-500mhz) Using A) Smith Chart. B) Rectangular Form.
Then we insert the impedance network for input and output equivalent circuit of the transistor, make optimization to get the best value for matching network component as shown in figure (15).
a) Input matching b) Output matching circuit
Fig .15. Matching network circuit for MRF321 transistor for frequency band 300MHz-350MHz after optimization
Figure (16) show the total input impedance for MRF321 transistor after adding designed matching circuit and we note that it closed to 50Ω that’s mean we satisfy matching condition.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Fig.16. Total Input Impedance Of MRF321 For Frequency
Band (300mhz-350mhz) A) Smith Chart B) Rectangure Form.
We note from Fig.17. That the VSWR that belong to the input circuit have constant value close or equal to 1 along the desired frequency band, that means there is no reflection on the input circuit.
Figure (18) show the total output impedance for MRF321 transistor after adding designed matching circuit and we note that it closed to 50Ω that’s mean we satisfy matching condition for output.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Fig.17. Total VSWR Of MRF321input Circuit For Frequency Band
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
Fig.18. Total Output Impedance Of MRF321 For Frequency
Band (300mhz-350mhz) A) Smith Chart B) Rectangure Form.
We note from Fig.19. That the VSWR that belong to the input circuit have constant value close or equal to 1 along the desired frequency band, that means there is no reflection on the output circuit
and we have maximum power transfer to the output.
Fig.19. Total VSWR of MRF321output circuit for frequency band (300MHz-350MHz)
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
7. Conclusion
Many designers believe that the analysis of the nonlinear characteristics of class C amplifiers is not practical with popular simulators such as microwave office2000. However, with the proper
modeling of the RF transistors and proper accounting of parasitic, virtually every aspect of the class C amplifier can be studied.
This paper explores some unique techniques and models for simulating amplifiers running in class C operation using the general purpose microwave office2000 circuit simulation program. Results of the
simulation of an 300MHz-350MHz amplifier including impedance and impedance matching circuit and waveform are given.
Journal of Engineering and Development, Vol. 15, No. 2, June (2011) ISSN 1813-7822
8. References
S. Saad, “DESIGN OF CLASS-E RADIO FREQUENCY POWER AMPLIFIER”
Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State
University, July 2001 Blacksburg, Virginia.
Steve; C. Jaime “Microwave Amplifier Design (part 1)”, Inderpreet Obhi December
15, 2003.
B. Manfred, “Design of Radio Frequency Power Amplifiers for Cellular Phones and
Base Stations in Modern Mobile Communication Systems”, Institut für Elektrische und Optische Nachrichtentechnik 2009.
S. Cripps, “RF power amplifier for wireless communications,” 1999 Artech House,
S. FATHI “POWER AMPLIFIER DESIGN”, GROUP PROJECT, PRESENTED
FOR THE COURSE ECE802-607 FS05.
W. Richard, “Matching Networks for Power Amplifiers Operating into High VSWR
Loads”, May 2004 High Frequency Electronics Copyright © 2004 Summit Technical
Media, LLC.
A.N. Riddle and R.J. Trew,’’ Abroad- band amplifier output network design’’, IEEE trans. On microwave theory and tech, vol.1, MTT.30, No.2, February, 1982. | {"url":"https://studylib.net/doc/18820133/impedance-matching-network-design-for-class-c-power-ampli...","timestamp":"2024-11-03T17:10:45Z","content_type":"text/html","content_length":"77539","record_id":"<urn:uuid:a775b9e7-1f68-4e65-b9dc-759e2d086de7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00785.warc.gz"} |
What is a circle inside a triangle called?
What is a circle inside a triangle called?
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle
center called the triangle’s incenter.
What does the circle in the corner of a triangle mean?
The circle stands for the whole world of AA, and the triangle stands for AA’s Three Legacies of Recovery, Unity and Service. The circle symbolizes serenity and perfection, and the source of unlimited
potential. Together they represent perfect union of mind and body.
When a circle is inscribed in a triangle?
When a circle inscribes a triangle, the triangle is outside of the circle and the circle touches the sides of the triangle at one point on each side. The sides of the triangle are tangent to the
circle. draw in the angle bisectors. The intersection of the angle bisectors is the center of the inscribed circle.
What is the symbolism of a triangle?
In most literary pieces, the triangle (with the number three) repersents perfectness, unity, and importance. It is the strongest unit. When a group/item in literature moves from three, the traingle,
to four, it foreshadows bad things and destruction.
What is the Orthocenter of a triangle?
An orthocenter can be defined as the point of intersection of altitudes that are drawn perpendicular from the vertex to the opposite sides of a triangle. The orthocenter of a triangle is that point
where all the three altitudes of a triangle intersect. Hence, a triangle can have three altitudes, one from each vertex.
How do you find a 30-60-90 Triangle?
30-60-90 Triangle Ratio
1. Short side (opposite the 30 degree angle) = x.
2. Hypotenuse (opposite the 90 degree angle) = 2x.
3. Long side (opposite the 60 degree angle) = x√3.
Is the center of the Triangle an inscribed circle?
A circle is inscribed in the triangle if the triangle’s three sides are all tangents to a circle. In this situation, the circle is called an inscribed circle, and its center is called the inner
center, or incenter. Imgur.
What are the properties of a circumscribed triangle?
In conclusion, the three essential properties of a circumscribed triangle are as follows: The segments from the incenter to each vertex bisects each angle. The distances from the incenter to each
side are equal to the inscribed circle’s radius. The area of the triangle is equal to 21 ×r× (the triangle’s perimeter),…
How are the four centers of a triangle constructed?
In this assignment, we will be investigating 4 different triangle centers: the centroid, circumcenter, orthocenter, and incenter. The centroid of a triangle is constructed by taking any given
triangle and connecting the midpoints of each leg of the triangle to the opposite vertex.
What is the area of the incircle of a triangle?
The distances from the incenter to each side are equal to the inscribed circle’s radius. The area of the triangle is equal to 21 ×r× (the triangle’s perimeter), where r is the inscribed circle’s
What is a circle inside a triangle called? In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides.
The center of the incircle is a triangle center called the triangle’s incenter. What does the circle in the corner of… | {"url":"https://bridgitmendlermusic.com/what-is-a-circle-inside-a-triangle-called/","timestamp":"2024-11-07T06:50:15Z","content_type":"text/html","content_length":"41729","record_id":"<urn:uuid:94dbf649-c1e3-4805-bdc8-da338b300fc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00270.warc.gz"} |
Inner/nested object term facet question (gisted)
Hi guys
I'm trying to do a term facet query over a nested object but I haven't been
able to make it work. I was wondering if someone can give me some insights
on my mistake?
I've gisted here https://gist.github.com/3407734 but I always got zero term
facets. I've done other facets many times but seems that there's a trick on
this one.
Bump!.. nobody? c'mon guys
On Monday, August 20, 2012 5:12:31 PM UTC-4, maverick wrote:
Hi guys
I'm trying to do a term facet query over a nested object but I haven't
been able to make it work. I was wondering if someone can give me some
insights on my mistake?
I've gisted here https://gist.github.com/3407734 but I always got zero
term facets. I've done other facets many times but seems that there's a
trick on this one.
There is not a single feed with data in the field -
On Wed, Aug 22, 2012 at 6:49 PM, maverick mauricio.alarcon@gmail.comwrote:
Bump!.. nobody? c'mon guys
On Monday, August 20, 2012 5:12:31 PM UTC-4, maverick wrote:
Hi guys
I'm trying to do a term facet query over a nested object but I haven't
been able to make it work. I was wondering if someone can give me some
insights on my mistake?
I've gisted here https://gist.github.com/**3407734https://gist.github.com/3407734 but
I always got zero term facets. I've done other facets many times but seems
that there's a trick on this one.
Thanks Vineeth,
actually there is one, check the mapping, is a multi_field one of them is
"name" the other is "nameFacet" that is analyzed with "keyword" analyzer..
In any case, not even the educationHistory.name works
Any other ideas?
On Wednesday, August 22, 2012 9:48:04 AM UTC-4, Vineeth Mohan wrote:
There is not a single feed with data in the field -
On Wed, Aug 22, 2012 at 6:49 PM, maverick <mauricio...@gmail.com<javascript:>
Bump!.. nobody? c'mon guys
On Monday, August 20, 2012 5:12:31 PM UTC-4, maverick wrote:
Hi guys
I'm trying to do a term facet query over a nested object but I haven't
been able to make it work. I was wondering if someone can give me some
insights on my mistake?
I've gisted here https://gist.github.com/**3407734https://gist.github.com/3407734 but
I always got zero term facets. I've done other facets many times but seems
that there's a trick on this one.
Your mapping is incorrect. The first field in the JSON mapping refers
to the type (which is redundant given the URL, but that's how it is).
Take a look at the PUT mapping example:
Also, your mapping redefines employmentHistory but the facet is on
On Wed, Aug 22, 2012 at 8:01 AM, maverick mauricio.alarcon@gmail.com wrote:
Thanks Vineeth,
actually there is one, check the mapping, is a multi_field one of them is
"name" the other is "nameFacet" that is analyzed with "keyword" analyzer..
In any case, not even the educationHistory.name works
Any other ideas?
On Wednesday, August 22, 2012 9:48:04 AM UTC-4, Vineeth Mohan wrote:
There is not a single feed with data in the field -
On Wed, Aug 22, 2012 at 6:49 PM, maverick mauricio...@gmail.com wrote:
Bump!.. nobody? c'mon guys
On Monday, August 20, 2012 5:12:31 PM UTC-4, maverick wrote:
Hi guys
I'm trying to do a term facet query over a nested object but I haven't
been able to make it work. I was wondering if someone can give me some
insights on my mistake?
I've gisted here https://gist.github.com/3407734 but I always got zero
term facets. I've done other facets many times but seems that there's a
trick on this one.
Thank you very much, I fixed part of it with your help. It still needs some
other tune up.. I'll post again with the current issue
On Wednesday, August 22, 2012 1:51:42 PM UTC-4, Ivan Brusic wrote:
Your mapping is incorrect. The first field in the JSON mapping refers
to the type (which is redundant given the URL, but that's how it is).
Take a look at the PUT mapping example:
Elasticsearch Platform — Find real-time answers at scale | Elastic
Also, your mapping redefines employmentHistory but the facet is on
On Wed, Aug 22, 2012 at 8:01 AM, maverick <mauricio...@gmail.com<javascript:>>
Thanks Vineeth,
actually there is one, check the mapping, is a multi_field one of them
"name" the other is "nameFacet" that is analyzed with "keyword"
In any case, not even the educationHistory.name works
Any other ideas?
On Wednesday, August 22, 2012 9:48:04 AM UTC-4, Vineeth Mohan wrote:
There is not a single feed with data in the field -
On Wed, Aug 22, 2012 at 6:49 PM, maverick mauricio...@gmail.com
Bump!.. nobody? c'mon guys
On Monday, August 20, 2012 5:12:31 PM UTC-4, maverick wrote:
Hi guys
I'm trying to do a term facet query over a nested object but I
been able to make it work. I was wondering if someone can give me
insights on my mistake?
I've gisted here https://gist.github.com/3407734 but I always got
term facets. I've done other facets many times but seems that
there's a
trick on this one. | {"url":"https://discuss.elastic.co/t/inner-nested-object-term-facet-question-gisted/8792","timestamp":"2024-11-02T22:16:02Z","content_type":"text/html","content_length":"40036","record_id":"<urn:uuid:5f943cc8-1896-4e4a-a79d-956d562962ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00275.warc.gz"} |
The Sequence Chat: Lewis Tunstall, Hugging Face, On Building the Model that Won the AI Math Olympiad
Details about NuminaMath, its architecture, training process and even things that didn't work.
Lewis Tunstall is a Machine Learning Engineer in the research team at Hugging Face and is the co-author of the bestseller “NLP with Transformers” book. He has previously built machine
learning-powered applications for start-ups and enterprises in the domains of natural language processing, topological data analysis, and time series. He holds a PhD in Theoretical Physics, was a
2010 Fulbright Scholar and has held research positions in Australia, the USA, and Switzerland. His current work focuses on building tools and recipes to align language models with human and AI
preferences through techniques like reinforcement learning.
This is your second interview at TheSequence. Please tell us a bit about yourself. Your background, current role and how did you get started in AI?
I currently lead the post-training team at Hugging Face, where we focus on providing the open-source community with robust recipes to fine-tune LLMs through libraries like TRL. In a previous life, I
was a theoretical physicist researching the strong nuclear force and its connection to dark matter. I accidentally stumbled into AI via my colleagues in experimental physics, who were very excited
about applying a (new at the time) technique called “deep learning” to particle collisions at the Large Hadron Collider. I was surprised to learn that around 100 lines of TensorFlow could train a
neural net to extract new signals from collision data, often much better than physics-derived features. This prompted me to take part in a Kaggle competition with a few physics friends, and I’ve been
hooked on AI ever since!
TheSequence is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber.
🛠 ML Work
You were part of the team that built NuminaMath, which recently won the AI Math Olympiad. Could you tell us more about the vision and inspiration behind this project?
This project was a collaboration between Hugging Face and Numina, a French non-profit that was inspired by the AI Math Olympiad (AIMO) to create high-quality datasets and tools for the open-source
community. Although there are several open weight models for mathematics, the training datasets are rarely, if ever, made public. We teamed up with Numina to bridge this gap by tackling the first
AIMO progress prize with a large-scale dataset of around 850,000 math problem-solution pairs that Numina had been developing prior to the competition. We saw that winning the AIMO competition would
be a great way to show the community the power of high-quality datasets and I’m very happy to see that it worked out well!
International Math Olympiads (IMO) cover a wide range of topics, from number theory to algebra and geometry. Recently, Google DeepMind published work in this area, using different models for
different types of problems. Do we need specialized models for specific math disciplines, or can a single model tackle the entire set of problems?
That’s a great question and I suspect the answer will depend on how straightforward it is to integrate multiple external verifiers like Lean4 and Wolfram Alpha for models to check their proofs. The
current systems are trained to interface with a single external solver, but I expect the next iteration of DeepMind’s approach will involve a generalist model that can use multiple solvers, like how
LLMs currently use tools for function calling.
NuminaMath is based on the DeepSeekMath-Base model. What led you to choose this model as the baseline for math reasoning in this project?
This model was largely chosen due to the constraints of the competition: onthe one hand, only pretrained models that were released before February 2024 could be used, and on the other hand, each
submission had to run on 2 T4 GPUs in under 9 hours (not easy for LLMs!). Both these constraints made DeepSeekMath 7B the best choice at the time, although there are now much better math models like
those from Qwen which would likely score even better in the competition.
Could you describe the various components of the NuminaMath recipe and explain how they work together? I am particularly curious about the application of techniques such as chain-of-thought and
tool-integrated reasoning.
We experimented with various fine-tuning recipes, but the one that gave the best results was based on an interesting paper called MuMath-Code.
This paper combines two insights that have been shown to improve the reasoning capabilities of LLMs: Chain of Thought (CoT) templates that provide the intermediate steps needed to obtain the correct
answer, and tool-integrated reasoning (TIR), where the LLM is given access to a tool like Python to run computations. Prior research had explored each one independently, but MuMath-Code showed that
you get best results by performing two-stage training: first on the CoT data (to learn how to solve problems step-by-step), followed by training on TIR data (to learn how outsource part of the
problem to Python). We applied this approach with DeepSeekMath 7B to the datasets Numina created and found it worked really well on our internal evals and the Kaggle leaderboard!
Do you think the NuminaMath architecture can be extended to other scientific fields, such as physics or chemistry?
Our training recipe was tailored to competitive mathematics, where one has a known answer for each problem. For competitive physics or chemistry problems, I suspect the method would generalise
provided one integrates the relevant tools for the domain. For example, I suspect that in chemistry one requires access to a broad range of tools in order to solve challenging problems.
What was the training process, pipeline, and dataset for NuminaMath?
Our winning recipe was quite simple in the end and involved applying two rounds of supervised fine-tuning (SFT) to the DeepSeekMath 7B model:
• Step 1: Fine-tune the base model on a dataset of 850,000 math problems, where each solution is annotatedin a CoT template.
• Step 2: Fine-tune the model from Stage 1 on a synthetic dataset of tool-integrated reasoning, where each math problem is decomposed into a sequence of rationales, Python programs, and their
outputs. We generated this dataset from GPT-4 to produce about 70,000 solutions.
For tools, we use the TRL library for training out models on one node of 8 x H100 GPUs. For evaluations and the Kaggle submissions we used vLLM which provides fast inference for LLMs.
What ideas did your team try but ultimately abandon during the implementation of NuminaMath?
The main ideas we tried but didn’t make it to the final submission were the following:
1. KTO: Using Kahneman-Tversky Optimisation (KTO) to boost the performance of our SFT models. Here the basic idea is to take your current model, generate multiple candidate solutions per problem,
and then label those solutions as correct or incorrect according to the ground truth. With this data, one can then optimise the model to boost the probability it produces more of the correct
versus incorrect tokens. Although we found this method worked quite well, we were unable to include it in the final submission due to various infrastructure issues on the Kaggle platform in the
final days of the competition.
2. Model merging: We explored a variety of model merging techniques like DARE, TIES, and WARP. Here we used mergekit to merge the SFT and KTO models, or the SFT models with the public DeepSeekMath
ones. Overall we found these merges led to either significant regressions on our internal evaluations and we ran out of time to explore this more deeply.
💥 Miscellaneous – a set of rapid-fire questions
What is your favorite area of research outside of generative AI?
As a former physicist, I am excited about the advances being made in AI4Science and specifically in simulating physical processes. I quite like the way Chris Bishop frames this as the “fifth
paradigm” of scientific discovery and am hopeful this will help us find answers to some rather thorny problems.
Will we achieve AGI through transformers and scaling laws?
It’s always risky to predict the future, but I think it’s clear that the current systems are lacking several capabilities like episodic memory and the ability to plan over long horizons. Whether
these capabilities will emerge at larger scales remains to be seen, but history generally shows that one shouldn’t bet against deep learning!
What will it take for AI to evolve from proving math theorems to formulating original theories, such as the Riemann Hypothesis or the Theory of General Relativity?
My current view is that the main barrier for models to generate novel ideas in mathematics and the natural sciences is having some means to verify correctness and compatibility with experiment. For
mathematics, I suspect that projects like Lean4 will play a key role in bridging the gap from proving what is known towards generating novel theorems which are then verified automatically by the
model for correctness. For the natural sciences, the challenge appears to be much harder since one needs to both synthesize vast amounts of experimental data and possibly run new experiments to
validate new ideas. Nevertheless, there are already hints from works like Sakana’s AI Scientist that formulating novel ideas in ML research is possible, albeit under rather stringent constraints. It
will be exciting to see how far this can be pushed into other domains!
Who is your favorite mathematician and computer scientist, and why?
My favorite mathematician is John von Neumann, mostly because I didn't really understand quantum mechanics until I read his excellent textbook on the subject.
TheSequence is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber. | {"url":"https://thesequence.substack.com/p/the-sequence-chat-lewis-tunstall","timestamp":"2024-11-10T11:46:57Z","content_type":"text/html","content_length":"185279","record_id":"<urn:uuid:71aa4c25-9c4b-42a7-8a4c-8171b921e0a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00408.warc.gz"} |
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